Congruences modulo 11 for broken 5-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule in 2007. For a fixed positive integer k, let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n. Recently, Paule and Radu conjectured two relations on \(\Delta _5(n)\) which were proved by Xiong and Jameson, respectively. In this paper, employing these relations, we prove that, for any prime p with \(p\equiv 1\ (\mathrm{mod}\ 4)\), there exists an integer \(\lambda (p)\in \{2,\ 3,\ 5,\ 6,\ 11\}\) such that, for \(n, \alpha \ge 0\), if \(p\not \mid (2n+1)\), then

$$\begin{aligned} \Delta _5\left( 11p^{\lambda (p)(\alpha +1)-1} n+\frac{11p^{\lambda (p)(\alpha +1)-1}+1}{2}\right) \equiv 0\ (\mathrm{mod}\ 11). \end{aligned}$$

Moreover, some non-standard congruences modulo 11 for \(\Delta _5(n)\) are deduced. For example, we prove that, for \(\alpha \ge 0\), \(\Delta _5\left( \frac{11\times 5^{5\alpha }+1}{2}\right) \equiv 7\ (\mathrm{mod}\ 11)\).

     

Augmentation quotients for real representation rings of cyclic groups

Denote by \(C_m\) the cyclic group of order m. Let \({\mathcal {R}}(C_m)\) be its real representation ring, and \(\Delta (C_m)\) its augmentation ideal. In this paper, we give an explicit \({\mathbb {Z}}\)-basis for the n-th power \(\Delta ^{n}(C_m)\) and determine the isomorphism class of the n-th augmentation quotient \(\Delta ^n(C_m)/\Delta ^{n+1}(C_m)\) for each positive integer n.     

The primitive idempotents and weight distributions of irreducible constacyclic codes

Let \({\mathbb {F}}_q\) be a finite field with q elements such that \(l^v||(q^t-1)\) and \(\gcd (l,q(q-1))=1\), where l, t are primes and v is a positive integer. In this paper, we give all primitive idempotents in a ring \(\mathbb F_q[x]/\langle x^{l^m}-a\rangle \) for \(a\in {\mathbb {F}}_q^*\). Specially for \(t=2\), we give the weight distributions of all irreducible constacyclic codes and their dual codes of length \(l^m\) over \({\mathbb {F}}_q\).     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

A new theorem on the prime-counting function

For \(x>0\), let \(\pi (x)\) denote the number of primes not exceeding x. For integers a and \(m>0\), we determine when there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). In particular, we show that, for any integers \(m>2\) and \(a\leqslant \lceil e^{m-1}/(m-1)\rceil \), there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). Consequently, for any integer \(m>4\), there is a positive integer n with \(\pi (mn)=m+n\). We also pose several conjectures for further research; for example, we conjecture that, for each \(m=1,2,3,\ldots \), there is a positive integer n such that \(m+n\) divides \(p_m+p_n\), where \(p_k\) denotes the k-th prime.     

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 \(\mathcal{H}(b)\).     

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 \(k-1\).     

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.     

Répartitio n simulta née de S( n) et $$ S( n+1)$$ S( n+1) da ns les progressio ns arithmétiques

If \(q\ge 2\) is an integer, we denote by \(S_q(n)\) the sum of the digits in base q of the positive integer n and by \(v_q(n)\) its q-adic valuation. The goal of this work is to study exponential sums of the form \(\displaystyle \sum \nolimits _{n\le x}\exp \big (2i\pi \big (\frac{l}{m} S_q(n)+\frac{k}{m'}S_q(n+1)+\theta n\big )\big )\) in order to prove some statistical properties of integers n for which \(S_q(n)\) and \(S_q(n+1)\) belong to given arithmetic progressions. This extends the results obtained by Gelfond in 1968 and those obtained by Mauduit–Sárközy in 1996.     

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

Estimates for the first eigenvalue of Jacobi operator on hypersurfaces with constant mean curvature in spheres

In this paper, we study the first eigenvalue of Jacobi operator on an n-dimensional non-totally umbilical compact hypersurface with constant mean curvature H in the unit sphere \(S^{n+1}(1)\). We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature H and the dimension n. This bound is attained if and only if, \(\varphi :\ M \rightarrow S^{n+1}(1)\) is isometric to \(S^1(r)\times S^{n-1}(\sqrt{1-r^2})\) when \(H\ne 0\) or \(\varphi :\ M \rightarrow S^{n+1}(1)\) is isometric to a Clifford torus \( S^{n-k}\left( \sqrt{\dfrac{n-k}{n}}\right) \times S^k\left( \sqrt{\dfrac{k}{n}}\right) \), for \(k=1, 2, \ldots , n-1\) when \(H=0\).     

Transcendence of zeros of Jacobi forms

A special case of a fundamental theorem of Schneider asserts that if \(j(\tau )\) is algebraic (where j is the classical modular invariant), then any zero z not in \(\mathbf{Q}.L_\tau := \mathbf{Q}\oplus \mathbf{Q}\tau \) of the Weierstrass function \(\wp (\tau ,\cdot )\) attached to the lattice \(L_\tau =\mathbf{Z}\oplus \mathbf{Z}\tau \) is transcendental. In this note we generalize this result to holomorphic Jacobi forms of weight k and index \(m\in \mathbf{N}\) with algebraic Fourier coefficients.     

