Bertrand C Barrois

BIRD'S-EYEVIEW OF THE SOLAR DYNAMO

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The dynamo mechanism by which turbulent convection in the outer core of the Earth and the outer layers of the Sun sustains the magnetic field deserves the appellation "MHD Hell". It poses all the difficulties of computational fluid dynamics, but resists experimental validation.

Recent computational approaches involving supercomputer simulations produce large volumes of detail but relatively little insight into the correlations that count. They bear out the adage, "You cannot see the forest for the trees."The goal of this paper is to map the forest at lower resolution from higher altitude.

After surveying the mathematical framework in which dynamo models must operate, we construct a ruthlessly thinned model that focuses on half-a-dozen key variables. It proves adequate to predict (1) oscillatory behavior, and (2) alignment with the axis of rotation. Itidentifies the key correlations and is amenable to calibration against the results of detailed simulations.

One might hope that such a God's-eye view of MHD Hell will serve to guide the computational investigators in their search for the needle in a haystack, if only by furnishing a description of the needle. But enough with the mixed metaphors. Let's get down to business. 1 E-mail: BBarrois@verizon.net

Mathematical Framework

At a fundamental level, MHD heat transport is governed by three interlockingequations, with a combination of linear, bilinear, and constant terms on the right hand side: .

2 2 2 11 2 2 11 22 1 1 ( )( ) ( ) ( ) ( ) 2 curl( ) ( ) ( ) ( ) curl( ) ( ) P P da HV dt C da V dt C d B dt H H H H H J sources PH                                          V g V V V V V Ω× V g J × B B B V × B B V B B V B V J B E + V × B      H =

P H C T g    

In the Sun, the convective shell is heated from below and cooled from above. In the Earth, the power source is thought to be the settling of heavier elements, which releases gravitational potential energy, but the effect is much the same. The heat transport equation therefore contains a time-independent source term, as well as the quadratic redeposition terms representing viscous and ohmic dissipation.

The evolution of the magnetic field is simply governed by the induction equation,  Expansions in VSH respect spherical symmetry explicitly, but every harmonic interacts with every other harmonic. The radial dependence is arbitrary and may be chosen for convenience. (Glatzmaier 1995; Clune 1999)  Simulations on a grid are computationally more efficient, because the equations are local. (But the Navier-Stokes equation is not quite as local as it looks, because the pressure must be calculated by solving Poisson's equation --a tedious step, even with Fast Fourier Transforms.)

whereas
Thinned models deal with a truncated set of VSH. Although any two harmonics may interact, selection rules restrict the outcomes:

 Clebsch-Gordan rules for vectors as well as scalars: ( , )( , ) ( ,

) L M L M L M        processes require that M M M     and L L L L L          . 2
These are divergence-free vector fields that transform as ordinary spherical harmonics under rotations. They are generated by repeatedly taking the curl of a purely radial vector field. Poloidal ("S") patterns have a radial component, and their tangential components form irrotational patterns. Toroidal("T") patterns have no radial component, and their tangential components form swirling patterns. We shall use idiosyncratic but self-explanatory notation: BS(L,M) poloidal field, BT(L,M) toroidal field, VS(L,M) poloidal flow, and VT(L,M) toroidal flow.Since density varies with depth, the divergence-free pattern VS(L,M) actually represents  V .For a good introduction, see (Backus 1996). One may finesse the need fordetailed simulation of heat transport by taking a free ride on convective instability analysis. Given the background temperature profile, one may use the linearized equations off heat transport to calculate the exponential growth rates of convective modes.

Mathematical Prototype

Thethinned model should copy the key features of the fundamental equations.

The prototype will be a system of coupled ODEs, with bilinear terms on the right hand side, and coefficients chosen to conserve the sum of squares, as well as a Liouville condition:

               i i i i i ij i i dt d k ij j ki i jk jk k j i jk i dt d X X f X f f f X X f X 0 / 0 0 ) ( 0 2 2 1 
The trajectories of such a system never leavethe hypersphere, which they almost always coverergodically and uniformly. However, when linear driver/damper terms are added, the trajectories tend to approach an attractor set of fractal character.

Turbulent transport processes that mimic diffusionmay be represented by linear damping terms, whereas interactions with omitted variables motivate the inclusion of additive noise terms with appropriate time-dependence. Poloidal flows driven by buoyancy may be modeled with linear drivers.

-5-

Thinned Model of the Dynamo

Dramatis personae: The process of hairpin formation is uncontroversial and is commonly known as the omega effect, but regeneration is more controversial. Mean-field theories of regeneration posit a quasi-local correlation between transient fields and flows that induces a JT(1,0) current loop. The literature mutters mystically ofalpha and beta effects (e.g., Schrinner 2008). 3 To avoid confusion with these preempted terms, we shall refer to regeneration as the zeta effect, making the non-local black-box mechanism an omegazetadynamo.

 ΩT(1,0) = overall rotation rate  BS(1,0) = poloidal dipole field  BT(2,0) = toroidal quadrupole field  VT(3,0) = differential rotation  VS(L,M) =
The black-box mechanism for regeneration posits correlations among three kinds of transient variables: convective eddies, their sidekicks, and resulting distortions of the fields. These correlations can only be determined via detailed simulations.

The overall sign of the black-box mechanism is still uncertain. Since stimulus and response should be positively correlated, the sign comes down to the couplings, which we now write in shorthand, with steady main-field components barred:

( : ) ( : ) ( : ) ( : ) ( : ) ( : ) f T VS VT f VS BT BS f VT BS BS VT BS f T VS VT f VT BT BT f VS BT BS VS BT          
The black-box mechanism is consistent with Cowling's Theorem, which maintains that no axisymmetric flow can sustain an axisymmetric field. Per Clebsch-Gordan rules, if

L L L odd       , as it is here, then ( , , ) ( , , ) M M M M M M           
, hence the axisymmetric process with

0 M M M     
 is forbidden.

Cyclical Behavior

Conservation of energy demands that the coefficients of bilinear terms obey the cyclic sum condition. The energy-conserving processes involving axisymmetric variablesare paired as follows:

(2, 0) It cannot legitimately be derived from any power of the rotation parameter Ω, iseven under parity, butit can be related to the helicity of turbulence. Helicity is also odd under parity, but simulations find local helicityto be an odd function of latitude, thereby resolving the paradox. Time reversal invariance is violated by dissipative systems.

(3, 0) (1, 0) (3, 0) (2, 0) (1, 0)
-7-Since VT(3,0) does not vary appreciably over the solar cycle, let us simply treat it as constant. We now throw ina one-line summary of the black box, which steals energy from convectively driven eddies:

(2, 0)

(3, 0) (1, 0) (1, 0) (1, 0) (2, 0) d dt d dt BT VT BS BS T BT       If (3, 0) W VT  

and

(1, 0)

ZT   have the same sign, 4 then these equations predictoscillatory behavior, as actually observed in the Sun.(On the other hand, if these quasi-constant coefficientswere to start out withopposite signs, the equations would predict runaway exponential growth of the fields, but the reactive force would put the brakes on differential rotation and quickly reverse it.)

But before we count this as triumph, we must deal with a paradox. So far, the equations predict that and BS(1,0) will oscillate exactly 90° out of phase, so that their product averages to zero and cannot possibly drive VT(3,0). This defect can be cured by including linear damping terms. If BS(1,0) is more strongly damped than BT(2,0), then it lags < 90° behind BT(2,0), and theircorrelation has the correct sign to -8-

Amplification by Hyperbolic Sloshing

Having included damping terms, we must also include driving terms, which might be modeled as additive noise, or more exotically, as hyperbolic sloshing. The latter mechanism is intriguing because it explains how a convectively driven eddy can pump up main field components that are not directly driven.

