Fabien Lauer

On the complexity of switching linear regression

This technical note extends recent results on the computational complexity of globally minimizing the error of piecewise-affine models to the related problem of minimizing the error of switching linear regression models. In particular, we show that, on the one hand the problem is NP-hard, but on the other hand, it admits a polynomial-time algorithm with respect to the number of data points for any fixed data dimension and number of modes.

Introduction

Hybrid system identification aims at estimating a model of a system switching between different operating modes from input-output data and is typically setup as a piecewise-affine (PWA) or switching regression problem (see [START_REF] Paoletti | Identification of hybrid systems: a tutorial[END_REF][START_REF] Garulli | A survey on switched and piecewise affine system identification[END_REF] for an overview). The present paper focuses on the issue of deterministically obtaining a global solution to the switching regression problem. In particular, we are interested in the rather theoretical question of the existence of an algorithm for this problem with a reasonable (i.e., polynomial) time complexity. Therefore, we will concentrate the discussion on computational complexity issues under the classical model of computation known as a Turing machine [START_REF] Garey | Computers and Intractability: a Guide to the Theory of NP-Completeness[END_REF]. In this framework, the time complexity of a problem is the lowest time complexity of an algorithm solving any instance of that problem, where the time complexity of an algorithm is the maximal number of steps occurring in the computation of the corresponding Turing machine program.

Let [n] = {1, . . . , n} denote the set of integers from 1 to n, x i ∈ Q d the regression vector of index i (possibly built from lagged inputs and outputs of a dynamical system) and y i ∈ Q the corresponding output. We consider the estimation of the parameters {w j } n j=1 ⊂ Q d of an arbitrarily switching linear model y i = w T qi x i + v i , where q i ∈ Q = [n] stands for the active mode at index i and v i ∈ Q is a noise term. We assume that the mode q i is independent of x i , that a data set {(x i , y i )} N i=1 ⊂ Q d × Q of size N significantly larger than the dimension d is available, and that the number of modes n is given. We concentrate on the most common approach minimizing the prediction error over the variables to be estimated, here the classification of the points into modes, i.e., q ∈ Q N , and the parameter vectors, {w j } n j=1 . Specifically, we formulate the problem in terms of a loss function ℓ : Q → Q + , assumed to be computable in polynomial time and to satisfy

     ℓ(0) = 0, ∀e ∈ Q, ℓ(-e) = ℓ(e), ∀(e, e ′ ) ∈ Q 2 , ℓ(e) < ℓ(e ′ ) ⇔ |e| < |e ′ |. (1)
Problem 1 (Switching linear regression). Given a data set

{(x i , y i )} N i=1 ⊂ Q d × Q and an integer n ∈ [2, N/d], find a global solution to min {wj ∈Q d } n j=1 ,q∈Q N 1 N N i=1 ℓ(y i -w T qi x i ). (2)
Other equivalent formulations based on mixed-integer programming with binary variables encoding q or on continuous optimization can be found in [START_REF] Paoletti | Identification of hybrid systems: a tutorial[END_REF][START_REF] Lauer | A continuous optimization framework for hybrid system identification[END_REF] and a number of heuristics subject to local minima [START_REF] Lauer | Estimating the probability of success of a simple algorithm for switched linear regression[END_REF][START_REF] Pham Dinh | A difference of convex functions algorithm for switched linear regression[END_REF] or only optimal under specific conditions [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF][START_REF] Bako | Identification of switched linear systems via sparse optimization[END_REF] have been proposed. Problem 1 can be solved explicitly with respect to (wrt.) q for fixed {w j } n j=1 by assigning each point to the model with minimum error as

q i ∈ argmin j∈{1,...,n} ℓ(y i -w T j x i ), i = 1, . . . , N. (3)
Conversely, Problem 1 can be solved wrt. to the w j 's for fixed q as n independent linear regression subproblems min

wj ∈Q d i∈{k:q k =j} ℓ(y i -w T j x i ), j = 1, . . . , n. (4)
Thus, two global optimization approaches can be readily formulated. The first one tests all possible classifications q and solves the problem wrt. the w j 's for each of them. But, this leads to n × n N linear regression subproblems (4) and quickly becomes intractable when N increases. The second approach applies a continuous global optimization strategy to directly estimate {w j } n j=1 under the optimal classification rule (3). However, global optimality cannot be guaranteed without constraints on the w j 's such as box bounds. And even so, the complexity remains exponential in the number of variables nd, for instance for a grid search to obtain a solution with an error that is only guaranteed to be close to the global optimum in finite time.

These straightforward observations illustrate the difficulty of the problem, which is here quantified more formally. In particular, we prove in Sect. 2 that Problem 1 is NP-hard. Nonetheless, we also show in Sect. 3 that the problem can be solved in polynomial time wrt. the number of data for fixed n and d. This result is obtained by generalizing ideas developed for PWA systems in [START_REF] Lauer | On the complexity of piecewise affine system identification[END_REF] and by deriving for the first time a clear connection between switching regression and linear classification.

NP-hardness

In computational complexity, an NP-hard problem is one that is at least as hard as any problem from the class NP of nondeterministic polynomial time decision problems [START_REF] Blondel | A survey of computational complexity results in systems and control[END_REF]. In particular, NP is the class of all decision problems for which a candidate solution can be certified in polynomial time. Under this definition, we have the following result.

Theorem 1. With ℓ as in (1), Problem 1 is NP-hard.

The proof is a direct consequence of the NP-completeness of the following decision form of Problem 1, where an NP-complete problem is one that is both NP-hard and in NP.

Problem 2 (Decision form of switching regression). Given a data set

{(x i , y i )} N i=1 ∈ (Q d × Q) N , an integer n ∈ [2, N/d]
and a threshold ǫ ≥ 0, decide whether there is a set of vectors

{w j } n j=1 ⊂ Q d and a labeling q ∈ Q N such that 1 N N i=1 ℓ(y i -w T qi x i ) ≤ ǫ. (5)
We show the completeness of Problem 2 by a reduction from the partition problem, known to be NP-complete [START_REF] Garey | Computers and Intractability: a Guide to the Theory of NP-Completeness[END_REF].

Problem 3 (Partition). Given a multiset (a set with possibly multiple instances of its elements) of d positive integers, S = {s 1 , . . . , s d }, decide whether there is a multisubset S 1 ⊂ S such that

si∈S1 s i = si∈S\S1 s i . Proposition 1. Problem 2 is NP-complete.
Proof. Since given a candidate solution {w j } n j=1 , q the condition ( 5) can be verified in polynomial time, Problem 2 is in NP. Then, the proof of its NP-completeness proceeds by showing that the Partition Problem 3 has an affirmative answer if and only if a particular instance of Problem 2 has an affirmative answer.

Given an instance of Problem 3, build an instance of Problem 2 with n = 2, N = 2d + 1, ǫ = 0 and a data set such that

(x i , y i ) =        (s i e i , s i ), if 1 ≤ i ≤ d (s i-d e i-d , 0), if d < i ≤ 2d s = d k=1 s k e k , 1 2
d k=1 s k , if i = 2d + 1,
where e k is the kth unit vector of the canonical basis for Q d . If Problem 3 has an affirmative answer, let I 1 be the set of indexes of the elements of S in S 1 and I 2 the set of indexes of the elements of S not in S 1 . Then we can set w 1 = i∈I1 e i and w 2 = i∈I2 e i , which gives

w T 1 x i =                s i = y i , if i ≤ d and i ∈ I 1 0, if i ≤ d and i ∈ I 2 s i-d = y i , if i > d and i -d ∈ I 1 0, if i > d and i -d ∈ I 2 k∈I1 s k = 1 2 d k=1 s k = y i , if i = 2d + 1 and w T 2 x i =                0, if i ≤ d and i ∈ I 1 s i = y i , if i ≤ d and i ∈ I 2 0, if i > d and i -d ∈ I 1 s i-d = y i , if i > d and i -d ∈ I 2 k∈I2 s k = 1 2 d k=1 s k = y i , if i = 2d + 1.
Therefore, for all points, either w T 1 x i = y i or w T 2 x i = y i , and (5) holds with q set as in (3), yielding an affirmative answer for Problem 2.

Assume now that Problem 2 has an affirmative answer with some {w j } n j=1 . Then, with ǫ = 0, the positivity of the loss function implies ℓ(y iw T qi x i ) = 0, i = 1, . . . , N , which, by (1) yields

w T 1 x i = y i or w T 2 x i = y i , i = 1, . . . , 2d + 1. (6)
We can always assume that s i = 0, since otherwise s i can be removed from the problem statement. Under this assumption, if w T 1 x i = y i for some i ≤ d, then w 1i = 1 and w T 1 x d+i = s i = y d+i , which further implies w T 2 x d+i = y d+i = 0 and w 2i = 0. Conversely, if w T 2 x i = y i for some i ≤ d, then w 2i = 1 and w 1i = 0. Therefore, (6) leads to w 1i ∈ {0, 1} and w 2i = 1 -w 1i , i = 1, . . . , d. In addition, recall that y 2d+1 = 1 2 d k=1 s k , such that, for i = 2d + 1, (6) yields at least one of the two equalities

w T 1 x 2d+1 = k∈{i≤d:w1i=1} s k = 1 2 d k=1 s k w T 2 x 2d+1 = k∈{i≤d:w1i=0} s k = 1 2 d k=1 s k ,
and a partition corresponding to an affirmative answer for Problem 3 is given by S 1 = {s i :

w 1i = 1, i ≤ d}.
Remark 1. The proof of Proposition 1 involves an instance of Problem 2 with ǫ = 0, i.e., a noiseless switching regression problem. However, it can be adapted to the restriction of Problem 2 to instances with ǫ > 0, implying the NP-hardness of Problem 1 even when excluding noiseless instances.

3 Polynomial time complexity wrt. N

We now turn to the analysis of the computational complexity of Problem 1 wrt. the number of data N , i.e., for fixed n and data dimension d, under the following assumptions.

Assumption 1. The points {x i } N i=1 are in general position, i.e., no hyperplane of Q d contains more than d points. Furthermore, the points Theorem 2. Under Assumptions 1-2, for given integers d and n, the time complexity of Problem 1 is no more than polynomial in the number of data N and in the order of T (N )O(N 2dn(n-1) ).

z i = [x T i , y i ] T are also in general position in Q d+1 . Assumption 2. Given {(x i , y i )} N i=1 ∈ (Q d × Q) N ,
Theorem 2 is a direct consequence of the existence of an exact algorithm that solves the problem in polynomial time, ensured by Corollary 1 at the end of this section. This algorithm relies on the enumeration of all classifications consistent with (3), which we prove to be in a number polynomial in N below.

For this, we will use results on the enumeration of all possible linear classifications of a set of N points. In the binary case with two categories, a linear classification is one that can be produced by a separating hyperplane dividing the space in two halfspaces. It is shown in [START_REF] Lauer | On the complexity of piecewise affine system identification[END_REF] that the number of different linear classifications of N points is on the order of O(N d ) in Q d and that these can be constructed efficiently. Here, we use an adaptation of these results for linear classifiers, while [START_REF] Lauer | On the complexity of piecewise affine system identification[END_REF] focused on affine classifiers. This minor difference amounts to the removal of a degree of freedom by forcing the hyperplane to pass through the origin. Since the results of [START_REF] Lauer | On the complexity of piecewise affine system identification[END_REF] are based on hyperplanes passing through sets of d points, they can be directly extended to linear classifiers by choosing one of these points to be the origin. Thus, we state the following without proof.

Proposition 2 (Adapted from Theorem 3 in [START_REF] Lauer | On the complexity of piecewise affine system identification[END_REF]). Let In PWA regression, the modes are typically assumed to be linearly separable in the regression space Q d and results in the flavor of Proposition 2 can readily be applied to find the optimal classification of the data points [START_REF] Lauer | On the complexity of piecewise affine system identification[END_REF]. In switching regression, the mode sequence {q i } is arbitrary and we cannot assume the modes to be linearly separable. However, the groups of data pairs (x i , y i ) associated to different linear models can be "linearly separated" in some sense. More precisely, we will show that the classification rule (3) implicitly entails a combination of two linear classifiers: one applying to the points z i = [x T i , y i ] T in Q d+1 and another one applying to the regression vectors x i in Q d . The equivalence between (3) and these linear classifiers will hold for all points with index not in

H S = b ∈ {-1, +1} N : b i = sign(h T x i ), i ∈ [N ], h ∈ Q d
E = {i ∈ [N ] : ∃(j, k) ∈ Q 2 , j = k, |y i -w j T x i | = |y i -w k T x i |}, ( 7
)
whose cardinality is bounded by the following lemma.

Lemma 1. Let E be defined as in [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF]. Under Assumption 1, |E| ≤ (2d + 1)n(n -1)/2.

