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).     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Estimating the Diffusion Coefficient for Diffusions Driven by fBm

When the Hurst coefficient of a fBm B _t ^H is greater than 1/2, it is possible to define a stochastic integral with respect to B _t ^H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type X_t = x₀ + ₀ ^T (X_s) dB_s ^H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X _t. We then study consistency in probability of this estimator and calculate convergence rates in probability.     

Exceedance probability of the integral of a stochastic process

Let X={X(s)}_s∈S be an almost sure continuous stochastic process (S compact subset of R^d) in the domain of attraction of some max-stable process, with index function constant over S. We study the tail distribution of ∫_SX(s)ds, which turns out to be of Generalized Pareto type with an extra ‘spatial’ parameter (the areal coefficient from Coles and Tawn (1996) [3]). Moreover, we discuss how to estimate the tail probability P(∫_SX(s)ds>x) for some high value x, based on independent and identically distributed copies of X. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. X represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall.The paper has two main purposes: first to formalize and justify the results of Coles and Tawn (1996) [3]; further we treat the problem in a non-parametric way as opposed to their fully parametric methods.     

Representation and approximation of solutions to semilinear Volterra equations with delay

In this paper we use a theorem of Crandall and Pazy to provide the product integral representation of the nonlinear evolution operator associated with solutions to the semilinear Volterra equation: x(?)(t) = W(t, τ) ?(0) + ∝_τ^tW(t, s)F(s, x_s(?)) ds.Here the kernel W(t, s) is a linear evolution operator on a Banach space X; I is an interval of the form [?r, 0] or (?∞, 0] and F is a nonlinear mapping of R × C(I, X) into X. The abstract theory is applied to examples of partial functional differential equations.     

Connection between random curves,changes of time,and regenerative times of stochastic processes

The product of spaces Φ × D is considered, where Φ is the set of all continuous, nondecreasing functions ?:[0,∞)→(0,∞), ?(0)=0, ?(t)→∞(t→∞), and D is the set of all right continuous functions ξ:(0,∞)→X; here X is some metric space. Two mappings are defined: the first is the projection q(?,ξ)=ξ, and the second is the change of time U(?,ξ)=ξº?. The following equivalence relation is defined on D: $$\xi _1 \sim \xi _2 \Leftrightarrow \exists _{\varphi _1 , \varphi _1 } \in \Phi :\xi _1 ^\circ \varphi _1 = \xi _2 ^\circ \varphi _2 $$ . Let? be the set of all equivalence classes, and let L be the mapping ξ₄～ξ₂, Lξ is called the curve corresponding to ξ. The following theorem is proved: two stochastic processes with probability measures P¹ and P² on D possess identical random curves (i.e.,P¹ºL^?1=P²ºL^?1) if and only if there exist two changes of time (i.e., probability measures Q¹ and Q² on ?×D for which P¹=Q¹ºq^?1, P²=Q²ºq^?1 which take these two processes into a process with measure \(\tilde P\) (i.e., Q¹ºu^?1=Q²ºu^?1,=～P) If (P _x ¹ )_x∈X and (P _x ² )_x∈X are two families of probability measures for which P _x ¹ ºL^?1=P _x ² ºL^?1?x∈X then for each x ε X the corresponding measures Q _X ¹ andQ _X ² can be found in the following manner. The set of regenerative times of the family \(\left( {\tilde P_x } \right)_{x \in X} \) contains all stopping times which are simultaneously regenerative times of the families (p _x ¹ )_x∈X and (P _x ² )_x∈X and possess a certain special property of first intersection.     

A Wiener–Wintner theorem for the Hilbert transform

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ,T _t ) and f∈L ^p (X,μ), there is a set X _f ⊂X of probability one, so that for all x∈X _f ,

A fixed point theorem in metric spaces

We discuss a fixed point theorem for a function f mapping a complete metric space X into itself. For all x ? X{x \in X} the iterates of f(x) are shown to converge to x_* = f(x_*){{x_{\star} = f(x_{\star})}} and an explicit estimate of the convergence rate is given.     

