On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

Ramanujan-type congruences for overpartitions modulo 16

Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\).     

On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.     

Congruences for $$\ell $$-regular overpartitions and Andrews’ singular overpartitions

Let \(\overline{A}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In a recent paper, Shen calls the overpartitions enumerated by the function \(\overline{A}_{\ell }(n)\) as \(\ell \)-regular overpartitions. In this paper, we find certain congruences for \(\overline{A}_{\ell }(n)\), when \(\ell =4, 8\), and 9. Recently, Andrews introduced the partition function \(\overline{C}_{k, i}(n)\), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i\pmod {k}\) may be over-lined. He also proved that \(\overline{C}_{3, 1}(9n+3)\) and \(\overline{C}_{3, 1}(9n+6)\) are divisible by 3. In this paper, we prove that \(\overline{C}_{3, 1}(12n+11)\) is divisible by 144 which was conjectured to be true by Naika and Gireesh.     

Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

Let \(\bar{p}(n)\) denote the number of overpartitions of n. Fortin et al. and Hirschhorn and Sellers established some congruences modulo powers of 2 for \(\bar{p}(n)\). Recently, Xia and Yao found several congruences modulo powers of 2 and 3. In particular, they proved that \(\bar{p}(96n+12)\equiv 0 \ (\mathrm{mod}\ 9)\) and \(\bar{p}(24n+19)\equiv 0\ (\mathrm{mod\ }27)\). In this paper, we generalize the two congruences and establish several new infinite families of congruences modulo 9 and 27 for \(\bar{p}(n)\). Furthermore, we prove some strange congruences modulo 9 and 27 for \(\bar{p}(n)\) by employing some results due to Cooper et al. For example, we prove that for \(k\ge 0\), \(\bar{p}(4^{k+1})\equiv 2^{k+3}+6(-1)^k\ (\mathrm{mod} \ 27) \) and \(\bar{p}\left( 7^{2k}\right) \equiv 2-2k\ (\mathrm{mod}\ 9)\). We also present two conjectures on congruences for \(\bar{p}(n)\).     

Some new Kirkman signal sets

A decomposition of the blocks of an \(\textsf {STS}(v)\) into partial parallel classes of size m is equivalent to a Kirkman signal set \(\textsf {KSS}(v,m)\). We give decompositions of \(\textsf {STS}(4v-3)\) into classes of size \(v-1\) when \(v \equiv 3 \pmod {6}\), \(v \not = 3\). We also give decompositions of \(\textsf {STS}(v)\) into classes of various sizes when v is a product of two arbitrary integers that are both congruent to \(3 \pmod {6}\). These results produce new families of \(\textsf {KSS}(v,m)\).     

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Measures under the flat norm as ordered normed vector space

The space of real Borel measures \(\mathcal {M}(S)\) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone \(\mathcal {M}_+(S)\) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of \(\mathcal {M}_+(S)\) are compact and semiflows on \(\mathcal {M}_+(S)\) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because \(\mathcal {M}(S)\) is rarely complete and \(\mathcal {M}_+(S)\) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on \(\mathcal {M}_+(S)\) and continuous semiflows. Both topics prepare for a dynamical systems theory on \(\mathcal {M}_+(S)\).     

Maximal essential extensions in the context of frames

We show that every frame can be essentially embedded in a Boolean frame, and that this embedding is the maximal essential extension of the frame in the sense that it factors uniquely through any other essential extension. This extension can be realized as the embedding \(L \rightarrow \mathcal {N}(L) \rightarrow \mathcal {B}\mathcal {N}(L)\), where \(L \rightarrow \mathcal {N}(L)\) is the familiar embedding of L into its congruence frame \(\mathcal {N}(L)\), and \(\mathcal {N}(L) \rightarrow \mathcal {B}\mathcal {N}(L)\) is the Booleanization of \(\mathcal {N}(L)\). Finally, we show that for subfit frames the extension can also be realized as the embedding \(L \rightarrow {{\mathrm{S}}}_\mathfrak {c}(L)\) of L into its complete Boolean algebra \({{\mathrm{S}}}_\mathfrak {c}(L)\) of sublocales which are joins of closed sublocales.     

