The Number of <Emphasis Type="Italic">k</Emphasis>-Sumsets in an Abelian Group

Let G be an abelian group of order n. The sum of subsets A₁,...,A_k of G is defined as the collection of all sums of k elements from A₁,...,A_k; i.e., A₁ + A₂ + · · · + A_k = {a₁ + · · · + a_k | a₁ ∈ A₁,..., a_k ∈ A_k}. A subset representable as the sum of k subsets of G is a k-sumset. We consider the problem of the number of k-sumsets in an abelian group G. It is obvious that each subset A in G is a k-sumset since A is representable as A = A₁ + · · · + A_k, where A₁ = A and A₂ = · · · = A_k = {0}. Thus, the number of k-sumsets is equal to the number of all subsets of G. But, if we introduce a constraint on the size of the summands A₁,...,A_k then the number of k-sumsets becomes substantially smaller. A lower and upper asymptotic bounds of the number of k-sumsets in abelian groups are obtained provided that there exists a summand A_i such that |A_i| = n log^qn and |A₁ +· · ·+ A_i-1 + A_i+1 + · · ·+A_k| = n log^qn, where q = -1/8 and i ∈ {1,..., k}.     

On the number of nonstationary bounded trajectories of a class of autonomous systems on the plane

We study the number of nonstationary bounded trajectories of autonomous systems of the form z′ = \(\overline {P_n (z)} \), z = x + iy ∈ C, where P _n (z) is a polynomial of degree n with complex coefficients that has k distinct roots, n, k > 1. We prove that the number N of nonstationary bounded trajectories of this system satisfies the following assertions (Theorem 1): (a) N = n + k ? N ₊, N ₊ = N _?, n + 1 ≤ N ₊ ≤ n + k, where N ₊ and N _? are the numbers of system trajectories unbounded as t → +∞ and t → ?∞, respectively; (b) if some r distinct roots \(c_{j_1 } \), ..., \(c_{j_r } \) of the polynomial P _n satisfy the relations V _n+1 (\(c_{j_1 } \)) = ··· = V _n+1 (\(c_{j_r } \)), where V _n+1 is the imaginary part of the indeterminate integral of P _n , then N ≥ \(m_{j_1 } \) + ··· + \(m_{j_r } \) + r ? n ? 1; (c) if k = 2, then the conditions N = 1 and V _n+1 (c ₁) = V _n+1 (c ₂) are equivalent. For n = k = 3, we derive a formula for the number of nonstationary bounded trajectories (Theorem 2).     

Block partitions of sequences

Given a sequence A = (a ₁, …, a _n ) of real numbers, a block B of A is either a set B = {a _i , a _i+1, …, a _j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a _i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B ₁, …, B _k with |b _i ?b _j | ≤ 1 for every i, j. We extend this result in several directions.     

Generalized competition index of primitive digraphs

For any positive integers k and m, the k-step m-competition graph C _m ^k (D) of a digraph D has the same set of vertices as D and there is an edge between vertices x and y if and only if there are distinct m vertices v₁, v₂, · · ·, v _m in D such that there are directed walks of length k from x to v _i and from y to v _i for all 1 ≤ i ≤ m. The m-competition index of a primitive digraph D is the smallest positive integer k such that C _m ^k (D) is a complete graph. In this paper, we obtained some sharp upper bounds for the m-competition indices of various classes of primitive digraphs.     

Bent functions with stronger nonlinear properties: K-bent functions

We introduce the notion of k-bent function, i.e., a Boolean functionwith evennumber m of variables υ ₁, …, υ _m which can be approximated with all functions of the form 〈u, v〉 _j ⊕ a in the equally poor manner, where u ∈ ? ₂ ^m , a ∈ ?₂, and 1 ? j ? k. Here 〈·, ·〉 _j is an analog of the inner product of vectors; k changes from 1 to m/2. The operations 〈·, ·〉 _k , 1 ? k ? m/2, are defined by using the special series of binary Hadamard-like codes A _m ^k of length 2 ^m . Namely, the vectors of values for the functions 〈u, v〉 _k ⊕ a are codewords of the code A _m ^k . The codes A _m ^k are constructed using subcodes of the ?₄-linear Hadamard-like codes of length 2 ^m+1, which were classified by D. S. Krotov (2001). At that the code A _m ¹ is linear and A _m ¹ , …, A _m ^m/2 are pairwise nonequivalent. On the codewords of a code A _m ^k , we define a group operation ? coordinated with the Hamming metric. That is why we can consider k-bent functions as functions that are maximal nonlinear in k distinct senses of linearity at the same time. Bent functions in usual sense coincide with 1-bent functions. For k > ? ? 1, the class of k-bent functions is a proper subclass of the class of ?-bent functions. In the paper, we give methods for constructing k-bent functions and study their properties. It is shown that there exist k-bent functions with an arbitrary algebraic degree d, where 2 ? d ? max {2, m/2 ? k + 1}. Given an arbitrary k, we define the subset \(\mathfrak{F}_m^k \mathcal{F}_m^k \) of the set \(\mathfrak{F}_m^k \mathcal{F}_m^k \) \(\mathcal{A}_m^k \mathcal{B}_m^k \) of all Boolean functions in m variables with the following property: on this subset k-bent functions and 1-bent functions coincide.     

