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Abstract Algebra/Clifford Algebras

In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry and theoretical physics. They are named for the English geometer William Clifford.

 * Some familiarity with the basics of multilinear algebra will be useful in reading this section.

Introduction and basic properties
Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q. The Clifford algebra Cℓ(V,Q) is the "freest" algebra generated by V subject to the condition1
 * $$v^2 = Q(v)\ \mbox{ for all } v\in V.$$

If the characteristic of the ground field K is not 2, then one can rewrite this fundamental identity in the form

 * $$uv + vu = \lang u, v\rang \mbox{ for all }u,v \in V,$$

where  = Q(u + v) − Q(u) − Q(v) is the symmetric bilinear form associated to Q. This idea of "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property (see below).

Clifford algebras are closely related to exterior algebras. In fact, if Q = 0 then the Clifford algebra Cℓ(V,Q) is just the exterior algebra Λ(V). For nonzero Q there exists a canonical linear isomorphism between Λ(V) and Cℓ(V,Q) whenever the ground field K does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication is strictly richer than the exterior product since it makes use of the extra information provided by Q. More precisely, they may be thought of as quantizations of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra.

Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if char K = 2 it is not true that a quadratic form is determined by its symmetric bilinear form, or that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not 2, and are false if this condition is removed.

Universal property and construction
Let V be a vector space over a field K, and let Q : V → K be a quadratic form on V. In most cases of interest the field K is either R or C (which have characteristic 0) or a finite field.

A Clifford algebra Cℓ(V,Q) is a unital associative algebra over K together with a linear map i : V → Cℓ(V,Q) defined by the following universal property: Given any associative algebra A over K and any linear map j : V → A such that

 * j(v)2 = Q(v)1 for all v ∈ V

(where 1 denotes the multiplicative identity of A), there is a unique algebra homomorphism f : Cℓ(V,Q) → A such that the following diagram commutes (i.e. such that f o i = j):

Working with a symmetric bilinear form <·,·> instead of Q (in characteristic not 2), the requirement on j is

 * j(v)j(w) + j(w)j(v) =  for all v, w ∈ V.

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains V, namely the tensor algebra T(V), and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal IQ in T(V) generated by all elements of the form

 * $$v\otimes v - Q(v)1$$ for all $$v\in V$$

and define Cℓ(V,Q) as the quotient

 * Cℓ(V,Q) = T(V)/IQ.

It is then straightforward to show that Cℓ(V,Q) contains V and satisfies the above universal property, so that Cℓ is unique up to isomorphism; thus one speaks of "the" Clifford algebra Cℓ(V, Q). It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of Cℓ(V,Q).

The universal characterization of the Clifford algebra shows that the construction of Cℓ(V,Q) is functorial in nature. Namely, Cℓ can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.

Basis and dimension
If the dimension of V is n and {e1,…,en} is a basis of V, then the set

 * $$\{e_{i_1}e_{i_2}\cdots e_{i_k} \mid 1\le i_1 < i_2 < \cdots < i_k \le n\mbox{ and } 0\le k\le n\}$$

is a basis for Cℓ(V,Q). The empty product (k = 0) is defined as the multiplicative identity element. For each value of k there are n choose k basis elements, so the total dimension of the Clifford algebra is

 * $$\dim C\ell(V,Q) = \sum_{k=0}^n\begin{pmatrix}n\\ k\end{pmatrix} = 2^n.$$

Since V comes equipped with a quadratic form, there is a set of privileged bases for V: the orthogonal ones. An orthogonal basis in one such that

 * $$\langle e_i, e_j \rangle = 0 \qquad i\neq j. \,$$

where <·,·> is the symmetric bilinear form associated to Q. The fundamental Clifford identity implies that for an orthogonal basis

 * $$e_ie_j = -e_je_i \qquad i\neq j. \,$$

This makes manipulation of orthogonal basis vectors quite simple. Given a product $$e_{i_1}e_{i_2}\cdots e_{i_k}$$ of distinct orthogonal basis vectors, one can put them into standard order by including an overall sign corresponding to the number of flips needed to correctly order them (i.e. the signature of the ordering permutation).

If the characteristic is not 2 then an orthogonal basis for V exists, and one can easily extend the quadratic form on V to a quadratic form on all of Cℓ(V,Q) by requiring that distinct elements $$e_{i_1}e_{i_2}\cdots e_{i_k}$$ are orthogonal to one another whenever the {ei}'s are orthogonal. Additionally, one sets
 * $$Q(e_{i_1}e_{i_2}\cdots e_{i_k}) = Q(e_{i_1})Q(e_{i_2})\cdots Q(e_{i_k})$$.

The quadratic form on a scalar is just Q(λ) = λ2. Thus, orthogonal bases for V extend to orthogonal bases for Cℓ(V,Q). The quadratic form defined in this way is actually independent of the orthogonal basis chosen (a basis-independent formulation will be given later).

Examples: Real and complex Clifford algebras
The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms.

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:
 * $$Q(v) = v_1^2 + \cdots + v_p^2 - v_{p+1}^2 - \cdots - v_{p+q}^2$$

where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp,q. The Clifford algebra on Rp,q is denoted Cℓp,q(R). The symbol Cℓn(R) means either Cℓn,0(R) or Cℓ0,n(R) depending on whether the author prefers positive definite or negative definite spaces.

A standard orthonormal basis {ei} for Rp,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1. The algebra Cℓp,q(R) will therefore have p vectors which square to +1 and q vectors which square to −1.

Note that Cℓ0,0(R) is naturally isomorphic to R since there are no nonzero vectors. Cℓ0,1(R) is a two-dimensional algebra generated by a single vector e1 which squares to −1, and therefore is isomorphic to C, the field of complex numbers. The algebra Cℓ0,2(R) is a four-dimensional algebra spanned by {1, e1, e2, e1e2}. The latter three elements square to −1 and all anticommute, and so the algebra is isomorphic to the quaternions H. The next algebra in the sequence is Cℓ0,3(R) is an 8-dimensional algebra isomorphic to the direct sum H ⊕ H called Clifford biquaternions.

One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form
 * $$Q(z) = z_1^2 + z_2^2 + \cdots + z_n^2$$

where n = dim V, so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on Cn with the standard quadratic form by Cℓn(C). One can show that the algebra Cℓn(C) may be obtained as the complexification of the algebra Cℓp,q(R) where n = p + q:
 * $$C\ell_n(\mathbb{C}) \cong C\ell_{p,q}(\mathbb{R})\otimes\mathbb{C} \cong C\ell(\mathbb{C}^{p+q},Q\otimes\mathbb{C})$$.

Here Q is the real quadratic form of signature (p,q). Note that the complexification does not depend on the signature. The first few cases are not hard to compute. One finds that
 * Cℓ0(C) = C
 * Cℓ1(C) = C ⊕ C
 * Cℓ2(C) = M2(C)

where M2(C) denotes the algebra of 2×2 matrices over C.

It turns out that every one of the algebras Cℓp,q(R) and Cℓn(C) is isomorphic to a matrix algebra over R, C, or H or to a direct sum of two such algebras. For a complete classification of these algebras see classification of Clifford algebras.

Relation to the exterior algebra
Given a vector space V one can construct the exterior algebra Λ(V), whose definition is independent of any quadratic form on V. It turns out that if F does not have characteristic 2 then there is a natural isomorphism between Λ(V) and Cℓ(V,Q) considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only if Q = 0. One can thus consider the Clifford algebra Cℓ(V,Q) as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on V with a multiplication that depends on Q (one can still define the exterior product independent of Q).

The easiest way to establish the isomorphism is to choose an orthogonal basis {ei} for V and extend it to an orthogonal basis for Cℓ(V,Q) as described above. The map Cℓ(V,Q) → Λ(V) is determined by
 * $$e_{i_1}e_{i_2}\cdots e_{i_k} \mapsto e_{i_1}\wedge e_{i_2}\wedge \cdots \wedge e_{i_k}.$$

Note that this only works if the basis {ei} is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism.

If the characteristic of K is 0, one can also establish the isomorphism by antisymmetrizing. Define functions fk : V × … × V → Cℓ(V,Q) by
 * $$f_k(v_1, \cdots, v_k) = \frac{1}{k!}\sum_{\sigma\in S_k}{\rm sgn}(\sigma)\, v_{\sigma(1)}\cdots v_{\sigma(k)}$$

where the sum is taken over the symmetric group on k elements. Since fk is alternating it induces a unique linear map Λk(V) → Cℓ(V,Q). The direct sum of these maps gives a linear map between Λ(V) and Cℓ(V,Q). This map can be shown to be a linear isomorphism, and it is natural.

A more sophisticated way to view the relationship is to construct a filtration on Cℓ(V,Q). Recall that the tensor algebra T(V) has a natural filtration: F0 ⊂ F1 ⊂ F2 ⊂ … where Fk contains sums of tensors with rank ≤ k. Projecting this down to the Clifford algebra gives a filtration on Cℓ(V,Q). The associated graded algebra
 * $$\bigoplus_k F^k/F^{k-1}$$

is naturally isomorphic to the exterior algebra Λ(V). Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of Fk in Fk+1 for all k), this provides an isomorphism (although not a natural one) in any characteristic, even two.