Constructive Approximation in de Branges–Rovnyak Spaces

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the de Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive side, we prove the following approximation theorem for Toeplitz operators on general de Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are dense in \(\mathcal{H}(b)\).     

Green rings of weak Hopf algebras based on generalized Taft algebras

In this paper, we first introduce two classes of weak Hopf algebras \(\mathfrak {w}^s_{n,d}\) of type \(s ~(s=0, 1)\), which are corresponding to generalized Taft Hopf algebras \(H_{n,d}(q)\). Then we compute their Green rings, and describe the structure of their Green rings by the generators and the generating relations. These results extend some known ones about Taft algebras and generalized Taft algebras.     

The Finite Hankel Transform Operator: Some Explicit and Local Estimates of the Eigenfunctions and Eigenvalues Decay Rates

For fixed real numbers \(c>0,\)\(\alpha >-\frac{1}{2},\) the finite Hankel transform operator, denoted by \(\mathcal {H}_c^{\alpha }\) is given by the integral operator defined on \(L^2(0,1)\) with kernel \(K_{\alpha }(x,y)= \sqrt{c xy} J_{\alpha }(cxy).\) To the operator \(\mathcal {H}_c^{\alpha },\) we associate a positive, self-adjoint compact integral operator \(\mathcal Q_c^{\alpha }=c\, \mathcal {H}_c^{\alpha }\, \mathcal {H}_c^{\alpha }.\) Note that the integral operators \(\mathcal {H}_c^{\alpha }\) and \(\mathcal Q_c^{\alpha }\) commute with a Sturm-Liouville differential operator \(\mathcal D_c^{\alpha }.\) In this paper, we first give some useful estimates and bounds of the eigenfunctions \(\varphi ^{(\alpha )}_{n,c}\) of \(\mathcal H_c^{\alpha }\) or \(\mathcal Q_c^{\alpha }.\) These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator \(\mathcal D_c^{\alpha }.\) If \((\mu _{n,\alpha }(c))_n\) and \(\lambda _{n,\alpha }(c)=c\, |\mu _{n,\alpha }(c)|^2\) denote the infinite and countable sequence of the eigenvalues of the operators \(\mathcal {H}_c^{(\alpha )}\) and \(\mathcal Q_c^{\alpha },\) arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer \(n\ge 0,\)\(\lambda _{n,\alpha }(c)\) is decreasing with respect to the parameter \(\alpha .\) As a consequence, we show that for \(\alpha \ge \frac{1}{2},\) the \(\lambda _{n,\alpha }(c)\) and the \(\mu _{n,\alpha }(c)\) have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.     

A Categorification of $\displaystyle {\mathfrak q} (2)$-Crystals

We provide a categorification of \(\mathfrak {q}(2)\)-crystals on the singular \(\mathfrak {gl}_{n}\)-category \({\mathcal O}_{n}\). Our result extends the \(\mathfrak {gl}_{2}\)-crystal structure on \(\text {Irr} ({\mathcal O}_{n})\) induced from the work of Bernstein-Frenkel-Khovanov. Further properties of the \({\mathfrak q}(2)\)-crystal \(\text {Irr} ({\mathcal O}_{n})\) are also discussed.     

A relative <Emphasis Type="Italic">m</Emphasis>-cover of a Hermitian surface is a relative hemisystem