On a question of Sárközy on gaps of product sequences

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b ₁ < b ₂ < …} of all products a _i a _j with a _i , a _j ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps b _n+1 ? b _n ? α ^?3 and that for t ≥ 2 there are infinitely many t-gaps b _n+t ? b _n ? t ² α ^?4. Furthermore, we prove that these estimates are best possible.We also discuss a related question about the cardinality of the quotient set A/A = {a _i /a _j , a _i , a _j ∈ A} when A ? {1, …, N} and |A| = αN.     

Real hypersurfaces in the complex quadric with commuting Ricci tensor

We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric Q^m = SO_m+2/SO_mSO₂. It is shown that the commuting Ricci tensor gives that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in Q^m = SO_m+2/SO_mSO₂ with commuting Ricci tensor.     

Counting sets with small sumset and applications

We study the number of k-element sets A? {1,...,N} with |A+A| ≤ K|A| for some (fixed) K > 0. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of 2 ^{o ( k )} N ^{o (1)} for most N and k. As a consequence of this and a further new result concerning the number of sets A??/N? with |A+A| ≤ c|A|², we deduce that the random Cayley graph on ?/N? with edge density ½ has no clique or independent set of size greater than (2+o(1)) log₂ N, asymptotically the same as for the Erd?s-Rényi random graph. This improves a result of the first author from 2003 in which a bound of 160log₂ N was obtained. As a second application, we show that if the elements of A ? ? are chosen at random, each with probability 1/2, then the probability that A+A misses exactly k elements of ? is equal to (2+O(1))^?k/2 as k → ∞.     

Ranks of matrices with few distinct entries

It is consistent that P(ω ₁) is the union of less than \({2^{{\aleph _1}}}\) parts such that if A ₀,..., A _n?1, B ₀,..., B _m?1 are distinct elements of the same part, then |A ₀ ∩ · · · ∩ A _n?1 ∩ (ω ₁ ? B ₀) ∩ · · ·∩ (ω ₁ ? B _m?1)| = N₁.     

Possible isolation number of a matrix over nonnegative integers

Let ?₊ be the semiring of all nonnegative integers and A an m × n matrix over ?₊. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.     

An extended class of stabilizable uncertain systems

The system of equations \(\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u\), where A(·) ∈ ?^{n × n}, B(·) ∈ ?^{n × m}, S(·) ∈ R^{n × m}, is considered. The elements of the matrices A(·), B(·), S(·) are uniformly bounded and are functionals of an arbitrary nature. It is assumed that there exist k elements \({\alpha _{{i_i}{j_l}}}\left( \cdot \right)\left( {l \in \overline {1,k} } \right)\) of fixed sign above the main diagonal of the matrix A(·), and each of them is the only significant element in its row and column. The other elements above the main diagonal are sufficiently small. It is assumed that m = n ?k, and the elements β_ij(·) of the matrix B(·) possess the property \(\left| {{\beta _{{i_s}s}}\left( \cdot \right)} \right| = {\beta _0} > 0\;at\;{i_s}\; \in \;\overline {1,n} \backslash \left\{ {{i_1}, \ldots ,{i_k}} \right\}\). The other elements of the matrix B(·) are zero. The positive definite matrix H = {h_ij} of the following form is constructed. The main diagonal is occupied by the positive numbers h_ii = h_i, \({h_{{i_l}}}_{{j_l}}\, = \,{h_{{j_l}{i_l}}}\, = \, - 0.5\sqrt {{h_{{i_l}}}_{{j_l}}} \,\operatorname{sgn} \,{\alpha _{{i_l}}}_{{j_l}}\left( \cdot \right)\). The other elements of the matrix H are zero. The analysis of the derivative of the Lyapunov function V(x) = x*H^–1x yields h_i\(\left( {i \in \overline {1,n} } \right)\) and λ_i ≤ 0 \(\left( {i \in \overline {1,n} } \right)\) such that for S(·) = H^?1ΛB(·), Λ = diag(λ₁, ..., λ_n), the system of the considered equations becomes globally exponentially stable. The control is robust with respect to the elements of the matrix A(·).     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

The boundary of the outer space of a free product

Let G be a countable group that splits as a free product of groups of the form G = G ₁ *···* G _k * F _N , where F _N is a finitely generated free group. We identify the closure of the outer space PO(G, {G ₁,..., G _k }) for the axes topology with the space of projective minimal, very small (G, {G ₁,..., G _k })-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the G _i ’s, and whose tripod stabilizers are trivial. Its topological dimension is equal to 3N + 2k ? 4, and the boundary has dimension 3N + 2k ? 5. We also prove that any very small (G, {G ₁,..., G _k })-tree has at most 2N + 2k?2 orbits of branch points.     