Grading
The linear map on V defined by $$v \mapsto -v$$ preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra automorphism
 * α : Cℓ(V,Q) → Cℓ(V,Q).

Since α is an involution (i.e. it squares to the identity) one can decompose Cℓ(V,Q) into positive and negative eigenspaces
 * $$C\ell(V,Q) = C\ell^0(V,Q) \oplus C\ell^1(V,Q)$$

where Cℓi(V,Q) = {x ∈ Cℓ(V,Q) | α(x) = (−1)ix}. Since α is an automorphism it follows that
 * $$C\ell^{\,i}(V,Q)C\ell^{\,j}(V,Q) = C\ell^{\,i+j}(V,Q)$$

where the superscripts are read modulo 2. This means that Cℓ(V,Q) is a Z2-graded algebra (also known as a superalgebra). Note that Cℓ0(V,Q) forms a subalgebra of Cℓ(V,Q), called the even subalgebra. The piece Cℓ1(V,Q) is called the odd part of Cℓ(V,Q) (it is not a subalgebra). This Z2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the main involution or grade involution.

Remark. In characteristic not 2 the algebra Cℓ(V,Q) inherits a Z-grading from the canonical isomorphism with the exterior algebra Λ(V). It is important to note, however, that this is a vector space grading only. That is, Clifford multiplication does not respect the Z-grading only the Z2-grading. Happily, the gradings are related in the natural way: Z2 = Z/2Z. The degree of a Clifford number usually refers to the degree in the Z-grading. Elements which are pure in the Z2-grading are simply said to be even or odd.

If the characteristic of F is not 2 then the even subalgebra Cℓ0(V,Q) of a Clifford algebra is itself a Clifford algebra. If V is the orthogonal direct sum of a vector a of norm Q(a) and a subspace U, then Cℓ0(V,Q) is isomorphic to Cℓ(U,−Q(a)Q), where −Q(a)Q is the form Q restricted to U and multiplied by −Q(a). In particular over the reals this implies that
 * $$C\ell_{p,q}^0(\mathbb{R}) \cong C\ell_{p,q-1}(\mathbb{R})$$ for q > 0, and
 * $$C\ell_{p,q}^0(\mathbb{R}) \cong C\ell_{q,p-1}(\mathbb{R})$$for p > 0.

In the negative-definite case this gives an inclusion Cℓ0,n−1(R) ⊂ Cℓ0, n(R) which extends the sequence
 * R ⊂ C ⊂ H ⊂ H⊕H ⊂ …

Likewise, in the complex case, one can show that the even subalgebra of Cℓn(C) is isomorphic to Cℓn−1(C).

Antiautomorphisms
In addition to the automorphism α, there are two antiautomorphisms which play an important role in the analysis of Clifford algebras. Recall that the tensor algebra T(V) comes with an antiautomorphism that reverses the order in all products:
 * $$v_1\otimes v_2\otimes \cdots \otimes v_k \mapsto v_k\otimes \cdots \otimes v_2\otimes v_1$$.

Since the ideal IQ is invariant under this reversal, this operation descends to an antiautomorphism of Cℓ(V,Q) called the transpose or reversal operation, denoted by xt. The transpose is an antiautomorphism: $$(xy)^t = y^t x^t$$. The transpose operation makes no use of the Z2-grading so we define a second antiautomorphism by composing α and the transpose. We call this operation Clifford conjugation denoted $$\bar x$$
 * $$\bar x = \alpha(x^t) = \alpha(x)^t.$$

Of the two antiautomorphisms, the transpose is the more fundamental.3

Note that all of these operations are involutions. One can show that they act as ±1 on elements which are pure in the Z-grading. In fact, all three operations depend only on the degree modulo 4. That is, if x is pure with degree k then
 * $$\alpha(x) = \pm x \qquad x^t = \pm x \qquad \bar x = \pm x$$

where the signs are given by the following table:

The Clifford scalar product
When the characteristic is not 2 the quadratic form Q on V can be extended to a quadratic form on all of Cℓ(V,Q) as explained earlier (which we also denoted by Q). A basis independent definition is
 * $$Q(x) = \lang x^t x\rang$$

where  denotes the scalar part of a (the grade 0 part in the Z-grading). One can show that
 * $$Q(v_1v_2\cdots v_k) = Q(v_1)Q(v_2)\cdots Q(v_k)$$

where the vi are elements of V — this identity is not true for arbitrary elements of Cℓ(V,Q).

The associated symmetric bilinear form on Cℓ(V,Q) is given by
 * $$\lang x, y\rang = \lang x^t y\rang.$$

One can check that this reduces to the original bilinear form when restricted to V. The bilinear form on all of Cℓ(V,Q) is nondegenerate if and only it is nondegenerate on V.

It is not hard to verify that the transpose is the adjoint of left/right Clifford multiplication with respect to this inner product. That is,
 * $$\lang ax, y\rang = \lang x, a^t y\rang,$$ and
 * $$\lang xa, y\rang = \lang x, y a^t\rang.$$

Structure of Clifford algebras
In this section we assume that the vector space V is finite dimensional and that the bilinear form of Q is non-singular. A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.

 * If V has even dimension then Cℓ(V,Q) is a central simple algebra over K.
 * If V has even dimension then Cℓ0(V,Q) is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K.
 * If V has odd dimension then Cℓ(V,Q) is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K.
 * If V has odd dimension then Cℓ0(V,Q) is a central simple algebra over K.

The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that U has even dimension and a non-singular bilinear form with discriminant d, and suppose that V is another vector space with a quadratic form. The Clifford algebra of U+V is isomorphic to the tensor product of the Clifford algebras of U and (−1)dim(U)/2dV, which is the space V with its quadratic form multiplied by (−1)dim(U)/2d. Over the reals, this implies in particular that
 * $$ Cl_{p+2,q}(\mathbb{R}) = M_2(\mathbb{R})\otimes Cl_{q,p}(\mathbb{R}) $$
 * $$ Cl_{p+1,q+1}(\mathbb{R}) = M_2(\mathbb{R})\otimes Cl_{p,q}(\mathbb{R}) $$
 * $$ Cl_{p,q+2}(\mathbb{R}) = \mathbb{H}\otimes Cl_{q,p}(\mathbb{R}) $$

These formulas can be used to find the structure of all real Clifford algebras.

The Clifford group Γ
In this section we assume that V is finite dimensional and the bilinear form of Q is non-singular.

The Clifford group Γ is defined to be the set of invertible elements x of the Clifford algebra such that
 * $$x v \alpha(x)^{-1}\in V$$

for all v in V. This formula also defines an action of the Clifford group on the vector space V that preserves the norm Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements r of V of nonzero norm, and these act on V by the corresponding reflections that take v to v − r/Q(r) (In characteristic 2 these are called orthogonal transvections rather than reflections.)

Many authors define the Clifford group slightly differently, by replacing the action xvα(x)−1 by xvx−1. This produces the same Clifford group, but the action of the Clifford group on V is changed slightly: the action of the odd elements Γ1 of the Clifford group is multiplied by an extra factor of −1. This action used here has several minor advantages: it is consistent with the usual superalgebra sign conventions, elements of V correspond to reflections, and in odd dimensions the map from the Clifford group to the orthogonal group is onto, and the kernel is no larger than K*. Using the action α(x)vx−1 instead of xvα(x)−1 makes no difference: it produces the same Clifford group with the same action on V.

The Clifford group Γ is the disjoint union of two subsets Γ0 and Γ1, where Γi is the subset of elements of degree i. The subset Γ0 is a subgroup of index 2 in Γ.

If V is finite dimensional with nondegenerate bilinear form then the Clifford group maps onto the orthogonal group of V and the kernel consists of the nonzero elements of the field K. This leads to exact sequences
 * $$ 1 \rightarrow K^* \rightarrow \Gamma \rightarrow O_V(K) \rightarrow 1,\,$$
 * $$ 1 \rightarrow K^* \rightarrow \Gamma^0 \rightarrow SO_V(K) \rightarrow 1.\,$$

In arbitrary characteristic, the spinor norm Q is defined on the Clifford group by
 * $$Q(x) = x^tx\,$$

It is a homomorphism from the Clifford group to the group K* of non-zero elements of K. It coincides with the quadratic form Q of V when V is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of −1, 2, or −2 on Γ1. The difference is not very important.

The nonzero elements of K have spinor norm in the group K*2 of squares of nonzero elements of the field K. So when V is finite dimensional and non-singular we get an induced map from the orthogonal group of V to the group K*/K*2, also called the spinor norm. The spinor norm of the reflection of a vector r has image Q(r) in K*/K*2, and this property uniquely defines it on the orthogonal group. This gives exact sequences:

 * $$ 1 \rightarrow \{\pm 1\} \rightarrow Pin_V(K) \rightarrow O_V(K) \rightarrow K^*/K^{*2},\,$$
 * $$ 1 \rightarrow \{\pm 1\} \rightarrow Spin_V(K) \rightarrow SO_V(K) \rightarrow K^*/K^{*2}.\,$$

Note that in characteristic 2 the group {±1} has just one element.

Spin and Pin groups
In this section we assume that V is finite dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.)

