Hölder continuity of interfaces for the porous medium equation with absorption

In this paper, we study the one-dimensional motion of viscous gas connecting to vacuum state with a jump in density when the viscosity depends on the density. Precisely, when the viscosity coefficient μ is proportional to ρ^θ and 0 < θ < 1/2, where ρ is the density, the global existence and the uniqueness of weak solutions are proved. This improves the previous results by enlarging the interval of θ.     

On tree census and the giant component in sparse random graphs

For each of the two models of a sparse random graph on n vertices, G(n, # of edges = cn/2) and G(n, Prob (edge) = c/n) define t_n(k) as the total number of tree components of size k (1 ≤ k ≤ n). the random sequence {[t_n(k) - nh(k)]n^?1/2} is shown to be Gaussian in the limit n →∞, with h(k) = k^k?2c^k?1e^?kc/k! and covariance function being dependent upon the model. This general result implies, in particular, that, for c> 1, the size of the giant component is asymptotically Gaussian, with mean nθ(c) and variance n(1 ? T)^?2(1 ? 2Tθ)θ(1 ? θ) for the first model and n(1 ? T)^?2θ(1 ? θ) for the second model. Here Te^?T = ce^?c, T<1, and θ = 1 ? T/c. A close technique allows us to prove that, for c < 1, the independence number of G(n, p = c/n) is asymptotically Gaussian with mean nc^?1(β + β²/2) and variance n[c^?1(β + β²/2) ?c^?2(c + 1)β²], where βe^β = c. It is also proven that almost surely the giant component consists of a giant two-connected core of size about n(1 ? T)β and a “mantle” of trees, and possibly few small unicyclic graphs, each sprouting from its own vertex of the core.     

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data.     

A Hard-Core Model on a Cayley Tree: An Example of a Loss Network

The paper is about a nearest-neighbor hard-core model, with fugacity λ>0, on a homogeneous Cayley tree of order k(with k+1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on ‘splitting’ Gibbs measures generating a Markov chain along each path on the tree. In this model, ?λ>0 and k≥1, there exists a unique translation-invariant splitting Gibbs measure μ^*. Define λ_c=1/(k?1)×(k/(k?1)) ^k . Then: (i) for λ≤λ_c, the Gibbs measure is unique (and coincides with the above measure μ^*), (ii) for λ>λ_c, in addition to μ^*, there exist two distinct translation-periodic measures, μ₊and μ_?, taken to each other by the unit space shift. Measures μ₊and μ_?are extreme ?λ>λ_c. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For $\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k $ , measure μ^*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λ_eand λ_o, for even and odd sites. We discuss open problems and state several related conjectures.     

Multiparameter acoustic imaging in the born approximation

Samples of biological tissue are modelled as inhomogeneous fluids with density ?(X) and sound speed c(x) at point x. The samples are contained in the sphere |x| ? δ and it is assumed that ?(x) ? ?₀ = 1 and c(x) ? c₀ = 1 for |x| ? δ, and |γ_n(x)| ? 1 and |?γ?(x)| ? 1 where γ?(x) = ?(x) ? 1 and γ_n(x) = c^?2(x) ? 1. The samples are insonified by plane pulses s(x · θ₀ – t) where x = |θ₀| = 1 and the scattered pulse is shown to have the form |x|^?1 e_s(|x| – t, θ, θ₀) in the far field, where x = |x| θ. The response e_s(τ, θ, θ₀) is measurable. The goal of the work is to construct the sample parameters γn and γ? from e_s(τ, θ, θ₀) for suitable choiches of s, θ and θ₀. In the limiting case of constant density: γ?(x)_? 0 it is shown that Where δ represents the Dirac δ and S² is the unit sphere |θ| = 1. Analogous formulas, based on two sets of measurements, are derived for the case of variable c(x) and ?(x).     

For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution

We consider linear equations y = Φx where y is a given vector in ?ⁿ and Φ is a given n × m matrix with n < m ≤ τn, and we wish to solve for x ∈ ?^m. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We prove the existence of ρ = ρ(τ) > 0 so that for large n and for all Φ's except a negligible fraction, the following property holds: For every y having a representation y = Φx₀ by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, the solution x₁ of the ??₁‐minimization problem is unique and equal to x₀. In contrast, heuristic attempts to sparsely solve such systems—greedy algorithms and thresholding—perform poorly in this challenging setting. The techniques include the use of random proportional embeddings and almost‐spherical sections in Banach space theory, and deviation bounds for the eigenvalues of random Wishart matrices. © 2006 Wiley Periodicals, Inc.     

Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability

We generalize Tollmien’s solutions of the Rayleigh problem of hydrodynamic stability to the case of arbitrary channel cross sections, known as the extended Rayleigh problem. We prove the existence of a neutrally stable eigensolution with wave number k _s ?>?0; it is also shown that instability is possible only for 0?<?k?<?k _s and not for k?>?k _s . Then we generalize the Tollmien–Lin perturbation formula for the behavior of c _i, the imaginary part of the phase velocity as the wave number k →k _s ? to the extended Rayleigh problem and subsequently, we use this formula to demonstrate the instability of a particular shear flow.     

Locating the peaks of solutions via the maximum principle II: A local version of the method of moving planes

Let Ω be a bounded, smooth domain in ?²ⁿ, n ≥ 2. The well‐known Moser‐Trudinger inequality ensures the nonlinear functional J_ρ(u) is bounded from below if and only if ρ ≤ ρ_2n := 2²ⁿn!(n ? 1)!ω_2n, where in , and ω_2n is the area of the unit sphere ??^{2n ? 1} in ?²ⁿ. In this paper, we prove the inf_u∈X J_ρ(u) is always attained for ρ ≤ ρ_2n. The existence of minimizers of J_ρ at the critical value ρ = ρ_2n is a delicate problem. The proof depends on the blowup analysis for a sequence of bubbling solutions. Here we develop a local version of the method of moving planes to exclude the boundary bubbling. The existence of minimizers for J_ρ at the critical value ρ = ρ_2n is in contrast to the case of two dimensions. © 2003 Wiley Periodicals, Inc.     

Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field

Assume that K ⁺: H _? → T _? is a bounded operator, where H _? and T _? are Hilbert spaces and ρ is a measure on the space H _?. Denote by ρ_K the image of the measure ρ under K ⁺. We study the measure ρ_K under the assumption that ρ is the spectral measure of a Jacobi field and obtain a family of operators whose spectral measure is equal to ρ_K. We also obtain an analog of the Wiener-Itô decomposition for ρ_K. Finally, we illustrate the results obtained by explicit calculations carried out for the case, where ρ_K is a Lévy noise measure.     

The limiting angle of certain riemannian brownian motions

Let M be a Cartan-Hadamard manifold of dimension d ≧ 3, let p ? M and x = exp {r(x)θ(x)} be geodesic polar coordinates with pole p and let X be the Brownian motion on M. Let Sect_M(x) denote the sectional curvature of any plane section in M_x. We prove that for each c > 2, there is a 0 < β < 1 such that if - L²r(x)^2β ≦ Sect_M(x) ≦ -cr(x)^?2 for all x in the complement of a compact set, then lim_{t → ∞} θ(X_t) exists a.s. and defines a nontrivial invariant random variable. The Dirichlet problem at infinity and a conjecture of Greene and Wu are also discussed.     

A dimension gap for continued fractions with independent digits—the non stationary case

We show there exists a constant 0 < c₀ < 1 such that the dimension of every measure on [0, 1], which makes the digits in the continued fraction expansion independent, is at most 1 ? c₀. This extends a result of Kifer, Peres and Weiss from 2001, which established this under the additional assumption of stationarity. For k ≥ 1 we prove an analogous statement for measures under which the digits form a *-mixing k-step Markov chain. This is also generalized to the case of f-expansions. In addition, we construct for each k a measure, which makes the continued fraction digits a stationary and *-mixing k-step Markov chain, with dimension at least 1 ? 2^3?k.     

The product of linear nonhomogeneous forms

We show that for an arbitrary unimodular lattice Λ of dimension n and an arbitrary point C =(c₁, c₂...c_n) ? Rⁿ a point Y = (y₁, y₂,..., y_n) ε Λ can be found and also a number h, satisfying the condition 1 ?h ? 2^?n/2 θ^?1 + 1 (0 < θ ? 2^?n/2), such that the inequality $$\prod\nolimits_{i = 1}^n {\left| {Y_i + hc_i } \right|}< \theta $$ will be satisfied.     

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions w_k ? 1. a(x) and φ(x) such that c(n, q) ? w_k(q^N)eⁿφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (N^q) exp(–ne^?2q/n). on the other hand, if k = o(n^1/2), this formula simplifies to c(n, n + k) ? 1/2 w_k (3/π)^1/2 (e/12k)^k/2nⁿ?^(3k?1)/2.     

A Note on the Asymptotic Normality of Sample Autocorrelations for a Linear Stationary Sequence

We consider a stationary time series {X_t} given byX_t=∑^∞_k=−∞ ψ_kZ_t−k, where {Z_t} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf(λ) of {X_t} is squared integrable andm^τ ∑_|k|?m ψ²_k→0 for someτ>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionm^τ ∑_|k|?m ψ²_k→0 for someτ>1/2 is equivalent to the squared integrability off(λ). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm ∑_|k|?m ψ²_k→0.     