Proof. Let us define, for all (j, k) such that 1 ≤ j < k ≤ n, the sets

I jk = {i ∈ [N ] : w j T x i = w k T x i } and M jk = {i ∈ [N ] : y i -w j T x i = -(y i -w k T x i )}. Then, we have E = 1≤j<k≤n I jk ∪ M jk . Since w j T x i = w k T x i ⇔ (w j -w k ) T x i =
0, all points x i with i ∈ I jk must lie on a hyperplane of Q d , and under Assumption 1 we have

|I jk | ≤ d. Similarly, since y i -w j T x i = -(y i -w k T x i ) ⇔ y i -(w j -w k ) T x i /2 = 0, all points z i = [x T i , y i ] T with i ∈ M jk must lie on a hyperplane of Q d+1 , and Assumption 1 implies that |M jk | ≤ d + 1. Hence, |E| ≤ 1≤j<k≤n |I jk | + |M jk | ≤ 1≤j<k≤n 2d + 1 ≤ (2d + 1)n(n -1)/2.
Proposition 3. Given a set of parameter vectors {w j } n j=1 ⊂ Q d , let E be defined as in [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF]. Then, for all i / ∈ E, the classification rule (3) with a loss function satisfying (1) is equivalent to the classification rule

q i = argmax j∈Q j-1 k=1 1 c kj (xi,yi)=-1 + n k=j+1 1 c jk (xi,yi)=+1 (8)
implementing a majority vote over a set of n(n -1)/2 pairwise classifiers {c jk } 1≤j<k≤n of Q d+1 , where each c jk is a product of binary linear classifiers defined as

∀(x, y) ∈ Q d × Q, c jk (x, y) = g jk (z)h jk (x), 1 ≤ j < k ≤ n,
with linear classifiers respectively operating in Q d+1 and Q d as

∀z ∈ Q d+1 , g jk (z) = sign([-w T jk , 1] T z), 1 ≤ j < k ≤ n, ∀x ∈ Q d , h jk (x) = sign( wT jk x), 1 ≤ j < k ≤ n
, where w jk = (w j + w k )/2 and wjk = w jw k .

Proof. Using the properties of the loss function (1), the classification rule (3) can be rewritten for any i / ∈ E as

q i = argmin k∈Q ℓ(y i -w T k x i ) ⇔ ∀k ∈ Q \ {q i }, |y i -w T qi x i | < |y i -w T k x i |. (9)
For any triplet (a, b, y

) ∈ Q 3 , we have |y -a| < |y -b| if and only if (a < b ∧ y < (a + b)/2) or (a > b ∧ y > (a + b)/2), i.e., |y -a| < |y -b| ⇔ (y -(a + b)/2)(a -b) > 0.
Thus, with the notations of Proposition 3, ( 9) is equivalent to

∀k ∈ Q \ {q i }, (y -w T qik x i ) wT qik x i > 0 ⇔ ∀k ∈ Q \ {q i }, g qik (z i )h qik (x i ) = +1 ⇔ k∈Q\{qi} 1 c q i k (xi,yi)=+1 = n -1. (10)
In addition, for all (j, k) ∈ {1, . . . , n} 2 , j = k, we have w jk = w kj and wjk = -wkj , so that c jk (x, y) = -g kj (z)h kj (x) = -c kj (x, y). Thus, the classification is entirely determined by a set of n(n -1)/2 pairs of linear classifiers g jk and h jk , 1 ≤ j < k ≤ n, and ( 10) is equivalent to

S(q i ) qi-1 k=1 1 c kq i (xi,yi)=-1 + n k=qi +1 1 c q i k (xi,yi)=+1 = n -1.
Given that max j∈Q S(j) ≤ n -1, we obtain that q i ∈ argmax j∈Q S(j). In addition, for all q = q i , q ∈ argmax j∈Q S(j) implies S(q) = S(q i ) = n -1, which implies by working the equivalences above backward that q ∈ argmin k∈Q ℓ(y iw T k x i ). However, this is not possible, since for all i / ∈ E, argmin k∈Q ℓ(y iw T k x i ) is a singleton. Thus, q i is the only element in argmax j∈Q S(j) as claimed in [START_REF] Bako | Identification of switched linear systems via sparse optimization[END_REF].

Thanks to Proposition 3, a bound on the number of classifications of N points by (3) can be formed from the product of the number of classifications of the z i 's and of the x i 's.

(x i , y i )} N i=1 ⊂ Q d × Q as Q S = q ∈ Q N : q i = argmin j∈Q ℓ(y i -w T j x i ), i ∈ [N ], w j ∈ Q d .
Then, under Assumption 1,

Π(N ) sup S∈(Q d ×Q) N |Q S | = O(N 2dn(n-1) )
and Q S can be computed in O(N 2dn(n-1) ) time.

Proof. By Proposition 3, any q ∈ Q S must result from a set of n(n -1)/2 products of linear classifications, possibly altered for all i ∈ E. Assuming |E| = l, n l such altered classifications can be generated from a "base classification" given by [START_REF] Bako | Identification of switched linear systems via sparse optimization[END_REF]. The number of base classifications returned by ( 8

(N ) ≤ L l=0 N l n l O(N p ) ≤ L l=0 O(N l )n l O(N p ) < LO(N L )n L O(N p ) = O(N L+p ).
Combining this with the value of the bound L given by Lemma 1 yields the desired result.

Theorem 3 implies the following for switching regression.

Corollary 1. Under Assumptions 1-2, there is an algorithm that exactly solves Problem 1 in T (N )O(N 2dn(n-1) ) time.

Proof. By Theorem 3, an algorithm can generate all classifications q ∈ Q S that are consistent with (3) and thus visit the optimal q for Problem 1 in O(N 2dn(n-1) ) iterations. Following the discussion of Sect. 1, the algorithm can compute the optimal w j 's for any q by solving (4), and thus find the global solution to Problem 1. Under Assumption 2, the cost of solving the n subproblems in (4) at each iteration is nT (N ), leading to the claimed time complexity for a fixed n.

Remark 2. The algorithm of Corollary 1 has a polynomial complexity in N but an exponential one in d. Therefore, by using the reduction from Problem 3 to Problem 2 of the proof of Proposition 1, its complexity remains exponential in the size of Problem 3, which is in line with the conjecture P =NP.

Remark 3. Assumption 1 clearly cannot hold without noise, since, in this case, the points z i precisely belong to the union of n hyperplanes. However, in the noiseless case, an algorithm that runs in time polynomial in N can be devised from the fact that each parameter vector w j can be determined from a subset of d points from the same mode. Assuming that such a subset exists for all modes, it suffices to find these subsets. A straightforward strategy is to consider all subsets of d data points among N for the first mode, d points among N -d for the second mode and so on. The number of collections of such disjoint subsets is

Conclusions

The paper showed that globally minimizing the error of a switching linear regression model is NP-hard, but also that, for fixed data dimension and number of modes, an exact algorithm with polynomial complexity in the number of data exists. This algorithm has an exponential complexity wrt. the data dimension and the number of modes, which strongly limits its practical applicability. Yet, the existence of an exact algorithm with polynomial complexity in the dimension is unlikely given the NP-hardness of the problem and the fact that it holds also with a fixed number of modes n = 2. Instead, future work will focus on polynomial-time approximation schemes.

  the problem min w∈Q d N i=1 ℓ(y iw T x i ) has a polynomial time complexity T (N ) for any fixed integer d ≥ 1.

  denote the set of binary linear classifications of N points S = {x i } N i=1 ⊂ Q d . Then, for any N > d, its cardinality is bounded as sup S∈(Q d ) N |H S | ≤ 2 d N d-1 and, for any set S of N points in general position, there is an algorithm that builds H S in O 2 d N d-1 iterations.

Theorem 3 .

 3 Let us define the set of minimum-error classifications of a data set S = {

  ) is bounded by the number of classifications generated by n(n -1)/2 pairs of linear classifiers (g jk , h jk ), 1 ≤ j < k ≤ n, on N -l points. Applying Proposition 2 twice, once in dimension d + 1 to bound the number of classifications returned by a g jk and once in dimension d to bound the number of classifications returned by a h jk , we can upper bound the number of classifications returned by a product classifier c jk by 2 2d+1 N -l d N -l d-1 = O(N 2d-1 ) and obtain an algorithm of a similar time complexity to compute these base classifications. With p = (2d -1)n(n -1)/2, this leads to n l O(N p ) classifications for each possible set E. Given an upper bound L on |E|, summing over all E of cardinality l ≤ L yields Π

  = O(N nd ) and thus polynomial in N . Since testing one of these collections amounts to solving n linear systems in constant time O(nd 3 ), the resulting algorithm runs in time polynomial in N .


M Cottrell

T S Turova

Use of an Hourglass Model in Neuronal Coding

Keywords: Ergodicity, Hourglass Model, Stochastic Neural Network AMS Classification: 60K20, 60K35

We study a system of interacting renewal processes which is a model for neuronal activity. We show that the system possesses an exponentially large number (with respect to the number of neurons in the network) of limiting configurations of the "firing neurons". These we call patterns. Furthermore, under certain conditions of symmetry we find an algorithm to control limiting patterns by means of the connection parameters.

Introduction

Definition of the model. We study here the hourglass model. This model is represented by a system of interacting renewal processes, numerated by the sites in a finite subset Λ ⊂ Z ν . Define for any z ∈ Λ its neighbourhood:

D(z) := {z ± l k , k = 1, . . . , K} ∩ Λ, (1.1)
where {l k , k = 1, . . . , K} ⊂ Z ν is a fixed but arbitrary set of non-zero vectors. Notice by our definition, that z ∈ D(z), and for any z, y ∈ Λ z ∈ D(y) iff y ∈ D(z).

Also let for any subset J ⊂ Λ D(J) := {j ± l k , k = 1, . . . , K, j ∈ J}.

Define on some probability space (Ω, Σ, P) some random variables X i (0), Y i , θ ij , i ∈ Λ, j ∈ D(i), to be independent with densities having a certain number of moments. The variables X i (0) and Y i , i ∈ Λ, are assumed to be positive and to represent the initial state and the self-characteristic of the ith neuron, respectively. We shall call the θ ij the connection parameters. For any fixed i ∈ Λ, j ∈ D(i), the distribution of θ ij is assumed to be concentrated either on R -or R + . We will specify the sign of θ ij in every example we treat below. Assume also θ ij ≡ 0 for any i ∈ Λ, j ∈ D(i). Finally let

Y (n) i , Y (n,z) i and θ (n) ij , θ (n,z) ij
, n ≥ 1, z ∈ Λ, be independent copies of the variables Y i and θ ij , respectively.

We shall define a Markov process X(t) = (X i (t), i ∈ Λ), t ≥ 0, with left-continuous trajectories in R Λ + as follows. For all i ∈ Λ and t > 0 define

X i (t) = X i (0) -t + 0<tn<t: X i (tn)=0 Y (n) i (1.2) + j∈D(i) 0<tn<t: X j (tn)=0 (Y (n,j) i -X i (t n ))I{X i (t n ) ≤ θ (n) ji } -θ (n) ji I{X i (t n ) > θ (n) ji } - j∈D(D(i))\{i}∪D(i) z∈D(i)∩D(j):θ zi <0 0 < t n < t : X j (t n ) = 0, X i (t n ) > θ (n) ji , and X z (t n ) ≤ θ (n,j) jz θ (n,j) zi .
Biological interpretation. Any moment t such that X z (t) = 0 for some z ∈ Λ we call the moment of firing of the zth neuron. Assume for a moment that θ ji ≡ 0 for any i, j ∈ Λ. Then the last two summation terms in the right-hand side of equation (1.2) turn into zero, meaning that X i (t) is merely a renewal process in this case. Thus each component X i (t) represents the duration of time before the next firing of the corresponding ith neuron assuming no interaction takes place meanwhile.

In the presence of non-zero interactions the dynamics of X(t) = (X i (t), i ∈ Λ), t ≥ 0, are described as follows. As long as all the components of X(t) are strictly positive, they decrease from the initial state X(0) linearly in time with rate one until the first time t z 1 that one of the components reaches zero for some z 1 ∈ Λ: X z 1 (t z 1 ) = 0. We say that at this moment t z 1 , the z 1 th neuron fires and sends impulse θ

(1) z 1 j to the jth neuron, if j ∈ D(z 1 ). This means the following. At the moment t z 1 , the trajectory X z 1 (t z 1 ) jumps to a random value Y (1) z 1 , i.e. X z 1 (t

z 1 +) = Y (1) z 1 , (1.3)
which corresponds to the first summation term in (1.2). At the same moment every trajectory X j (t) with j ∈ D(z 1 ) receives an increment θ

z 1 j independent of the other processes, more precisely, according to the second summation term in (1.2)

X j (t z 1 +) =        X j (t z 1 ) -θ (1) z 1 j , if X j (t z 1 ) -θ (1) z 1 j > 0, Y (1) j ,
otherwise.