The Pin group PinV(K) is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the Spin group SpinV(K) is the subgroup of elements of Dickson invariant 0 in PinV(K). When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group.

Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group. We define the special orthogonal group to be the image of Γ0. If K does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. If K does have characteristic 2, then all elements of the orthogonal group have determinant 1, and the special orthogonal group is the set of elements of Dickson invariant 0.

There is a homomorphism from the Pin group to the orthogonal group. The image consists of the elements of spinor norm 1 ∈ K*/K*2. The kernel consists of the elements +1 and −1, and has order 2 unless K has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V.

In the common case when V is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when V has dimension at least 3. Warning: This is not true in general: if V is Rp,q for p and q both at least 2 then the spin group is not simply connected and does not map onto the special orthogonal group. In this case the algebraic group Spinp,q is simply connected as an algebraic group, even though its group of real valued points Spinp,q(R) is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.

Spinors
Suppose that p+q=2n is even. Then the Clifford algebra Cℓp,q(C) is a matrix algebra, and so has a complex representation of dimension 2n. By restricting to the group Pinp,q(R) we get a complex representation of the Pin group of the same dimension, called the spinor representation. If we restrict this to the spin group Spinp,q(R) then it splits as the sum of two half spin representations (or Weyl representations) of dimension 2n-1.

If p+q=2n+1 is odd then the Clifford algebra Cℓp,q(C) is a sum of two matrix algebras, each of which has a representation of dimension 2n, and these are also both representations of the Pin group Pinp,q(R). On restriction to the spin group Spinp,q(R) these become isomorphic, so the spin group has a complex spinor representation of dimension 2n.

More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the structure of the corresponding Clifford algebras: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra. For examples over the reals see the article on spinors.

Differential geometry
One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry.

Physics
Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra spanned by matrices γ1,…,γn called Dirac matrices which have the property that

 * $$\gamma_i\gamma_j + \gamma_j\gamma_i = 2\eta_{ij}\,$$

where η is the matrix of a quadratic form of signature (p,q) — typically (1,3) when working in Minkowski space. These are exactly the defining relations for the Clifford algebra Cl1,3(C) (up to an unimportant factor of 2), which by the classification of Clifford algebras is isomorphic to the algebra of 4 by 4 complex matrices.

The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the Dirac equation. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears.

title: Abstract Algebra/Clifford Algebras


text:
Eiffel Programming/Unfeatures

Goto, break or the equivalent
Ever since the advent of structure programming the goto has been considered evil, yet many programming languages support it. Eiffel does not. The language is designed to uphold the principle of Single Entry/Single Exit.

Some people consider this a gross inefficiency and will labor endlessly to demonstrate arcane algorithms that are allegedly faster with a well-placed goto, or it requires fewer variables, or fewer lines of code. Others consider is a serious crimp to their writing style.

Yet there are two serious advantages to the principle of Single Entry/Single Exit. First, it encourages the programmer to think clearly about the implementation. In other languages, if you find yourself in getting into trouble deep inside a multiply nested loop, you bail out with a goto. In Eiffel, you have to rethink the structure of the code, often to the benefit of the final program.

Second, source code is read much more often than it is written. The absence of a goto statement makes it easier to grasp the implementation. There are no subtle tricks buried in the inner loop that can skip to some arbitrary location.

Globals
State shared via the once statement Allows precise control over sharing data

CamelCase
Underscore_notation_is_preferred_for_better_readability Vs. CamelCaseNotationWhichCanBeSeemsSmart

Case sensitivity
True = true = TRUE

Overloaded Names
Overloaded names, also known as operator overloading, introduces ambiguity for the human programmer and reader of source code, and in some cases, for the compiler as well. Eiffel avoids this source of errors by not implementing it.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Eiffel Programming/Unfeatures


text:
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User talk:Ryoske

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Good luck! -- Herby talk thyme 08:20, 19 January 2007 (UTC)

title: User talk:Mishalla


text:
User talk:Lolit cbls

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Welcome
Hi, and welcome to wikibooks. If you need any help or advice of any kind, please feel free to ask me or one of the other editors at any time. I will be more than happy to help, as will the other editors. Urbane User (Talk)   (Contributions)  10:15, 21 January 2007 (UTC)

title: User talk:Lolit cbls


text:
User talk:Tintina

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thank you! Tintina 19:24, 19 January 2007 (UTC)

You are welcome! -- Herby talk thyme 19:33, 19 January 2007 (UTC)

title: User talk:Tintina


text:
User talk:AlanClifford

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Possible dupe
Hi - looking at ERP Internals/Accounts Payable & Accounts Payable they appear to be duplicates - does one need deleting - thanks -- Herby talk thyme 08:23, 19 January 2007 (UTC)
 * No problem - I take it Accounts Payable is the one to go? Let me know & I'll do it and ask around if you need any help - regards -- Herby  talk thyme 18:27, 19 January 2007 (UTC)

Lost page?
Hi - found this one Cash on Delivery. I guess you may want to move it? Regards -- Herby talk thyme 12:27, 1 February 2007 (UTC)

Nice work on ERP
Howdy. Just thought I'd drop you a line and thank you for your contributions to ERP. 74.97.43.10 01:06, 15 May 2007 (UTC)

title: User talk:AlanClifford


text:
User talk:Electrice

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title: User talk:Electrice


text:
User talk:Fg2

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title: User talk:Fg2


text:
Talk:Freistil

Article wishes & feedback

Untitled
Longing for a text about that particular interest of yours? Seen something on TV that you'd now like to read about in German? Got a soft spot for some local German speciality? Then this wishing well is for you. Click on the "+" button on the navigation bar on top of this page, drop in your thoughts and we'll see what we can do.

This is also the place to give general feedback about what you like and dislike about this project. As all wikibooks, Freistil is a "living" project and you as a reader can directly influence its development. Since we're not getting any feedback from sales figures, as do authors and publishing houses, we're glad about any comment that shows us this book is actually being read.

German dialects
For me, it's interesting that German has so many different dialects, even in one country. I know it's a very broad scope, so perhaps there may be a series of articles, each one about one region (not neccessarily for every land, but for regions which have their own dialect). I imagine these articles could give some of most interesting expressions in local dialect compared with High German equivalents, together with some historical background. If a text gets too long, there could be one day break after it (so that the author won't have to write too much and readers have time to read the whole ;-). I would like to see Lower German, Saksisch and Berlin dialect plus Austria and Switzerland (when I learned German in school noone told me that they don't use sharp "s" in Switzerland). Oh, when it comes to writing - I have heard something about a special way German people were once taught to write by hand and you can recognise native German speaker by the way he or she writes "z" letter (with horizontal line like z ).

Plus perhaps small languages like Romansh, Frisian, Luxembourgish. I'm also interested in Sorbs in Germany. --Derbeth talk 18:27, 19 January 2007 (UTC)

I have contacted Wikipedian groups about the dialect issue. As I live in Heidelberg, I can only provide correct information about my home dialect Kurpfälzisch. There are indeed several differences when it comes to writing: the "z" always with horizontal bar, the "1" always with diagonal upper end, varying quotation marks „ and “ instead of " and ', different lengths of dashes – — and usage of blanks before and after them and last but not least the additional letters ä, ö, ü and ß. Over the years, script was subject to various reforms. Today, young people learn a simplified basic script and can barely read their grandfathers' Sütterlin-cursive. We will certainly have a set of texts about local German differences in language. I am not familiar with Sorbs though, and will look out for someone else who could do that. --Bitbert 16:25, 22 January 2007 (UTC)

Look
I think that marking words having translation with blue colour is a bit embarrasing, as usually hyperlinks look like this. I find italic text or light gray underline a better solution. --Derbeth talk 09:56, 21 January 2007 (UTC)

On Talk:Freistil/Participate there has already been a discussion on this topic:

I would like to suggest some other visual identifier (probably color) in addition to (or possibly instead of) italics that would encourage people to mouseover the words while they read. This would aid in learning how to use the book without having to read the "how to use this book" page (which I suspect would often be skipped). Kellen T 17:14, 15 January 2007 (UTC)

I don't stick to italics, only the markup must not distract the reader. I could imagine either a dark colour similar to that of links, i. e. dark blue or dark purple, or some sort of underlining, both of which could encourage people to hover their cursor. Maybe some changing visual key like the regular link markup would do as well. --Bitbert 20:23, 15 January 2007 (UTC)

I have tried underline and dark blue color, and I think color markup works better. Let me know about your thoughts. --Bitbert 20:26, 15 January 2007 (UTC)

In the German mirror project, the item lately has been addressed, too, with no conclusion up till now. --Bitbert 08:12, 22 January 2007 (UTC)

 * See Talk:Freistil/Participate. --Derbeth talk 18:33, 22 January 2007 (UTC)

Advertising
If you know any internet portals for learning German, you can try to add link to Freistil there. You can also add it to some web directories. --Derbeth talk 09:56, 21 January 2007 (UTC)

Article
I don't see any sense in giving license of image used in article. It only produces information noise and distracts readers from what is really important. You can find all neccessary details about a photo by entering its description page. I also don't see much sense in informing what license Wikinews is using. People who are curious about it can follow a link to the original text. --Derbeth talk 15:23, 22 January 2007 (UTC)

The licence information is intended to be on the safe side with the licenses' requirements, not to help the reader access any data. Both Creative Commons licenses require the instance reusing a piece of work to name the licence itself. I am not an expert in copyright law, so I decided to rather give too much information than to risk a legal row. If you find that piece of text too disturbing, we can either change the layout or try and see someone at Wikibooks or Wikipedia who is more proficient in legal issues. In my eyes, the surplus rather makes the text more trustworthy, as it shows that someone actually spared a thought on the copyright issue. --Bitbert 16:13, 22 January 2007 (UTC)
 * But it's completely absolete to provide license for images if it can be checked by clicking on the image. --Derbeth talk 18:33, 22 January 2007 (UTC)

title: Talk:Freistil


text:
User:Rwst~enwikibooks/Eq

 This template displays a centered equation and, optionally, an equation label or equation number. This equation label is also used as a tag name and thus can be a link target, so that a reader can click on a reference to an equation and the browser will jump to that equation. (The equation label does not have to be a number, and text labels might be more useful in a collaboratively edited environment, unless an automatic renumbering system existed that could also update all references to a link target when it's name changed.)

