Search

In this paper, we formulate best proximity pair theorems for noncyclic relatively ρ-nonexpansive mappings in modular spaces in the setting of proximal ρ-admissible sets. As a companion result, we establish a best proximity pair theorem for pointwise noncyclic contractions in modular spaces. To that end, we provide some examples throughout the paper to illustrate the validity of the obtained results. Keywords: Best proximity pair, Modular spaces, Relatively ρ-nonexpansive mappings, ρ-admissible sets, ρ-normal structure, Mathematics Subject Classification: 47H09, 41A65

The paper is devoted to the study of conformally flat Lie groups with left-invariant (pseudo) Riemannianmetric of an algebraic Ricci soliton. Previously conformally flat algebraic Ricci solitons on Lie groups have been studied in the case of small dimension and under an additional diagonalizability condition on the Ricci operator. The present paper continues these studies without the additional requirement that the Ricci operator be diagonalizable. It is proved that any nontrivial conformally flat algebraic Ricci soliton on a Lie group must be steady and have Ricci operator of Segre type {(1... 1 2)} with a unique eigenvalue (equal to 0).

For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

In this paper, we consider the square function(Sf)(x)=(∫0∞|(f⊗Φt)(x)|2dtt)1/2$$\begin{array}{} \displaystyle (\mathcal{S}f)(x)=\left( \int\limits_{0}^{\infty }|(f\otimes {\it\Phi}_{t})\left( x\right) |^{2}\frac{dt}{t}\right) ^{1/2} \end{array} $$associated with the Bessel differential operatorBt=d2dt2+(2α+1)tddt,$\begin{array}{} B_{t}=\frac{d^{2}}{dt^{2}}+\frac{(2\alpha+1)}{t}\frac{d}{dt}, \end{array} $α> −1/2,t> 0 on the half-line ℝ+= [0, ∞). The aim of this paper is to obtain the boundedness of this function inLp,α,p> 1. Firstly, we provedL2,α-boundedness by means of the Bessel-Plancherel theorem. Then, its weak-type (1, 1) andLp,α,p> 1 boundedness are proved by taking into account vector-valued functions.

Here we show existence of numerous subsets of Euclidean and metric spaces that, despite having empty interior, still support Poincar\'e inequalities. Most importantly, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the notion of uniform domain of Martio and Sarvas. Our condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components. In doing so, our results hold true for metric spaces equipped with doubling measures and Poincar\'e inequalities in general, and for the Heisenberg groups in particular. To our knowledge, these are the first examples of such subsets on any step-2 Carnot group. Such subsets also give, in general, new examples of Sobolev extension domains on doubling metric measure spaces. When specialized to the plane, we give general sufficient conditions for planar subsets, possibly with empty interior, to be Ahlfors 2-regular and to satisfy a (1,2)-Poincar\'e inequality. In the Euclidean case, our construction also covers the non-self-similar Sierpi\'nski carpets of Mackay, Tyson, and Wildrick, as well as higher dimensional analogues not treated in the literature. The analysis of the Poincar\'e inequality with exponent p=1, for these carpets and their higher dimensional analogues, includes a new way of proving an isoperimetric inequality on a space without constructing Semmes families of curves., Comment: 67 pages, 3 figures. A number of typos fixed, and Section 5 of the paper was removed. It appears now in a much expanded form in a paper "Isoperimetric and Poincar\'e inequalities on non-self-similar Sierpi\'nski sponges: the borderline case''

In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated.

In the paper we use the notion of weakly weighted sharing and relaxed weighted sharing to investigate the uniqueness problems when two difference products of entire functions share a small function. The results of the paper improve and extend some recent results due to the present first author [Commu. Math. Stat., 3 (2015), 227-238].

