Heat kernel fluctuations for a resistance form with non-uniform volume growth

In this article, we consider the problem of estimating the heatkernel on measure-metric spaces equipped with a resistance form.Such spaces admit a corresponding resistance metric that reflectsthe conductivity properties of the set. In this situation, ithas been proved that when there is uniform polynomial volumegrowth with respect to the resistance metric the behaviour ofthe on-diagonal part of the heat kernel is completely determinedby this rate of volume growth. However, recent results haveshown that for certain random fractal sets, there are globaland local (point-wise) fluctuations in the volume as r 0 andso these uniform results do not apply. Motivated by these examples,we present global and local on-diagonal heat kernel estimateswhen the volume growth is not uniform, and demonstrate thatwhen the volume fluctuations are non-trivial, there will benon-trivial fluctuations of the same order (up to exponents)in the short-time heat kernel asymptotics. We also provide boundsfor the off-diagonal part of the heat kernel. These resultsapply to deterministic and random self-similar fractals, andmetric space dendrites (the topological analogues of graph trees).     

Diffusion-dominated asymptotics of solution to chemotaxis model

The paper contains results on the asymptotic behavior, as t → +∞, of small solutions to simplified Keller–Segel problem modeling chemotaxis in the whole space \mathbb R²{\mathbb R^2}. We prove that the multiple of the heat kernel is a surprisingly good approximation of solutions.     

Large holes in sparse random graphs

We show that there is a function α(r) such that for each constantr≧3, almost everyr-regular graph onn vertices has a hole (vertex induced cycle) of size at least α(r)n asn→∞. We also show that there is a function β(c) such that forc>0 large enough,G _{n, p} ,p=c/n almost surely has a hole of size at least β(c)n asn→∞.     

Application of a Bernstein type inequality to rational interpolation in the Dirichlet space

We prove a Bernstein type inequality involving the Bergman and Hardy norms for rational functions in the unit disk \mathbb D {\mathbb D} that have at most n poles all of which are outside the disk \frac1r \mathbb D \frac{1}{r} {\mathbb D} , 0 < r < 1. The asymptotic sharpness of this inequality is shown as n → ∞ and r → 1—. We apply our Bernstein type inequality to an efficient Nevanlinna–Pick interpolation problem in the standard Dirichlet space constrained by the H2-nom. Bibliography: 14 titles.     

Sharp metastability threshold for two-dimensional bootstrap percolation

 In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p→0 and L→∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p)→1 if , and I(L,p)→0 if , where λ=π²/18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ^′=π²/6. The existence of the thresholds λ,λ^′ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2]. Received: 12 May 2002 / Revised version: 12 August 2002 / Published online: 14 November 2002 Research funded in part by NSF Grant DMS-0072398 Mathematics Subject Classification (2000): Primary 60K35; Secondary 82B43 Key words or phrases: Bootstrap percolation – Cellular automaton – Metastability – Finite-size scaling     

A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

Asymptotic of the heat kernel in general Benedicks domains

 Using a new inequality relating the heat kernel and the probability of survival, we prove asymptotic ratio limit theorems for the heat kernel (and survival probability) in general Benedicks domains. In particular, the dimension of the cone of positive harmonic measures with Dirichlet boundary conditions can be derived from the rate of convergence to zero of the heat kernel (or the survival probability). Received: 31 March 2002 / Revised version: 12 August 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 60J65, 31B05 Key words or phrases: Positive harmonic functions – Ratio limit theorems – Survival probability     

Heat equation on the arithmetic von Koch snowflake

We investigate the asymptotic behaviour of the heat content as the time t→ 0 for an s-adic von Koch snowflake generated by a square. We show that the heat content satisfies a functional equation which, after appropriate transformations, takes the form of an inhomogeneous renewal equation. We obtain the structure of the solution of this equation in the arithmetic case up to an exponentially small remainder in t. <!-ID="Mathematics Subject Classification (2000): 35K05, 60J65, 28A80--> <!-ID="Key words: Heat equation – Arithmetic – Snowflake--> Received: 24 March 1999 / Revised version: 14 October 1999 / Published online : 8 August 2000     

Contributions to the Hypergeometric Function Theory of Heckman and Opdam: Sharp Estimates, Schwartz Space, Heat Kernel

Under the assumption of positive multiplicity, we obtain basic estimates of the hypergeometric functions F _λ and G _λ of Heckman and Opdam, and sharp estimates of the particular functions F ₀ and G ₀. Next we prove the Paley–Wiener theorem for the Schwartz class, solve the heat equation and estimate the heat kernel. Received: June 2006 Revision: December 2006 Accepted: March 2007     

