Fourier series of half-range functions by smooth extension

This paper considers Fourier series approximations of one- and two-dimensional functions over the half-range, that is, over the sub-interval [0, L] of the interval [−L, L] in one-dimensional problems and over the sub-domain [0, L_x] × [0, L_y] of the domain [−L_x, L_x] × [−L_y, L_y] in two-dimensional problems. It is shown how to represent these functions using a Fourier series that employs a smooth extension. The purpose of the smooth extension is to improve the convergence characteristics otherwise obtained using the even and odd extensions. Significantly improved convergence characteristics are illustrated in one-dimensional and two-dimensional problems.     

A note on morphic characterization of languages

It is noted, by using ideas of Culik, Fich and Salomaa [3], that for each language L and each regular language R the language L ∩ R is obtained from L# by applying the operation ‘a morphism followed by an inverse morphism' twice. As a consequence new purely morphic characterizations, based on the Dyck language D₂ and the twin-shuffle language L₂, are derived for the families of context-free and recursively enumerable languages, respectively. Also a morphic representation result for rational transductions follows.     

Derivations of polynomial algebras without Darboux polynomials

We present several new examples of homogeneous derivations of a polynomial ring k[X]=k[x₁,…,x_n] over a field k of characteristic zero without Darboux polynomials. Using a modification of a result of Shamsuddin, we produce these examples by induction on the number n of variables, thus more easily than the previously known example multidimensional Jouanolou systems of ?o?a?dek.     

Simple Lie algebras having extremal elements

Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fact of classical type.     

Modularly complemented geometric lattices

A geometric lattice L is strongly uniform if the quotients [x₁, 1] and [x₂, 1] are isomorphic for all x₁, x₂εL of the same rank. It is shown that if L is a simple geometric lattice in which each element has a modular complement then L is strongly uniform.     

Hypergeometric series solutions of linear operator equations

Let K be a field and L:K[x]→K[x] be a linear operator acting on the ring of polynomials in x over the field K. We provide a method to find a suitable basis {b_k(x)} of K[x] and a hypergeometric term c_k such that is a formal series solution to the equation L(y(x))=0. This method is applied to construct hypergeometric representations of orthogonal polynomials from the differential/difference equations or recurrence relations they satisfied. Both the ordinary cases and the q-cases are considered.     

Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions

On the interval (0, π), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) ∈ L ₁ (0, π) [q(x) ∈ L ₂(0, π)], we establish exact asymptotic formulas for the eigenvalues of this problem.     

An Engel condition with skew derivations

Let R be a prime ring and set [x, y]₁ = [x, y] = xy ? yx for ${x,y\in R}$ and inductively [x, y] _k = [[x, y] _k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x] _k  = 0 for all ${x \in L}$ , where k is a fixed positive integer, then charR = 2 and ${R\subseteq M_{2}(F)}$ for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne.     

Cycloidal algebras

In this paper we introduce for an arbitrary algebra (groupoid, binary system) (X; *) a sequence of algebras (X; *) _n = (X; °), where x ° y = [x * y] _n = x * [x * y] _n?1, [x * y]₀ = y. For several classes of examples we study the cycloidal index (m, n) of (X; *), where (X; *) _m = (X; *) _n for m > n and m is minimal with this property. We show that (X; *) satisfies the left cancellation law, then if (X; *) _m = (X; *) _n , then also (X; *) _m?n = (X; *)₀, the right zero semigroup. Finite algebras are shown to have cycloidal indices (as expected). B-algebras are considered in greater detail. For commutative rings R with identity, x * y = ax + by + c, a, b, c ∈ ? defines a linear product and for such linear products the commutativity condition [x * y] _n = [y * x] _n is observed to be related to the golden section, the classical one obtained for ?, the real numbers, n = 2 and a = 1 as the coefficient b.     

Holomorphic Solutions of a Functional Equation Related to Nonlinear Difference Systems

Here we consider the following functional equation, $$\Psi(X(x,\Psi(x)))=Y(x, \Psi(x)),$$ where X(x, y) and Y(x, y) are holomorphic functions in |x| < δ ₁, |y| < δ ₁. When we consider a nonlinear simultaneous system of two variables difference equations, we can reduce it to a single difference equation of first order by a solution Ψ of the above functional equation. We obtain a matrix by the linear terms of functions X and Y. When the all eigenvalues of the matrix are equal to 1, it is difficult to have a solution of the above functional equation. In the present paper, we derive a formal solution of the above functional equation under the condition. Further we prove the existence of a solution which is holomorphic and have an asymptotically expansion of the formal solution. Moreover, we will show an example of nonlinear difference system such that our results are applicable.     

