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In our previous paper we have elaborated a certain signed count of real lines on real projective n-dimensional hypersurfaces of degree 2n-1. Contrary to the honest "cardinal" count, it is independent of the choice of a hypersurface, and by this reason provides a strong lower bound on the honest count. In this count the contribution of a line is its local input to the Euler number of a certain auxiliary vector bundle. The aim of this paper is to present other, in a sense more geometric, interpretations of this local input. One of them results from a generalization of Segre species of real lines on cubic surfaces and another from a generalization of Welschinger weights of real lines on quintic threefolds., Comment: 20 pages, typos are corrected (most essential, in Proposition 4.3.3)

Let $f\in\mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in\mathbb{Z}[X_{0},...,X_{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we give a bound for \[ N(f,F,B):=|\{\textbf{x}\in\mathbb{Z}^{n+1}:\max_{0\leq i\leq n}|x_{i}|\leq B,\exists t\in\mathbb{Z}\text{ such that }f(t)=F(\textbf{x})\}|, \] To do this, we introduce a generalization of the Heath-Brown and Munshi's power sieve and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables., Theorem 1 has been improved. The paper has been reorganized to improve the exposition

We investigate a steady planar flow of an ideal fluid in a bounded simply connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.

An endofunction on an Abelian group A is a fonction f: An → A such that φf (x1, …, xn) = f (φ(x1), …, φ(xn)) for all endomorphisms φ of group A and all n from ℕ. If each endofunction has the form $$f\left(x_{1}, \ldots, x_{n}\right)=\sum\nolimits_{i=1}^{n} \lambda_{i} x_{i}$$ for some central endomorphisms λ1, …, λn of group A, then such a group is called generalized endoprimal (GE) group. In this paper, we find GE-groups in the class of nonreduced Abelian groups. In addition, results paper, we find GE-group in the class of nonreduced Abelian groups. In addition, results concerning connections of GE-groups with Abelian groups whose endomorphism rings are unique addition rings have been obtained.

The paper presents an efficient numerical technique for 3D-modeling of seismic stability of large buried structures. This technique allows considering the subsoil-structure contact interaction, gravity field effects, the inhomogeneous structure of soil and the varieties location of the earthquake hypocenter. As part of this technique is substantiated sizing of the soil-structure computational domain and continuum model for hard and soft soil foundations describing. The numerical technique for determining the kinematic conditions at the lower boundary of the computational domain from the experimental accelerations at the soil surface is given, offered special non-reflecting waves boundary conditions. The methods described above and algorithms for seismic resistance solving have been implemented in certified software package “Dynamica-3”, parallelization of the algorithm has been held according to the spatial domain decomposition principle. The developed numerical technique allows correctly posing the problem of seismic vibrations of buried structure, reducing computing costs and increasing the efficiency of numerical studies of Earthquake Engineering. Through this, the multiple recalculation of the task with different action scenarios generated from experimental seismograms by probabilistic methods became technically possible. The results of these calculations allow reflecting the experience of many earthquakes, which increases reliability of the estimates. The paper presents the model calculations results of seismic stability of large-sized buried structure—the mutual vertical and horizontal displacement of soil and building walls—which can then be used to assess strength of adjacent underground pipelines. A number of numerical experiments on seismic vibrations of nearby large-sized structures have been carried out to assess the mutual influence of structures on their behavior in an earthquake. It is established that the influence of nearby large-sized objects on seismic vibrations of structures differs for different buildings.

The purpose of this paper is to show well-posedness results in L 2 -based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m ≥ 2 . The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L ∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L ∞ coefficients and its mixed variational counterpart, and then the well posedness in L 2 -based Sobolev spaces by using the Necas-Babuska-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L ∞ coefficients in L 2 -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m ∈ { 2 , 3 } .

The main goal of this paper is to demonstrate that the newest generation of NEC SX-Aurora TSUBASA architecture can perform large-scale graph processing extremely efficiently. This paper proposes approaches, which can be used for the development of high-performance vector-oriented implementations of page rank and shortest paths algorithms, including vectorised graph storage format, efficient vector-friendly graph traversals, optimised cache-aware memory accesses and efficient load-balancing. The developed implementations are optimised according to the most important features and properties of SX-Aurora architecture, which allows them achieve up to 15 times better performance compared to the optimised Intel Skylake parallel implementations and up to 5 times better performance compared to NVGRAPH library implementations for Pascal GPU architecture.

