Beatty sequences and prime numbers with restrictions on strongly <Emphasis Type="Italic">q</Emphasis>-additive functions

We estimate exponential sums over a non-homogenous Beatty sequence with restriction on strongly q-additive functions. We then apply our result in a few special cases to obtain an asymptotic formula for the number of primes \(p=\lfloor \alpha n +\beta \rfloor \) and \(f(p)\equiv a (\mathrm{mod\,}b)\), with \(n\ge N \), where \(\alpha \), \(\beta \) are real numbers and f is a strongly q-additive function (for example, the sum of digits function in base q is a strongly q-additive function). We also prove that for any fixed integer \(k\ge 3 \), all sufficiently large \(N\equiv k (\mathrm{mod\,}2) \) could be represented as a sum of k prime numbers from a Beatty sequence with restriction on strongly q-additive functions.     

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.     

Universal Cycle Packings and Coverings for <Emphasis Type="Italic">k</Emphasis>-Subsets of an <Emphasis Type="Italic">n</Emphasis>-Set

A cyclic sequence of elements of [n] is an (n, k)-Ucycle packing (respectively, (n, k)-Ucycle covering) if every k-subset of [n] appears in this sequence at most once (resp. at least once) as a subsequence of consecutive terms. Let \(p_{n,k}\) be the length of a longest (n, k)-Ucycle packing and \(c_{n,k}\) the length of a shortest (n, k)-Ucycle covering. We show that, for a fixed \(k,p_{n,k}={n\atopwithdelims ()k}-O(n^{\lfloor k/2\rfloor })\). Moreover, when k is not fixed, we prove that if \(k=k(n)\le n^{\alpha }\), where \(0<\alpha <1/3\), then \(p_{n,k}={n\atopwithdelims ()k}-o({n\atopwithdelims ()k}^\beta )\) and \(c_{n,k}={n\atopwithdelims ()k}+o({n\atopwithdelims ()k}^\beta )\), for some \(\beta <1\). Finally, we show that if \(k=o(n)\), then \(p_{n,k}={n\atopwithdelims ()k}(1-o(1))\).     

Well-Quasi-Order of Relabel Functions

We define NLC\(_k^{\mathcal{F}}\) to be the restriction of the class of graphs NLC _k , where relabelling functions are exclusively taken from a set \(\mathcal{F}\) of functions from {1,...,k} into {1,...,k}. We characterize the sets of functions \(\mathcal{F}\) for which NLC\(_k^{\mathcal{F}}\) is well-quasi-ordered by the induced subgraph relation ≤? _i . Precisely, these sets \(\mathcal{F}\) are those which satisfy that for every \(f,g\in \mathcal{F}\), we have Im(f?°?g)?=?Im(f) or Im(g?°?f)?=?Im(g). To obtain this, we show that words (or trees) on \(\mathcal{F}\) are well-quasi-ordered by a relation slightly more constrained than the usual subword (or subtree) relation. A class of graphs is n-well-quasi-ordered if the collection of its vertex-labellings into n colors forms a well-quasi-order under ≤? _i , where ≤? _i respects labels. Pouzet (C R Acad Sci, Paris Sér A–B 274:1677–1680, 1972) conjectured that a 2-well-quasi-ordered class closed under induced subgraph is in fact n-well-quasi-ordered for every n. A possible approach would be to characterize the 2-well-quasi-ordered classes of graphs. In this respect, we conjecture that such a class is always included in some well-quasi-ordered NLC\(_k^{\mathcal{F}}\) for some family \(\mathcal{F}\). This would imply Pouzet’s conjecture.     

Subgroups of cyclic groups and values of the Riemann zeta function

Let k be a positive integer, x a large real number, and let \(C_n\) be the cyclic group of order n. For \(k\le n\le x\) we determine the mean average order of the subgroups of \(C_n\) generated by k distinct elements and we give asymptotic results of related averaging functions of the orders of subgroups of cyclic groups. The average order is expressed in terms of Jordan’s totient functions and Stirling numbers of the second kind. We have the following consequence. Let k and x be as above. For \(k\le n\le x\), the mean average proportion of \(C_n\) generated by k distinct elements approaches \(\zeta (k+2)/\zeta (k+1)\) as x grows, where \(\zeta (s)\) is the Riemann zeta function.     

On the sumsets of exceptional units in $$\mathbb {Z}_n$$

Let R be a commutative ring with \(1\in R\) and \(R^{*}\) be the multiplicative group of its units. In 1969, Nagell introduced the concept of an exceptional unit, namely a unit u such that \(1-u\) is also a unit. Let \({\mathbb {Z}}_n\) be the ring of residue classes modulo n. In this paper, given an integer \(k\ge 2\), we obtain an exact formula for the number of ways to represent each element of \( \mathbb {Z}_n\) as the sum of k exceptional units. This generalizes a recent result of J. W. Sander for the case \(k=2\).     

