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In Ruscheweyh and Salinas (Math Z 263:607–617, 2009) and Ruscheweyh et al. (Isreal J Math 171:285–304, 2009) the researchers introduced universally convex,universally starlike and universally prestarlike functions in the slit domain $$ {\mathbb {C}}\backslash [1,\infty {)}.$$ These papers extended the corresponding notions from the unit disc to other discs and half-planes containing the origin. In this paper, we consider universally prestarlike generalized functions of order $$\alpha $$ with $$\alpha \le 1$$ and we obtain upper bounds of the second Hankel determinant $$|a_{2}a_{4}-a_{3}^{2}|$$ for such functions related with exponential function.

In this paper, we introduce the notion of $$\chi $$ -pure exact sequence. An exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$$ is said to be $$\chi $$ -pure-exact if $$0\longrightarrow A\bigotimes M\longrightarrow B\bigotimes M\longrightarrow C\bigotimes M\longrightarrow 0$$ is again an exact sequence, where A, B, C are right R-modules and $$M\simeq R/I$$ is a left R-module for $$I\in \chi $$ , where $$\chi $$ denotes the collection of left ideals of R. In this paper, we establish several equivalent conditions for a pure exact sequence to be $$\chi $$ -pure exact. We further define a $$\chi $$ -pure injective module and study the topological aspects of this module and introduce a condition under which a $$\chi $$ -pure injective module coincides with an algebraically compact module.

Let R be a commutative ring with identity and M be an R-module. Let Z(R) be the set of zero-divisors of R and Z(M) be the set of annihilators over M in R. The total graph of idealization, \(T\left( \Gamma (R(+)M)\right) \) was defined as the graph with all elements of the ring \(R(+)M =\lbrace (r,m)\mid r \in R, m \in M\rbrace \) as vertices where any two distinct vertices \((x,m), (y,n) \in R(+)M\) are adjacent if and only if \((x,m) + (y,n) \in Z(R(+)M)\), the set of zero-divisors of \(R(+)M\). In this paper we define the line graph of total graph of idealization, denoted by \(L\left( T(\Gamma (R(+)M))\right) \) as the graph with all the edges of \(T\left( \Gamma (R(+)M)\right) \) as vertices and any two distinct vertices are adjacent if and only if their corresponding edges share a common vertex in the graph \(T\left( \Gamma (R(+)M)\right) \). In this paper we discuss the diameter, girth and clique number of the graph \(L\left( T(\Gamma (R(+)M))\right) \). We also find condition for the graph \(L\left( T(\Gamma (R(+)M))\right) \) to be connected when \(Z(R(+)M)\) is not an ideal of the \(R(+)M\).

In this paper, we introduce the notion of power ternary semiring as a generalization of power semirings introduced by Sen and Bhuniya (Semigr Forum 82:131–140, 2011). The main aim of this paper is twofold. The first one is to study some properties of power ternary semirings with the help of corresponding properties of ternary semigroups. The another important aspect is—if \(S_1\) and \(S_2\) are isomorphic ternary semigroups, then the corresponding power ternary semirings \(P(S_1)\) and \(P(S_2)\) are obviously isomorphic. It is quiet natural to ask whether the converse is true, i.e. is it true that for any ternary semigroups \(S_1\) and \(S_2\) it holds: \(P(S_1)\cong P(S_2)\) implies that \(S_1\cong S_2\)? In this paper, we try to answer this question partially.

The first and second status connectivity indices of a connected graph G are defined as $$S_{1}(G) = \sum _{uv \in E(G)}[\sigma _G(u)+ \sigma _G(v)]$$ and $$S_{2}(G) = \sum _{uv \in E(G)}\sigma _G(u)\sigma _G(v)$$ respectively, where E(G) is the edge set of G and $$\sigma _G(u)$$ is the sum of distances between the vertex u and all other vertices of G. In this paper we compute the status connectivity indices of line graphs of trees. Further we give the bounds for these indices of line graphs of connected graphs.

