Revisiting Fryszkowski’s problem

Professor Andrzej Fryszkowski formulated, at the 2nd Symposium on Nonlinear Analysis in Toruń, September 13–17, 1999, the following problem: given \(\alpha \in (0,1)\), an arbitrary non-empty set \(\Omega \) and a set-valued mapping \(F:\Omega \rightarrow 2^{\Omega }\), find necessary and (or) sufficient conditions for the existence of a (complete) metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d. Com?neci (Stud. Univ. Babe?-Bolyai Math. 62:537–542, 2017) provided necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d, in case that \(\alpha \in (0,\frac{1}{2})\) and there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) . We improve Com?neci’s result by allowing \(\alpha \) to belong to the interval (0, 1). In addition, we provide necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) such that F is a Nadler set-valued \(\alpha \)-similarity with respect to d, in case that \(\alpha \in (0,1)\), there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) and F is non-overlapping.     

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let \(A_q(n,d)\) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on \(A_q(n,d)\). For any k, let \(\mathcal{C}_k\) be the collection of codes of cardinality at most k. Then \(A_q(n,d)\) is at most the maximum value of \(\sum _{v\in [q]^n}x(\{v\})\), where x is a function \(\mathcal{C}_4\rightarrow {\mathbb {R}}_+\) such that \(x(\emptyset )=1\) and \(x(C)=\!0\) if C has minimum distance less than d, and such that the \(\mathcal{C}_2\times \mathcal{C}_2\) matrix \((x(C\cup C'))_{C,C'\in \mathcal{C}_2}\) is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds \(A_4(6,3)\le 176\), \(A_4(7,3)\le 596\), \(A_4(7,4)\le 155\), \(A_5(7,4)\le 489\), and \(A_5(7,5)\le 87\).     

Ill-Posedness for the <Emphasis Type="Italic">b</Emphasis>-Family of Equations

For \(s<3/2\), it is shown that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces \(H^s\) on both the torus and the line when \(b>1\). The proof of ill-posedness depends on the value of b, where the most interesting case arises for \(b=3\), the Degasperis–Procesi equation. Considering that the b-family of equations is locally well-posed in \(H^s\) for \(s>3/2\), this work establishes 3 / 2 as the critical index of well-posedness in Sobolev spaces for \(b>1\).     

Hermitian rank distance codes

Let \(X=X(n,q)\) be the set of \(n\times n\) Hermitian matrices over \(\mathbb {F}_{q^2}\). It is well known that X gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study d-codes in this scheme, namely subsets Y of X with the property that, for all distinct \(A,B\in Y\), the rank of \(A-B\) is at least d. We prove bounds on the size of a d-code and show that, under certain conditions, the inner distribution of a d-code is determined by its parameters. Except if n and d are both even and \(4\le d\le n-2\), constructions of d-codes are given, which are optimal among the d-codes that are subgroups of \((X,+)\). This work complements results previously obtained for several other types of matrices over finite fields.     

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.     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

Motivated by stochastic 0–1 integer programming problems with an expected utility objective, we study the mixed-integer nonlinear set: \(P = \big \{(w,x)\in \mathbb {R}\times \left\{ 0,1\right\} ^N: w \le f(a'x + d), b'x \le B\big \}\) where N is a positive integer, \(f:\mathbb {R}\mapsto \mathbb {R}\) is a concave function, \(a, b \in \mathbb {R}^N\) are nonnegative vectors, d is a real number and B is a positive real number. We propose a family of inequalities for the convex hull of P by exploiting submodularity of the function \(f(a'x + d)\) over \(\{0,1\}^N\) and the knapsack constraint \(b'x \le B\). Computational effectiveness of the proposed inequalities within a branch-and-cut framework is illustrated using instances of an expected utility capital budgeting problem.     

The spectral excess theorem for distance-regular graphs having distance-<Emphasis Type="Italic">d</Emphasis> graph with fewer distinct eigenvalues

Let \(\Gamma \) be a distance-regular graph with diameter d and Kneser graph \(K=\Gamma _d\), the distance-d graph of \(\Gamma \). We say that \(\Gamma \) is partially antipodal when K has fewer distinct eigenvalues than \(\Gamma \). In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues) and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with \(d+1\) distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a more general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.     

On the estimates of the upper and lower bounds of Ramanujan primes

For \(n\ge 1\), the nth Ramanujan prime is defined as the least positive integer \(R_{n}\) such that for all \(x\ge R_{n}\), the interval \((\frac{x}{2}, x]\) has at least n primes. Let \(p_{i}\) be the ith prime and \(R_{n}=p_{s}\). Sondow, Laishram, and other scholars gave a series of upper bounds of s. In this paper we establish several results giving estimates of upper and lower bounds of Ramanujan primes. Using these estimates, we discuss a conjecture on Ramanujan primes of Sondow–Nicholson–Noe and prove that if \(n>10^{300}\), then \(\pi (R_{mn})\le m\pi (R_{n})\) for \(m\ge 1\).     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

