NONTRIVIAL SOLUTIONS FOR SEMILINEAR DIRICHLET FORMS VIA MORSE THEORY

Using variational methods and Morse theory, we obtain some existence results of multiple solutions for certain semilinear problems associated with general Dirichlet forms.

The Existence of Solutions for a Class of Schr¨odinger Equations via Morse Index Theory

In this paper,with the relative Morse index,we will study the existence of solutions of(1.1)under the assumptions that V satisfies some weaker conditions than those in[2].

MULTIPLICITY RESULTS FOR THE TWO-POINT BOUNDARY VALUE PROBLEMS AT RESONANCE

In this article several existence theorems on multiple solutions for the twopoint boundary value problem with resonance at, both infinity and zero are proved.

A LARGE-SCALE ESTIMATE OF THE NUMBEROF PERIODIC SMALL OSCILLATIONS OFNONLINEAR SYSTEMS

A conservative system performing a small oscillation near every equilibrium position is analysed in classical way. The paper tries to answer the following question: How many types of the periodic small oscillation in the whole configuration space of the system are there? Making some hypotheses, it expresses the lower bounds of the number of the types for two cases where critical points of the potential function are nondegenerate and degenerate respectively by the Betti numbers and dimension of the constraint manifold only.

MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR ELLIPTIC EQUATIONS INVOLVING NATURAL GROWTH TERM

In this article, under some conditions on the behaviors of the perturbed function f（x, s） or its primitive F（x,s） =∫so f（x,t）dt near infinity and near zero, a class of asymptotically linear elliptic equations involving natural growth term is studied. By computing the critical group, the existence of three nontrivial solutions is proved.

MULTIPLE SOLUTIONS OF SOME ELLIPTIC SYSTEMS WITH LINEAR COUPLINGS

In this paper,we study the existence of nontrivial solutions to the elliptic system {-△u=λv+Fu(x,u,v),x∈Ω,-△v=λu+Fv(x,u,v),x∈Ω,u=v=0,x∈∂Ω,where Ω■R^(N) is bounded with a smooth boundary.By the Morse theory and the Gromoll-Meyer pair,we obtain multiple nontrivial vector solutions to this system.

Lefschetz Thimbles and Wigner Functions

The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.

The Realization Problem for Discrete Morse Functions on Trees

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram.We compare this notion of equivalence to other notions of equivalent discrete Morse functions.Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree.This is a version of the"realization problem"of the persistence map.We conclude with an example illustrating our construction.

Min-Max Theory for Cell Complexes

In the study of smooth functions on manifolds,min-max theory provides a mechanism for identifying critical values of a function.We introduce a discretized version of this theory associated to a discrete Morse function on a(regular)cell complex.As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik-Schnirelmann theorem.

Multiple Solutions for a Fourth-order Asymptotically Linear Elliptic Problem

Under simple conditions, we prove the existence of three solutions for a fourth-order asymptotically linear elliptic boundary value problem. For the resonance case at infinity, we do not need to assume any more conditions to ensure the boundedness of the （PS） sequence of the corresponding functional.

Polygonal Shape Blending with Topological Evolutions

This paper presents a new general approach to blend 2D shapes with differenttopologies. All possible topolog-ical evolutions are classified into three types by attaching threedifferent topological cells. This formalism is resulted from Morse theory on the behavior of the 3Dsurface around a non-degenerate critical point. Also we incorporate degenerate topologicalevolutions into our framework which produce more attractive morphing effects. The user controls themorph by specifying the types of topological evolutions as well as the feature correspondencesbetween the source and target shapes. Some techniques are also provided to control the vertex pathduring the morphing process. The amount of user input required to produce a morph is directlyproportional to the amount of control the user wishes to impose on the process. The user may allowthe system to automatically generate the morph as well. Our approaches are totally geometric basedand are easy and fast enough in fully interactive time. Many experimental results show theapplicability and flexibility of our approaches.

Some Abstract Critical Point Theorems for Self-adjoint Operator Equations and Applications

By using the index theory for linear bounded self-adjoint operators in a Hilbert space related to a fixed self-adjoint operator A with compact resolvent,the authors discuss the existence and multiplicity of solutions for(nonlinear) operator equations,and give some applications to some boundary value problems of first order Hamiltonian systems and second order Hamiltonian systems.

A standard form of incompressible surfaces in 3-dimensional handlebodies

Applying the Morse theory,we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies.We also give a necessary and sufficient condition to determine the incompressibility of such surfaces placed in our standard form.Our algorithm is practical.Several examples are given to test the algorithm.

Symmetric Closed Characteristics on Symmetric Compact Convex Hypersurfaces in R^8

Let ∑be a C^3 compact symmetric convex hypersurface in R^8.We prove that when ∑ carries exactly four geometrically distinct closed characteristics,then all of them must be symmetric.Due to the example of weakly non-resonant ellipsoids,our result is sharp.

Multiple Solutions for a Fractional p-Laplacian Equation with Concave Nonlinearities

We investigate a fractional p-Laplacian equation with right-hand-side nonlinearity which exhibits (p-1)-sublinear term of the formλ|u|q-2,q < p (concave term),and a continuous term f(x,u) which is respectively (p-1)-superlinear or asymptotically (p-1)-linear at infinity.Some existence results for multiple nontrivial solutions are established by using variational methods combined with the Morse theory.

Existence Theorem for a Class of Nonlinear Fourth-order Schrodinger-Kirchhoff-Type Equations

This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type {Δ^2u-（a＋b∫R^N｜▽u｜^2dx）Δu＋λv（x）u=f（x,u）,x∈R^N,u∈H^2（R^N）,where a,b are positive constants, λ≥ 1 is a parameter, and the nonlinearity f is either superlinear or sublinear at infinity in u. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.

The 2‑Width of Embedded 3‑Manifolds

We discuss a possible definition for“k-width”of a closed d-manifold Md,and on embedding Md e↪ℝn,n>d≥k,generalizing the classical notion of width of a knot.We show that for every 3-manifold 2-width(M3)≤2 but that there are embeddings ei∶T3↪ℝ4 with 2-width(ei)→∞.We explain how the divergence of 2-width of embeddings offers a tool to which might prove the Goeritz groups Gg infinitely generated for g≥4.Finally we construct a homomorphismg∶Gg→MCG(#g S2×S2),suggesting a potential application of 2-width to 4D mapping class groups.

MULTIPLICITY OF SOLUTIONS TO ASYMPTOTICALLY LINEAR SECOND-ORDER ORDINARY DIFFERENTIAL SYSTEM

In this paper,we consider an asymptotically linear second-order ordinary differential system with Dirchlet boundary value conditions. Under some conditions,we show the multiplicity of solutions to the system by the Morse theory and an index theory.

EXISTENCE OF SOLUTIONS TO 2m-ORDER PERIODIC BOUNDARY VALUE PROBLEM

In this paper, we are concerned with the existence of nontrivial solutions to a 2m-order nonlinear periodic boundary value problem. By the infinite dimensional Morse theory, under some conditions on nonlinear term, we obtain that there exist at least two nontrivial solutions.