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In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level $$O\left( {1 \over k^4}\right) $$ O 1 k 4 , where k is the number of iterations. This is very close to the lower bound of the order $$O\left( {1 \over k^5}\right) $$ O 1 k 5 , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces $$X\subset \mathbb {P}^{n+1}$$ . After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if $$X\subset \mathbb {P}^{n+1}$$ is a very general hypersurface of degree $$d\geqslant 2n+2$$ , the least gonality of a curve $$C\subset X$$ passing through a general point of X is $$\mathrm {gon}(C)=d-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor $$ , apart from some exceptions we list.

We are interested in the McKay quiver $\Gamma(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $G \subseteq \mathrm{GL}(V)$ and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups $G(r,p,n)$. We first look at the case $G(1,1,n)$, which is isomorphic to the symmetric group $S_n$, followed by $G(r,1,n)$ for $r >1$. Then, using Clifford theory, we can determine the McKay quiver for any $G(r,p,n)$ and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde A(G)$ of a finite group $G \subseteq \mathrm{GL}(V)$, which is Morita equivalent to the skew group ring $A*G$. This description gives us an embedding of the basic algebra Morita equivalent to $A*G$ into a matrix algebra over $A$., Comment: v2: minor revision, final version to appear in Algebr. Represent. Theory

Let E∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗ G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗ G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group, This paper is partially supported by the Spanish Ministry of Science and Innovation through the grant MTM2017-82690-P and the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S4). I would like to thank Dawid Kielak and an anonymous referee for useful suggestions and comments

This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies. This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies.

In the present paper we consider a boundary homogenization problem for the Poisson’s equation in a bounded domain and with a part of the boundary conditions of highly oscillating type (alternating between homogeneous Neumman condition and a nonlinear Robin type condition involving a small parameter). Our main goal in this paper is to investigate the asymptotic behavior as ε → 0 of the solution to such a problem in the case when the length of the boundary part, on which the Robin condition is specified, and the coefficient, contained in this condition, take so-called critical values. We show that in this case the character of the nonlinearity changes in the limit problem. The boundary homogenization problems were investigate for example in [1, 2, 4]. For the first time the effect of the nonlinearity character change via homogenization was noted for the first time in [5]. In that paper an effective model was constructed for the boundary value problem for the Poisson’s equation in the bounded domain that is perforated by the balls of critical radius, when the space dimension equals to 3. In the last decade a lot of works appeared, e.g., [6–10], in which this effect was studied for different geometries of perforated domains and for different differential operators. We note that in [6–10] only perforations by balls were considered. In papers [11, 12] the case of domains perforated by an arbitrary shape sets in the critical case was studied.

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the Dirichlet series $\sum_{n=1}^{\infty} \chi(n) n^{-s}$ can be continued meromorphically to the half-plane $\operatorname{Re} s>\alpha$, and denote by $\zeta_{\chi}(s)$ the corresponding meromorphic function in $\operatorname{Re} s>\sigma(\chi)$. We construct $\zeta_{\chi}(s)$ that have $\sigma(\chi)\le 1/2$ and are universal for zero-free analytic functions on the half-critical strip $1/21$. Finally, we show that there exists $\zeta_{\chi}(s)$ with $\sigma(\chi) \le 1/2$ and zeros at any discrete multiset in the strip $1/21/2$; on the Riemann hypothesis, this strip may be replaced by the half-critical strip $1/2 < \operatorname{Re} s < 1$., Comment: This is the final version of the paper which has been accepted for publication in Journal d'Analyse Math\'{e}matique

In this note we show that multiple solutions exist for the production-inventory example in the seminal paper on adjustable robust optimization in Ben-Tal et al. (Math Program 99(2):351---376, 2004). All these optimal robust solutions have the same worst-case objective value, but the mean objective values differ up to 21.9 % and for individual realizations this difference can be up to 59.4 %. We show via additional experiments that these differences in performance become negligible when using a folding horizon approach. The aim of this paper is to convince users of adjustable robust optimization to check for existence of multiple solutions. Using the production-inventory example and an illustrative toy example we deduce three important implications of the existence of multiple optimal robust solutions. First, if one neglects this existence of multiple solutions, then one can wrongly conclude that the adjustable robust solution does not outperform the nonadjustable robust solution. Second, even when it is a priori known that the adjustable and nonadjustable robust solutions are equivalent on worst-case objective value, they might still differ on the mean objective value. Third, even if it is known that affine decision rules yield (near) optimal performance in the adjustable robust optimization setting, then still nonlinear decision rules can yield much better mean objective values.

