A degenerate and strongly coupled quasilinear parabolic system not in divergence form

This paper deals with positive solutions of degenerate and strongly coupled quasi-linear parabolic system not in divergence form: u_t=v^p(u+au), v_t=u^q (v+bv) with null Dirichlet boundary condition and positive initial condition, where p, q, a and b are all positive constants, and p, q 1. The local existence of positive classical solution is proved. Moreover, it will be proved that: (i) When min {a, b} ₁ then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm (we can not prove the uniqueness result in general); (ii) When min {a, b} > ₁, there is no global positive classical solution (we can not still prove the uniqueness result), if in addition the initial datum (u₀v₀) satisfies u₀ + au₀ 0, v₀+bv₀ 0 in , then the positive classical solution is unique and blows up in finite time, where ₁ is the first eigenvalue of – in with homogeneous Dirichlet boundary condition.This project was supported by PRC grant NSFC 19831060 and 333 Project of JiangSu Province.     

Embedded continued fractals and their Hausdorff dimension

A continued fractal is a curve which is associated to a real number[0, 1]. Properties of the continued fraction expansion of appear as geometrical properties ofQ . It is shown how number theoretic properties of affect topological and geometric properties ofQ such as existence, continuity, Hausdorff dimension, and embeddedness.Communicated by Michael F. Barnsley.     

Spectral synthesis for Euler type operators

A=(a _ij) _i ^j=1 — k-o ,a _ij . :

The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S _n ) _n0 on ^d there exists a smallest measurable subgroup of ^d , called minimal subgroup of (S _n ) _n0, such that P(S _n )=1 for all n1. can be defined as the set of all x ^d for which the difference of the time averages n ^{–1 n} _k=1 P(S _k ) and n ^{–1 n} _k=1 P(S _k +x) converges to 0 in total variation norm as n. The related subgroup * consisting of all x ^d for which lim _n P(S _n )–P(S _n +x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S _n ) _n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on ^d . The main result shows that, up to regular linear transformations, the only subgroups of ^d admitting a quasi-invariant measure are those of the form ₁×...× _k × ^l–k ×{0} ^d–l , 0kld, with ₁,..., _k being countable subgroups of . The proof is based on a result recently proved by Kharazishvili⁽³⁾ which states no uncountable proper subgroup of admits a quasi-invariant measure.     

Blocks of orthogonal random variables and the strong law of large numbers

(X _k ),k=1,2,... — _k ² >1; (X _k ) , E(X _k X _t )=0 p k<>(p+1) (p,k,l=1, 2, ...) , , ,

Maximal polynomial subordination to univalent functions in the unit disk

Let C be a simply connected domain, 0, and let _n,nN, be the set of all polynomials of degree at mostn. By _n() we denote the subset of polynomials p _n withp(0)=0 andp(D), whereD stands for the unit disk {z: |z|<1}, and=" by=">we denote the maximal range of these polynomials. Letf be a conformal mapping fromD onto ,f(0)=0. The main theme of this note is to relate _n (or some important aspects of it) to the imagesf _s (D), wheref _s (z):=f[(1–s)z], 0s<1. for=" instance=" we=" prove=" the=" existence=" of=" a=" universal=">c ₀ such that, forn2c ₀,     

Euler Characteristics of Local Systems on \mathcal{M}> 2

We calculate the Euler characteristics of the local systems S ^k S ² on the moduli space ₂ of curves of genus 2, where is the rank 4 local system R ¹ _* .     

Some remarks on Turán's inequality. III: The completion

. 0pq, 1–1/p+1/p0. f(x) — n, [–1,1],

A class of strongly nonlinear functional differential equations

Let H be a real Hilbert space, :H [0, + ] a proper l.s.c., convex function with L_k:={u H; u² + (u) k} compact for every k > 0, let > 0 be a given constant and . We prove an existence result for strong solutions to a class of functional differential equations of the form

Large-time behavior of perturbed diffusion markov processes with applications to the second eigenvalue problem for fokker-planck operators and simulated annealing

We study the large-time behavior and rate of convergence to the invariant measures of the processes dX (t)=b(X) (t)) dt + (X (t)) dB(t). A crucial constant appears naturally in our study. Heuristically, when the time is of the order exp( – )/² , the transition density has a good lower bound and when the process has run for about exp( – )/², it is very close to the invariant measure. LetL =(²/2) – U · be a second-order differential operator on ^d. Under suitable conditions,L ^z has the discrete spectrum

Fubini theorem w.r.t. stochastic measure on product measurable space

ＦＵＢＩＮＩＴＨＥＯＲＥＭｗ．ｒ．ｔ．ＳＴＯＣＨＡＳＴＩＣＭＥＡＳＵＲＥＯＮＰＲＯＤＵＣＴＭＥＡＳＵＲＡＢＬＥＳＰＡＣＥＣＨＥＮＰＥＩＤＥ（陈培德）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ,ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳ...     

Necessary and sufficient conditions for interpolation with functions having monotoner-th derivatives

,

Generalized solution to a semilinear hyperbolic system with a non-Lipshitz nonlinearity

Let

Isomorphisms of Steinberg groups over commutative rings

LetA andR be commutative rings, andm andn be integers3. It is proved that, if :St _m (A)St _n (R) is an isomorphism, thenm=n. Whenn4, we have: (1) Every isomorphism :St _n(A)St _n(R) induces an isomorphism:E _n (A)E _n (R), and is uniquely determined by; (2) IfSt _n (A) St _n (R) thenK _2.n (A)K _2.n (R); (3) Every isomorphismE _n (A) E _n (R) can be lifted to an isomorphismSt _n(A)St _n(R); (4)St _n(A) St _n(R) if and only ifAR. For the casen=3, ifSt ₃(A) andSt ₃(R) are respectively central extensions ofE ₃(A) andE ₃ (R), then the above (1) and (2) hold.The Project supported by National Natural Science Foundation of China     

Survival time of a random graph

LetV _n ={1, 2, ...,n} ande ₁,e ₂, ...,e _N ,N= be a random permutation ofV _n ⁽²⁾. LetE _t={e ₁,e ₂, ...,e _t} andG _t=(V _n ,E _t ). If is a monotone graph property then the hitting time() for is defined by=()=min {t:G _t }. Suppose now thatG starts to deteriorate i.e. loses edges in order ofage, e ₁,e ₂, .... We introduce the idea of thesurvival time =() defined by t = max {u:(V _n, {e _u,e _u+1, ...,e _T }) }. We study in particular the case where isk-connectivity. We show that

A New Plethysm Formula for Symmetric Functions

This paper gives a new formula for the plethysm of power-sum symmetric functions and Schur symmetric functions with one part. The form of the main result is that for b,