A characterization of $$A_5$$ by same-order type

For a group G, write \(g \sim h\) if \(g, h \in G\) have the same order. The set of sizes of the equivalence classes with respect to this relation is called the same-order type of G; thus it is the set with numbers of elements of each order. In this article we prove that a group is isomorphic to the alternating group \(A_5\) if and only if the same-order type of G is \(\{1,pq,4p,8q\}\) with the p and q primes.     

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).     

Every 3-connected $$\{K_{1,3},N_{1,2,3}\}$$-free graph is Hamilton-connected

A graph G is \(\{X,Y\}\)-free if it contains neither X nor Y as an induced subgraph. Pairs of connected graphs X, Y such that every 3-connected \(\{X,Y\}\)-free graph is Hamilton-connected have been investigated recently in (2002, 2000, 2012). In this paper, it is shown that every 3-connected \(\{K_{1,3},N_{1,2,3}\}\)-free graph is Hamilton-connected, where \(N_{1,2,3}\) is the graph obtained by identifying end vertices of three disjoint paths of lengths 1, 2, 3 to the vertices of a triangle.     

Existence Criteria and Expressions of the (<Emphasis Type="Italic">b</Emphasis>, <Emphasis Type="Italic">c</Emphasis>)-Inverse in Rings and Their Applications

Let R be a ring. Existence criteria for the (b, c)-inverse are given. We present explicit expressions for the (b, c)-inverse by using inner inverses. We answer the question when the (b, c)-inverse of \(a\in R\) is an inner inverse of a. As applications, we give a unified theory of some well-known results of the \(\{1,3\}\)-inverse, the \(\{1,4\}\)-inverse, the Moore–Penrose inverse, the group inverse and the core inverse.     

A spectral-like decomposition for transitive Anosov flows in dimension three

Given a (transitive or non-transitive) Anosov vector field X on a closed three dimensional manifold M, one may try to decompose (M, X) by cutting M along tori and Klein bottles transverse to X. We prove that one can find a finite collection \(\{S_1,\dots ,S_n\}\) of pairwise disjoint, pairwise non-parallel tori and Klein bottles transverse to X, such that the maximal invariant sets \(\Lambda _1,\dots ,\Lambda _m\) of the connected components \(V_1,\dots ,V_m\) of \(M-(S_1\cup \dots \cup S_n)\) satisfy the following properties:

• each \(\Lambda _i\) is a compact invariant locally maximal transitive set for X;
• the collection \(\{\Lambda _1,\dots ,\Lambda _m\}\) is canonically attached to the pair (M, X) (i.e. it can be defined independently of the collection of tori and Klein bottles \(\{S_1,\dots ,S_n\}\));
• the \(\Lambda _i\)’s are the smallest possible: for every (possibly infinite) collection \(\{S_i\}_{i\in I}\) of tori and Klein bottles transverse to X, the \(\Lambda _i\)’s are contained in the maximal invariant set of \(M-\cup _i S_i\).

To a certain extent, the sets \(\Lambda _1,\dots ,\Lambda _m\) are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition \(V_1,\dots ,V_m\), equipped with the restriction of the Anosov vector field X, are “almost unique up to topological equivalence”.

     

On a theorem of Rigby

Let \({\Delta = BAG(2, q)}\) denote the classical biaffine plane of order q, that is, the symmetric \({((q^2 - 1)_q)}\) configuration obtained from the classical affine plane \({\Sigma = AG(2, q)}\) of order q by omitting a point of \({\Sigma}\) together with all lines through this point. Now let \({q \geq 4}\) be a power of a prime p and assume that \({\Delta}\) admits an embedding into the projective plane \({\Pi = PG(2, F)}\), where F is a (not necessarily commutative) field. Then this embedding extends to a projective subplane \({\Pi_0 \cong PG(2, q)}\) of \({\Pi}\); in particular, F has characteristic p. Consequently, \({BAG(2, q)}\) with \({q\geq 4}\) admits an embedding into \({PG(2, q')}\) if only if q′ is a power of q. This strengthens a result of Rigby (Canad J Math 17:977–1009, 1965) in a special case while simultaneously providing a more elegant proof.     

A note on the size of the set $$\varvec{A^2+A}$$

Let \(F(x,y,z)=xy+z\). We consider some properties of expansion of the polynomial F in different settings, namely in the integers and in prime fields. The main results concern the question of covering \(\{0,1,\ldots , N\}\) (resp. \(\mathbf {F}_p\)) by \(A^2+A\) with some thin sets A.     

Sequences of Weighted Composition Operators on the Hardy–Hilbert Space

Let \(\varphi \) be an analytic self map of the open unit disc \(\mathbb {D}\). Assume that \(\psi \) is an analytic map of \(\mathbb {D}\). Suppose that f is in the Hardy–Hilbert space of the open unit disc \(H^2\). The operator that takes f into \(\psi \cdot f \circ \varphi \) is a weighted composition operator, and is denoted by \(C_{\psi ,\varphi }\). In this paper we relate the convergence of the sequence \(\{ C_{\psi _n,\varphi _n}\}\) in different operator topologies to the convergence of the two sequences of maps \(\{\varphi _n \}\) and \(\{ \psi _n\}\).     