In this paper, we address the additivity of $n$-multiplicative isomorphisms and $n$-multiplicative derivations on Gamma rings. We proved that, if $\M$ is a $\Gamma$-ring satisfying the some conditions, then any $n$-multiplicative isomorphism $\left(\varphi, \phi\right)$ of $\M$ onto an arbitrary gamma ring is additive., Comment: 10 pages

In this paper, we first introduce a generalized K-algebra (briefly, gK-algebra) $$(P, \cdot , \odot , e, ^{-1})$$ which is constructed on a non-symmetric polygroup $$(P, \cdot , e, ^{-1})$$ , by adjoining binary hyperoperation $$\odot $$ defined by $$x \odot y = x \cdot y^{-1}$$ , for all $$x, y \in P$$ . Then, we show that every gK-algebra associated to a canonical hypergroup is a hyper B-algebra with constant e. Finally, we define generalized gK-subalgebras, (prime) generalized gK-ideals, homomorphisms of $$(P, \cdot , \odot , e, ^{-1})$$ and quotient generalized gK-algebras.

In this paper, we generalize the concept of interval valued $$(\in ,\in \vee q_{\widetilde{k}})$$ -fuzzy left (right) filters in ordered semigroups introducing and studying the notion of an interval valued $$(\in ,\in \vee q_{\widetilde{ k}}^{_{\widetilde{\delta }}})$$ -fuzzy left (right) filter. Several properties and characterizations are investigated and obtained on ordered semigroups by an interval valued $$ (\in ,\in \vee q_{\widetilde{k}}^{_{\widetilde{\delta }}})$$ -fuzzy left (right) filter. Characterizations of interval valued fuzzy left (resp. right) filters and interval valued $$(\in ,\in \vee q_{\widetilde{k} }^{_{\widetilde{\delta }}})$$ -fuzzy left (resp. right) filters are considered by using implication operators and the notion of implication-based interval valued fuzzy left (resp. right) filters.

In this paper, we prove that the $$(\omega +1)$$ -projective $$\Sigma $$ -modules are direct sums of countably generated modules each of which has length at most $$(\omega +1)$$ . We also give an example that this is not true for all other $$(\omega +k)$$ -projectives with $$2\le k< \omega $$ . Certain related assertions are established as well. These results are then used to discuss the class of highly essentially finitely indecomposable modules.

In this paper, the concept of co-ordinated (log, (s, m))-preinvex functions is introduced. Some new fractional Hermite–Hadamard type inequalities based on new integral identity are established.

Let R be a commutative ring with identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R and two distinct vertices x and y are adjacent if and only if $$x+y$$ is a unit of R. In this paper, perfect unit graphs of Artinian rings are investigated.

Let $$\mathbb {D}$$ be a simple connected digraph with n vertices and m arcs and let $$W(\mathbb {D})=(\mathbb {D},\omega )$$ be the weighted digraph corresponding to $$\mathbb {D}$$ , where the weights are taken from the set of non-zero real numbers. In this paper, we define the skew Laplacian matrix $$SL(W(\mathbb {D}))$$ and skew Laplacian energy $$SLE(W(\mathbb {D}))$$ of a weighted digraph $$W(\mathbb {D})$$ , which is defined as the sum of the absolute values of the skew Laplacian eigenvalues, that is, $$SLE(W(\mathbb {D}))=\sum _{i=1}^{n}|\rho _i|$$ , where $$\rho _1,\rho _2, \ldots ,\rho _n$$ are the skew Laplacian eigenvalues of $$W(\mathbb {D})$$ . We show the existence of the real skew Laplacian eigenvalues of a weighted digraph when the weighted digraph has an independent set such that all the vertices in the independent set have the same out-neighbors and in-neighbors. We obtain a Koolen type upper bound for $$SLE(W(\mathbb {D}))$$ . Further, for a connected weighted digraph $$W(\mathbb {D})$$ , we obtain bounds for $$SLE(W(\mathbb {D}))$$ , in terms of different digraph parameters associated with the digraph structure $$\mathbb {D}$$ . We characterize the extremal weighted digraphs attaining these bounds.

The solution of $$A\otimes x{=}B{\otimes} y$$ has been considered in the literature and various methods have been established. However, a solution such that the resulting product is a vector having all its components equal has not been treated to the best of our knowledge. In this paper, we study a synchronization problem $$A\otimes x{=}B\otimes y{=}\alpha $$ and proposed an $$O(mn+mk)$$ algorithm for its solution. Where $$m$$ is the number of rows of the matrices and $$n \, and \, k$$ are the number of columns of $$A \, and \, B$$ respectively. That is, given any Two matrices that have the same number of rows, $$m$$ we introduce some algorithm that generates two column vectors $$x\, and \, y$$ such that $$A\otimes x=B\otimes y=\alpha $$ , where $$A \, and \, B$$ assumed to be P-doubly G-astic matrices having the same number of rows and $$\alpha $$ is a column vector having all its components equal.