In a recent paper (Cartis et al. in Math Prog A 144(2):93---106, 2014), the evaluation complexity of an algorithm to find an approximate first-order critical point for the general smooth constrained optimization problem was examined. Unfortunately, the proof of Lemma 3.5 in that paper uses a result from an earlier paper in an incorrect way, and indeed the result of the lemma is false. The purpose of this corrigendum is to provide a modification of the previous analysis that allows us to restore the complexity bound for a different, scaled measure of first-order criticality.

Let $R$ be a commutative Noetherian ring and $\alpha$ an automorphism of $R$. This paper addresses the question: when does the skew polynomial ring $S = R[\theta; \alpha]$ satisfy the property $(\diamond)$, that for every simple $S$-module $V$ the injective hull $E_S(V)$ of $V$ has all its finitely generated submodules Artinian. The question is largely reduced to the special case where $S$ is primitive, for which necessary and sufficient conditions are found, which however do not between them cover all possibilities. Nevertheless a complete characterisation is found when $R$ is an affine algebra over a field $k$ and $\alpha$ is a $k$-algebra automorphism - in this case $(\diamond)$ holds if and only if all simple $S$-modules are finite dimensional over $k$. This leads to a discussion, involving close study of some families of examples, of when this latter condition holds for affine $k$-algebras $S = R[\theta;\alpha]$. The paper ends with a number of open questions., Comment: 23 pages; comments welcome

Polynomial chaos-based methods have been extensively applied in electrical and other engineering problems for the stochastic simulation of systems with uncertain parameters. Most of the implementations are based on either the intrusive stochastic Galerkin method or on non-intrusive collocation approaches, of which a very common example is the pseudo-spectral method based on Gaussian quadrature rules. This paper shows that, for the important class of linear differential algebraic equations, the latter can be cast as an approximate factorization of the stochastic Galerkin approach, thus generalizing recent discussions in literature in this regard. Consistently with this literature, we show that the factorization turns out to be exact for first-order random inputs, and hence the two methods coincide under this assumption. Further, the presented results also generalize recent work in the field of electrical circuit simulation, in which a similar decomposition was derived ad hoc, via error minimization, for the case of Hermite chaos. We demonstrate that the factorization stems from the general properties of orthogonal polynomials and the error introduced by the approximation—or in other terms, the error of the stochastic collocation method in comparison with the stochastic Galerkin method—is carefully quantified and assessed. An illustrative example concerning the stochastic analysis of an RLC circuit is used to illustrate the main findings of this paper. In addition, a more complex and real-life example allows emphasizing the generality of the achieved results.

In this paper, we study the notions of accretive operators, contraction mappings, non-expansive mappings using the idea of n (>1)-normed structures for some relevant results. Further, we define \(\left( \lambda ,q\right) \)-transform by utilizing the definition of the generating function of q-Genocchi polynomials with weight zero to construct interesting properties related to q-Genocchi polynomials with weight zero and Frobenius–Euler polynomials. Furthermore, we describe p-adic q-\( \lambda \)-transform of higher order and construct a link between q-Genocchi polynomials of higher order with weight zero and higher order Frobenius–Euler polynomials using this transform. Our applications in this paper provide a link between analytic numbers theory and n-normed spaces.

We introduce a new tuberculosis (TB) mathematical model, with $25$ state-space variables where $15$ are evolution disease states (EDSs), which generalises previous models and takes into account the (seasonal) flux of populations between a high incidence TB country (A) and a host country (B) with low TB incidence, where (B) is divided into a community (G) with high percentage of people from (A) plus the rest of the population (C). Contrary to some beliefs, related to the fact that agglomerations of individuals increase proportionally to the disease spread, analysis of the model shows that the existence of semi-closed communities are beneficial for the TB control from a global viewpoint. The model and techniques proposed are applied to a case-study with concrete parameters, which model the situation of Angola (A) and Portugal (B), in order to show its relevance and meaningfulness. Simulations show that variations of the transmission coefficient on the origin country has a big influence on the number of infected (and infectious) individuals on the community and the host country. Moreover, there is an optimal ratio for the distribution of individuals in (C) versus (G), which minimizes the reproduction number $R_0$. Such value does not give the minimal total number of infected individuals in all (B), since such is attained when the community (G) is completely isolated (theoretical scenario). Sensitivity analysis and curve fitting on $R_0$ and on EDSs are pursuit in order to understand the TB effects in the global statistics, by measuring the variability of the relevant parameters. We also show that the TB transmission rate $\beta$ does not act linearly on $R_0$, as is common in compartment models where system feedback or group interactions do not occur. Further, we find the most important parameters for the increase of each EDS., Comment: This is a preprint of a paper whose final and definite form is with 'Ricerche di Matematica', ISSN 0035-5038 (Print) 1827-3491 (Online), available at [http://link.springer.com/journal/11587]. Paper Submitted 10-Feb-2016; Revised and Accepted 31-Jan-2017