(1.4)

Notice, that by (1.4) the negative connections θ

(1) z 1 j delay the moment when X j (t) hits zero, i.e. the moment of firing of the j-th neuron. Whereas positive connections can shorten the time-interval until the next firing. That is why we shall call negative connections inhibitory and the positive ones excitatory.

Case 1. Assume that all the interactions θ ij , j ∈ D(i), are negative. Then definition (1.2) simply becomes

X inh i (t) = X i (0)-t+ 0<tn<t: X i (tn)=0 Y (n) i + j∈D(i) 0<tn<t: X j (tn)=0 |θ (n) ji |, (1.5)
for i ∈ Λ and t ≥ 0. We use a notation X inh (t) for this particular case. This model is due to Cottrell (1992) [START_REF] Cottrell | Mathematical analysis of a neural network with inhibitory coupling[END_REF] and it has been intensively studied (see an account of the previous results in Section 2 below).

Case 2. Assume, some of the interactions θ ij are positive. Notice that, only if θ z 1 j 1 > 0, it may happen that X j 1 (t z 1 )θ

(1) z 1 j 1 ≤ 0. In this case the trajectory X j 1 (t) is reset instantaneously to an independent value Y (1) j according to (1.4). Furthermore, we say that the j 1 th neuron also fires at the moment t z 1 and changes instanteneously the trajectories of the neighbouring neurons according to the last term in (1.2), which means the following.

Given X z 1 (t z 1 ) = 0, define the set of the firing at the same moment t z 1 neurons:

F 1 (z 1 , t z 1 ) := {j 1 ∈ D(z 1 ) : X j 1 (t z 1 ) -θ (1)
z 1 j 1 ≤ 0}. Then at time (t z 1 +) the state of the system is defined as follows:

X j (t z 1 +) (1.6) =                  Y (1) j , if j ∈ {z 1 } ∪ F 1 , X j (t z 1 ) -θ (1)
z 1 j -i θ (1) ij , if j ∈ D(z 1 ) \ F 1 , X j (t z 1 ) -i θ (1) ij , if j ∈ D(F 1 ) \ (D(z 1 ) ∪ F 1 ∪ {z 1 }) ,
where

F 1 = F 1 (z 1 , t z 1 )
, the summation i runs over the set {i ∈ F 1 : i ∈ D(j) and θ ij < 0}, and any summation over an empty set equals zero. The rest of the trajectories X i (t) with i ∈ D(F 1 ) ∪ F 1 ∪ D(z 1 ) ∪ {z 1 } remain unchanged. After moment t z 1 the foregoing dynamics are repeated.

To end this description recall the well-known fact from physiology that a neuron does not react to the incoming impulses during a certain period right after it's own firing. This period is called a refractory period. Observe that in our model a neuron receives an impulse only in the moments when it does not fire itself. This reflects the property of the refractory period.

This type of neural network has been proved to be equivalent, in a sense, to the "classical" neural model, which describes the interacting membrane potentials (see [START_REF] Turova | Analysis of a biologically plausible neural network via an hourglass model[END_REF]).

Plan of the paper. Our paper is organized as follows. In Section 2 we find the critical values of the parameters which separate the ergodic and transient cases. Also we provide some historical comments on our model in Section 2. In Section 3 we analyze a fully connected network with inhibitory connections only, in which case we solve a problem related to the memory capacity.

2 Critical parameters.

Results.

Here we formulate our results on the critical values of the parameters which separate the ergodic and transient cases of the networks with excitatory and inhibitory connections.

In order to eliminate boundary effects, we shall assume here that Λ = {-N, . . . , N} ν is a ν-dimensional torus, i.e. we identify any two points (i 1 , . . . , i ν ) and (j 1 , . . . , j ν ) in Z ν whenever |i mj m | ∈ {0, 2N} for every 1 ≤ m ≤ ν. Hence, any point in Λ has the same number of neighbours. Further we assume that N > 1 and ν ∈ N + are fixed but arbitrary (although only the cases ν = 1, 2, 3 are relevant for our context). Define for any i, j

∈ Λ i -j Λ := ν k=1 |i k -j k | 2N ,
where |x -y| 2N := min{|x -y|, |xy -2N|, |xy + 2N|} for any x, y ∈ Z. Let Λ 0 be the subset of the points in Λ such that: i) the origin (0, . . . , 0) ∈ Λ 0 , ii) xy Λ = 1 for any x, y ∈ Λ 0 .

Consider now the network X(t) = (X i (t), i ∈ Λ) defined in (1.2) with the following connection architecture.

Assumption 1 Let the neighbourhood D(i) defined in (1.1) be such that

D(i) = D I (i) ∪ D E (i), where D I (i) = {j ∈ Λ : i -j Λ = 1}
and

D E (i) ⊆ Λ 0 iff i ∈ Λ 0 , D E (i) ⊆ Λ \ Λ 0 iff i ∈ Λ \ Λ 0 , (2.1)
with |D I (i)| = 2ν and |D E (i)| = K E . The constants 0 < K E < N are fixed but arbitrary.

Then we set

θ ij =      -w I η ij 1 , if j ∈ D I (i), w E η ij 2 , if j ∈ D E (i), (2.2)
where η ij k , i = j are independent copies of positive independent random variables η k , respectively, k = 1, 2, with Eη k = 1, and where w I and w E are positive parameters of the inhibitory and excitatory connections, respectively.

In words the condition (2.2) means that the nearest connections are inhibitory while the more distant ones are excitatory.

Assume further that X(0), X i (0), i ∈ Λ, are i.i.d., and also Y, Y i , i ∈ Λ, are i.i.d. with EY = 1.

Assumption 2 Assume, that the distributions of Y, η 1 , η 2 , and X(0) have densities g 0 (u), g 1 (u), g 2 (u) and g 3 (u), respectively, which are positive differentiable functions, such that for some positive constants a and α g k (u) ≤ ae -αu for all u > 0 and k = 0, 1, 2, 3.

(

We note that the assumption of the exponential decay, though seemingly rather restrictive, arises naturally from the physiology (see, for example [START_REF] Turova | Analysis of a biologically plausible neural network via an hourglass model[END_REF] and the references therein).

Theorem 1 Under Assumptions 1 and 2 for any w E ≥ 0 there exists a positive w cr I (w E ) such that the system X(t) with parameters (2.2) is transient if

w I > w cr I (w E ), (2.4)
and ergodic if w I < w cr I (w E ).

(2.5)

We specify the critical function w cr I (w E ) in (2.15) below. Generally speaking, this function depends on N. However, its asymptotic behaviour when w E → 0 is uniform in N, as we establish in the following theorem.

Theorem 2 There exist positive constants C and C 0 independent of N such that | w cr I (w E ) -

1 2ν - K E 2ν w E |≤ Cw 2 E (2.6)
for all 0 ≤ w E ≤ C 0 .

We postpone the proofs of these theorems to the next section. So far only the case w E = 0 has been studied analytically. Cottrell [START_REF] Cottrell | Mathematical analysis of a neural network with inhibitory coupling[END_REF] proved that when w E = 0, the network X(t) = X inh (t) (see (1.5) ) is ergodic, whenever w I < 1/(2ν), which is, clearly a particular case of (2.6). In [START_REF] Cottrell | Mathematical analysis of a neural network with inhibitory coupling[END_REF] sufficient conditions for convergence and transience were found for the finite network X(t) with Λ ⊂ Z 2 and a specific structure of the connections. Further Piat [START_REF] Piat | Modélisation du système olfactif par une approche neuromimétique: aspects cognitifs et théoriques[END_REF] (1994) extended these results to a more general connection structure. [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF] provided a complete analysis of the evolution of inhibitory networks when the matrix of the expected values of the connections defines a self-adjoint operator.

Paper [START_REF] Cottrell | A stochastic model for interconnected neurons[END_REF] presents simulated results for the model, in which both inhibitory and excitatory connections were incorporated. Here for the first time we study analytically a rigorous mathematical model for such a network. Our result explains the slope of the diagram given on Fig. 4 [START_REF] Cottrell | A stochastic model for interconnected neurons[END_REF] in the neighbourhood of the critical value (0, 1/(2ν)). The most attractive feature of the hourglass network for neuromodelling is that in the transient case the system splits as t ↑ ∞ into two subsets. These are a subset of active (i.e. infinitely often firing) neurons and one of inactive neurons, which can be recognized as dark and white areas, respectively, in the simulations, see e.g. figures 3 and 6 of [START_REF] Cottrell | A stochastic model for interconnected neurons[END_REF]. A rigorous definition of the possible limiting patterns of active and inactive neurons, called traps (see Section 2.2.1 below) was given by Karpelevich, Malyshev and Rybko in [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF]. Thus the set of all possible traps for a network should be naturally thought of as a system of patterns, which the network can hold, that is, memorize and recognize, hopefully. Recently [START_REF] Malyshev | Gibbs Measures on Attractors in Biological Neural Networks. Gibbs measures on attractors in biological neural networks[END_REF] obtained a result on phase transitions in the thermodynamical limit, which establishes the large memory capacity of these networks.

Further (in Section 3) besides a description of the possible patterns, we solve an inverse problem. This is the problem of how to determine the connections in order to get a system which stores a given system of patterns.

2.2 Proofs of Theorem 1 and Theorem 2.

Preliminary definitions and results.

We shall use in our proofs the results of [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF]. Therefore firstly we introduce the terminology of [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF] for our model.

For any non-empty W ⊂ Λ we call X W (t) the restriction of process X(t) on the set W . This is defined as X(t), but with initial conditions

X W i (0) := X i (0), if i ∈ W, ∞, if i ∈ W,
in which case we assume that

X W i (t) = ∞ if i ∈ W , for all t > 0, i.e. the nodes Λ \ W are "deleted".
Assume, the process (X W i (t), i ∈ W ) is ergodic for some non-empty W ⊂ Λ. In this case we define for any i ∈ W and j ∈ D E (i) ∩ W the following limiting frequencies:

π W,0 i := lim T →∞ 1 T #{0 < t n < T : X W i (t n ) = 0}, (2.7)
and

π W,e ij := lim T →∞ 1 T #{0 < t n < T : X W j (t n ) = 0 and X W i (t n ) -θ (n) ji ≤ 0}. (2.8) Thus π W,0 i
is the limiting frequency of firing of the ith neuron due to the hitting 0 by the trajectory X W i (t), while π W,e ij is the limiting frequency of firing of the ith neuron due to an immediate excitatory impulse from the jth neuron. Then it is natural to call the (total) limiting frequency of firing of the ith neuron the following sum of the defined above limits:

π W i := π W,0 i + j∈D E (i)∩W π W,e ij , i ∈ W.
(2.9)

Next we define the second vector field v W = (v W j , j ∈ Λ \ W ) by the following formula for its components:

v W j = -1 - i∈D I (j)∩W Eθ ij π W i - i∈D E (j)∩W Eθ ij π W,0 i , j ∈ Λ \ W. (2.10)
Formally v W j is the limiting mean drift of the jth component X W j (t). Indeed, according to our definition (1.2) the ith neuron sends inhibitory impulses to the corresponding neighbours with the limiting frequency π W i , but it sends excitatory impulses with the limiting frequency π W,0 i . For the details on the application of the theory of the second vector field we refer to [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF] and the references therein.

We shall call a trap for the process X(t) any non-empty set M ⊂ Λ such that the process (X Λ\M i (t), i ∈ Λ \ M) is ergodic, while any coordinate of the second vector field v Λ\M is positive, i.e. v Λ\M j > 0 for all j ∈ M.

(2.11)

For future reference let us rewrite now in our notations the criteria from [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF] on inductive ergodicity and transience conditions.

Theorem A (See Theorem 2.1, [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF].) I. Inductive ergodicity. If for any W ⊂ Λ the process (X W i (t), i ∈ W ) is ergodic and v W j < 0 for all j ∈ Λ \ W then the process X(t) is ergodic.

II. Sufficient transient conditions. If there exists a set M which is a trap for the process X(t), then the process X(t) is transient.

Subsystems with excitatory connections only.

Consider now the restriction X Λ 0 (t). Notice that due to the definition of Λ 0 and Assumption 1 the only connections between the components of (X Λ 0 i (t), i ∈ Λ 0 ) are excitatory. Then by the definition of the process X Λ 0 (t) we have for all i ∈ Λ 0 :

X Λ 0 i (t) = X i (0) -t + 0<sn<t: X Λ 0 i (sn)=0 Y (n) (2.12) - j∈D E (i) 0<tn<t: X Λ 0 j (tn)=0 w E η (n,j) 2 I{X Λ 0 i (t n ) > w E η (n,j) 2 } +(X Λ 0 i (t n ) -Y (n,j) )I{X Λ 0 i (t n ) ≤ w E η (n,j) 2 } ,
where η

(n,j) 2

, n ≥ 1, j ∈ Λ 0 , are independent copies of the variable η 2 . Clearly, for any i ∈ Λ 0 the one-dimensional X {i} i (t) is ergodic. Further for any w E > 0 and for any W ⊂ Λ 0 such that (X W i (t), i ∈ W ) is ergodic, the second vector field defined in (2.10) has negative coordinates only:

v W j = -1 - i∈D E (j)∩W w E π W,0 i , j ∈ Λ 0 \ W.
Hence, by Theorem A on inductive ergodicity (X Λ 0 i (t), i ∈ Λ 0 ) is ergodic for any fixed w E > 0. Taking into account translation invariance, we derive from (2.12) and (2.9)

π Λ 0 i =: π + (w E ) = π +,0 (w E ) + K E π +,e (w E ) > 0 (2.13)
for any i ∈ Λ 0 . In particular,

π + (0) = π +,0 (0) = 1/EY = 1, (2.14)
since in the case w E = 0 we have a system of independent renewal processes numerated by the sites of Λ 0 . Thus one can view the system X(t) as a result of inhibitory interactions between two (independent at initial moment) excitatory networks:

(X i (t), i ∈ Λ 0 ) and (X i (t), i ∈ Λ \ Λ 0 ).

