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In this paper, we establish coincidence point and common fixed point theorems involving two pairs of weakly compatible mapping satisfying contraction condition with complex coefficient are proved in complex valued metric space. The presented theorems generalize, extend and improve many existing results in the literature. An example is given at the end of the paper.

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

In this paper, we present a refinement of the global Langlands correspondence for discrete symplectic motives of rank 2n over Q. To such a motive Langlands conjecturally associates a generic, automorphic representation of the split orthogonal group SO2n+1 over Q, which appears with multiplicity one in the cuspidal spectrum. Using the local theory of generic representations of odd orthogonal groups, we define a new vector F in this representation, which is the tensor product of local test vectors for the Whittaker functionals [9]. I hope that the defining properties ofF will make it easier to investigate the Langlands correspondence computationally, especially for the cohomology of algebraic curves. Our refinement is similar to the refinement that Weil [24] proposed for the conjecture that elliptic curves over Q are modular. Namely, Weil proposed that such a curve should be associated with a homomorphic newform F = P anq n of weight 2 on 0(N), where N is equal to the conductor of the curve. This paper expands on a letter that I wrote to Serre in 2010. It was motivated by a question Serre posed at my 60th birthday conference, and a suggestion Brumer made of a family of discrete subgroups generalizing 0(N). I would like to thank them, and to thank Deligne for his comments.

We study etale extensions of rings that have FIP., Comment: The paper entitled FIP and FCP products of ring morphisms (arXiv: 1312.1250 [math.AC]) is now split into three papers. The present paper contains the last section of the original paper and many other results on etale FIP extensions

In this paper, we study some uniqueness problems of meromorphic functions when certain nonlinear differential polynomials generated by them share a nonconstant polynomial. The results of the paper improve the concerning results due to Xu et al. (Mat Vesnik 64:1–16, 2012).

The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.

In this paper, we give some properties of the generalized derivation algebra \(\mathrm{GDer}(T)\) of a Hom–Lie triple systems T. In particular, we prove that \(\mathrm{GDer}(T) = \mathrm{QDer}(T) + \mathrm{QC}(T)\), the sum of the quasiderivation algebra and the quasicentroid. We also prove that \(\mathrm{QDer}(T)\) can be embedded as derivations in a larger Hom–Lie triple system. General results on centroids of Hom–Lie triple systems are also developed in this paper.

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (fi nitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial functors from (finitely generated free) P-algebras to abelian groups and from (finitely generated free) groups to abelian groups are described explicitely by their cross-e ffects and maps relating them which satisfy a list of relations., Comment: 58 pages

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants ofmcopies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|.

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety X ⊂ P d of an arbitrary codimension l with respect to a real l − 1 -dimensional linear subspace V ⊂ P d and study its basic properties. We also consider a class of determinantal representations that we call Livsic-type and a nice subclass of these that we call very reasonable. Much like in the case of hypersurfaces ( l = 1 ), the existence of a definite Hermitian very reasonable Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a very reasonable Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in P d hyperbolic with respect to some real d − 2 -dimensional linear subspace admits a definite Hermitian, or even definite real symmetric, very reasonable Livsic-type determinantal representation.

A Lie algebra L over a field $$\mathbb {F}$$ is said to be zero product determined (zpd) if every bilinear map with the property that $$f(x,y)=0$$ , whenever x and y commute, is a coboundary. The main goal of the paper is to determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra , where V is a simple $$\mathfrak {sl}_2$$ -module, is zpd if and only if $$\dim V =2$$ or $$\dim V$$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $$\mathscr {L}_q$$ and $$\mathscr {L}^+_q$$ , the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most 3, while the class of non-zpd Lie algebras includes the (4-dimensional) aging Lie algebra and all Lie algebras of dimension more than 3 in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of zpd Lie algebra by using it in the study of commutativity preserving linear maps.

A closed linear relation T in a Banach space X is called left (resp. right) Fredholm if it is upper (resp. lower) semiFredholm and its range (resp. null space) is topologically complemented in X . We say that T is left (resp. right) Browder if it is left (resp. right) Fredholm and has a finite ascent (resp. descent). In this paper, we analyze the stability of the left (resp. right) Fredholm and the left (resp. right) Browder linear relations under commuting Riesz operator perturbations. Recent results of Zivkovic et al. to the case of bounded operators are covered.

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given., 19 pages, to appear in Advances in Mathematics

In this paper we pose some questions about superderivations on $${\mathbb {Z}}_{2}$$ -graded rings. Then we consider the quaternion rings and upper triangular matrix rings with special $${\mathbb {Z}}_{2}$$ -gradings and we check the answer to these questions about them.

The present paper is devoted to the study of some subclasses of H-rings, i.e., rings in which all subrings are ideals. In the description of H-rings an important role is played by almost null rings. We classify, up to isomorphism, torsion almost null rings of bounded exponent.

