Search

This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea-Pandharipande. We first prove a structural result expressing the equivariant virtual $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: (i) arbitrary surfaces for the zero curve class, (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class, (iii) minimal surfaces of general type with $p_g>0$ for any curve classes. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual $\chi_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea-Pandharipande. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula., Comment: 48 pages. Reformulation of Theorem 1 for readability. Extra explanation for the mixed terms in section 2.2.3. Reference updates

In Li et al. (J Aust Math Soc 97:383–390, 2014), the authors discussed two classes of mappings in Banach spaces, which are $$\varphi $$ -quasiisometric mappings in the distance ratio metric (briefly, $$\varphi $$ -DRQI mappings) and fully $$\varphi $$ -quasiisometric mappings in the distance ratio metric (briefly, $$\varphi $$ -FDRQI mappings), where $$\varphi :$$ $$[0,\infty )\rightarrow [0,\infty )$$ denotes a quasiisometric control function. In this paper, we continue this discussion by introducing a new class of mappings, that is, $$\varphi $$ -PDRQI mappings. The purpose of this paper is twofold. Let f be a homeomorphism between two proper domains in a Banach space. First, we prove that f being $$\varphi $$ -PDRQI is equivalent to f being $$\varphi $$ -FDRQI. Then, as an application of the obtained equivalent relation, we show that f being $$\varphi $$ -DRQI, f being $$\varphi $$ -PDRQI and f being $$\varphi $$ -FDRQI are equivalent to each other when the quasiisometric control functions have the form $$\varphi (t)=Mt$$ .

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun only recently. In this paper we obtain a conditional theorem for all integral norms $$L_q$$ , $$1\le q

This paper investigates the asset-liability management problem for an ordinary robust insurance system between a reinsurer and an insurer under the Heston model with delay. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the Heston model. The insurer, as the follower of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. Under the consideration of the performance-related capital inflow/outflow, the wealth processes of the insurer and reinsurer are modeled by stochastic differential delay equations (SDDEs). This paper aims to find the equilibrium strategies for the reinsurer and insurer by maximizing the expected utility of the player’s terminal wealth with delay under the worst-case scenario of the alternative measures. By using the idea of backward induction and dynamic programming approach, the explicit expressions of the robust equilibrium strategies and value functions are derived. Finally, some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the robust optimal reinsurance and investment strategies are performed.

A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics

Full-waveform inversion (FWI) is today a standard process for the inverse problem of seismic imaging. PDE-constrained optimization is used to determine unknown parameters in a wave equation that represent geophysical properties. The objective function measures the misfit between the observed data and the calculated synthetic data, and it has traditionally been the least-squares norm. In a sequence of papers, we introduced the Wasserstein metric from optimal transport as an alternative misfit function for mitigating the so-called cycle skipping, which is the trapping of the optimization process in local minima. In this paper, we first give a sharper theorem regarding the convexity of the Wasserstein metric as the objective function. We then focus on two new issues. One is the necessary normalization of turning seismic signals into probability measures such that the theory of optimal transport applies. The other, which is beyond cycle skipping, is the inversion for parameters below reflecting interfaces. For the first, we propose a class of normalizations and prove several favorable properties for this class. For the latter, we demonstrate that FWI using optimal transport can recover geophysical properties from domains where no seismic waves travel through. We finally illustrate these properties by the realistic application of imaging salt inclusions, which has been a significant challenge in exploration geophysics.

In this paper, the global exponential stability and periodicity are investigated for delayed neural network models with continuous coefficients and piecewise constant delay of generalized type. The sufficient condition for the existence and uniqueness of periodic solutions of the model is established by applying Banach’s fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with generalized piecewise constant delay, some sufficient conditions for the global exponential stability of the model are obtained. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. This paper ends with a brief conclusion.

We classify the Seifert fibrations of lens spaces where the base orbifold is non-orientable. This is an addendum to our earlier paper ‘Seifert fibrations of lens spaces’. We correct Lemma 4.1 of that paper and fill the gap in the classification that resulted from the erroneous lemma.