A note on Huijsmans-de Pagter problem in ordered Banach algebras

We give an example of a positive element a in some ordered Banach algebra \(\mathcal A\) such that its spectrum is equal to \(\{1\}\) and it is not greater than or equal to the unit element of \(\mathcal A\).     

Integral Modular Categories of Frobenius-Perron Dimension <Emphasis Type="Italic">pq</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Integral modular categories of Frobenius-Perron dimension pq ⁿ , where p and q are primes, are considered. It is already known that such categories are group-theoretical in the cases of 0 ≤ n ≤ 4. In the general case we determine that these categories are either group-theoretical or contain a Tannakian subcategory of dimension q ⁱ for i > 1. We then show that all integral modular categories \(\mathcal {C}\) with \(\text {FPdim}(\mathcal {C})=pq^{5}\) are group-theoretical, and, if in addition p < q, all with \(\text {FPdim}(\mathcal {C})=pq^{6}\) or pq ⁷ are group-theoretical. In the process we generalize an existing criterion for an integral modular category to be group-theoretical.     

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 Dual Form of ‘Low <Emphasis Type="Italic">M</Emphasis><Superscript>*</Superscript> Estimate’ in the Quasi-convex Case

Let\(B_{2}^{n}\) denote the Euclidean ball in\({\mathbb R}^n\), and, given closed star-shaped body\(K \subset {\mathbb R}^{n}, M_{K}\) denote the average of the gauge of K on the Euclidean sphere. Let\(p \in (0,1)\) and let\(K \subset {\mathbb R}^{n}\) be a p-convex body. In [17] we proved that for every\(\lambda \in (0,1)\) there exists an orthogonal projection P of rank\((1 - \lambda)n\) such that

$\frac{f(\lambda)}{M_K} PB^{n}_{2} \subset PK,$

where\(f(\lambda)=c_p\lambda^{1+1/p}\) for some positive constant c _p depending on p only. In this note we prove that\(f(\lambda)\) can be taken equal to\(C_p\lambda^{1/p-1/2}\). In terms of Kolmogorov numbers it means that for every\(k \leq n\)

$d_k (\hbox{Id}:\ell^{n}_{2} \to ({\mathbb R}^{n},\|\cdot\|_{K})) \leq C_p \frac{n^{1/p-1}}{k^{1/p-1/2}} \ell(\hbox{ID}: \ell^{n}_{2} \to ({\mathbb R}^{n}, \|\cdot\|_{K})),$

where\(\ell(\hbox{Id})={\bf E}\|\sum\limits^{n}_{i=1}g_i e_i\|_K\) for the independent standard Gaussian random variables\(\{g_i\}\) and the canonical basis\(\{e_i\}\) of\({\mathbb R}^n\). All results do not require the symmetry of K.

     

A generalization of the normal rational curve in $$\mathop {\mathrm{PG}}(d,q^n)$$ and its associated non-linear MRD codes

Let A and B be two points of \(\mathop {\mathrm{PG}}(d,q^n)\) and let \(\Phi \) be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if \(d=2\) or \(d\ge 3\) and the lines \(\Phi ^{-1}(AB), AB, \Phi (AB) \) are not in a common plane, then the set \(\mathcal{C}\) of points of intersection of corresponding lines under \(\Phi \) is the union of \(q-1\) scattered \({\mathbb {F}}_{q}\)-linear sets of rank n together with \(\{A,B\}\). As an application we will construct, starting from the set \(\mathcal{C}\), infinite families of non-linear \((d+1, n, q;d-1)\)-MRD codes, \(d\le n-1\), generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).     

On complementary dual quasi-twisted codes

A linear complementary-dual (LCD) code C is a linear code whose dual code \(C^{\perp }\) satisfies \(C \cap C^{\perp }=\{0\}\). In this work we characterize some classes of LCD q-ary \((\lambda , l)\)-quasi-twisted (QT) codes of length \(n=ml\) with \((m,q)=1\), \(\lambda \in F_{q} \setminus \{0\}\) and \(\lambda \ne \lambda ^{-1}\). We show that every \((\lambda ,l)\)-QT code C of length \(n=ml\) with \(dim(C)<m\) or \(dim(C^{\perp })<m\) is an LCD code. A sufficient condition for r-generator QT codes is provided under which they are LCD. We show that every maximal 1-generator \((\lambda ,l)\)-QT code of length \(n=ml\) with \(l>2\) is either an LCD code or a self-orthogonal code and a sufficient condition for this family of codes is given under which such a code C is LCD. Also it is shown that every maximal 1-generator \((\lambda ,2)\)-QT code is LCD. Several good and optimal LCD QT codes are presented.     

