Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle

We consider the class P _n ^* of algebraic polynomials of a complex variable with complex coefficients of degree at most n with real constant terms. In this class we estimate the uniform norm of a polynomial P _n ∈ P _n ^* on the circle Γ_r = z ∈ ?: ¦z¦ = r of radius r = 1 in terms of the norm of its real part on the unit circle Γ₁ More precisely, we study the best constant μ(r, n) in the inequality ||P_n||C(Γ_r) ≤ μ(r,n)||Re P_n||C(Γ₁). We prove that μ(r,n) = rⁿ for rⁿ⁺² ? r ⁿ ? 3r² ? 4r + 1 ≥ 0. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality μ(r, n) = r ⁿ is valid for r sufficiently close to 1.     

On the Generalized Derivation Induced by Two Projections

We prove that

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.     

A Skorohod representation theorem for uniform distance

Let??? _n be a probability measure on the Borel ??-field on D[0, 1] with respect to Skorohod distance, n ?? 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X _n such that X _n ~ ?? _n for all n ?? 0 and ||X _n ? X ₀|| ?? 0 in probability, where ||·|| is the sup-norm. Such conditions do not require??? ₀ separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.     

A stability result using the matrix norm to bound the permanent

We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose A is an n × n matrix over C (resp. R), and let P denote the set of n × n matrices over C (resp. R) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of A satisfies |per(A)| ≤ ||A|| ⁿ ₂ with equality iff A/||A||₂ ∈ P (where ||A||₂ is the operator 2-norm of A). We show a stability version of this result asserting that unless A is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of n) than ||A|| ⁿ ₂. In particular, for any fixed α, β > 0, we show that |per(A)| is exponentially smaller than ||A|| ⁿ ₂ unless all but at most αn rows contain entries of modulus at least ||A||₂(1?β).     

Norms of Moore-Penrose Inverses of Fredholm Toeplitz Operators

In this paper we estimate the norm of the Moore-Penrose inverse T(a)⁺ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈L_{N × N}^∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)⁺|| > ||a⁻¹||_∞ holds. Moreover, we discuss the asymptotic behavior of ||T(t^ra)⁺|| for . The results are illustrated by numerical experiments.     

Bounding partial sums of Fourier series in weighted L2-norms,with applications to matrix analysis

For integrable functions f defined on the interval [−π,π], we denote the partial sums of the corresponding Fourier series by S_n(f) (n=0,1,2,…). In this paper, we deal with the problem of bounding sup_n||S_n||, where ||·|| denotes an operator norm induced by a weighted L²-norm for functions f on [−π,π]. For weight functions w belonging to a class introduced by Helson and Szegö (Ann. Mat. Pura Appl. 51 (1960) 107), we present explicit upper bounds for sup_n||S_n|| in terms of w.The authors were led to the problem of deriving explicit upper bounds for sup_n||S_n||, by the need for such bounds in an analysis related to the Kreiss matrix theorem—a famous result in the fields of linear algebra and numerical analysis. Accordingly, the present paper highlights the relevance of bounds on sup_n||S_n|| to these fields.     

Powers of Rational Numbers Modulo 1 Lying in Short Intervals

Let p > q > 1 be two coprime integers. We construct some positive numbers ξ such that the numbers ξ(p/q) ⁿ , n = 0, 1, 2, . . . , modulo 1 all lie in a short interval. Our results imply, for instance, that there exist three positive real numbers ξ, ζ, τ such that the inequalities ||ξ(5/3) ⁿ || < 2/5, ||ζ (5/3) ⁿ || > 1/10 and ||τ (3/2)²ⁿ || < 14/45 hold for each integer ${n \geqslant 0}$ .     