Example:

or if this is moved to Wikipedia's global templates library, the template name would simplify to just "Eq":

Euler's Equation is one of the most beautiful and elegant of all equations, because it shows a special relation between 5 of the most important numbers in mathematics, as shown in Eq-Euler.

generates:

 (Eq-Euler) $$e^{i \pi} + 1 = 0$$

Euler's Equation is one of the most beautiful and elegant of all equations, because it shows a special relation between 5 of the most important numbers in mathematics, as shown in Eq-Euler.

Caveats:

(1) It appears that apostrophes aren't allowed in link names, so in this example I used "Eq-Euler" as the label, instead of "Euler's Equation", which might have been more natural.

(2) One could imagine a somewhat more modular template that could be called after an equation is formatted (instead of taking the equation as an argument to the template). This template would just generate a right-aligned label, and an associated tag that could be a link target. But I haven't yet figured out how to get this label on the same line as the equation.

title: User:Rwst~enwikibooks/Eq


text:
User talk:ChenJunZhou

Hello and welcome to Wikibooks!

Here are some tips to help you get started:
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Good luck! -- Herby talk thyme 14:38, 19 January 2007 (UTC)

Wikipedia copy
Looking at the page you created (Abstract algebra/Clifford Algebras), we cannot keep this. Copying erases the contributors history and so violates licensing. If you want the page to be on Wikibooks you need to look at having it transwiki'd - WB:TW and Special:Import will give you some pointers. If you need help please ask -- Herby talk thyme 14:38, 19 January 2007 (UTC)

title: User talk:ChenJunZhou


text:
User:JAltman752~enwikibooks

My name is James. For more information about me, please see my Wikipedia user page.

title: User:JAltman752~enwikibooks


text:
User talk:201.240.197.52

Please do not add inappropriate external links to Wikibooks. Wikibooks is intended to be read without the need for external reference and requires content rather than external links particularly those which are for advertising or promotion. Inappropriate links may include (but are not limited to) links to personal web sites, links to web sites with which you are affiliated, and links that exist to attract visitors to a web site or promote a product. If you feel the link should be added to the article, then please discuss it on the article's talk page rather than re-adding it. See the welcome page to learn more about Wikibooks. Thank you. -- Herby talk thyme 16:01, 19 January 2007 (UTC)

title: User talk:201.240.197.52


text:
Talk:Self-Replicating Automata

Various Issues
Could this be made more accessible to people who are intrested but who are not such low level coders.

What OS and version's do each variant work on, presumably each relise of a service pach 'fixes' most of thease.

Is there any self replicating code that mutates sightly at random on making a copy? is this possible?

Alan2here 16:46, 19 January 2007 (UTC)

 * This is possible. I'm not at home, but if I was I'd find the name of the experiment, but in short, years ago someone did an experiment with assembly in which every 'critter' started off with the same code that copied itself and also mutated itself randomly for the next iteration. Improper programs were killed outright, and the ones that perpetuated had interesting properties. The experiment was done in such a way that the shorter the program was, the quicker it would spawn, so that parasites developed that used code other than it's own. User:albinofrenchy

title: Talk:Self-Replicating Automata


text:
LaTeX/Export To Other Formats

\documentclass{article}

\usepackage{amsmath}

\title{Dosage Calculations}

\date{}

\begin{document}

\maketitle

\section*{Problem 1: Converting between units (mg to g)}

\subsection*{Question}

Convert 750 mg to grams.

\subsection*{Solution}

\[

750\, \text{mg} \times \left( \frac{1\, \text{g}}{1000\, \text{mg}} \right) = 0.75\, \text{g}

\]

\subsection*{Rationale}

To convert milligrams to grams, you divide by 1000 because there are 1000 milligrams in a gram.

\section*{Problem 2: Calculating dosages based on body weight (mg/kg)}

\subsection*{Question}

A patient weighs 85 kg and is prescribed a medication at 4 mg/kg. What is the total dose?

\subsection*{Solution}

\[

85\, \text{kg} \times 4\, \text{mg/kg} = 340\, \text{mg}

\]

\subsection*{Rationale}

Multiply the patient’s weight in kilograms by the prescribed dose per kilogram to find the total dose.

\section*{Problem 3: Adjusting doses when the available drug concentration differs from the prescribed dose}

\subsection*{Question}

A patient is prescribed 1 g of a medication. The medication is available as 250 mg tablets. How many tablets should the nurse administer?

\subsection*{Solution}

\[

1\, \text{g} = 1000\, \text{mg}

\]

\[

\text{Tablets to administer} = \left( \frac{1000\, \text{mg}}{250\, \text{mg/tablet}} \right) = 4\, \text{tablets}

\]

\subsection*{Rationale}

Convert the prescribed dose from grams to milligrams (since 1 g = 1000 mg). Then, divide by the concentration per tablet to find the number of tablets needed.

\section*{IV Flow Rate Calculations}

\subsection*{Problem 4: Calculating the flow rate in mL/hr}

\subsection*{Question}

An order is given to infuse 1500 mL of IV fluid over 10 hours. What is the flow rate in mL/hr?

\subsection*{Solution}

\[

\text{Flow rate} = \left( \frac{1500\, \text{mL}}{10\, \text{hours}} \right) = 150\, \text{mL/hr}

\]

\subsection*{Rationale}

Divide the total volume of fluid by the number of hours to find the flow rate in mL/hr.

\subsection*{Problem 5: Determining the drop rate in drops per minute (gtt/min) using drop factors}

\subsection*{Question}

An IV is ordered to infuse 600 mL over 5 hours. The drop factor is 20 gtt/mL. What is the flow rate in gtt/min?

\subsection*{Solution}

\[

\text{Flow rate} = \left( \frac{600\, \text{mL} \times 20\, \text{gtt/mL}}{5\, \text{hours} \times 60\, \text{minutes/hour}} \right) = \left( \frac{12000\, \text{gtt}}{300\, \text{minutes}} \right) = 40\, \text{gtt/min}

\]

\subsection*{Rationale}

Multiply the total volume by the drop factor to find the total drops. Then, divide by the total time in minutes to find the drop rate.

\section*{Infusion Time Calculations}

\subsection*{Problem 6: Determining how long an IV will run based on the volume and flow rate}

\subsection*{Question}

An IV of 800 mL is running at 100 mL/hr. How long will it take for the IV to complete?

\subsection*{Solution}

\[

\text{Time} = \left( \frac{800\, \text{mL}}{100\, \text{mL/hr}} \right) = 8\, \text{hours}

\]

\subsection*{Rationale}

Divide the total volume by the flow rate to determine the infusion time.

\section*{Pediatric Dosages}

\subsection*{Problem 7: Calculating safe dose ranges for children based on weight or body surface area (BSA)}

\subsection*{Question}

A child weighs 15 kg and the safe dosage range for a medication is 8-10 mg/kg/day, divided into 3 doses. What is the safe dosage range per dose?

\subsection*{Solution}

\[

\text{Total daily dose range} = 15\, \text{kg} \times (8\, \text{mg/kg to 10\, mg/kg}) = 120\, \text{mg to 150\, mg}

\]

\[

\text{Per dose} = \left( \frac{120\, \text{mg to 150\, mg}}{3} \right) = 40\, \text{mg to 50\, mg}

\]

\subsection*{Rationale}

Multiply the child’s weight by the safe dosage range to find the total daily dose. Divide the total daily dose by the number of doses per day to find the dose per administration.

\section*{More Practice Problems with Rationale}

\subsection*{Problem 8: Converting units (mcg to mg)}

\subsection*{Question}

Convert 5000 mcg to mg.

\subsection*{Solution}

\[

5000\, \text{mcg} \times \left( \frac{1\, \text{mg}}{1000\, \text{mcg}} \right) = 5\, \text{mg}

\]

\subsection*{Rationale}

To convert micrograms to milligrams, divide by 1000 because there are 1000 micrograms in a milligram.

\subsection*{Problem 9: Dosage calculation based on body weight}

\subsection*{Question}

A patient weighs 60 kg and is prescribed a medication at 3 mg/kg. What is the total dose?

