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Every Finsler metric naturally induces a spray but not so for the converse. The notion for sprays of scalar (resp. isotropic) curvature has been known as a generalization for Finsler metrics of scalar (resp. isotropic) flag curvature. In this paper, a new notion, sprays of constant curvature, is introduced and especially it shows that a spray of isotropic curvature is not necessarily of constant curvature even in dimension n ≥ 3 . Further, complete conditions are given for sprays of isotropic (resp. constant) curvature to be Finsler metrizable. Based on this result, the local structure is determined for locally projectively flat Berwald sprays of isotropic (resp. constant) curvature which are Finsler metrizable, and some more sprays of isotropic curvature are discussed for their metrizability. Besides, the metrizability problem is also investigated for sprays of scalar curvature under certain curvature conditions. [ABSTRACT FROM AUTHOR]

The A ^ 2 n (2) -hierarchy can be constructed from a splitting of the Kac–Moody algebra of type A ^ 2 n (1) by an involution. By choosing certain cross section of the gauge action, we obtain the A ^ 2 n (2) -KdV hierarchy. They are the equations for geometric invariants of isotropic curve flows of type A, which gives a geometric interpretation of the soliton hierarchy. In this paper, we construct Darboux and Bäcklund transformations for the A ^ 2 n (2) -hierarchy, and use it the construct Darboux transformations for the A ^ 2 n (2) -KdV hierarchy and isotropic curve flows of type A. Moreover, explicit soliton solutions for these hierarchies are given. [ABSTRACT FROM AUTHOR]

In mathematics, the term "differential" refers to several related notions derived from the early days of mathematical analysis, which became rigorous later, such as infinitesimal differences and the derivatives of functions. The notion is used in various areas of mathematics such as algebraic geometry, algebraic topology, calculus, differential geometry etc. The term differential is used non rigorously in calculus referring to an infinitely small ("infinitesimal") variation change in a quantity. For example, if one considers x as a variable, then a "bigger" change in the value of x is often denoted by x. The differential dx is an infinitesimal change of the variable x. The concept of an infinitely small or infinitely slow change is very useful, and there are several of mathematical tools to make it precise. Using calculus, it is possible to relate the infinitesimal changes of several variables to each other using function derivatives. In this article we present the notions of Gateaux and Fréchet differentials of a multivariable function with their properties, geometric interpretation and applications to function approximations. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for solving the Poisson-Boltzmann (PB) equation based on a rigorous treatment of geometric singularities of the dielectric interface and a Green’s function formulation of charge singularities. Geometric singularities, such as cusps and self-intersecting surfaces, in the dielectric interfaces are bottleneck in developing highly accurate PB solvers. Based on an advanced mathematical technique, the matched interface and boundary (MIB) method, we have recently developed a PB solver by rigorously enforcing the flux continuity conditions at the solvent-molecule interface where geometric singularities may occur. The resulting PB solver, denoted as MIBPB-II, is able to deliver second order accuracy for the molecular surfaces of proteins. However, when the mesh size approaches half of the van der Waals radius, the MIBPB-II cannot maintain its accuracy because the grid points that carry the interface information overlap with those that carry distributed singular charges. In the present Green’s function formalism, the charge singularities are transformed into interface flux jump conditions, which are treated on an equal footing as the geometric singularities in our MIB framework. The resulting method, denoted as MIBPB-III, is able to provide highly accurate electrostatic potentials at a mesh as coarse as 1.2 Å for proteins. Consequently, at a given level of accuracy, the MIBPB-III is about three times faster than the APBS, a recent multigrid PB solver. The MIBPB-III has been extensively validated by using analytically solvable problems, molecular surfaces of polyatomic systems, and 24 proteins. It provides reliable benchmark numerical solutions for the PB equation. [ABSTRACT FROM AUTHOR]

Abstract Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC) was proposed: Whenever (1 , h 1 , ... , h i , ... , h e − i , ... , h e − 1 , 1) and (1 , h 1 , ... , h i + α , ... , h e − i + α , ... , h e − 1 , 1) are both Gorenstein Hilbert functions for some α ≥ 2 , then (1 , h 1 , ... , h i + β , ... , h e − i + β , ... , h e − 1 , 1) is also Gorenstein, for all β = 1 , 2 , ... , α − 1. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such basic functions in commutative algebra. Before now, very little progress was made on the GIC. The main goal of this note is to prove the case e ≤ 5 , in arbitrary codimension. Our arguments will be in part constructive, and will combine several different tools of commutative algebra and classical algebraic geometry. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Hyers-Ulam stability of the following additive functional equations ... in paranormed spaces. [ABSTRACT FROM AUTHOR]

This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as x' = a1x - y - a3x² + (2a2 + a5) xy + a6y², y' = x + a1y + a2x² + (2a3 + a4) xy - a2y². In particular, we study the semi-varieties defined in terms of the parameters a1, a2, ..., a6 where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry. [ABSTRACT FROM AUTHOR]