This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem { u t = u x x + f ( t , u ) , x ∈ R , t > 0 , u ( x , 0 ) = u 0 , x ∈ R , where u 0 is a nonnegative bounded function with compact support and f is a rather general nonlinearity that is periodic in t and satisfies f ( ⋅ , 0 ) = 0 . In the autonomous case where f = f ( u ) , the convergence of every bounded solution to an equilibrium has been established by Du and Matano (2010). However, the presence of periodic forcing makes the problem significantly more difficult, partly because the structure of time periodic solutions is much less understood than that of steady states. In this paper, we first prove that any ω-limit solution is either spatially constant or symmetrically decreasing. Furthermore, we show that the set of ω-limit solutions either consists of a single time-periodic solution or it consists of multiple time-periodic solutions and heteroclinic connections among them. Next, under a mild non-degenerate assumption on the corresponding ODE, we prove that the ω-limit set is a singleton, which implies the solution converges to a time-periodic solution. Lastly, we apply these results to equations with bistable nonlinearity and combustion nonlinearity, and specify more precisely which time-periodic solutions can possibly be selected as the limit.

The word maps $$ \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) $$ and $$ \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } $$ for a word $$ w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], $$ where D is a division algebra over a field K, are considered. It is proved that if $$ \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], $$ then $$ \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), $$ where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then $$ \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). $$ The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.

This paper proves the boundedness of the generalized Riesz potentials I ρ , μ , τ ⁢ f {I_{\rho,\mu,\tau}f} of functions in the Morrey space L ( 1 , φ ; κ ) ⁢ ( G ) {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is X ) {X)} , as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on x ∈ X {x\in X} . An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK–Ising percolation to chordal $$\mathrm {SLE}_\kappa ( \kappa -6)$$ with $$\kappa =16/3$$. Our proof follows the classical excursion construction of $$\mathrm {SLE}_\kappa (\kappa -6)$$ processes in the continuum, and we are thus led to introduce suitable cut-off stopping times in order to analyse the behaviour of the driving function of the discrete system when Dobrushin boundary condition collapses to a single point. Our proof is very different from that of Kemppainen and Smirnov (Conformal invariance of boundary touching loops of FK–Ising model. arXiv:1509.08858, 2015; Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE. arXiv:1609.08527, 2016) as it only relies on the convergence to the chordal $$\mathrm {SLE}_{\kappa }$$ process in Dobrushin boundary condition and does not require the introduction of a new observable. Still, it relies crucially on several ingredients: One important emphasis of this paper is to carefully write down some properties which are often considered folklore in the literature but which are only justified so far by hand-waving arguments. The main examples of these are: We end the paper with a detailed sketch of the convergence to radial $$\mathrm {SLE}_\kappa ( \kappa -6)$$ when $$\kappa =16/3$$ as well as the derivation of Onsager’s one-arm exponent 1 / 8.

Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X \in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$, consisting of parameterized spectrum objects over $X$. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is $\mathcal{T}_X\mathcal{C}$. When $\mathcal{C}$ consists of algebras over a nice $\infty$-operad in a stable $\infty$-category, $\mathcal{T}_X\mathcal{C}$ is equivalent to the $\infty$-category of operadic modules, by work of Basterra--Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper., Comment: The section concerning stabilization of model categories was separated into an independent paper, appearing now as arXiv:1802.08031

The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L ( Y , W ) is strongly unique. In this paper we show (in some circumstances) that if 1 λ ( Y , X ) , then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation.

It is well known that developing methods for solving Dirichlet problems is important and relevant for various areas of mathematical physics related to the Laplace equation, the Helmholtz equation, the Stokes equation, the Maxwell equation, the Dirac equation, and others. The author in previous papers studied the solvability of Dirichlet boundary value problems of the first and second orders in quaternionic analysis. In the present paper, we study a higher-order Dirichlet boundary value problem associated with the two-dimensional Helmholtz equation with complex potential. The existence and uniqueness of a solution to the Dirichlet boundary value problem in the two-dimensional case is proved and an appropriate representation formula for the solution of this problem is found. Most Dirichlet problems are solved for the case in three variables. Note that the case of two variables is not a simple consequence of the three-dimensional case. To solve the problem, we use the method of orthogonal decomposition of the quaternion-valued Sobolev space. This orthogonal decomposition of the space is also a tool for the study of many elliptic boundary value problems that arise in various areas of mathematics and mathematical physics. An orthogonal decomposition of the quaternion-valued Sobolev space with respect to the high-order Dirac operator is also obtained in this paper.

Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63] ), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by e ≥ 0 . Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with e = 0 ) plus the inner layer as e → 0 , where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63] , and open a new window in the future theoretical study of chemotaxis models.

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042

A great number of theoretical results are known about log Gromov-Witten invariants, but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov-Witten invariants. The first such calculation by Gross-Pandharipande-Siebert deals with multiple covers over rigid curves in the log Calabi-Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi-Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of "logarithmic" 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps., Comment: Added two example sections including a comparison with tropical multiplicity. 53 pages, 4 figures

In this paper, we present some fixed point theorems of Krasnosel’skii’s type in Banach spaces. The involved operators need not to be compact nor weakly continuous. The results are obtained and formulated with the use of the measures of weak noncompactness and a large classes of contractions (strict contractions, nonlinear contractions, as well as nonexpansive or pseudocontractive mappings). Throughout the paper, we use the hypothesis $$\mathsf {(H1)}$$ and $$\mathsf {(H2)}$$ , which are one of the main ingredients of the proofs. Finally, with the obtained fixed point results, we discuss the existence of solutions to a stationary transport equation with delayed neutrons.

In the present paper, we investigate the existence and regularity of weak solutions for $$\psi $$ -Hilfer fractional boundary value problem in $$\mathbb {C}^{\alpha ,\beta ;\psi }_{2}$$ and $$\mathcal {H}$$ (Hilbert space) spaces, using extension of the Lax–Milgram theorem. In this sense, to finalize the paper, we discuss the integration by parts for $$\psi $$ -Riemann–Liouville fractional integral and $$\psi $$ -Hilfer fractional derivative.

In the paper a second order differential system with a periodic boundary data is considered. Using some variational methods sufficient conditions for the continuous dependence of trajectories on controls are proved. The obtained results are applied then to the optimal control problem governed by the above system and the cost functional of a Bolza-type. In the end of the paper, an example of periodic optimal control problem demonstrating the applicability of the results is presented.

Purpose Besse first conjectured that the solution of the critical point equation (CPE) must be Einstein. The CPE conjecture on some other types of Riemannian manifolds, for instance, odd-dimensional Riemannian manifolds has considered by many geometers. Hence, it deserves special attention to consider the CPE on a certain class of almost contact metric manifolds. In this direction, the authors considered CPE on almost f-cosymplectic manifolds. Design/methodology/approach The paper opted the tensor calculus on manifolds to find the solution of the CPE. Findings In this paper, in particular, the authors obtained that a connected f-cosymplectic manifold satisfying CPE with \lambda=\tilde{f} is Einstein. Next, the authors find that a three dimensional almost f-cosymplectic manifold satisfying the CPE is either Einstein or its scalar curvature vanishes identically if its Ricci tensor is pseudo anti‐commuting. Originality/value The paper proved that the CPE conjecture is true for almost f-cosymplectic manifolds.

In this paper, we obtain the existence of positive solution for a system of quasilinear Schrodinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrodinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrodinger equations.

In a companion paper to this one, we proved that the Gromov--Witten theory of a Fano orbifold line of type $D$ is governed by a system of Hirota Bilinear Equations. The goal of this paper is to prove that every solution to the Hirota Bilinear Equations determines a solution to a new integrable hierarchy of Lax equations. We suggest the name extended D-Toda hierarchy for this new system of Lax equations, because it should be viewed as the analogue of Carlet's extended bi-graded Toda hierarchy, which is known to govern the Gromov--Witten theory of Fano orbifold lines of type $A$, 73 pages

In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate a plausible conjecture under which the said three results in [3] remain true and the method of proof goes through using the same arguments. However, these results are now only conditional and still await a full proof.

