AFPP vs FPP

The main theme of this paper is that almost fixed point properties of discrete structures and fixed point properties of (topological) spaces are interdeducible via a suitable category which contains both graphs and spaces as objects. To carry out the program, we have to consider (almost) fixed points of multifunctions, and for this we need a preliminary discussion of power structures for graphs and simplicial complexes. Specific applications developed are: a digital convexity (discrete) version of Kakutani's fixed point theorem for convex-valued multifunctions; and fixed point properties of dendrites in terms of those of finite discrete trees.     

On the nonexistence of finite time bubble trees in symmetric harmonic map heat flows from the disk to the 2-sphere

Let and be the unit disk and the unit sphere, and let be a radially symmetric harmonic map heat flow, whose singularities coincide with downward energy jumps. Then its finite time singularities are simple in the sense that precisely one harmonic sphere separates at a time.     

On Universal Representation of Random Graphs

It is shown that every probability measure on the interval [0, 1] gives rise to a unique infinite random graph g on vertices {v₁, v₂, . . .} and a sequence of random graphs gn on vertices {v₁, . . . , v_n} such that . In particular, for Bernoulli graphs with stable property Q, can be strengthened to: probability space (, F, P), set of infinite graphs G(Q) , F with property Q such that .AMS Subject Classification: 05C80, 05C62.     

Sparse pseudo‐random graphs are Hamiltonian

In this article we study Hamilton cycles in sparse pseudo‐random graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d‐regular graph G on n vertices satisfies and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for every c >0 and large enough n a Cayley graph X (G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 17–33, 2003     

Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.     

Light paths with an odd number of vertices in polyhedral maps

Let P _k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2-manifold with Euler characteristic contains a path P _k such that each vertex of this path has, in G, degree . Moreover, this bound is attained for k = 1 or k 2, k even. In this paper we prove that for each odd , this bound is the best possible on infinitely many compact 2-manifolds, but on infinitely many other compact 2-manifolds the upper bound can be lowered to .     

THE HEAT FLOWOF HARMONIC MAPS FROM NONCOMPACT MANIFOLDS

1.IntroductionLetMandNbetwoRiemannianmanifoldsofdimensionmandn.Supposetheirmetricsaregivenbydski~giid-c'dxianddsX~h.odu"duo.Letu:M~Nbeasmoothmap.Theenergydensityfunctionofuisgivenbye(u)~g'jCZh.,~Ical'.ThetotalenergyisdefinedbyE(u)=[e(u)dx.iMAmappingu:M~NiscalledaharmonicmapifitisaclassicalsolutionoftheEulerLagrangeequationofE(u)whichcanbewrittenasT"(U(X))~AU"(X) r3.(U(X))ZZg.j~0,whereT(u)iscalledthetensionfieldofu.Thecorrespondingparabolicsystemwithinitialdata"o(x)knownastheheatequ…     

Global Optimization: On Pathlengths in Min-Max Graphs

Let f be a smooth nondegenerate real valued function on a finite dimensional, compact and connected Riemannian manifold. The bipartite min-max graph is defined as follows. Its nodes are formed by the set of local minima and the set of local maxima. Two nodes (a local minimum and a local maximum) are connected in by means of an edge if some trajectory of the corresponding gradient flow connects them. Given a natural number k, we construct a function f such that the length of the shortest path in between two specific local minima exceeds k. The latter construction is independent of the underlying Riemannian metric.     

An Elementary Approach to an Eigenvalue Estimate for Matrices

A celebrated result of Johnson, Maurey, König and Retherford from 1977 states that for every complex matrix satisfies the following eigenvalue estimate:

A 250?GHz, 50?W, CW Second Harmonic Gyrotron

A design concept of a 250 GHz, ≈ 50 W, CW, second harmonic gyrotron for spectroscopy will be presented in this paper. Mode competition, in particular with regard to competition with first harmonic modes, and mode selection procedures are carefully investigated by considering various candidate modes and the TE_6,2 and TE_6,3 modes are chosen as probable operating modes. Cold cavity and self-consistent calculations of power and efficiency are carried out to determine optimum parameters for these two modes. The results indicate that an output power of well over 50 W, CW can be achieved.