Kisin and Pappas constructed integral models of Hodge-type Shimura varieties with parahoric level structure at $p>2$, such that the formal neighbourhood of a mod~$p$ point can be interpreted as a deformation space of $p$-divisible group with some Tate cycles (generalising Faltings' construction). In this paper, we study the central leaf and the closed Newton stratum in the formal neighbourhoods of mod~$p$ points of Kisin-Pappas integral models with parahoric level structure; namely, we obtain the dimension of central leaves and the almost product structure of Newton strata. In the case of hyperspecial level strucure (i.e., in the good reduction case), our main results were already obtained by Hamacher, and the result of this paper holds for ramified groups as well., 33 pages; section 2.5 added to fill in the gap in the earlier version

A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of when two algebras are homotopically Morita equivalent. We discuss this special case in the first part of the paper, extending previously known results. To approach the general question, we introduce the notion of a 'braided bimodule' and show that adjunctions between A-Mod and B-Mod that lift to adjunctions between (A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between (A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when a braided bimodule induces a Quillen equivalence. In particular, we obtain criteria for when a morphism of corings induces a Quillen equivalence, providing a homotopic generalization of results by Hovey and Strickland on Morita equivalences of Hopf algebroids. To illustrate the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring., Comment: 46 pages

The CMV matrix is the five-diagonal matrix that represents the operator of multiplication by the independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle C. The article by Cantero, Moral, and Velazquez, published in 2003 and describing this matrix, has attracted much attention because it implies that the conventional orthogonal polynomials on C can be interpreted as the characteristic polynomials of the leading principal submatrices of a certain five-diagonal matrix. The present paper recalls that finite-dimensional sections of the CMV matrix appeared in papers on the unitary eigenvalue problem long before the article by Cantero et al. was published. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety $\mathbf{B}$ of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety $\mathbf{B}_{R}$ of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices ${\mathcal{L}}(\mathbf{B})$ and ${\mathcal{L}}(\mathbf{B}_{R})$ of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval $[\mathbf{D},\mathbf{D}\vee \mathbf{M}]$, every subvariety of $\mathbf{B}$ is induced from a member of $\mathbf{B}_{R}$ and vice versa. Here $\mathbf{D}$ is generated by the three-element left restriction semigroup $D$ and $\mathbf{M}$ is the variety of monoids. The analogues hold for pseudovarieties.

In this paper a general fixed point theorem for mappings with a new type of common limit range property satisfying a mixed implicit relation is proved. In the last part of the paper, as application, some fixed point results for mappings satisfying contractive conditions of integral type and for φ-contractive mappings are obtained.

The value distribution and, in particular, the numbers of a-points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a-points and Ahlfors islands, which play to a certain extend a role similar to that of the numbers of a-points, have been considered in some recent papers. In this paper, we consider meromorphic functions in a given domain, which are the solutions of some higher order equations and largely generalize the solutions of Painlevé equations 3–6. We give the upper bounds for the numbers of good a-points and Ahlfors islands of similar solutions.

In this paper we study non-compact self-shrinkers first in general codimension and then in codimension 1. We respectively prove some vanishing theorems giving rise to rigidity of the self-shrinker and then estimates involving the higher order mean curvatures for the oriented case. The paper ends with some results on their index when considered as appropriate $$\bar f$$ -minimal hypersurfaces.

In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions for this branch. At the end of the paper we study the numerical behavior of the approximants of these expansions.

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.

In 2002, in the paper entitled “A subspace theorem approach to integral points on curves”, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt’s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt’s subspace in Nevanlinna theory is H. Cartan’s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan’s original theorem. We call such method “a Cartan’s Second Main Theorem approach”. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the (1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are Kahler Calabi–Yau surfaces and Hopf surfaces.

This paper establishes a version of Nevanlinna theory based on Askey–Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane C . A second main theorem that we have derived allows us to define an Askey–Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole-scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey–Wilson type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey–Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey–Wilson. This paper concludes with an application to difference equations generalising the Askey–Wilson second-order divided difference equation.

The present paper modifies and studies jumping finite automata so they always perform two simultaneous jumps according to the same rule. For either of the two simultaneous jumps, it considers three natural directions – (1) to the left, (2) to the right, and (3) in either direction. According to this jumping-direction three-part classification, the paper investigates the mutual relation between the language families resulting from jumping finite automata performing the jumps in these ways and the families of regular, linear, context-free, and context-sensitive languages. It demonstrates that most of these language families are pairwise incomparable. In addition, many closure and non-closure properties of the resulting language families are established.

In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Holder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.

