Vanishing Conditions Research Articles

On Weakly Cyclic Generalized Z-Symmetric Manifolds

In this paper, we define a (0,2)-type symmetric tensor Z and called it a generalized Z tensor. We study weakly cyclic generalized Z-symmetric manifold and prove that it is a generalized quasi-Einstein manifold. We have obtained a condition for vanishing of sum of 1-forms. We further find a condition to be a Ricci semisymmetric manifold. Finally, it is shown that the semisymmetry and Weyl semisymmetry are equivalent in such a manifold.

Finite quotients of three-dimensional complex tori

We provide a characterization of quotients of three-dimensional complex tori by finite groups that act freely in codimension one via a vanishing condition on the first and second orbifold Chern class. We also treat the case of actions free in codimension two, using instead the "birational" second Chern class, as we call it. Both notions of Chern classes are introduced here in the setting of compact complex spaces with klt singularities. In such generality, this topic has not been treated in the literature up to now. We also discuss the relation of our definitions to the classical Schwartz-MacPherson Chern classes.

Financial cooperation initiatives in Latin America: Conditions of origins, subsistence and eventual vanishing

Why do countries engage in Regional Financial Cooperation (RFC) initiatives and why they may give up on them? Under which conditions are those mechanisms born and how may changes affect their performance? Although comparative studies have been a prolific strategy to investigate RFC the focus on the experiences of a specific region may reveal new insights. Therefore the aim of this paper is to map the existing RFC mechanisms in Latin America, seeking to identify the demand, supply and conjectural conditions behind the processes of their creation and evolution. The theoretical framework provides concepts from International Relations’ theories concerning regional institution building. Empirically fourteen Latin American RFC initiatives are surveyed. As a result important variables explaining RFC mechanisms in Latin America are presented in the paper: demand for greater participation (sense of belonging), material and political capacity from a paymaster and macroeconomic coordination.

Creeping flow of fluid sphere contained in a spherical envelope: magnetic effect

The characteristic of creeping flow past a fluid sphere enclosed in a spherical envelope bearing fluid of different viscosity has been studied under the impact of transverse magnetic field. Stream functions related to modified Bessel functions are used in order to calculate the solution in closed form. The problem is considered to be parted into two flow regions as inner fluid and outer fluid region respectively, which are supposed to be governed using Stokes equations with different Hartmann number. At the contact layer of outer and inner fluid sphere, we assume the vanishing of normal components of velocity along with continuity of tangential components of velocity and stress respectively as boundary conditions. The condition of vanishing of vorticity (Kuwabara model) is considered to be applicable at the outer layer of fluid envelope. Expression for drag acting on the inner fluid sphere is presented. In limiting cases, several significant results accessible in literature are evaluated.

A (3,2) reduced degree-of-freedom unified zigzag laminated beam theory

A (3,2) unified zigzag beam theory is developed with a reduced number of degree-of-freedom. Comparing to previous methods in the field of zigzag beam theory, the main novelty in this paper's method is that a more general non-vanishing top/bottom surface's shear stress boundary conditions are satisfied automatically in strong form. The bottom surface shear stress condition and the interface shear stress continuity conditions are used to uniquely determine the coefficients of zigzag functions. For the top surface shear stress condition, it is used to eliminate one degree-of-freedom, changing the 7°-of-freedom (3,2) zigzag beam to a 6°-of-freedom (3,2) zigzag beam. The zigzag coefficients are derived with an explicit formulation. Since the proposed method's formula is based on the unified beam theory, the formulation can be applied to any specific beam theory. The corresponding zigzag coefficients are also dependent on the specific beam theory's thickness basis function.In the numerical test section, several benchmark problems are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for both thick and thin beams. The shear stress accuracy is also good for both vanishing and non-vanishing shear stress boundary conditions on top/bottom surfaces.

A nonlocal SIS epidemic problem with double free boundaries

In this paper, we investigate a nonlocal SIS epidemic model with double free boundaries. The existence, uniqueness and some estimates of the global solution are discussed first. Then, with the help of studying the long-time behavior of the solution to a Cauchy problem for a nonhomogeneous heat equation, the long-time behavior of the solution to the SIS free boundary problem is obtained for the disease vanishing case. At last, some sufficient conditions for the disease vanishing are established.

