Presentations of alternating semigroups

One presentation of the alternating groupA _n hasn?2 generatorss ₁,…,s_n?2 and relationss ₁ ³ =s _i ² =(s_1?1s_i)³=(s_js_k)²=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA _n ^c )A _n is introduced: It hasn?1 generatorss ₁,…,S _n?2,e, theA _n-relations (above), and relationse ²=e, (es ₁)⁴, (es _j)²=(es _j)⁴,es _i=s _i s ₁ ^-1 es ₁, wherej>1 andi≥1.     

Group pursuit in Pontryagin’s nonstationary example

On the interval [t ₀, ∞), we consider the following group pursuit problem with one evader: 1 $$ z_i^{(l)} + a_1 (t)z_i^{(l - 1)} + a_2 (t)z_i^{(l - 2)} + \cdots + a_l (t)z_i = u_i - v, u_i ,v \in V, z_i^{(q)} (t_0 ) = z_i^q , $$ where z _i , u _i , v ∈ R ^v , (v ≥ 2), V is a strictly convex compact set in R ^v , the functions a ₁(t), a ₂(t), …, a _l (t) are continuous, i = 1, 2, …, n and q = 0, 1, …, l ? 1. Let ? _q (t, s) be the solution of the Cauchy problem $$ \begin{gathered} \omega ^{(l)} + a_1 (t)\omega ^{(l - 1)} + a_2 (t)\omega ^{(l - 2)} + \cdots + a_l (t)\omega = 0, \omega ^{(q)} (s) = 1, \hfill \\ \omega ^{(r)} (s) = 0, r = 0, \ldots q - 1,q + 1, \ldots ,l - 1, \hfill \\ \end{gathered} $$ and let $$ \xi _\iota (t) = \varphi _0 (t,t_0 )Z_i^0 + \varphi _1 (t,t_0 )Z_i^1 + \cdots + \varphi _{l - 1} (t,t_0 )Z_i^{l - 1} . $$ We prove that if there exist continuous functions α _i (t) and ξ _i ¹ (t) such that the ξ _i ¹ (t) are Bohr almost periodic on [t ₀, ∞), α _i (t) > 0 for all t ≥ t ₀, lim _t→∞(ξ _i ¹ (t) ? α _i (t)ξ _i (t)) = 0, lim _t→∞(min _i α _i (t) ∝ _t0 ^t |? _l?1(t, s)| ds) = ∞, and there exist points h _i ⁰ ∈ H _i ¹ = {ξ _i ¹ (t), t ∈ [0, ∞)} such that 0 ∈ Int co{h _i ⁰ }, then the pursuit problem with evader discrimination is solvable.     

On σ-algebras related to the measurability of compositions

Given a measurable space (T, F), a set X, and a map ?: T → X, the σ-algebras N _Ф = ?_?∈Φ N _?, and M _Φ = ?_?∈Φ N _?, where G _?(t) = (t, ?(t)) and Φ ? X ^T , are considered. These σ-algebras are used to characterize the (F, B, ?)-measurability of the compositions g ○ ? and f о G _?, where g: X → Y, f: T × X → Y, and (Y, ?) is a measurable space. Their elements are described without using the operations ? ^?1 and G _? ^?1 .     

Combinational properties of sets of residues modulo a prime and the Erdős—Graham problem

Consider an arbitrary ε > 0 and a sufficiently large prime p > 2. It is proved that, for any integer a, there exist pairwise distinct integers x ₁, x ₂, ..., x _N , where N = 8([1/ε + 1/2] + 1)² such that 1 ≤ x _i ≤ p ^ε, i = 1, ..., N, and $$a \equiv x_1^{ - 1} + \cdots + x_N^{ - 1} (\bmod p)$$ , where x _i ^?1 is the least positive integer satisfying x _i ^?1 x _i ≡ 1 (modp). This improves a result of Sparlinski.     

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations(P2):(Mu)(t)+α(Mu)(t) = Au(t)+ft-∞ a(ts)Au(s)ds + f(t),0t2π,with periodic boundary conditions M u(0)=Mu(2π),(Mu)(0) =(M u)(2π),in periodic Lebesgue-Bochner spaces Lp(T,X),periodic Besov spaces B s p,q(T,X) and periodic Triebel-Lizorkin spaces F s p,q(T,X),where A and M are closed linear operators on a Banach space X satisfying D(A) D(M),a∈L1(R+) and α is a scalar number.Using known operatorvalued Fourier multiplier theorems,we completely characterize the well-posedness of(P2) in the above three function spaces.     

Proof of Gromov’s theorem on homogeneous nilpotent approximation for vector fields of class C 1

The article is devoted to the asymptotic properties of the vector fields $\tilde X_i^g $ , i = 1, …, N, θ _g -connected with C ¹-smooth basis vector fields {X _i } _i=1,…,N satisfying condition (+ deg). We prove a theorem of Gromov on the homogeneous nilpotent approximation for vector fields of classC ¹. Nontrivial examples are constructed of quasimetrics induced by vector fields {X _i } _{i=1, …, N} .     

