On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve \(E/\mathbb {Q}\), which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval \((X, X + cX^{\frac{1}{4}})\) for all \(X \gg 0\) and for some \(c > 0\). We use this fact to produce non-CM cuspidal eigenforms f of level \(N>1\) and weight \(k > 2\) such that \(i_f(n) \ll n^{\frac{1}{4}}\) for all \(n \gg 0\).     

<Emphasis Type="Italic">M</Emphasis>-curves and symmetric products

Let \((X\, , \sigma )\) be a geometrically irreducible smooth projective M-curve of genus g defined over the field of real numbers. We prove that the n-th symmetric product of \((X\, , \sigma )\) is an M-variety for \(n\,=\,2\, ,3\) and \(n \,\ge \, 2g -1\).     

Interassociates of the bicyclic semigroup

An interassociate of a semigroup \((S,\cdot )\) is a semigroup \((S, *)\) such that for all \(a, b, c \in S\), \(a\cdot (b*c)=(a\cdot b) *c\) and \(a*(b\cdot c)=(a*b) \cdot c\). We investigate the bicyclic semigroup C and its interassociates. In particular, if p and q are the generators of the bicyclic semigroup and m and n are fixed nonnegative integers, the operation \(a*_{m,n} b= aq^mp^n b\) is known to be an interassociate. We show that for distinct pairs (m, n) and (s, t), the interassociates \((C, *_{m,n})\) and \((C, *_{s,t})\) are not isomorphic. We also generalize a result regarding homomorphisms on C to homomorphisms on its interassociates.     

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

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.     

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 \(k-1\).     

Regulating Hartshorne’s connectedness theorem

The tensor square conjecture states that for \(n \ge 10\), there is an irreducible representation V of the symmetric group \(S_n\) such that \(V \otimes V\) contains every irreducible representation of \(S_n\). Our main result is that for large enough n, there exists an irreducible representation V such that \(V^{\otimes 4}\) contains every irreducible representation. We also show that tensor squares of certain irreducible representations contain \((1-o(1))\)-fraction of irreducible representations with respect to two natural probability distributions. Our main tool is the semigroup property, which allows us to break partitions down into smaller ones.     

$${\varvec{}}$$-Jorda homomorphisms ad pseudo $${\varvec{}}$$-Jorda homomorphisms o Baach algebras

Let \(n\in \mathbb {N}\), A and B be Banach algebras and let B be a right A-module. We say that a linear mapping \(\varphi :A\longrightarrow B\) is a pseudo n-Jordan homomorphism if there exists an element \(w\in A\) such that \(\varphi (a^nw)=\varphi (a)^n\cdot w\), for every \(a\in A\) and \(n\ge \) 2. In this paper, among other things, we show that under some conditions if a linear mapping \(\varphi \) is a (pseudo) n-Jordan homomorphism, then it is a (pseudo) \((n + 1)\)-Jordan homomorphism. Additionally, we investigate automatic continuity of surjective pseudo n-Jordan homomorphisms under some conditions.     

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 \(q=2\).     

Cliques in $$C_4$$-free graphs of large minimum degree

A graph G is called \(C_4\)-free if it does not contain the cycle \(C_4\) as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd?s) a peculiar property of \(C_4\)-free graphs: \(C_4\)-free graphs with n vertices and average degree at least cn contain a complete subgraph (clique) of size at least \(c'n\) (with \(c'= 0.1c^2\)). We prove here better bounds \(\big ({c^2n\over 2+c}\) in general and \((c-1/3)n\) when \( c \le 0.733\big )\) from the stronger assumption that the \(C_4\)-free graphs have minimum degree at least cn. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2k-regular \(C_4\)-free graphs on \(4k+1\) vertices contain a clique of size \(k+1\). This is the best possible as shown by the kth power of the cycle \(C_{4k+1}\).     

Hyperinvariant Subspace Problem for Some Classes of Operators

An n-normal operator may be defined as an \(n \times n\) operator matrix with entries that are mutually commuting normal operators and an operator \(T \in \mathcal {B(H)}\) is quasi-nM-hyponormal (for \(n \in \mathbb {N}\)) if it is unitarily equivalent to an \(n \times n\) upper triangular operator matrix \((T_{ij})\) acting on \(\mathcal {K}^{(n)}\), where \(\mathcal {K}\) is a separable complex Hilbert space and the diagonal entries \(T_{jj}\) \((j = 1,2,\ldots , n)\) are M-hyponormal operators in \(\mathcal {B(K)}\). This is an extended notion of n-normal operators. We prove a necessary and sufficient condition for an \(n \times n\) triangular operator matrix to have Bishop’s property \((\beta )\). This leads us to study the hyperinvariant subspace problem for an \(n \times n\) triangular operator matrix.     

