Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: u_t=Δu+u^p+u^q(x^*,t) in B(R) where p,q>0,B(R)={x∈R^N:|x|<R} and x^*≠0. If max(p,q)>1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.

(i)

If or p=q>2, then single point blow-up occurs whenever solutions blow up.

(ii)

If 1<p<q, both phenomena, total blow-up and single point blow-up, occur depending on the initial data.

(iii)

If p?1<q, total blow-up occurs whenever solutions blow up.

(iv)

If max(p,q)?1, every solution exists globally in time.

     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.

     

Structural analysis of a fractional matching problem

Mixed Software Programming refers to a novel software development paradigm resulting from efforts to combine two different programming approaches: Solo Programming and Pair Programming. Solo Programming refers to the traditional practice of assigning a single developer to develop a software module and Pair Programming refers to a relatively new approach where two developers work simultaneously on developing a module. In Mixed Programming, given a set of modules to be developed, a chosen subset of modules may be developed using Solo Programming and the remaining modules using Pair Programming.Motivated by applications in Mixed Software Programming, we consider the following generalization of classical fractional 1-matching problem: Given an undirected simple graph G=(V;E), and a positive number F, find values for x_e,e∈E, satisfying the following:

1.

.

2.

, where δ(i)={e∈E:e=(i,j)},i∈V.

3.

Maximize {2∑_e∈Ex_e−F|{i∈V:∑_e∈δ(i)x_e=1}|}.

We show that this problem is solvable in strongly polynomial time. Our primary focus in this paper is on obtaining the structure of the optimal solution for an arbitrary instance of the problem.     

Uniqueness theorems for meromorphic mappings with few hyperplanes

Let f,g be linearly nondegenerate meromorphic mappings of Cm into CPn. Let be hyperplanes in CPn in general position, such that

(a)

f−1(Hj)=g−1(Hj), for all 1?j?q,

(b)

dim(f−1(Hi)∩f−1(Hj))?m−2 for all 1?i<j?q, and

(c)

f=g on .

It is well known that if q?3n+2, then f≡g. In this paper we show that for every nonnegative integer c there exists positive integer N(c) depending only on c in an explicit way such that the above result remains valid if q?(3n+2−c) and n?N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n?0. At the same time, a big number of recent uniqueness theorems are generalized considerably.     

Dissipative operators on Banach spaces

A bounded linear operator T on a Banach space is said to be dissipative if ‖etT‖?1 for all t?0. We show that if T is a dissipative operator on a Banach space, then:

(a)

.

(b)

If σ(T)∩iR is contained in [−iπ/2,iπ/2], then

     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0<δ?secM?1. We prove that M is homeomorphic to a spherical space form if one of the following conditions holds:

(i)

The center of the fundamental group has index ?w(δ), a constant depending on δ;

(ii)

and the fundamental group is a non-cyclic group of order ?C, a constant;

(iii)

The volume is less than ?(δ) and the fundamental group is either isomorphic to a spherical 5-space group or has an odd order, and it has a center of index ?w, a constant.

     

Constructions of unitals in Desarguesian planes

We present a new construction of non-classical unitals from a classical unital U in . The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π of with the following three properties:

(i)

points of Π are those of ;

(ii)

lines of Π are certain lines and conics of ;

(iii)

the points in U form a non-classical B-M unital in Π.

Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.     

Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:

(i)

AM is relatively weakly compact;

(ii)

B is a strict contraction;

(iii)

.

Then A+B has at least one fixed point in M.This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.     

K-group and similarity classification of operators

Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist Ω, a connected open subset of complex plane C, and n, a positive integer, such that

(a)

(b)

  for z in Ω;

(c)

; and

(d)

for z in Ω.

