Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.     

The Erdös-Kac theorem for additive arithmetical semigroups

We show that the Erdös-Kac theorem for additive arithmetical semigroups can be proved under the condition that the counting function of elements has the asymptotics G(n) = q ⁿ (A + O(1/(lnn)^k) as n → ∞ with A > 0, q > 1, and arbitrary k ∈ ? and that P(n) = O(q ⁿ /n) for the number of prime elements of degree n. This improves a result of Zhang.     

On finite groups with automorphisms whose fixed points are Engel

The main result of the paper is the following theorem. Let q be a prime, n a positive integer, and A an elementary abelian group of order q². Suppose that A acts coprimely on a finite group G and assume that for each \({a \in A^{\#}}\) every element of C_G(a) is n-Engel in G. Then the group G is k-Engel for some \({\{n,q\}}\)-bounded number k.     

Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

If all prime closed geodesics on (Sⁿ, F) with an irreversible Finsler metric F are irrationally elliptic, there exist either exactly 2 \(\left[ {\frac{{n + 1}}{2}} \right]\) or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler (S³, F) if any prime closed geodesic has non-zero Morse index.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups

We study 2-primary parts ⅢX(E~((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E~((n)): y~2= x~3-n~2x, n ∈Q~×/Q~(×2). Previous results focused on finding sufficient conditions for ⅢX(E~((n))/Q)[2~∞]trivial or isomorphic to(Z/2Z)~2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E~((n))is isomorphic to(Z/2Z)～2 and the Mordell-Weil rank of E~((n)) is zero,provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for ⅢX(E~((n))/Q)[2~∞]■(Z/2Z)~(2k), where k≥2.     

The normalizers of free subgroups in free burnside groups of odd period <Emphasis Type="Italic">n ≥</Emphasis> 1003

Let B(m, n) be a free periodic group of arbitrary rank m with period n. In this paper, we prove that for all odd numbers n ≥ 1003 the normalizer of any nontrivial subgroup N of the group B(m, n) coincides with N if the subgroup N is free in the variety of all n-periodic groups. From this, there follows a positive answer for all prime numbers n > 997 to the following problem set by S. I. Adian in the Kourovka Notebook: is it true that none of the proper normal subgroups of the group B(m, n) of prime period n > 665 is a free periodic group? The obtained result also strengthens a similar result of A. Yu. Ol’shanskii by reducing the boundary of exponent n from n > 10⁷⁸ to n ≥ 1003. For primes 665 < n ≤ 997, the mentioned question is still open.     

Automorphisms of prime order of smooth cubic <Emphasis Type="Italic">n</Emphasis>-folds

In this paper we give an effective criterion as to when a prime number p is the order of an automorphism of a smooth cubic hypersurface of \({\mathbb{P}^{n+1}}\), for a fixed n ≥ 2. We also provide a computational method to classify all such hypersurfaces that admit an automorphism of prime order p. In particular, we show that p < 2 ⁿ⁺¹ and that any such hypersurface admitting an automorphism of order p > 2 ⁿ is isomorphic to the Klein n-fold. We apply our method to compute exhaustive lists of automorphism of prime order of smooth cubic threefolds and fourfolds. Finally, we provide an application to the moduli space of principally polarized abelian varieties.     

On generalized Heawood inequalities for manifolds: a van Kampen–Flores-type nonembeddability result

The fact that the complete graph K₅ does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M (other than the Klein bottle) if and only if (n?3)(n?4) ≤ 6b₁(M), where b₁(M) is the first Z₂-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_n+1) embeds in R^2k if and only if n ≤ 2k + 1.Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k ? 1)-connected 2k-manifold with kth Z₂-Betti number b_k only if the following generalized Heawood inequality holds: ( _k+1 ^n?k?1 ) ≤ ( _k+1 ^2k+1 )b_k. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem.In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z₂-Betti number bk, then n ≤ 2b_k( _k ^2k+2 )+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k?1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.     

Some variations on Tverberg’s theorem

Define T(d, r) = (d + 1)(r - 1) + 1. A well known theorem of Tverberg states that if n ≥ T(d, r), then one can partition any set of n points in R^d into r pairwise disjoint subsets whose convex hulls have a common point. The numbers T(d, r) are known as Tverberg numbers. Reay added another parameter k (2 ≤ k ≤ r) and asked: what is the smallest number n, such that every set of n points in R^d admits an r-partition, in such a way that each k of the convex hulls of the r parts meet. Call this number T(d, r, k). Reay conjectured that T(d, r, k) = T(d, r) for all d, r and k. In this paper we prove Reay’s conjecture in the following cases: when k ≥ [d+3/2], and also when d < rk/r-k - 1. The conjecture also holds for the specific values d = 3, r = 4, k = 2 and d = 5, r = 3, k = 2.     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

Facets of the <Emphasis Type="Italic">r</Emphasis>-Stable (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-Hypersimplex

Let k, n, and r be positive integers with k < n and \({r \leq \lfloor \frac{n}{k} \rfloor}\). We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every \({r < \lfloor \frac{n}{k} \rfloor}\). We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart \({\delta}\)-vectors.     