This paper is devoted, first of all, to give a complete unified proof of the characterization theorem for compact generalized Kahler manifolds. The proof is based on the classical duality between “closed” positive forms and “exact” positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where “exact” positive forms seem to play a more significant role than “closed” forms. In this setting, we state the appropriate characterization theorems and give some interesting applications.

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree $N$. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials $\{x^N\}_{N=0}^\infty$. We propose a family of solvable $N$-body problems such that their stable equilibria are the zeros of certain Ulam polynomials., This version contains clarifications of the exposition to match the published version of the paper

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.

In the papers [1] and [6], for an inverse sequence of pointed topological spaces and fibrations preserving the base points E = E 1 ← p 1 E 2 ← p 2 ⋯ ← p m E m + 1 , E=E_{1}\xleftarrow{p_{1}}E_{2}\xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}E_{m+1}, there exists an exact sequence * → lim ← ( 1 ) [ X , Ω E m ] → [ X , lim ← E ] → lim ← ( 1 ) [ X , E m ] → * . *\rightarrow{\varprojlim}^{(1)}[X,\Omega E_{m}]\rightarrow[X,\varprojlim E]% \rightarrow{\varprojlim}^{(1)}[X,E_{m}]\rightarrow*. In the present paper, for an inverse sequence of pointed topological semi-simplicial sets and fibrations preserving base points E ¯ = E ¯ ← p 1 1 E ¯ ← p 2 2 ⋯ ← p m E ¯ ← m + 1 ⋯ , \underline{E}=\underline{E}{}_{1}\xleftarrow{p_{1}}\underline{E}{}_{2}% \xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}\underline{E}{}_{m+1}\xleftarrow{% \hphantom{p_{1}}}\cdots, an analogous formula is proved.

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of π 0 {\pi_{0}} .

In the present paper we study the generic ranks of special matrix-valued maps defined by certain systems of parameters via Kronecker products. We introduce the notions of minimal superabundant, balanced and reducible systems. The main result of the paper is a theorem for maps with minimal superabundant systems of parameters. For such systems it associates the value of the generic rank with the balancedness. The proof of this theorem is based on a reduction by the parameters and consists of verifying the fact of reducibility.

In this paper we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r >= 1). In the case of the C1-topology we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum C1- robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of f are continuous in the weak*-topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The later can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures., 17 pages, 3 figures. In this new version, due to a mistake on the proof of Theorem 4 of the first version of the paper, we remove Section 2.4. (Regularity of conjugacy classes). Moreover, we introduce a new result on the Cr-topology (see Theorem 1 on this new version) in the same line of the ones obtained in the C1-topology

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

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**

A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks (see [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I, 350, 2012, 861–865].

In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket { − , − } and Turaev cobracket δ on the K -span of homotopy classes of free loops on Σ. Applying an expansion θ : K π → K 〈 x 1 , … , x n 〉 yields an algebraic description of the operations { − , − } and δ in terms of non-commutative variables x 1 , … , x n . If Σ is a surface of genus g = 0 the lowest degree parts { − , − } − 1 and δ − 1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31] . It was conjectured by the second and the third authors that one can define an expansion θ such that { − , − } = { − , − } − 1 and δ = δ − 1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24] , Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2] ).

We investigate N\'eron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a priori) real numbers between 0 and 1, called "jumps". The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions, and generalize Raynaud's description of the identity component of the N\'eron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence which decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of N\'eron models. Previously, only split semiabelian varieties were known to have this property., Comment: 37 pages. Corrected two minor inaccuracies (added a factor of 1/[K':K] in the definition of Chai's base change conductor, and added the condition "purely wild" in Theorem 2.11)