Extrapolation in Grand Lebesgue Spaces with <Emphasis Type="Italic">A</Emphasis>∞ Weights

Results on extrapolation withA∞ weights in grand Lebesgue spaces are obtained. Generally, these spaces are defined with respect to the productmeasure μ₁ ×· · ·×μ_n onX₁ ×· · ·×X_n, where (X_i, d_i, μ_i), i = 1,..., n, are spaces of homogeneous type. As applications of the obtained results, new one-weight estimates with A_∞ weights for operators of harmonic analysis are derived.     

New algorithm for computing the Hermite interpolation polynomial

Let x ₀, x ₁,? , x _n , be a set of n + 1 distinct real numbers (i.e., x _i ≠ x _j , for i ≠ j) and y _{i, k} , for i = 0,1,? , n, and k = 0 ,1 ,? , n _i , with n _i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p _{N ? 1}(x) of degree N ? 1 where \(N={\sum }_{i=0}^{n}(n_{i}+1)\), such that \(p_{N-1}^{(k)}(x_{i})=y_{i,k}\), for i = 0,1,? , n and k = 0,1,? , n _i . P _N?1(x) is the Hermite interpolation polynomial for the set {(x _i , y _{i, k} ), i = 0,1,? , n, k = 0,1,? , n _i }. The polynomial p _N?1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n _i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.     

On the nonexistence of solutions of some anisotropic elliptic and backward parabolic inequalities

We study the nonexistence of weak solutions of higher-order elliptic and parabolic inequalities of the following types: \(\sum {_{i = 1}^N\sum\nolimits_{{e_i} \leqslant {\alpha _i} \leqslant {m_i}} {D_{{x_i}}^{{\alpha _i}}\left( {{A_{{\alpha _i}}}\left( {x,u} \right)} \right)} \geqslant f\left( {x,u} \right),} x \in {\mathbb{R}^N}\), and \({u_t} + \sum {_{i = 1}^N\sum\nolimits_{{k_i} \leqslant {\beta _i} \leqslant {n_i}} {D_{{x_i}}^{{\beta _i}}\left( {{B_{{\beta _i}}}\left( {x,t,u} \right)} \right)} > g\left( {x,t,u} \right),\left( {x,t} \right)} \in {\mathbb{R}^N} \times {\mathbb{R}_ + }\), where l _i , m _i , k _i , n _i ∈ N satisfy the condition l _i , k _i > 1 for all i = 1,..., N, and A _αi (x, u), B _βi (x, t, u), f(x, u), and g(x, t, u) are some given Carathéodory functions. Under appropriate conditions on the functions A _αi , B _βi , f, and g, we prove theorems on the nonexistence of solutions of these inequalities.     

Paranormal Elements in Normed Algebra

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P _k (A). We show that P₁(A) is a subset of P _k (A) for all k. If T in P₁(A), then Tⁿ lies in P₁(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P _k (A), then UTV lies in P _k (A) for all natural k. Let A be unital, then 1) if an element T in P₁(A) is right invertible, then any right inverse element T^?1 lies in P₁(A); 2) for ßßIßß = 1 the class P₁(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P₁(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P₁(A) coincides with the set of all paranormal operators on a Hilbert space H.     

Some new sums related to D. H. Lehmer problem

About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N(k; p) denote the number of all 1 ? a _i ? p ? 1 such that a ₁ a ₂ … a _k ≡ 1 mod p and 2 | a _i + ā _i + 1, i = 1, 2, …, k. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N(k; p), and give an interesting asymptotic formula for it.     

Precise large deviations for sums of random vectors with dependent components of consistently varying tails

Let {X _i = (X _1,i ,...,X _m,i )^?, i ≥ 1} be a sequence of independent and identically distributed nonnegative m-dimensional random vectors. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Here, the components of X ₁ are allowed to be generally dependent. Moreover, let N(·) be a nonnegative integer-valued process, independent of the sequence {X _i , i ≥ 1}. Under several mild assumptions, precise large deviations for S _n = Σ _i=1 ⁿ X _i and S _N(t) = Σ _i=1 ^N(t) X _i are investigated. Meanwhile, some simulation examples are also given to illustrate the results.     

On the evaluation of highly oscillatory finite Hankel transform using special functions

In this paper, an efficient Clenshaw–Curtis–Filon–type method is presented for approximation of the highly oscillatory finite Hankel transform \({{\int }_{0}^{1}}f(x)H_{\nu }^{(1)}(\omega x)dx\), which arises in acoustic and electromagnetic scattering problems. This method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments by using Meijer G–function and Lommel function. Moreover, the method shares the property that the higher the frequency ω, the higher the precision. In particular, for each fixed ω the method is uniformly convergent as N tends to infinity, where (N+1) is the number of Clenshaw–Curtis points c_i=(1+ cos(iπ/N))/2,i=0,? ,N. Also, the corresponding error bound in inverse powers of ω for this method for the integral is presented. The efficiency and accuracy of the proposed method are illustrated by numerical examples.