This paper deals with the pulsating waves and entire solutions for a spatially periodic nonlocal dispersal model with a quiescent stage. By the method of super- and subsolutions together with the comparison principle, we establish the existence of pulsating waves with any speed larger than the spreading speed. The construction of the super- and subsolutions depends on the principal eigenvalue theory for a periodic eigenvalue problem with partially degenerate nonlocal dispersal. The joint results of this paper and our recent work (Wang J B, Li W T, Sun J W. Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats. Proc Roy Soc Edinburgh Sect A, 2018, 148: 849–880) show that the spreading speed coincides with the minimal wave speed of pulsating waves for the considered system. Moreover, combining the rightward and leftward pulsating waves with different speeds and a spatially periodic solution, we prove the existence and qualitative properties of entire solutions other than pulsating waves, which provide some new spreading ways in a heterogeneous habitat.

The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of denumerable Markov processes. We present a detailed analytic proof of this extremely useful tool and explain its clear probabilistic interpretation. We then apply this tool to investigate the basic problems of existence and uniqueness criteria for denumerable Markov processes with instantaneous states to which few results have been obtained even until now. Although the complete answers regarding these existence and uniqueness criteria will be given in a subsequent paper, we shall, in this paper, present part solutions of these very important problems that are closely linked with the subtle Williams S and N conditions.

The aim of this paper is twofold. Firstly, we study the non-existence of finite order meromorphic solutions to the Cubic type of Fermat functional equation $$f(z)^3-3\tau f(z)f(z+c)+f(z+c)^3=1$$. In addition, the paper is concerned with the description of finite order entire solutions of the Quadratic type of Fermat functional equation $$f(z)^2-2\mu f(z)f(z+c)+f(z+c)^2=1$$.

In this paper, we study the uniqueness problem of meromorphic functions sharing two finite sets ignoring multiplicities. This paper is inspired from an article due to Chen (Open Math 15:1244–1250, 2017).

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

After the description by E. Cartan in 1932 holomorphically homogeneous real hypersurfaces of two-dimensional complex spaces, a similar study in the 3-dimensional case remains incomplete. In a series of works performed by several international teams of authors, the problem is reduced to describing homogeneous surfaces that are non-degenerate in Levi sense and have exactly 5-dimensional Lie algebras of holomorphic vector fields. In this paper, precisely such homogeneous surfaces are investigated. At the same time, a significant part of the extensive list of abstract 5-dimensional Lie algebras does not provide new examples of homogeneity. A complete description of the orbits of 5-dimensional non-solvable Lie algebras in a three-dimensional complex space, given in the paper, includes examples of new homogeneous hypersurfaces. The presented results bring to finish a large-scale scientific study of interest to various branches of mathematics.

The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of d-cycles, d-hypertrees and d-hypercuts are, respectively, (d +1)-, d-and (n − d − 1)-vertex-connected. It is also shown that the facet graph of a d-cycle cannot be split into more than s connected components by removing at most s vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.

The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory forquantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.

We define the concepts of commutator (Krylov) transitive and strongly commutator (Krylov) transitive Abelian p-groups. These two innovations are respectively non-trivial generalizations of the notions of commutator fully transitive and strongly commutator fully transitive p-groups from a paper of Chekhlov and Danchev (J. Group Theory, 2015). They are also commutator socle-regular in the sense of Danchev and Goldsmith (J. Group Theory, 2014). Various results from there and from a paper of Goldsmith and Strüngmann (Houston J. Math., 2007) are considerably extended to this new point of view. We also define and explore the concept of a commutator weakly transitive Abelian p-group, comparing its properties with those of the aforementioned two group classes. Some affirmations, sounding quite curiously, are detected in order to illustrate the pathology of the commutators in the endomorphism rings of p-primary Abelian groups.

In this paper, we study the concomitants of m-dual generalized order statistics (and consequently m-generalized order statistics) from Huang–Kotz Farlie–Gumble–Morgenstern bivariate distribution as an extension of several recent papers. Some well-known information measures, the Shannon entropy, the Kullback–Leibler distance and the Fisher information number, are derived. Moreover, some useful recurrence relations between single and product moments of concomitants are obtained. Finally, an application of these results is given for bivariate generalized exponential distribution.

This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator ℓ p → X {\ell_{p}\to X} has the BAP when X has it and 0 < p ≤ 1 {0 , which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.