Consider any three components coupled so as to conserve energy:

where 0

d dt d dt d dt A BC B AC C AB             If ()
Ct is directly driven, it may be considered an autonomous process that is not influenced by donating energy to A and/or B.

If 0   , then ( , )
AB is driven along a hyperbolic path, 22 As a rule, hyperbolic sloshing tends to pump up the most durable component ("Methuselah mode") of a system. Paradoxically, other components must be killed off. (Methuselah's wife had to die, for him to live so long.)

B A -9-
The sine qua non of hyperbolic sloshing is the requirement that 0   , and the geomagnetic dynamo furnishes a real-life example. Radial flows driven by buoyancy can induce hyperbolic sloshing that may pump up the main magnetic field. Rising plumes create divergence zones as they approach the surface, thereby stretching tangential field lines longitudinally, squeezing them transversely, and amplifying the field strength, say by a factor of exp( ) a  . Falling plumes elsewhere do the opposite, but the overall effect amounts to amplification because 11 22 exp( ) exp( ) 1 aa     .

In the jargon of vector spherical harmonics, we may say that the real poloidal flow VS(L,M)+VS(L,-M) distorts the toroidal quadrupole field BT(2,0) to BS(L±1,M) and back. A similar mechanism can amplify the poloidal dipole field BS(1,0) via BT(L,M).

Periodic vs. Erratic Behavior

With linear damping terms and additive noise drivers included, the system of coupled equations reads as follows:

()

() S S S S d dt T T T T B g Z B N t B W g B N t                           
The systemresonates at 2 WZ g   with bandwidth g   , where 1 2 ST g g g   and 1 2 () ST g g g  

. We may distinguish three kinds of behavior:

 Weakly damped ("narrowband") oscillations: Reversals are obviously periodic, but erratic variations of amplitude may be mistaken for beats. A good model for the Sun.  Strongly damped ("wideband") oscillations: Erratic reversals obscureresonant periodicity. A possible model for the Earth.

 Suppressed oscillation: 2 WZ g   . Another possible model for the Earth.

The BB covariance matrix could in principle be calculated from the unknown NN matrix, which must in practice be extracted from the results of detailed simulations.

Alignment with the Axis of Rotation

We must now repent of a mortal sin, one that could send us straight to actual Hell. In formulating the thinned model, wesingled out main-field components aligned with the axis of rotation. If we wish to investigate how and why the main field aligns with the -10-axis, we must respect rotational symmetry and treat all components evenhandedly. We may do thisexplicitly by using tensor representations of BS, BT, and VT:

 BS(1,...) is denoted i S  BT(2,...) is denoted ij T (symmetric traceless)  VT(3,...) is denoted ijk D (symmetric traceless)
Using fancy brackets to denote symmetrization and trace removal, we may rewrite theequations as follows:

  () () () d k kl l S k S dt d ij ijk k T ij T dt d ijk ij k D ijk D dt S T g S N t T D S g T N t D T S g D N t                     
If these equations were driven by isotropic additive noise, the main field would not naturally align. However, there may be third-ordereffects.

Magnetic fields tend to oppose transverse fluid flows. 6 Since strong tangential fields act as a barrier to radial convection currents, we argue that the latitude-dependent magnitude of VT(2,0) modulates convection. Since the Coriolis effect is also latitudedependent, there could well be extra



 F E + V × B B , but kinetic energy may either be transferred to the magnetic field or dissipated as heat. Elastic transfers set the stage for Alfvén waves. The magnetic term is inherently dissipative, but the electric term of the force equation tends to offset it, either due to prompt charge separation (e.g., if currents diverge or dead-end on a boundary) or due to induction by the changing magnetic field.

  time-average enthalpy profile * ρ= density of fluid H  = enthalpy fluctuation * a = thermal expansion P  = pressure fluctuation C P = specific heat * V = velocity of fluid X H = heat diffusion Σ = shear-rate tensor X V = kinematic viscosity B = magnetic field X B = magnetic diffusion J = electrical current σ = electrical conductivity g = acceleration of gravity μ = magnetic permeability * Per unit mass Convective instability is induced by a super-adiabatic gradient:

  fluid flow is influenced by Coriolis, buoyant, and magnetic forces. The conservation of energy may be verified as an exercise in double-entry bookkeeping. The energy flows are shown in Figure 1. At the three-way junction that diverts energy into electromagnetic

Figure 1 .

 1 Figure 1. Energy Flows in MHD Hell



  Parity rules: As vectors, ( , ) S L M and ( , )T L M have opposite parity. It follows that TT→T, SS→T, ST→S and TS→S processes require L L L odd TT→Sprocesses arealways forbidden, for lack of a radial component.  There are further selection rules for hydrodynamic processes in a flat (or very thin) layer with hard walls. They exploit the fact that the flow patterns would have definite parity under exchange of the upper and lower boundaries, VT being even, VS odd. Magnetic boundary conditions are never so symmetric.

  a representative poloidal eddy  VT(L±1,M) = its toroidal Coriolis sidekick  BS(L±1,M) = transient distortion of the main field  BT(L,M) = transient distortion of the main field The equations of the model must capture four key effects:  Coriolis effect: A linear process twists VS(L,M) eddies to VT(L±1,M). This process would be forbidden by parity selection rules in a flat layer with hard walls, but the convective shells of the Earth and Sun are neither thin nor flat.  Hairpin formation: VT(3,0) distorts BS(1,0) to generate BT(2,0). This process does not draw energy from BS(1,0).  Regeneration: A complex black-box process by which transient eddies and their sidekicks act in tandem on BT(2,0) to regenerate BS(1,0). Figure 2 lifts the lid of the black box to show how regeneration might work.  Reactive forces: A further process, required by conservation of energy, by which BT(2,0) and BS(1,0) may either drive or oppose VT(3,0).

Figure

  Figure 2. Black-Box Mechanism for Regeneration

  coefficientαis odd underparity (P) symmetry.

  Diagram furnishes visible evidence that this is precisely what happens. Figure 3 reveals the latitudes at which sunspots occur at various phases of the solar cycle. By one interpretation, 5 sunspots occur predominantly at the latitudes where tangential fields are strongest: BT(2,0) at mid-latitudes, BS(1,0) at the equator. If the phase lag were ±90°, then no forward-backward asymmetry would be expected. The observed asymmetry indicates that BS(1,0) is lagging roughly 45° behind BT(2,0).

Figure 3 .

 3 Figure 3. Latitudes at which Sunspots Occur

Figure 4 .

 4 Figure 4. Pumping by hyperbolic sloshing



  demons in MHD Hell, to wit, the many unsolved problems regarding dynamo phenomena, a few of which the author hopes to address future publications: Calibration of coefficients using detailed simulations  Validation of the black-box mechanism for regeneration  Erratic reversals of Earth's field vice regular oscillations  Partition of energy among BS, BT, VS, VT components 6 And conversely, transverse fluid flows tend to distort magnetic field lines. The opposing force is ()


(1) To verify that p,d,a,b sends an integer not divisible by d to another such integer, observe that, for any n (d) , there exists a unique j, 0 ≤ j ≤ s -1, such that n r j mod p ; thus, according to (6), for any such n , p,d,a,b (n) = a j n + b j = a j . (r j + m . p) + b j = (a j . r j + b j ) + a j .m.p (d) .

(2) The actual number of parameters (a j , b j ) does not need to be as big as 2s. Thus, [START_REF] Belaga | [END_REF]] (cf. also [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x+d Context[END_REF]]) defines a pwpl-function W acting in a two-to-one way on the set (6) of positive integers not divisible by 6. The function W is closely related to the 3x+1 transform T , ÇcorrectingÈ its unpleasant property to be an [either-two-or-one]-to-one function : cf. the formula (4) above. W is formally characterized by the parameters p = 96 , d = 6 , 2s = 64, but is actually defined by 12 rational numbers on 8 classes mod 12, 24, 48, 96. conjecture implies an occasional but persisting appearance of very long primitive T d -cycles whose existence depends on a huge common factor (the number C below) of a pair of Çvery bigÈ numbers (A and B) related to the structure of the cycle.