2.2.3

Proof of Theorem 1.

We shall show that under condition (2.4), with

w cr I (w E ) = 1 2νπ + (w E ) (2.15)
the set Λ 0 is a trap for the system X(t). Then the first statement on transience in our theorem follows from Theorem A.

As we have shown above, (X Λ 0 (t), i ∈ Λ 0 ) is ergodic. Computing the second vector field with respect to formula (2.10), and taking into account (2.2) and (2.13) we get

v Λ 0 j = -1 + i∈D I (j) w I π Λ 0 i = -1 + 2νw I π + (w E ), j ∈ Λ \ Λ 0 . (2.16)
Hence condition (2.4) together with (2.15) is equivalent to

v Λ 0 j > 0, j ∈ Λ \ Λ 0 , (2.17)
i.e. inequality in (2.11) is fulfilled for all j ∈ Λ \ Λ 0 , which implies that Λ 0 is a trap. Hence, our statement on transience follows by Theorem A. Assume now that condition (2.5) with (2.15) holds. Clearly, for any i ∈ Λ one-dimensional X {i} i (t) is ergodic. Next we will show that for any ergodic face W the components of the second vector field v W defined by (2.10) are negative. It obviously follows from our definition of the model, that the limiting firing frequency is the highest when the neuron has the excitatory connections only and the maximal number of them, i.e.

π W i ≤ π Λ 0 j = π + (w E ) for any W ∈ Λ, any i ∈ Λ \ W and j ∈ Λ \ Λ 0 .
From here and (2.10) we derive for any i

∈ Λ \ W v W i ≤ -1 + 2νw I π + (w E ) < 0,
where the last inequality is due to (2.5). Hence, using Theorem A (Theorem 2.1 [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF]) on inductive ergodicity, we readily derive our statement on ergodicity. Theorem 1 is proved.

Proof of Theorem 2.

Let p

A (t, u A ), u A ∈ R + |A| , denote for any finite A ⊆ Λ 0 the density of the process X Λ 0 A (t) = (X Λ 0 i (t), i ∈ A).
In the particular case w E = 0, we denote the corresponding density by p 0 A (t, u A ). Due to ergodicity the following limit lim

t→∞ p A (t, u A ) =: p A (u A ) (2.18)
exists for any A ⊆ Λ 0 . In particular, we have for the limiting density of an independent renewal process:

lim t→∞ p 0 A (t, u A ) = p 0 A (u A ) = j∈A p 0 j (u j ) (2.19)
where

p 0 j (u) = ∞ u g 0 (v)dv EY =: p 0 (u).
(2.20) Further we will use the following lemma stating that the limiting density in (2.18) is a small perturbation of the function in (2.19) when the parameter of interaction w E is sufficiently small. Lemma 1 There exist positive and independent of N constants C, β and c such that for any 0 < w E ≤ c and for any subset

A ⊆ Λ 0 |p A (t, u A ) -p 0 A (t, u A )| ≤ w E C |A| e -β i∈A u i , (2.21)
for any t > 0 and u A ∈ R A + .

Proof. Recall that g 0 and g 3 are the density functions of the variables Y and X(0), respectively. Let p(t, u Λ 0 , v Λ 0 ), u Λ 0 , v Λ 0 ∈ R + Λ 0 , denote the transition density of the process (X Λ 0 i (t), i ∈ Λ 0 ). Then for any A ⊆ Λ 0 the density p A (t, u A ), u A ∈ R + |A| , is given by the formula

p A (t, u A ) = R + |Λ 0 \A| R + Λ 0   z∈Λ 0 g 3 (u z )   p(t, u Λ 0 , v Λ 0 ) d u Λ 0 d v Λ 0 \A . (2.22)
In the particular case w E = 0, we denote the transition density by p 0 (t, u Λ 0 , v Λ 0 ). Clearly, in this case

p 0 (t, u Λ 0 , v Λ 0 ) = j∈Λ 0 p 0 (t, u j , v j )
due to the independence of the components. Notice, that the transition density of each of the independent renewal processes is Using the results of Stone [START_REF] Stone | On moment generating functions and renewal theory[END_REF] one can show that under Assumption 2 there exist positive constants C ′ and α ′ such that

p 0 (t, u, v) = (2.23) = δ(u -t -v), if 0 ≤ t < u, g 0 (t -u + v) + ∞ k=1 t-u 0 p S k (x)g 0 (t -u + v -x) dx, if 0 ≤ u ≤ t,
|p 0 (t + t ′ , 0, v) -p 0 (t, 0, v)| ≤ C ′ e -α ′ (t+v) (2.24)
for all v ∈ R + and t, t ′ > 0 (for the details see for example, [START_REF] Turova | Exponential rate of convergence of an infinite neuron model with local connections[END_REF] Section 4). Combining (2.24) and (2.19), we get:

|p 0 (v) -p 0 (t, 0, v)| ≤ C ′ e -α ′ (t+v) .
(2.25) (For the general theory of piece-wise linear stochastic processes we refer to the book [START_REF] Davis | Markov Models and Optimization Chapman and Hall[END_REF] by Davis, 1993.) After these preliminaries the proof of Lemma 1 follows by standard techniques using so-called cluster expansions (see, e.g. [START_REF] Turova | Exponential rate of convergence of an infinite neuron model with local connections[END_REF] for a similar model and [START_REF] Ignatyuk | Stability of infinite systems of stochastic equations[END_REF] for a class of stochastic processes with local weak interactions) as soon as the Kolmogorov equation (see (2.26) below) is written down for the transition density p(t, u Λ 0 , v Λ 0 ).

Let M (R Λ 0 + ) denote a class of densities µ on R Λ 0 + which possess all continuous partial derivatives. Define also

p(t, u Λ 0 , Γ) = Γ p(t, u Λ 0 , v Λ 0 )dv Λ 0 for any Borel set Γ ∈ R Λ 0 + . Denote here O(i) = D E (i)
and w E = ǫ. Then for our model we derive for any density µ ∈ M (R Λ 0 + ) and for any Borel set

Γ ∈ R Λ 0 + ∂ ∂t R Λ 0 + µ(u Λ 0 ) p(t, u Λ 0 , Γ) du Λ 0 (2.26) = - R Λ 0 + i∈Λ 0 µ(u Λ 0 ) ∂ ∂u i p(t, u Λ 0 , Γ) du Λ 0 + R Λ 0 + µ(u Λ 0 ) i∈Λ 0 δ(u i ) R 1+2|O(i)| + p Y (w i ) j∈O(i) p Y (y j ) p ǫη 2 (w j ) × [ p(t, u Λ 0 + U Λ 0 (u Λ 0 , w i O(i) , y O(i) ), Γ) -p(t, u Λ 0 , Γ) ] × d y O(i) d w i O(i) du Λ 0 ,
where δ(•) is the Dirac measure concentrated at 0,

p Y (u) = g 0 (u), p ǫη 2 (u) = g 2 (u/ǫ)/ǫ, if ǫ > 0, δ(u), if ǫ = 0, u ≥ 0;
and the vector U Λ 0 (u Λ 0 , w i O(i) , y O(i) ) has components:

U j (u Λ 0 , w i O(i) , y O(i) ) =      w i , if j = i, -w j I{w j < u j } + (y j -u j )I{w j ≥ u j }, if j ∈ O(i ), 0, otherwise.
Further for any w i ≥ 0 let the vector wi = ( wi j , j ∈ Λ 0 ) have the following components:

wi j = w i , if j = i, 0, otherwise.
Let us rewrite now formula (2.26) in the following operator form:

∂ ∂t R Λ 0 + µ(u Λ 0 ) p(t, u Λ 0 , Γ) du Λ 0 (2.27) = R Λ 0 + µ(u Λ 0 ) i∈Λ 0 - ∂ ∂u i p(t, u Λ 0 , Γ) + δ(u i ) R + p Y (w i )[p(t, u Λ 0 + wi , Γ) -p(t, u Λ 0 , Γ)] dw i du Λ 0 + R Λ 0 + µ(u Λ 0 ) i∈Λ 0 δ(u i ) R 1+2|O(i)| + p Y (w i ) j∈O(i) p Y (y j ) p ǫη 2 (w j ) ×[ p(t, u Λ 0 + U Λ 0 (u Λ 0 , w i O(i) , y O(i) ), Γ) -p(t, u Λ 0 + wi , Γ) ] × d y O(i) d w i O(i) d u Λ 0 =: R Λ 0 + µ(u Λ 0 ) i∈Λ 0 (H i,Λ 0 0 + H i,Λ 0 1 ) p(t, u Λ 0 , Γ)d u Λ 0 .
Notice, that when ǫ = 0 the operator H i,Λ 0 1 ≡ 0 for any i. In this case (2.27) simply becomes

∂ ∂t R Λ 0 + µ(u Λ 0 ) p 0 (t, u Λ 0 , Γ) du Λ 0 = R Λ 0 + µ(u Λ 0 ) i∈Λ 0 H i,Λ 0 1 p 0 (t, u Λ 0 , Γ) d u Λ 0
(2.28) i.e., the equation for the density of the process whose components are independed renewal processes. Hence the formula (2.23) gives a solution to (2.28):

p 0 (t, u Λ 0 , Γ) = Γ p 0 (t, u Λ 0 , v Λ 0 ) d v Λ 0 .
Now we can find the solution to equation (2.27) by solving the equivalent integral equation (for the reference see also [START_REF] Turova | Exponential rate of convergence of an infinite neuron model with local connections[END_REF] and [START_REF] Ignatyuk | Stability of infinite systems of stochastic equations[END_REF]):

R Λ 0 + µ(u Λ 0 ) p(t, u Λ 0 , Γ) du Λ 0 = R Λ 0 + µ(u Λ 0 ) p 0 (t, u Λ 0 , Γ) du Λ 0 (2.29) + t 0 R Λ 0 + R Λ 0 + µ(u Λ 0 ) p 0 (t -s, u Λ 0 , u 1 Λ 0 ) i∈Λ 0 H i,Λ 1 p(s, u 1 Λ 0 , Γ) du 1 Λ 0 du Λ 0 ds.
Finally from (2.29) we obtain the formula for the density defined by (2.22):

p A (t, v A ) = p 0 A (t, v A ) (2.30) + ∞ k=1 (i 1 ,...,i k )∈Λ k 0 t 0 R Λ 0 + R Λ 0 + z∈Λ 0 g 3 (u z ) p 0 (t -s 1 , u Λ 0 , u 1 Λ 0 ) H i 1 ,Λ 0 1 s 1 0 R Λ 0 + p 0 (s 1 -s 2 , u 1 Λ 0 , u 2 Λ 0 ) . . . H i k ,Λ 0 1 s k-1 0 R Λ + p 0 (s k , u k Λ 0 , v Λ 0 ) dv Λ 0 \A du k Λ 0 ds k . . . du 1 Λ 0 du Λ 0 ds 1 ,
Making use of the formula (2.23) and the bounds (2.3) and ( 2.24), one can prove the convergence of the series in (2.30) and get the necessary bounds by following the the formulae in Section 3 of [START_REF] Turova | Exponential rate of convergence of an infinite neuron model with local connections[END_REF] (p.179 and further) with only minor modifications due to (2.26). Therefore for the sake of brevity we skip the rest of the straightforward part of the proof of Lemma 1.

Consider now the embedded Markov chain (x i (n) := X Λ 0 i (τ n ), i ∈ Λ 0 ), n = 1, 2, . . . , where τ 1 , τ 2 , . . . denote the consecutive moments when at least one of the components of X Λ 0 reaches zero, i.e. X Λ 0 j (τ n ) = 0 for some j ∈ Λ 0 . Due to ergodicity of (X Λ 0 i (t), i ∈ Λ 0 ), there exists a limiting distribution, call it F , of x(n) as n → ∞. Let x ∞ (n) be the stationary version of the Markov chain x(n), i.e. whose initial distribution is F . Further for any i ∈ Λ 0 and j ∈ D E (i) let Z ij be a random variable with distribution function

F ij (u) := P {x ∞ i (n) < u | x ∞ j (n) = 0}. Clearly, the density function f ij for each F ij is given by f ij (u) = p {ij} (u, 0) p j (0) (2.31)
as long as p j (0) > 0. Observe that p 0 j (0) = 1 EY = 1 by (2.20). Hence, according to Lemma 1 condition p j (0) > 0 is satisfied at least for all small values of w E .