On the derivatives of X-ray functions

We show that for every n \geqq 4, 0 \leqq k \leqq n - 3, p ? (0, 3] n \geqq 4, 0 \leqq k \leqq n - 3, p \in (0, 3] and every origin-symmetric convex body K in \mathbbRⁿ \mathbb{R}^n , the function ||x ||^-k₂ ||x ||^-n+k+p_K \parallel x \parallel^{-k}_{2} \parallel x \parallel^{-n+k+p}_{K} represents a positive definite distribution on \mathbbRⁿ \mathbb{R}^n , where ||·||₂ \parallel \cdot \parallel_{2} is the Euclidean norm and ||·||_K \parallel \cdot \parallel_{K} is the Minkowski functional of K. We apply this fact to prove a result of Busemann-Petty type that the inequalities for the derivatives of order (n - 4) at zero of X-ray functions of two convex bodies imply the inequalities for the volume of average m-dimensional sections of these bodies for all 3 \leqq m \leqq n 3 \leqq m \leqq n . We also prove a sharp lower estimate for the maximal derivative of X-ray functions of the order (n - 4) at zero.     

Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.     

Optimal extended optical flow subject to a statistical constraint

This work is motivated by the necessity to improve heart image tracking. This technique is related to the ability of generating an apparent continuous motion, which is observable through the variation of intensity from a starting image to an ending one. Given two images ρ₀ and ρ₁, we calculate an evolution process ρ(t,⋅) which transports ρ₀ to ρ₁ by using the optimal extended optical flow. Such a strategy is found to be well suited for heart image tracking, provided the motion is controlled by a statistical model. In this paper we use viability theory to give sufficient conditions to handle the optimal extended optical flow subject to a point-wise statistical constraint by using Parzen’s approximation. The strategy is implemented in a 1D case and the numerical results which are presented show the efficiency of the proposed strategy.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

The minimum increment of f-divergences given total variation distances

Let (P _i , Q _i ), i = 0, 1, be two pairs of probability measures defined on measurable spaces (Ω _i ,F _i ) respectively. Assume that the pair (P₁, Q₁) is more informative than (P₀,Q₀) for testing problems. This amounts to say that If (P₁,Q₁) ≥ If (P₀,Q₀), where If (·, ·) is an arbitrary f-divergence. We find a precise lower bound for the increment of f-divergences I_f(P₁,Q₁) ? I_f(P₀,Q₀) provided that the total variation distances ||Q₁ ? P₁|| and ||Q₀ ? P₀|| are given. This optimization problem can be reduced to the case where P₁ and Q₁ are defined on the space consisting of four points, and P₀ and Q₀ are obtained from P₁ and Q₁ respectively by merging two of these four points. The result includes the well-known lower and upper bounds for I_f(P,Q) given ||Q ? P||.     

Homogenization of transport equations: A simple PDE approach to the Kubo formula

We are interested in the behavior with respect to the small parameter ?>0 of solutions ρ? of the conservative transport(-diffusion) equation t_∂ρ?+∇x(ρ?u?)=ηΔxρ?, with η?0, driven by a large random velocity field: |u?|=O(1/?). Assuming that the velocity does not have long-time memory we justify the convergence of the expectation Eρ? to the solution of a diffusion equation. This question has been widely investigated; here we present a simple proof which only relies on PDE tools.     

The polynomial Daugavet property for atomless L 1(μ)-spaces

For any atomless positive measure μ, the space L ₁(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial ${P:L_1(\mu)\longrightarrow L_1(\mu)}For any atomless positive measure μ, the space L ₁(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial P:L₁(m)? L₁(m){P:L_1(\mu)\longrightarrow L_1(\mu)} satisfies the Daugavet equation ||Id + P||=1 + ||P||{\|{\rm Id} + P\|=1 + \|P\|}. The same is true for the vector-valued spaces L ₁(μ, E), μ atomless, E arbitrary.     