More precisely, let p be the length of a T d -cycle, k be the number of odd members of the cycle, n be its first and minimal member (always odd), and let p j , 1 £ j £ k-1 , 0 < p 1 < p 2 ... < p k-1 < p be the index of the (j+1)-th odd member of the cycle [START_REF] Belaga | Counting Primitive 3x+d Cycles[END_REF]. One associates with the cycle two ÇbigÈ positive integers, A and B , as follows

A = 2 p -3 k , B = 3 k-1 + 3 k-2 .2 + 3 k-3 .2 + ... + 3.2
+ 2 , and one defines C = gcd (A, B).

Apparently, there are no a priori reasons to believe that the two numbers A and B should have a ÇbigÈ common divisor C. (As a matter of fact, the arithmetics of the above numbers is extremely obscure [START_REF] Wirsching | [END_REF]).

However, it is easy to verify that the numbers A, B, C are related to ÇsmallÈ parameters d and n of the cycle by the formulae [Lagarias 1990], [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x+d Context[END_REF]] :

d = A/C , n = B/C .
Which implies that C needs to be a Çvery bigÈ number, too ! Here again, for a rarefied (and, thus, fast growing) sub-sequence of integers d, the related growth of the corresponding ÇprimitiveÈ parameters p, k, and n could induce such a ÇfastÈ growth of the numbers A , B , C , that the aforementioned phenomenon of Çfast growingÈ improvability might inescapably emerge.

Annex : Piecewise Periodically Linear Functions

In many cases, the condition that a periodically linear function acts on the full set of integers is too restrictive. We generalize here Definition 1 to piecewise periodically linear functions acting on integers not divisible by a nontrivial divisor d of the period p : Definition 2. Let p ³ 2 be an integer and d ³ 2 be its divisor. Let (d) be the set of integers relatively prime to d, and let { r 0 , r 1 , ... , r s -1 } be the subset of positive integers Çcut offÈ from (d) by the period p, through transfinite numbers up to e 0 , and, thus, cannot be carried out in Peano arithmetics [Kirby, Paris 1982] ! Also, Harvey Friedman discovered a remarkably transparent combinatorial theorem which (very roughly) states that in any sufficiently long finite sequence of finite trees, there is a pair of trees, with one tree embeddable into the second one [Harrington, Morley, Scedrov, Simpson 1985]. (This is a finitistic version, called FFF, of KruskalÕs classical theorem, dealing in its original form with infinite sequences of finite trees). FriedmanÕs construction is elementary and predicative. However, the proof of FFF demonstrably requires mathematical induction up to the first impredicative denumerable ordinal G 0 [Gallier 1991].

1 £ r 0 < r 1 < ... < r s -1 £ p-1 ; r j 0 mod d, 0 £ j £ s -1 ; s = pf (d)/d
Those and similar phenomena [Simpson 1987] confirm (or, if one prefers, make Çaxiomatic senseÈ of) our Theses 2 and 3, opening the way for their formalization, as follows :

Thesis 4.In many interesting cases, the termination can be proved only in the axiomatic domain which vastly, and unpredictably, outstrips the original domain to which the problem has been directly related.

Going now back to 3x+1 and 3x+d iterations, take the above ÇcollapsingÈ formulation of the 3x+d conjecture and. It asserts that any initially divergent trajectory would finally collapse into some (possibly, growing very fast with d) N d -vicinity of 1. Now, among possible scenarios of such a collapse, the following one, being quite realistic, would create serious difficulties for any (imaginable at present) proof :

Suppose there exists such a fast growing sequence of integers n d,q , q ³ 1, that the T dtrajectories starting at n = n d,q would begin their N d -descent only after a very long ascent, with the ascent span K d,n growing very fast with d and n.

In other words, the collapse in question might depend quantitatively on so fast growing functions N d = N(d), n d,q = n(d,q) , and K d,n = K(d,n) , that its proof would require axiomatic means vastly outstripping those available now in Mathematics (technically, one talks here about the second order arithmetics Z 2 ). -Which would bring us back to the above 3x+1 paradox !

Primitive Cycles and Fast Growing Factors of Numbers 2 p -3 k

Another outstanding conjecture related to 3x+d iterations stumbles apparently on a similar fast growing obstacle.

Jeff Lagarias has defined a T d -cycle to be primitive if the greatest common divisor of its members is 1 ; he also conjectured that, for any odd and not divisible by 3 integer d ³ -1, there exists a primitive T d -cycle [Lagarias 1990]. The conjecture is verified for all d £ 1000 [START_REF] Belaga | Counting Primitive 3x+d Cycles[END_REF].

However, both theoretical considerations and computations show that, to be true, the the same role as {1 2 1} plays for the iterates of T = T 1 . However, here the similarity ends, and the high balancing act of the 3x+1 case breaks down :

Calculations show that, for any d ¹ 1 defined as in ( 5), the function T d always has at least few nontrivial cycles4 . For example, T 13 has at least 9 such cycles, of the lengths, respectively , 4 (1 cycle), 8 (7), and24 (1); T 71 has 7, 10 (2), 27 (5); T 91 has 13, 4 (2), 8 (7), 12 (2), 24 (1), 48 (1) .

Thus, it is impossible to literally transpose the 3x+1 conjecture into the 3x+d context. Still, calculations show that, for a given d, the number of different cycles is finite, and that ÇbigÈ 3x+d trajectories finally ÇcollapseÈ, as do their 3x+1 analogues. The following reformulation of the 3x+1 conjecture better fits this new dynamics :

The 3x+1 Conjecture*.

(1) There are no divergent trajectories T(n), T(T(n)), ..., of iterates of T.

(2) The only cycle is that of {1 2 1}.

The generalization to the 3x+d case : change (2) to ÇThe number of cycles is finiteÈ [Lagarias 1990]. Or state it in the following ÇcollapsingÈ way [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x+d Context[END_REF]] :

The

> K d,n , T d (k) (n) £ N d .
This generalization illustrates how subtle, unique, and, apparently, extremely difficult (and possibly, improvable!) is the condition of the 3x+1 conjecture demanding that all the iterates of T run ultimately into the single cycle {1 2 1}.

On the other hand, the experimentally confirmed plausibility of the existence of a Çcollapsing mechanismÈ, common to all 3x+d cases, holds out some hope for a future proof of the ÇdivergentÈ part of the 3x+1 conjecture.

Still, the availability of such a proof could be impeded by Çfast growingÈ obstacles, to the description of which we now proceed.

Fast Growing Functions and Termination Proofs

The unsolvability of TuringÕs halting problem does not block, of course, resolution of some individual termination problems. Thus, for example, Reuben L. Goodstein [Goodstein 1944] has constructed a truly elementary function n g(n) , definable in Peano arithmetics, whose iterates g k (n) ultimately terminate at 0 for any n. However, with n growing, it takes them very long indeed to arrive at 0, which means that the function

K(n) = min (k, g k (n) = 0
) is growing so fast, that any proof of this fact necessary uses a mathematical induction Edward G. BELAGA page: 6

A proof that the complement T*( 1) is not recursively enumerable would imply unsolvability of the 3x+1 conjecture (cf. Note 2 above). Even in the absence of a definitive proof, the reasons to believe T*( 1) is not recursively enumerable (and thus, the conjecture is unsolvable) might become overwhelming.

