Map determined by rank-$$\varvec{}$$ matrice for relatively mall $$\varvec{}$$

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \).     

O $$\varvec{}$$-orm preservers ad the Aleksadrov coservative $$\varvec{}$$-distace problem

The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for \(n \ge 3\) any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine (\(n \ge 2\)) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and ?emrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions (\(n\ge 2\)) is automatically an n-isometry.     

$$\varvec{}$$-Classes in finite groups of conjugate type $$\varvec{(n,1)}$$(n,1)

Two elements in a group G are said to be z-equivalent or to be in the same z-class if their centralizers are conjugate in G. In a recent work, Kulkarni et al. (J. Algebra Appl., 15 (2016) 1650131) proved that a non-abelian p-group G can have at most \(\frac{p^k-1}{p-1} +1\) number of z-classes, where \(|G/Z(G)|=p^k\). Here, we characterize the p-groups of conjugate type (n, 1) attaining this maximal number. As a corollary, we characterize p-groups having prime order commutator subgroup and maximal number of z-classes.     

$$\varvec{}$$-Polar Fuzzy Labeling Graphs with Application

In this article, we apply the concept of m-polar fuzzy sets in labeling graphs and introduce the concept of m-polar fuzzy labeling graphs. We present properties of m-polar fuzzy labeling cycle and m-polar fuzzy labeling with bridge and cut nodes. We introduce several different types of distances, including w-distance, strongest strong distance, strong geodesic distance, \(\delta \)-distance in m-polar fuzzy labeling graph and investigate some of their properties. We also present an application of m-polar fuzzy labeling graphs in image processing.     

Soe fractional integral inequalities involving $$\varvec{}$$-convex functions

In this paper, some fractional integral inequalities involving m-convex functions are established. The presented results are generalizations of the obtained inequalities in Dragomir and Toader (Babe?-Bolyai Math 38:21–28, 1993).     

A gradient estimate on graphs with the $$\varvec{}$$-condition

Let \(\omega \) be an unbounded radial weight on \(\mathbb {C}^d\), \(d\ge 1\). Using results related to approximation of \(\omega \) by entire maps, we investigate Volterra type and weighted composition operators defined on the growth space \(\mathcal {A}^\omega (\mathbb {C}^d)\). Special attention is given to the operators defined on the growth Fock spaces.     

On $$\mathcal {}$$-subgroups of finite groups

Let G be a finite group. A subgroup H of G is s-permutable in G if H permutes with every Sylow subgroup of G. A subgroup H of G is called an \(\mathcal {SSH}\)-subgroup in G if G has an s-permutable subgroup K such that \(H^{sG} = HK\) and \(H^g \cap N_K (H) \leqslant H\), for all \(g \in G\), where \(H^{sG}\) is the intersection of all s-permutable subgroups of G containing H. We study the structure of finite groups under the assumption that the maximal or the minimal subgroups of Sylow subgroups of some normal subgroups of G are \(\mathcal {SSH}\)-subgroups in G. Several recent results from the literature are improved and generalized.     

O the volume growth of the hyperbolic regular $$\varvec{}$$-simplex

In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the n-dimensional volume of a regular simplex and the \((n-1)\)-dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for \(n=3\).     

A nested family of $$\varvec{}$$-total effective rewards for positional games

We consider Gillette’s two-person zero-sum stochastic games with perfect information. For each \(k \in \mathbb {N}=\{0,1,\ldots \}\) we introduce an effective reward function, called k-total. For \(k = 0\) and 1 this function is known as mean payoff and total reward, respectively. We restrict our attention to the deterministic case. For all k, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that k-total reward games can be embedded into \((k+1)\)-total reward games.     

Indecomosability of modular standard modules of schurian association schemes and $$\varvec{}$$-schemes

Let p be a prime number. It is known that the standard module of a p-association scheme over a field of characteristic p is indecomposable. By examples, the converse of the above fact is not true. We will prove that, for a schurian association scheme, the standard module is indecomposable if and only if the association scheme is a p-scheme.     

rthogonality Property $$\mathcal {}$$ and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.     

Most small $${\varvec{}}$$-grous have an automorhism of order 2

Let f(p, n) be the number of pairwise nonisomorphic p-groups of order \(p^n\), and let g(p, n) be the number of groups of order \(p^n\) whose automorphism group is a p-group. We prove that the limit, as p grows to infinity, of the ratio g(p, n) / f(p, n) equals 1/3 for \(n=6,7\).     