In this paper , we obtain initial coefficient bounds for functions belonging to the subclass of bi-univalent functions $$ B^{\lambda }_{\Sigma }(f,g,h;t)$$ by using the Chebyshev polynomials. We find Fekete–Szego inequalities and also seek the upper bound for the second Hankel determinant for functions belonging to this subclass.

In this paper, the Fischer–Clifford matrices and associated character table of a maximal subgroup $$(2^9{:}(L_3(4)){:}3$$ of one of the automorphism groups $$U_6(2){:}3$$ of the unitary group $$U_6(2)$$ are constructed.

The purpose of this paper is to explore the question of embedding a zero-divisor graph and its iterated line graphs on the plane such that the resulting graphs are generalized outerplanar graphs. We give a full characterization of all zero-divisor graphs with respect to their generalized outerplanar index.

By counting on the multiset partitions, we give in this paper a combinatorial interpretation of a class of the generalized Stirling numbers generalizing the Stirling, r-Stirling, Jacobi–Stirling, r-Jacobi–Stirling and two classes of p-Stirling numbers of the second kind.

In this paper, soft intersection $$\varGamma $$ -hyperideals of ordered $$\varGamma $$ -semihypergroups over an initial universe $$\mathcal {V}$$ are defined and some properties of them with soft intersection product are obtained. Further, some characterizations of regular and intra-regular ordered $$\varGamma $$ -semihypergroups in terms of soft intersection $$\varGamma $$ -hyperideals are provided. We also introduce the notion of semisimple ordered $$\varGamma $$ -semihypergroups. Finally, we give some characterizations of semisimple ordered $$\varGamma $$ -semihypergroups in terms of soft prime and soft semiprime $$\varGamma $$ -hyperideals.

In this paper, we introduce the class $$ {\mathbf{M}}_{b,\,\lambda ,\,n}^{q} (\varphi ,\,\phi ) $$ of quasi subordinations defined by Salagean operator, which generalizes known classes and contains new classes. Also we will obtain the coefficient bounds for functions belonging to the class $$ {\mathbf{M}}_{b,\,\lambda ,\,n}^{q} (\varphi ,\,\phi ). $$ A number of known or new results are shown to follow upon specializing the parameters involved in our main results.

In this paper, we extend two efficient computational algorithms for the determinant evaluation of general cyclic heptadiagonal matrices with Toeplitz structure. We try to design two numerical algorithms by a certain type of matrix reordering in matrix partition and another algorithm by using the transformation of a block upper triangular transformation for the cyclic heptadiagonal Toeplitz matrices. The cost of these algorithms is about $$11n+O(\hbox {log}\,n)$$ for computing $$n\hbox {th}$$ order cyclic heptadiagonal Toeplitz determinants. Some numerical experiments are presented to demonstrate the performance and effectiveness of the proposed algorithms with other published algorithms.

In this paper, we prove various inclusion relationships among different classes of algebraic structures and hyperstructures of type U on the right of finite size. In particular, we consider in detail the inclusion properties $$\mathcal{G}_n\subseteq \mathcal{PUR}_n\subseteq \mathcal{HUR}_n\subseteq \mathcal{SUR}_n$$ between the classes of groups $$\mathcal{G}_n$$, polygroups $$\mathcal{PUR}_n$$, hypergroups $$\mathcal{HUR}_n$$ and semihypergroups $$\mathcal{SUR}_n$$ of type U on the right of size n and provide conditions such that the equality holds. As a particular result, we prove that every polygroup of type U on the right is a group.

Let G be a finite group which is not a cyclic p-group, p is a prime number. The undirected simple graph $$\varGamma (G)$$ whose vertices are the proper subgroups of G which are not contained in the Frattini subgroup of G and two vertices $$H_1$$ and $$H_2$$ are joined by an edge if and only if $$G=\left\langle H_1,H_2 \right\rangle $$ . In this paper, we classify all finite abelian groups G for which $$\varGamma (G)$$ has genus one.