In this article, we survey some results on geometric methods to study quiver representations, and applications of these results to sheaves, equivariant sheaves and parabolic bundles. [ABSTRACT FROM AUTHOR]

An embedding theorem of weighted spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space in weighted Lebesgue spaces is established. The theorem is formulated depending on geometric parameters of the domain of functions. [ABSTRACT FROM AUTHOR]

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower. [ABSTRACT FROM AUTHOR]

Let M be a hypermodule over a hyperring R such that the intersection of any two subhypermodules of M, is a subhypermodule of M. We introduce the concept of an essential subhypermodule in M relative to an arbitrary subhypermodule T of M, which is called a T-essential subhypermodule of M. Our main goal in this work is to investigate properties of (relative) essential subhypermodules. We apply this concept to introduce extending hypermodules. Examples are provided to illustrate different concepts. [ABSTRACT FROM AUTHOR]

A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules. [ABSTRACT FROM AUTHOR]

This note is a summary of our work [OO], which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kähler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not necessarily "maximally degenerating". Our results also give a proof of Kontsevich-Soibelman [KS06,Conjecture 1] (cf., [GW00, Conjecture 6.2]) in the case of K3 surfaces as a byproduct [ABSTRACT FROM AUTHOR]

We give an introduction to superalgebra, founded on the difference between even (commuting) and odd (anti-commuting) variables. We give a sketch of Graßmann's work, and show how derivations of those structures induce various superalgebra structures, Lie superalgebras of Cartan type being obtained with even derivations, while odd derivations induce Jordan-type superalgebras. [ABSTRACT FROM AUTHOR]

Any virtually free group H containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing H as a verbally closed subgroup. [ABSTRACT FROM AUTHOR]

We have found an infinite family of nonisomorphic hyperelliptic curves of genus two over the field of rational numbers whose Jacobian varieties contain rational points of order 28. Previously, only 10 such curves were known. [ABSTRACT FROM AUTHOR]

Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation. [ABSTRACT FROM AUTHOR]

Equivariant twisted K theory classes on compact Lie groups G can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra L G using a supersymmetric Wess–Zumino–Witten model. The aim of the present paper is to extend the construction to higher loop algebras using an abelian extension of a 3D current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to 3D gauge transformations but do not define true Hilbert space operators. [ABSTRACT FROM AUTHOR]

We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose maximal nowhere degenerate abelian subvariety is isogenous to a constant abelian variety. To prove the results, we investigate closed subvarieties of abelian schemes over constant varieties, where constant varieties are varieties over a function field which can be defined over the constant field of the function field. [ABSTRACT FROM AUTHOR]

Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some conjectures on the usual topological closure of an algebraic flow in A. The main result is a proof of these conjectures when the algebraic flow is given by an algebraic curve. [ABSTRACT FROM AUTHOR]

Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry. [ABSTRACT FROM AUTHOR]

Since the work of Mikhail Kapranov in [Compos. Math. 115 (1999), no. 1, 71–113], it is known that the shifted tangent complex 𝕋 X \mathbb{T}_{X} [-1] of a smooth algebraic variety X is endowed with a weak Lie structure. Moreover, any complex of quasi-coherent sheaves E on X is endowed with a weak Lie action of this tangent Lie algebra. This Lie action is given by the Atiyah class of E. We will generalise this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to singular schemes. This work uses tools of both derived algebraic geometry and ∞ {\infty} -category theory. [ABSTRACT FROM AUTHOR]

We construct the dagger realization functor for analytic motives over non-archimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a discrete field of positive characteristic. In particular, the motivic language on the classic étale site provides a new direct definition of the overconvergent de Rham cohomology and rigid cohomology and shows that their finite dimensionality follows formally from one of Betti cohomology for smooth projective complex varieties. [ABSTRACT FROM AUTHOR]

Let X be an irreducible smooth projective curve, defined over an algebraically closed field k , of genus at least three and L a line bundle on X . Let M X ( r , L ) be the moduli space of stable vector bundles on X of rank r and determinant L with r ≥ 2 . We prove that the Brauer group Br ( M X ( r , L ) ) is cyclic of order g . c . d . ( r , degree ( L ) ) . We also prove that Br ( M X ( r , L ) ) is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in [1] under the assumption that k = C . [ABSTRACT FROM AUTHOR]

We study the Poincaré series of modules over a fiber product of commutative local rings. We introduce the notion of a weak complete intersection ideal; these are the ideals with the property that every differential in their minimal free resolutions can be represented by a matrix whose entries are in the ideal itself. We show that many properties of the Poincaré series that are known to hold for a fiber product over the maximal ideal also hold for those over weak intersection ideals. [ABSTRACT FROM AUTHOR]

The main goal of this article is to construct some geometric invariants for the topology of the set [ABSTRACT FROM AUTHOR]