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.     

Approximations of spectra of Schrödinger operators with complex potentials on ℝd

We study spectral approximations of Schrödinger operators T = ?Δ+Q with complex potentials on Ω = ?^d, or exterior domains Ω??^d, by domain truncation. Our weak assumptions cover wide classes of potentials Q for which T has discrete spectrum, of approximating domains Ω_n, and of boundary conditions on ?Ω_n such as mixed Dirichlet/Robin type. In particular, Re Q need not be bounded from below and Q may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of T by those of the truncated operators T_n without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for d = 1,2,3, illustrate our results.     

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.     

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.     

Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Let T be an integral domain with a maximal ideal M, ?: T → K: = T/M the natural surjection, and R the pullback ?^?1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x ₁]]…[x _n ]] to be catenarian, where each [x _i ]] is fixed as either [x _i ] or [[x _i ]]. We also give a complete answer to the question of determining the field extensions k ? K such that the contraction map Spec(K[x ₁]]…[x _n ]]) → Spec(k[x ₁]]…[x _n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x ₁]]…[x _n ]] is catenarian.     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

On Pseudo-Almost Valuation Domains

Let R be an integral domain with quotient field K and integral closure R ^′. Anderson and Zafrullah called R an “almost valuation domain” if for every nonzero x ∈ K, there is a positive integer n such that either x ⁿ  ∈ R or x ^?n  ∈ R. In this article, we introduce a new closely related class of integral domains. We define a prime ideal P of R to be a “pseudo-strongly prime ideal” if, whenever x, y ∈ K and xyP ? P, then there is a positive integer m ≥ 1 such that either x ^m  ∈ R or y ^m P ? P. If each prime ideal of R is a pseudo-strongly prime ideal, then R is called a “pseudo-almost valuation domain” (PAVD). We show that the class of valuation domains, the class of pseudo-valuation domains, the class of almost valuation domains, and the class of almost pseudo-valuation domains are properly contained in the class of pseudo-almost valuation domains; also we show that the class of pseudo-almost valuation domains is properly contained in the class of quasilocal domains with linearly ordered prime ideals. Among the properties of PAVDs, we show that an integral domain R is a PAVD if and only if for every nonzero x ∈ K, there is a positive integer n ≥ 1 such that either x ⁿ  ∈ R or ax ^?n  ∈ R for every nonunit a ∈ R. We show that pseudo-almost valuation domains are precisely the pullbacks of almost valuation domains, we characterize pseudo-almost valuation domains of the form D + M, and we use this characterization to construct PAVDs that are not almost valuation domains. We show that if R is a Noetherian PAVD, then R has Krull dimension at most one and R ^′ is a valuation domain; we show that every overring of a PAVD R is a PAVD iff R ^′ is a valuation domain and every integral overring of R is a PAVD.     

On Generalizations of Prime Ideals (II)

Let R be a commutative ring with identity. We say that a proper ideal P of R is (n ? 1, n)-weakly prime (n ≥ 2) if 0 ≠ a ₁…a _n  ∈ P implies a ₁…a _i?1 a _i+1…a _n  ∈ P for some i ∈ {1,…, n}, where a ₁,…, a _n  ∈ R. In this article, we study (n ? 1, n)-weakly prime ideals. A number of results concerning (n ? 1, n)-weakly prime ideals and examples of (n ? 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n ? 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n ? 1, n)-weakly prime ideals and (n ? 1, n)-prime ideals coincide.     

On Skew Triangular Matrix Rings

For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring T _n (R, α). By using an ideal theory of a skew triangular matrix ring T _n (R, α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x; α]/ ? x ⁿ  ?, for each positive integer n, where R[x; α] is the skew polynomial ring, and ? x ⁿ  ? is the ideal generated by x ⁿ .     

Projective Varieties Swept Out by Rational Normal Curves

Let X ? ? ⁿ be a complex nondegenerate projective variety of dimension m ≥ 2. For t ≤ n ? m and a general q ∈ ? ⁿ , the linear space L _q spanned by q and t general points of X meets X in a finite set of points. We classify those X ? ? ⁿ for which there exists a point q ∈ ? ⁿ such that L _q meets X in a positive dimensional variety. If this occurs, there exists d ≤ n ? m such that a degree d rational normal curve through d general points of X is contained in X. Examples of this situation are provided. An infinitesimal generalization of part of the main result is also stated.     

On n-Absorbing Ideals of Commutative Rings

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x ₁…x _n+1 ∈ I for x ₁,…, x _n+1 ∈ R (resp., I ₁…I _n+1 ? I for ideals I ₁,…, I _n+1 of R), then there are n of the x _i 's (resp., n of the I _i 's) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.     

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.     

The Leading Ideal of a Complete Intersection of Height Two in a 2-Dimensional Regular Local Ring

Let (S,𝔫) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I* be the leading ideal of I in the associated graded ring gr𝔫(S), and set R = S/I and 𝔪 = 𝔫/I. In Goto et al. (2007 Goto , S. , Heinzer , W. , Kim , M.-K. ( 2007 ). The leading ideal of a complete intersection of height two, II . J. Algebra 312 : 709 – 732 . [Google Scholar]), we prove that if μ _G (I*) = n, then I* contains a homogeneous system {ξ _i }_1≤i≤n of generators such that deg ξ _i  + 2 ≤ deg ξ _i+1 for 2 ≤ i ≤ n ? 1, and ht _G (ξ₁, ξ₂,…, ξ _n?1) = 1, and we describe precisely the Hilbert series H(gr𝔪(R), λ) in terms of the degrees c _i of the ξ _i and the integers d _i , where d _i is the degree of D _i  = GCD(ξ₁,…, ξ _i ). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c ₁ + 1, an ascending sequence of positive integers (c ₁, c ₂,…, c _n ), and a descending sequence of integers (d ₁ = c ₁, d ₂,…, d _n  = 0) such that c _i+1 ? c _i  > d _i?1 ? d _i  > 0 for each i with 2 ≤ i ≤ n ? 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.     

Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

We consider the Aharonov–Bohm effect for the Schrödinger operator H = (?i? _x  ? A(x))² + V(x) and the related inverse problem in an exterior domain Ω in R ² with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω ^c is convex.     

Localization and Catenarity in Iterated Differential Operator Rings

ABSTRACT

Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ₁, δ₁] ··· [Θ _n , δ _n ] an iterated differential operator k-algebra over R such that δ _j (Θ _i ) ∈ R[Θ₁, δ₁] ··· [Θ _i?1, δ _i?1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the Δ-prime ideals of R, the local ring A _P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ _i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U(g) the enveloping algebra of g. Then the crossed product of R by U(g) is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero.     

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ? ^d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ? _R ≈(??λ I)^?1, where ? is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ? ^d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e ^{?i κ‖x‖}/‖x‖, x∈? ^d , κ∈?₊, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ?|log?ε|²? n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n ²), even in the case κ=O(n). Numerical illustrations demonstrate the efficiency of low tensor-rank approximation for Green’s kernels e ^?λ‖x‖/‖x‖, x∈?³, in the case of Newton (λ=0), Yukawa (λ∈?₊), and Helmholtz (λ=i κ,?κ∈?₊) potentials, as well as for the kernel functions 1/‖x‖ and 1/‖x‖ ^d?2, x∈? ^d , in higher dimensions d>3. We present numerical results on the iterative calculation of the minimal eigenvalue for the d-dimensional finite difference Laplacian by the power method with the rank truncation and based on the approximate inverse ? _R ≈(?Δ)^?1, with 3≤d≤50.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.