We shall derive now an equation for π Λ 0 ,0 i = π +,0 (w E ) (see definition (2.7) and (2.13) ). From (2.12) and the above arguments on ergodicity we obtain for all i ∈ Λ 0 :

lim t→∞ 1 t X Λ 0 i (t) = 0 = -1 + π +,0 (w E )EY -K E π +,0 (w E )w E (2.32) - j∈D E (i) π +,0 (w E )E(Z ij -Y -w E η 2 )I{Z ij ≤ w E η 2 },
where Z ij , Y and η 2 are independent by their definitions. Therefore taking into account that EY = 1, we obtain from (2.32) the following equation:

π +,0 (w E )   1 -K E w E + j∈D E (i) P{Z ij ≤ w E η 2 } + j∈D E (i) E(w E η 2 -Z ij )I{Z ij ≤ w E η 2 }   = 1. (2.

33)

Lemma 2 There exists a positive constant C 1 independent of N such that for any i ∈ Λ 0 and for any j ∈ D E (i)

|P{Z ij ≤ w E η 2 } -w E | ≤ w 2 E C 1 (2.34) and E(w E η 2 -Z ij )I{Z ij ≤ w E η 2 } ≤ w 2 E C 1 (2.35)
for all 0 ≤ w E ≤ c (with c defined in Lemma 1).

Proof of Lemma 2. In the case

w E = 0 let Z ij = Z 0 ij . Then the density of Z 0 ij (call it correspondingly f 0 ij (u) ) is given by f 0 ij (u) = p 0 (u) = ∞ u g 0 (y)dy (2.36) due to (2.

31) and (2.20).

First let us obtain the following bound as a corollary of Lemma 1:

|f ij (u) -f 0 ij (u)| ≤ w E C 2 , u ∈ R + , (2.37)
where C 2 is some positive constant independent of N. Notice, that from (2.21) and (2.18) we immediately derive:

|p A (u A ) -p 0 A (u A )| ≤ w E C 3 exp{-β i∈A u i }, (2.38)
for any u A ∈ R A + , where C 3 is some positive constant independent of N. Then the bound (2.37) readily follows from (2.38) and the continuity of the transformation (2.31).

Let us prove now (2.34). Consider

P{Z ij ≤ w E η 2 } = ∞ 0 g 2 (y) w E y 0 f 0 ij (u)du dy (2.39) + ∞ 0 g 2 (y) w E y 0 (f ij (u) -f 0 ij (u)) du dy.
Substituting formula (2.36) into the first integral in (2.39) we derive after simple calculations:

∞ 0 g 2 (y) w E y 0 f 0 ij (u)du dy (2.40) = ∞ 0 P{η 2 ≥ y}w E (1 -P{Y ≤ w E y}) dy = w E -w E ∞ 0 P{η 2 ≥ y}P{Y ≤ w E y} dy,
where we used the condition Eη 2 = 1. Taking into account (2.3) it is easy to see that

w E ∞ 0 P{η 2 ≥ y}P{Y ≤ w E y} dy ≤ C ′ w 2 E (2.41)
for some positive constant C ′ . Next, making use of (2.37) we readily derive

| ∞ 0 g 2 (y) w E y 0 (f ij (u) -f 0 ij (u)) du dy| ≤ w 2 E C 2 . ( 2
E(w E η 2 -Z ij )I{Z ij ≤ w E η 2 } = ∞ 0 g 2 (y) w E y 0 (w E y -u)f 0 ij (u) du dy + ∞ 0 g 2 (y) w E y 0 (w E y -u)(f ij (u) -f 0 ij (u)) du dy.
Using again (2.3) and (2.37) we immediately derive from here:

E(w E η 2 -Z ij )I{Z ij ≤ w E η 2 } ≤ w 2 E C 3 + w 3 E C 4 , (2.43)
where C 3 and C 4 are some positive constants, which implies (2.35). The lemma is proved. Lemma 2 allows us to derive from equation (2.33) that

|π +,0 (w E ) -1| ≤ w 2 E C 5 (2.44)
for some positive constant C 5 . Finally let us consider π +,e ij (w E ) for i ∈ Λ 0 , j ∈ D E (i). According to definition (2.8) we derive similarly (2.32)

π W,e ij = lim t→∞ 1 t #{0 < t n < t : X W j (t n ) = 0 and X W i (t n ) -θ (n) ji ≤ 0} (2.45) = lim t→∞ 1 t 0<tn<t: X Λ 0 j (tn)=0 I{X Λ 0 i (t n ) ≤ w E η (n) 2 } = π +,0 (w E ) P{Z ij ≤ w E η 2 }.
Substituting (2.45) into definition (2.13) we obtain

π + (w E ) = π +,0 (w E )(1 + j∈D E (i) P{Z ij ≤ w E η 2 }), (2.46)
which together with (2.44) and (2.34) gives us the following bound:

|π + (w E ) -(1 + K E w E )| ≤ w 2 E C 6 (2.47)
for all w E ≤ C 0 , where C 6 and C 0 are some positive constants. Substituting the bound (2.47) into (2.15) we get (2.6). This completes the proof of Theorem 2.

3 Fully connected finite networks.

The model.

Let Λ = {1, . . . , N}. Consider an N-neuron network with inhibitory connections only, i.e. X(t) = X inh (t) as defined in (1.5). Here we assume that D(i) = Λ\{i} for any i ∈ Λ, which means that our network is fully connected.

Let N = 2pk, where k, p ∈ Z + are fixed arbitrary. Suppose a network consists of 2p symmetric subnets (blocks), determined by the values of the connection constants Eθ ij as follows. We divide set Λ = {1, ..., N} into 2p subsets W i , i = 1, ..., 2p, so that

|W i | = k and ∪ 2p i=1 W i = Λ. ( 3.1)
Notice that we keep k > 1 fixed but arbitrary. This is simply a size of one block or unit of our network. The case k = 1 is trivial. Let us divide also the set {1, ..., 2p} into a set of p non-intersecting pairs, i.e., {1, ...,

2p} = ∪ p n=1 C n = ∪ p n=1 {C 1 n , C 2 n }, (3.2)
where

|C n | = 2, and C i ∩ C j = ∅.
Then we set

EY x = a > 0, Eθ xy = -c ij < 0, if x ∈ W i , y ∈ W j , (3.3)
for all x, y ∈ Λ and 1 ≤ i, j ≤ 2p, where

c ij = c, if {i, j} = C n for some n, b, otherwise, (3.4)
for some arbitrary but fixed constants 0 < b < a < c.

Thus our network consists of 2p connected blocks of interacting identical neurons. The connections between the neurons of different blocks are different in general from those between the neurons of the same block. In the following Theorem 3 below, we describe all the possible traps for our network, i.e. all the possible limiting states in the transient case. Notice that the total number of the patterns is |T | = 2 p = 2 N 2k , which is exponentially large with respect to N. This description will allow us to find a method to reconstruct or "memorize", in some particular cases, a given system of traps (see Corollary 1 below). More precisely, we will show how to choose the appropriate connection constants.

Description of patterns.

Theorem 3 Let N = 2pk, where k > 1, p > 1, and decompositions (3.1) and (3.2) be fixed but arbitrary. Call L the set of all the decompositions of {1, ..., 2p} into a pair of subsets such that

L := {(L 1 , L 2 ) : L 1 ∪ L 2 = {1, ..., 2p}, L 1 ∩ L 2 = ∅, (3.6)
L i ∩ C n = 1, i = 1, 2, n = 1, ..., p}.
Then a set A ⊂ Λ is a trap for the network with the connection constants (3.3)-(3.5), if and only if

A = {x ∈ Λ : x ∈ ∪ i∈B W i }, (3.7)
for some B ⊂ Λ such that (B, {1, ..., 2p} \ B) ∈ L.

(3.8)

Proof. First we will show that any set A satisfying conditions (3.7) and (3.8) is a trap for the above defined system. Indeed, consider a restriction X Λ\A (t). In this case (X Λ\A i (t), i ∈ Λ \ A) is a completely connected system, and for any x, y ∈ Λ \ A, y = x

-Eθ xy = b < a = EY x , (3.9)
which implies ergodicity of (X

Λ\A i (t), i ∈ Λ \ A) due to Proposition 2.2 in [7].
Assume, (X W i (t), i ∈ W ) is ergodic for some W ⊂ Λ. Observe that in the case of a fully connected network with only negative connections definition (2.9) becomes

π W i = π W,0 i , i ∈ W. (3.10)
Furthermore it follows by ergodicity from (1.5) and definition (2.7) that

π W i =   EY i + j∈W \{i} E|θ ji |   -1 , i ∈ W. (3.11)
Correspondingly, we derive from (2.10)

v W j = -1 + i∈W E|θ ij |π W i , j ∈ Λ \ W. (3.12)
Notice that definitions (3.1)-(3.2) imply |B| = p for the set B satisfying conditions (3.7)- (3.8). Then for any x ∈ Λ \ A we derive from (3.11) and condition (3.4) (see also [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF]), that Hence, we conclude that A is a trap. Next we will show that any trap satisfies (3.7)- (3.8). Clearly, any subset A ⊆ Λ can be represented as

π Λ\A x = 1 a + (|Λ \ A| -1)b = 1 a + (pk -1)b . ( 3
A = {x ∈ Λ : x ∈ ∪ i∈B A i }, (3.15
:= Λ \ A = {∪ i∈ B W i } ∪ {∪ i∈B W i \ A i }. (3.19)
Define the subset B 1 ⊆ B so that

W i \ A i = ∅ iff i ∈ B 1 , (3.20)
and thus

Ā = {∪ i∈ B W i } ∪ {∪ i∈B 1 W i \ A i }.
Notice that B 1 can be empty. Further define I 1 so that

C n ∈ B ∪ B 1 iff n ∈ I 1 . (3.21) (a) Suppose I 1 = ∅. Let B 0 := {C 1 n , n ∈ I 1 } ∪ { B ∪ B 1 \ {C n , n ∈ I 1 }}. (3.22) Denote ĀB 0 = {∪ i∈ B∩B 0 W i } ∪ {∪ i∈B 1 ∩B 0 W i \ A i }.
Then the subsystem (X ĀB 0 i (t), i ∈ ĀB 0 ) is ergodic, since it is completely connected, and (3.9) holds for any x, y ∈ ĀB 0 , x = y. Also, we can compute as in (3.13)

π ĀB 0 x = 1 a + (| ĀB 0 | -1)b , (3.23)
Taking into account that |I 1 | ≥ 1, we easily derive from (3.23) the following upper bound for the components of the second vector field v ĀB 0 :

v ĀB 0 z ≥ -1 + c + (| ĀB 0 | -1)b a + (| ĀB 0 | -1)b > 0 (3.24)
for any z ∈ Ā\ ĀB 0 due to condition (3.5). Hence, (X Ā i (t), i ∈ Ā) is transient according to Theorem A. This contradicts our assumption that A is a trap.

(b) Suppose now that I 1 = ∅. Then it follows from (3.21) and (3.17) that we also have

I ′ = ∅. (3.25)
This together with our assumption (3.18) implies that I = ∅. Thus in this case we have

A = {x ∈ Λ : x ∈ ∪ i∈B A i }, (3.26)
where B ∋ {C n , n ∈ I}, and

Ā = {∪ i∈ B W i } ∪ {∪ i∈B 1 W i \ A i }, (3.27)
where { B ∪B 1 }∩C n ≤ 1 for any n, according to (3.21) and (3.25). The latter implies that (3.9) holds for any y ∈ Ā which in turn implies ergodicity of (X Ā i (t), i ∈ Ā). Furthermore, we can find π Ā x , analogously to (3.13), namely: The latter contradicts our assumption, that A is a trap. Hence we conclude that if A is a trap then necessarily condition (3.8) holds. Assume now situation (II). More precisely, suppose that A satisfies (3.8) but does not satisfy (3.7), i.e.

π Ā x = 1 a + (| Ā| -1)b , x ∈ Ā. ( 3
A = {x ∈ Λ : x ∈ ∪ i∈B A i }, (3.30)
for some B ⊂ Λ such that (3.8) holds, while

W i \ A i = ∅ for i ∈ B 1 ⊆ B for some nonempty B 1 (see definition (3.20) ). The latter implies Ā = {∪ i∈ B W i } ∪ {∪ i∈B 1 W i \ A i },
where according to the assumption (3.8)

|B 1 ∩ C n | = 1 (3.31)
for at least one C n . The latter implies C n ∈ B ∪ B 1 , i.e. the set I 1 defined in (3.21) is non-empty. But as we have seen, this situation contradicts the assumption that A is a trap. This finishes our argument that conditions (3.7) and (3.8) are necessary for a set A to be a trap. This completes the proof of Theorem 3.