Theta divisors and Ulrich bundles on geometrically ruled surfaces

We consider the following question: for which invariants $g$ and $e$ is there a geometrically ruled surface $S \rightarrow C$ over a curve $C$ of genus $g$ with invariant $e$ such that $S$ is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper Theta divisors on some moduli spaces of vector bundles on $C$ with the existence of Ulrich line bundles on $S$ will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles to specific geometric conditions on the symmetric powers of the defining vector bundle of a given ruled surface. This general principle leads to some finer existence results of Ulrich line bundles in particular cases. Another focus is on the rank two case where, with very few exceptions, we show the existence of large families of special Ulrich bundles on arbitrary polarized ruled surfaces.

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

Inverse Problem for an Equation with A Nonstandard Growth Condition

This paper describes an inverse problem for determining the right side of a parabolic equation with a nonstandard growth condition and integral overdetermination condition. The Galerkin method is used to prove the existence of two solutions of the inverse problem and their uniqueness, one of them being local and the other one being global in time. Sufficient blow-up conditions for the local solution for a finite time in a limited region with a homogeneous Dirichlet condition on its boundary are obtained. The blow-up of the solution is proven using the Kaplan method. The asymptotic behavior of the inverse problem solutions for large time values is investigated. Sufficient conditions for vanishing of the solution for a finite time are obtained. Boundary conditions ensuring the corresponding behavior of the solutions are considered.

Asymptotic spreading of a diffusive competition model with different free boundaries

Recently, the authors of [22] studied a diffusive prey–predator model with two different free boundaries. They first obtained the existence, uniqueness, regularity, uniform estimates and long time behaviors of global solution, and then established the conditions for spreading and vanishing. Especially, when spreading occurs, they provided accurate limits of two species as t→+∞, and gave some estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries. Motivated by the paper [22], in this paper we discuss the diffusive competition model with two different free boundaries, which had been investigated by [7,11,15,21]. The main purpose of this paper is to establish much sharper estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries when spreading occurs. Furthermore, how the solution approaches the semi-wave when spreading happens is also described.

Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an [Formula: see text] linear system of ODEs with an irregular singularity of Poincaré rank 1 at [Formula: see text] and Fuchsian singularity at [Formula: see text], holomorphically depending on parameter [Formula: see text] within a polydisk in [Formula: see text] centered at [Formula: see text]. The eigenvalues of the leading matrix at [Formula: see text], which is diagonal, coalesce along a coalescence locus [Formula: see text] contained in the polydisk. Under minimal vanishing conditions on the residue matrix at [Formula: see text], we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .] that isomonodromic deformations can be extended to the whole polydisk, including [Formula: see text], in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of [Formula: see text], where it simplifies. Conversely, if the [Formula: see text]-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting [Formula: see text], and if suitable entries of the Stokes matrices vanish, then [Formula: see text] is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.

Solutions for a nonlinear fractional Euler–Lagrange type equation

The aim of this paper is the study of a nonlinear fractional Euler–Lagrange type equation with a nonlocal condition by means of lower and upper solutions method. For this purpose, we begin by solving an auxiliary problem by using Laplace transform, then we convert the posed problem to an equivalent right Caputo fractional differential equation with a vanishing terminal boundary condition. After constructing the lower and upper solutions, we define a sequence of modified problems that we solve by Schauder fixed point theorem. Finally, two numerical examples are given to illustrate the obtained results.

The diffusive Beddington‐DeAngelis predator‐prey model with nonlinear prey‐taxis and free boundary

The diffusive Beddington‐DeAngelis predator‐prey model with nonlinear prey‐taxis and free boundary is considered. We investigate the existence, uniqueness, regularity, uniform estimates, and long‐time behavior of the global solution. Some sufficient conditions for both spreading and vanishing are also established. To our best knowledge, this is the first work concerning the dynamics of the cross‐diffusion systems with free boundaries.

Expansion-free self-gravitating dust dissipative fluids

The present study deals with the spherically symmetric distributions of dust dissipative fluids with a central vacuum cavity, evolving under the condition of vanishing of expansion scalar. The work is based on two aspects—on the one hand the description of model in terms of structures scalars emerging in the orthogonal splitting of Riemann tensor. Secondly, to obtain the solutions of dynamical equations under some special cases and assumption on implicit function of variables. The solutions here discussed, model the evolution of the vacuum cavity of fluid distribution, thereby showing the potential of expansion free motion.