In the paper, we give a similarity classification of Cowen-Douglas operators by using the ordered K-group of the commutant algebra as an invariant, and characterize the maximal ideals of the commutant algebras of Cowen-Douglas operators. The theorem greatly generalizes the main result in (Canada J. Math. 156(4) (2004) 742) by simply removing the restriction of strong irreducibility of the operators. The research is also partially inspired by the recent classification theory of simple AH algebras of Elliott-Gong in (Documenta Math. 7 (2002) 255; On the classification of simple inductive limit C^*-algebras, II: The isomorphism theorem, preprint.) (also see (J. Funct. Anal. (1998) 1; Ann. Math. 144 (1996) 497; Amer. J. Math. (1996) 187)).     

Optimum quantization and its applications

Minimum sums of moments or, equivalently, distortion of optimum quantizers play an important role in several branches of mathematics. Fejes Tóth's inequality for sums of moments in the plane and Zador's asymptotic formula for minimum distortion in Euclidean d-space are the first precise pertinent results in dimension d?2. In this article these results are generalized in the form of asymptotic formulae for minimum sums of moments, resp. distortion of optimum quantizers on Riemannian d-manifolds and normed d-spaces. In addition, we provide geometric and analytic information on the structure of optimum configurations. Our results are then used to obtain information on

(i)

the minimum distortion of high-resolution vector quantization and optimum quantizers,

(ii)

the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures,

(iii)

the minimum error of numerical integration formulae for classes of Hölder continuous functions and optimum sets of nodes,

(iv)

best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes, and

(v)

the minimum isoperimetric quotient of convex polytopes in Minkowski spaces and the form of the minimizing polytopes.

     

The space of adding machines generated by continuous self maps of manifolds

Let M be the Cantor space or an n-dimensional manifold with C(M,M) the set of continuous self-maps of M, and . We prove the following:

(1)

If α≠∞, then Sα(M) is a nowhere dense subset of M×C(M,M) that contains no isolated points.

(2)

If α?β, then .

     

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that and let Γ be a uniform lattice in G.

(a)

If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.

(b)

If CH fails, then Γ has 2^2ω asymptotic cones up to homeomorphism.

     

The Mukai pairing—II: the Hochschild-Kostant-Rosenberg isomorphism

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are:

•

we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds;

•

we give a proof of the fact that the natural Chern character map K₀(X)→HH₀(X) becomes, after the HKR isomorphism, the usual one ; and

•

we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.

     

Strong zero-dimensionality of hyperspaces

For a space X, X² denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of X². The following are known:

•

ω² is not normal, where ω denotes the discrete space of countably infinite cardinality.

•

For every non-zero ordinal γ with the usual order topology, K(γ) is normal iff whenever cf γ is uncountable.

In this paper, we will prove:

(1)

ω² is strongly zero-dimensional.

(2)

K(γ) is strongly zero-dimensional, for every non-zero ordinal γ.

In (2), we use the technique of elementary submodels.     

Spectral characterization of some weighted rooted graphs with cliques

The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which

(1)

the edges connecting vertices at consecutive levels have the same weight,

(2)

each set of children, in one or more levels, defines a weighted clique, and

(3)

cliques at the same level are isomorphic.

These eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. Finally, we apply the results to the unweighted case and some particular graphs are studied.     

Banach spaces with the Daugavet property, and the centralizer

We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties:

(1)

Every member of R has the Daugavet property.

(2)

It Y is a member of R, then, for every Banach space X, both the space L(X,Y) (of all bounded linear operators from X to Y) and the complete injective tensor product lie in R.

(3)

If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, and for most vector space topologies τ on Y, the space C(K,(Y,τ)) (of all Y-valued τ-continuous functions on K) is a member of R.

(4)

If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, most C(K,Y)-superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195-206]) are members of R.

(5)

All dual Banach spaces without minimal M-summands are members of R.

     

Exponential stability and inequalities of solutions of abstract functional differential equations

With the Lyapunov second method, we study the abstract functional differential equation, . We obtain inequalities of solutions and exponential stability with conditions like:

(i)

,

(ii)

.

Applications in ordinary and partial functional differential equations are given.