Learning rule for the almost symmetric case

We will show here that the Hebbian rule (see [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF]) of learning patterns which successfully works in the case of Hopfield networks ( [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF]), is applicable for our network at least for the following particular case. For any A ⊂ Λ define configuration ξ(A) = (ξ i (A), i ∈ Λ) ∈ {-1, +1} Λ such that

ξ i (A) = +1, if i ∈ A, -1, otherwise. (3.32)
We shall also call a configuration ξ a trap for X(t) if and only if ξ = ξ(A), where A is a trap. Suppose that we are given 2 p , p = N 2k , binary vectors (images) ξ 1 , . . . , ξ 2 p . We shall find the connection constants Eθ xy such that X(t) satisfying (3.3) with these parameters, possesses a system of traps consisting exactly of the given 2 p vectors.

Notice the difference between this task and the problem of stability of patterns for Hopfield neural model (see for example, [START_REF]Mathematical Aspects of Spin Glasses and Neural Networks[END_REF] for a recent account on the relevant results). Recall, that the capacity of Hopfield network is determined by the number of given i.i.d. patterns, which are stable with respect to the dynamics of the system. This means that starting from an arbitrary initial state the system should converge with a large probability to one of the given patterns, which is the closest to the initial state. It was conjectured that the number of such patterns for Hopfield network of N neurons is at most a fraction of N. Here we construct a network which posesses given exponentially large (with respect to N) number of the limiting patterns and only them. These patterns are stable in a trivial sense, i.e. if the initial state of the system is one of the given traps (patterns), then the system stays at this trap forever. However, we do not predict which state (out of p possible) the system converges to, starting from an arbitrary initial condition. The problem of determination of the basins of attraction of the limiting patterns for the hourglass model will be a subject of a separate study.

Corollary 1 Suppose the collection of N-dim vectors {ξ µ , µ = 1, . . . , M} where M = 2 p , and ξ µ

x ∈ {-1, 1} for any µ and 1 ≤ x ≤ N, has the following properties:

1. ξ µ x ξ µ y = 1 for any µ if x, y ∈ W n for some n ∈ {1, ..., 2p}, 2. N x=1 ξ µ x = 0 for all µ, 3. for any n ∈ {1, ..., 2p} there exists unique l = l(n) ∈ {1, ..., 2p} \ {n} such that ξ µ x ξ µ y = -1 for any µ whenever x ∈ W n and y ∈ W l . Then the N-neuron system with a i = a and the connection constants

  for every u, v ∈ R + and t > 0, where δ(u -•) is the Dirac distribution concentrated at u; Y, Y k , k ≥ 1, are i.i.d.; S k := k l=1 Y l , k ≥ 1, and p S k is the density of the distribution S k .

. 13 )

 13 Substituting (3.13) into definition (3.12) and taking into account (3.3)-(3.4), we obtain for any y ∈ A v Λ\A y := -1 + ck + (p -1)bk a + (pk -

  ) where B ⊆ {1, ..., 2p} and A i ⊆ W i for any i ∈ B. Suppose a subset A does not satisfy conditions (3.7)-(3.8), which happens if and only if at least one of the following situations (I) or (II) takes place: (I) the set B does not satisfy (3.8), which means that B = {C n , n ∈ I} ∪ {C l n , n ∈ J} (3.16) and B := {1, . . . , 2p} \ B = {C n , n ∈ I ′ } ∪ {C l ′ n , n ∈ J}, (3.17) where {l, l ′ } = {1, 2} and sets I and I ′ are such that I ∪ I ′ = ∅, I ∩ I ′ = ∅, and I ∪ I ′ ∪ J = {1, . . . , 2p}; (3.18) (II) W i \ A i = ∅ at least for some i ∈ B, i.e. (3.7) is not satisfied. Suppose situation (I) takes place. Without loss of generality let l = 1 and l ′ = 2. Consider Ā

  .28) Let us compute now the xth component of the second vector field v Ā x for x ∈ A i with i ∈ I. According to (3.28) and assumption (3.4) we get v Ā x := -1 + | Ā|b a + (| Ā| -1)b < 0. (3.29)

  Eθ xy = b(x, y), if b(x, y) = min (x ′ ,y ′ ) b(x ′ , y ′ ) max (x ′ ,y ′ ) b(x ′ , y ′ ), otherwise,(3.33)where b(x, y):= Aa 1 M M µ=1 ξ µ x ξ µ y -Ba,(3.34)with the constants A and B satisfying the conditions0 < B -A < 1, 1 < B + A,(3.35)has a system of traps, which is {ξ µ , µ = 1, . . . , 2 p }.Proof. Indeed, having conditions 1-3 of the corollary satisfied, we derive from(3.34) that b(x, y) = -(A + B)a, if x ∈ W n , y ∈ W l(n) , -(B -A)a, if x, y ∈ W n ,(3.36) and -(A + B)a < b(x, y) < -(B -A)a otherwise. (3.37) Substituting (3.36) snd (3.37) into (3.33), and taking into account condition (3.35), we obtain-Eθ xy = (A + B)a > a, if x ∈ W n , y ∈ W l(n) , -Eθ xy = (B -A)a < a, otherwise,(3.38)which shows that conditions (3.4)-(3.5) are satisfied. Hence we can use Theorem 3 to conclude that {ξ µ , µ = 1, . . . , 2 p } is a system of traps for the defined network.

Acknowledgements We thank the referee for useful remarks.


Michelle Garcia-Vega
email: michellegv@bellsouth.netmachonis@fiu.edu

Peter A Machonis

Keywords: lexicon-grammar, light verbs, support verb constructions, support verb take

Using the lexicon-grammar framework as developed by Maurice Gross (1994), we systematically analyzed 245 verbs that can be used as nominals with the light or support verb take. The data was divided into three main tables: purely intransitive verbs (e.g. bathe  take a bath), transitive (e.g. drink N 1  take a drink of N 1 ), and intransitive with prepositional complements (e.g. look at N 1  take a look at N 1 ). A final table demonstrates a set of converse take constructions as initially observed by Gaston Gross (1989), (e.g. N 0 confess N 1 to N 2 ↔ N 0 take confession ↔ N 2 take the confession).

Key words:

In (4) N 0 indicates the subject, N 1 indicates the first complement, and the arrow signifies relative synonymy. The V-n indicates a deverbal nominal, i.e. a verb derived from a noun.

All of the verbs in our corpus may undergo the transformation in (4). This criterion distinguishes the support verb take from its more ordinary uses such as in (1). It also distinguishes it from take in idiomatic expressions as in (2), as well as expressions in which the support verb construction cannot appear in simple verb form, i.e. Max took hold of the situation → *Max held the situation. Although it is at times difficult to clearly differentiate support verbs from idiomatic verbal expressions - [START_REF] Freckleton | Une taxonomie des expressions idiomatiques anglaises[END_REF], for example, lists (give + take) umbrage as an idiom, whereas others might see this as a support verb constructionfor this preliminary analysis we only considered take followed by a noun clearly related to a verb in the sense of (4) above.

The 245 verbs appearing as nominals in light verb take constructions were divided into three main tables: (1) purely intransitive verbs (e.g., bathe), (2) transitive verbs (e.g., chug N 1 ), and (3) intransitive with prepositional complement (e.g., swing at N 1 ). All the deverbal nominals in the data were also specified for determiner and quantificational phrases. These expressions are highly productive in language and, as will be shown, require the development of a formal lexicon or lexicon-grammar to best describe the not always predictable properties of support verb constructions. Nevertheless, as [START_REF] Newman | Give: A Cognitive Linguistics Study[END_REF] points out in his cognitive study of the verb give, the literal meaning of a support verb may influence which figurative extensions are acceptable. For example, the light verb take "is compatible with complements relating to an assumption of control," by the taker argues Newman (1996:245), "without any necessary implication that control was passed to that person from someone else," as in the case of give.

Intransitive verbs with support take

Intransitive verbal constructions that may appear in expressions with the light verb take consist of 62 of the 245 entries. Many of these may be categorized into a subclass of [START_REF] Levin | English Verb Classes and Alternations: A Preliminary Investigation[END_REF] These support verb take constructions, however, are not entirely predictable based solely on semantic class as seen by the unacceptability of the manner of motion verbs *take a (dart + roam + race + scurry + scram), the following sleep verbs *take a (drowse + sleep + slumber) and the ungrammaticality of the bodily function verbs in *take a (drool + vomit + weep + bleed). Although Newman's "implication of control" criterion might help to explain the acceptability and unacceptability to some of these verbs, there still seems to be an element of randomness, for example between the acceptability of take a puke, but not *take a vomit. For the verb race, the person who races might not have full control, and thus might help to explain the unacceptability of *take a race, yet the control factor does not sufficiently explain why roam and scurry are also unacceptable.

It is also worth mentioning that the support verb may contribute an aspectual interpretation not fully synonymous with the single verb expression. Thus a construction may be ambiguous between the semantically empty take and a more contentive verb, as in The lawyer took the foreclosure where the expression may be interpreted as "did, accept, choose." This is similar to the difficulty we saw in trying to delimit lexical make "bring into existence" from the support verb make in such expressions as make an estimate (Machonis 1991:146).

Finally, the choice of determiner may be responsible for the difference between the support verb take interpretation (e.g. Ty took a bath) and the literal meaning (e.g. I then took the bath to a panel beater and they sprayed the thing with baked enamel for $250), where the indefinite determiner a forces a more natural light verb interpretation, whereas the definite determiner the forces a literal one. However, even with the indefinite determiner a, there is also the possibility of the idiomatic expression take a bath on N, as in Max took a bath on that deal (i.e. "obtain a severe financial loss").

Determiners and quantifiers

All of the take constructions were systematically analyzed for the indefinite determiner a, the definite determiner the, zero article represented as E, and general quantificational phrases such as number terms and quantifiers like several, many, much, and the first. Over 90% of the data may take the indefinite determiner a in the take construction:

(6) Max took a (drive + swim + plunge + jog) There are, however, 30 expressions that do not accept the determiner as part of the light verb take construction, and prefer the zero article, noted E:

(7) The security officer took (E + *a) (watch + guard + patrol)

Although some might argue that these should be construed as idiomatic expressions, we point out that these also appear with the support verb keep, as well as part of be + Prep expressions:

(8) The security officer (keeps + is on) (watch + guard + patrol) Other verbs reject both the indefinite determiner a, as well as the zero article, but prefer the definite determiner the instead, like (9). Table 1 illustrates a sample of the intransitive support verb constructions with take and some of the possible combinatorial properties of determiners and quantificational phrases these expressions exhibit.

For each verb, a potential subject N 0 is marked for the properties of human (Nhum) and non-human (Nhum) indicated by a plus or minus sign. The next column illustrates the verb, while the following columns show the acceptability of the indefinite article a, zero article E, the definite article the, a quantifier, and the plus a quantificational phrase, marked by a plus or minus sign. While all the verbs in the data were specified for determiner and quantificational phrases, for exposition purposes, they have been excluded from the subsequent tables. Finally, the V-n indicates the nominalizing suffix of the verb, where -E represents empty morpheme. This column is followed by the intransitive construction and the semantic properties of the verb.

Transitive verbs with support take

Table 2 demonstrates a sample of the transitive verbs that may appear as a nominal complement to the light verb take. The first two columns are specified for human/non-human properties, followed by the verb, the determiners and quantifiers, and the nominalzing suffix, V-n. This column is followed by the intransitive N 0 V and transitive N 0 V N 1 features, and a potential first complement N 1 marked for Nhum and N-hum. The next three columns demonstrate the N 1 complement optionally reappearing in the light verb construction as part of a case or prepositional phrase such as of, at, with, on, in, over or around. The five columns of determiners, as seen in N 0 take V-n of N 1 was the most common transformation seen in our data. In fact, 75% of the constructions derived from the transitive verbs in Table 2 can introduce the second complement N 1 into the take expression by means of an optional case particle of phrase, as exemplified in ( 14) and ( 15): ( 14) Verbs of protecting: The officer is (patrolling + watching + observing + guarding) ↔ The officer is (patrolling + watching + observing + guarding) the area ↔ The officer is taking (patrol + watch + observations + guard) (E + of the area) (15) Verbs of recording: The director is (photographing + recording + videotaping) ↔ The director is (photographing + recording + videotaping) the artist ↔ The director is taking a (photograph + recording + videotape) (E + of the artist)

N 1 Semantic Properties + -observe + + + ---ation + + engine + -+ -----observe + -pardon -+ --- -E -+ his sins -+ ------forgive + + patrol -+ --- -E
+ + the area + + + ---+ + patrol +photocopy + + + + + -E + + paper -+ + ----photocopy + + photograph + + + + + -E + + Max + + + ----photograph + + pick + -----E + + the shirt + + + --+ -pick +pity -+ -+ --E -+ the man + ---+ --feel bad This highlights not only the difficulties of support verb constructions, but also of the need to further classify English ordinary verbs in a lexicon-grammar, similar to the work of Boons, Guillet & Leclère (1976). In fact, some of these verbs have both an intransitive and transitive use as ordinary verbs, while others are purely transitive even though they form part of the same semantic class:

+ + profile + -+ -- -E -+ Max + -+ -----profile + -profit + + + -- -E + + money -+ + -----profit + -puke + ---- -E + + the food -+ ------bodily function + -pull + --+ - -E -+ the door + + -+ ----pull + -punch + + -+ - -E -+ Max + + -+ ----punch + -quote + -+ + + -E -+ the car -+ + -+ ---estimate
(17) Verbs of inhaling: Max (breathed + sniffed + gasped + *smelled + *whiffed) ↔ Max (breathed + sniffed + gasped + smelled + whiffed) the air ↔ Max took a (breath + sniff + gasp + smell + whiff) (E + of the air)

Intransitive take constructions with prepositional complements

Table 3 of our data represents 70 light verb take entries that can be derived from intransitive verbs that take a prepositional phrase, but reject a direct object N 1 . These intransitive verbs with prepositional complements are marked with a plus or minus in the appropriate column syntactically represented as ( 18): The sample Table 3 shows the variety of prepositional complements these verbs may take and the variation exhibited when part of the light verb construction. Again, the five columns of determiners, as seen in Table 1, are not shown in this table due to space considerations.

