The Hardy–Littlewood theorem for multiple fourier series with monotone coefficients

It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classicalHardy–Littlewood theorem is not valid for p ≤ 2m/(m+1), where m is the dimension of the space. We establish how the theorem must be modified in this case.     

NP completeness conditions for verifying the consistency of several kinds of systems of linear Diophantine congruences and equations

Three series of number-theoretic problems with explicitly marked parameters that concerning systems of modulo m congruences and systems of Diophantine equations with solutions from the given segment are proposed. Parameter constraints such that any problem of each series is NP complete when they are met are proved. For any m₁ and m₂ (m₁ < m₂ and m1 is not a divisor of m₂), the verification problem for the consistency of a system of linear congruences modulo m₁ and m₂ simultaneously, each containing exactly three variables, is proved to be NP complete. In addition, for any m > 2, the verification problem for the consistency on the subset, containing at least two elements, of the set {0, …, m–1} for the system of linear congruences modulo m, each of which contains exactly three variables, is proved to be NP complete. If P ≠ NP, one cannot replace the term 3-discongruence with the term 2-discongruence in the statement of the theorem. For systems of Diophantine linear equations, each of which contains exactly three variables, the verification problem for their consistency on the given segment of integers is proved to be NP complete. If P ≠ NP, one cannot replace the term 3-equation with the term 2-equation in the statement of the theorem. This problem can also have a simple geometrical interpretation concerning the NP completeness of the verification problem on whether there an integer point of intersection of the given hyperplanes exists that cuts off equivalent segments on three axes and are parallel to other axes inside of a multidimensional cube. The problems of the stated series include practically useful problems. Since the range of values for an integer computer variable can be considered integer values from a segment, if P ≠ NP, theorem 5 proves that any algorithm that solves these systems in the set of numbers of the integer type is nonpolynomial [6].     

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.     

Functional Identities and Rings of Quotients

The fundamental theorem on functional identities states that a prime ring R with \(\deg (R)\ge d\) is a d-free subset of its maximal left ring of quotients Q _{m l} (R). We consider the question whether the same conclusion holds for symmetric rings of quotients. This indeed turns out to be the case for the maximal symmetric ring of quotients Q _{m s} (R), but not for the symmetric Martindale ring of quotients Q _s (R). We show, however, that if the maps from the basic functional identities have their ranges in R, then the maps from their standard solutions have their ranges in Q _s (R). We actually prove a more general theorem which implies both aforementioned results. Its proof is somewhat shorter and more compact than the standard proof used for establishing d-freeness in various situations.     

P-positions in modular extensions to Nim

In this paper, we consider a modular extension to the game of Nim, which we call m-Modular Nim, and explore its optimal strategy. In m-Modular Nim, a player can either make a standard Nim move or remove a multiple of m tokens in total. We develop a winning strategy for all m with 2 heaps and for odd m with any number of heaps.     

On positive quaternionic Kähler manifolds with certain symmetry rank

Let M be a positive quaternionic Kähler manifold of real dimension 4m. In this paper we show that if the symmetry rank of M is greater than or equal to [m/2] + 3, then M is isometric to HP ^m or Gr2(C ^m+2). This is sharp and optimal, and will complete the classification result of positive quaternionic Kähler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kähler manifolds with a group action and the induction arguments on the dimension of the manifold.     

Non-homogeneous <Emphasis Type="Italic">Tb</Emphasis> Theorem for Bi-parameter <Emphasis Type="Italic">g</Emphasis>-function

The main result of this paper is a bi-parameter Tb theorem for Littlewood–Paley g-function, where b is a tensor product of two pseudo-accretive function. Instead of the doubling measure, we work with a product measure μ = μ_n × μ_m, where the measures μ_n and μ_m are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter b-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.     

On the k-polygonal numbers and the mean value of Dedekind sums

For any positive integer k ≥ 3, it is easy to prove that the k-polygonal numbers are a_n(k) = (2n+n(n?1)(k?2))/2. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S(a_n(k)ā_m(k), p) for k-polygonal numbers with 1 ≤ m, n ≤ p ? 1, and give an interesting computational formula for it.     

Word maps and word maps with constants of simple algebraic groups

In the present paper, we consider word maps w: G ^m → G and word maps with constants w _Σ: G ^m → G of a simple algebraic group G, where w is a nontrivial word in the free group F _m of rank m, w _Σ = w ₁ σ ₁ w ₂ ··· w _r σ _r w _{r + 1}, w ₁, …, w _{r + 1} ∈ F _m , w ₂, …, w _r ≠ 1, Σ = {σ ₁, …, σ _r | σ _i ∈ G Z(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γ_w, G) of the group Γ_w = F _m /<w>.     

Normal families of meromorphic functions and shared functions

The paper is devoted to the normal families of meromorphic functions and shared functions. Generalizing a result of Chang (2013), we prove the following theorem. Let h (≠≡ 0,∞) be a meromorphic function on a domain D and let k be a positive integer. Let F be a family of meromorphic functions on D, all of whose zeros have multiplicity at least k + 2, such that for each pair of functions f and g from F, f and g share the value 0, and f^(k) and g^(k) share the function h. If for every f ∈ F, at each common zero of f and h the multiplicities m_f for f and m_h for h satisfy m_f ≥ m_h + k + 1 for k > 1 and m_f ≥ 2m_h + 3 for k = 1, and at each common pole of f and h, the multiplicities nf for f and nh for h satisfy n_f ≥ n_h + 1, then the family F is normal on D.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.     

Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative

In this paper, the estimate of the radius of the convergence ball of the modified Halley’s method for finding multiple zeros of nonlinear equations is provided under the hypotheses that the derivative f ^(m?+?1) of function f is Hölder continuous, and f ^(m?+?1) is bounded. The uniqueness ball of solution is also established. Finally, some examples are provided to show applications of our theorem.     

Engel-type subgroups and length parameters of finite groups

Let g be an element of a finite group G. For a positive integer n, let E _n (g) be the subgroup generated by all commutators [...[[x, g], g],..., g] over x ∈ G, where g is repeated n times. By Baer’s theorem, if E _n (g) = 1, then g belongs to the Fitting subgroup F(G). We generalize this theorem in terms of certain length parameters of E _n (g). For soluble G we prove that if, for some n, the Fitting height of E _n (g) is equal to k, then g belongs to the (k+1)th Fitting subgroup F_k+1(G). For nonsoluble G the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F_h* (H) = H, where F₀* (H) = 1, and F_i+1(H)* is the inverse image of the generalized Fitting subgroup F*(H/F*_i (H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of E _n (g) is equal to k, then g belongs to F*_f(k,m)(G), where f(k, m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(E _n (g)) = k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

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.     

Interpolation of vector measures

Let (Ω, Σ) be a measurable space and m ₀: Σ → X ₀ and m ₁: Σ → X ₁ be positive vector measures with values in the Banach Köthe function spaces X ₀ and X ₁. If 0 < α < 1, we define a new vector measure [m ₀, m ₁] _α with values in the Calderón lattice interpolation space X ₀ ^1?ga X ₁ ^α and we analyze the space of integrable functions with respect to measure [m ₀, m ₁] _α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L ^p -spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.     

Connection between the clique number and the Lagrangian of 3-uniform hypergraphs

There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus (J Math 17:533–540, 1965). It would be useful in practice if similar results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus’ result to hypergraphs is false. Frankl and Füredi conjectured that the r-uniform hypergraph with m edges formed by taking the first m sets in the colex ordering of \({\mathbb N}^{(r)}\) has the largest Lagrangian of all r-uniform hypergraphs with m edges. For \(r=2\), Motzkin and Straus’ theorem confirms this conjecture. For \(r=3\), it is shown by Talbot that this conjecture is true when m is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for 3-uniform hypergraphs. As an application of this connection, we confirm that Frankl and Füredi’s conjecture holds for bigger ranges of m when \(r=3\). We also obtain two weaker versions of Turán type theorem for left-compressed 3-uniform hypergraphs.     

Linear preservers of row-dense matrices

Let M_m,n be the set of all m × n real matrices. A matrix A ∈ M_m,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_m,n → M_m,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ M_n,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.     

Length-type parameters of finite groups with almost unipotent automorphisms

Let α be an automorphism of a finite group G. For a positive integer n, let E _G,n (α) be the subgroup generated by all commutators [...[[x,α],α],…,α] in the semidirect product G 〈α〉 over x ∈ G, where α is repeated n times. By Baer’s theorem, if E _G,n (α)=1, then the commutator subgroup [G,α] is nilpotent. We generalize this theorem in terms of certain length parameters of E _G,n (α). For soluble G we prove that if, for some n, the Fitting height of E _G,n (α) is equal to k, then the Fitting height of [G,α] is at most k + 1. For nonsoluble G the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F _h* (H) = H, where F ₀* (H) = 1, and F _i+1* (H) is the inverse image of the generalized Fitting subgroup F*(H/F _i *(H)). Let m be the number of prime factors of the order |α| counting multiplicities. It is proved that if, for some n, the generalized Fitting height E _G,n (α) of is equal to k, then the generalized Fitting height of [G,α] is bounded in terms of k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λE _G,n (α)= k, then the nonsoluble length of [G,α] is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

New Applications of the <Emphasis Type="Italic">p</Emphasis>-Adic Nevanlinna Theory

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fⁿ(x)f^m(ax + b), gⁿ(x)g^m(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n ? m|_∞ ≥ 5, then fⁿ(x)f^m(ax + b) ? w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fⁿ(x)f^m(ax + b).     

Linear homotopy method for computing generalized tensor eigenpairs

Let m,m′, n be positive integers such that m ≠ m′. Let A be an mth order n-dimensional tensor, and let ? be an m′th order n-dimensional tensor. λ ∈ ? is called a ?-eigenvalue of A if A x^m?1 = λ?x^m′?1 and ?xm′= 1 for some x ∈ ?ⁿ\{0}. In this paper, we propose a linear homotopy method for solving this eigenproblem. We prove that the method finds all isolated ?-eigenpairs. Moreover, it is easy to implement. Numerical results are provided to show the efficiency of the proposed method.