On complexity of three-dimensional hyperbolic manifolds with geodesic boundary

The nonintersecting classes ? _p,q are defined, with p, q ?? ? and p ?? q ?? 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ?? ? _p,q , then the complexity c(M) and the Euler characteristic ??(M) of M are related by the formula c(M) = p???(M). The classes ? _q,q , q ?? 1, and ?_2,1 are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from ?_3,1 and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ?-invariants of manifolds.     

On deep holes of standard Reed-Solomon codes

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes [p-1, k]p with p a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes [q-1, k]q with q a power of the prime p. Let q≥4 and 2≤k≤q-2. We show that the received word u is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. So there are at least 2(q-1)qk deep holes if k q-3.     

Basic Fourier Series: Convergence on and Outside the?q-Linear Grid

A q-type H?lder condition on a function f is given in order to establish (uniform) convergence of the corresponding basic Fourier series S _q [f] to the function itself, on the set of points of the q-linear grid. Furthermore, by adding other conditions, one guarantees the (uniform) convergence of S _q [f] to f on and ??outside?? the set points of the q-linear grid.     

Critical exponents for the evolution p-Laplacian equation with a localized reaction

This paper deals with the large time behavior of nonnegative solutions to the equation $$u_t = div\left( {\left| {\nabla u} \right|^{p - 2} \nabla u} \right) + a\left( x \right)u^q ,\left( {x,t} \right) \in R^N \times (0,T),$$ where p > 2, q > 0, and the function a(x) ?? 0 has a compact support. We obtain the critical exponent for global existence q ₀ and the Fujita exponent q _c . In one-dimensional case N = 1, we have $q_0 = \frac{{2(p - 1)}} {p}$ and q _c = 2(p ? 1). Particularly, all solutions are global in time if 0 < q ?? q _o, but blow up if q ₀ < q ?? q _c ; while if q > q _c both blowing up solutions and global solutions exist. However, for the case N ?? p > 2, these two critical exponents are exactly the same. Namely, q ₀ = p ? 1 = q _c .     

Expansion in perfect groups

Let ?? be a subgroup of ${{\rm GL}_d(\mathbb{Z}[1/q_0])}$ generated by a finite symmetric set S. For an integer q, denote by ?? _q the projection map ${\mathbb{Z}[1/q_0] \to \mathbb{Z}[1/q_0]/q \mathbb{Z}[1/q_0]}$ . We prove that the Cayley graphs of ?? _q (??) with respect to the generating sets ?? _q (S) form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of ?? is perfect, i.e. it has no nontrivial Abelian quotients.     

Column continuous transition functions

A column continuous transition function is by definition a standard transition function P(t) whose every column is continuous for t?0 in the norm topology of bounded sequence space l_∞. We will prove that it has a stable q-matrix and that there exists a one-to-one relationship between column continuous transition functions and increasing integrated semigroups on l_∞. Using the theory of integrated semigroups, we give some necessary and sufficient conditions under which the minimal q-function is column continuous, in terms of its generator (of the Markov semigroup) as well as its q-matrix. Furthermore, we will construct all column continuous Q-functions for a conservative, single-exit and column bounded q-matrix Q. As applications, we find that many interesting continuous-time Markov chains (CTMCs), say Feller-Reuter-Riley processes, monotone processes, birth-death processes and branching processes, etc., have column continuity.     

Pauli graphs, Riemann hypothesis, and Goldbach pairs

We consider the Pauli group P_q generated by unitary quantum generators X (shift) and Z (clock) acting on vectors of the q-dimensional Hilbert space. It has been found that the number of maximal mutually commuting sets within P_q is controlled by the Dedekind psi function ??(q) and that there exists a specific inequality involving the Euler constant ?? ?? 0.577 that is only satisfied at specific low dimensions q ?? A = {2, 3, 4, 5, 6, 8, 10, 12, 18, 30}. The set A is closely related to the set A??{1, 24} of integers that are totally Goldbach, i.e., that consist of all primes p < n ? 1 with p not dividing n and such that n?Cp is prime. In the extreme high-dimensional case, at primorial numbers N_r, the Hardy-Littlewood function R(q) is introduced for estimating the number of Goldbach pairs, and a new inequality (Theorem 4) is established for the equivalence to the Riemann hypothesis in terms of R(N_r). We discuss these number-theoretical properties in the context of the qudit commutation structure.     

�����֧�֧�ߧڧܧ� �� �ߧѧڧݧ���֧� ���ڧҧݧڧا֧ߧڧ� �ܧݧѧ���� B p,�� r ��֧�ڧ�էڧ�֧�ܧڧ� ���ߧܧ�ڧ� �ާߧ�ԧڧ� ��֧�֧ާ֧ߧߧ��

Exact estimates with respect to the order of magnitude are obtained for the ortho-projective and linear diameters of the classes B _p,?? ^r periodic functions of several variables in the spaces L _q , 1 ?? p, q ?? ??. The order of magnitude of the best approximation is established in the space Leo of the classes B _??,?? ^r of periodic functions of two variables with trigonometric polynomials with harmonics from a hyperbolic cross.     

