2-Arc-transitive cyclic covers of $$K_{n,n}$$

In this paper, a complete classification is achieved of all the regular covers of the complete bipartite graphs \(K_{n,n}\) with cyclic covering transformation group, whose fibre-preserving automorphism group acts 2-arc-transitively. All these covers consist of one threefold covers of \(K_{6,6}\), one twofold cover of \(K_{12, 12}\) and one infinite family X(r, p) of p-fold covers of \(K_{p^r,p^r}\) with p a prime and r an integer such that \(p^r\ge 3\). This infinite family X(r, p) can be derived by a very simple and nice voltage assignment f as follows: \(X(r, p)=K_{p^r, p^r}\times _f \mathbb {Z}_p\), where \(K_{p^r, p^r}\) is a complete bipartite graph with the bipartition \(V=\{ \alpha \bigm |\alpha \in V(r,p)\}\cup \{ \alpha '\bigm |\alpha \in V(r,p)\}\) for the r-dimensional vector space V(r, p) over the field of order p and \(f_{\alpha ,\beta '}=\sum _{i=1}^ra_ib_i,\,\, \mathrm{for\,\,all}\,\,\alpha =(a_i)_r, \beta =(b_i)_r\in V(r,p)\).     

Study of LG-Holling type III predator–prey model with disease in predator

Let \(\mathbb F_{q}\) be a finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{n})\in \mathbb F_{q}^{n}\backslash \{(0,0,\ldots )\}: Tr(x_{1}^{p^{k_{1}}+1}+x_{2}^{p^{k_{2}}+1}+\cdots +x_{n}^{p^{k_{n}}+1})=c\}\), where \(c\in \mathbb F_p\), Tr is the trace function from \(\mathbb F_{q}\) to \(\mathbb F_{p}\) and each \(m/(m,k_{i})\) ( \(1\le i\le n\) ) is odd. we define a p-ary linear code \(C_{D}=\{c(a_{1},a_{2},\ldots ,a_{n}):(a_{1},a_{2},\ldots ,a_{n})\in \mathbb F_{q}^{n}\}\), where \(c(a_{1},a_{2},\ldots ,a_{n})=(Tr(a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}))_{(x_{1},x_{2},\ldots ,x_{n})\in D}\). We present the weight distributions of the classes of linear codes which have at most three weights.     

Triviality of Iwasawa module associated to some abelian fields of prime conductors

Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

Differentiability of arithmetic Fourier series arising from Eisenstein series

Let \(k\in \mathbb {N}^*\) be even. We consider two trigonometric series \( F_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \sin (2\pi n x)\) and \(G_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \cos (2\pi n x),\) where \(\sigma _{k-1}\) is the divisor function. They converge on \(\mathbb {R}\) to continuous functions. In this paper, we examine the differentiability of \(F_k\) and \(G_k\). These functions are related to Eisenstein series, and their (quasi-)modular properties allow us to apply the method proposed by Itatsu in 1981 in the study of the Riemann series. We focus on the case \(k=2\) and we show that the sine series exhibits a different behaviour with respect to differentiability than the cosine series. We prove that the differentiability of \(F_2\) at an irrational x is related to the continued fraction expansion of x. We estimate the modulus of continuity of \(F_2\). We formulate a conjecture concerning differentiability of \(F_k\) and \(G_k\) for any k even.     

Structural Properties of Word Representable Graphs

Given a word \(w=w_1w_2\cdots w_n\) of length n over an ordered alphabet \(\Sigma _k\), we construct a graph \(G(w)=(V(w), E(w))\) such that V(w) has n vertices labeled \(1, 2,\ldots , n\) and for \(i, j \in V(w)\), \((i, j) \in E(w)\) if and only if \(w_iw_j\) is a scattered subword of w of the form \(a_{t}a_{t+1}\), \(a_t \in \Sigma _k\), for some \(1 \le t \le k-1\) with the ordering \(a_t<a_{t+1}\). A graph is said to be Parikh word representable if there exists a word w over \(\Sigma _k\) such that \(G=G(w)\). In this paper we characterize all Parikh word representable graphs over the binary alphabet in terms of chordal bipartite graphs. It is well known that the graph isomorphism (GI) problem for chordal bipartite graph is GI complete. The GI problem for a subclass of (6, 2) chordal bipartite graphs has been addressed. The notion of graph powers is a well studied topic in graph theory and its applications. We also investigate a bipartite analogue of graph powers of Parikh word representable graphs. In fact we show that for G(w), \(G(w)^{[3]}\) is a complete bipartite graph, for any word w over binary alphabet.     