In this paper, we introduce some classes of R-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set ℍ = { M ∣ M ⁢ is an ⁢ R ⁢ -module and ⁢ Nil ⁢ ( M ) ⁢ is a divided prime submodule of ⁢ M } . \mathbb{H}=\bigl{\{}M\mid M\text{ is an }R\text{-module and }\mathrm{Nil}(M)% \text{ is a divided prime submodule of }M\bigr{\}}. For an R-module M ∈ ℍ {M\in\mathbb{H}} , set T = ( R ∖ Z ⁢ ( R ) ) ∩ ( R ∖ Z ⁢ ( M ) ) {T=(R\setminus Z(R))\cap(R\setminus Z(M))} , 𝔗 ⁢ ( M ) = T - 1 ⁢ M {\mathfrak{T}(M)=T^{-1}M} and P = ( Nil ( M ) : R M ) {P=(\mathrm{Nil}(M):_{R}M)} . In this case, the mapping Φ : 𝔗 ⁢ ( M ) → M P {\Phi:\mathfrak{T}(M)\to M_{P}} given by Φ ⁢ ( x / s ) = x / s {\Phi(x/s)=x/s} is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M into M P {M_{P}} given by Φ ⁢ ( x ) = x / 1 {\Phi(x)=x/1} for every x ∈ M {x\in M} . A nonnil submodule N of M is said to be Φ-invertible if Φ ⁢ ( N ) {\Phi(N)} is an invertible submodule of Φ ⁢ ( M ) {\Phi(M)} . Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if Φ ⁢ ( N ) {\Phi(N)} is a principal ideal of Φ ⁢ ( M ) {\Phi(M)} for every finitely generated submodule N of the R-module M. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R-modules.

Since its invention in 1979, the Feichtinger algebra has become a very useful Banach space of functions with applications in time-frequency analysis, the theory of pseudo-differential operators and several other topics. It is easily defined on locally compact abelian groups and, in comparison with the Schwartz(-Bruhat) space, the Feichtinger algebra allows for more general results with easier proofs. This review paper gives a linear and comprehensive deduction of the theory of the Feichtinger algebra and its favourable properties. The material gives an entry point into the subject, but it will also give new insight to the expert. A main goal of this paper is to show the equivalence of the many different characterizations of the Feichtinger algebra known in the literature. This task naturally guides the paper through basic properties of functions that belong to the space, over operators on it and to aspects of its dual space. Further results include a seemingly forgotten theorem by Reiter on operators which yield Banach space isomorphisms of the Feichtinger algebra; a new identification of the Feichtinger algebra as the unique Banach space in $L^{1}$ with certain properties; and the kernel theorem for the Feichtinger algebra. A historical description of the development of the theory, its applications and related function space constructions is included., noteable changes, additions and corrections in section 5-9

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H2(D2). A closed subspace M in H2(D2) is called a submodule if z i M ⊂ M (i = 1, 2). An associated integral operator (defect operator) C M captures much information about M. Using a Kreĭn space indefinite metric on the range of C M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.

In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten--von Neumann norm $\boldsymbol S_p$, $1\le p\le\infty$, for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$. In other words, we prove that for $p\in[1,\infty]$, there is no constant $K>0$ such that the inequality \begin{align*} \|f(A_1,B_1,C_1)&-f(A_2,B_2,C_2)\|_{\boldsymbol S_p}\\[.1cm] &\le K\|f\|_{B_{\infty,1}^1} \max\big\{\|A_1-A_2\|_{\boldsymbol S_p},\|B_1-B_2\|_{\boldsymbol S_p},\|C_1-C_2\|_{\boldsymbol S_p}\big\} \end{align*} holds for an arbitrary function $f$ in $B_{\infty,1}^1({\Bbb R}^3)$ and for arbitrary finite rank self-adjoint operators $A_1,\,B_1,\,C_1,\,A_2,\,B_2$ and $C_2$., 14 pages. arXiv admin note: substantial text overlap with arXiv:1606.08961

It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra $A=S(V^*)$ fails if the vector space $V$ is infinite-dimensional. In the present paper, we study the corresponding obstructions. We construct an $L_\infty$ structure on polyvector fields on $V$ having the even degree Taylor components, with the degree 2 component given by the Schouten-Nijenhuis bracket, but having as well higher non-vanishing Taylor components. We prove that this $L_\infty$ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our $L_\infty$ algebra is $L_\infty$ quasi-isomorphic to the Lie algebra of polyvector fields on $V$ with the Schouten-Nijenhuis bracket, if $V$ is finite-dimensional., 40 pages, v7. The paper is essentially edited. The exposition in Section 1 is improved. Appendices A and B are added