Local properties of gaussian random fields on compact symmetric spaces and theorems of the Jackson-Bernstein type

We consider local properties of smaple functions of Gaussian isotropic random fields on compact Riemannian symmetric spacesM of rank 1. We give conditions under which the sample functions of a field almost surely possess logarithmic and power modulus of continuity. As a corollary, we prove a theorem of the Bernstein type for optimal approximations of functions of this sort by harmonic polynomials in the metric of the spaceL ₂(M). We use theorems of the Jackson-Bernstein-type to obtain sufficient conditions for the sample functions of a field to almost surely belong to the classes of functions associated with the Riesz and Cesàro means. International Mathematical Center, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 60–68, January, 1999.     

On the existence of entire functions of boundedl-index andl-regular growth

We prove that, under certain conditions on a positive functionl continuous on [0, +∞], there exists an entire transcendental functionf of boundedl-index such that lnlnM _f(r)lnL(r),r→∞, whereM _f (r)=max {|f(z)|: |z|=r} andL(r)=∫ ₀ ^r l(t)dt. Ifl(r)=r ^p-1 forr≥1, 0<ρ<∞, then there exists an entire functionf of boundedl-index such thatM _f (r)≈r ^p . Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1166–1182, September, 1996.     

Time periodic solutions of smooth convergent dissipative ε-approximations of the modified Navier-Stikes equations

In this paper, we prove the global existence of time periodic classical solutions v' of dissipative ε-approximations (4)–(6) for the three-dimensional modified Navier-Stokes Eqs. (1)–(3) that satisfy the first boundary condition. We also study the convergence for ε → 0 of solutions {v'} to time periodic classical solutions v of Eqs. (1)–(3). Bibliography: 21 titles. Dedicated to V. A. Solonnikov on his sixtieth anniversary Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 116–130.     

Lyapunov functions and convergence to steady state for differential equations of fractional order

We study the asymptotic behaviour, as t → ∞, of bounded solutions to certain integro-differential equations in finite dimensions which include differential equations of fractional order between 0 and 2. We derive appropriate Lyapunov functions for these equations and prove that any global bounded solution converges to a steady state of a related equation, if the nonlinear potential occurring in the equation satisfies the Łojasiewicz inequality.      

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

The Mean and Variance of the Numbers of r-Pronged Nodes and r-Caterpillars in Yule-Generated Genealogical Trees

The Yule model is a frequently-used evolutionary model that can be utilized to generate random genealogical trees. Under this model, using a backwards counting method differing from the approach previously employed by Heard (Evolution 46: 1818–1826), for a genealogical tree of n lineages, the mean number of nodes with exactly r descendants is computed (2 ≤ r ≤ n − 1). The variance of the number of r-pronged nodes is also obtained, as are the mean and variance of the number of r-caterpillars. These results generalize computations of McKenzie and Steel for the case of r = 2 (Math. Biosci. 164: 81–92, 2000). For a given n, the two means are largest at r = 2, equaling 2n/3 for n ≥ 5. However, for n ≥ 9, the variances are largest at r = 3, equaling 23n/420 for n ≥ 7. As n→∞, the fraction of internal nodes that are r-caterpillars for some r approaches (e² − 5)/4≈ 0.59726. Received August 23, 2004     

Remarks on the Cwikel–Lieb–Rozenblum and?Lieb–Thirring Estimates for Schr?dinger Operators on?Riemannian Manifolds

Let M be a general complete Riemannian manifold and consider a Schr?dinger operator −Δ+V on L ²(M). We prove Cwikel–Lieb–Rozenblum as well as Lieb–Thirring type estimates for −Δ+V. These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schr?dinger operators with complex-valued potentials.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

Qualitative investigation of a singular Cauchy problem for a functional differential equation

We consider the singular Cauchy problem

Amemiya Norm Equals Orlicz Norm in Musielak-Orlicz Spaces

Let （Ω,μ） be a a-finite measure space and Φ ： Ω × [0,∞） → [0, ∞] be a Musielak-Orlicz function. Denote by L^Φ（Ω） the Musielak-Orlicz space generated by Φ. We prove that the Amemiya norm equals the Orlicz norm in L^Φ（Ω）.     

An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim _{r → ∞} e^2r s(r) = 0, then (M, g) has to be isometric to ℍ ⁿ . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim _{r → ∞} r ² s(r) = 0, then (M, g) is isometric to ℝ ⁿ , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball B _p (R) in M and K = a on ∂B _p (R), then K = a on B _p (R).