Factorization of $$\mathbb {}$$-Valued Functions Induced by Graphs

In this paper, we study certain \(\mathbb {C}\)-valued functions induced by given graphs called graph zeta functions. In particular, we are interested in factorizations of such functions. We show that the factorizations of graph zeta functions are characterized by certain subgraphs of given graphs. Based on this factorization property on graph zeta functions, we define-and-study the non-factorizability of graphs.     

On maximal green sequences for type $$\mathbb {}$$ quivers

Given a framed quiver, i.e., one with a frozen vertex associated with each mutable vertex, there is a concept of green mutation, as introduced by Keller. Maximal sequences of such mutations, known as maximal green sequences, are important in representation theory and physics as they have numerous applications, including the computations of spectrums of BPS states, Donaldson–Thomas invariants, tilting of hearts in derived categories, and quantum dilogarithm identities. In this paper, we study such sequences and construct a maximal green sequence for every quiver mutation equivalent to an orientation of a type \(\mathbb {A}\) Dynkin diagram.     

Actions of $$\mathbf {SAut}\varvec{(F_n)}$$

We study actions of SAut\((F_n)\), the unique subgroup of index two in the automorphism group of a free group of rank n, and obtain rigidity results for its representations. In particular, we show that every smooth action of SAut\((F_n)\) on a low dimensional torus is trivial.     

A Characterisation of Some $$\mathbf {}$$-Like Logics

In Béziau (Log Log Philos 15:99–111, 2006) a logic \(\mathbf {Z}\) was defined with the help of the modal logic \(\mathbf {S5}\). In it, the negation operator is understood as meaning ‘it is not necessary that’. The strong soundness–completeness result for \(\mathbf {Z}\) with respect to a version of Kripke semantics was also given there. Following the formulation of \(\mathbf {Z}\) we can talk about \(\mathbf {Z}\)-like logics or Beziau-style logics if we consider other modal logics instead of \(\mathbf {S5}\)—such a possibility has been mentioned in [1]. The correspondence result between modal logics and respective Beziau-style logics has been generalised for the case of normal logics naturally leading to soundness–completeness results [see Marcos (Log Anal 48(189–192):279–300, 2005) and Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 34(4):229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 37(3–4):185–196, 2008), (Bull Sect Log 38(3–4):189–203, 2009) some partial results for non-normal cases are given. In the present paper we try to give similar but more general correspondence results for the non-normal-worlds case. To achieve this aim we have to enrich original Beziau’s language with an additional negation operator understood as ‘it is necessary that not’.     

On the image,characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ )-derivations

The first main theorem of this paper asserts that any \((\sigma , \tau )\)-derivation d, under certain conditions, either is a \(\sigma \)-derivation or is a scalar multiple of (\(\sigma - \tau \)), i.e. \(d = \lambda (\sigma - \tau )\) for some \(\lambda \in \mathbb {C} \backslash \{0\}\). By using this characterization, we achieve a result concerning the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras which reads as follows. Let \(\mathcal {A}\) be a unital, commutative, semi-simple Banach algebra, and let \(\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}\) be two distinct endomorphisms such that \(\varphi \sigma (\mathbf e )\) and \(\varphi \tau (\mathbf e )\) are non-zero complex numbers for all \(\varphi \in \Phi _\mathcal {A}\). If \(d : \mathcal {A} \rightarrow \mathcal {A}\) is a \((\sigma , \tau )\)-derivation such that \(\varphi d\) is a non-zero linear functional for every \(\varphi \in \Phi _\mathcal {A}\), then d is automatically continuous. As another objective of this research, we prove that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every Jordan \(\sigma \)-derivation \(d:\mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$

For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where P denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and Q denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).     

Pair frames in Hilbert $$\varvec{C^*}$$-modules

In this paper, we introduce pair frames in Hilbert \(C^*\)-modules and show that they share many useful properties with their corresponding notions in Hilbert spaces. We also obtain the necessary and sufficient conditions for a standard Bessel sequence to construct a pair frame and get the necessary and sufficient conditions for a Hilbert \(C^*\)-module to admit a pair frame with a symbol and two standard Bessel sequences. Moreover by generalizing some of the results obtained for Bessel multipliers in Hilbert \(C^*\)-modules to pair frames and considering the stability of pair frames under invertible operators, we construct new pair frames and show that pair frames are stable under small perturbations.