The purpose of this paper is to present some characterizations of almost fuzzy prime ideals and sub modules; and to introduce certain innovative fuzzy prime ideals and sub modules respectively. Moreover, every fuzzy ideal in R is an almost prime ideal for a quasi-local ring \( (R, M) \) with \( M^{2} = 0 \). Moreover, prime sub module µN in a finitely generated multiplication R − module M and fuzzy prime ideal \( I_{\mu } \) in R have been constructed.

The warped product $$M_1 \times _F M_2$$ of two Riemannian manifolds $$(M_1,g_1)$$ and $$(M_2,g_2)$$ is the product manifold $$M_1 \times M_2$$ equipped with the warped product metric $$g=g_1 + F^2 g_2$$ , where F is a positive function on $$M_1$$ . The notion of warped product manifolds is one of the most fruitful generalizations of Riemannian products. Such a notion plays very important roles in differential geometry as well as in physics, especially in general relativity. In this paper we study warped product manifolds endowed with a Tripathi connection. We establish some relationships between the Tripathi connection of the warped product M to those $$M_1$$ and $$M_2$$ .

S. Ramanujan has recorded many theta-function identities in his notebooks and ‘Lost’ notebook. Inspired by the works of Ramanujan, recently M. Somos discovered several new theta-function identities using PARI/GP scripts without offering the proof, which are analogous to Ramanujan’s theta-function identities. In this paper, we give proofs for some theta-function identities of level 8 discovered by Somos. Furthermore, we extract some partition identities from them.

In the present paper, we determine necessary and sufficient conditions for zF(a, b; c; z) and $$z(2-F(a,b;c;z))$$ where $$F(a,b;c;z)=\sum \nolimits _{n=0}^{\infty }[(a)_{n}(b)_{n}/(c)_{n}(1)_{n}]z^{n}$$ to be in a certain class of analytic functions with negative coefficients. Furthermore, we consider an integral operator related to hypergeometric functions.

In this paper, we are mainly interested to find the sufficient conditions on parameters A, B, b and c that will ensure the generalized Struve function $$ u_{v,b,c}$$ satisfies the subordination $$u_{v,b,c}\left( z\right) \prec \left( 1+Az\right) /\left( 1+Bz\right) $$ .

An ai-semiring is a semiring S such that $$a+a=a$$ for all $$a \in S$$ . In the present paper, we characterize minimal k-bi-ideals and minimal strong quasi k-ideals in ai-semirings. We show that every minimal k-bi-ideal is a product of a minimal right k-ideal and a minimal left k-ideal of S, and conversely. Also we prove a number of equivalent characterizations for a k-bi-ideal to be minimal. Minimal strong quasi k-ideals are precisely intersections of a minimal left k-ideal and a minimal right k-ideal of S. As a consequence, we show that a strong quasi k-ideal is minimal if and only if it is an $${\mathcal {{\overline{H}}}}$$ -class.

This paper is devoted to introduce the notions of n-fold left-(right-)Heyting, n-fold left-(right-)Boolean and n-fold left-(right-)MV filters in residuated lattices and to investigate their properties. Several characterizations of these notions are derived. The relations between n-fold left-(right-)Boolean filters and n-fold left-(right-)Heyting filters are investigated and we prove that an n-fold left-(right-)Boolean filter is an n-fold left-(right-)Heyting filter, respectively, and this implication is strict. Also, the relations between n-fold left-(right-)Boolean filters and n-fold left-(right-)MV filters are investigated and we show that a normal n-fold left-(right-)Boolean filter is a normal n-fold left-(right-)MV filter and each n-fold left-(right-)Heyting and n-fold left-(right-)MV filter is an n-fold left-(right-)Boolean filter.

In this paper, we introduce certain new subclasses of the bi-univalent function class $$\sigma $$ in which both f and $$f^{-1}$$ are m-fold symmetric analytic with their derivatives in the class $${\mathcal {P}}$$ of analytic functions. Furthermore, we obtain coefficient bounds of $$|a_{m+1}|$$ and $$|a_{2m+1}|$$ for these new subclasses.