Geometric Constructibility of Polygons Lying on a Circular Arc

For a positive integer n, an n-sided polygon lying on a circular arc or, shortly, an n-fan is a sequence of \(n+1\) points on a circle going counterclockwise such that the “total rotation” \(\delta \) from the first point to the last one is at most \(2\pi \). We prove that for \(n\ge 3\), the n-fan cannot be constructed with straightedge and compass in general from its central angle \(\delta \) and its central distances, which are the distances of the edges from the center of the circle. Also, we prove that for each fixed \(\delta \) in the interval \((0, 2\pi ]\) and for every \(n\ge 5\), there exists a concrete n-fan with central angle \(\delta \) that is not constructible from its central distances and \(\delta \). The present paper generalizes some earlier results published by the second author and Á. Kunos on the particular cases \(\delta =2\pi \) and \(\delta =\pi \).     

The non-abelian tensor square of residually finite groups

Let m, n be positive integers and p a prime. We denote by \(\nu (G)\) an extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). We prove that if G is a residually finite group satisfying some non-trivial identity \(f \equiv ~1\) and for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) such that \([x,y^{\varphi }]^q = 1\), then the derived subgroup \(\nu (G)'\) is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) dividing \(p^m\) such that \([x,y^{\varphi }]^q\) is left n-Engel, then the non-abelian tensor square \(G \otimes G\) is locally virtually nilpotent (Theorem B).     

Construction of a (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-visual cryptography scheme

In this paper we give an explicit construction of basis matrices for a (k, n)-visual cryptography scheme \((k,n){\hbox {-}}\mathrm{VCS}\) for integers k and n with \(2\le k \le n\). In balanced VCS every set of participants with equal cardinality has same relative contrast. The VCS constructed in this paper is a balanced \((k,n){\hbox {-}}\mathrm{VCS}\) for general k. Also we obtain a formula for pixel expansion and relative contrast. We also prove that our construction gives optimal contrast and minimum pixel expansion when \(k=n\) and \(n-1\).     

A Compactness Theorem of the Space of Free Boundary <Emphasis Type="Italic">f</Emphasis>-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

Let \((M^3,g,e^{-f}d\mu _M)\) be a compact three-dimensional smooth metric measure space with nonempty boundary. Suppose that M has nonnegative Bakry–Émery Ricci curvature and the boundary \(\partial M\) is strictly f-mean convex. We prove that there exists a properly embedded smooth f-minimal surface \(\Sigma \) in M with free boundary \(\partial \Sigma \) on \(\partial M\). If we further assume that the boundary \(\partial M\) is strictly convex, then we prove that \(M^3\) is diffeomorphic to the 3-ball \(B^3\), and a compactness theorem for the space of properly embedded f-minimal surfaces with free boundary in such \((M^3,g,e^{-f}d\mu _M)\), when the topology of these f-minimal surfaces is fixed.     

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

In this paper, we study the harmonic equation involving subcritical exponent \((P_{\varepsilon })\): \( \Delta u = 0 \), in \(\mathbb {B}^n\) and \(\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }\) on \( \mathbb {S}^{n-1}\) where \(\mathbb {B}^n \) is the unit ball in \(\mathbb {R}^n\), \(n\ge 5\) with Euclidean metric \(g_0\), \(\partial \mathbb {B}^n = \mathbb {S}^{n-1}\) is its boundary, K is a function on \(\mathbb {S}^{n-1}\) and \(\varepsilon \) is a small positive parameter. We construct solutions of the subcritical equation \((P_{\varepsilon })\) which blow up at two different critical points of K. Furthermore, we construct solutions of \((P_{\varepsilon })\) which have two bubbles and blow up at the same critical point of K.     

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 \(A_5(7,5)\le 87\).     

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 \((-1)^n\mu _{2k}(-1,n)\).     

Continued fractions and irrational rotations

Let \(\alpha \in (0, 1)\) be an irrational number with continued fraction expansion \(\alpha =[0; a_1, a_2, \ldots ]\) and let \(p_n/q_n= [0; a_1, \ldots , a_n]\) be the nth convergent to \(\alpha \). We prove a formula for \(p_nq_k-q_np_k\) \((k<n)\) in terms of a Fibonacci type sequence \(Q_n\) defined in terms of the \(a_n\) and use it to provide an exact formula for \(\{n\alpha \}\) for all n.     

On a conjecture of Stein

Stein (Pac J Math 59:567–575, 1975) proposed the following conjecture: if the edge set of \(K_{n,n}\) is partitioned into n sets, each of size n, then there is a partial rainbow matching of size \(n-1\). He proved that there is a partial rainbow matching of size \(n(1-\frac{D_n}{n!})\), where \(D_n\) is the number of derangements of [n]. This means that there is a partial rainbow matching of size about \((1- \frac{1}{e})n\). Using a topological version of Hall’s theorem we improve this bound to \(\frac{2}{3}n\).