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures., 15 pages, to appear in European Journal of Mathematics

Aronhold’s classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e., special 7-tuples of bitangents such that the six points at which any sub-triple of bitangents touches the quartic do not lie on the same conic in the projective plane. Lehavi (cf. [L05]) proved that a smooth plane quartic can be explicitly reconstructed from its 28 bitangents; this result improved Aronhold’s method of recovering the curve. In a 2011 paper [PSV11] Plaumann, Sturmfels and Vinzant introduced an 8 × 8 symmetric matrix that parametrizes the bitangents of a nonsingular plane quartic. The starting point of their construction is Hesse’s result for which every smooth quartic curve has exactly 36 equivalence classes of linear symmetric determinantal representations. In this paper we tackle the inverse problem, i.e., the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic, and we provide a Sage script intended for computing the bitangent matrix of a given curve.

This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly K\"ahler manifold. Finally, we prove that there exist no nearly Kenmotsu hypersurface of nearly K\"ahler manifold. It is shown that a normal nearly Kenmotsu manifold is Kenmotsu manifold.

In this paper we study infinitesimal and finite flexibility for generic semidiscrete surfaces. We prove that generic 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of existence. Further we find a necessary condition of 3-ribbon infinitesimal flexibility. For an arbitrary $n\ge 3$ we prove that every generic $n$-ribbon surface has at most one degree of finite/infinitesimal flexibility. Finally, we discuss the relation between general semidiscrete surface flexibility and 3-ribbon subsurface flexibility. We conclude this paper with one surprising property of isometric deformations of developable semidiscrete surfaces.

In the earlier paper (Itoh and Kiyohara, Manuscr Math 114:247–264, 2004), we showed that the cut locus of a general point on two-dimensional ellipsoid is a segment of a curvature line and proved “Jacobi’s last geometric statement” on the singularities of the conjugate locus. In the present paper, we show that a wider class of Liouville surfaces possess such simple cut loci and conjugate loci. The results include the determination of cut loci and the set of poles on two-sheeted hyperboloids and elliptic paraboloids.

The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curvatures for this problem will be determined: we will assume that the radial curvature $K_{\rm rad.}$ of an end converges to a constant -1 at infinity with the decay order $K_{\rm rad.} + 1 = o(r^{-1})$ and prove the absence of eigenvalues embedded in the essential spectrum. Furthermore, in order to show that this decay order $K_{\rm rad.} + 1 = o(r^{-1})$ is sharp, we will construct a manifold with the radial curvature decay $K_{\rm rad.} + 1 = O(r^{-1})$ and with an eigenvalue $\frac{(n-1)^2}{4} + 1$ embedded in the essential spectrum $[ \frac{(n-1)^2}{4}, \infty)$ of the Laplacian., Comment: 26 pages; changed content; added reference; typos added; changed content; Proposition 3.1 is made better;correct mistake, easy to read;correct careless mistake

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

WOS: 000351527900020, According to the principle of the three-dimensional linearized theory of elastic waves in initially stressed bodies, a dynamical stress field in a pre-stressed bi-layered plate-strip under the action of an arbitrary inclined force resting on a rigid foundation is studied. It is assumed that the force applied to upper free surface of the plate-strip is time-harmonic and the materials used are linearly elastic, homogenous, and isotropic. By employing finite element method, the governing system of partial differential equations of motion is approximately solved. The different dependencies of the problem such as the ratio of height of plates and initial stress of the materials are numerically investigated. Particularly the effect of arbitrary inclined force is analyzed. It is observed that the numerical results obtained according to various angles converge to the ones in the previous studies., Research Fund of Sakarya University [2012-02-00-001], The authors would like to thank the anonymous referees for their very helpful and detailed comments, which have significantly let to improve the paper. The works presented in this paper were supported by Research Fund of Sakarya University under project Number 2012-02-00-001.