In this paper, we introduce the new concept isoclinism of two non-commuting graphs. We describe it with this hope to determine the properties of the graph with large number of vertices and edges more easier by use of its smaller correspondence graph in its isoclinic class. In 1939, Hekster classified the groups by n-isoclinism which was weaker than isomorphism, where n is a positive integer. The abelian groups are in the same class by group-isoclinism, although their intrinsic properties are not the same. The notion of isoclinic groups is the inspiration to define the isoclinism of two graphs. The isoclinism of two graphs is a pair of significant special isomorphism between the quotient graphs of the given graphs. We observe that all complete 3-partite non-commuting graphs are in the same isoclinic class and the non-commuting graph associated to the dihedral group of order 8, $$D_{8}$$ is the representative of this class.

A dominating set D of a connected graph $$G = (V, E)$$ is said to be bi-connected dominating set if the induced subgraphs of both $$\langle D \rangle $$ and $$\langle V-D \rangle $$ are connected. The bi-connected domination number $$\gamma _{bc}(G)$$ is the minimum cardinality of a bi-connected dominating set. A $$\gamma _{bc}$$ -set is a minimum bi-connected dominating set of G. In this paper, we obtain the Partially Balanced Incomplete Block (PBIB)-designs with m = 1, 2, 3, 4 and $$\lfloor \frac{p}{2}\rfloor $$ association schemes arising from $$\gamma _{bc}$$ -sets of some special classes of graphs.

The purpose of the present paper is to establish connections between various subclasses of analytic univalent functions by applying certain convolution operator involving Poisson distribution series. To be more precise,we investigate such connections with the classes of analytic univalent functions of $$\beta $$ -starlike and $$\beta $$ -uniformly convex functions of order $$\alpha $$ in the open unit disk $${\mathbb {U}}$$ . Further we point out some consequences of our main results.

In this paper we study separation axioms and connected properties on (semi)topological quotient BCK-algebras. We bring some conditions which under a (semi)topological quotient BCK-algebra have at least one of the topological properties $$T_1,$$ Hausdorff, regular, normal, connected, locally connected, totally disconnected space.

This paper is dealing with two split extensions of the form $$2^{8}{:}A_{9}.$$ We refer to these two groups by $$\overline{G}_{1}$$ and $$\overline{G}_{2}.$$ For $$\overline{G}_{1},$$ the 8-dimensional GF(2)-module is in fact the deleted permutation module for $$A_{9}.$$ We firstly determine the conjugacy classes of $$\overline{G}_{1}$$ and $$\overline{G}_{2}$$ using the coset analysis technique. The structures of inertia factor groups were determined for the two extensions. The inertia factor groups of $$\overline{G}_{1}$$ are $$A_{9},\,A_{8},\, S_{7},\,(A_{6} \times 3){:}2 $$ and $$(A_{5} \times A_{4}){:}2,$$ while the inertia factor groups of $$\overline{G}_{2}$$ are $$A_{9},\, PSL(2,8){:}3$$ and $$2^{3}{:}GL(3,2).$$ We then determine the Fischer matrices for these two groups and apply the Clifford–Fischer theory to compute the ordinary character tables of $$\overline{G}_{1}$$ and $$\overline{G}_{2}.$$ The Fischer matrices of $$\overline{G}_{1}$$ and $$\overline{G}_{2}$$ are all integer valued, with sizes ranging from 1 to 9 and from 1 to 4 respectively. The full character tables of $$\overline{G}_{1}$$ and $$\overline{G}_{2}$$ are $$84 \times 84$$ and $$40 \times 40$$ complex valued matrices respectively.

Let X be a Banach lattice and $$\chi ^{2}_{f}$$ be an double gai Orlicz sequence space associated to an Orlicz function with the $$\Delta _{2}$$ - condition. In this paper we define the Ces $$\grave{\mathbf{a }}$$ ro $$\chi ^{2}$$ sequence space Ces $$_{p}^{q}\left( \chi ^{2}_{f}\right) $$ generated by a Orlicz sequence space and exhibit some general properties of the spaces.

The zero divisor graph of a partially ordered set (poset, briefly) $$(P, \le )$$ with the least element 0, which is denoted by $$G^*(P)$$ , is an undirected graph with vertex set $$P^*= P{\setminus } \{0\}$$ and, for two distinct vertices x and y, x is adjacent to y in $$G^*(P)$$ if and only if $$\{x,y\}^l=\{0\}$$ , where, for a subset S of P, $$S^l$$ is the set of all elements $$x\in P$$ with $$x\le s$$ , for all $$s\in S$$ . In this paper we completely characterize all posets P with projective zero divisor graphs $$G^*(P)$$ .