矩阵方程X-A*X-2A=Q的正定解及其扰动分析

研究非线性矩阵方程X-A*X-2A=Q(Q>0)的Hermite正定解及其扰动问题.给出了该方程存在唯-Hermite正定解的充分条件及解的迭代计算公式.在此条件下,给出了该唯一解的扰动界及正定解条件数的一种表达式,并用数值例子对所得结果进行了说明.     

An exact Daugavet type inequality for small into isomorphisms in L 1

The well known Daugavet property for the space L ₁ means that || I  +  K || = 1+ || K || for any weakly compact operator K : L ₁ → L ₁, where I is the identity operator in L ₁. We generalize this theorem to the case when we consider an into isomorphism J : L ₁ → L ₁ instead of I and a narrow operator T. Our main result states that , where d  =  || J|| || J ⁻¹||. We also give an example which shows that this estimate is exact. Received: 21 August 2007     

Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier-Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006) [3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique global classical solution (ρ,u) where ρ∈C¹([0,T];H¹([0,1])) and u∈H¹([0,T];H²([0,1])) for any T>0. As it is pointed out in Xin (1998) [31] that the smooth solution (ρ,u)∈C¹([0,T];H³(R¹)) (T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ²utt, such as Lemmas 3.2-3.6. This leads to further regularities of (ρ,u) where ρ∈C¹([0,T];H³([0,1])), u∈H¹([0,T];H³([0,1])). It is still open whether the regularity of u could be improved to C¹([0,T];H³([0,1])) with the appearance of vacuum, since it is not obvious that the solutions in C¹([0,T];H³([0,1])) to the initial boundary value problem must blow up in finite time.     

The completeness of systems of functions of the Mittag-Leffler type for weighted uniform approximation in a complex domain

For a givenρ(1/2 <ρ < + ∞) let us set L _ρ = {z: |arg z| = π/(2ρ)} and assume that a real valued measurable function ?(t) such that ?(t) ≥ 1(t ∈ L _ρ ) and \(\mathop {\lim }\limits_{|t| \to + \infty } \varphi (t) = + \infty (t \in L_\rho )\) is defined on L _ρ . Let C _? (L _ρ ) denote the space of continuous functionsf(t) on L _ρ such that \(\lim \tfrac{{f(t)}}{{\varphi (t)}} = 0\) , where the norm of an elementf is defined as: \(\parallel f\parallel = \mathop {\sup }\limits_{t \in L_\rho } \tfrac{{|f(t)|}}{{\varphi (t)}}\) . In this note we pose the question about the completeness of the system of functions of the Mittag-Leffler type {Eρ(ut; μ)} (μ ≥ 1, 0 ≤ u ≤a) or, what is the same thing, of the system of functions \(p(t) = \int_0^a {E_\rho (ut;\mu )d\sigma (u)} \) in C _? (L _ρ ). The following theorem is proved: The system of functions of the Mittag-Leffler type is complete in C _? (L _ρ ) if and only if sup |p(z)| ≡ +∞, z ∈ L _ρ , where the supremum is taken over the set of functions p(t) such that ∥p(t) (t + 1)^?1 ∥ ≤ 1.     

Self-intersection local times of random walks: exponential moments in subcritical dimensions

Fix p?>?1, not necessarily integer, with p(d ? 2)?< d. We study the p-fold self-intersection local time of a simple random walk on the lattice ${\mathbb Z^d}$ up to time t. This is the p-norm of the vector of the walker??s local times, ? _t . We derive precise logarithmic asymptotics of the expectation of exp{?? _t ||? _t || _p } for scales ?? _t >?0 that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of t and ?? _t , and the precise rate is characterized in terms of a variational formula, which is in close connection to the Gagliardo?CNirenberg inequality. As a corollary, we obtain a large-deviation principle for ||? _t || _p /(tr _t ) for deviation functions r _t satisfying ${t r_t\gg \mathbb E[||\ell_t||_p]}$ . Informally, it turns out that the random walk homogeneously squeezes in a t-dependent box with diameter of order ? t ^1/d to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.