WOS: 000358675000001, For a positive real number let the Balancing distance be the distance from to the closest Balancing number. The Balancing sequence is defined by the initial values , and by the binary recurrence relation , . In this paper, we show that there exist only one positive integer triple such that the Balancing distances , and all are exactly 1., TUBITAK-BIDEB, This research is supported by TUBITAK-BIDEB. The paper was essentially prepared when the third author visited Department of Mathematics, Nigde University in August, 2013. The third author is grateful for the support of TUBITAK-BIDEB, and for the hospitality of Department of Mathematics, especially he is greatly indebted to Professor Murat Alp, Tarik Atay and Nurettin Irmak.

The theoretical model outlined in this paper, has been experimentally validated by: H. Kim ,J-Y. Moon ,G.A. Mashour & U. Lee, ''Mechanisms of hysteresis in human brain networks during transitions of consciousness and unconsciousness: Theoretical principles and empirical evidence'', PLOS-Computational Biology, August 30, 2018, https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1006424; International audience; In this brief note, we focus attention on a possible implementation of a basic hysteretic pattern (the Preisach one), suitably generalized, into a formal model of unconscious-conscious interconnection and based on representation of mental entities by m-adic numbers. PS. The theoretical model described in this paper, has been experimentally confirmed by: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1006424

In this paper, the Poincare (or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain Open image in new window In particular, another proof of a recent result of Gardiner and Lakic [7] is given with explicit constant. This and some other constants in this paper involve particular values of complete elliptic integrals and related special functions. A concrete estimate for the hyperbolic distance near a boundary point is also given, from which refinements of Littlewood’s theorem are derived.

We obtain a discrete universality theorem on the approximation of analytic functions by shifts of L-functions of normalized Hecke-eigen cusp forms with respect to congruence subgroups. Such a result is already proved in the authors’ previous paper [A. Lauriňcikas, K. Matsumoto, and J. Steuding, Discrete universality of L-functions for new forms, Math. Notes, 78(4):551–558, 2005] under a certain arithmetical condition, which we remove in this paper. Also, we present discrete analogues of results of [A. Lauriňcikas, K. Matsumoto, and J. Steuding, Universality of some functions related to zeta-functions of certain cusp forms, Osaka J. Math., 50(4):1021–1037, 2013].

Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the N\'eron model of $J_K$. We study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres. We show that $q$ is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties., Comment: This paper arises from the confluence and the comparison of the results contained in the preprints arXiv:1106.1540v2 and arXiv:1005.0462v1 of the two authors. Final version and to appear in Math. Zeit. We thank the referee for the very detail review of our paper

In this paper, we describe a novel approach to classical approximation theory of periodic univariate and multivariate functions by trigonometric polynomials. While classical wisdom holds that such approximation is too sensitive to the lack of smoothness of the target functions at isolated points, our constructions show how to overcome this problem. We describe applications to approximation by periodic basis function networks, and indicate further research in the direction of Jacobi expansion and approximation on the Euclidean sphere. While the paper is mainly intended to be a survey of our recent research in these directions, several results are proved for the first time here.

This paper deals with exponential neighborhoods for combinatorial optimization problems. Exponential neighborhoods are large sets of feasible solutions whose size grows exponentially with the input length. We are especially interested in exponential neighborhoods over which the TSP (respectively, the QAP) can be solved in polynomial time, and we investigate combinatorial and algorithmical questions related to such neighborhoods.¶First, we perform a careful study of exponential neighborhoods for the TSP. We investigate neighborhoods that can be defined in a simple way via assignments, matchings in bipartite graphs, partial orders, trees and other combinatorial structures. We identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms, and we also provide variants that slightly violate these properties and lead to NP-complete optimization problems. Whereas it is relatively easy to find exponential neighborhoods over which the TSP can be solved in polynomial time, the corresponding situation for the QAP looks pretty hopeless: Every exponential neighborhood that is considered in this paper provably leads to an NP-complete optimization problem for the QAP.