On the other hand, ever widening evidence might indicate that this complement is empty, T*(1) = ¯, thus, ipso facto, confirming the conjecture.

The paradox can be also reformulated and specified in terms of Çfast growingÈ functions,see below Thesis 4 and the related deliberations. In either way, the possible, -and desirable, -outcome of these two developing and conflicting mathematical accounts should be an eventual deepening of our understanding of what constitutes a mathematical proof, -beyond the acknowledged now limits of provability. (This might be one of the possible interpretations of the aforementioned remark of P. Erdšs : «Mathematics is not yet ready for such problems»).

In the absence of such a new understanding, one should proceed both ways, ÇnegativeÈ and ÇpositiveÈ : trying to confirm the conjecture, without losing the sight on the efforts of proving its unsolvability.

Methodological Aside. The vagueness of the above wording of our aspirations (Çdeepening of our understanding of what constitutes a mathematical proof, -beyond the acknowledged now limits of provabilityÈ) is intentional. We talk here about the basic paradigm of mathematical proof, of which, say, the axiomatic aspect is only a part (albeit, one of the most important, -cf. Thesis 4 below ; still more important might be the universal cultural Çabstraction ascentÈ). Clearly, the Greeks didnÕt invent non-Euclidian geometry (and would have a great difficulty to understand it) just because they were not aware of the independence of the V-th postulate from other (known to them) axioms of Euclidian geometry. 3

Widening the 3x+1 Case to Its 3x+d Analogue

The unsolvability case being discussed above, let us try now the other way. There follows a brief description of a simple generalization of the conjecture which permits to better appreciate its particularity, beauty, and complexity ; it also shows that the formalism of periodically linear functions can work both ways. Besides, the generalization will supply us with few hints on how might evolve the interpretation of the above paradox.

We widen the 3x+1 definition (2) by introducing one free parameter [Lagarias 1990], [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x+d Context[END_REF]] :

for any odd d ³ -1 , d 0 mod 3, (5)

T d : ; T d (n) = n/2 if n is even ; (3n+d)/2 if n is odd .
Edward G. BELAGA page: 5

Reflecting on the 3x +1 Mystery 12/11/98........3:15 PM 3 A possible future paradigm shift in axiomatics is discussed in [Friedman 1992], p. 51 : «The fundamental issue is this : Is there a basic mathematical problem about standard finite objects such as, say, natural numbers or rational numbers ..., etc., with a clear and intuitive meaning, conveying interesting mathematical information, that is readily graspable, and which is independent of ZFC ? We speculate that sometime during the twenty-first century, someone will answer the above question in affirmative ... » Restricting ourselves to this axiomatic aspect, we submit below that the 3x+1 conjecture is independent of the second order arithmetics Z 2 . g (k) (n) = (a j . (a j . ...(a j . ((a j . n) mod p) modp) ... ) mod p) modp,

(3) where 0 £ j 1 , ..., j k £ p-1 , j 1 n mod p, j 2 a j .n mod p , ... , j k a j .n mod p.

Imitating Kurt GšdelÕs classical number-theoretical coding technique used in the proof of GšdelÕs incompleteness theorem, Conway has shown that, for any partial recursive function f (including that which encodes TuringÕs halting problem), a proper choice of parameters p and a 0 , ... , a p -1 guarantees that the interplay (3) between iterates of the corresponding g simulate f, in the following sense (see for details [Conway 1972], [Lagarias 1985]) :

(a) g (k) (n) = 2 s , for some k ³ 1 and some s, if and only if f is defined on n;

(b) g (k) (n) = 2 f(n) , for the minimal k ³ 1 such that g(n) is a power of 2.
(2) In some cases, the condition that the function acts on the full set of integers is too restrictive [START_REF] Belaga | [END_REF]]. It can be weakened in many ways. A generalisation of the above definition to piecewise periodically linear functions acting on integers not divisible by a nontrivial divisor d of the period p can be found in the Annex below.

The 3x+1 Paradox

ConwayÕs discovery confirms the difficulty of both CollatzÕs questions : predicting the behaviour of the trajectory g (k) (8) and proving the 3x+1 conjecture might well be unsolvable problems.

Unfortunately, such a distant unsolvability prospect sheds not much light on the 3x+1 case, flying in the face, so to say, of the remarkable wealth of its experimental and heuristic confirmations.

The conflict (already anticipated in our Thesis 2) is a sign, as the present author is convinced, that Thesis 3. We are confronted here with not just a difficult question about terminations of iterates, but with a paradox of a new type, for which there exists as yet no proper formal discourse.

In what follows, we shall try to make this vague assertion as precise as possible. Let us start with a tentative informal description of the paradox in question, with the term paradox understood in its simple original sense, i. e., as a conjunction of two separately plausible and mutually incompatible statements :

The 3x+1 Conjectural Paradox. (i) It might be well that the evidence in the favour of unsolvability of the 3x+1 conjecture would steadily grow until «almost» become a certainty. (ii) It also might be that all direct investigations and calculations would confirm the conjecture with ever growing evidence.

Or, in a more formal way : Let T*(1) be the set of all iterated inverse images T (-k) (1) of 1, i. e., the set of all positive integers n, whose T-iterates land ultimately at 1, and then run the cycle {1 2 1}. The set T*(1) is recursively enumerable; it is enumerated by repeated recursive applications, starting with T (-1) (1) = {2}, of the inversion formula :

(4) Thesis 2.There is no doubt that the «unsolvability rank» of problems related to iterations of number-theoretical functions is, by their very iterative nature, much «higher» than, say, of diophantine problems. It means that the 3x+1 related landscape should be incomparably more «steep», irregular, unpredictable, and, as a rule, inaccessible than the diophantine one. In practice, the 3x+1 inspired research testifies in a most unambiguous way to at least three aforementioned phenomena : discontinuity, meaningfulness, complexity. Wanted : formal methods of noticeable power.

T (-1) (n) = {(2n-1)/3, 2n}, if 2n 1 mod 3 ; {2n}, otherwise .

Periodically Linear Functions

There is, however, in the last comparison an important gap : whereas the diophantine ÇlandscapeÈ is formed by all diophantine equations, no such a formal and existing «by definition» context is at hand in the 3x+1 case. At least if one disregards as too abstract and unnecessary powerful the full formalism of TuringÕs machines, with its theorem of unsolvability of the halting problem.

Fortunately, a more narrow framework, that of periodically linear functions, perfectly fits in with our purposes. Introduced by John Conway [Conway 1972 , with the iterates encoding TuringÕs halting problem. Thus, the Çperiodically linear landscapeÈ is rich enough to naturally accommodate both the 3x+1 case and some explicit unsolvability phenomena.

Note. (1)

As a matter of fact, ConwayÕs construction does not use the full strength of the above definition : his are homogeneous periodically linear functions, g = a,0 , with a = (a 0 , ... , a p-1 ) , a j > 0, for j = 0, ... , p -1, and b = 0 = (0, ..., 0) , whose iterates are repeated multiplications by rational numbers a j , followed by mod p reductions : Since the problem became known about sixty years ago, many interesting and deep facts concerning the iterations of T have been discovered [Lagarias 1985], [START_REF] Wirsching | [END_REF]], all confirming or otherwise ÇjustifyingÈ the 3x+1-conjecture. However, no definite proof is as yet in sight, and the current state of affairs in the 3x+1 industry tends to confirm the old, startling remark of the late Paul Erdšs : «Mathematics is not yet ready for such problems» (quoted in [Lagarias 1985]). One can only marvel at how such a straightforward and primitive in extremis iterative rule can produce such an immensely rich and mysteriously balanced dynamical pattern !

Edward G.