An element in a ring R with identity is said to be strongly nil clean if it is the sum of an idempotent and a nilpotent that commute, R is said to be strongly nil clean if every element of R is strongly nil clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. Then R is said to be strongly g(x)-nil clean if every element in R is a sum of a nilpotent and a root of g(x) that commute. In this paper, we give some relations between strongly nil clean rings and strongly g(x)-nil clean rings. Various basic properties of strongly g(x) -nil cleans are proved and many examples are given.

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

In this paper we identify the structure of complex finite-dimensional Leibniz algebras with associated Lie algebras $$sl_2^1\oplus sl_2^2\oplus \dots \oplus sl_2^s\oplus R,$$ where R is a solvable radical. The classifications of such Leibniz algebras in the cases $$\mathrm{dim} R=2, 3$$ and $$\mathrm{dim} I\ne 3$$ have been obtained. Moreover, we classify Leibniz algebras with $$L/I\cong sl_2^1\oplus sl_2^2$$ and some conditions on ideal $$I=\mathrm{id} $$ .

Let $a,b$ be two commutative generalized Drazin invertible elements in a Banach algebra; the expressions for the generalized Drazin inverse of the product $ab$ and the sum $a+b$ were studied in some current literature on this subject. In this paper, we generalize these results under the weaker conditions $a^{2}b=aba$ and $b^{2}a=bab$. As an application of our results, we obtain some new representations for the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra, extending some recent works.

In this paper, some additive properties of the pseudo Drazin inverse are obtained in a Banach algebra. In addition, we find some new conditions under which the pseudo Drazin inverse of the sum a + b can be explicitly expressed in terms of a, az, b, bz. In particular, necessary and sufficient conditions for the existence as well as the expression for the pseudo Drazin inverse of the sum a+b are obtained under certain conditions. Also, a result of Wang and Chen [Pseudo Drazin inverses in associative rings and Banach algebras, LAA 437(2012) 1332-1345] is extended.

The main purpose of this paper is to introduce and investigate the concepts of lacunary strong summability of order and lacunary statistical convergence of order of real-valued functions which are measurable (in the Lebesgue sense) in the interval (1,∞). Some relations between lacunary statistical convergence of order and lacunary strong summability of order are also given.

In this paper, we introduce and study the notions of Gorenstein n-FP-injective and Gorenstein n-flat complexes, which are special cases of Gorenstein FP-injective and Gorenstein flat complexes respectively. We prove that over a left n-coherent ring R the class of all Gorenstein n-FP-injective (resp., Gorenstein n-flat) complexes is injectively (resp., projectively) resolving and we discuss the relationship between Gorenstein n-FP-injective and Gorenstein n-flat complexes.

This paper concerns mainly on various ring properties of some subrings of a skew triangular matrix ring. Necessary and sufficient conditions are obtained for a skew triangular matrix ring with constant diagonal over a ring to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, clean, (weakly) nil clean, exchange, uniquely (nil) clean, Hermite ring, stably finite, semiregular and I-ring. It is also proved that the projective-free property of a ring preserves by a skew triangular matrix ring with constant diagonal.

In this paper, we investigate the uniform convergence of continuous linear set-valued functions on compact sets. We also consider conditions under which the family of continuous linear extensions of a differential iteration semigroup of continuous linear set-valued functions is a differentiable iteration semigroup. In particular, since the cones and normed spaces are not supposed to be complete our main results generalize some recent results on Hukuhara’s derivative of set-valued functions.

We study some properties of the generalized local cohomology modules [Formula: see text] of [Formula: see text] with respect to a pair of ideals [Formula: see text] in Serre subcategories. Some results concerning to [Formula: see text]-cofinite modules are also given in this paper.

In the present paper, we determine the complex-valued solutions (f, g) of the functional equation $$f(x\sigma(y))+f(\tau(y)x)=2f(x)g(y),$$ in the setting of groups and monoids that need not be abelian, where \({\sigma,\tau}\) are involutive automorphisms. We prove that their solutions can be expressed in terms of multiplicative and additive functions.

In this paper, we deal with a new approach to quasipolarity notion for rings, namely an element a of a ring R is called weakly nil-quasipolar if there exists \(p^2 = p\in comm^2(a)\) such that \(a + p\) or \(a-p\) is nilpotent, and the ring R is called weakly nil-quasipolar if every element of R is weakly nil-quasipolar. The class of weakly nil-quasipolar rings lies properly between the classes of nil-quasipolar rings and quasipolar rings. Although it is an open problem whether strongly clean (even quasipolar) rings have stable range one, we show that there is an affirmative answer for weakly nil-quasipolar rings. It is also proved that if R is a weakly nil-quasipolar NI ring, then R / N(R) is commutative. Moreover, we consider the question of when certain \(2 \times 2\) matrices over a commutative local ring is weakly nil-quasipolar.