The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole N\'eron--Severi space, thus giving rise to a basic invariant of the underlying projective variety. Analogously, one can also define the volume function of a possibly non-complete multigraded linear series. In this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties. In the multigraded setting, relying on the work of Lazarsfeld and Musta\c{t}\u{a} (2009) on Okounkov bodies, we show that any continuous, homogeneous, and log-concave function appears as the volume function of a multigraded linear series. By contrast we show that there exists countably many functions which arise as the volume functions of projective varieties. We end the paper with an example, where the volume function of a projective variety is given by a transcendental formula, emphasizing the complicated nature of the volume in the classical case., Comment: 16 pages, minor revision

At present noone knows an unconditional proof that between two consecutive squares there is always a prime number. In a previous paper the author proved that, under the assumption of the Lindelof hypothesis, each of the intervals [n 2, (n+1)2] ⊂ [1,N], with at most O(N ɛ) exceptions, contains the expected number of primes, for every constant ɛ > 0. In this paper we improve the result by weakening the hypothesis in two different ways.

The aim of this paper is to present and discuss some equivalent characterizations of p-parabolicity in terms of existence of special exhaustion functions. In particular, Khas'minskii in [K] proved that if there exists a 2-superharmonic function k defined outside a compact set such that $\lim_{x\to \infty} k(x)=\infty$, then R is 2-parabolic, and Sario and Nakai in [SN] were able to improve this result by showing that R is 2-parabolic if and only if there exists an Evans potential, i.e. a 2-harmonic function $E:R\setminus K \to \R^+$ with $\lim_{x\to \infty} \E(x)=\infty$. In this paper, we will prove a reverse Khas'minskii condition valid for any p>1 and discuss the existence of Evans potentials in the nonlinear case., final version of the article available at http://www.springer.com

In this paper we discuss and prove an analogy of the Carleson–Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin–Fourier series of$$f\in L_p(G_m)$$f∈Lp(Gm)for$$p>1$$p>1in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for$$p=1$$p=1and construct a function$$f\in L_1(G_m)$$f∈L1(Gm)such that the partial sums with respect to Vilenkin systems diverge everywhere.

In this paper we show how it is possible to construct an efficient Migration models in the study of credit risk problems presented in Jarrow et al. (Rev Financ Stud 10:481–523, 1997) with Markov environment. Recently it was introduced the semi-Markov process in the migration models (D’Amico et al. Decis Econ Finan 28:79–93, 2005a). The introduction of semi-Markov processes permits to overtake some of the Markov constraints given by the dependence of transition probabilities on the duration into a rating category. In this paper, it is shown how it is possible to take into account simultaneously backward and forward processes at beginning and at the end of the time in which the credit risk model is observed. With such a generalization, it is possible to consider what happens inside the time after the first transition and before the last transition where the problem is studied. This paper generalizes other papers presented before. The model is presented in a discrete time environment.

In this paper, we study variational solutions to parabolic equations of the type $$\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0$$ ∂ t u - div x ( D ξ f ( D u ) ) + D u g ( x , u ) = 0 , where u attains time-independent boundary values $$u_0$$ u 0 on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values $$u_0$$ u 0 admit a modulus of continuity $$\omega $$ ω and the estimate $$|u(x,t)-u_0(\gamma )| \le \omega (|x-\gamma |)$$ | u ( x , t ) - u 0 ( γ ) | ≤ ω ( | x - γ | ) holds, then u admits the same modulus of continuity in the spatial variable.

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite., Comment: 33 pages

This paper is concerned with the almost sure partial practical stability of stochastic differential equations with general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functions. Finally, we provide a numerical example to demonstrate the efficiency of the obtained results.

In this paper, we investigate the partial practical exponential stability of neutral stochastic functional differential equations with Markovian switching. The main tool used to prove the results is the Lyapunov method. We analyze an illustrative example to show the applicability and interest of the main results.