\subsection*{Solution}

\[

60\, \text{kg} \times 3\, \text{mg/kg} = 180\, \text{mg}

\]

\subsection*{Rationale}

Multiply the patient’s weight in kilograms by the prescribed dose per kilogram to find the total dose.

\subsection*{Problem 10: Adjusting doses}

\subsection*{Question}

A patient is prescribed 2 g of a medication. The medication is available as 500 mg tablets. How many tablets should the nurse administer?

\subsection*{Solution}

\[

2\, \text{g} = 2000\, \text{mg}

\]

\[

\text{Tablets to administer} = \left( \frac{2000\, \text{mg}}{500\, \text{mg/tablet}} \right) = 4\, \text{tablets}

\]

\subsection*{Rationale}

Convert the prescribed dose from grams to milligrams (since 1 g = 1000 mg). Then, divide by the concentration per tablet to find the number of tablets needed.

\section*{IV Flow Rate Calculations}

\subsection*{Problem 11: Calculating the flow rate in mL/hr}

\subsection*{Question}

An order is given to infuse 1200 mL of IV fluid over 6 hours. What is the flow rate in mL/hr?

\subsection*{Solution}

\[

\text{Flow rate} = \left( \frac{1200\, \text{mL}}{6\, \text{hours}} \right) = 200\, \text{mL/hr}

\]

\subsection*{Rationale}

Divide the total volume of fluid by the number of hours to find the flow rate in mL/hr.

\subsection*{Problem 12: Determining the drop rate in drops per minute (gtt/min) using drop factors}

\subsection*{Question}

An IV is ordered to infuse 800 mL over 8 hours. The drop factor is 10 gtt/mL. What is the flow rate in gtt/min?

\subsection*{Solution}

\[

\text{Flow rate} = \left( \frac{800\, \text{mL} \times 10\, \text{gtt/mL}}{8\, \text{hours} \times 60\, \text{minutes/hour}} \right) = \left( \frac{8000\, \text{gtt}}{480\, \text{minutes}} \right) = 16.67\, \text{gtt/min} \approx 17\, \text{gtt/min}

\]

\subsection*{Rationale}

Multiply the total volume by the drop factor to find the total drops. Then, divide by the total time in minutes to find the drop rate.

\section*{Infusion Time Calculations}

\subsection*{Problem 13: Determining how long an IV will run based on the volume and flow rate}

\subsection*{Question}

An IV of 900 mL is running at 75 mL/hr. How long will it take for the IV to complete?

\subsection*{Solution}

\[

\text{Time} = \left( \frac{900\, \text{mL}}{75\, \text{mL/hr}} \right) = 12\, \text{hours}

\]

\subsection*{Rationale}

Divide the total volume by the flow rate to determine the infusion time.

\section*{Pediatric Dosages}

\subsection*{Problem 14: Calculating safe dose ranges for children based on weight or body surface area (BSA)}

\subsection*{Question}

A child weighs 25 kg and the safe dosage range for a medication is 6-8 mg/kg/day, divided into 4 doses. What is the safe dosage range per dose?

\subsection*{Solution}

\[

\text{Total daily dose range} = 25\, \text{kg} \times (6\, \text{mg/kg to 8\, mg/kg}) = 150\, \text{mg to 200\, mg}

\]

\[

\text{Per dose} = \left( \frac{150\, \text{mg to 200\, mg}}{4} \right) = 37.5\, \text{mg to 50\, mg}

\]

\subsection*{Rationale}

Multiply the child’s weight by the safe dosage range to find the total daily dose. Divide the total daily dose by the number of doses per day to find the dose per administration.

\section*{More Practice Problems with Rationale}

\subsection*{Problem 15: Converting units (mcg to mg)}

\subsection*{Question}

Convert 2000 mcg to mg.

\subsection*{Solution}

\[

2000\, \text{mcg} \times \left( \frac{1\, \text{mg}}{1000\, \text{mcg}} \right) = 2\, \text{mg}

\]

\subsection*{Rationale}

To convert micrograms to milligrams, divide by 1000 because there are 1000 micrograms in a milligram.

\subsection*{Problem 16: Dosage calculation based on body weight}

\subsection*{Question}

A patient weighs 70 kg and is prescribed a medication at 5 mg/kg. What is the total dose?

\subsection*{Solution}

\[

70\, \text{kg} \times 5\, \text{mg/kg} = 350\, \text{mg}

\]

\subsection*{Rationale}

Multiply the patient’s weight in kilograms by the prescribed dose per kilogram to find the total dose.

\subsection*{Problem 17: Adjusting doses}

\subsection*{Question}

A patient is prescribed 1.5 g of a medication. The medication is available as 500 mg tablets. How many tablets should the nurse administer?

\subsection*{Solution}

\[

1.5\, \text{g} = 1500\, \text{mg}

\]

\[

\text{Tablets to administer} = \left( \frac{1500\, \text{mg}}{500\, \text{mg/tablet}} \right) = 3\, \text{tablets}

\]

\subsection*{Rationale}

Convert the prescribed dose from grams to milligrams (since 1 g = 1000 mg). Then, divide by the concentration per tablet to find the number of tablets needed.

\subsection*{Problem 18: IV Flow Rate Calculation}

\subsection*{Question}

An order is given to infuse 1000 mL of IV fluid over 5 hours. What is the flow rate in mL/hr?

\subsection*{Solution}

\[

\text{Flow rate} = \left( \frac{1000\, \text{mL}}{5\, \text{hours}} \right) = 200\, \text{mL/hr}

\]

\subsection*{Rationale}

Divide the total volume of fluid by the number of hours to find the flow rate in mL/hr.

\subsection*{Problem 19: Drop Rate Calculation}

\subsection*{Question}

An IV is ordered to infuse 500 mL over 4 hours. The drop factor is 15 gtt/mL. What is the flow rate in gtt/min?

\subsection*{Solution}

\[

\text{Flow rate} = \left( \frac{500\, \text{mL} \times 15\, \text{gtt/mL}}{4\, \text{hours} \times 60\, \text{minutes/hour}} \right) = \left( \frac{7500\, \text{gtt}}{240\, \text{minutes}} \right) = 31.25\, \text{gtt/min} \approx 31\, \text{gtt/min}

\]

\subsection*{Rationale}

Multiply the total volume by the drop factor to find the total drops. Then, divide by the total time in minutes to find the drop rate.

\section*{Infusion Time Calculations}

\subsection*{Problem 20: Infusion Time Calculation}

\subsection*{Question}

An IV of 700 mL is running at 50 mL/hr. How long will it take for the IV to complete?

\subsection*{Solution}

\[

\text{Time} = \left( \frac{700\, \text{mL}}{50\, \text{mL/hr}} \right) = 14\, \text{hours}

\]

\subsection*{Rationale}

Divide the total volume by the flow rate to determine the infusion time.

\section*{Final Tips}

\begin{itemize}

\item Practice Regularly: Consistent practice helps reinforce your skills.

\item Use Dimensional Analysis: This method helps ensure units cancel out correctly.

\item Double-Check Your Work: Always verify your calculations to avoid errors.

\item Memorize Key Conversions: Knowing common conversions can save time and reduce mistakes.

\end{itemize}

By understanding the rationale behind each step, you'll improve your ability to perform these calculations accurately and confidently during the NCLEX-RN exam.

\end{document}

Strictly speaking, LaTeX source can be used to directly generate two formats: Using other software freely available on Internet, you can easily convert DVI and PDF to other document formats. In particular, you can obtain the PostScript version using software which is included in your LaTeX distribution. Some LaTeX IDE will give you the possibility to generate the PostScript version directly (even if it uses internally a DVI mid-step, e.g. LaTeX → DVI → PS). It is also possible to create PDF from DVI and vice versa. It doesn't seem logical to create a file with two steps when you can create it straight away, but some users might need it because, as you remember from the first chapters, the format you can generate depends upon the formats of the images you want to include (EPS for DVI, PNG and JPG for PDF). Here you will find sections about different formats with description about how to get it.
 * DVI using latex, the first one to be supported;
 * PDF using, more recent.

Other formats can be produced, such as RTF (which can be used in Microsoft Word) and HTML. However, these documents are produced from software that parses and interprets the LaTeX files, and do not implement all the features available for the primary DVI and PDF outputs. Nonetheless, they do work, and can be crucial tools for collaboration with colleagues who do not edit documents with LaTeX.

Tools installation
This chapter features a lot of third-party tools; most of them are installed independently of your TeX distribution.

Some tools are Unix-specific (*BSD, GNU/Linux and Mac OS X), but it may be possible to make them work on Windows. If you have the choice, it is often easier with Unix systems for command line tools.

Some tools may already be installed. For instance, you can check if dvipng is installed and ready to use (Unix only): type dvipng

Most of these tools are installable using your package manager or portage tree (Unix only).

Preview mode
This section describes how to generate a screenshot of a LaTeX page or of a specific part of the page using the LaTeX package. Screenshots are useful, for example, if you want to include a LaTeX generated formula on a presentation using your favorite slideware like Powerpoint, Keynote or LibreOffice Impress. First, start by making sure you have. See Installing Extra Packages.