A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients

<p style='text-indent:20px;'>This paper is devoted to a spatial heterogeneous SIS model with the infected group equipped with a free boundary. Our main aim is to determine whether the disease is spreading forever or extinct eventually, and to illustrate, under the nonhomogeneous spatial environment, free boundaries can have a large influence on the infected behavior at the large time. For this purpose, we first introduce a basic reproduction number and then establish a spreading-vanishing dichotomy. Then by investigating the effect of the diffusion rate, initial domain and spreading speed on the asymptotic behavior of the infected group, we establish some sufficient conditions and even necessary and sufficient conditions for disease spreading or vanishing.

Spatial‐temporal basic reproduction number and dynamics for a dengue disease diffusion model

A reaction‐diffusion system with free boundary is proposed to describe the transmission of the dengue disease from mosquitoes to humans. In addition to the classical basic reproduction number R0, the spatial‐temporal basic reproduction number is introduced to determine the persistence and eradication of the disease. Some sufficient conditions for the disease vanishing or spreading are obtained. The disease will go extinct under one of the conditions: the classical basic reproduction number R0 &lt; 1 and the spatial‐temporal basic reproduction number with small expanding capability. The spread of the disease in the whole area is possible if for some t≥0. Numerical simulations are also given to illustrate the theoretical results.

Rational singularities of $G$-saturation

Let $G$ be a semisimple algebraic group defined over an algebraically closed field of characteristic 0 and $P$ a parabolic subgroup of $G$. Let $M$ be a $P$-module and $V$ a $P$-stable closed subvariety of $M$. We show in this paper that, if the varieties $V$ and $G\cdot M$ have rational singularities, and the induction functor $R^i ind _P^G(-)$ satisfies certain vanishing conditions, then the variety $G\cdot V$ has rational singularities. This generalizes a result of Kempf on the collapsing of homogeneous bundles. As an application, we prove the property of having rational singularities for nilpotent commuting varieties over $3\times 3$ matrices.

On the Unstable Hurewicz Homomorphism and Toda Brackets

The aim of this note is to examine the Curtis conjecture in the light of existing structural results about the 2-primary part of the stable homotopy ring. Motivated by Joel Cohen’s result on generating stable stems using higher Toda brackets, we obtain sufficient conditions for vanishing of the unstable Hurewicz homomorphism $${_2\pi _*^s}\simeq {_2\pi _*}QS^0\rightarrow H_*(QS^0;\mathbb {Z}/2)$$ . We also record some partial results on the relation between EHP sequence and the behaviour of Hurewicz homomorphism.

Topological strings on singular elliptic Calabi-Yau 3-folds and minimal 6d SCFTs

We apply the modular approach to computing the topological string partition function on non-compact elliptically fibered Calabi-Yau 3-folds with higher Kodaira singularities in the fiber. The approach consists in making an ansatz for the partition function at given base degree, exact in all fiber classes to arbitrary order and to all genus, in terms of a rational function of weak Jacobi forms. Our results yield, at given base degree, the elliptic genus of the corresponding non-critical 6d string, and thus the associated BPS invariants of the 6d theory. The required elliptic indices are determined from the chiral anomaly 4-form of the 2d worldsheet theories, or the 8-form of the corresponding 6d theories, and completely fix the holomorphic anomaly equation constraining the partition function. We introduce subrings of the known rings of Weyl invariant Jacobi forms which are adapted to the additional symmetries of the partition function, making its computation feasible to low base wrapping number. In contradistinction to the case of simpler singularities, generic vanishing conditions on BPS numbers are no longer sufficient to fix the modular ansatz at arbitrary base wrapping degree. We show that to low degree, imposing exact vanishing conditions does suffice, and conjecture this to be the case generally.

Dynamics of the Smooth Positons of the Wadati-Konno-Ichikawa Equation* *Supported by the National Natural Science Foundation of China under Grant No. 11671219, the K. C. Wong Magna Fund in Ningbo University

We discuss a modified Wadati-Konno-Ichikawa (mWKI) equation, which is equivalent to the WKI equation by a hodograph transformation. The explicit formula of degenerated solution of mWKI equation is provided by using degenerate Darboux transformation with respect to the eigenvalues, which yields two kinds of smooth solutions possessing the vanishing and nonvanishing boundary conditions respectively. In particular, a method for the decomposition of modulus square is operated to the positon solution, and the approximate orbits before and after collision of positon solutions are displayed explicitly.