Every word $w$ in a free group naturally induces a probability measure on every compact group $G$. For example, if $w=\left[x,y\right]$ is the commutator word, a random element sampled by the $w$-measure is given by the commutator $\left[g,h\right]$ of two independent, Haar-random elements of $G$. Back in 1896, Frobenius showed that if $G$ is a finite group and $\psi$ an irreducible character, then the expected value of $\psi\left(\left[g,h\right]\right)$ is $\frac{1}{\psi\left(e\right)}$. This is true for any compact group, and completely determines the $\left[x,y\right]$-measure on these groups. An analogous result holds with the commutator word replaced by any surface word. We prove a converse to this theorem: if $w$ induces the same measure as $\left[x,y\right]$ on every compact group, then, up to an automorphism of the free group, $w$ is equal to $\left[x,y\right]$. The same holds when $\left[x,y\right]$ is replaced by any surface word. The proof relies on the analysis of word measures on unitary groups and on orthogonal groups, which appears in separate papers, and on new analysis of word measures on generalized symmetric groups that we develop here., Comment: 16 pages, fixed the proof of Theorem 3.6, updated references

In this paper we characterize the boundedness, compactness, and weak compactness of the integration operators Tg(f)(z) = ˆ z 0 f(w)g (w) dw acting on the average radial integrability spaces RM(p, q). For these purposes, we develop different tools such as a description of the bidual of RM(p, 0) and estimates of the norm of these spaces using the derivative of the functions, a family of results that we call Littlewood–Paley type inequalities. Ministerio de Economía y Competitividad, Spain, and the European Union (FEDER) PGC2018-094215-13-100 Junta de Andalucía, Spain FQM133 Junta de Andalucía, Spain FQM-104

Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay, depending on both the time and the state variable. The case when the lower limit of the Caputo fractional derivative is fixed at the initial time, and the case when the lower limit of the fractional derivative is changed at the end of each interval of action of the impulse are studied. Practical stability properties, based on the modified Razumikhin method are investigated. Several examples are given in this paper to illustrate the results. published

Let $$\Omega \subset M$$ be an open subset of a Riemannian manifold M and let $$V:M\rightarrow \mathbb {R}$$ be a Kato decomposable potential. With $$W^{1,2}_{0}(M;V)$$ the natural form domain of the Schrodinger operator $$-\Delta +V$$ in $$L^2(M)$$ , in this paper we study systematically the following question: Under which assumption on $$\Omega $$ is the statement $$\begin{aligned} \text { for all } f\in W^{1,2}_{0}(M;V) \text { with } f=0 \text { a.e. in } M{\setminus } \Omega \text { one has } f|_\Omega \in W^{1,2}_{0}(\Omega ;V) \end{aligned}$$ true for every such V? Generalizing a classical result by Herbst and Zhao, who treat the Euclidean $$\mathbb {R}^m$$ and $$V=0$$ , we prove that without any further assumptions on V, the above property is satisfied, if $$\Omega $$ is Kac regular, a probabilistic property which means that the first exit time of Brownian motion on M from $$\Omega $$ is equal to its first penetration time to $$M{\setminus } \Omega $$ . In fact, we treat more general covariant Schrodinger operators acting on sections in metric vector bundles, allowing new results concerning the harmonicity of Dirac spinors on singular subsets. Finally, we prove that locally Lipschitz regular $$\Omega $$ ’s are Kac regular.

In this paper we consider the class of mathematical programs with complementarity constraints (MPCC). Under an appropriate constraint qualification of Mangasarian–Fromovitz type we present a topological and an equivalent algebraic characterization of a strongly stable C-stationary point for MPCC. Strong stability refers to the local uniqueness, existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. This concept of strong stability was originally introduced by Kojima for standard nonlinear optimization; here, its generalization to MPCC demands a sophisticated technique which takes the disjunctive properties of the solution set of MPCC into account.

In this paper, we give some new generalizations to the Hardy-type integral inequalities for functions of two variables by using weighted mean operators $$S_{1}:=S_{1}^{w}f$$ and $$S_{2}:=S_{2}^{w}f$$ defined by $$\begin{aligned}S_{1}(x,y)=\displaystyle \frac{1}{W(x)W(y)}\int _{\frac{x}{2}}^{x}\int _{\frac{y }{2}}^{y}w(t)w(s)f(t,s)dsdt,\end{aligned}$$ and $$\begin{aligned}S_{2}(x,y)=\displaystyle \int _{\frac{x}{2}}^{x}\int _{\frac{y}{2}}^{y}\frac{ w(t)w(s)}{W(t)W(s)}f(t,s)dsdt,\end{aligned}$$ with $$\begin{aligned}W(z)=\displaystyle \int _{0}^{z}w(r)dr\quad for\, \,z\in (0,+\infty ),\end{aligned}$$ where w is a weight function.