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).

This paper presents an approach, based on interpolation theory of operators, to the study of interpolating sequences for interpolation Banach spaces between Hardy spaces. It is shown that the famous Carleson result forH∞can be lifted to a large class of abstract Hardy spaces. A description is provided of the range of the Carleson operator defined on interpolation spaces between the classical Hardy spaces in terms of uniformly separated sequences. A key role in this description is played by some general interpolation results proved in the paper. As by-products, novel results are obtained which extend the Shapiro–Shields result on the characterisation of interpolation sequences for the classical Hardy spacesHp. Applications to Hardy–Lorentz, Hardy–Marcinkiewicz and Hardy–Orlicz spaces are presented.

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of $k$-th mean curvatures with $k=1,\cdots,n$, and positive powers of $p$-th power sums $S_p$ with $p>0$. We prove that if the initial hypersurface $M_0$ is smooth and closed and has positive sectional curvatures, then the solution $M_t$ of the flow has positive sectional curvature for any time $t>0$, exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov-Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature. In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and homogeneous degree one function $f$ of the shifted principal curvatures $\lambda_i=\kappa_i-1$, plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where $f$ is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov-Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.

In this paper, we consider boundary-value problems for a new class of hybrid systems of differential equations whose coefficients contain the Dirac delta-function. Hybrid systems are systems that contain both ordinary and partial differential equations; such systems appear, for example, when equations of motion of mechanical systems of rigid bodies attached to a beam by elastic bonds are derived from the Hamilton–Ostrogradsky variational principle. We present examples that lead to such systems and introduce the notions of generalized solutions and eigenvalues of a boundary-value problem. We also compare results of numerical simulations based on methods proposed in this paper with results obtained by previously known methods and show that our approach is reliable and universal.

This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. We then use our estimates for the Green function to prove that $p_n\asymp R^{-n}n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.

In this paper, we study the concomitants of m-generalized order statistics (m-GOSs) and m-dual generalized order statistics (m-DGOSs) from bivariate Cambanis family with nonzero parameter values as an extension of several recent papers. Moreover, we derive some information measures, namely the Shannon entropy, Kullback–Leibler (KL) divergence and Fisher information number (FIN) for the concomitants of m-GOSs, when $$m>-1,$$ and record values, for $$m=-1.$$ Also, the joint distribution of concomitants of m-GOSs and record values for this family are studied. Besides, some useful recurrence relations between moments of concomitants are obtained. Finally, the ordinary order statistics (OOSs), record values and sequential order statistics (SOSs) as subclasses of m-GOSs, as well as the progressive type II censored order statistics (POSs) as a more general subclass of GOSs, are separately discussed.

Non-monotonicity height is an important index to describe the complexity of dynamics for piecewise monotone functions. Although it is used extensively in the theory of iterative roots, its calculation is still difficult especially in the infinite case. In this paper, by introducing the concept of spanning interval, we first present a sufficient condition for piecewise monotone functions to have height infinity and then an algorithm for finding the spanning intervals is given. We further investigate the density of all piecewise monotone functions with infinite and finite height, respectively, and the results indicate the instability of height. At the end of this paper, the variance of height under composition, especially for functions of height 1 and infinity, are also discussed.

In this paper, generalized versions of Hadamard and Fejér–Hadamard type fractional integral inequalities are obtained. By using generalized fractional integrals containing Mittag-Leffler functions, some well-known results for convex and harmonically convex functions are generalized. The results of this paper are connected with various published fractional integral inequalities.

In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.

The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages

In this paper, we prove that if a nonconstant finite order meromorphic function f and its n-th order difference operator $$\Delta ^n_{\eta }f$$ share $$a_1,$$ $$a_2,$$ $$a_3$$ CM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, and $$a_1,$$ $$a_2,$$ $$a_3$$ are three distinct finite complex values, then $$f(z)=\Delta ^n_{\eta }f(z)$$ for each $$z\in \mathbb {C}.$$ The main results in this paper improve and extend many known results concerning a conjecture posed by Chen and Yi in 2013.