Vertex-fault-tolerant cycles embedding on enhanced hypercube networks

In this paper, we focus on the vertex-fault-tolerant cycles embedding on enhanced hypercube, which is an attractive variant of hypercube and is obtained by adding some complementary edges from hypercube. Let F _v be the set of faulty vertices in the n-dimensional enhanced hypercube Q _n,k (1 ≤ k ≤ n?1). When |F _v | = 2, we showed that Q _n,k ? F _v contains a fault-free cycle of every even length from 4 to 2 ⁿ ?4 where n (n ≥ 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2 ⁿ ? 4, simultaneously, contains a cycle of every odd length from n ? k + 2 to 2 ⁿ ? 3 where n(≥ 3) and k have the different parity. Furthermore, when |F ^v | = f ^v ≤ n ? 2, we proof that there exists the longest fault-free cycle, which is of even length 2 ⁿ ? 2f ^v whether n(n ≥ 3) and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2 ⁿ ? 2f ^v ? 1 in Q ^n,k ? F ^v where n(≥ 3) and k have the different parity.     

On the tree structure of the power digraphs modulo n

For any two positive integers n and k ? 2, let G(n, k) be a digraph whose set of vertices is {0, 1, …, n ? 1} and such that there is a directed edge from a vertex a to a vertex b if a ^k ≡ b (mod n). Let \(n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} \) be the prime factorization of n. Let P be the set of all primes dividing n and let P ₁, P ₂ ? P be such that P ₁ ∪ P ₂ = P and P ₁ ∩ P ₂ = ?. A fundamental constituent of G(n, k), denoted by \(G_{{P_2}}^*(n,k)\), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \(\prod\nolimits_{{p_i} \in {P_2}} {{p_i}} \) and are relatively prime to all primes q ∈ P ₁. L. Somer and M. K?i?ek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic.     

Conditional connectivity of bubble sort graphs

A subset F ? V (G) is called an R ^k -vertex-cut of a graph G if G ? F is disconnected and each vertex of G ? F has at least k neighbors in G ? F. The R ^k -vertex-connectivity of G, denoted by κ ^k (G), is the cardinality of a minimum R ^k -vertex-cut of G. Let B _n be the bubble sort graph of dimension n. It is known that κ _k (B _n ) = 2 ^k (n ? k ? 1) for n ≥ 2k and k = 1, 2. In this paper, we prove it for k = 3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.     

Complexities of normal bases constructed from Gauss periods

Let q be a power of a prime p, and let \(r=nk+1\) be a prime such that \(r\not \mid q\), where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\); the complexity of the resulting normal basis of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\) is denoted by C(n, k; p). Recent works determined C(n, k; p) for \(k\le 7\) and all qualified n and q. In this paper, we show that for any given \(k>0\), C(n, k; p) is given by an explicit formula except for finitely many primes \(r=nk+1\) and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes \(r=nk+1\). Our numerical results cover C(n, k; p) for \(k\le 20\) and all qualified n and q.     

On the additive complexity of GCD and LCM matrices

In the paper, the additive complexity of matrices formed by positive integer powers of greatest common divisors and least common multiples of the indices of the rows and columns is considered. It is proved that the complexity of the n × n matrix formed by the numbers GCD^r(i, k) over the basis {x + y} is asymptotically equal to rn log²n as n→∞, and the complexity of the n × n matrix formed by the numbers LCM^r(i, k) over the basis {x + y,?x} is asymptotically equal to 2rn log₂n as n→∞.     

Finite configurations in sparse sets

Let E ? ?ⁿ be a closed set of Hausdorff dimension α. For m > n, let{B₁, …, B_k} be n × (m ? n) matrices. We prove that if the system of matrices B_j is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration {B₁y, …, B_ky}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in ?ⁿ and isosceles right triangles in ?²). This can be viewed as a multidimensional analogue of the result of [25] on 3-term arithmetic progressions in subsets of ?.     

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.     

On eigenvalues of random complexes

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam model X^k(n, p) of random k-dimensional simplicial complexes on n vertices. We show that for p = Ω(logn/n), the eigenvalues of each of the matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of (k - 2)-dimensional faces. Garland’s result concerns the Laplacian; we develop an analogous result for the adjacency matrix.