This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos’ results on irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.

In this paper we consider a second-order Sturm-Liouville operator of the form $$l(y): = - [p(t)y^\Delta (t)]^\nabla + q(t)y(t)$$ on bounded time scales. In this study, we construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self-adjoint and other extensions of the dissipative Sturm-Liouville operators in terms of boundary conditions. Using Krein’s theorem, we proved a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators on bounded time scales.

In this paper, we prove a Tauberian theorem for the product of the Abel method and the Cesàro method of order α, which improves some classical Tauberian theorems for the Abel and Cesàro summability methods.

In this paper, we discuss the analytic representations of q-Euler sums which involve q-harmonic numbers through q-polylogarithms, either linearly or nonlinearly, and give explicit formulae for several classes of q-Euler sums in terms of q-polylogarithms and q-special functions. Furthermore, we develop new closed form representations of sums of quadratic and cubic parametric q-Euler sums. Finally, we can find that the q-Euler sums are reducible to the classical Euler sums when q approaches 1.

In this paper we study the spectral properties of (m, C)-isometric operators. In particular, if $$T\in \mathcal{{L(H)}}$$ is (m, C)-isometric operators, then the power of (m, C)-isometric operators is also (m, C)-isometric operators. Moreover, if $$T^{*}$$ has the single-valued extension property, then T has the single-valued extension property. Finally, we investigate conditions for (m, C)-isometric operators to be (1, C)-isometric operators.

It is well-known that a linear stochastic operator (a Markov operator) associated with a square stochastic matrix is a surjection of the simplex if and only if it is bijective. The similar problem was open for nonlinear stochastic operators (nonlinear Markov operators) associated with stochastic hyper-matrices (higher dimensional matrices). In this paper, we solved this problem for quadratic stochastic operators acting on the simplex. Namely, we showed that a quadratic stochastic operator associated with a cubic stochastic matrix is a surjection of the simplex if and only if it is bijective. Moreover, we also described all surjective quadratic stochastic operators of the simplex.

This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra Sα. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.

In this paper, we study mod-retractable modules, CSL-modules, fully Kasch modules, and their interrelations. Right fully Kasch rings are described. It is proved that for a module M of finite length, the following conditions are equivalent. (1) In the category σ(M), every module is retractable. (2) In the category σ(M), every module is coretractable. (3) M is a CSL-module. (4) Ext 1 (S 1 , S 2) = 0 for any two simple nonisomorphic modules S 1 , S 2 ∈ σ(M). (5) M is a fully Kasch module.

In this paper, we extend the definition of weakly second submodule of a module over a commutative ring to a module over an arbitrary ring. First, we investigate some properties of weakly second submodules. We define the notion of weakly second radical of a submodule and determine the weakly second radical of some modules. We also define the notion of weak m*-system and characterize the weakly second radical of a submodule in terms of weak m*-systems. Then we introduce and study a topology on the set of all weakly second submodules of a module. We give some results concerning irreducible subsets, irreducible components and compactness of this topological space. Finally, we investigate this topological space from the point of view of spectral spaces.

As is well known, for any operator T on a complex separable Hilbert space, T has the polar decomposition T = U |T | , where U is a partial isometry and |T | is the nonnegative operator (T ∗T ) 1 2 . In 2014, Tian et al. proved that on a complex separable infinite dimensional Hilbert space, any operator admits a polar decomposition in a strongly irreducible sense. More precisely, for any operator T and any e > 0, there exists a decomposition T = (U +K)S , where U is a partial isometry, K is a compact operator with ||K|| < e , and S is strongly irreducible. In this paper, we will answer the question for operators on two-dimensional Hilbert spaces.

In this paper we establish some new Tauberian theorems for the statistical weighted mean method of summability via the weighted general control modulo of the oscillatory behavior of nonnegative integer order of a real sequence. The main results improve the well-known classical Tauberian theorems which are given for weighted mean method of summability and statistical convergence.

In this paper, by establishing free-probabilistic models on the Hecke algebras \(\mathcal{H}\left(GL_{2}(\mathbb{Q}_{p})\right)\) induced by \(p\)-adic number fields \(\mathbb{Q}_{p}\), we construct free probability spaces for all primes \(p\). Hilbert-space representations are induced by such free-probabilistic structures. We study \(C^{*}\)-algebras induced by certain partial isometries realized under the representations.

In this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.

In the present paper we introduce the notion of a positive linear relation and we investigate the class of such operators. As well as placing the theory of positive operators in a natural setting, this structure seems to be interesting for the study of abstract boundary value problems.