In this paper, we consider the problem of planarity and outerplanarity of iterated line graphs of the zero divisor graphs for finite commutative rings. We give a full characterization of all zero divisor graphs with respect to their planarity and outerplanarity indexes.

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithms for solving symmetric optimization problems. In this paper we present a new kernel function for which interior point method yields iteration bounds $${\mathcal {O}}(\sqrt{r}\log r\log \frac{r}{\epsilon })$$ and $${\mathcal {O}}(\sqrt{r}\log \frac{r}{\epsilon })$$ for large-and small-update methods, respectively, which matches currently the best known bounds for such methods.

In this paper, we prove that there is no number with the Lehmer property in the sequence of Pell numbers.

Let \(K_{n}=\mathbb {Q}( \alpha _{2^{0}},\ldots ,\alpha _{2^{n-3}},~\mu ~\alpha _{2^{n-2}},~\mu ^{\prime }\alpha _{2^{n-1}}) \) be the compositum of the family \((\mathbb {Q} ( \alpha _{2^{k}}) ) _{0\le k\le n-1}\) of n quadratic fields which are distinct in pairs. \(K_{n}\) is so called n-quadraticnumberfield. In our paper, using Danielle Chatelain’s method, we calculate an integral basis of the integral ring \( \mathbb {Z}_{K_{3}}\) of thetriquadraticfield\(K_{3}= \mathbb {Q} ( \sqrt{dm},\sqrt{dn},\sqrt{d^{\prime }m^{\prime }n^{\prime }l}) . \) We give both an integral basis \(\mathfrak {B}_{K_{3}}\)\(\left( i.e\text { a }\mathfrak { \mathbb {Z} }\text {-basis of } \mathbb {Z} _{K_{3}}\right) \) called the\(\mathfrak { \mathbb {Z} }\text {-}basis\)of Chatelain for \( \mathbb {Z} _{K_{3}}\) and the discriminant \(\mathfrak {D}_{K_{3}/ \mathbb {Q} }\) of \(K_{3}\) over \( \mathbb {Q} \) in the following complete and general cases: \(( dm,dn,d^{\prime }m^{\prime }n^{\prime }l) \equiv \left( 1,1,1\right) ,\left( 1,1,2\right) ,\left( 1,1,3\right) \) and \(\left( 1,2,3\right) \)\((\mathrm {mod} \ 4),\) where the square-free integers dm, dn and \(d^{\prime }m^{\prime }n^{\prime }l\) are such that: \(( dm,dn) =d,\)\(\left( dmn,l\right) =1,\)\(( dm,d^{\prime }m^{\prime }n^{\prime }) =d^{\prime }m^{\prime }\) and \(( dn,d^{\prime }m^{\prime }n^{\prime }) =d^{\prime }n^{\prime }.\)

In this paper, we will investigate some hyperstability results of an n-dimensional Jensen type functional equation $$\begin{aligned} \sum _{i=1}^{n} p_{i} f(x_{i})=f\left( \sum _{i=1}^{n} p_{i} x_{i}\right) , \end{aligned}$$ where \(n>1\) is an integer, and \(p_{1},\ldots ,p_{n}\) are positive rational numbers with $$\begin{aligned} \sum _{i=1}^{n} p_{i}=1. \end{aligned}$$

In this present paper, we determine the Fischer-Clifford matrices and the associated character table of a subgroup $$\overline{G}=2^8{:}(U_4(2){:}2)$$ of the orthogonal group $$O_{10}^+(2)$$ , where $$2^8$$ is an irreducible module for $$U_4(2){:}2\cong G O_{6}^-(2)$$ . The group $$\overline{G}$$ has order 13,271,040.

In the present paper, we investigate some characterization for Poisson distribution series to be in the unified subclasses $$P_{\lambda }(\alpha )$$ and $$D_{\lambda }(\alpha )$$ of analytic functions.

In this paper, we prove the generalization of Titchmarsh’s theorem for the Fourier transform for functions satisfying the \((n,\alpha )\)-Fourier Lipschitz condition in the space \(\mathrm {L}^{2}(\mathbb {R}^{n})\).