International audience; This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Bérard-Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer-Persson-Sundqvist for the Neumann problem to the case where h > 0 is small. MSC classification (2010): 35P99, 58J50, 58J37.

In the present paper, we introduce a certain subclass 𝒦q(λ, γ, h) of analytic functions by means of a quasi-subordination. Sharp bounds of the Fekete-Szegő functional for functions belonging to the class 𝒦q(λ, γ, h) are obtained. The results presented in the paper give improved versions for the certain subclasses involving the quasi-subordination and majorization.

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space R n , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25] , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.

Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \dfrac{(L \cdot A) \: (L \cdot B)}{A \cdot B}$, where $A \cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{\l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric., Comment: 50 pages, 14 figures. Final version

Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $$\gamma $$, and it has long been believed that when $$\gamma = \sqrt{8/3}$$, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other’s structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other’s structure and showing that the resulting laws agree. The present work considers a growth process called quantum Loewner evolution (QLE) on a $$\sqrt{8/3}$$-LQG surface $${\mathcal {S}}$$ and defines $$d_{{\mathcal {Q}}}(x,y)$$ to be the amount of time it takes QLE to grow from $$x \in {\mathcal {S}}$$ to $$y \in {\mathcal {S}}$$. We show that $$d_{{\mathcal {Q}}}(x,y)$$ is a.s. determined by the triple $$({\mathcal {S}},x,y)$$ (which is far from clear from the definition of QLE) and that $$d_{{\mathcal {Q}}}$$ a.s. satisfies symmetry (i.e., $$d_{{\mathcal {Q}}}(x,y) = d_{{\mathcal {Q}}}(y,x)$$) for a.a. (x, y) pairs and the triangle inequality for a.a. triples. This implies that $$d_{{\mathcal {Q}}}$$ is a.s. a metric on any countable sequence sampled i.i.d. from the area measure on $${\mathcal {S}}$$. We establish several facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric a.s. extends uniquely and continuously to the entire $$\sqrt{8/3}$$-LQG surface and that the resulting measure-endowed metric space is TBM.

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1-dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function $${{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R -{{\,\mathrm{pd}\,}}I^{(t)} - 1$$ , where $$I^{(t)}$$ denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and $${{\,\mathrm{pd}\,}}$$ denotes the projective dimension. It has been an open question whether the function $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is non-increasing if I is a squarefree monomial ideal. We show that $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is almost non-increasing in the sense that $${{\,\mathrm{depth}\,}}R/I^{(s)} \ge {{\,\mathrm{depth}\,}}R/I^{(t)}$$ for all $$s \ge 1$$ and $$t \in E(s)$$ , where $$\begin{aligned} E(s) = \bigcup _{i \ge 1}\{t \in {\mathbb {N}}|\ i(s-1)+1 \le t \le is\} \end{aligned}$$ (which contains all integers $$t \ge (s-1)^2+1$$ ). The range E(s) is the best possible since we can find squarefree monomial ideals I such that $${{\,\mathrm{depth}\,}}R/I^{(s)} < {{\,\mathrm{depth}\,}}R/I^{(t)}$$ for $$t \not \in E(s)$$ , which gives a negative answer to the above question. Another open question asks whether the function $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is always constant for $$t \gg 0$$ . We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that $$I^{(t)}$$ is integrally closed for $$t \gg 0$$ (e.g. if I is a squarefree monomial ideal), then $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is constant for $$t \gg 0$$ with $$\begin{aligned} \lim _{t \rightarrow \infty }{{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R - \dim \oplus _{t \ge 0}I^{(t)}/{\mathfrak {m}}I^{(t)}. \end{aligned}$$ Our last result (which is the main contribution of this paper) shows that for any positive numerical function $$\phi (t)$$ which is periodic for $$t \gg 0$$ , there exist a polynomial ring R and a homogeneous ideal I such that $${{\,\mathrm{depth}\,}}R/I^{(t)} = \phi (t)$$ for all $$t \ge 1$$ . As a consequence, for any non-negative numerical function $$\psi (t)$$ which is periodic for $$t \gg 0$$ , there is a homogeneous ideal I and a number c such that $${{\,\mathrm{pd}\,}}I^{(t)} = \psi (t) + c$$ for all $$t \ge 1$$ .