Multipliers of classes of cauchy transforms

The analytic map g on the unit disk D is said to induce a multiplication operator L from the Banach space X to the Banach space Y if L(f)=f·g∈Y for all f∈X. For z ∈ D and α>0 the families of weighted Cauchy transforms F_α are defined by ?(z) = ∫_T K_x ^α (z)dμ(x) where μ(x) is complex Borel measures, x belongs to the unit circle T and the kernel K_x (z) = (1- xz)^?1. In this article we will explore the relationship between the compactness of the multiplication operator L acting on F ₁ and the complex Borel measures μ(x). We also give an estimate for the essential norm of L     

Kernel-type density and failure rate estimation for associated sequences

Let {X _n ,n1} be a strictly stationary sequence of associated random variables defined on a probability space (,B, P) with probability density functionf(x) and failure rate functionr(x) forX ₁. Letf _n (x) be a kerneltype estimator off(x) based onX ₁,...,X _n . Properties off _n (x) are studied. Pointwise strong consistency and strong uniform consistency are established under a certain set of conditions. An estimatorr _n (x) ofr(x) based onf _n (x) andF _n (x), the empirical survival function, is proposed. The estimatorr _n (x) is shown to be pointwise strongly consistent as well as uniformly strongly consistent over some sets.     

Laws of large numbers for weighted sums of random variables and random elements

Consider the weighted sums of a sequence {X _n} of independent random variables or random elements inD [0,1]. For convergence ofS _n in probability and with probability one, in [2],[3] etc., the following stronger condition is required: {X _n} is uniformly bounded by a random variableX,i.e.P(¦X _n¦x)P(¦X¦x) for allx>0. Our paper aims at trying to drop this restriction.The Project supported by National Natural Science Foundation of China     

Global solutions for operator Riccati equations with unbounded coefficients: A non-linear semigroup approach

Let X be a Banach space of real-valued functions on [0, 1] and let ?(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ?(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the B_i(t)'s are bounded linear integral operators on X. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L¹(0, 1). Conditions on X, R(0), T, and the coefficients are found such that the theory of non-linear semigroups may be used to prove global existence of strong solutions in ?(X) that also satisfy R(t) ? ?(L¹(0,1)) for all t ≥ 0.     

New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow

In this note, we prove an ?‐regularity theorem for the Ricci flow. Let (Mⁿ,g(t)) with t ? [?T,0] be a Ricci flow, and let H_x0(y,s) be the conjugate heat kernel centered at some point (x₀,0) in the final time slice. By substituting H_x0(?,s) into Perelman's W‐functional, we obtain a monotone quantity W_x0(s) that we refer to as the pointed entropy. This satisfies W_x0(s) ≤ 0, and W_x0(s) = 0 if and only if (Mⁿ,g(t)) is isometric to the trivial flow on Rⁿ. Then our main theorem asserts the following: There exists ? > 0, depending only on T and on lower scalar curvature and μ‐entropy bounds for the initial slice (Mⁿ,g(?T)) such that W_x0(s) ≥ ?? implies |Rm| ≤ r^?2 on P_{? r}(x₀,0), where r² ≡ |s| and P_ρ(x,t) ≡ B_ρ(x,t) × (t?ρ²,t] is our notation for parabolic balls. The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s‐average of W_x(s). To accomplish this, we require a new log‐Sobolev inequality. Perelman's work implies that the metric measure spaces (Mⁿ,g(t),dvol_g(t)) satisfy a log‐Sobolev; we show that this is also true for the heat kernel weighted spaces (Mⁿ,g(t),H_x0(?,t)dvol_g(t)). Our log‐Sobolev constants for these weighted spaces are in fact universal and sharp. The weighted log‐Sobolev has other consequences as well, including certain average Gaussian upper bounds on the conjugate heat kernel. © 2014 Wiley Periodicals, Inc.     