In this paper, we introduce a new difference operator \(B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})\), defined by \((B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})x)_k=r_kx_k+s_{k-1}x_{k-1}+t_{k-2}x_{k-2}+u_{k-3}x_{k-3},\) where \(\tilde{r},\tilde{s},\tilde{t},\tilde{u}\) are convergent sequences of real numbers satisfying certain conditions and any term with negative subscript is equal to zero. In fact, as the generalizations of most of the difference operators, the operator \(B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})\) includes the difference operators \(\Delta ^{1}, \Delta ^{2}, \Delta ^{3}, \Delta _\nu ,\Delta _{uv},B(r,s)\) and B(r, s, t). We determine certain Toeplitz matrices such as Fibonacci and other summable matrices which can be obtained immediately by taking the inverse of the difference operator \(B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})\) under some limiting conditions. Moreover, their spectral and other basic properties have been studied.

In this paper, we propose a primal-dual interior point method for linear optimization (LO) based on a new parameterized kernel function. The proposed kernel function is a generalization of the one used recently in Bai et al. (SIAM J Optim 15:101–128, 2004) for LO. The investigation according to it yields the best known iteration bound \(O\left( \sqrt{n} \log n \log \frac{n}{\epsilon }\right) \) for large-update algorithm and thus improves the iteration bound obtained in Bai et al. (SIAM J Optim 15:101–128, 2004) for large-update algorithm. Finally, we present few numerical results to demonstrate the efficiency of the proposed algorithm.

This paper treats superstability problem of the generalized Wilson’s equation $$\begin{aligned} \underset{\varphi \in \Phi }{\sum }\int \limits _{G}\int \limits _{K}f(xtk \varphi (y)k^{-1})dw_{K}(k)d\mu (t)=\left| \Phi \right| f(x)g(y),~\ \ \ x,y\in G, \end{aligned}$$ where G is an arbitrary locally compact group, that need not be abelian, K is a compact subgroup of G, \(\omega _{K}\) is the normalized Haar measure of K, \(\Phi \) is a finite group of K-invariant morphisms of G, \(\mu \) is a complex measure with compact support and f, g\( :G\longrightarrow \mathbb {C}\) are continuous complex-valued functions. We dont impose any condition on the continuous function f. In addition, superstability problem for a large class of related functional equations are considered.

The aim of this paper is to study $$\mathsf{H}_v\mathsf{MV}$$ -algebras from fuzzy set theory point of view. Some types of fuzzy ideals are introduced and many properties, characterizations and related results are given. Particularly, the fuzzy weak $$\mathsf{H}_v\mathsf{MV}$$ -ideal generated by a fuzzy subset is characterized. In the sequel, some kinds of fuzzy congruences are introduced and characterizations of them are obtained. Finally, some homomorphism theorems are stated and proved.

Let \(P(z)=a_{0}+\sum _{\nu =\mu }^{n}a_{\nu }z^{\nu },~1\le \mu \le n,\) be a polynomial of degree \(n\) such that \(P(z)\ne 0\) in \(|z| 0,\) then for \(0 1\) by proving $$\begin{aligned} \max _{|z|=1}|P(Rz)-\beta P(rz)|\le & {} \left[ (|\beta |+|1-\beta |)\left( \frac{R^{\mu }+k^{\mu }}{r^{\mu }+k^{\mu }}\right) ^{\frac{n}{\mu }}-|\beta |\right] \max _{|z|=r}|P(z)|\\&-\left[ \left( \frac{R^{\mu }+k^{\mu }}{r^{\mu }+k^{\mu }}\right) ^{\frac{n}{\mu }}-1\right] \min _{|z|=k}|P(z)| \end{aligned}$$ and $$\begin{aligned} \max _{|z|=R}|P(\ell z)-P(z)|\le \frac{R^{\mu }(\ell ^{n}-1)}{r^{\mu }+k^{\mu }}\left( \frac{R^{\mu }+k^{\mu }}{r^{\mu }+k^{\mu }}\right) ^{\frac{n}{\mu }-1}\left\{ \max _{|z|=r}|P(z)|-\min _{|z|=k}|P(z)|\right\} . \end{aligned}$$ In this paper, we obtain improvements of the above inequalities by involving some of the coefficients of the polynomial \(P(z).\) Besides this, our results also generalize and refines the results of Barchand and Dewan (J Math Anal Appl 336:171–179, 2007).