The aim of this paper is to generalize the notion of pseudo-valuation to modules over arbitrary commutative rings. We generalize the notion of strongly prime ideal, as defined in Badawi et al. (Lecture Notes in Pure and Applied Mathematics 185:57–67, 1997, to the notion of strongly prime submodule. We define a module M to be a pseudo-valuation module if every prime submodule of M is strongly prime. It is shown that if M has a maximal submodule N, then M is pseudo-valuation if and only if N is strongly prime. Also, we characterize strongly prime submodules in pseudo-valuation modules. We investigate some properties of these modules, and study relations between some structures and these modules.

For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\mathrm{End}(V)$ has an inner inverse, i.e., an element $y\in\mathrm{End}(V)$ satisfying $xyx=x.$ We show here that a large class of such $x$ have inner inverses $y$ that satisfy with $x$ an infinite family of additional monoid relations, making the monoid generated by $x$ and $y$ what is known as an inverse monoid (definition recalled). We obtain consequences of these relations, and related results. P. Nielsen and J. \v{S}ter, in a paper to appear, show that a much larger class of elements $x$ of rings $R,$ including all elements of von Neumann regular rings, have inner inverses satisfying arbitrarily large finite subsets of the abovementioned set of relations. But we show by example that the endomorphism ring of any infinite-dimensional vector space contains elements having no inner inverse that simultaneously satisfies all those relations. A tangential result proved is a condition on an endomap $x$ of a set $S$ that is necessary and sufficient for $x$ to belong to an inverse submonoid of the monoid of all endomaps of $S.$, Comment: 18pp. The main change from the preceding version is the discussion of three questions posed by the referee, two on p.10, starting on line 6, and one starting at the top of p.16. There are also many small revisions of wording etc

It has been shown that in characteristic zero the generators of the minimal supervarieties of finite basic rank belong to the class of minimal superalgebras introduced by Giambruno and Zaicev (Trans Am Math Soc 355:5091–5117, 2003). In the present paper the complete list of minimal supervarieties generated by minimal superalgebras whose maximal semisimple homogeneous subalgebra is the sum of three graded simple algebras is provided. As a consequence, we negatively answer the question of whether any minimal superalgebra generates a minimal supervariety.

In this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between normed spaces and Cesaro convergence. Also, we obtain a new version of the Orlicz–Pettis Theorem by means of Cesaro summability.

Let (L, λ) be a locally solid Riesz space. (L, λ) is said to have the Levi property if for every increasing λ-bounded net (xα) ⊂ L+, sup xα exists. The Levi property, appearing in literature also as weak Fatou property (Luxemburg and Zaanen), condition (B) or monotone completeness (Russian terminology), is a classical object of investigation. In this paper we are interested in some variations of the property, their mutual relationships and applications in the theory of topological Riesz spaces. In the first part of the paper we clarify the status of two problems of Aliprantis and Burkinshaw. In the second part we study ideal-injective Riesz spaces.

In the paper, we study the uniqueness problems of meromorphic functions whose certain nonlinear differential polynomials share certain value or fixed point counting multiplicity and obtain two theorems which improve some previous results as well as two recent results due to Bhoosnurmath and Kabbur (Tamkang J. Math. 44: 11–22 2013).

This paper gives a characterization of the asymptotic limit AT associated to a contraction T that is similar to a normal operator (Theorem 2). Extensions from contractions to power bounded operators intertwined to a contraction with a \({\mathcal{C}_{0}}\). completely nonunitary part (not necessarily a normaloid contraction) are considered as well (Theorem 1). It is also given a characterization of the asymptotic limit AT for a hyponormal contraction T, and it is shown that if a hyponormal contraction has no nontrivial invariant subspace, then one of the defect operators is not finite-rank (Corollary 1).

In this paper we prove, using inequalities between infinite cardinals, that, if R is an hereditary ring, the contravariant derived functor $$\mathrm {Ext}^{1}_{R}(-,G)$$ commutes with direct products if and only if G is an injective R-module.

The well-known results of M. G. Kreĭn concerning the description of selfadjoint contractive extensions of a hermitian contraction \(T_1\) and the characterization of all nonnegative selfadjoint extensions \({{\widetilde{A}} }\) of a nonnegative operator A via the inequalities \(A_K\le {{\widetilde{A}} } \le A_F\), where \(A_K\) and \(A_F\) are the Kreĭn–von Neumann extension and the Friedrichs extension of A, are generalized to the situation, where \({{\widetilde{A}} }\) is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators \(I-T_1^*T_1\) and A, respectively; these conditions are automatically satisfied if \(T_1\) is contractive or A is nonnegative, respectively. The approach developed in this paper starts by establishing first a generalization of an old result due to Yu. L. Shmul’yan on completions of nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M. G. Kreĭn and, in addition, to solve some related lifting problems for J-contractive operators in Hilbert, Pontryagin and Kreĭn spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.