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space.

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

Let $G$ be a connected complex semi-simple Lie group, and let $Z_{\bf u}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure $\pi_n$ on $Z_{\bf u}$ defined by a standard multiplicative Poisson structure $\pi_{\rm st}$ on $G$. We explicitly express $\pi_n$ on each of the $2^n$ affine coordinate charts, one for every subexpression of ${\bf u}$, in terms of the root strings and the structure constants of the Lie algebra of $G$. We show that the restriction of $\pi_n$ to each affine coordinate chart gives rise to a Poisson structure on the polynomial algebra ${\mathbb{C}}[z_1, \ldots, z_n]$ which is an {\it iterated Poisson Ore extension} of $\mathbb{C}$ compatible with a rational action by a maximal torus of $G$. For canonically chosen $\pi_{\rm st}$, we show that the induced Poisson structure on ${\mathbb{C}}[z_1, \ldots, z_n]$ for every affine coordinate chart is in fact defined over ${\mathbb Z}$, thus giving rise to an iterated Poisson Ore extension of any field ${\bf k}$ of arbitrary characteristic. The special case of $\pi_n$ on the affine chart corresponding to the full subexpression of ${\bf u}$ yields an explicit formula for the standard Poisson structures on {\it generalized Bruhat cells} in Bott-Samelson coordinates. The paper establishes the foundation on generalized Bruhat cells and sets up the stage for their applications, some of which are discussed in the Introduction of the paper.

In 1964, A. N. Sharkovskii published a paper in which he introduced an ordering relation on the set of positive integers. His ordering had the property that if a continuous self-map of an interval has a periodic point of some period p, then it also has periodic points of any period larger than p in this ordering. The least number in this ordering is 3. Thus, if a continuous self-map of an interval has a point of period 3, then it has points of any period. In 1975, this result was rediscovered by Lie and Yorke, who published it in their paper “Period three implies chaos.” Their work has led to the international recognition of Sharkovskii’s theorem. Since then, numerous papers on properties of self-maps of an interval have appeared. In 1994, even a conference named “Thirty Years after Sharkovskii’s Theorem: New Perspectives” was held. One of the research directions is estimating the number of periodic trajectories which a map satisfying the conditions of Sharkovskii’s theorem must have. In 1985, Bau-Sen Du published a paper in which he obtained an exact lower bound for the number of periodic trajectories of given period. In the present paper, a new, significantly shorter and more natural, proof of this result is given.

Geometric properties of spaces of Keplerian orbits are of interest for celestial mechanics problems related to the search for groups of celestial bodies with close orbits. Those groups include asteroid families and meteor streams. Studying these groups provides important information on the evolution of the Solar System, as well as on the characteristics of objects within a family and their parent bodies. The local properties of a distance function between orbits are of primary importance for the problems of the search for families of related celestial bodies, because orbits of family members cluster together in a small region of the orbit space. Several metrics on the set of Keplerian orbits $$\mathbb{H}$$ and its quotient sets are considered in this paper. For each of these metrics we solve the question: is there a normed vector space that is locally isometric to the orbit metric space? In two of the considered cases, the answer turns out to be positive: the quotient space of $$\mathbb{H}$$ by the equivalence relation neglecting the magnitude of the pericenter argument of the orbit can be isometrically embedded into $${{\mathbb{R}}^{4}}$$ . The embedding into $${{\mathbb{R}}^{3}}$$ also exists for the quotient space by the pair of elements: the longitude of ascending node and the pericenter argument. It is shown in this paper that for other metrics, the answer to the stated question is negative. The possibility of an isometric embedding of the orbit space or its part into Euclidean space is useful in application to the aforementioned problems of celestial mechanics. The isometric map helps to define the mean of the orbit family in a natural way: the arithmetic mean of images corresponds to the orbit with the minimum square deviation of distances from the orbits of the family.

Moment functions play a basic role in probability theory. A natural generalization can be defined on hypergroups which leads to the concept of generalized moment function sequences. In a former paper we studied some function classes on hypergroup joins which play a basic role in spectral synthesis. Moment functions are also important basic blocks of spectral synthesis. All these functions can be characterized by well-known functional equations. In this paper we describe generalized moment function sequences on hypergroup joins.

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.