Remarkable Ramifications of Unsolvability

We attempt here to build on the subtle difference in the quality of our ignorance in the two examples above. Our purpose will be to informally conceptualize some intuitive insights related to the ultimate (un)solvability of the 3x+1 conjecture. To this end, we need to view the conjecture in a more general formal context, where unsolvability appears on a regular, well-understood basis. Then we should compare the emerging 3x+1 related ÇlandscapeÈ with similar unsolvability phenomena in other branches of mathematics.

In practice, this order will be reversed. Our first informal thesis is inspired by the theory of diophantine equations and HilbertÕs tenth problem : we want to capture here, as if from an outer space, the mountainous diophantine landscape, with its lovely oases, its deserts, and its forbidding impenetrable peaks, -unsolvable diophantine equations. 1 Our second thesis (its very intuitive presentation will be made more formal in the following sections) carries out the comparison with the 3x+1 case. Then a proper technical context will be proposed.

As a matter of fact, the following statement spells out a well-known but not so well reported mathematical experience (cf., e. g., [Cherlin 1998], p. 1 2 ) :

Thesis 1. The presence of unsolvability phenomena in a mathematical field brings with it the emergence of sometimes isolated and mutually almost unrelated «islands», -beautiful, deep, rich in consequences, and difficult «local» theories, with extremely complicated and / or lengthy proofs of the most important results. Or , for short, the proximity of unsolvability leads to (i) abrupt conceptual discontinuity, (ii) novel meaningfulness, (iii) startling logical complexity. Against such a background, the development of (iv) new powerful problem oriented methods is stimulated. 2 « It will be seen that these two results, like other classification theorems, require a surprisingly lengthy argument, even though the classifications themselves are not very complex. It is not at all clear as yet why this is the case. The most obvious possibility is that our combinatorial tools need to be refined. A more intriguing possibility is that the classification of more general structures cannot be carried out effectively. This effectivity problem can be posed quite precisely, and is discussed in some detail ... below.»

  p,d,a,b : (d) (d) ; p,d,a,b (n) = a j n + b j , if n (d) and n r j mod p . Remark.

  , where f (d) is EulerÕs function. Finally, let a = (a 0 , ... , a s -1 ) , b = (b 0 , ... , b s -1 ), be a couple of s-tuples of rational numbers satisfying the following 2s = 2pf (d)/d conditions : (6) a j p/d , a j r j + b j (d) , for all j, 0 £ j £ s -1 . Then the parameters p, d, a, b define a piecewise periodically linear (pwpl-) function p,d,a,b , as follows : Edward G. BELAGA page: 8 Reflecting on the 3x +1 Mystery 12/11/98........3:14 PM

  Edward G. BELAGA page: 7 Reflecting on the 3x +1 Mystery 12/11/98........3:14 PM Note that {d 2d d}, call it the trivial, or basic d-cycle, plays for for the iterates of T d

  3x+d Conjecture. For any odd and not divisible by 3 integer d ³ -1, there exists such an integer N d > 0 that, for any positive integer n, the sequence T d (n), T d (T d (n)), ..., of iterates of T d ultimately collapses «under» N d . More precisely, for any n > 0, there exists such an integer K d,n > 0 that, for all k

  Edward G. BELAGA page: 4 Reflecting on the 3x +1 Mystery 12/11/98........3:15 PM as follows :

  ], periodically linear functions represent a most natural and parsimonious generalization (more precisely, a straightforward parametrization) of the functions g and T : Definition 1. Let p ³ 2 be an integer, and let a = (a 0 , ... , a p-1 ) , b = (b 0 , ... , b p-1 ), be a couple of p-tuples of rational numbers satisfying the following 2p conditions : a j p , a j j + b j , for j = 0, ... , p -1 . Then the couple (a, b) defines on the set a periodically linear function a,b of integers, as follows : a,b (n) = a j n + b j , if n j modp . (Choosing, in particular, p = 3 , a = (2/3 , 4/3 , 4/3) , b = (0 , -1/3 , 1/3) , p = 2 , a = (1/2 , 3/2) , b = (0 , 1/2) , one gets, respectively, the functions g and T .) J. Conway has also explicitly constructed a periodically linear function h, acting, as T, on positive integers, h:

  BELAGA page: 3 Reflecting on the 3x +1 Mystery 12/11/98........3:15 PM For any positive integer n, the sequence T(n), T(T(n)), ..., of iterates of T runs ultimately into the cycle {1 2 1}.

  An informal comparison of the diophantine and ÇiterativeÈ landscapes could be carried out, Edward G. BELAGA page: 2 Reflecting on the 3x +1 Mystery 12/11/98........3:15 PM 1 Another question is what sort of a formal mechanism permits to discover and ultimately prove the unsolvability. As Robert Soare writes, «the frequent occurrence of r. e. <recursively enumerable> sets in <some> branches of mathematics and the existence of nonrecursive r. e. sets ... have yielded numerous undecidability results, such as the Davis-Matijasevich-Putnam-Robinson resolution of Hilbert's tenth problem on the solution of certain Diophantine equations, and the Boone-Novikov theorem on the unsolvability of the word problem for finitely presented groups.»

More precisely, primitive cycles, -see below.[Lagarias 1990] conjectures that the number of primitive cycles is positive for any d.

December 1998

Reflecting on the 3x+1 Mystery.

Outline of a Scenario -Improbable or Realistic ?

Abstract. Guessing the outcome of iterations of even most simple arithmetical functions could be an extremely hazardous experience. Not less harder, if at all possible, might be to prove the veracity of even a ÇsureÈ guess concerning iterations : this is the case of the famous 3x+1 conjecture. Our purpose here is to study and conceptualize some intuitive insights related to the ultimate (un)solvability of this conjecture.

The 3x+1 Conjecture

Guessing the outcome of iterations of even most simple arithmetical functions could be an extremely hazardous experience. Take, for example, the following elementary function defined by Lothar Collatz in 1932 [Lagarias 1985] :

It is easy to verify that 1 is the fixed point of g , g(1) = 1, and that the iterations of g starting at 2 and 4 yield the cycles of the length, respectively, two and five. On the other hand, it remains an open question whether the trajectory staring at 8 is divergent or ultimately cyclic. And yet, in the light of the unsolvability of TuringÕs halting problem, the impossibility to make even a plausible guess in a similar situation comes as no surprise to us.

It does surprise us, however, that not less harder, if at all possible, might be to prove the veracity of even a ÇsureÈ guess concerning iterations, that is, to prove the validity of an extremely simple and transparent outcome which is Çforced upon usÈ, so to say, by vast numerical checks and a host of plausible heuristic arguments. This is the case of the famous 3x+1 conjecture (its origins are traced to L. Collatz, too [Lagarias 1985]) :

The 3x+1 Conjecture. Define the function T, acting on positive integers, as follows :

(2) T :

;


Is Eastern Europe Uniformly

AntiImmigrant? Not so fast.

Understanding immigration policy positions and policy change in Eastern Europe*

As the European Union struggled to address an unprecedented influx of refugees in 2015, four Eastern European governments rejected a proposal for European Union refugee quotas. Within each country, however, there are different views on the migrant crisis and immigration in general that are overshadowed by this uniform policy response. My research on the political divisions in each country explains that these differences are related to how political camps developed after communism. Through an analysis of the causes of immigration salience and the reasons behind immigration and integration policy positions of various parties in Eastern European countries, this research finds that which party -left or rightadopts more socially liberal policy positions depends on its relationships to communist federalism and the most politically notable ethnic group in the country. My work finds three distinct political patterns in Eastern Europe.