We determine the image of the (strongly) parabolic Hitchin map for all parabolics in classical groups and $G_2$. Surprisingly, we find that the image is isomorphic to an affine space in all cases, except for certain "bad parabolics" in type $D$, where the image can be singular., In this paper we compute the base of the strongly parabolic Hitchin map. In the companion paper arXiv:1608.05454 we prove that this gives a completely integrable system

Let $d >1$. In this paper we show that for an irreducible permutation $\pi$ which is not a rotation, the set of $[\lambda]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([\lambda],\pi)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates on the Hausdorff dimension of the set of "bad" parameters in the context of fast decaying cocycles, following a strategy developed by V. Delecroix and the first author., Comment: 29 pages, 1 figure

Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper by Lee-Park. Also, some exceptional bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing are constructed based on the idea of Hacking. In the case if Dolgachev surfaces were of type $(2,3)$, we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exist a nontrivial phantom category in the derived category., Comment: 35 pages; 3 figures; exposition improved; Adv. Math. (to appear)

In the work we derive regularized trace formulas which were established in papers of Kanguzhin and Tokmagambetov for the Laplace and m-Laplace operators in a punctured domain with the fixed iterating order m 2 N. By using techniques of Sadovnichii and Lyubishkin, the authors in that papers described regularized trace formulae in the spatial dimension d = 2. In this note one claims that the formulas are also true for more general operators in the higher spatial dimensions, namely, 2 ? d ? 2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called ?nonharmonic? analysis.

International Conference on Approximation and Computation - Theory and Applications (ACTA) -- NOV 30-DEC 02, 2017 -- Belgrade, SERBIA KUCUKOGLU, IREM/0000-0001-9100-2252 WOS: 000463447500004 The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results. Serbian Acad Sci & Arts, Math Inst, Univ Belgrade, Fac Mech Engn, Univ Belgrade, Fac Math, Univ Belgrade, Sch Elect Engn, Univ Nis, Univ Kragujevac, Fac Sci, Univ Novi Sad, Fac Sci Scientific Research Project Administration of Akdeniz UniversityAkdeniz University [FBA-2018-3292] The present paper was supported by Scientific Research Project Administration of Akdeniz University (with Project Number: FBA-2018-3292).

In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of `tail densities' defined for subsets of the integers.

The study of CR-submanifolds of a Kaehler manifold was initiated by Bejancu [1]. Since then many papers have appeared on CR-submanifolds. The aim of the present paper is to study generalised CR-submanifolds of a LP-Sasakian manifolds.

The paper studies the open-hereditary property of semi-separation axioms and applies it to the study of digital topological spaces such as an n-dimensional Khalimsky topological space, a Marcus-Wyse topological space and so on. More precisely, we study various properties of digital topological spaces related to low-level and semi-separation axioms such as T1/2 , semi-T1/2 , semi-T1, semi-T2, etc. Besides, using the finite or the infinite product property of the semi-Ti-separation axiom, i ? {1,2}, we prove that the n-dimensional Khalimsky topological space is a semi-T2-space. After showing that not every subspace of the digital topological spaces satisfies the semi-Ti-separation axiom, i ?{1,2}, we prove that the semi-Tiseparation property is open-hereditary, i ? {1,2}. All spaces in the paper are assumed to be nonempty and connected.

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics

Embeddings of maximal tori into classical groups over global fields of characteristic not 2 are the subject matter of several recent papers, with special attention to the Hasse principle. The present paper gives necessary and sufficient conditions for this embedding problem, and in particular for the Hasse principle to hold. Using work of Borovoi, this is interpreted as a Brauer-Manin type obstruction.

It is well known that the geometric genus and multiplicity are two important invariants for isolated singularities. In this paper we give a sharp lower estimate of the geometric genus in terms of the multiplicity for isolated hypersurface singularities. In 1971, Zariski asked whether the multiplicity of an isolated hypersurface singularity depends only on its embedded topological type. This problem remains unsettled. In this paper we answer positively Zariski’s multiplicity question for isolated hypersurface singularity if Milnor number or geometric genus is small.

Alfred Schild has established conditions that Lorentz transformationsmap world-vectors (ct, x, y, z) with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in SL(z, ℂ). We consider the lattice of subgroups of the group originated in Schild’s paper and obtain generating sets for the full group and its subgroups.