Приближение всплесками и поперечники Фурье классов периодических функций многих переменных. II

Estimates sharp in order for Fourier widths of the classes $ B_{pq}^{sm} (\mathbb{T}^k ) $ and $ L_{pq}^{sm} (\mathbb{T}^k ) $ of Nikol??skii-Besov and Lizorkin-Triebel types, respectively, in the space $ L_r (\mathbb{T}^k ) $ are established for a certain range of the parameters s, p, q, r (here s ?? (0,??) ⁿ , 1 ??p, r, q ???, 1 ?? n ?? k, m = (m ₁, ??,m _n ) ?? ? ⁿ : m ₁ + ?? + m _n = k).     

2-Cell Embeddings with Prescribed Face Lengths and Genus

Let n be a positive integer, let d ₁, . . . , d _n be a sequence of positive integers, and let ${{q = \frac{1}{2}\sum^{n}_{i=1} d_{i}\cdot}}$ . It is shown that there exists a connected graph G on n vertices, whose degree sequence is d ₁, . . . , d _n and such that G admits a 2-cell embedding in every closed surface whose Euler characteristic is at least n ? q?+?1, if and only if q is an integer and q ?? n ? 1. Moreover, the graph G can be required to be loopless if and only if d _i ?? q for i = 1, . . . , n. This, in particular, answers a question of Skopenkov.     

On the divergence of expansions in systems of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville operator with an arbitrary complex-valued potential q(x) ?? L ₁(0, ??) and with degenerate boundary conditions. We show that, under some additional condition, the system of root functions of that operator is not a basis in the space L ₂(0, ??).     

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let T _g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T _g has a Cantor-type attractor F and a unique invariant measure ??₀ supported on F. The corresponding unitary operator (U _g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? _n,r , n ?? 1, 0 ?? r ?? 2 ^n?1 ? 1 with eigenfunctions e _r ⁽ⁿ⁾ (x). Suppose that f ?? C ¹([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L ^p ([?1, 1], d??₀), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T _g we prove that S _n (f) ?? 2^?n q ⁿ , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.     

Construction of nonseparable orthonormal compactly supported wavelet bases for L2（Rd）

Suppose M and N are two r × r and s × s dilation matrices,respectively.Let ΓM and ΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-TZr/Zr and N-TZs/Zs,respe...     

Uniform approximation of Poisson integrals of functions from the class H ω by de la Vallée Poussin sums

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C _?? ^q H _?? of Poisson integrals of functions from the class H _?? generated by convex upwards moduli of continuity ??(t) which satisfy the condition ??(t)/t ?? ?? as t ?? 0. As an implication, a solution of the Kolmogorov-Nikol??skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H ^??, 0 < ?? < 1, is obtained.     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001     

Extensions with estimates of cohomology classes

We prove an extension theorem of ??Ohsawa-Takegoshi type?? for Dolbeault q-classes of cohomology (q??? 1) on smooth compact hypersurfaces in a weakly pseudoconvex K?hler manifold.     

On the Comparison of Cryptographical Properties of Two Different Families of Graphs with Large Cycle Indicator

The paper is devoted to the implementations of the public key algorithms based on simple algebraic graphs A(n, K) and D(n, K) defined over the same finite commutative ring K. If K is a finite field both families are families of graphs with large cycle indicator. In fact, the family D(n, F _q ) is a family of graphs of large girth (f.g.l.g.) with c =?1, their connected components CD(n, F _q ) form the f.g.l.g. with the speed of growth 4/3. Family A(n, q), char F _q ?? 2 is a family of connected graphs with large cycle indicator with the largest possible speed of growth. The computer simulation demonstrates the advantage (better density which is the number of monomial expressions) of public rules derived from A(n, q) in comparison with symbolic algorithm based on graphs D(n, q).     

The shadowing chain lemma for singular Hamiltonian systems involving strong forces

We consider a planar autonomous Hamiltonian system :q+?V(q) = 0, where the potential V: ?² \{??}?? ? has a single well of infinite depth at some point ?? and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ?? we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits ?? the shadowing chain lemma ?? via minimization of action integrals and using simple geometrical arguments.     

Congruence modularity at 0

Given a variety ${\mathcal{V}}$ with a constant 0 in its type and a lattice identity p ?? q, we say that p ?? q holds for congruences in ${\mathcal{V}}$ at 0 if the p-block of 0 is included in the q-block of 0 for all substitutions of congruences of ${\mathcal{V}}$ -algebras for the variables of p and q. Varieties that are congruence modular at 0 are characterized by a Mal??tsev condition. This result generalizes the classical characterization of congruence modularity by Day terms.