Newly appointed editors

Let k be a field and \(k(x_0,\ldots ,x_{p-1})\) be the rational function field of p variables over k where p is a prime number. Suppose that \(G=\langle \sigma \rangle \simeq C_p\) acts on \(k(x_0,\ldots ,x_{p-1})\) by k-automorphisms defined as \(\sigma :x_0\mapsto x_1\mapsto \cdots \mapsto x_{p-1}\mapsto x_0\). Denote by P the set of all prime numbers and define \(P_0=\{p\in P:\mathbb {Q}(\zeta _{p-1})\) is of class number one\(\}\) where \(\zeta _n\) a primitive n-th root of unity in \(\mathbb {C}\) for a positive integer n; \(P_0\) is a finite set by Masley and Montgomery (J Reine Angew Math 286/287:248–256, 1976). Theorem. Let k be an algebraic number field and \(P_k=\{p\in P: p\) is ramified in \(k\}\). Then \(k(x_0,\ldots ,x_{p-1})^G\) is not stably rational over k for all \(p\in P\backslash (P_0\cup P_k)\).     

Packing chromatic number versus chromatic and clique number

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in [k]\), where each \(V_i\) is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that \(\omega (G) = a\), \(\chi (G) = b\), and \(\chi _{\rho }(G) = c\). If so, we say that (a, b, c) is realizable. It is proved that \(b=c\ge 3\) implies \(a=b\), and that triples \((2,k,k+1)\) and \((2,k,k+2)\) are not realizable as soon as \(k\ge 4\). Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on \(\chi _{\rho }(G)\) in terms of \(\Delta (G)\) and \(\alpha (G)\) is also proved.     

Coloring Complete and Complete Bipartite Graphs from Random Lists

Assign to each vertex v of the complete graph \(K_n\) on n vertices a list L(v) of colors by choosing each list independently and uniformly at random from all f(n)-subsets of a color set \([n] = \{1,\dots , n\}\), where f(n) is some integer-valued function of n. Such a list assignment L is called a random (f(n), [n])-list assignment. In this paper, we determine the asymptotic probability (as \(n \rightarrow \infty \)) of the existence of a proper coloring \(\varphi \) of \(K_n\), such that \(\varphi (v) \in L(v)\) for every vertex v of \(K_n\). We show that this property exhibits a sharp threshold at \(f(n) = \log n\). Additionally, we consider the corresponding problem for the line graph of a complete bipartite graph \(K_{m,n}\) with parts of size m and n, respectively. We show that if \(m = o(\sqrt{n})\), \(f(n) \ge 2 \log n\), and L is a random (f(n), [n])-list assignment for the line graph of \(K_{m,n}\), then with probability tending to 1, as \(n \rightarrow \infty \), there is a proper coloring of the line graph of \(K_{m,n}\) with colors from the lists.     

Skew <Emphasis Type="Italic">n</Emphasis>-derivation on prime and semi prime rings

Let \(n \ge 2\) be a fixed integer, R be a noncommutative n!-torsion free ring and I be any non zero ideal of R. In this paper we have proved the following results; (i) If R is a prime ring and there exists a symmetric skew n-derivation \(D: R^n \rightarrow R\) associated with the automorphism \(\sigma \) on R,  such that the trace function \(\delta : R \rightarrow R \) of D satisfies \([\delta (x), \sigma (x)] =0\), for all \(x\in I,\) then \(D=0;\,\)(ii) If R is a semi prime ring and the trace function \(\delta ,\) commuting on I,  satisfies \([\delta (x), \sigma (x)]\in Z\), for all \(x \in I,\) then \([\delta (x), \sigma (x)] = 0 \), for all \(x \in I.\) Moreover, we have proved some annihilating conditions for algebraic identity involving multiplicative(generalized) derivation.     

On Convex Geometric Graphs with no $$$$ Pairwise Disjoint Edges

A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on n vertices that does not contain \(k+1\) pairwise disjoint edges is kn (provided \(n>2k\)). For \(k=1\) and \(k=n/2-1\), the extremal examples are completely characterized. For all other values of k, the structure of the extremal examples is far from known: their total number is unknown, and only a few classes of examples were presented, that are almost symmetric, consisting roughly of the kn “longest possible” edges of CK(n), the complete CGG of order n. In order to understand further the structure of the extremal examples, we present a class of extremal examples that lie at the other end of the spectrum. Namely, we break the symmetry by requiring that, in addition, the graph admit an independent set that consists of q consecutive vertices on the boundary of the convex hull. We show that such graphs exist as long as \(q \le n-2k\) and that this value of q is optimal. We generalize our discussion to the following question: what is the maximal possible number f(n, k, q) of edges in a CGG on n vertices that does not contain \(k+1\) pairwise disjoint edges, and, in addition, admits an independent set that consists of q consecutive vertices on the boundary of the convex hull? We provide a complete answer to this question, determining f(n, k, q) for all relevant values of n, k and q.     

A geometric approach to alternating <Emphasis Type="Italic">k</Emphasis>-linear forms

Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.     