Hyperbolicity in median graphs

If X is a geodesic metric space and x ₁,x ₂,x ₃?∈?X, a geodesic triangle T?=?{x ₁,x ₂,x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., ${\delta}(X)=\inf\{{\delta}\ge 0: \, X \, \text{is $\delta$-hyperbolic}\}. $ In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin.     

PiecewiseL-splines

Leta=x ₀<x ₁<...<x _N =b be a partition of the interval [a, b] and letL be a normalm-th order linear differential operator. The purpose of this note is to point out that spline functions in one variable need not be excluded to piecewise fits of functions belonging to the null spaceN(L ^* L) on each closed subinterval [x _i,x _i+1], 0in-1 but may be extended to piecewise fits of functions belonging toN(L _i ^* L _i) on each subinterval [x _i,x _i+1] provided theL _i's are selected from a uniformly bounded family of normal linear differential operators. Furthermore when theL _i's are so selected one obtains the usual integral relations and error estimates obtained for splines [2, 8 and 9] including the extended error estimates obtained by Swartz and Varga [10].     

Preface

The following morphic characterization of EOL languages is established. The family of EOL languages equals the family of all languages of the form h(L∩R) where h is a morphism, R is a regular language and L is the maximal solution of an equation $\text{?(X) = g(X)}$, where ? is a morphism, g is a coding and X is a language variable. It is shown that if g is allowed to be a weak coding, then a larger family of languages is obtained, which however is strictly contained in the family of ETOL languages.     

Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x ₁, x ₂) at the vertices of a regular triangulation T _h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Π _h [u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T _h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite element spaces of linear triangular elements from [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644]. The second-order approximation of the gradient is extended from the vertices to the whole domain and applied to the a posteriori error estimates of the finite element solutions of the planar elliptic boundary-value problems of second order. Numerical illustrations of the accuracy of the averaging method and of the quality of the a posteriori error estimates are also presented.     

On analytic equivalence of functions at infinity

In this paper we define the relation of analytic equivalence of functions at infinity. We prove that if the ?ojasiewicz exponent at infinity of the gradient of a polynomial f∈R[x₁,…,xn] is greater or equal to k−1, then there exists ε>0 such that for every polynomial P∈R[x₁,…,xn] of degree less or equal to k, whose coefficients of monomials of degree k are less or equal ε, the polynomials f and f+P are analytically equivalent at infinity.     

Polynomial dynamical systems and the Korteweg—de Vries equation

We explicitly construct polynomial vector fields L_k, k = 0, 1, 2, 3, 4, 6, on the complex linear space C⁶ with coordinates X = (x₂, x₃, x₄) and Z = (z₄, z₅, z₆). The fields L_k are linearly independent outside their discriminant variety Δ ? C⁶ and are tangent to this variety. We describe a polynomial Lie algebra of the fields L_k and the structure of the polynomial ring C[X,Z] as a graded module with two generators x₂ and z₄ over this algebra. The fields L₁ and L₃ commute. Any polynomial P(X,Z) ∈ C[X,Z] determines a hyperelliptic function P(X,Z)(u₁, u₃) of genus 2, where u₁ and u₃ are the coordinates of trajectories of the fields L₁ and L₃. The function 2x₂(u₁, u₃) is a two-zone solution of the Korteweg–de Vries hierarchy, and ?z₄(u₁, u₃)/?u₁ = ?x₂(u₁, u₃)/?u₃.     

On the complexity of constrained VC-classes

Sauer's lemma is extended to classes H_N of binary-valued functions h on [n]={1,…,n} which have a margin less than or equal to N on all x∈[n] with h(x)=1, where the margin μ_h(x) of h at x∈[n] is defined as the largest non-negative integer a such that h is constant on the interval I_a(x)=[x-a,x+a]⊆[n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S⊆[n].     

Recognizing splicing languages: Syntactic monoids and simultaneous pumping

We use syntactic monoid methods, together with an enhanced pumping lemma, to investigate the structure of splicing languages. We obtain an algorithm for deciding whether a regular language is a reflexive splicing language, but the general question remains open.     

Minimizing bumps for posets of width two

Let L={x₁<…< x_n} be a linear extension of a finite partially ordered set P. A pair (x_i, x_i+1) forms a bump in L whenever x_i< x_i+1 in P. We give an effective solution for the problem of finding a linear extension with a minimum number of bumps when the width ofP is two.     

Contractive properties for the heat equation in Sobolev spaces

Consider the heat equation ?_ru ? Δ_xu = 0 in a cylinder Ω × [0,T] ? $\text{R}$ⁿ⁺¹ smooth lateral boundary under zero Neumann or Dirichlet conditions. Geometric conditions for Ω are given that guarantee that for a given P, 6▽_xu(·, t)6_L^p will be non-increasing for any solution. Decay rates are also given. For arbitrary Ω and p, it is shown how to construct an equivalent L^p-norm, such that ▽_x(·, t) is non-increasing in this norm.