Toeplitz Operators from One Fock Space to Another

In this paper, we study Toeplitz operators T _μ from one Fock space \({F^{p}_{\alpha}}\) to another \({F^{q}_{\alpha}}\) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of \({T_\mu: F^{p}_{\alpha} \to F^{q}_{\alpha}}\) in terms of the averaging function \({\widehat{\mu}_r}\) and the t-Berezin transform \({\widetilde{\mu}_t}\) respectively. Quite differently from the Bergman space case, we show that T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for some p ≤ q if and only if T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for all p ≤ q. In order to prove our main results on T _μ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C ⁿ .     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

Iterated local search with Trellis-neighborhood for the partial Latin square extension problem

A partial Latin square (PLS) is a partial assignment of n symbols to an \(n\times n\) grid such that, in each row and in each column, each symbol appears at most once. The PLS extension problem is an NP-hard problem that asks for a largest extension of a given PLS. We consider the local search such that the neighborhood is defined by (p, q)-swap , i.e., the operation of dropping exactly p symbols and then assigning symbols to at most q empty cells. As a fundamental result, we provide an efficient \((p,\infty )\)-neighborhood search algorithm that finds an improved solution or concludes that no such solution exists for \(p\in \{1,2,3\}\). The running time of the algorithm is \(O(n^{p+1})\). We then propose a novel swap operation, Trellis-swap, which is a generalization of (p, q)-swap with \(p\le 2\). The proposed Trellis-neighborhood search algorithm runs in \(O(n^{3.5})\) time. The iterated local search (ILS) algorithm with Trellis-neighborhood is more likely to deliver a high-quality solution than not only ILSs with \((p,\infty )\)-neighborhood but also state-of-the-art optimization solvers such as IBM ILOG CPLEX and LocalSolver.     

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

We study inverse scattering problems at a fixed energy for radial Schrödinger operators on \({\mathbb{R}^n}\), \({n \geq 2}\). First, we consider the class \({\mathcal{A}}\) of potentials q(r) which can be extended analytically in \({\Re z \geq 0}\) such that \({\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}}\), \({\rho > \frac{3}{2}}\). If q and \({\tilde{q}}\) are two such potentials and if the corresponding phase shifts \({\delta_l}\) and \({\tilde{\delta}_l}\) are super-exponentially close, then \({q=\tilde{q}}\). Second, we study the class of potentials q(r) which can be split into q(r) = q ₁(r) + q ₂(r) such that q ₁(r) has compact support and \({q_2 (r) \in \mathcal{A}}\). If q and \({\tilde{q}}\) are two such potentials, we show that for any fixed \({a>0, {\delta_l - \tilde{\delta}_l \ = \ o \left(\frac{1}{l^{n-3}}\ \left({\frac{ae}{2l}}\right)^{2l}\right)}}\) when \({l \rightarrow +\infty}\) if and only if \({q(r)=\tilde{q}(r)}\) for almost all \({r \geq a}\). The proofs are close in spirit with the celebrated Borg–Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in \({\Re z \geq 0}\) with \({\mid q(z)\mid \leq C (1+ \mid z \mid)^{-\rho}}\), \({\rho >1}\), we show that the Regge poles are confined in a vertical strip in the complex plane.     

On the asymptotics of supremum distribution for some iterated processes

In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes \(\{X(Y(t)) : t \in [0, \infty )\}\), where \(\{X(t) : t \in \mathbb {R} \}\) is a centered Gaussian process and \(\{Y(t): t \in [0, \infty )\}\) is an independent of {X(t)} stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of \(\mathbb {P}(\sup _{s\in [0,T]} X(Y (s)) > u)\) as \(u \to \infty \), where T>0, as well as \(\lim _{u\to \infty } \mathbb {P}(\sup _{s\in [0,h(u)]} X(Y (s)) > u)\), for some suitably chosen function h(u) are analyzed. As an illustration, we study the asymptotic behavior of the supremum distribution of iterated fractional Brownian motion process.     

On integer programming with bounded determinants

Let A be an \((m \times n)\) integral matrix, and let \(P=\{ x :A x \le b\}\) be an n-dimensional polytope. The width of P is defined as \( w(P)=min\{ x\in \mathbb {Z}^n{\setminus }\{0\} :max_{x \in P} x^\top u - min_{x \in P} x^\top v \}\). Let \(\varDelta (A)\) and \(\delta (A)\) denote the greatest and the smallest absolute values of a determinant among all \(r(A) \times r(A)\) sub-matrices of A, where r(A) is the rank of the matrix A. We prove that if every \(r(A) \times r(A)\) sub-matrix of A has a determinant equal to \(\pm \varDelta (A)\) or 0 and \(w(P)\ge (\varDelta (A)-1)(n+1)\), then P contains n affine independent integer points. Additionally, we present similar results for the case of k-modular matrices. The matrix A is called totally k-modular if every square sub-matrix of A has a determinant in the set \(\{0,\, \pm k^r :r \in \mathbb {N} \}\). When P is a simplex and \(w(P)\ge \delta (A)-1\), we describe a polynomial time algorithm for finding an integer point in P.