Spectral multiplier theorems and averaged <Emphasis Type="Italic">R</Emphasis>-boundedness

Let A be a 0-sectorial operator with a bounded \(H^\infty (\Sigma _\sigma )\)-calculus for some \(\sigma \in (0,\pi ),\) e.g. a Laplace type operator on \(L^p(\Omega ),\, 1< p < \infty ,\) where \(\Omega \) is a manifold or a graph. We show that A has a \(\mathcal {H}^\alpha _2(\mathbb {R}_+)\) Hörmander functional calculus if and only if certain operator families derived from the resolvent \((\lambda - A)^{-1},\) the semigroup \(e^{-zA},\) the wave operators \(e^{itA}\) or the imaginary powers \(A^{it}\) of A are R-bounded in an \(L^2\)-averaged sense. If X is an \(L^p(\Omega )\) space with \(1 \le p < \infty \), R-boundedness reduces to well-known estimates of square sums.     

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.     

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.     

<Emphasis Type="Italic">X</Emphasis>-coordinates of Pell equations as sums of two tribonacci numbers

In this paper, we find all positive squarefree integers d such that the Pell equation \(X^2 - dY^2 = \pm 1\) has at least two positive integer solutions (X, Y) and \((X^{\prime },Y^{\prime })\) such that both X and \(X^{\prime }\) are sums of two Tribonacci numbers.     

On the semistability of certain Lazarsfeld–Mukai bundles on abelian surfaces

Let \(X=\mathscr {J}(\widetilde{\mathscr {C}})\), the Jacobian of a genus 2 curve \(\widetilde{\mathscr {C}}\) over \({\mathbb {C}}\), and let Y be the associated Kummer surface. Consider an ample line bundle \(L=\mathscr {O}(m\widetilde{\mathscr {C}})\) on X for an even number m, and its descent to Y, say \(L'\). We show that any dominating component of \({\mathscr {W}}^1_{d}(|L'|)\) corresponds to \(\mu _{L'}\)-stable Lazarsfeld–Mukai bundles on Y. Further, for a smooth curve \(C\in |L|\) and a base-point free \(g^1_d\) on C, say (A, V), we study the \(\mu _L\)-semistability of the rank-2 Lazarsfeld–Mukai bundle associated to (C, (A, V)) on X. Under certain assumptions on C and the \(g^1_d\), we show that the above Lazarsfeld–Mukai bundles are \(\mu _L\)-semistable.     

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 \(p_{n,k}={n\atopwithdelims ()k}(1-o(1))\).     

<Emphasis Type="Italic">b</Emphasis>-Generalized Skew Derivations on Lie Ideals

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, \(F\ne 0\) an b-generalized skew derivation of R, L a non-central Lie ideal of R, \(0\ne a\in R\) and \(n\ge 1\) a fixed integer. In this paper, we prove the following two results:

1. 1.
   If R has characteristic different from 2 and 3 and \(a[F(x),x]^n=0\), for all \(x\in L\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\), the standard identity of degree 4, and there exist \(\lambda \in C\) and \(b\in Q\), such that \(F(x)=bx+xb+\lambda x\), for all \(x\in R\).
    
2. 2.
   If \(\mathrm{{char}}(R)=0\) or \(\mathrm{{char}}(R) > n\) and \(a[F(x),x]^n\in Z(R)\), for all \(x\in R\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\).
    

     

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.     

Spectral Properties of <Emphasis Type="Italic">k</Emphasis>-Quasi-<Emphasis Type="Italic">M</Emphasis>-hyponormal Operators

In this paper, we study the k-quasi-M-hyponormal operator and mainly prove that if T is a k-quasi-M-hyponormal operator, then \(\sigma _{ja}(T)\backslash \{0\}=\sigma _{a}(T)\backslash \{0\}\), and the spectrum is continuous on the class of all k-quasi-M-hyponormal operators; let \(d_{AB}\in B(B(H))\) denote either the generalized derivation \(\delta _{AB}= L_{A}-R_{B}\) or the elementary operator \(\Delta _{AB} =L_{A}R_{B}- I\), we show that if A and \(B^{*}\) are k-quasi-M-hyponormal operators, then \(d_{AB}\) is polaroid and generalized Weyl’s theorem holds for \(f(d_{AB})\), where f is an analytic function on \(\sigma (d_{AB})\) and f is not constant on each connected component of the open set U containing \(\sigma (d_{AB})\). In additon, we discuss the hyperinvariant subspace problem for k-quasi-M-hyponormal operators.     

On the number of monic integer polynomials with given signature

In this paper, we show that the number of monic integer polynomials of degree \(d \ge 1\) and height at most H which have no real roots is between \(c_1H^{d-1/2}\) and \(c_2 H^{d-1/2}\), where the constants \(c_2>c_1>0\) depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying \(0 \le s < d/2\) we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to \(\lambda (d,s)H^{d}\) as \(H \rightarrow \infty \). The constants \(\lambda (d,s)\) are all positive and come from a recent paper of Bertók, Hajdu, and Peth?. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.