An m-cover of the Hermitian surface \(\mathrm {H}(3,q^2)\) of \(\mathrm {PG}(3,q^2)\) is a set \(\mathcal {S}\) of lines of \(\mathrm {H}(3,q^2)\) such that every point of \(\mathrm {H}(3,q^2)\) lies on exactly m lines of \(\mathcal {S}\), and \(0<m<q+1\). Segre (Annali di Matematica Pura ed Applicata Serie Quarta 70:1–201, 1965) proved that if q is odd, then \(m=(q+1)/2\), and called such a set \(\mathcal {S}\) of lines a hemisystem. Penttila and Williford (J Comb Theory Ser A 118(2):502–509, 2011) introduced the notion of a relative hemisystem of a generalised quadrangle \(\varGamma \) with respect to a subquadrangle \(\varGamma '\): a set of lines \(\mathcal {R}\) of \(\varGamma \) disjoint from \(\varGamma '\) such that every point P of \(\varGamma \setminus \varGamma '\) has half of its lines (disjoint from \(\varGamma '\)) lying in \(\mathcal {R}\). In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of generalised quadrangles of order \((q^2,q)\) with respect to a subquadrangle and proving that m must be q / 2 when the subquadrangle is doubly subtended. In particular, a relative m-cover of \(\mathrm {H}(3,q^2)\) with respect to a symplectic subgeometry \(\mathrm {W}(3,q)\) is a relative hemisystem.     

On Irreducible Representations of the Zassenhaus Superalgebras with p-Characters of Height 0

Let n be a positive integer, and \(\mathfrak {A}(n)=\mathbb {F}[x]/(x^{p^{n}})\), the divided power algebra over an algebraically closed field \(\mathbb {F}\) of prime characteristic p >?2. Let π(n) be the tensor product of \(\mathfrak {A}(n)\) and the Grassmann superalgebra \(\bigwedge (1)\) in one variable. The Zassenhaus superalgebra \(\mathcal {Z}(n)\) is defined to be the Lie superalgebra of the special super derivations of the superalgebra π(n). In this paper we study simple modules over the Zassenhaus superalgebra \(\mathcal {Z}(n)\) with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is obtained.     

On symplectic semifield spreads of $$\hbox {PG}(5, q^2)$$PG(5, q2), q even

We prove that the only symplectic semifield spreads of \(\hbox {PG}(5,q^2)\), \(q\ge 2^{14}\) even, whose associated semifield has center containing \({\mathbb F}_q\), is the Desarguesian spread, by proving that the only \({\mathbb F}_q\)-linear set of rank 6 disjoint from the secant variety of the Veronese surface of \(\hbox {PG}(5,q^2)\) is a plane with three points of the Veronese surface of \(\hbox {PG}(5,q^6){\setminus } \hbox {PG}(5,q^2)\).     

Overpartition function modulo powers of 2

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, congruences modulo powers of 2 for \(\overline{p}(n)\) were widely studied. In this paper, we prove several new infinite families of congruences modulo powers of 2 for \(\overline{p}(n)\). For example, for \(\alpha \ge 1\) and \(n\ge 0\),

$$\begin{aligned} \overline{p}(8\cdot 3^{4\alpha +4}n+5\cdot 3^{4\alpha +3})\equiv 0 \quad (\mathrm{mod}\,\,{2^8}). \end{aligned}$$

     

<Emphasis Type="Italic">q</Emphasis>-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form \(C_{R,T}(S)=RST\) to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map \(C_{R}(S)=RSR^*\) is shown to be q-frequently hypercyclic on the space \(\mathcal {K}(H)\) of all compact operators and the real topological vector space \(\mathcal {S}(H)\) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for \(C_{R,T}\) to be q-frequently hypercyclic on the Schatten von Neumann classes \(S_p(H)\). We also characterize frequent hypercyclicity of \(C_{M^*_\varphi ,M_\psi }\) on the trace-class of the Hardy space, where the symbol \(M_\varphi \) denotes the multiplication operator associated to \(\varphi \).     

Negative Ricci curvature on some non-solvable Lie groups

We show that for any non-trivial representation \((V, \pi )\) of \(\mathfrak {u}(2)\) with the center acting as multiples of the identity, the semidirect product \(\mathfrak {u}(2) \ltimes _\pi V\) admits a metric with negative Ricci curvature that can be explicitly obtained. It is proved that \(\mathfrak {u}(2) \ltimes _\pi V\) degenerates to a solvable Lie algebra that admits a metric with negative Ricci curvature. An n-dimensional Lie group with compact Levi factor \(\mathrm {SU}(2)\) admitting a left invariant metric with negative Ricci is therefore obtained for any \(n \ge 7\).