Alors que l'Union Européenne a fait face à un afflux de réfugiés sans précédent en 2015, quatre gouvernements d'Europe de l'Est ont rejeté la proposition de mise en place de quotas de réfugiés au sein de l'Union. Toutefois, à l'intérieur de ces pays, les positions diffèrent concernant la crise des migrants et l'immigration en général mais ces différences sont éclipsées par cette réponse uniforme à la réforme proposée. Ma recherche sur les divisions politiques propres à chacun de ces pays montre que ces différences sont liées au développement de différents partis politiques après la chute du communisme. À travers une analyse des causes de l'importance de l'immigration et des raisons expliquant les positionnements politiques des différents partis en Europe de l'Est sur ces questions, cette recherche démontre que l'adoption de politiques socialement libérales par un parti -qu'il soit de gauche ou de droite -dépend de sa relation au communisme fédéral, mais également du lien qu'il entretient avec le groupe ethnique national le plus important d'un point de vue électoral. Mon travail fait apparaître trois modèles politiques distincts en Europe de l'Est.

picture of an Eastern Europe unified by an antiimmigrant policy stance. These seemingly united viewpoints, however, mask the fact that there is significant policy differentiation among political parties at the country-level throughout the region. In fact, some Eastern European politicians are proponents of supporting the migrants [START_REF]Pomozme potřebným, trvejme však na dodržování pravidel[END_REF] and voluntarily increasing acceptance of refugees [START_REF]Poland considers accepting higher number of immigrants[END_REF] from conflict zones. Others oppose welcoming non-European refugees altogether, and would only accept migrants they deem easily assimilable. [START_REF]Premier: Privital by, keby sa prestalo s moralizovanim na adresu V4[END_REF] So, what accounts for these varying stances? And how do attitudes towards immigration in Eastern Europe map onto existing political divisions?

Existing scholarship on Eastern European politics predicts that party competition in the region is determined by various past experiences with communism, pitting the old state-centric authoritarianism against a new liberal market economic system. As such, the region's experience with communism is thought to tie left-wing economics to social conservatism and the economic right to pro-democratic social liberalism. [START_REF] Rovny | Communism, Federalism, and Ethnic Minorities[END_REF] Recent empirical evidence, however, exposes significant variance in party competition patterns across East European countries that are not accounted for by the dominant explanatory frameworks for party competition in the regioncommunist regime types and democratic transition experience.

My research supports the argument that party competition in Eastern Europe is determined by the interaction of two factors: a country's past experience with communist federalism and partisan responses to ethnic minorities. The argument follows in three steps:

(1) Due to historical institutional determinants, politically notable ethnic minorities associate with specific political forces; [START_REF]Pomozme potřebným, trvejme však na dodržování pravidel[END_REF] Given their interest in group rights, including citizenship, and their skepticism towards the repressive capacity of the (majority controlled) state, ethnic minorities hold liberal preferences [START_REF] Rovny | The sociocultural dimension combines preferences on noneconomic issues, juxtaposing traditional, paternalistic, authoritative, and nationalist views with egalitarian, liberal, and alternative outlooks; Kitschelt[END_REF]; (3) the liberal interests of ethnic minorities are consequently translated into the general socially liberal positions of the political actors affiliated with or tolerant of these minorities. [START_REF] Rovny | Communism, Federalism, and Ethnic Minorities[END_REF] Following this argument, I theorize and show the conditions under which political parties throughout Eastern Europe adopt socially liberal versus conservative views, thereby demonstrating the role ethnicity plays in forming the underlying ideological structure of party competition.

Applying this argument to policy positions on the migrant crisis and immigration being taken throughout Eastern Europe reveals a more complex ideological structuring of politics than generally accepted. In some Eastern European countries, it is the left-wing that adopts socially liberal positions paving the way toward greater openness to immigration. In other countries, it is the right-wing that takes this position. The determining factor in all cases, as my research shows, is the party's past relationship to communist federalism and to the most politically notable ethnic group in the country. Thus, I find three distinct political patterns across Eastern Europe that underpin the divergent policy positions taken on immigration.

1.Patterns of leftright positions on immigration in Eastern Europe

1.1 Relation to Communist Federationssecession matters.

The first pattern emerges in Eastern European countries that began their transition to democracy by seceding from a communist federation such as the Soviet Union or Yugoslavia. These secessionist countries include Croatia, Estonia, Latvia and Slovenia. Breaking away from the communist federal center often led to the newly formed state harboring a significant number of ethnic nationals from the old federal center --Russians or Serbs. Politically speaking, this federal diaspora tends to identify with the communist legacy, and subsequently with the Communist Party, its successors, or other left-wing parties. These left parties become either tacit supporters or the active representatives of the diaspora ethnic interest. Thus, in these cases, there is a departure from the expected party competition framework that would predict leftist parties to adopt ethnonational conservatism.

Over time, these left party affinities with ethnic minorities promote left-wing multiculturalism and the relationship between economic left-right placement on the political spectrum and immigration is similar to what is www.sciencespo.fr/liepp found in Western Europe --the left has more liberal views on immigration policies. In fact, in these countries the association between ethnic interests and the economic left is such that ethnic minorities frequently support and vote for the economic left-wing parties (see Figure 1). These subsequently support more liberal immigration and encourage multicultural approaches for integrating immigrants. In contrast, the economic right supports anti-immigrant policies and is predominantly assimilationist.[8] 1.2 Presence of prominent Ethnic Minorities not from the old federal center.

Turning to countries that did not break away from a communist federal center, the second pattern occurs in countries that have dominant ethnic minorities not originating from an ex-federal center, such as Bulgaria, Lithuania, Romania and Slovakia. Because the ethnic minorities in these countries are not tied to a former federal center, the left parties that have emerged since 1989 do not exhibit particular ethnic affinities. They are not prone to the same cultural openness characteristic of left economic parties in countries following the first pattern. In fact, these second pattern leftist parties either sideline ethnic issues, or mobilize national chauvinism to rejuvenate their left-wing ideology, which had been compromised by the fall of communism. Consequently, the left tends towards conservative views regarding all "others." In contrast, the opposition to communism in these countries supports market economic outlooks coupled with social liberalism, which in turn predisposes right-wing parties to be more sympathetic to the concerns of ethnic minorities.

As in the first case, these left-right positions inform the views of respective political parties towards immigrants and policy. As such, left-wing voters are prone to oppose immigration (see Figure 2), which is the opposite of what we find in Western Europe (see Figure 3) where the competition pattern is such that left-wing economics espouses social liberalism and those holding positions on the economic right hold socially conservative views.[9] And instead, it is the political right-wing that supports immigration and multiculturalism.

An additional observation from these initial two patterns is that in both country groups, ethnic minorities tend overall to be more open to immigration than the majority of the population, regardless of the country's relation to communist federalism (See Figure 4 on the next page). These findings emphasize the intimate ideological relationship between views on ethnic minority rights and immigration in Eastern Europe, which can then be understood as an indicator for policy positions.