PurposeIn this paper, Picard–S hybrid iterative process is defined, which is a hybrid of Picard and S-iterative process. This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid and Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.Design/methodology/approachThis new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings.FindingsShowed the fastest convergence of this new iteration and then other iteration defined in this paper. The author finds the solution of delay differential equation using this hybrid iteration. For new iteration, the author also proved a theorem for nonexpansive mapping.Originality/valueThis new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.

In this paper, we present some new identities for multiple polylogarithm functions by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish several duality formulas for Arakawa–Kaneko zeta values and Kaneko–Tsumura $$\eta $$ -values. At the end of the paper, we study a variant of Kaneko–Tsumura $$\eta $$ -function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto.

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics exhibit unbounded minimizing properties. We also compute the length of the shortest closed geodesic on doubled odd-gons and show that this length approaches 4 times the diameter., Comment: This paper is a result of a SUMRY (REU) project at Yale

In 1936 W. Burau discovered an interesting family of n×n matrices that give a linear representation of Artin’s classical braid group Bn, n=1,2,…. A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for n≤3, nonfaithful for n≥5, but the case of n=4 remains open to this day, in spite of many papers on the topic. This paper introduces braid groups, describes the problem in ways that make it accessible to readers with a minimal background, reviews the literature, and makes a contribution that reinforces conjectures that the Burau representation of B4 is faithful.

Let $\mathbb{R}^n$ be a Euclidean space of dimension $n\geq 2$ . For a domain $G\subset \mathbb{R}^n$ , we denote by $V_r(G)$ the set of functions $f\in L_{\mathrm{loc}}(G)$ having zero integrals over all closed balls of radius r contained in G (if domain G does not contain such balls, we set $V_r(G)=L_{\mathrm{loc}}(G)$ ). Let E be a nonempty subset of $\mathbb{R}^n$ . In this paper we study the following questions related to the extension problem. 1) Which conditions guarantee the extension of a continuous function defined on E to a continuous function of class $V_r(\mathbb{R}^n)$ defined on the whole $\mathbb{R}^n$ ? 2) If the above extension exists, obtain growth estimates of the extended function at infinity. Theorem 1 of this paper shows that for a wide class of continuous functions on segment E defined in terms of the modulus of continuity, there exists an extension to a bounded function of class $(V_r\cap C)(\mathbb{R}^n)$ regardless of the length of segment E. A similar result is not true for open sets E with a diameter greater than 2r, even without conditions for extension growth. Theorem 1 also contains an estimate of the velocity decrease of the extended function at infinity in directions orthogonal to the segment E.

This paper is devoted to the stochastic properties of dynamical systems preserving an infinite measure. More precisely we prove central limit theorems for Birkhoff sums of observables of ℤ2-extensions of dynamical systems (satisfying some nice spectral properties). The motivation of our paper is the ℤ2-periodic Lorentz process for which we establish a functional central limit theorem for Holder continuous observables (in discrete time as well as in continuous time).

In this paper, we develop a new method for studying the inhomogeneous vector Riemann–Hilbert boundary value problem (which is also called the Riemann boundary value problem) in the Wiener algebra of order two. The method consists in reducing the Riemann problem to a truncated Wiener–Hopf equation (to a convolution equation on a finite interval). The idea of the method was proposed by the author in a previous work. Here the method is applied to the inhomogeneous Riemann boundary value problem and to matrix functions of a more general form. The efficiency of the method is shown in the paper: new sufficient conditions for the existence of a canonical factorization of the matrix function in the Wiener algebra of order two are obtained. In addition, it was established that for the correct solvability of the inhomogeneous vector Riemann boundary value problem, it is necessary and sufficient to prove the uniqueness of the solution to the corresponding truncated homogeneous Wiener–Hopf equation.

The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations., 20 pages, 1 figure. The condition of finite generation in Thm 1.9 was replaced by countability. The proof of Thm 1.9 has been simplified. The notation used in 5 has been modified. Several minor corrections across the paper

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems., 50 pages. Originally part of arXiv:1703.09945 . We decided to split that paper following the recommendations of a referee. Updated subsequent to acceptance by Transactions of the American Mathematical Society

Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity.