Monochromatic Solutions for Multi-Term Unknowns

Given integers \(k\ge 2\), \(n \ge 2\), \(m \ge 2\) and \( a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}\), and let \(f(z)= \sum _{j=0}^{n}c_jz^j\) be a polynomial of integer coefficients with \(c_n>0\) and \((\sum _{i=1}^ma_i)|f(z)\) for some integer z. For a k-coloring of \([N]=\{1,2,\ldots ,N\}\), we say that there is a monochromatic solution of the equation \(a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)\) if there exist pairwise distinct \(x_1,x_2,\ldots ,x_m\in [N]\) all of the same color such that the equation holds for some \(z\in \mathbb {Z}\). Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if \(a_i>0\) for \(1\le i\le m\), then there exists an integer \(N_0=N(k,m,n)\) such that for \(N\ge N_0\), each k-coloring of [N] contains a monochromatic solution \(x_1,x_2,\ldots ,x_m\) of the equation \(a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)\). Moreover, if n is odd and there are \(a_i\) and \(a_j\) such that \(a_ia_j<0\) for some \(1 \le i\ne j\le m\), then the assertion holds similarly.     

On the eigenvalues and spectral radius of starlike trees

Let \(k\ge 1\) and \(n_1,\ldots ,n_k\ge 1\) be some integers. Let \(S(n_1,\ldots ,n_k)\) be a tree T such that T has a vertex v of degree k and \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1},\ldots ,P_{n_k}\), that is \(T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}\) so that every neighbor of v in T has degree one or two. The tree \(S(n_1,\ldots ,n_k)\) is called starlike tree, a tree with exactly one vertex of degree greater than two, if \(k\ge 3\). In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if \(k\ge 4\) and \(n_1,\ldots ,n_k\ge 2\), then \(\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}\), where \(\lambda _1(T)\) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval \((-2,2)\).     

Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let \(I_1\) be the interval where f is supported, and \(I_2\) be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is \(\left. {\mathcal {H}}_1 f=g\right| _{I_2}\), where \({\mathcal {H}}_1\) is the FHT that integrates over \(I_1\) and gives the result on \(I_2\), i.e. \({\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)\). In the case of complete data, \(I_1\subset I_2\), and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), \(I_1\) is no longer a subset of \(I_2\), and the inversion problems becomes severely unstable. By using a differential operator L that commutes with \({\mathcal {H}}_1\), one can obtain the singular value decomposition of \({\mathcal {H}}_1\). Then the rate of decay of singular values of \({\mathcal {H}}_1\) is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals \(I_{1,2}\) are possible. The cases when \(I_1\) and \(I_2\) are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator \({\mathcal {H}}_1^*{\mathcal {H}}_1\) is discrete, and the asymptotics of its eigenvalues \(\sigma _n\) as \(n\rightarrow \infty \) has been obtained. In this paper we consider the case when the intervals \(I_1=(a_1,0)\) and \(I_2=(0,a_2)\) are adjacent. Here \(a_1 < 0 < a_2\). Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations \(U_1\), \(U_2\), such that \(U_2{\mathcal {H}}_1 U_1^*\) is the multiplication operator with the function \(\sigma (\lambda )\), \(\lambda \ge (a_1^2+a_2^2)/8\). Here \(\lambda \) is the spectral parameter. Then we show that \(\sigma (\lambda )\rightarrow 0\) as \(\lambda \rightarrow \infty \) exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators \(U_1\), \(U_2\) as \(\lambda \rightarrow \infty \). When the intervals are symmetric, i.e. \(-a_1=a_2\), the operators \(U_1\), \(U_2\) are obtained explicitly in terms of hypergeometric functions.     

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 \(k-1\).     

Gravitational Field Equations on Fefferman Space–Times

The total space \({\mathfrak M} \approx {\mathbb H}_1 \times S^1\) of the canonical circle bundle over the 3-dimensional Heisenberg group \({\mathbb H}_1\) is a space–time with the Lorentzian metric \(F_{\theta _0}\) (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form \(\theta _0\) on \({\mathbb H}_1\). The matter and energy content of \(\mathfrak M\) is described by the energy-momentum tensor \({T}_{\mu \nu }\) (the trace-less Ricci tensor of \(F_{\theta _0}\)) as an effect of the non flat nature of Feferman’s metric \(F_{\theta _0}\). We study the gravitational field equations \(R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }\) on \({\mathfrak M}\). We consider the first order perturbation \(g = F_{\theta _0} + \epsilon \, h\), \(\epsilon<< 1\), and linearize the field equations about \(F_{\theta _0}\). We determine a Lorentzian metric g on \({\mathfrak M}\) which solves the linearized field equations corresponding to a diagonal perturbation h.     