(18) N 0 V ↔ *N 0 V N 1 ↔ N 0 V Prep N 1 ↔ N 0 take a V-n (E +
(19) *N 0 V ↔ *N 0 V N 1 ↔ N 0 V Prep N 1 ↔ N 0 take a V-n (E + Prep N 1 )

Residual data.

In addition to the three tables analyzed in this article so far, we also noticed two smaller, restricted, but rather interesting tables. Some of the verbs in these tables do not appear in Tables 1-3 (e.g. appeal, confess, criticize), while others (e.g. sail, hike) appear in other tables. Table 4 consists of 22 intransitive verbs that can also be transitive (e.g., sail (E + the ocean)). Due to space considerations, these residual tables are not shown, but briefly, the verbs in Table 4 can all introduce a prepositional phrase complement Prep N 1 that may be an implicit part of the construction when in its transitive form N 0 V N 1 . These transitive verbs with implicit locative prepositional phrase complements vary as to the type of preposition accepted by the construction, even if part of the same semantic class, as exemplified in ( 25):

(25) N 0 V N 1 ↔ N 0 take V-n (E + Prep N 1 ) a. We sailed the Atlantic ↔ We took a sail ↔ We took a sail (to +*of +*at + in + on + around + up) the Atlantic b. The pirate walked the plank ↔ ?The pirate took a walk ↔ The pirate took a walk (*to + *of + *at + on + *around + up) the plank c. The old man hiked the hill ↔ The old man took a hike ↔ The old man took a hike (to + *of + at + on + around + up) the hill

  verbs of manner of motion which we further divide into run, jump, and walk verbs, but also include other categories, like wash and sleep verbs, and bodily function verbs: (5) a. Run verbs: The runner (hiked + jogged + ran) ↔ The runner took a (hike + jog + run) b. Jump verbs: The tiger (hopped + leaped) ↔ The tiger took a (hop + leap) c. Walk verbs: The old man (walked + strolled) ↔ The old man took a (walk + stroll) d. Wash verbs: Ty (bathed + showered) ↔ Ty took a (bath + shower) e. Sleep verbs: The woman (napped + snoozed) ↔ The woman took a (nap + snooze) f. Bodily function verbs: The old man (peed + puked) ↔ The old man took a (pee + puke)

( 9 )

 9 Max took (*E + *a + the ) (rebound + reign) Few examples freely accept any determiner or quantificational expression as part of the construction: (10) The artist took (E + a + the + several + two + the two) (sketches + photographs + flights) Some only accept the indefinite article and reject all the other phrases, while other expressions accept, or reject, a combination of possible phrases that precede the nominal. Even verbs from the same semantic class vary in the type of determiner or quantificational phrase the construction accepts: (11) Bodily function verbs a. The girl took (a + *the + *E + *several + *the two + *the first) pee(s) b. The boy took (a + *the + *E + several + the two + the first) poop(s) c. The man took (a + *the + *E + *several + *the two + *the first) puke(s) (12) Drink verbs a. The man took (a + *the + E + several + *the two + the first) sip(s) b. The woman took (a + *the + *E + *several + *the two + *the first) drink(s) c. The girl took (a + *the + *E + several + *the two + *the first) chug(s) (13) Wash verbs a. The baby took (a + *the + E + several + two + *the two) bath(s) b. The girl took (a + *the + E + several + two + *the two) shower(s) c. The boy took (a + the + *E + *several + *two + *the two) rinse(s)

  a. *The surfer barreled ↔ *The surfer barreled the wave ↔ The surfer barreled in the ocean ↔ The surfer took a barrel (E + in the ocean) b. *The class broke ↔ *The class broke 5 minutes ↔ The class broke for 5 minutes ↔ The class took a break (E + for 5 minutes) Finally, again highlighting the importance of constructing lexicon-grammars, other examples show that the preposition changes in the prepositional phrase between the ordinary verb and the support verb construction, as in the following cases: (20) The mother cared for the baby ↔ The mother took care of the baby (21) Max obsessed over his girlfriend ↔ Max took an obsession for his girlfriend (22) The hockey player went for the goal ↔ The hockey player took a go at the goal (23) The scientist accounted for the situation ↔ The scientist took an account of the situation And in one particular case, the verb seems to not allow the prepositional phrase when part of the support verb construction: (24) *Max sided ↔ *Max sided friends ↔ Max sided with friends ↔ Max took (a side + sides) (E + *with friends)

Table 1 : Sample of Intransitive Verbs with take specified for determiner and quantificational phrases

 1

	Semantic
	Properties

Table 2 : Sample of Transitive Verbs with take 33%

 2 of the verbs classified in Table2take have no intransitive ordinary verb use, but can appear in both intransitive and transitive support verb constructions, as in (16):

	(16) Verbs of controlling: Max (*controlled + *surveyed + *gripped)
	↔ Max (controlled + surveyed + gripped) the situation
	↔ Max took (control + surveillance + grip) (E + of the situation)

  Some verbs in this table, however, take a prepositional phrase complement, but reject both the purely intransitive and transitive uses, as exemplified in (19):

	Prep N 1 )
	a. The man (winked + gazed + glanced + glimpsed + looked + stared + smirked)
	↔ *The man (winked + gazed + glanced + glimpsed + looked + stared + smirked) the woman
	↔ The man (winked + gazed + glanced + glimpsed + looked + stared + smirked) at the woman
	↔ The man took a (wink + gaze + glance + glimpse + look + stare + smirk) (E + at the woman)

N 1

Semantic Property + + account -E --+ ------situation + + + -----note + + advance -ment + + + -+ --+ -career + + + -+ ---improve +act -ion + ----+ ---the plan + + ---+ --act +audition -E + -+ ------the play + + + -----audition +bake -E + ---+ ----the sun -+ --+ ---get sun

Table 3: Sample of light verb take constructions with prepositional complements

We also found another small table of N 0 V N 1 Prep N 2 expressions that involve a symmetrical relationship such that either N 1 take V-n (e.g. criticize) or N 2 take V-n exists (e.g. confess). These types of support verbs were first referred to as "converse support verb constructions" by Gross 1989. The 16 verbs in Table 5 (not shown) reveal a variety of transformational properties in the sense of [START_REF] Harris | Introduction to Transformations[END_REF], in which there is a potential relationship of relative synonymy between the transitive verbal expression and the take construction. In the converse support verb construction, however, take implies "accept" or "receive" in the abstract or metaphorical sense, or perhaps better seen as a type of passive as described in Newman 1996:245-5). In the first of these transformations, the complement N 1 becomes the subject of the take construction, as in ( 26):

(26) N 0 V N 1 Prep N 2 ↔ N 1 take V-n a. The people criticized the president for the speech ↔ The president took the criticism b. The mother blamed the child for the error ↔ The child took the blame

In others, the complement N 2 becomes the subject of the take construction. In a few cases, however, both the N 0 and N 2 complement may occupy the subject position of the support take expression, as in ( 27), which shows that take may be ambiguous between two interpretations: one being the light verb Max confessed ↔ Max took confession, and the other being the converse support verb, where take means something like "perform" as in Max confessed to the priest ↔The priest took the confession.

(27) N 0 V N 1 Prep N 2 ↔ N 0 take V-n ↔ N 2 take V-n a. Lawyers appealed the case to the court ↔ Lawyers took the appeal ↔ The court took the appeal b. Max confessed his sins to the priest ↔ Max took confession ↔ The priest took the confession Further research will show if these last two tables are more productive in English, and perhaps also reflective of other support verbs. Nevertheless, we can see that take support verb expressions are highly productive in English, but due to high variability of acceptable expressions, even among verbs in semantically related verb classes, they are best described through a formal lexicon or lexicon-grammar.


Daniel Morgan

Muddled Math and its Implications for Early Chinese Manuscript Culture

My purpose in this presentation is to take my experience working with astronomy texts and discuss their implications for broader issues of early

to the anomalous is generally to bracket or correct it as the first order of business, after which the text as it is actually written disappears from discussion about the text. With these texts it is easy both to make and justify corrections because the anomalies they present are of a different sort than those of, say, poetry-they are, for the most part, unambiguous mistakes in numbers or tiāngān dìzhī within coherent formulae or patterns of distribution. 1 In this paper I attempt to apply the approach of the former to the materials of the latter, and tease a story from muddled math. My focus is the planetary astronomy of the second century BC silk manuscript Wǔxīng zhān recovered from Mǎwángduì tomb 3, which provides for Jupiter, Saturn, and Venus a planetary model and 70-year "ephemeris" from the first year of Qín Shǐhuáng's reign as king in 246 BC to the third year of Hàn Wéndì in 177 BC. For reasons I will touch upon in passing, Wǔxīng zhān appears to be a Hàn copy of a Qín Dynasty work.

On your handout I have provided for you a transcription and translation of the Wǔxīng zhān Venus model. At a glance, the number of brackets is a clear indication of the degree to which such texts can and must be corrected before meaningful discussion.

First a word of explanation. Up until the Sui, planetary models for inferior planets like Venus were comprised of three elements:

• One, a start-point, or "system origin".

• Two, on page 2, a synodic period-that is, the time it takes for the object to reappear in the same position relative to the sun. Thus, for example, Wǔxīng zhān tells us that Venus makes 5 first morning risings in 8 years, for a synodic period of 584.4 days.

• Three, a model that divides each iteration of the synodic period into stages of angular velocity and visibility phenomena such as first risings and settings. This is best represented in table format, as I have provided on page 2. These models are predicated on two principles:

• One, symmetry.

• Two, a fixed angle from the sun at which risings and settings occur, which in early models is half a cì, or 15°. For convenience, I will call this the "zone of invisibility."

The consequence of this principle is that from rising to setting, the angular distance traveled by the planet must equal the angular distance traveled by the sun over the same period.2

Now that we understand the principles involved, let us see how they were violated. In Venus' seventh stage, the slow stretch before setting in the west, Wǔxīng zhān records a velocity of 40 fen. This is impossible. It violates both the principle of symmetry and of the "zone of invisibility." For the model to work at all, the value must be 120. Before photographs of the manuscript were published, Hé Yòuqí argued that the graph for 40 was actually a compound of bǎi niàn, 120.3 However, if you turn to page 3, you can tell from the pixelated photograph at the top that the graph can only be 40. Thus, it appears to be a mistake.

Next, the velocity of stages 5, 6, and 7, when Venus is in the west, present a progression from fast to slow symmetrical to stages 1, 2, and 3, when Venus is in the east. However, the durations of 5, 6, and 7 do not-they are in the same order as 1, 2, and 3. This violates the principle of symmetry. It also violates the principle of the "zone of invisibility," because whereas the total distance traveled by Venus in the east equals the distance traveled by the Sun in the same time-the Sun traveling one dù per day-in the west, Venus overshoots it. This too is clearly a mistake.

Lastly, the Wǔxīng zhān is silent on Mercury and Mars, but gives the start-points of Venus,

Jupiter and Saturn all as a first morning rising with the Sun in Yíngshì on lìchūn, the first day of the first month of the first year of King Zhèng of Qín, later Qín Shǐhuáng-that is, approximately February 8 th , 246 BC. Later descriptions of the Qín's Zhuānxū astronomical system confirm that its system origin was a five-planet conjunction at lìchūn. This conjunction was fudged. If you look at the table at the bottom half of the page you will see that:

• Mars was out of the picture entirely, half-way round the sky and having risen some nine months prior.