Say you want to take a screenshot of

\pi = \sqrt{12}\sum^\infty_{k=0} \frac{(-3)^{-k}}{2k+1}. $$ Write this formula in the environment:

Note the option in the package declaration and the  environment around the equation's code. Without any of these two, you won't get any output.

This package is also very useful to export specific parts to other format, or to produce graphics (e.g. using PGF/TikZ) and then including them in other documents. You can also automate the previewing of specific environments:

This will produce a PDF containing only the listing content, the page layout will depend on the shape of the source code.

Directly
pdflatex my_file

DVI to PDF
dvipdfm my_file.dvi will create my_file.pdf. Another way is to pass through PS generation: dvi2ps myfile.dvi ps2pdf myfile.ps you will get also a file called my_file.ps that you can delete.

Merging PDF
If you have created different PDF documents and you want to merge them into one single PDF file you can use the following command-line command. You need to have Ghostscript installed:

Using Windows
gswin32 -dNOPAUSE -sDEVICE=pdfwrite -sOUTPUTFILE=Merged.pdf -dBATCH 1.pdf 2.pdf 3.pdf

Using Linux
gs -dNOPAUSE -sDEVICE=pdfwrite -sOUTPUTFILE=Merged.pdf -dBATCH 1.pdf 2.pdf 3.pdf

Alternatively, PDF-Shuffler is a small python-gtk application, which helps the user to merge or split pdf documents and rotate, crop and rearrange their pages using an interactive and intuitive graphical interface. This program may be available in your Linux distribution's repository.

Another option to check out is pdftk (or PDF toolkit), which is a command-line tool that can manipulate PDFs in many ways. To merge one or more files, use:

pdftk 1.pdf 2.pdf 3.pdf cat output 123.pdf

Using pdfLaTeX
Note: If you are merging external PDF documents into a LaTeX document which is compiled with, a much simpler option is to use the  package, e.g.:

Three simple shell scripts using the package are provided in the pdfjam bundle by D. Firth. They include options to merge several pdf files (pdfjoin), put several pages in one physical sheet (pdfnup) and rotate pages (pdf90).

See also Modular Documents

XeTeX
You can also use XeTeX (or, more precisely, XeLaTeX), which works in the same way as pdflatex: it creates a PDF file directly from LaTeX source. One advantage of XeTeX over standard LaTeX is support for Unicode and modern typographic technologies such as TrueType/OpenType fonts. See its Wikipedia entry for more details.

Customization of PDF output in XeTeX (setting document title, author, keywords etc.) is done using the configuration of hyperref package.

Convert to PostScript
pdf2ps my_file.pdf
 * from PDF:

dvi2ps my_file.dvi
 * from DVI:

Convert to RTF
LaTeX can be converted into an RTF file, which in turn can be opened by a word processor such as LibreOffice Writer or Microsoft Word. This conversion is done through latex2rtf, which may run on any computer platform, however is only actively supported on Windows, Linux and BSD, with the last mac update being from 2001 (a recent version for OSX is available via MacPorts). The program operates by reading the LaTeX source, and mimicking the behaviour of the LaTeX program. supports most of the standard implementations of LaTeX, such as standard formatting, some math typesetting, inclusion of EPS, PNG or JPG graphics, and tables. As well, it has some limited support for packages, such as varioref, and natbib. However, many other packages are not supported.

latex2rtf is simple to use. The Windows version has a GUI (l2rshell.exe), which is straightforward to use. The command-line version is offered for all platforms, and can be used on an example mypaper.tex file:

latex mypaper bibtex mypaper # if you use bibtex latex2rtf mypaper

Both  and (if needed)   commands need to be run before , because the .aux and .bbl files are needed to produce the proper output. The result of this conversion will create <tt>myfile.rtf</tt>, which you may open in many word processors such as Microsoft Word or LibreOffice.

Convert to HTML
There are many converters to HTML. Some of them use an intermediate file which then will be converted to the destination format.

hevea mylatexfile
 * HEVEA:

latex2html -html_version 4.0,latin1,unicode -split 1 -nonnavigation -noinfo -title "MyDocument" MyDocument.tex
 * latex2html:

latexmlc paper.tex --destination=paper.html
 * LaTeXML:

pdf2htmlEX [options] <input.pdf> [<output.html>] pdf2htmlEX can convert PDF to HTML without losing text or format. It is designed as a general PDF to HTML converter, not only restricted to the PDF generated by LaTeX source. LaTeX users can compile the LaTeX source code to PDF, and then convert the PDF to HTML via pdf2htmlEX. Some introductions of pdf2htmlEX can be found on its own wiki page. More technical details can be found on the paper published on TUGboat: Online publishing via pdf2htmlEX HTML / PDF. The Figure 3 of the paper gives different work-flows of publishing HTML online.
 * pdf2htmlEX:

TeX4ht has many options and possible configurations, but for a basic conversion, htlatex myfile.tex will usually result in a reasonable HTML approximation. An introduction by the original author was published in TUGboat.
 * TeX4ht

For exporting the BibTeX file only. bibtex2html mybibtexfile
 * bibtex2html:

Convert to image formats
It is sometimes useful to convert LaTeX output to image formats for use in systems that do not support DVI nor PDF files, such as Wikipedia.

There are two families of graphics:
 * Vector graphics can be scaled to any size, thus do not suffer from quality loss. SVG is a vector format.
 * Raster graphics define every pixel explicitly. PNG is a raster format.

So vector graphics are usually preferred. There is still some cases where raster graphics are used:
 * The target system does not handle vector graphics, only raster graphics are supported.
 * SVG can not embed fonts. So either the font will be rendered using a local .ttf or .otf font (which will mostly change the output), or all characters must be turned to vector graphics. This last method makes the SVG big and slow. If the input LaTeX file contains a lot of text which formatting must be preserved, SVG is not that great.

So SVG is great for drawings and a small amount of text. JPG is a well known raster formats, however it is usually not as good as PNG for text.

In some cases it may be sufficient to simply copy a region of a PDF (or PS) file using the tools available in a PDF viewer (for example using LaTeX to typeset a formula for pasting into a presentation). This however will not generally have sufficient resolution for whole pages or large areas.

Multiple formats
There is <tt>pdftocairo</tt> featured in the poppler toolset. pdftocairo -svg latexdoc.pdf output.svg <tt>pdftocairo</tt> also supports various raster graphic formats.
 * pdftocairo

Vector graphics
Direct conversion from PDF to SVG can be done using the command line tool pdf2svg. pdf2svg file.pdf file.svg
 * pdf2svg

Alternatively DVI or PDF can be converted to PS as described before, then the bash script ps2svg.sh can be used (as all the software used by this script is multiplatform, this is also possible in Windows, a step-by-step guide could be written).
 * ps2svg

One can also use dvisvgm, an open source utility that converts from DVI to SVG. dvisvgm -n file.dvi
 * dvisvgm

Inkscape is able to convert to SVG, PDF, EPS, and other vector graphic formats. inkscape --export-area-drawing --export-ps=OUTPUT INPUT inkscape --export-area-page --export-plain-svg=OUTPUT INPUT
 * Inkscape

Raster graphics

 * JPEG
 * Run ghostscript on the PostScript file created by pdf2ps as follows: MacOS: macTex distribution come with handy cli for "printing": pdftoppm is flexible on the manipulation - you can provide quality, dimention etc. which fits most needs of typical user. It can also print PDF into PNG and PPM files, read more in the manual for the utility. It is best on non interactive batch jobs.

 * GIMP
 * Open your file with GIMP. It will ask you which page you want to convert, whether you want to use anti-aliasing (choose strong if you want to get something similar to what you see on the screen). Try different resolutions to fit your needs, but 100 dpi should be enough. Once you have the image within GIMP, you can post-process it as you like and save it to any format supported by GIMP, as PNG for example.

 * dvipng
 * A method for DVI files is dvipng. Usage is the same as <tt>dvipdfm</tt>. Run <tt>latex</tt> as usual to generate the dvi file. Now, we want an X font size formula, where X is measure in pixels. You need to convert this, to dots per inch (dpi). The formula is: <tt> = <font_px>*72.27/10</tt>. If you want, for instance, X = 32, then the size in dpi corresponds to 231.26. This value will be passed to <tt>dvipng</tt> using the flag <tt>-D</tt>. To generate the desired png file run the command as follows:

The flag <tt>-T</tt> sets the size of the image. The option <tt>tight</tt> will only include all ink put on the page. The option <tt>-o</tt> sends the output to the file name <tt>foo.png</tt>.

convert input.pdf output.png
 * ImageMagick
 * The <tt>convert</tt> command from the ImageMagick suite can convert both DVI and PDF files to PNG.

 * optipng
 * You can optimize the resulting image using optipng so that it will take up less space.

Convert to plain text
If you are thinking of converting to plain text for spell-checking or to count words, there may be an easier way — read Tips and Tricks first. Following are available tools

 * [detex] - comes with latex distribution

Most LaTeX distributions come with <tt>detex</tt> program, which strips LaTeX commands. It can handle multi-file projects, so all you need is to give one command: detex yourfile (note the omission of .tex extension). Outputs result to the standard output. Redirect output to file using detex yourfile > yourfile.txt

Output of <tt>detex</tt> can contain undesired elements, the tool does not claims perfect conversion - make sure you use latest verion opendetex, or use HTML conversion first and then copy text from your browser.