Let $$S=\{p_{1},\ldots ,p_{t}\}$$ be a fixed finite set of prime numbers listed in increasing order. In this paper, we prove that the Diophantine equation $$(F_n^{(k)})^s=p_{1}^{a_{1}}+\cdots +p_{t}^{a_{t}}$$ , in integer unknowns $$n\ge 1$$ , $$s\ge 1,~k\ge 2$$ and $$a_i\ge 0$$ for $$i=1,\ldots ,t$$ such that $$\max \left\{ a_{i}: 1\le i\le t\right\} =a_t$$ has only finitely many effectively computable solutions. Here, $$F_n^{(k)}$$ is the nth k–generalized Fibonacci number. We compute all these solutions when $$S=\{2,3,5\}$$ . This paper extends the main results of [15] where the particular case $$k=2$$ was treated.

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation$$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$F(x+y)-F(x)-F(y)=yf(x)+xf(y)considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.

This paper is devoted to the problem of where the critical points of a polynomial are relative to their zeros. Classical and new developments are surveyed along with illustrative examples. The paper finishes with a short proof of the sector theorem of Sendov and Sendov.

"The main object of this paper is to give sufficient conditions for the general integral operator Tn, to be univalent in the open disk U, when gi, hi, ki ∈ Gbi for all i = 1, n. This general integral operator was considered in a recent work [Barbatu, C. and Breaz, D., ˘ Classes of an univalent integral operator, Studia Univ. Babes¸-Bolyai Math., accepted]. The results derived in this paper are shown to follow upon specializing the parameters involved in our results. Several corollaries of the main results are also considered."

Since the appearance of Robbins’s paper (1948) the central limit theorem for a sum of a random number of independent identically distributed random variables is one of the most fundamental results in probability, and explains the appearance of the normal distribution in a whole host of diverse applications in mathematics, physics, biology and the social sciences. Compound random sums are extensions of classical random sums when the numbers of independent summands in sums are partial sums of independent identically distributed positive integer-valued random variables, assumed independent of summands of sums. The main aim of this paper is to introduce central limit theorems for normalized compound random sums of independent random variables and establish the convergence rates in types of small-o and large- $$\mathcal{O}$$ estimates, in term of Trotter-distance. The obtained results in this paper are extensions of several known ones.

In this paper we compute the rates of convergence of power series statistical convergence of sequences of positive linear operators. We also investigate some Korovkin type approximation properties of the $$q$$ -Meyer–Konig and Zeller operators and Durrmeyer variant of the $$q$$ -Meyer–Konig and Zeller operators via power series statistical convergence. We show that the approximation results obtained in this paper expand some previous approximation results of the corresponding operators.

In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower box dimension at least $n-0.5$ and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper., 9 pages. arXiv admin note: text overlap with arXiv:1711.06533

In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial., Comment: 29 pages

The first aim of this paper is to investigate the growth of the entire function defined by the Laplace-Stieltjes transform converges on the whole complex plane. By introducing the concept of generalized order, we obtain two equivalence theorems of Laplace-Stieltjes transforms related to the generalized order, A * and λn. The second purpose of this paper is to study the problem on the approximation of this Laplace-Stieltjes transform. We also obtain some theorems about the generalized order, the error, and the coefficients of Laplace-Stieltjes transforms, which are generalization and improvement of the previous results.

In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.

In this paper, we introduce a Halpern algorithm and a nonconvex combination algorithm to approximate a solution of the split common fixed problem of quasi- ϕ -nonexpansive mappings in Banach space. In our algorithms, the norm of linear bounded operator does not need to be known in advance. As the application, we solve a split equilibrium problem in Banach space. Finally, some numerical examples are given to illustrate the main results in this paper and compare the computed results with other ones in the literature. Our results extend and improve some recent ones in the literature.

