Existence of travelling waves with their minimal speed for a diffusing Lotka–Volterra system

Travelling waves are natural phenomena ubiquitously for reaction–diffusion systems in many scientific areas, such as in biophysics, population genetics, mathematical ecology, chemistry, chemical physics, and so on. It is pretty well understood for a diffusing Lotka–Volterra system that there exist travelling wave solutions which propagate from an equilibrium point to another one. In this paper, we prove there exists, at least, a wave front—the monotone travelling wave—with its minimal speed.     

Uniqueness for inverse Sturm–Liouville problems with a finite number of transmission conditions

We establish various uniqueness results for inverse spectral problems of Sturm–Liouville operators with a finite number of discontinuities at interior points at which we impose the usual transmission conditions. We consider both the cases of classical Robin and of eigenparameter dependent boundary conditions.     

Existence of positive weak solutions for fractional Lane–Emden equations with prescribed singular sets

In this paper, we consider the problem of the existence of positive weak solutions of

$$\begin{aligned}{\left\{ \begin{array}{ll} (-\Delta )^s{u}=u^p&{}\text {in}~\Omega \\ u=0&{}\text {on}~{\mathbb R}^n \backslash \Omega \end{array}\right. }\\\end{aligned}$$

having prescribed isolated interior singularities. We prove that if \(\frac{n}{n-2s}<p<p_1\) for some critical exponent \(p_1\) defined in the introduction which is related to the stability of the singular solution \(u_s\), and if S is a closed subset of \(\Omega \), then there are infinitely many positive weak solutions with S as its singular set. We also show the existence of solutions to the fractional Yamabe problem with singular set to be the whole space \({\mathbb R}^n\). These results are the extension of Chen and Lin’s result (Chen and Lin in J Geom Anal 9(2):221–246, 1999) to the fractional case.

     

Existence for a Class of Non-Newtonian Fluids with a Nonlocal Friction Boundary Condition

We deal with a variational inequality describing the motion of incompressible fluids, whose viscous stress tensors belong to the subdifferential of a functional at the point given by the symmetric part of the velocity gradient, with a nonlocal friction condition on a part of the boundary obtained by a generalized mollification of the stresses. We establish an existence result of a solution to the nonlocal friction problem for this class of non-Newtonian flows. The result is based on the Faedo-Galerkin and Moreau Yosida methods, the duality theory of convex analysis and the Tychonov-Kakutani-Glicksberg fixed point theorem for multi-valued mappings in an appropriate functional space framework.     

Source algebra equivalences for blocks¶of finite general linear groups over a fixed field

T. Jost proved that Donovan's conjecture holds for the unipotent blocks of the finite general linear groups over a fixed field. In this paper, we show that the Morita equivalences exhibited by Jost are in fact equivalences between the source algebras of the corresponding blocks, and thus that Puig's conjecture holds for the unipotent blocks of finite general groups over a fixed field. Received: 3 October 2000     

On Chern?s problem for rigidity of minimal hypersurfaces in the spheres

For a compact minimal hypersurface M in Sn+1 with the squared length of the second fundamental form S we confirm that there exists a positive constant δ(n) depending only on n, such that if n?S?n+δ(n), then S≡n, i.e., M is a Clifford minimal hypersurface, in particular, when n?6, the pinching constant .     

Existence of solutions for a class of Navier–Stokes equations with infinite delay

In this paper, a class of Navier–Stokes equations with infinite delay is considered. It includes delays in the convective and the forcing terms. We discuss the existence of mild and classical solutions for the problem. We establish the results for an abstract delay problem by using the fact that the Stokes operator is the infinitesimal generator of an analytic semigroup of bounded linear operators. Finally, we apply these abstract results to our particular situation.     

Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

Acta Mathematicae Applicatae Sinica, English Series - We study the following quasilinear Schrödinger equation $$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in...     

Existence of positive solutions for a system of second-order m-point BVPs with variable parameters

In this paper, we study the existence of positive solutions for a class of second-order nonlinear m-point boundary value problems of differential system. By using the fixed point theory of cone expansion and compression with norm type, we show the sufficient conditions for the existence results.     

A Teichmüller theoretical construction of¶high genus singly periodic minimal surfaces invariant¶under a translation

We prove the existence of singly periodic minimal surfaces invariant under a translation such that a fundamental piece has arbitrarily many parallel planar ends and arbitrarily high genus. These surfaces generalize the Callahan-Hoffman-Meeks surface. We also discuss briefly the effective computation of the periods and techniques to parameterize these surfaces.     

An elementary proof of the Karush–Kuhn–Tucker theorem in normed linear spaces for problems with a finite number of inequality constraints

We present an elementary proof of the Karush–Kuhn–Tucker theorem for the problem with a finite number of nonlinear inequality constraints in normed linear spaces under the linear independence constraint qualification. Most proofs in the literature rely on advanced concepts and results such as the convex separation theorem and Farkas, lemma. By contrast, the proofs given in this article, including a proof of the lemma, employ only basic results from linear algebra. The lemma derived in this article represents an independent theoretical result.     

Manin’s conjecture for a class of singular cubic hypersurfaces

Let l > 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in l variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u³ = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work.     

The Global Existence of Solutions for a Cross-Diffusion System

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Existence of positive solution for a sixth–order differential system with variable parameters

This paper investigates the existence of positive solutions for a sixth-order differential system with three variable parameters. Using a fixed point theorem and an operator spectral theorem we give some new existence results.     

Moduli for pairs of elliptic curves¶with isomorphic N-torsion

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a “determinant” map from this moduli surface to (Z/N Z)^*; its fibers are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences between them, and study how those spaces of modular forms behave as modules for the Hecke algebra. We discover that the component with determinant −1 is somehow the “dominant” one; we characterize the difference between its spaces of modular forms and the spaces of modular forms on the other components using forms with complex multiplication. In addition, we prove Atkin–Lehner-style results about these spaces of modular forms. Finally, we show some simplifications that arise when N is prime, including a complete determination of such CM-forms, and give numerical examples. Received: 20 September 2000 / Revised version: 7 February 2001     

The Global Existence of Solutions for a Cross-diffusion System

This paper is concerned with the global existence of solutions for a class of quasilinear cross-diffusion system describing two species competition under self and cross population pressure. By establishing and using more detailed interpolation results between several different Banach spaces, the global existence of solutions are proved when the self and cross diffusion rates for the first species are positive and there is no self or cross-diffusion for the second species.