Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type

We consider the quasi-linear Keller-Segel system of singular type, where the principal part Δum represents a fast diffusion like 0<m<1. We first construct a global weak solution with small initial data in the scaling invariant norm for all dimensions N?2 and all exponents q?2. As for the large initial data, we show that there exists a blow-up solution in the case of N=2. In the second part, the decay property in Lr with 1<r<∞ for with the mass conservation is shown. On the other hand, in the case of , the extinction phenomenon of solution is proved. It is clarified that the case of exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across . For the borderline case of , our solution decays in Lr exponentially as t→∞.     

Large energy entire solutions for the Yamabe equation

We consider the Yamabe equation in Rn, n?3. Let k?1 and . For all large k we find a solution of the form , where , for n?4, for n=3 and o(1)→0 uniformly as k→+∞.     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .     

Scattering above energy norm of solutions of a loglog energy-supercritical Schrödinger equation with radial data

We prove global well-posedness and scattering of -solutions of the loglog energy-supercritical Schrödinger equation , 0<c<cn, n={3,4}, with radial data , . This is achieved, roughly speaking, by extending Bourgain's argument in Bourgain (1999) [1] (see also Grillakis, 2000 [5]) and Tao's argument in Tao (2005) [10] in high dimensions.     

The Cauchy problem for the Schrödinger-KdV system

In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.     

New applications of Besov-type and Triebel-Lizorkin-type spaces

In this paper, the authors prove that Besov-Morrey spaces are proper subspaces of Besov-type spaces and that Triebel-Lizorkin-Morrey spaces are special cases of Triebel-Lizorkin-type spaces . The authors also establish an equivalent characterization of when τ∈[0,1/p). These Besov-type spaces and Triebel-Lizorkin-type spaces were recently introduced to connect Besov spaces and Triebel-Lizorkin spaces with Q spaces. Moreover, for the spaces and , the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos-Torres by taking τ=0.     

Exact rates in log law for positively associated random variables

Let be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set , Mn=maxk?n|Sk|, n?1. Suppose . In this paper, we study the exact convergence rates of a kind of weighted infinite series of , and as ε↘0, respectively.     

Derivative of power series attached to Γ-transform of p-adic measures

We extend the result of Anglès (2007) [1], namely for the Iwasawa power series . For the derivative , a numerical polynomial Q on Zp, and a prime π in over p, we show that if and only if i.e. for all x∈Zp. This result comes from a similar assertion for the power series attached to the Γ-transform of a p-adic measure which is related to a certain rational function in .     

On the computation of the jdeg of blowup algebras

For a graded algebra , its is a global degree that can be used to study issues of complexity of the normalization . Here some techniques grounded on Rees algebra theory are used to estimate . A closely related notion, of divisorial generation, is introduced to count numbers of generators of .     

How to play the Majority game with a liar

The Majority game is played by a questioner () and an answerer (). holds n elements, each of which can be labeled as 0 or 1. is trying to identify some element holds as having the Majority label or, in the case of a tie, claim there is none. To do this asks questions comparing whether two elements have the same or different label. ’s goal is to ask as few questions as possible while ’s goal is to delay as much as possible. Let q^∗ denote the minimal number of questions needed for to identify a Majority element regardless of ’s answers.In this paper we investigate upper and lower bounds for q^∗ in a variation of the Majority game, where is allowed to lie up to t times. We consider two versions of the game, the adaptive (where questions are asked sequentially) and the oblivious (where questions are asked in one batch).     

More Efficient Algorithm for Ordered Tree Inclusion

Given two ordered trees and , the tree inclusion problem is to determine whether it is possible to obtain from by deleting nodes. Recently, this problem has been recognized as an important primitive in query processing for structured text databases. In this paper we present anO(|leaves()| ||) time andO(|leaves()|min(depth(), |leaves()|)) space algorithm for ordered tree inclusion, by means of a sophisticated bottom-up-matching strategy. Our algorithm improves the previous best one (Kilpeläinen, 1992, Ph.D. thesis, Dept. Computer Science, Univ. Helsinki) that requiresO(|| ||) time andO(||min(depth(), |leaves()|)) space.     

Kirillov-Reshetikhin crystals for nonexceptional types

We provide combinatorial models for all Kirillov-Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types , , we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type we use a Dynkin diagram folding and for types , a similarity construction. We also show that for types and the analog of the Dynkin diagram automorphism exists on the level of crystals.     

A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees

We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property .In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber ℵ₁ and for forcing notions with the property fails. This negatively answers a part of one of the classical problems about implications between fragments of .     

Well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in Triebel-Lizorkin space and Besov space with negative indices

This paper is concerned with the well-posedness of the Navier-Stokes-Nerst-Planck-Poisson system (NSNPP). Let sp=−2+n/p. We prove that the NSNPP has a unique local solution for in a subspace, i.e., Vu1×Vv1×Vv1, of with . We also prove that there exists a unique small global solution for any small initial data with .     

Well-posedness and ill-posedness for a fifth-order shallow water wave equation

In this paper we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the [k;Z] multiplier norm method of Tao (2001) [2]; then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in with is obtained by the Fourier restriction norm method. And some ill-posedness in with is derived from a general principle of Bejenaru and Tao.     

Classical predicative logic-enriched type theories

A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named and , which we claim correspond closely to the classical predicative systems of second order arithmetic and . We justify this claim by translating each second order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics.The two LTTs we construct are subsystems of the logic-enriched type theory , which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system has also been claimed to correspond to Weyl’s foundation. By casting and as LTTs, we are able to compare them with . It is a consequence of the work in this paper that is strictly stronger than .The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.     

Problems arising from jackknifing the estimate of a Kaplan-Meier integral

Stute and Wang (1994) considered the problem of estimating the integral S^θ = ∫ θ dF, based on a possibly censored sample from a distribution F, where θ is an F-integrable function. They proposed a Kaplan-Meier integral to approximate S^θ and derived an explicit formula for the delete-1 jackknife estimate . differs from only when the largest observation, X_(n), is not censored (δ_(n) = 1 and next-to-the-largest observation, X_(n-1), is censored (δ_(n-1) = 0). In this note, it will pointed out that when X_(n) is censored is based on a defective distribution, and therefore can badly underestimate . We derive an explicit formula for the delete-2 jackknife estimate . However, on comparing the expressions of and , their difference is negligible. To improve the performance of and , we propose a modified estimator according to Efron (1980). Simulation results demonstrate that is much less biased than and and .