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 \(n\equiv 0\pmod {4}\).     

Arithmetic properties of 7-regular partitions

Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\),

$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$

We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).

     

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 \(\mathcal{H}(b)\).     

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.     

Overpartitions and ternary quadratic forms

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, Mahlburg showed that \(\overline{p}(n) \equiv 0 \pmod {64}\) and Kim showed that \(\overline{p}(n) \equiv 0 \pmod {128}\) for almost all integers n. In this paper, with the help of some ternary quadratic forms, we prove that \(\overline{p}(n) \equiv 0 \pmod {256}\) for almost all integers n, which was conjectured by Mahlburg.     

Some results on generalized strong external difference families

A generalized strong external difference family (briefly \((v, m; k_1,\dots ,k_m; \lambda _1,\dots ,\lambda _m)\)-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we give some nonexistence results for GSEDFs. In particular, we prove that a \((v, 3;k_1,k_2,k_3; \lambda _1,\lambda _2,\lambda _3)\)-GSEDF does not exist when \(k_1+k_2+k_3< v\). We also give a first recursive construction for GSEDFs and prove that if there is a \((v,2;2\lambda ,\frac{v-1}{2};\lambda ,\lambda )\)-GSEDF, then there is a \((vt,2;4\lambda ,\frac{vt-1}{2};2\lambda ,2\lambda )\)-GSEDF with \(v>1\), \(t>1\) and \(v\equiv t\equiv 1\pmod 2\). Then we use it to obtain some new GSEDFs for \(m=2\). In particular, for any prime power q with \(q\equiv 1\pmod 4\), we show that there exists a \((qt, 2;(q-1)2^{n-1},\frac{qt-1}{2};(q-1)2^{n-2},(q-1)2^{n-2})\)-GSEDF, where \(t=p_1p_2\dots p_n\), \(p_i>1\), \(1\le i\le n\), \(p_1, p_2,\dots ,p_n\) are odd integers.     

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$-modules

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in (0,1)\cup (1,+\,\infty )\). As an application, we prove that for an adjontable operator A if \(\mathscr {R}(A)\) is nonclosed, then \(\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty \). Finally, we show that for an adjointable operator A if \( \overline{\mathscr {R}(A^*) } \) is orthogonally complemented in \( \mathscr {E} \), then under certain coditions there exists an idempotent C and a unique operator X such that \( XAX=X, AXA=CA, AX=C \) and \( XA=P_{A^*} \), where \( P_{A^*} \) is the orthogonal projection of \( \mathscr {E} \) onto \( \overline{\mathscr {R}(A^*)}\).     

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 \(\bar{p}(n)\).     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.     

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.     

Improved upper bounds for partial spreads

A partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) is a collection of \((k-1)\)-dimensional subspaces with trivial intersection. So far, the maximum size of a partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) was known for the cases \(n\equiv 0\pmod k\), \(n\equiv 1\pmod k\), and \(n\equiv 2\pmod k\) with the additional requirements \(q=2\) and \(k=3\). We completely resolve the case \(n\equiv 2\pmod k\) for the binary case \(q=2\).     

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 7 and 49-regular partitions modulo powers of 7

Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$

and

$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$

     

The crank moments weighted by the parity of cranks

In this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\). When \(k=0\), the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\), where \(M_e(n)\) (resp. \(M_o(n)\)) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\).     

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 \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

Inequalities and infinite product formula for Ramanujan generalized modular equation function

We present several inequalities for the Ramanujan generalized modular equation function \(\mu _{a}(r)=\pi F(a,1-a;1;1-r^2)/\) \([2\sin (\pi a)F(a,1-a;1;r^2)]\) with \(a\in (0,1/2]\) and \(r\in (0,1)\), and provide an infinite product formula for \(\mu _{1/4}(r)\), where \(F(a,b;c;x)={}_{2}F_{1}(a,b;c;x)\) is the Gaussian hypergeometric function.     

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 \(n \ge 7\).     

New upper bounds for parent-identifying codes and traceability codes

In the last two decades, parent-identifying codes and traceability codes are introduced to prevent copyrighted digital data from unauthorized use. They have important applications in the scenarios like digital fingerprinting and broadcast encryption schemes. A major open problem in this research area is to determine the upper bounds for the cardinalities of these codes. In this paper we will focus on this theme. Consider a code of length N which is defined over an alphabet of size q. Let \(M_{IPPC}(N,q,t)\) and \(M_{TA}(N,q,t)\) denote the maximal cardinalities of t-parent-identifying codes and t-traceability codes, respectively, where t is known as the strength of the codes. We show \(M_{IPPC}(N,q,t)\le rq^{\lceil N/(v-1)\rceil }+(v-1-r)q^{\lfloor N/(v-1)\rfloor }\), where \(v=\lfloor (t/2+1)^2\rfloor \), \(0\le r\le v-2\) and \(N\equiv r \mod (v-1)\). This new bound improves two previously known bounds of Blackburn, and Alon and Stav. On the other hand, \(M_{TA}(N,q,t)\) is still not known for almost all t. In 2010, Blackburn, Etzion and Ng asked whether \(M_{TA}(N,q,t)\le cq^{\lceil N/t^2\rceil }\) or not, where c is a constant depending only on N, and they have shown the only known validity of this bound for \(t=2\). By using some complicated combinatorial counting arguments, we prove this bound for \(t=3\). This is the first non-trivial upper bound in the literature for traceability codes with strength three.     

<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 \(\varphi \).