 * [catdvi]

If you want to keep the formatting, you can use a DVI-to-plain text converter, like <tt>catdvi</tt>. Example: catdvi yourfile.dvi | fmt -u The use of fmt -u (available on most Unices) will remove the justification.

LaTeX/Отпремање у друге формате

title: LaTeX/Export To Other Formats


text:
User talk:84.70.67.85

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 18:32, 19 January 2007 (UTC)

title: User talk:84.70.67.85


text:
User talk:207.28.162.253

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 20:25, 19 January 2007 (UTC)

title: User talk:207.28.162.253


text:
A-level Biology/Cells & Cell Structure

Cells are the base units of organisms, some organisms are unicellular (they only consist of one cell), others, like humans, are multicellular (consist of many cells). In humans, similar cells are grouped together into tissues which perform a particular function, different tissues are joined together into organs to complete a more complicated function, and organs are arrangned into interdependant organ systems which together form an organism.

Cells themselves can be divided into two types, prokaryotic and eukaryotic. Prokaryotes have no nucleus (pro is the ancient Greek for 'before' and karyote for 'nucleus), eukaryotic cells do have a nucleus (eu is ancient Greek for 'true' or 'good'). There are other differences between prokaryotes and eukaryotes as well, such as eukaryotes are much bigger, have membrane bound organelles and linear DNA, prokaryotes are smaller, they don't have membrane bound organelles and their DNA is circular (and not contained in a nucleus).

Organelles. Eukaryotic cells contain membrane bound organelles, such and a nucleus and mitochondria (singular = mitochondrion. You need to know what some of these organelles look like and what their function is.

Mitochondria. These are the energy producers of the cell, they produce ATP, which is used by cells to release energy. They have a double membrane, the one on the inside is folded to provide a large surface area.

title: A-level Biology/Cells & Cell Structure


text:
Civ/Civilization IV/Diplomacy

=Civ IV Diplomacy=

Basics
Diplomacy is the means by which the player in Civ IV can interact with other factions, negotiate trade agreements, trade technology, build alliances and gain suppoprt for a war. The mechanics are similar to those used in Civ III: things you can trade are displayed on the right of the screen, and items you may ask for in return lie on the left.

Any ongoing deal may be canceled after 10 turns without a declaration of war. An ongoing item, such as money per turn, may not be traded for an immediate item, such as a city.

Whom to attack during democracy, when and how
When your civilization has become a democracy, declaring war on a nonagressive competitor can be very costly in terms of discontent citizens. Instead of bribing your citizens with luxuries and entertainers to keep them content, you can try to offer money to a third competitor to declare war against the nonagressive competitor. Your citizens will never find out and remain happy at a much lower cost.

If you are the strongest civilization you should do this: Bribe as many civilizations as possible to attack the weakest competitor (let's call him John). When John's army is weakened, he still has a lot of valuable land and cities you can take. That is worth the cost of temporary discontent. The new land and its undefended cities will become permanent assets compared to the discontent that is only temporary.

If a civilization threatens your position as number one in terms of military power, you should bribe the other competitors to attack the threatening civilization instead of bribing them to attack the weakest competitor. When you become number one, you start bribing the other civilizations to declare war against the weakest competitor. Repeat this process until you have assimilated all the other civilizations one by one.

Trading
This is the list of things, to my knowledge, one can trade in a diplomacy screen. During War, a cease fire or peace treaty may be offered.

Potential Trade Items

 * 1) Gold, lump sum.
 * 2) Gold per turn
 * 3) World map
 * 4) Open Boarders Agreement
 * 5) Defense Pact
 * 6) Permanent alliance
 * 7) Technology
 * 8) Resources
 * 9) Cities
 * 10) 'Make peace with'
 * 11) 'Declare War on'
 * 12) 'Stop Trading With'
 * 13) Adopt (civics)
 * 14) Convert (religion)

Gold
Requires: Currency This can either be in the form of a one off payment, or a series of payments. The lump sum offered or demanded cannot exceed the total available to that civilisation at that time. This prevents you from actually bankrupting anyone. Likewise, Gold Per Turn cannot exceed the faction’s current net gain per turn.

World Map
Requires: Paper

Etc. (watch this space)

Modifiers
These are very important. Your interactions with other civilisations, as well as your civics, juxtaposition and religion can affect their opinion of you. This is measured with modifiers. Positive modifiers make them like you more, negative ones make them dislike you.

title: Civ/Civilization IV/Diplomacy


text:
8th Grade Science/Section 2: Scientific Problem Solving

=As You Read=
 * Explain the steps taken in scientific methods.
 * Compare and contrast scientific variables and constants.
 * Explain how a control is used during an experiment.
 * Vocabulary: scientific methods, observation, inference, hypothesis, independent variable, dependent variable, constant, and control.

=Scientific Methods= Several steps were taken to learn about the pottery found at York Middle School. When the pottery was found, a decision was made to stop construction at the site. One adult guessed that the pottery was old. An expert was called to verify the guess made about the pottery. Based on prior knowledge and further testing, it was concluded that the pottery was from a prehistoric culture.

Think about the last time you had a problem that took several steps or actions to solve. Step-by-step procedures of scientific problem solving are called scientific methods. Solving any problem scientifically involves several steps. The steps used can vary from situation to situation and aren’t always done in the same order. But for now, take a look at each step in turn.

=Recognize the Problem= Ben thought about all the science he had learned over the past few months. He was eager to find out more about the world around him. As he looked around his bedroom, he wondered what he could explore. It was then that Ben noticed that the plant on his windowsill was droopy. He quickly watered the wilting plant. Later in the day, Ben observed that the droopy plant had perked up. He concluded that he should water the plant on a regular basis. Every day after school, he watered the plant in his room.

After a few weeks, Ben noticed that the leaves on his plant had turned yellow and brown. He knew from science class that plants need water, so why was this plant not doing well. He talked to his teacher about the plant. She suggested that Ben use what he learned in science class to solve his problem. She pointed out that this problem might make a good project for the upcoming science fair.

Ben already has completed the first step in using a scientific approach to solving a problem - he recognizes a problem. A scientific problem is simply a question you don’t know the answer to. To solve his problem, Ben must do research about his plant. Using sources of information, Ben identified his plant as a fig. In his Science Journal, he drew a picture of the plant and listed some facts about it.

Observe
In order for Ben to be able to answer the question about why his plant was not thriving, he needed to plan and carry out an experiment. First, he made and recorded careful observations about his plant. Observations can be bits of information you gather with your senses. Most scientific observations are made with your eyes and ears. You also can observe with your senses of touch, taste, and smell. Ben observed that many of the leaves had fallen off his plant. The stem, in places, was peeling. Ben also noticed that some white, powdery, smelly stuff was covering the soil in the pot. He stuck his finger into the soil. It was wet.

Infer
Observations like Ben’s often lead to inferences. An inference is a conclusion about an observation. Ben inferred that perhaps he was watering his plant too often. Can you make any other inferences about why Ben’s plant wasn’t thriving?

=Form a Hypothesis= After a problem is identified, a scientist might make a hypothesis. A hypothesis is a statement that can be tested. Hypotheses are based on observations, research, and prior knowledge of a problem. Sometimes more than one hypothesis can be developed.

Example 1 compares and contrasts hypotheses with two other scientific statements - scientific theories and scientific laws. Ben decided to use his inference about watering too often as his hypothesis. His hypothesis was the following: Fig plants grow best when they are watered only once a week.

 * Example 1
 * Hypothesis: A hypothesis is a statement that is tested with experiments. Hypotheses supported by repeated tests are used to form theories.
 * Theory: A theory is an explanation supported by results of many experiments. Theories attempt to explain why something happens.

=Test Your Hypothesis= To test his hypothesis, Ben will carry out an experiment using three plants. An experimental investigation is a series of carefully planned steps used to test a hypothesis. In any experiment, it’s important to carefully consider what resources you will use and how you can take steps to prevent wasting those resources. It’s also important to keep everything the same except for the item or variable you are testing so that you’ll know which variable caused the results. The one factor that you change in an experiment is called the independent variable.

In Ben’s proposed experiment, the independent variable is the number of times he waters each plant in a week. He then will observe how well each plant grows based on the amount of water it receives. The growth of the plants is the dependent variable in Ben’s experiment. A dependent variable is the factor, or outcome, that will be measured in an experiment.

Plan the Experiment
In order to test only one variable at a time, scientists often use constants. Constants are factors in an experiment that stay the same. In his experiment, Ben will use the same species and size of plants, which will be potted with the same kinds and amounts of soil. His teacher pointed out that Ben also must put his plants into identical containers. Another constant in Ben’s experiment will be the amount of light each plant will get.

Experimental investigations also have a control. A control is a standard used for comparison. Ben knows that all plants, even cacti, need water. His control will be that one of the plants receives no water during the experiment.