The fundamental constituents of iteration digraphs of finite commutative rings

For a finite commutative ring R and a positive integer k ? 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = a ^k . Let R = R ₁ ⊕ … ⊕ R _s , where s > 1 and R _i is a finite commutative local ring for i ∈ {1, …, s}. Let N be a subset of {R ₁, …, R _s } (it is possible that N is the empty set \(\not 0\) ). We define the fundamental constituents G _N ^* (R, k) of G(R, k) induced by the vertices which are of the form {(a ₁, …, a _s ) ∈ R: a _i ∈ D(R _i ) if R _i ∈ N, otherwise a _i ∈ U(R _i ), i = 1, …, s}, where U(R) denotes the unit group of R and D(R) denotes the zero-divisor set of R. We investigate the structure of G* _N (R, k) and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.     

The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

The Dirichlet problem on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter ? taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost ?-uniformly, i.e., with an error that weakly depends on the parameter ?: |u(x, t) ? z(x, t)| ≤ M[N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀ + ?^?1 N ₁ ^?K ln ^K?1 N ₁], (x, t) ε ? _h , where N ₁ + 1 and N ₀ + 1 are the numbers of grid points in x and t, respectively; K is the number of refinement iterations (with respect to x) in the adapted grid; and M = M(K). Outside the σ-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges ?-uniformly at a rate O(N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀), where σ ≤ MN ₁ ^{?K + 1} ln ^K?1 N ₁ for K ≥ 2.     

Polynomials orthogonal on the semicircle,II

Generalizing previous work [2], we study complex polynomials {π _k },π _k (z)=z ^k +?, orthogonal with respect to a complex-valued inner product (f,g)=∫ ₀ ^π f(e ^iθ)g(e ^iθ)w(e ^iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ ₀ ^π w(e ^iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π _k } in the case of the Jacobi weight functionw(z)=(1?z)^α(1+z)^β, α>?1, and its special case \(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ _n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ _n (z). It has regular singular points atz=1, ?1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.     

Existence of positive solutions for the singular fractional differential equations

In this paper, we consider the following two-point fractional boundary value problem. We provide sufficient conditions for the existence of multiple positive solutions for the following boundary value problems that the nonlinear terms contain i-order derivative where n?1<α≤n is a real number, n is natural number and n≥2, α?i>1, i∈N and 0≤i≤n?1. ${}^{c}D_{0^{+}}^{\alpha}$ is the standard Caputo derivative. f(t,x ₀,x ₁,…,x _i ) may be singular at t=0.     

Computer difference scheme for a singularly perturbed convection-diffusion equation

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ? (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an \(\mathcal{O}\) (δ _st ) rate, where δ _st = (? + N ^?1)^?1 N ^?1 and N + 1 is the number of grid nodes; the scheme is not ?-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing ? down to its complete loss for small ? (namely, for ? = \(\mathcal{O}\) (δ^?2max _i,j |δa _i ^j | + δ^?1 max _{i, j} |δb _i ^j |), where δ = δ _st and δa _i ^j , δb _i ^j are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for ? ∈ (0, 1] at the same rate as the standard scheme in the absence of perturbations.     

The integral representation of the exponential product of stochastic semigroups

An integral representation is obtained for the exponential product of stochastic semigroups $$X_s^t \otimes Z_s^t = X_s^t + \mathop \smallint \limits_{s< u< t} X_u^t dV_u X_s^u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} X_{u_2 }^t dV_{u_2 } X_{u_1 }^{u_2 } dV_{u_1 } X_s^{u_1 } + \cdots ,$$ whereV _t is the generating process of the semigroupZ _s ^t and the integrals are understood in the sense of mean-square limits of the Riemann-Stieltjes sums. This representation is different from the traditional representation $$X_s^t \otimes Z_s^t = E + \mathop \smallint \limits_{s< u< t} dW_u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} dW_{u_2 } dW_{u_1 } + \cdots ,$$ in which the integration extends over the processW _t=Y_t+V_t that is the generating process of the exponential productX _s ^t ?Z _s ^t andY _t is the generator of the semigroupX _s ^t .     

The Hopfian property of n-periodic products of groups

LetH be a subgroup of a groupG. A normal subgroupN _H ofH is said to be inheritably normal if there is a normal subgroup N _G of G such that N _H = N _G ∩ H. It is proved in the paper that a subgroup $N_{G_i }$ of a factor G _i of the n-periodic product Π _i∈I ⁿ G _i with nontrivial factors G _i is an inheritably normal subgroup if and only if $N_{G_i }$ contains the subgroup G _i ⁿ . It is also proved that for odd n ≥ 665 every nontrivial normal subgroup in a given n-periodic product G = Π _i∈I ⁿ G _i contains the subgroup G ⁿ . It follows that almost all n-periodic products G = G _{1 *} ⁿ G ₂ are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents.     