The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from $n$-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{T}/\mathcal{T}^{fd}$. Let $k$ be a field, $n\geq 3$ an integer and $\mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $\mathcal{T}^{fd}$ and a subcategory $\mathcal{M}=\text{add}(M)$ such that $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$ is an $n$-Calabi-Yau triple. In this paper, we prove some properties of the triangulated categories $\mathcal{T}$ and $\mathcal{T}/\mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $\mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $\mathcal{T}/\mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{T}$ and over $\mathcal{T}/\mathcal{T}^{fd}$ are isomorphic. Note that these properties make $\mathcal{T}/\mathcal{T}^{fd}$ a generalisation of the cluster category., Comment: 17 pages. Final accepted version to appear in the Pacific Journal of Mathematics

A sequence $$(v_1,\ldots ,v_k)$$ of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex $$v_i$$ in the sequence totally dominates at least one vertex that was not totally dominated by $$\{v_1,\ldots , v_{i-1}\}$$ and $$\{v_1,\ldots ,v_k\}$$ is a total dominating set of G. The length of a shortest such sequence is the total domination number of G ( $$\gamma _{t}(G)$$ ), while the length of a longest such sequence is the Grundy total domination number of G ( $$\gamma _{gr}^t(G)$$ ). In this paper we study graphs with equal total and Grundy total domination numbers. We characterize bipartite graphs with both total and Grundy total dominations number equal to 4, and show that there is no connected chordal graph G with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=4$$ . The main result of the paper is a characterization of bipartite graphs with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=6$$ proved by establishing a surprising correspondence between the existence of such graphs and a classical but still open problem of the existence of certain finite projective planes.

In this paper we study a connective commutative differential graded algebra (CDGA) which is piecewise Noetherian. The principal aim is to analyze a dualizing complex of CDGA’s and investigate a Gorenstein CDGA. We also establish many other results which can be expected to play basic roles in the study of a CDGA, such as the Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolutions for ordinary modules. In the paper, we show that sppj and ifij resolutions are powerful tools to study DG-modules.

This paper concerns on two types of integral boundary value problems of a nonlinear fractional differential system, i . e ., nonlocal strip integral boundary value problems and coupled integral boundary value problems. With the aid of the monotone iterative method combined with the upper and lower solutions, the existence of extremal system of solutions for the above two types of differential systems is investigated. In addition, a new comparison theorem for fractional differential system is also established, which is crucial for the proof of the main theorem of this paper. At the end, an example explaining how our studies can be used is also given.

The main aim of this paper is to give refinement of bounds of fractional integral operators involving extended generalized Mittag-Leffler functions. A new definition, namely, strongly α , m -convex function is introduced to obtain improvements of bounds of fractional integral operators for convex, m -convex, and α , m -convex functions. The results of this paper will provide simultaneous generalizations as well as refinements of various published results.

This paper concerns the density of the Hartman--Watson law. Yor (1980) obtained an integral formula that gives a closed-form expression of the Hartman--Watson density. In this paper, based on Yor's formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a Dufresne's result (2001) that exhibits remarkably simple representations for the laws of exponential additive functionals of Brownian motion., Comment: 21 pages; this is an abridged version of the previous one, accepted for publication in the Journal of Theoretical Probability, with shortened Section 4

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers' second moment estimates. In this paper we establish such results in a very general framework. Given any subhomogeneous function (a notion to be defined) $f: \mathbb{R}^n \to \mathbb{R}$, we derive a necessary and sufficient condition on the approximating function $\psi$ for guaranteeing that a generic element $f\circ g$ in the $G$-orbit of $f$ is $\psi$-approximable; that is, $|f\circ g(\mathbf{v})| \le \psi(\|\mathbf{v}\|)$ for infinitely many $\mathbf{v} \in \mathbb{Z}^n$. We also deduce a sufficient condition in the case of uniform approximation. Here, $G$ can be any closed subgroup of $\rm{ASL}_n(\mathbb{R})$ satisfying certain axioms that allow for the use of Rogers-type estimates., Comment: 26 pages; misprints corrected, concluding remarks added