Do the Experiment
Ben gathered all the materials he would need to test his hypothesis. Before he started, Ben knew from Ms. Garcia’s labs that he must write down a plan to follow. In his Science Journal, he wrote that he would use three fig plants. Plant A would not be watered. This would be his control. A second fig plant, Plant B, would get watered every day. The third fig plant, Plant C, would get watered only once each week. His experiment would last one month.

Ben then made a table in which to record his observations. He listed each plant and the number of times it was to get watered. Ben made room in the table for his measurements. He also made a plan to record his observations, which would include the height of each plant, the color of its leaves, and the number of leaves it dropped, if any.

=Analyze Your Data= Data are collected during any scientific study. Some data are numeric values such as the length of an object or the temperature of a liquid. Other data you collect may include observations that use adjectives and phrases such as “faster”, “smaller”, “not as well as”, and “greener”. An experimenter must record and study the data collected before he or she can draw conclusions about an experiment.

By the end of the month, Ben observed that the leaves still left on the plant that received no water were brown and shriveled. It had lost most of its leaves. The plant that was watered every day had a few leaves left on its branches, but these leaves didn’t look too healthy. A white, smelly substance covered the soil. Ben noticed that the plant that was watered once each week had grown the tallest. Many healthy green-and-white leaves hung from its branches.

=Draw Conclusions and Communicate= After studying his data, Ben was ready to draw some conclusions. A conclusion is a statement based on what was observed. Ben concluded that not watering a plant causes the leaves to dry out and die. Watering a plant too much also causes the leaves to die. Watering the plant once a week seems to be the best schedule, of those tested, for a fig plant.

Ben told his teacher about his results. She reminded him that in order to make sure his conclusions were valid, he should repeat his experiment. Ben agreed and did the same experiment again. Based on the results of his second experiment, Ben was able to conclude confidently that watering a fig plant once a week made it grow well in the conditions he used. His hypothesis was supported. An important step in the scientific process is to communicate the results of an investigation. Ben entered his project in his school’s science fair. He seemed to have won the science fair with his project and his teacher congratulated him and he thanked her for her help.


text:
ERP Internals/Modules/Systems Administration

The System Administration module controls data and tables that are used by more than one module. Typically, this data tailors the ERP to a particular enterprise. System Administration sets up new companies as well as user names, passwords and access to ERP data and functions. It starts up and shuts down the ERP, archives old data, backs up and restores data, and monitors performance. An example of multi-module data is the company name and address. Every ERP has a Systems Administration module.

 * Functions
 * Reports

Tables
The first group of tables consists of the User table which contains one record for every user and the User Access table which contains one record for every function the user can perform. There is at least one access record for every user and there can be many.

Data Exchange
Systems Administration does not send or receive transactions to/from other modules.

Transactions
Systems Administration


text:
ERP Internals/Modules/Accounts Receivable

Every business sells products and/or services to its customers. Order Entry creates invoices when a customer orders something and Accounts Receivable keeps track of what payments have been made to which invoices. Commonly, it also prepares a report ("Aged Analysis") which shows how long each invoice has been outstanding. Every ERP has an Accounts Receivable module.

 * Functions
 * Reports

Collection
If Accounts Receivable invoices become overdue, Collection initiates processes to collect the money. Usually this means a telephone call followed by a series of dunning letters of increasing severity. If these efforts fail, Legal may become involved or the invoice may be written off, or sold to a collection agency for a fraction of its face value. As part of CRM, details of each contact with the customer including date, employee, contact person, and activity are stored in Customer.

Self-service
Customers can see their invoices, checks, and basic information via the web and update some portions of it (name, address, and contact information for example).

Tables
There are two major groups of tables in Accounts Receivable. The first consists of the Customer table which contains one record for every customer and the Customer Contact table which contains one record for every contact at each customer. There is at least one contact record for every customer and there can be several. The second group consists of the Invoice table which contains one record for every customer invoice and the Invoice Detail table which contains one record for every line on the customer invoice. There is at least one detail record for every invoice and there can be many.

Data Exchange
Accounts Receivable receives invoices from Order Entry/Invoicing and sends transactions to Bank Reconciliation and General Ledger.

Transactions
The first transaction records an invoice. This is done by Order Entry/Invoicing rather than Accounts Receivable.

The second transaction records a customer payment. One customer payment can pay one or more invoices. Payment can be in cash, by debit card, credit card, or check. Sometimes payments are received without information about what invoices are being paid.

The vast majority of the module's transactions consist of the above two. The third fundamental transaction is that of adjustment when things go wrong. We will document this later.

Accounts Receivable


text:
ERP Internals/Modules/Bank Reconciliation

Every enterprise has at least one bank account. Enterprises can have several accounts in different currencies at different banks. Most enterprises make a daily bank deposit. This consists of the checks the enterprise has received plus any cash or credit/debit card receipts. Periodically (depending on the account activity), the enterprise receives a bank statement for each account it has. Statements can be received electronically or in the mail. Bank Reconciliation assists the enterprise in reconciling the balance shown on the bank statement with the amount shown in the corresponding General Ledger cash accounts. This involves flagging Accounts Payable, Accounts Receivable, and Payroll checks as cleared, flagging deposits as recorded, and entering service charges and other miscellaneous debits and credits. Bank Reconciliation also handles non-AR deposits such as owner investment. Every ERP has a Bank Reconciliation module.

 * Functions
 * Reports

Tables
The first table is the Bank Account table which contains one record for every bank account. The second table is the Transaction table which contains one record for every bank transaction (check, deposit, interest, service charge, adjustment).

Transactions

 * The first type of transaction is a check drawn on the account.
 * The second type of transaction is a deposit made to the account.
 * The third type of transaction is a service charge made by the bank.
 * The fourth type of transaction is interest paid into the account by the bank.
 * The fifth type of transaction is a transfer of funds from one account to another.
 * The last transaction records miscellaneous adjustments made to the account.

Bank Reconciliation


text:
Russian/Print version

=Main Contents=
 * Introduction
 * Lesson 1
 * Lesson 2
 * Lesson 3
 * Lesson 4
 * Lesson 5
 * Grammar
 * Vocabulary
 * Names
 * Useful Words and Expressions
 * Numbers
 * Declensions
 * Adjectives
 * Conjugations
 * Prepositions
 * Verbal Aspect
 * Interrogative Pronouns
 * Personal Pronouns
 * Possessive Pronouns
 * Appendix
 * Alphabet
 * Internet Resources

=Introduction=

=Lesson 1=

=Lesson 2=

=Lesson 3=

=Lesson 4=

=Lesson 5=

title: Russian/Print version


text:
Talk:SQL Dialects Reference/Data structure definition/Data types/Numeric types

I just want to add that SAS also has SQL features. I am no expert, just had been taught a few SQL topics in a SAS programming class.


text:
Cookbook:Palm Sugar

Palm sugar is an ambiguous term. It could mean:

 * Coconut Sugar
 * Gula Jawa
 * Jaggery

title: Cookbook:Palm Sugar


text:
ERP Internals/Modules/Cash Management

Cash Management tracks the cash the business will be receiving from customer payments in Accounts Receivable against the cash the business will be spending through Accounts Payable, Purchasing (a promise to spend), Payroll, bond interest, and stock dividends. It prepares a cash forecast week by week to allow businesses to anticipate cash shortages. Many low-end ERP packages do not have a Cash Management module.

 * Functions
 * Reports

Tables
Cash Management is unique in that it has no tables of its own. It uses tables from other modules to create its forecast.

Data Exchange
Although Cash Management uses data from other modules, it does not send or receive transactions.

Transactions
Cash Management

title: ERP Internals/Modules/Cash Management


text:
ERP Internals/Modules/Contacts

Contacts keeps track of the names and addresses of everyone the enterprise has dealings with who isn't a customer, supplier, or employee. In addition to the usual information, Contacts often keeps personal information such as birthday on the contact. Most of these names and addresses are leads or prospects from Sales & Marketing. Normally, Contacts keeps track of every meeting anyone from the enterprise has with the contact including meetings scheduled for the future. Every ERP has a Contacts module.

 * Functions
 * Reports

Tables
The first group of tables consists of the Contact table which contains one record for every contact and the Personal Contact table which contains one record for every personal contact at each business contact. There is at least one personal contact record for every contact and there can be several. The second table is the Interaction table which contains one record for every interaction (meeting, telephone call, etc.). There is at least one interaction for every contact and there can be many.

Data Exchange
Contacts does not send or receive transactions to/from other modules.

Transactions
Contacts

title: ERP Internals/Modules/Contacts


text:
ERP Internals/Modules/Fixed Assets

A "fixed asset" is something the enterprise has purchased that has a useful life of more than one year. Fixed Assets keeps track of all these objects and allocates part of their purchase price to each year of their life. There are a variety of ways in which this allocation can be done. For some assets, the allocation or "depreciation" is specified by law. Not all ERP packages have a Fixed Assets module.

 * Functions
 * Reports

Tables
The first table is the Fixed Asset table which contains one record for every fixed asset. The second table is the Transaction table which contains one record for every transaction (purchase, sale, depreciate, adjustment). There is at least one transaction for every fixed asset and there can be many.

Data Exchange
Fixed Assets sends transactions to General Ledger.

Transactions
Fixed Assets

title: ERP Internals/Modules/Fixed Assets