$${\varvec{}}$$-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.     

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 \(k-1\).     

The Gray image of constacyclic codes over the finite chain ring $$F_{p^m}[u]/\langle u^k\rangle $$

Let \(\mathbb {F}_{p^m}\) be a finite field of cardinality \(p^m\), where p is a prime, and k, N be any positive integers. We denote \(R_k=F_{p^m}[u]/\langle u^k\rangle =F_{p^m}+uF_{p^m}+\cdots +u^{k-1}F_{p^m}\) (\(u^k=0\)) and \(\lambda =a_0+a_1u+\cdots +a_{k-1}u^{k-1}\) where \(a_0, a_1,\ldots , a_{k-1}\in F_{p^m}\) satisfying \(a_0\ne 0\) and \(a_1=1\). Let r be a positive integer satisfying \(p^{r-1}+1\le k\le p^r\). First we define a Gray map from \(R_k\) to \(F_{p^m}^{p^r}\), then prove that the Gray image of any linear \(\lambda \)-constacyclic code over \(R_k\) of length N is a distance preserving linear \(a_0^{p^r}\)-constacyclic code over \(F_{p^m}\) of length \(p^rN\). Furthermore, the generator polynomials for each linear \(\lambda \)-constacyclic code over \(R_k\) of length N and its Gray image are given respectively. Finally, some optimal constacyclic codes over \(F_{3}\) and \(F_{5}\) are constructed.     

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.     

A generalization of the normal rational curve in $$\mathop {\mathrm{PG}}(d,q^n)$$ and its associated non-linear MRD codes

Let A and B be two points of \(\mathop {\mathrm{PG}}(d,q^n)\) and let \(\Phi \) be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if \(d=2\) or \(d\ge 3\) and the lines \(\Phi ^{-1}(AB), AB, \Phi (AB) \) are not in a common plane, then the set \(\mathcal{C}\) of points of intersection of corresponding lines under \(\Phi \) is the union of \(q-1\) scattered \({\mathbb {F}}_{q}\)-linear sets of rank n together with \(\{A,B\}\). As an application we will construct, starting from the set \(\mathcal{C}\), infinite families of non-linear \((d+1, n, q;d-1)\)-MRD codes, \(d\le n-1\), generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).     

On the family of elliptic curves $$\varvec{y^2=x^3-m^2x+p^2}$$

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of \(E_{m,p} : y^2=x^3-m^2x+p^2\), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of \(E_{m,p}(\mathbb {Q})\) is trivial for both the cases {\(m\ge 1\), \(m\not \equiv 0\pmod 3\)} and {\(m\ge 1\), \(m \equiv 0 \pmod 3\), with \(gcd(m,p)=1\)}. We also show that given any odd prime p and for any positive integer m with \(m\not \equiv 0\pmod 3\) and \(m\equiv 2\pmod {32}\), the lower bound for the rank of \(E_{m,p}(\mathbb {Q})\) is 2. Finally, we find curves of rank 9 in this family.     

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.     

A characterization of the desarguesian and the Figueroa planes of order $$q^{3}$$

Let \(\Pi \) be a plane of order \(q^{3}\), \(q>2\), admitting \(G\cong PGL(3,q)\) as a collineation group. By Dempwolff (Geometriae Dedicata 18:101–112, 1985) the plane \(\Pi \) contains a G-invariant subplane \(\pi _{0}\) isomorphic to PG(2, q) on which G acts 2-transitively. In this paper it is shown that, if the homologies of \(\pi _{0}\) contained in G extend to \(\Pi \) then \(\Pi \) is either the desarguesian or the Figueroa plane.     

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 \(\mathbb {Z}_4\).     

Revisiting Fryszkowski’s problem

Professor Andrzej Fryszkowski formulated, at the 2nd Symposium on Nonlinear Analysis in Toruń, September 13–17, 1999, the following problem: given \(\alpha \in (0,1)\), an arbitrary non-empty set \(\Omega \) and a set-valued mapping \(F:\Omega \rightarrow 2^{\Omega }\), find necessary and (or) sufficient conditions for the existence of a (complete) metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d. Com?neci (Stud. Univ. Babe?-Bolyai Math. 62:537–542, 2017) provided necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d, in case that \(\alpha \in (0,\frac{1}{2})\) and there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) . We improve Com?neci’s result by allowing \(\alpha \) to belong to the interval (0, 1). In addition, we provide necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) such that F is a Nadler set-valued \(\alpha \)-similarity with respect to d, in case that \(\alpha \in (0,1)\), there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) and F is non-overlapping.     

Some infinite families of Ramsey ($${\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}$$)-minimal trees

For any given two graphs G and H, the notation \(F\rightarrow \) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if \(F\rightarrow \) (G, H) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The class of all Ramsey (G, H)-minimal graphs is denoted by \(\mathcal {R}(G,H)\). In this paper, we construct some infinite families of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n=8\) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n\ge 10\).     

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.     

Characterization of finite $$\varvec{}$$-grous by their Schur multilier

Let G be a finite p-group of order \(p^n\) and M(G) be its Schur multiplier. It is a well known result by Green that \(|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}\) for some \(t(G) \ge 0\). In this article, we classify non-abelian p-groups G of order \(p^n\) for \(t(G)=\log _p(|G|)+1\).     

A note on compact Weingarten hypersurfaces embedded in $$\mathbb {R}^{n + 1}$$

We prove that the round sphere is the only compact Weingarten hypersurface embedded in the Euclidean space such that \(H_r = aH + b\), for constants \(a, b \in \mathbb {R}\). Here, \(H_r\) stands for the r-th mean curvature and H denotes the standard mean curvature of the hypersurface.     

On the 2-adic valuation of the cardinality of elliptic curves over finite extensions of $$\mathbb {F}_{\varvec{q}} $$

In this paper we study the difference between the 2-adic valuations of the cardinalities \( \# E( \mathbb {F}_{q^k} ) \) and \( \# E( \mathbb {F}_q ) \) of an elliptic curve E over \( \mathbb {F}_q \). We also deduce information about the structure of the 2-Sylow subgroup \( E[ 2^\infty ]( \mathbb {F}_{q^k} ) \) from the exponents of \( E[ 2^\infty ]( \mathbb {F}_q ) \).     

Matrix Gegenbauer Polynomials: The $$2\times 2$$ Fundamental Cases

In this paper, we exhibit explicitly a sequence of \(2 \times 2\) matrix valued orthogonal polynomials with respect to a weight \(W_{p,n}\), for any pair of real numbers p and n such that \(0<p<n\). The entries of these polynomiales are expressed in terms of the Gegenbauer polynomials \(C_k^\lambda \). The corresponding three-term recursion relations are also given, and we make some studies of the algebra of differential operators associated with the weight \(W_{p,n}\).     

On the sumsets of exceptional units in $$\mathbb {Z}_n$$

Let R be a commutative ring with \(1\in R\) and \(R^{*}\) be the multiplicative group of its units. In 1969, Nagell introduced the concept of an exceptional unit, namely a unit u such that \(1-u\) is also a unit. Let \({\mathbb {Z}}_n\) be the ring of residue classes modulo n. In this paper, given an integer \(k\ge 2\), we obtain an exact formula for the number of ways to represent each element of \( \mathbb {Z}_n\) as the sum of k exceptional units. This generalizes a recent result of J. W. Sander for the case \(k=2\).     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in  \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in  \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in  \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in  \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.