On $$(\ell,m)$$-regular partitions with distinct parts

Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\)-regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\). For example,

$$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$

where \(\alpha , \beta \ge 0\).

     

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

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 \(p\in P\backslash (P_0\cup P_k)\).     

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with \(n\ge 2\)), there is a natural construction of a semigroup of transformations \(\langle D\rangle \). For any edge (a, b) of D, let \(a\rightarrow b\) be the idempotent of rank \(n-1\) mapping a to b and fixing all vertices other than a; then, define \(\langle D\rangle \) to be the semigroup generated by \(a \rightarrow b\) for all \((a,b) \in E(D)\). For \(\alpha \in \langle D\rangle \), let \(\ell (D,\alpha )\) be the minimal length of a word in E(D) expressing \(\alpha \). It is well known that the semigroup \(\mathrm {Sing}_n\) of all transformations of rank at most \(n-1\) is generated by its idempotents of rank \(n-1\). When \(D=K_n\) is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate \(\ell (K_n,\alpha )\), for any \(\alpha \in \langle K_n\rangle = \mathrm {Sing}_n\); however, no analogous non-trivial results are known when \(D \ne K_n\). In this paper, we characterise all simple digraphs D such that either \(\ell (D,\alpha )\) is equal to Howie–Iwahori’s formula for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {fix}(\alpha )\) for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {rk}(\alpha )\) for all \(\alpha \in \langle D\rangle \). We also obtain bounds for \(\ell (D,\alpha )\) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank \(n-1\) of \(\mathrm {Sing}_n\)). We finish the paper with a list of conjectures and open problems.     

On the peripheral spectrum of positive elements

Let A be an ordered Banach algebra with a unit \(\mathbf{e}\) and a cone \(A^+\). An element p of A is said to be an order idempotent if \(p^2 = p\) and \(0 \le p\le \mathbf{e}\). An element \(a\in A^+\) is said to be irreducible if the relation \((\mathbf{e}-p)ap = 0\), where p is an order idempotent, implies \(p = 0\) or \(p = \mathbf{e}\). For an arbitrary element a of A the peripheral spectrum \(\sigma _\mathrm{per}(a)\) of a is the set \(\sigma _\mathrm{per}(a) = \{\lambda \in \sigma (a):|\lambda | = r(a)\}\), where \(\sigma (a)\) is the spectrum of a and r(a) is the spectral radius of a. We investigate properties of the peripheral spectrum of an irreducible element a. Conditions under which \(\sigma _\mathrm{per}(a)\) contains or coincides with \(r(a)H_m\), where \(H_m\) is the group of all \(m^\mathrm{th}\) roots of unity, and the spectrum \(\sigma (a)\) is invariant under rotation by the angle \(\frac{2\pi }{m}\) for some \(m\in {\mathbb N}\), are given. The correlation between these results and the existence of a cyclic form of a is considered. The conditions under which a is primitive, i.e., \(\sigma _\mathrm{per}(a) = \{r(a)\}\), are studied. The necessary assumptions on the algebra A which imply the validity of these results, are discussed. In particular, the Lotz–Schaefer axiom is introduced and finite-rank elements of A are defined. Other approaches to the notions of irreducibility and primitivity are discussed. Conditions under which the inequalities \(0 \le b < a\) imply \(r(b) < r(a)\) are studied. The closedness of the center \(A_\mathbf{e}\), i.e., of the order ideal generated by \(\mathbf{e}\) in A, is proved.     

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 \(A_5(7,5)\le 87\).     

Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Let \(X_n = \{x^j\}_{j=1}^n\) be a set of n points in the d-cube \({\mathbb {I}}^d:=[0,1]^d\), and \(\Phi _n = \{\varphi _j\}_{j =1}^n\) a family of n functions on \({\mathbb {I}}^d\). We consider the approximate recovery of functions f on \({{\mathbb {I}}}^d\) from the sampled values \(f(x^1), \ldots , f(x^n)\), by the linear sampling algorithm \( L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. \) The error of sampling recovery is measured in the norm of the space \(L_q({\mathbb {I}}^d)\)-norm or the energy quasi-norm of the isotropic Sobolev space \(W^\gamma _q({\mathbb {I}}^d)\) for \(1 < q < \infty \) and \(\gamma > 0\). Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces \(B^{\alpha ,\beta }_{p,\theta }\) of a “hybrid” of mixed smoothness \(\alpha > 0\) and isotropic smoothness \(\beta \in {\mathbb {R}}\), and spaces \(B^a_{p,\theta }\) of a nonuniform mixed smoothness \(a \in {\mathbb {R}}^d_+\). We constructed asymptotically optimal linear sampling algorithms \(L_n(X_n^*,\Phi _n^*,\cdot )\) on special sparse grids \(X_n^*\) and a family \(\Phi _n^*\) of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in \(B^{\alpha ,\beta }_{p,\theta }\) and \(B^a_{p,\theta }\). As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.