Ethnic Homogeneity

The final pattern emerges in countries that are ethnically homogenous, and do not have any politically relevant ethnic minority groups. Subsequently, the structure of party competition in these countries is not significantly influenced by ethnic minority issues. This group includes Hungary, Poland and the Czech Republic (which was a federal center). Research shows that what matters in these particular contexts is the extent to which the former communist party reformed in terms of party ideology and subsequent positions. Nonetheless, social liberalism, and, with it, the positions on immigration, are again rooted in views on nationalism and ethnic minorities. On the one hand, the Czech Republic is home to an intransigent Communist party that represents a left-wing rousable by ethnic nationalism aimed against the historical, but no longer present, German minority. Today, the party takes a negative stance on the refugee crisis devoid of any solidaristic language, rather proposing to take the European Commission to the Court of Justice. [START_REF] Vojtěch | [END_REF] On the other hand, Polish and Hungarian politics are defined by reformist post-communists who adopted liberal stances in the 1990s, only to be eclipsed by their nationalist right-wing opponents in the 2000s. Here, it is the conservative right that vocally opposes immigration, while the left (quietly) presents a more conciliatory stance. Furthermore, in Hungary, the right-wing led by Victor Orban draws much of its current political identity from its nationalistic support of extraterritorial Hungarian nationals. He and his political party have extensively opposed the acceptance of refugees, which resonates with the preferences of the right-wing supporters, while the supporters of the left-wing hold significantly more liberal views. [START_REF]Hungarians Fear of Migrants and Terrorism[END_REF] Interestingly, and in line with the general trends, the Hungarian minority party in Slovakia (Most-Hid) presents one of the most supportive positions towards migrants, pointing out that migration might bring benefits to the receiving countries. [START_REF]MIGRÁCIA: Solidarita a zodpovednosť sa nevylučujú" Most-Hid official website[END_REF] The recent 2015 national election campaign in Poland extensively contested immigration policy, with the leader of the winning right-wing nationalist party playing up xenophobic sentiments. [START_REF] Cienski | Migrants carry 'parasites and protozoa,' warns Polish opposition leader[END_REF]

Policy Outcomes Based on these Patterns

There are thus diverse views on immigration policy and the current migration flow in Eastern Europe. This variance is strongly related to views of ethnic minorities, and nationalism. The ongoing migration crisis has put the issue of immigration, which was hitherto only marginal in the political discourses in the region, into the spotlight. As such, the crisis has greatly heightened the salience of immigration and asylum policy, and made it a significant political topic. Simultaneously, the real or potential inflow of migrants of culturally different backgrounds has created a significant political opportunity for political entrepreneurs. While parts of Eastern Europe have witnessed a shift towards conservative and nationalist politics prior to the migration crisis (the continuing strength of Victor Orban's Fidesz party, and the waning support of the liberal government in Poland prior to the 2015 election), the migration crisis offered nationalist politicians a potent topic for mobilizing voters. Victor Orban, as well as Jaroslaw Kaczynski, have seized this opportunity.

In terms of specific policy changes related to the migration crisis, there are two factors to consider. First is the political color of the government, which determines the likelihood that the government adopts liberal or conservative stances on the migration issue, in line with the three political patterns discussed above. Second factor is whether or not the country lies on the Balkan migration route. Table 1 provides an overview of immigration policy changes implemented by eastern European governments in the fall of 2015. It considers the restriction of movement by imposing border controls, or the adjustment of legislation concerning migration. In addition, the table also presents the rhetoric of the government concerning migration, as this may differ from actual policy implementation.

The table highlights two key patterns. First, all countries on the Balkan migration route closed their borders, no matter their political Second, the table demonstrates an association between the political pattern expected in a country, and which color of government enacts restrictive policy changes. For example, the technocratic liberal government of Romania and the centrist government of Lithuania amend national legislation so that it is in line with EU's regulations. On the contrary, and in line with theoretical expectations, the left-wing conservative government of Slovakia introduces border controls. In Latvia, it is the right-wing conservative government that enacts restrictive migration laws. In sum, the political patterns of Eastern Europe indicate the type of policy change that different colors of government are likely to follow.

In the long-run, the most obvious expectation concerning immigration policies in Eastern Europe is that, like in the west, these policies will be extensively contested by the competing liberal and conservative forces within each country. Who falls on which side of this divide will be dominantly determined by their longstanding views of ethnic minorities. Secondly, given the potent political opportunity offered by the migration crisis to the conservative camps, and the assertive use of this opportunity by some key actors, we should expect a significant rise of opposition to immigration on the part of the public, which will further strengthen the conservative side of the political divide.

Conclusion

The seemingly simple story of policy uniformity among Eastern European countries on the issue of immigration is now revealed as a much more complex aggregation of diverse political views based on attitudes towards ethnic minorities. In contexts where the left generally supports ethnic minorities due to its past affinity with the federal diaspora, these parties currently have more liberal immigration positions. Left parties who do not have this connection, however, are tending toward nationalist conservatism, particularly on the issue of immigration. And finally, in countries where there is no notable ethnic minority, positions on immigration hinge on whether or not the excommunist party reformed. In these contexts, including the Czech Republic, Hungary, and Poland, ideological principles about ethnic minorities now have a target group that was www.sciencespo.fr/liepp 5 Table 1 *An election occured in November 2015, however, the new government did not assume office until January 22, 2016. Source: This table was created by the author and references can be provided upon request.

previously absent in these relatively homogenous countries. Incoming immigrants and refugees have become the new "other." Subsequently, the real or potential influx of migrants in the context of the 2015 migration crisis has heightened the salience of the immigration issue. While all countries on the Balkan migration route imposed border controls, the general rhetorical, as well as policy response to the crisis, is importantly determined by the color of government and the political patterning of eastern European countries. As concerns over the migration remain, policy stances throughout Eastern Europe will continue to be driven by political views based on attitudes towards ethnic minorities .

3 [ 8 ]

 38 Rovny (2014), "The other 'other': Party responses to immigration in eastern Europe", p. 647.[9] Rovny (2014), "Communism, Federalism, and Ethnic Minorities", p. 672 : Van de Brug and van Spanje 2009.

  Figure 1:

Figure 3 :

 3 Figure 3:

Figure 4 :

 4 Figure 4:

Is Eastern Europe Uniformly Anti-Immigrant? Not so fast.www.sciencespo.fr/liepp

Jan Rovny is an Assistant Professor of Political Science at Sciences Po (Paris) attached to the Centre d'études 1] on its member states painted a *This work was supported by a public grant overseen by the French National Research Agency (ANR) as part of the ''Investissements d'Avenir'' program LIEPP (ANR-11-LABX-0091, ANR-11-IDEX-0005-02), and by the Swedish Research Council (Vetenskapsrådet), grant number 421-2012-1188. The author would like to thank Jessica Flakne for research assistance and editorial support. [1] Alison Smale and Dan Bilefsky, "Quota Proposal Fails to Gain Traction as Germany Prepares for More Arrivals," in New York Times (12/09/2015) : http://www.nytimes.

Le LIEPP (Laboratoire interdisciplinaire d'évaluation des politiques publiques) est un laboratoire d'excellence (Labex). Ce projet est distingué par le jury scientifique international désigné par l'Agence nationale de la recherche (ANR). Il est financé dans le cadre des investissements d'avenir. (ANR11LABX0091, ANR11IDEX000502)


Ana Gogava

FORTHCOMING EVENTS AT THE LSE

Keywords:

The government's own numbers show that alcohol is under-taxed

Hungary and Poland's anti-democratic turn: a new era or more of the same?

Reforms affecting the independence of courts and the media in Hungary and Poland have received significant attention in recent months. But to what extent do these developments constitute a genuine shift in the nature of Hungarian and Polish politics? Jan Rovny writes that while both countries have witnessed a rise in support for parties with anti-democratic tendencies, the dynamics of party competition remain consistent with the liberal-conservative political divide that has characterised the politics of these countries since the fall of communism.

Hungary and Poland have recently witnessed constitutional changes that have limited the independence of courts and the media. As these developments attract international attention, they beg the question of whether the politics of eastern Europe is being fundamentally altered, and the democratic achievements of the region reversed. My research suggests that while the recent changes in Hungary and Poland are a cause for concern, they do not represent a departure from traditionally entrenched contests within these societies. Instead, the current politics of Hungary and Poland continue to replicate, and even deepen, historical political divides.

Hungary and Poland: A new era or more of the same?

In a recent study, I illustrate that the main political divide in Hungary and Poland is one that combines economic preferences together with general socio-cultural visions of society. In both countries, reformed ex-communist socialist parties (MSZP in Hungary and SLD in Poland) initially coalesce with liberals around socially-liberal, pro-market policies, while deepening their countries' cooperation with western partners, particularly the European Union. These forces are consistently opposed by traditional, conservative, nationalist, and generally protectionist opposition.