Estimation of the Location of the Maximum of a Regression Function Using Extreme Order Statistics

In this paper, we consider the problem of approximating the location,x₀∈C, of a maximum of a regresion function,θ(x), under certain weak assumptions onθ. HereCis a bounded interval inR. A specific algorithm considered in this paper is as follows. Taking a random sampleX₁, …, X_nfrom a distribution overC, we have (X_i, Y_i), whereY_iis the outcome of noisy measurement ofθ(X_i). Arrange theY_i's in nondecreasing order and take the average of ther X_i's which are associated with therlargest order statistics ofY_i. This average,x₀, will then be used as an estimate ofx₀. The utility of such an algorithm with fixed r is evaluated in this paper. To be specific, the convergence rates ofx₀tox₀are derived. Those rates will depend on the right tail of the noise distribution and the shape ofθ(·) nearx₀.     

Spectral Criteria of Abstract Sequences; Application to Difference Equations

We suppose that M is a closed subspace of l ^∞(J, X), the space of all bounded sequences {x(n)} _n?J ? X, where J ? {Z⁺,Z} and X is a complex Banach space. We define the M-spectrum σ_M (u) of a sequence u ? l ^∞(J,X). Certain conditions will be supposed on both M and σ_M (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σ_M (u,) is at most countable and, for every λ ? σ_M (u), the sequence e^?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ ^r _k=1 a_k (u(n + k) ? u(n)) + Σ _{s ? Z}?(n ? s)u(s) = h(n), n?J, where ψ?l ^∞(Z,X) such that ψ| _J ? M, A is the generator of a C ₀-semigroup of linear bounded operators {T(t)} _t>0 on X, h ? M, ? ? l ¹(Z) and a_k ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.     

A general duality theorem for marginal problems

Summary Given probability spaces (X _i ,A _i ,P _i ),i=1, 2 letM(P ₁,P ₂) denote the set of all probabilities on the product space with marginalsP ₁ andP ₂ and leth be a measurable function on (X ₁×X ₂,A ₁ A ₂). In order to determine supfh dP where the supremum is taken overP inM(P ₁,P ₂), a general duality theorem is proved. Only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions. An example without any further topological or tightness assumptions. An example is given to show that the assumption of perfectness is essential. Applications to probabilities with given marginals and given supports, stochastic order and probability metrics are included.     

Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

Asymptotics of random sums of negatively dependent random variables in the presence of dominatedly varying tails

Let X ₁ , X ₂ , . . . be a sequence of negatively dependent and identically distributed random variables, and let N be a counting random variable independent of X _i ’s. In this paper, we study the asymptotics for the tail probability of the random sum S_N = ?_{k = 1}^N X_k {S_N} = \sum\nolimits_{k = 1}^N {{X_k}} in the presence of heavy tails. We consider the following three cases: (i) P(N > x) = o(P(X ₁  > x)), and the distribution function (d.f.) of X ₁ is dominatedly varying; (ii) P(X ₁  > x) = o(P(N > x)), and the d.f. of N is dominatedly varying; (iii) the tails of X ₁ and N are asymptotically comparable and dominatedly varying.     

Upper semicontinuity of parametric projections

In this paper we study upper semicontinuity of the metric projection P _(p)(x) with respect to (x, p), where x is a point in a normed linear space X and (p) is an approximatively compact subset of X depending on a parameter p. An application to parametric spline approximation is given.     

No TVD Fields for 1-D Isentropic Gas Flow

We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set ?(r ₀, s ₀) = {r ₀ < r < s < s ₀}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation Var _X (?(τ(t, X), u(t, X))) is non-increasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking shock-rarefaction interaction gives the asymptotic states in the exact solution.

Little is known about global existence of large-variation solutions to hyperbolic systems of conservation laws u _t  + f(u) _x  = 0. In particular it is not known if isentropic gas flow admits a priori BV bounds which apply to all BV data.

In the few cases where such results are available (scalar case, Temple class, systems satisfying Bakhvalov's condition, isothermal gas dynamics) there are TVD fields which play a key role for existence. Our results show that the same approach cannot work for isentropic flow.