Many contemporary HPC systems expose their jobs to substantial amounts of interference, leading to significant run-to-run variation. For example, application runtimes on Theta, a Cray XC40 system at Argonne National Laboratory, vary by up to 70 $$\%$$ , caused by a mix of node-level and system-level effects, including network and file-system congestion in the presence of concurrently running jobs. This makes performance measurements generally irreproducible, heavily complicating performance analysis and modeling. On noisy systems, performance analysts usually have to repeat performance measurements several times and then apply statistics to capture trends. First, this is expensive and, second, extracting trends from a limited series of experiments is far from trivial, as the noise can follow quite irregular patterns. Attempts to learn from performance data how a program would perform under different execution configurations experience serious perturbation, resulting in models that reflect noise rather than intrinsic application behavior. On the other hand, although noise heavily influences execution time and energy consumption, it does not change the computational effort a program performs. Effort metrics that count how many operations a machine executes on behalf of a program, such as floating-point operations, the exchange of MPI messages, or file reads and writes, remain largely unaffected and—rare non-determinism set aside—reproducible. This paper addresses initial stage of an ExtraNoise project, which is aimed at revealing and tackling key questions of system noise influence on HPC applications.

We develop Stein’s method for $$\alpha $$ -stable approximation with $$\alpha \in (0,1]$$ , continuing the recent line of research by Xu (Ann Appl Probab 29(1):458–504, 2019) and Chen et al. (J Theor Probab, 2018. https://doi.org/10.1007/s10959-020-01004-1 ) in the case $$\alpha \in (1,2)$$ . The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, the proof strategy is significantly different from that in the integrable case. We rely on fine regularity estimates of the solution to Stein’s equation established in this paper.

LetX= (X(t))(t >= 0)(X(0) = 0) be a continuous centered Gaussian process on a probability space (omega,F,P), and let (Y-t)(t is an element of)[0,1] (Y-0= 0) be a continuous process (on the same probability space) with nondecreasing paths, independent ofX. Define the time-changed Gaussian processZ(t)=X(Y-t),t is an element of [0,1]. In this paper, we investigate a problem of finite-dimensional large deviations and a problem of pathwise large deviations for time-changed continuous Gaussian processes. As applications, we considered subordinated Gaussian processes.

In 2008, M. Kaneko made several interesting observations about the values of the modular j invariant at real quadratic irrationalities. The values of modular functions at real quadratics are defined in terms of their cycle integrals along the associated geodesics. In this paper we prove some of the conjectures of M. Kaneko for a general modular function.

The paper proposes a combination of the subdomain deflation method and local algebraic multigrid as a scalable distributed memory preconditioner that is able to solve large linear systems of equations. The implementation of the algorithm is made available for the community as part of an open source AMGCL library. The solution targets both homogeneous (CPU-only) and heterogeneous (CPU/GPU) systems, employing hybrid MPI/OpenMP approach in the former and a combination of MPI, OpenMP, and CUDA in the latter cases. The use of OpenMP minimizes the number of MPI processes, thus reducing the communication overhead of the deflation method and improving both weak and strong scalability of the preconditioner. The examples of scalar, Poisson-like, systems as well as non-scalar problems, stemming out of the discretization of the Navier-Stokes equations, are considered in order to estimate performance of the implemented algorithm. A comparison with a traditional global AMG preconditioner based on a well-established Trilinos ML package is provided., Comment: 21 pages, 7 figures

In this paper we establish a strong comparison principle for a nonautonomous differential inclusion with a forcing term of Heaviside type. Using this principle, we study the structure of the global attractor in both the autonomous and nonautonomous cases. In particular, in the last case we prove that the pullback attractor is confined between two special bounded complete trajectories, which play the role of nonautonomous equilibria.

This paper presents a novel computational model to measure the entropy value of 2-dimensional geometric patterns. The model takes an image file as input, it automatically identifies shapes and calculates entropy values for the given patterns based on the adjacency relationships of the detected shape tokens and the repetition frequencies of the shapes in the whole. The model utilizes Shannon’s definition of entropy to calculate shape entropy (SE), horizontal relationship entropy (HRE), and vertical relationship entropy (VRE) of 2-dimensional muqarnas patterns. In this study, the proposed model was tested in the muqarnas patterns in the portal of 12 mosques by architect Sinan which were built in the Classical Ottoman Period (sixteenth century). Initial findings of the implementation of the model show that the proposed automatized calculations are applicable to the selected sample group and have the potential to be adapted into different contexts.