Simplified Generalized Gauss-Newton Iterative Method under Morozove Type Stopping Rule

Recently, Mahale and Nair considered a simplified generalized Gauss-Newton iterative method for getting an approximate solution for the nonlinear ill-posed operator equation under the modified general source condition. The advantage of this method and the source condition over the classical Gauss-Newton iterative method is that the iterations and source condition involve calculation of the Fréchet derivative only at the point x ₀, i.e., at the initial approximation for the exact solution x ^? of the nonlinear ill-posed operator equation F(x) = y. Motivated by the work of Qinian Jin and Tautenhan, error analysis of the simplified Gauss-Newton iterative method is done in this article under a Morozove-type stopping rule, which is much simpler than the stopping rule considered in the article of Mahale and Nair. An order optimal error estimate is obtained under a modified general source condition which also involves calculation of the Fréchet derivative at the point x ₀.     

Asymptotic Behavior of Gaussian Samples in Fréchet Spaces

Abstract

In this article we investigate unnormalized samples of Gaussian random elements in a separable Fréchet space 𝕄. First we describe a connection between shifts of a Gaussian measure μ in a separable Fréchet space and the infinite product of standard normal distributions in ?^∞, and on the basis of this result we derive the so‐called self‐sufficient expansion for Gaussian random elements in a Fréchet space. Moreover, we find lower bounds for the Gaussian measure μ of shifted balls in 𝕄 and estimate the metric entropy of balls in the Hilbert space ? ? 𝕄 which generates μ. Finally, applying the Brunn–Minkowski inequality we prove a kind of the logarithmic law of large numbers. The last result is an extension of the analogous theorem obtained by Goodman (Characteristics of normal samples. Ann. Probab. 1988, 16, 1281–1290), for a sequence of Gaussian random elements in a separable Banach space.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

On the local existence of solutions to a class of nonconvex evolution inclusions

We prove the local existence of solutions to the Cauchy problemx'∈-?_F V(x)+F(x+f(t,x),x(0)=x ₀, where? _FV is the Fréchet subdifferential of a functionV with aψ-monotone subdifferential of order 2,F is an upper semicontinuous set-valued map contained in the Fréchet subdifferential of aφ- convex function of order two andf is a Carathéodory mapping.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

This paper studies the behavior under iteration of the maps T _ab (x,y) = (F _ab (x) ? y, x) of the plane ?², in which F _ab (x) = ax if x ≥ 0 and bx if x < 0. These maps are area-preserving homeomorphisms of ?² that map rays from the origin to rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x _n+2 = 1/2(a ? b)|x _n+1|+1/2(a+b)x _n+1 ? x _n . This difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type ? x _n+2+2x _n+1 ? x _n +V _μ(x _n+1)x _n+1 = Ex _n+1, in which μ = (1/2)(a ? b) is fixed, and V _μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 ? 1/2(a+b). We study the set Ω _SB of parameter values where the map T _ab has at least one bounded orbit, which correspond to l _∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ the set Spec_∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions n ≥ 4

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L ^∞(R ⁿ ), n ≥ 4, satisfying V(x) = O(?x?^?(n+1)/2?ε), ε > 0.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

Generalized Skew Derivations with Nilpotent Values in Prime Rings

Let ? be a prime ring, 𝒞 the extended centroid of ?, ? a Lie ideal of ?, F be a nonzero generalized skew derivation of ? with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) ? yx) ⁿ  = 0 for all x, y ∈ ?, then ? is commutative and one of the following statements holds:

(1) Either ? is central;

(2) Or ? ? M ₂(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2.     

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r ^k ), r ^k ] _n  = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c ? α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.     

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X ⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q ₁ ⊕ Q ₂ ⊕ Q ₃ such that Q ₁ is a ring satisfying S _2n?2, the standard identity of degree 2n ? 2, Q ₂ ? M _n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x ^q  = x for all x ∈ E, and Q ₃ is a both faithful S _2n?2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.     

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n?1 x ^n?1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 ? |i ? j|/n, for i = 0,…, n ? 1 and j = 0,…, n ? 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.     

Radicals of Skew Polynomial and Skew Laurent Polynomial Rings Over Skew Armendariz Rings

In this note we study radicals of skew polynomial ring R[x; α] and skew Laurent polynomial ring R[x, x ^?1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J(R[x; α]) = N ₀(R[x; α]) = Ni?_*(R)[x; α] and J(R[x, x ^?1; α]) = N ₀(R[x, x ^?1; α]) = Ni?_*(R)[x, x ^?1; α].     

Some Results on p-Nilpotence and Supersolvability of Finite Groups

Let G be a finite group. A subgroup K of a group G is called an ?-subgroup of G if N _G (K) ∩ K ^x  ≦ K for all x ? G. The set of all ?-subgroups of G will be denoted by ?(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P ₀ ? P ₁ ? ··· ? P _n  = P is called a maximal chain of P provided that |P _i : P _i?1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P ₀ ? P ₁ ? ··· ? P _i  ? ··· ? P _n  = P such that P _i  ? ?(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups.     

AA-Rings

Abstract

Let R be a ring with identity such that R ⁺, the additive group of R, is torsion-free of finite rank (tffr). The ring R is called an E-ring if End(R ⁺) = {x ? ax : a ∈ R} and is called an A-ring if Aut(R ⁺) = {x ? ux : u ∈ U(R)}, where U(R) is the group of units of R. While E-rings have been studied for decades, the notion of A-rings was introduced only recently. We now introduce a weaker notion. The ring R, 1 ∈ R, is called an AA-ring if for each α ∈ Aut(R ⁺) there is some natural number n such that α ⁿ  ∈ {x ? ux : u ∈ U(R)}. We will find all tffr AA-rings with nilradical N(R) ≠ {0} and show that all tffr AA-rings with N(R) = {0} are actually E-rings. As a consequence of our results on AA-rings, we are able to prove that all tffr A-rings are indeed E-rings.     

A Time-Independent Approach for the Study of the Spectral Shift Function and an Application to Stark Hamiltonians

In this paper we develop a time-independent approach for the study of the spectral shift function (SSF for short). We apply this method for the perturbed Stark Hamiltonian. We obtain a weak and a Weyl-type asymptotics with optimal remainder estimate of the SSF of the operator pair (P = P₀ + V(x), P₀ = ? h²Δ +x₁), x = (x₁,…, x_n) where V(x) ∈ 𝒞^∞(?ⁿ, ?) decays sufficiently fast at infinity, and h is a small positive parameter. Near a non-trapping energy λ, we give a pointwise asymptotic expansions in powers of h of the derivative of the SSF, and we compute explicitly the two leading terms.