A new composite implicit iterative algorithm for nonself asymptotically nonexpansive mappings

Let E be a real uniformly convex and smooth Banach space with P as a sunny nonexpansive retraction, K be a nonempty closed convex subset of E. Let {S _i } _i=1 ^N , {T _i } _i=1 ^N :K→K be two finite families of weakly inward and asymptotically nonexpansive mappings with respect to P. It is proved that the composite implicit iteration process converges weakly and strongly to a common fixed point of {S _i } _i=1 ^N , {T _i } _i=1 ^N under certain conditions. The results of this paper improve and extend some well known corresponding results.     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [?π,π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y: = {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that f is nondecreasing on [y _i ,y _i?1] if i is odd and not increasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \parallel f - P_n \parallel \leqslant c(s) \omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω₂(f,·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Some results on the local cohomology of minimax modules

Let R be a commutative Noetherian ring with identity and I an ideal of R. It is shown that, if M is a non-zero minimax R-module such that dim Supp H _I ⁱ (M) ? 1 for all i, then the R-module H _I ⁱ (M) is I-cominimax for all i. In fact, H _I ⁱ (M) is I-cofinite for all i ? 1. Also, we prove that for a weakly Laskerian R-module M, if R is local and t is a non-negative integer such that dim Supp H _I ⁱ (M) ? 2 for all i < t, then Ext _R ^j (R/I,H _I ⁱ (M)) and Hom _R (R/I,H _I ^t (M)) are weakly Laskerian for all i < t and all j ? 0. As a consequence, the set of associated primes of H _I ⁱ (M) is finite for all i ? 0, whenever dim R/I ? 2 and M is weakly Laskerian.     

Asymptotics for transportation cost in high dimensions

LetX ₁,...,X _n ,Y ₁,...,Y _n be i.i.d. with the law μ on the cube [0, 1] ^d ,d?3. LetL _n (μ)=inf_πΣ _i=1 ⁿ ||X _i ?Y _π(i)|| denote the optimal bipartite matching of theX andY points, where π ranges over all permutations of the integers 1, 2,...,n, and where ‖·‖ is a norm on ? ^d . If μ is Lebesgue measure it is shown that $$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha {\text{a}}{\text{.s}}{\text{.}}$$ where α is a finite constant depending on ‖ ‖ andd only. More generally, for arbitrary μ it is shown that $$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha \int {(f{\text{(}}x{\text{)}})^{(d - 1)/d} dxa.s.} $$ wheref is the density of the absolutely continuous part of μ. We also find the rate of convergence.     

Existence of multiple positive solutions for one-dimensional p-Laplacian operator

In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian $$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$ subject to the boundary value conditions: $$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$ where φ _p (s)=|s| ^p?2?s,p>1;ξ _i ∈(0,1) with 0<ξ ₁<ξ ₂<???<ξ _n?2<1 and α _i ,β _i satisfy α _i ,β _i ∈[0,∞),0≤∑ _i=1 ^n?2 α _i <1 and 0≤∑ _i=1 ^n?2 β _i <1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.     

Evaluation of the Riemann-Mellin integral

A method for evaluating the Riemann-Mellin integral $$ f(t) = \frac{1} {{2\pi i}}\int\limits_{c - i\infty }^{c + i\infty } {e^{zt} F(z)dz,c > 0,} $$ which determines the inverse Laplace transform, is considered; the method consists in reducing the integral to the form I = ∝ _?∞ ^∞ g(u) by means of a suitable deformation of the contour of integration and applying the trapezoidal quadrature formulas with an infinite number of nodes (I _h = hΣ _k=?∞ ^∞ g(kh)) or with a finite number 2N + 1 of nodes (I _{h, N} = hΣ _{k = ?N} ^N g(kh)). For parabolic and hyperbolic contours of integration, procedures for choosing the step size h in numerical integration and the summation limits ±N for truncating the infinite sum in the trapezoidal formula, which depend on the arrangement of the singular points of the image, are suggested. Errors are estimated, and their asymptotic behavior with increasing N is described.     

A Classification of Graphs Whose Subdivision Graph is Locally Distance Transitive

The subdivision graph S(Σ) of a connected graph Σ is constructed by adding a vertex in the middle of each edge. In a previous paper written with Cheryl E. Praeger, we characterised the graphs Σ such that S(Σ) is locally (G, s)-distance transitive for s ≤ 2 diam(Σ) ? 1 and some G?≤ Aut(Σ). In this paper, we solve the remaining cases by classifying all the graphs Σ such that S(Σ) is locally (G, s)-distance transitive for some s?≥ 2 diam(Σ) and some G?≤ Aut(Σ). As a corollary, we get a classification of all the graphs whose subdivision graph is locally distance transitive.