• Mercury was indeed nearby, but had already risen and set by system origin.

• The other planets were spread out over 38° and several different lodges, and their first morning risings were weeks apart.

The result of synchronizing the planetary models of Venus, Jupiter, and Saturn to this fudged start-point is miscalibration-the accuracy of the models vis-à-vis the planets is compromised by beginning several weeks too early or too late. The graphs on page 4 help visualize this. PAUSE The yaxis is right ascension in degrees, and the x-axis is time-the graph for Venus covering its first 8 years The story of the system origin is clear. A five-planet conjunction at the beginning of a dynasty was considered an auspicious omen, and many dynasties purport to have begun with one: not only the Qín, for example, but also the Zhōu and Hàn. Huáng Yīnóng has shown that the fabrication of celestial omens was something of a cottage industry such that the majority of five-planet conjunction records in Chinese history were fabricated to glorify good emperors, while actual conjunctions during the reigns of bad emperors went unrecorded. 4 The Wǔxīng zhān, however, is the first instance I know of in which the data of mathematical astronomy is grounded in propaganda to the detriment of its accuracy.

Following Hé Yòuqí, I argue that our system origin is a case of misrepresentation.

The two mistakes tell a different story.

First off, when we encounter mistakes in mathematical manuscripts it is crucial that we distinguish between issues of numeracy, literacy, expertise in subject matter, and care of copying, as well as the roles of author, owner, and copyist.

Numeracy, as Christopher Cullen has forcefully argued in several recent articles, 5 was quite prevalent among the early elite, the most evident reason for which was the crucial role of accounting in the administration of a bureaucratic empire. Not only do the biographies of great men tell us as much, the majority of excavated writings from the period happen to be meticulous accounts of people, land, taxes, and goods, and laws and administrative code concerning accounting. On the other hand, Yabuuti Kiyosi, Cullen, and Chén Měidōng have shown that most of the work of mathematical astronomy was done outside the context of government service by highly numerate enthusiasts. 6 Obviously, the Qín author was competent on all charges-when corrected, the Wǔxīng zhān Venus model works to predict the position and motion of Venus more accurately than any other system known in China until Líu Xīn's Sāntǒng lì more than two centuries later. We cannot know the relationship between the copyist and the owner-who was a young male relative of Lǐ Cāng 李 蒼 , Marquis of Daì-they could well be the same person, thus I will treat them separately, but not as mutually exclusive. 7 5 Cullen, "Huo Rong's observation programme of AD 102 and the Han li solar table"; Cullen, "Actors, networks, and 'disturbing spectacles' in institutional science: 2nd century Chinese debates on astronomy"; Cullen, "People and numbers in early imperial China: locating 'mathematics' and 'mathematicians' in Chinese space"; Cullen, "Numbers, numeracy and the cosmos." 6 Yabuuti Kiyosi 藪內清, Chūgoku no temmon rekihō 中國の天文曆法; Cullen, Astronomy and mathematics in ancient China; Cullen, "Actors, networks, and 'disturbing spectacles' in institutional science: 2nd century Chinese debates on astronomy"; Chen Meidong 陳美東, Zhongguo gudai tianwenxue sixiang 中國古代天文學思想. 7 In tomb 3 was a wooden placard records the internment of the occupant on -year 12, month 2, new moon at yisi (42),

If the aesthetic quality of his calligraphy is any reliable indicator, the copyist was clearly literate. Anyways, his mistakes involve only basic numbers-simple and common graphs which someone with even the barest level of literacy could not mistake. Rather than illiteracy, these mistake appear to the product of the copyist's carelessness and/or inexpertise with the subject matter.

Substituting "40" for "120" is clearly a mistake. The visual similarity between "20" and "40", as well as the fact that the mistake appears at the beginning of a new line suggest a process of visual copying from another manuscript rather than oral transmission.

Mistakes like these are endemic in mathematical texts. In many, if not the majority of, cases, it is evident that complex formulae were calculated correctly, but in copying, one or more numbers were omitted, reversed, or mistaken. For example, we all know that in the transcription of ancient texts into Hàn clerical script, the numbers 7 and 10 were frequently confused. But even within clerical script, numbers continued to be a problem. In the Suànshù shū mathematics primer from Zhāngjiāshān, for example, and government accounting such as the Húběi river dyke surveys purchased by the Chinese University of Hong Kong, graphically similar numbers like • one, and two, • two and three, and even • one and seven are frequently confused, and the numbers 1 and 100 are often dropped. 8 The inversion of the durations of stages 5 and 7 suggests a similar explanation. It appears that in day wuchen (5), which accords with Wendi 文帝 12 (-167). See Hunan sheng bowuguan 湖南省博物館 and Zhongguo kexueyuan kaogu yanjiusuo 中國科學院考古研究所, "Changsha Mawangdui er, san hao Han mu fajue jianbao 長沙馬 王堆二、三號漢墓發覺簡報." 8 Suanshu shu mistakes: s4 8 for 30; s5 …; s23 1 left off; s27 1 left off; s41 2 for 1; s48 1 for 2; s48 7 for 8; s50 2 for 1; s 68 1 for 7; s71 40 for 20; s76 2 for 5; s84 10 left off; s84 100 left off; s85 1 left off; s104 4 for 5; s108 5 left off; s127 7 for 1; s141 5 for 3; s141 6 for 9; s142 section left off; s150 1 left off; s155 7 for 1; s162 7 for 1; s173 之 for 5; s176 40 for 30; s178 9 for 7; s181 9 for 8. See Cullen, The Suàn shù shū 筭數書 'Writings on reckoning': a translation of a Chinese mathematical collection of the second century BC, with explanatory commentary. For the river dyke surveys, see Peng Hao 彭浩, "'Hedi jian' jiaodu "河堤簡"校讀"; Lander, "Han dynasty texts on the surveying of river dykes"; Chen Songchang 陳松長, Xianggang Zhongwen daxue wenwuguan cang jiandu 香港中文大學文物館藏簡牘. Lander argued that the numerical disarray in calculations of land-area therein was evidence of poor math skills amongst local officials. In discussion, I showed that even there, it was evident that most areas were calculated correctly, but that the occasional number was miscopied due to graphic similarity and carelessness. See also the issue of 古 度 lodge extensions in Sun Zhanyu. (More?) copying from another manuscript, the copyist grasped that there was some symmetry at play in the array of data, but confused the correct order, either because he skipped to the wrong line, or set out to correct the master copy with a weak understanding of the principles of the mathematical model.

As for the owner of Wǔxīng zhān, the fact that the ephemerides were filled out until nine years prior to his internment in the twelfth year of Hàn Wéndì suggests that he or someone he knew used and added to the manuscript decades after the text's authorship in the Qín. From his expansive tomb library, we can tell that the owner was irrefutably literate, and was a connoisseur of natural and occult knowledge. He does not, however, appear to have been an expert in astronomy. Unlike other manuscripts of his library, all on other subjects, the owner made no corrections to Wǔxīng zhān: he did not correct the wrong and inverted numbers; nor did he attempt by observation and recalculation to unanchor it from the propaganda of a despised dynasty and make it accurate for his day. He simply used it as is.

Stepping back, I think that the case of Wǔxīng zhān highlights an important double-function of manuscript culture in the transmission of technical knowledge. On the one hand, as Donald Harper has written about at length, manuscript culture facilitated the horizontal dissemination of technical knowledge, both within circles of specialists and beyond, to a larger audience of literate and numerate connoisseurs.9 On the other hand, manuscript culture also acted to dilute and scramble that knowledge, as well as, vertically, to transmit knowledge that had gone out of date, and out of fashion.

Furthermore, I think that the example of mathematical texts can be informative to the study of manuscripts on philosophy, literature, and other topics at the core of modern sinology. For example, technical literature is often intensely visual, featuring diagrams, illustrations, and tables, and instances of textual corruption are almost always at the level of layout, and simple data like numbers and tiāngān dìzhī. On the one hand, this is strong evidence for the prevalence of the written, rather than oral, transmission of knowledge. On the other hand, it also reminds us that reducing the issues of anomaly and textual corruption to the literacy of scribes is too simplistic. Instead, we should also consider problems independent of literacy, like inexpertise with subject matter, and carelessness in copying. If copyists were sloppy with with government accounts and imperial edicts, as A. F. P.

秦始皇帝元年正月,大白出東方,【日】行百廿分,百日。行益【疾】,【日行一度】, 【六】十日。行有(又)益疾,日行一度百八十七半[分]以從日,六十四日而復遝日,晨 入東方,凡二百廿四日。浸行百廿日,夕出西方。太白出西【方】,【日行一度百八十七 半】,【百日】。行益徐,日行一度以侍(待)之,六十日。行有(又)益徐,日行( ) [百廿]分,六十四日而入西方,凡二百廿四日。伏十六日九十六分。【太白一復】爲日五 【百八十四日九十六分日】。【凡出入東方各五】,【復】與營室晨出東方,爲八歲。
In the first month of the first year of Emperor Qin Shihuang (r. 246-210 BC) Venus emerges from the east. It travels 120 fen (out of 240) [per day] for 100 days; then, its movement [accelerates, and it travels 1 du per day for 6]0 days; then, its movement further accelerates, and it travels 1 du and 187½ (fen) per day, whereupon it charges toward the sun, catching up with it again in 64 days and setting in the morning to the east-224 days in all.

Then, it travels submerged for 120 days, and emerges at sunset from the west.

[Venus rises in the west traveling 1 du and 187 and a half fen per day for 100 days;] then, its movement decelerates, and it travels 1 du per day and waits for it (the sun) for 60 days; then, its movement further decelerates, and it travels (40) [120] fen per day for 64 days before setting in the west-224 days in all.

Then, it hides for 16 days and 96 fen.

[One return of Venus] takes 5[84 days and 96 fen in days. In total it rises and sets in the east and west each five times, then again] emerges in the morning in the east with [the Sun in] Yingshi, constituting 8 years. The two principles of motion models:  Symmetry  Fixed 15° (半次) -zone of invisibility‖ from Sun determining risings and settings.

Synodic period:

The time it takes for the planet to reappear in the same position relative to the sun, e.g. from inferior conjunction to inferior conjunction, or first morning rising to first morning rising.

  and 5 synodic periods, and the graphs for Jupiter and Saturn covering the entire 70-year period of the Wǔxīng zhān ephemerides. The gray lines represent true right ascension, the black, that calculated from the Wǔxīng zhān, and the faint line at the bottom, the difference between the two. The breaks in the lines represent periods of invisibility. Now, for Venus, note how the Wǔxīng zhān starts considerably behind the true, a difference of up to several weeks. Saturn starts fairly accurately because its true first morning rising was only one week after system origin; its right ascension and periods of invisibility then fall out of sync over the course of 70 years. However, Jupiter, whose true first morning rising occurred 33 days prior to system origin, begins 20 to 40° off of true, and falls gradually into sync over the years. So what's the story?



  The three elements of early planetary astronomy (for inferior planets):  System origin (yuan 元): all planets experience first morning rising with Sun in Yingshi (α Peg) on lichun, first day of the first month of the first year of Qin Shihuang (ca. February 2 nd , 246 BC) o 曆記始於顓頊上元太始閼蒙攝提格之歲,畢陬之月,朔日己巳立春,七曜俱在 營室五度。(《新唐書•曆三上》引《洪範傳》) o 顓頊曆術曰:"天元正月己巳朔旦立春,俱以日月起於天廟營室五度。"今月 令孟春之月,日在營室。(《後漢書•律曆志中》李賢注引蔡邕) o 顓頊以今之孟春正月為元,其時正月朔旦立春,五星會于天廟,營室也。(《晉書• 律曆志中》引董巴議) MORGAN -March 2011 -Muddled Math and its Implications for Early Chinese Manuscript Culture 2 Synodic period: days  (Angular) motion model (xingdu 行度):

  Hulsewé has shown, then why should philosophy or science have fared any better? 10 Thank you. Daniel Morgan, University of Chicago Ph.D. Student American Oriental Society 221 st Meeting, Chicago 2011 Wuxing zhan 五星占 Venus model (lines 143-146):

Examples of eminent scholars: Liu Lexian 劉樂賢, Jianbo shushu wenxian tanlun 簡帛數術文獻探論 ; Cullen, The Suàn shù shū 筭數書 'Writings on reckoning': a translation of a Chinese mathematical collection of the second century BC, with explanatory commentary. Cullen has several dozen examples, mostly leaving numbers off, confusing 7 and 1.

Teboul, Les premières théories planétaires chinoises.

He Youqi 何幼琦, "Shi lun Wu xing zhan de shidai he neirong 試論《五星占》的時代和内容."

Huang Yi-Long 黃一農, Shehui tianwenxue shi shi jiang 社會天文學史十講.

Harper, Early Chinese medical literature: the Mawangdui medical manuscripts; Harper, "Warring States natural philosophy and occult thought."

Hulsewé, "Qin and Han Legal Manuscripts."