In both Hungary and Poland, this core divide has been present since the early 1990s, and remains unaltered, even as it witnesses a flux of party organisations on either side. First, the original liberal parties in both countries -that is the Union of Freedom (UW) in Poland, and the Alliance of Free Democrats (SzDSz) in Hungary -lost parliamentary representation, only to be replaced by similar socially-liberal political organisations in later elections, such as Your Movement (RP) in Poland, and Politics Can Be Different (LMP) and other minor formations in Hungary.

Second, the socialist parties (MSZP in Hungary and SLD in Poland) in both countries suffered significant electoral setbacks -largely due to corruption scandals -which strengthened the conservative pole of the political divide. In Hungary the great electoral beneficiary has been the conservative nationalist Fidesz party. In Poland, the collapse of the socialists focused competition on the contest between the liberal-conservative Civic Platform (PO), which takes up the liberal pole of Polish politics, and the conservative nationalist Law and Justice (PiS).

Third, in both countries minor radical nationalist parties -such as the League of Polish Families (LPR) and Self-Defense in Poland, and the Justice and Life Party (MIEP) in Hungary -have given way to electorally more powerful organisations. In the case of Poland, Law and Justice has come to represent the nationalist pole, while in Hungary, this pole is represented jointly by the radical Jobbik, and the governing Fidesz. Consequently, despite a number of dramatic shifts in terms of party organisation and governing coalition options, the key divide in both Hungary and Poland remains intact. This is an important finding, further supported by ongoing research I am conducting with Jonathan Polk, as it underlines the fact that even when political parties have a weak organisational basis, as is typically the case in eastern Europe, they nonetheless tend to represent stable political divides and electorates. Although they come and go, they replicate stable patterns of political competition.

What then has changed in Hungary and Poland?

The key change in both countries has been the continuous weakening of the socially-liberal pole of the main political divide, coupled with increasingly bold assertions of conservative nationalism from the opposite side. Since the electoral undermining of socialist parties, the liberal bloc in Hungary contains a multiplicity of weak organisations struggling to form an effective anti-Fidesz coalition. In Poland, the liberal-conservative Civic Platform (PO) is under pressure from the conservative Law and Justice (PiS), the minor liberals, as well as from anti-system populists.

Simultaneously, the politicians of the conservative pole in both countries are pursuing further goals on the liberalconservative divide with greater confidence. Shifting away from simply championing conservative values in the current political framework, the Hungarian and, more recently Polish, governments are reformulating their political regimes by enacting legal changes deemed as limiting the independence of constitutional courts, and of the media. The Prime Minister of Hungary, Viktor Orban, whose tactics have been closely observed by the current government in Warsaw, argues for the need to "construct a new state built on illiberal and national foundations".

Finally, the conservative pole in both countries has been strengthened by the migration wave, which saw hundreds of thousands of refugees arriving in Europe from primarily Syria and Afghanistan in 2015. As many unregistered migrants passed through Hungary on their way to Germany and Scandinavia, Viktor Orban seized the political opportunity by sealing the Hungarian-Serbian border, and by vocally criticising immigration, as well as European Union plans to resettle the refugees throughout the EU.

The 2015 Polish election took place in the context of this migration wave, giving the opposition Law and Justice (PiS) party the opportunity to criticise the incumbent Civic Platform (PO) for accepting EU refugee quotas. Thus, the conservatives of both countries profited from this exogenous shock by integrating the political issue of immigration -a hitherto novel issue in Hungarian and Polish politics -into the existing liberal-conservative divide, and taking a vocal position on it.

The significant changes that have occurred in Poland and Hungary amount to worrying attempts at the hollowing out of liberal democracy, and the fueling of xenophobic nationalism. These changes are, however, taking place within the historical liberal-conservative political divide which has shaped the politics of these countries over the last quarter of a century. Despite the rise of various political organisations, and despite the increasingly anti-democratic character of the conservative political camps, as well as the increased salience of immigration, the core political division in Poland and Hungary remains stable.

Figure :

 : Figure: Visual representation of the Hungarian and Polish party systems (2014)

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  Please read our comments policy before commenting. Note: This article gives the views of the author, and not the position of EUROPP -European Politics and Policy, nor of EUROPP -Hungary and Poland's anti-democratic turn: a new era or more of the same? http://blogs.lse.ac.uk/europpblog/2016/03/02/hungary-and-polands-anti-democratic-turn-a-new-era-or-more-of-the-same/[09/03/2016 16:58:54] March 2nd, 2016 | current-affairs, EU Politics, Jan Rovny | 3 Comments 3 Comments  Previous post the London School of Economics. Shortened URL for this post: http://bit.ly/1LSSbSm _________________________________ About the author Jan Rovny -Sciences Po / University of Gothenburg Jan Rovny is an Assistant Professor in Political Science at the CEE and LIEPP at Sciences Po, Paris, and at CERGU at the University of Gothenburg. His research concentrates on political competition in Europe with the aim of uncovering the ideological conflict lines in different countries. He is also one of the principal investigators of the Chapel Hill Expert Survey. He has published in World Politics, Party Politics, European Union Politics, European Political Science Review, East European Politics and Societies and elsewhere. Share this: Next post  Leto March 2, 2016 at 9:25 pm -Reply anti-democratic = liberals don't like it zygmunt March 3, 2016 at 5:03 am -Reply This article is ideologically driven. PIS is not a nationalist party. PiS introduced a welfare scheme that is more generous that that of Norway.        LSE BrexitVote Blog -informing the debate on UK's EU referendum EU Events Calendar Click here for this month's upcoming EU related events Euro Crisis in the Press  European Council on Foreign Relations

The problem with the West that you are self-righteous, superficial and hypocritical. You call Central and Eastern European countries undemocratic when their governments are not social-liberal and/or do not let your countries to exploit them unscrupulously (free market economy -privatization -profit making). In other words, the West does not tolerate their desire for self determination and independence. They have to obey and play according to Western rules, even if often it is against their own (economic) interest. You also call these countries 'nationalist', meanwhile in practice you are more nationalist than those people (i.e. British-Brexit, USA). The only difference that you call yourselves 'patriotic', it sounds better. Do you 'worry' about democracy? In your countries democracy is a nice notion, nothing else, not even in the UK (i.e. lying about the Iraq war, bombing Syria without parliamentary approval) or in the US (i.e. lobbying -elections). You worry about free media too (a nice notion again), although, in your countries it does not even exist, for example, BBC or ZDF (in Germany) are not neutral at all or I can mention the deal between Murdoch and Blair. Lying and manipulating freely without any legal consequences and dividing people, this is what your 'free media' does (i.e. did in Hungary under the socialist government). Media freedom = dominance of social liberal ideas or spreading the word of the West. If the media is being regulated strictly (it is necessary!), you immediately start worrying about freedom. Xenophobic nationalism? What the heck is that? The British colonized a quarter of the world with brute force. You ruined millions of lives and created several failed states. Your master, the USA too. You went to Iraq and murdered hundreds of thousands of people for selfish reasons, therefore, you are responsible for this mess in the Middle-East. You also helped to build a fence in Calais because you do not want 'refugees' and other migrants to 'invade' your country illegally. No one can know who these people are. However, you condemn poorer Central and Eastern Europeans for worrying about them and doing the same? In addition, this is one of the reasons why you want to leave the EU. On this basis people are racist in the UK too………and so on. You should ask sometimes questions.

What happened? Why do they do this and that? What are their reasons? You do not really care what lies beneath the surface, do you? As a researcher you work for someone (for money) and you do your best to satisfy your master. You are also driven by your own conviction. In other word, you present your and your master's views as 'facts'. It is a shame and not science.