The Gray images of $$$$ constacyclic codes over $$F_{2^m}[u]/\langle u^{k} \rangle $$F2m[u]/⟨uk⟩

Let \(R_{k}\) denote the polynomial residue ring \(F_{2^m}[u]/\langle u^{k} \rangle \), where \(2^{j-1}+1\le k\le 2^{j}\) for some positive integer \(j\). Motivated by the work in [1], we introduce a new Gray map from \(R_{k}\) to \(F_{2^m}^{2^{j}}\). It is proved that the Gray image of a linear \((1+u)\) constacyclic code of an arbitrary length \(N\) over \(R_{k}\) is a distance invariant linear cyclic code of length \(2^{j}N\) over \(F_{2^m}\). Moreover, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over \(F_{2}\) and \(F_{4}\) are constructed under this Gray map.     

Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case

Let M be a 3-manifold with torus boundary components \(T_{1}\) and \(T_2\). Let \(\phi :T_{1} \rightarrow T_{2}\) be a homeomorphism, \(M_\phi \) the manifold obtained from M by gluing \(T_{1}\) to \(T_{2}\) via the map \(\phi \), and T the image of \(T_{1}\) in \(M_\phi \). We show that if \(\phi \) is “sufficiently complicated” then any incompressible or strongly irreducible surface in \(M_\phi \) can be isotoped to be disjoint from T. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.     

Appell polynomial sequences with respect to some differential operators

We present a study of a specific kind of lowering operator, herein called \(\Lambda \), which is defined as a finite sum of lowering operators and might be presented by various configurations. We characterize the polynomial sequences fulfilling an Appell relation with respect to \(\Lambda \), and considering a concrete cubic decomposition of a simple Appell sequence, we prove that the polynomial component sequences are \(\Lambda \)-Appell, with \(\Lambda \) defined as previously, although by a three term sum. Ultimately, we prove the non-existence of orthogonal polynomial sequences which are also \(\Lambda \)-Appell, when \(\Lambda \) is the lowering operator \(\Lambda =a_{0}D+a_{1}DxD+a_{2}\left( Dx\right) ^2D\), where \(a_{0}\), \(a_{1}\) and \(a_{2}\) are constants and \(a_{2} \ne 0\). The case where \(a_{2}=0\) and \(a_{1} \ne 0\) is also naturally recaptured.     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

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

Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation

On the Hilbert space \(\widetilde{L}_{2}(\mathbb {T})\) the singular integral operator with non-Carleman shift and conjugation \(K=P_{+}+(aI+AC)P_{-}\) is considered, where \(P_{\pm }\) are the Cauchy projectors, \(A=\sum \nolimits _{j=0}^{m}a_{j}U^{j}\), \(a,a_{j}\), \(j=\overline{1,m}\), are continuous functions on the unit circle \(\mathbb {T}\), U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K. The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.     

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.     

On a Generalization for Tribonacci Quaternions

Let \(V_{n}\) denote the third order linear recursive sequence defined by the initial values \(V_{0}\), \(V_{1}\) and \(V_{2}\) and the recursion \(V_{n}=rV_{n-1}+sV_{n-2}+tV_{n-3}\) if \(n\ge 3\), where r, s, and t are real constants. The \(\{V_{n}\}_{n\ge 0}\) are generalized Tribonacci numbers and reduce to the usual Tribonacci numbers when \(r=s=t=1\) and to the 3-bonacci numbers when \(r=s=1\) and \(t=0\). In this study, we introduced a quaternion sequence which has not been introduced before. We show that the new quaternion sequence that we introduced includes the previously introduced Tribonacci, Padovan, Narayana and third order Jacobsthal quaternion sequences. We obtained the Binet formula, summation formula and the norm value for this new quaternion sequence.     

Sandwich semigroups in locally small categories I: foundations

Fix (not necessarily distinct) objects i and j of a locally small category S, and write \(S_{ij}\) for the set of all morphisms \(i\rightarrow j\). Fix a morphism \(a\in S_{ji}\), and define an operation \(\star _a\) on \(S_{ij}\) by \(x\star _ay=xay\) for all \(x,y\in S_{ij}\). Then \((S_{ij},\star _a)\) is a semigroup, known as a sandwich semigroup, and denoted by \(S_{ij}^a\). This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on \(S_{ij}^a\) and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set \({\text {Reg}}(S_{ij}^a)\) of all regular elements of \(S_{ij}^a\) is a subsemigroup of \(S_{ij}^a\). Under this condition, we carefully analyse the structure of the semigroup \({\text {Reg}}(S_{ij}^a)\), relating it via pullback products to certain regular subsemigroups of \(S_{ii}\) and \(S_{jj}\), and to a certain regular sandwich monoid defined on a subset of \(S_{ji}\); among other things, this allows us to also describe the idempotent-generated subsemigroup \(\mathbb E(S_{ij}^a)\) of \(S_{ij}^a\). We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups \(S_{ij}^a\), \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\); we give lower bounds for these ranks, and in the case of \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\) show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.