A Fujita-type global existence—global non-existence theorem for a system of reaction diffusion equations with differing diffusivities

In this paper we condiser non-negative solutions of the initial value problem in ?^N for the system where 0 ? δ ? 1 and pq > 0. We prove the following conditions. Suppose min(p,q)≥1 but pq1.

• (a) If δ = 0 then u=v=0 is the only non-negative global solution of the system.
• (b) If δ>0, non-negative non-globle solutions always exist for suitable initial values.
• (c) If 0<?1 and max(α, β) ≥ N/2, where qα = β + 1, pβ = α + 1, then the conclusion of (a) holds.
• (d) If N > 2, 0 < δ ? 1 and max (α β) < (N - 2)/2, then global, non-trivial non-negative solutions exist which belong to L^∞(?^N×[0, ∞]) and satisfy 0 < u(X, t) ? c∣x∣^?2α and 0 < v(X, t) ? c ∣x∣^?2bT for large ∣x∣ for all t > 0, where c depends only upon the initial data.
• (e) Suppose 0 > δ 1 and max (α, β) < N/2. If N> = 1,2 or N > 2 and max (p, q)? N/(N-2), then global, non-trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert space H¹ (K) where K(x) ? exp(¼∣x∣²). They decay like e^{[max(α,β)-(N/2)+ε]t} for every ε > 0. These solutions are classical solutions for t > 0.
• (f) If max (α, β) < N/2, then threre are global non-tivial solutions which satisfy, in the hot spot variables where where 0 < ε = ε(u₀, v₀) < (N/2)?;max(α, β). Suppose min(p, q) ? 1.
• (g) If pq ≥ 1, all non-negative solutions are global. Suppose min(p, q) < 1.
• (h) If pg > 1 and δ = 0, than all non-trivial non-negative maximal solutions are non-global.
• (i) If 0 < δ ? 1, pq > 1 and max(α,β)≥ N/2 all non-trivial non-negative maximal solutions are non-global.
• (j) If 0 < δ ≥ 1, pq > 1 and max(α,β) < N/2, there are both global and non-negative solutions.

We also indicate some extensions of these results to moe general systems and to othere geometries.     

A Positive Fraction Erdos - Szekeres Theorem

We prove a fractional version of the Erdős—Szekeres theorem: for any k there is a constant c _k > 0 such that any sufficiently large finite set X⊂ R ² contains k subsets Y ₁ , ... ,Y _k , each of size ≥ c _k |X| , such that every set {y ₁ ,...,y _k } with y _i ε Y _i is in convex position. The main tool is a lemma stating that any finite set X⊂ R ^d contains ``large' subsets Y ₁ ,...,Y _k such that all sets {y ₁ ,...,y _k } with y _i ε Y _i have the same geometric (order) type. We also prove several related results (e.g., the positive fraction Radon theorem, the positive fraction Tverberg theorem). <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p335.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received March 8, 1996, and in revised form June 24, 1996.     

Parameter-dependent operator equations of the Riccati type with application to transport theory

We obtain an existence result for global solutions to initial-value problems for Riccati equations of the form R′(t) + TR(t) + R(t)T = T^ρ A(t)T^1?ρ + T^ρ B(t)T^1?ρ R(t) + R(t)T^ρC(t) T^1?ρ + R(t)T^ρD(t)T^1?ρ R(t), R(0)=R₀, where 0 ? ρ ? 1 and where the functions R and A through D take on values in the cone of non-negative bounded linear operators on L¹ (0, W; μ). T is an unbounded multiplication operator. This problem is of particular interest in case ρ = 1 since it arisess in the theories of particle transport and radiative transfer in a slab. However, in this case there are some serious difficulties associated with this equation, which lead us to define a solution for the case ρ = 1 as the limit of solutions for the cases 0 < ρ < 1.     

A Markov-Nikolskii type inequality for absolutely monotone polynomials of order <Emphasis Type="Italic">K</Emphasis>

A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q′(x) ≥ 0, …, Q(k)(x) ≥ 0, for all x ε I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in L _p [−1, 1], p > 0, is established. One may guess that the right Markov factor is cn ²/k, and this indeed turns out to be the case. Similarly sharp results hold in the case of higher derivatives and Markov-Nikolskii type inequalities. There is also a remarkable connection between the right Markov inequality for absolutely monotone polynomials of order k in the supremum norm and essentially sharp bounds for the largest and smallest zeros of Jacobi polynomials. This is discussed in the last section of the paper.