Algorithms for positive semidefinite factorization

This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an m-by-n nonnegative matrix X and an integer k, the PSD factorization problem consists in finding, if possible, symmetric k-by-k positive semidefinite matrices \(\{A^1,\ldots ,A^m\}\) and \(\{B^1,\ldots ,B^n\}\) such that \(X_{i,j}=\text {trace}(A^iB^j)\) for \(i=1,\ldots ,m\), and \(j=1,\ldots ,n\). PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and two algorithms based on the coordinate descent framework. The main application of PSD factorization is the computation of semidefinite extensions, that is, the representations of polyhedrons as projections of spectrahedra, for which the matrix to be factorized is the slack matrix of the polyhedron. We compare the performance of our algorithms on this class of problems. In particular, we compute the PSD extensions of size \(k=1+ \lceil \log _2(n) \rceil \) for the regular n-gons when \(n=5\), 8 and 10. We also show how to generalize our algorithms to compute the square root rank (which is the size of the factors in a PSD factorization where all factor matrices \(A^i\) and \(B^j\) have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality \(A^i=B^i\) is required for all i).     

<Emphasis Type="Italic">k</Emphasis>-Extreme Points in Symmetric Spaces of Measurable Operators

Let \({\mathcal{M}}\) be a semifinite von Neumann algebra with a faithful, normal, semifinite trace \({\tau}\) and E be a strongly symmetric Banach function space on \({[0,\tau({\bf 1}))}\) . We show that an operator x in the unit sphere of \({E(\mathcal{M}, \tau)}\) is k-extreme, \({k \in {\mathbb{N}}}\) , whenever its singular value function \({\mu(x)}\) is k-extreme and one of the following conditions hold (i) \({\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}\) or (ii) \({n(x)\mathcal{M}n(x^*) = 0}\) and \({|x| \geq \mu(\infty, x)s(x)}\) , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever \({\mathcal{M}}\) is non-atomic. The global k-rotundity property follows, that is if \({\mathcal{M}}\) is non-atomic then E is k-rotund if and only if \(E(\mathcal{M}, \tau)\) is k-rotund. As a consequence of the noncommutative results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement \({\mu(f)}\) is k-extreme and \({|f| \geq \mu(\infty,f)}\) . We conclude with the corollary on orbits Ω(g) and Ω′(g). We get that f is a k-extreme point of the orbit \({\Omega(g),\,g \in L_1 + L_{\infty}}\) , or \({\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}\) , if and only if \({\mu(f) = \mu(g)}\) and \({|f| \geq \mu(\infty, f)}\) . From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.     

Critical vertices in k-connected digraphs

It is proved that every non-complete, finite digraph of connectivity number k has a fragment F containing at most k critical vertices. The following result is a direct consequence: every k-connected, finite digraph D of minimum out- and indegree at least \(2k+ m- 1\) for positive integers k, m has a subdigraph H of minimum outdegree or minimum indegree at least \(m-1\) such that \(D - x\) is k-connected for all \(x \in V(H)\). For \(m = 1\), this implies immediately the existence of a vertex of indegree or outdegree less than 2k in a k-critical, finite digraph, which was proved in Mader (J Comb Theory (B) 53:260–272, 1991).     

Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset P, denoted \(\dim (P)\), is the least positive integer d for which P is the intersection of d linear extensions of P. The maximum dimension of a poset P with \(|P|\le 2n+1\) is n, provided \(n\ge 2\), and this inequality is tight when P contains the standard example \(S_n\). However, there are posets with large dimension that do not contain the standard example \(S_2\). Moreover, for each fixed \(d\ge 2\), if P is a poset with \(|P|\le 2n+1\) and P does not contain the standard example \(S_d\), then \(\dim (P)=o(n)\). Also, for large n, there is a poset P with \(|P|=2n\) and \(\dim (P)\ge (1-o(1))n\) such that the largest d so that P contains the standard example \(S_d\) is o(n). In this paper, we will show that for every integer \(c\ge 1\), there is an integer \(f(c)=O(c^2)\) so that for large enough n, if P is a poset with \(|P|\le 2n+1\) and \(\dim (P)\ge n-c\), then P contains a standard example \(S_d\) with \(d\ge n-f(c)\). From below, we show that \(f(c)={\varOmega }(c^{4/3})\). On the other hand, we also prove an analogous result for fractional dimension, and in this setting f(c) is linear in c. Here the result is best possible up to the value of the multiplicative constant.     

On the Number of Irreducible Points in Polyhedra

An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in n-dimensional polytope P is at most \(O(m^{\lfloor \frac{n}{2}\rfloor }\log ^{n-1}\gamma )\), where n is fixed and P is given by a system of m linear inequalities with integer coefficients not exceeding (by absolute value) \(\gamma \). This bound is tight. Using this result we prove the conjecture asserting that the teaching dimension in the class of threshold functions of k-valued logic in n variables is \(\varTheta (\log ^{n-2} k)\) for any fixed \(n\ge 2\).     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

<Emphasis Type="Italic">M</Emphasis>-curves and symmetric products

Let \((X\, , \sigma )\) be a geometrically irreducible smooth projective M-curve of genus g defined over the field of real numbers. We prove that the n-th symmetric product of \((X\, , \sigma )\) is an M-variety for \(n\,=\,2\, ,3\) and \(n \,\ge \, 2g -1\).