On the singularities of solutions of partial differential equations with constant coefficients

LetP be a differential operator with constant coefficients in ℝ ⁿ . Ifu is a distribution, the singular support ofu is the complement of the largest set whereu ∈C ^∞. Necessary and sufficient conditinos are obtained for a closed convex set Γ to be equal to the singular support ofu for someu withPu ∈C ^∞ or, equivalently, for Γ to contain the singular support ofu for someu withPu ∈C ^∞ butu ∉C ^∞. Related local uniqueness theorems analogous to the Holmgren theorem with supports replaced by singular supports are also given, as well as applications concerningP-convexity with respect to singular supports.     

The four-block Adamjan-Arov-Krein problem

This paper is concerned with the mixed sensitivity H^∞ design in the most general, i.e., the four-block, case. This problem involves in a crucial manner the so-called four-block operator Γ, the norm of which is the achievable feedback tolerance. Our objective in this paper is to provide an interpretation for all singular values of Γ. These singular values are given an Adamjan-Arov-Krein interpretation in terms of the L^∞ distance of an L^∞ function to H^∞(l). Intuitively, the singular values of Γ are the various tolerance levels that can be achieved if we allow a various number of unstable poles in the closed loop. We finally provide an upper bound on the number of singular values.     

A twisted factorization method for symmetric SVD of a complex symmetric tridiagonal matrix

This paper presents an O(n²) method based on the twisted factorization for computing the Takagi vectors of an n‐by‐n complex symmetric tridiagonal matrix with known singular values. Since the singular values can be obtained in O(n²) flops, the total cost of symmetric singular value decomposition or the Takagi factorization is O(n²) flops. An analysis shows the accuracy and orthogonality of Takagi vectors. Also, techniques for a practical implementation of our method are proposed. Our preliminary numerical experiments have verified our analysis and demonstrated that the twisted factorization method is much more efficient than the implicit QR method, divide‐and‐conquer method and Matlab singular value decomposition subroutine with comparable accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

Singular Set of a Convex Potential in Two Dimensions

Let u be a convex potential of the optimal transfer map from a convex open set X to a nonconvex open set Y in the plane. If u only has singularities whose sets of supports are one dimension, then under some mild assumptions on Y, we show that the singular set of u are disjoint union of countably many C ¹ curves. Or we can say that the singular set is a C ¹ manifold.     

Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

The bilinear finite element methods on appropriately graded meshes are considered both for solving singular and semisingular perturbation problems. In each case, the quasi-optimal order error estimates are proved in the ε-weighted H¹-norm uniformly in singular perturbation parameter ε, up to a logarithmic factor. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

The monodromy of a series of hypersurface singularities

Let {f=0} be a hypersurface inC ⁿ⁺¹ with a 1-dimensional singular set Σ. We consider the series of hypersurfaces {f+ɛx ^N=0} wherex is a generic linear form. We derive a formula, which relates the characteristic polynomials of the monodromies off andf+ɛx ^N. Other ingredients in this formula are the horizontal and the vertical monodromies of the transversal (isolated) singularities on each branch of the singular set. We use polar curves and the carrousel method in the proof. The formula is a generalization of the Iomdin formula for the Milnor numbers: μ(f+ɛx ^N )=μ _n (f)−μ _n −1(f)+Ne ₀(Σ)     

Singular hyperbolicity for transitive attractors with singular points of 3-dimensional <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript>-flows

In the context of C^r-flows on 3-manifolds (r ≥ 1), the notion of singular hyperbolicity, inspired on the Lorenz Attractor, is the right generalization of hyperbolicity (in the sense of Smale) for C¹-robustly transitive sets with singularities. We estabish conditions (on the associated linear Poincaré flow and on the nature of the singular set) under which a transitive attractor with singularities of a C²-flow on a 3-manifold is singular hyperbolic.     

Inclusion sets for the singular values of a square matrix

The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (a_ij) ∈ ℂ ^n×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n, whose diagonal entries are σ² − | a_ii|², i = 1, …, n, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain inclusion sets for the singular values of A. Scaled versions of the inclusion sets, allowing one, in particular, to obtain Ky Fan type results for the singular values, are derived by passing to the conjugated matrix D⁻¹C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 52–77.     

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999     

Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators

We consider nonself-adjoint nondissipative trace class additive perturbations L=A+iV of a bounded self-adjoint operator A in a Hilbert space ,H. The main goal is to study the properties of the singular spectral subspace N _i ⁰ of L corresponding to part of the real singular spectrum and playing a special role in spectral theory of nonself-adjoint nondissipative operators.To some extent, the properties of N _i ⁰ resemble those of the singular spectral subspace of a self-adjoint operator. Namely, we prove that L and the adjoint operator ,L ^* are weakly annihilated by some scalar bounded outer analytic functions if and only if both of them satisfy the condition N _i ⁰ =H. This is a generalization of the well-known Cayley identity to nonself-adjoint operators of the above-mentioned class.     

Left and right Blaschke-Potapov products and Arov-singular matrix-valued functions

IfJ is an indefinite signature matrix, then there exists aJ contractive holomorphic matrix valued functionW(z) in the open unit disc which can be expressed as a left Blaschke-Potapov product:W(z)=B ^(l)(z), but not as a right Blaschke-Potapov product:W(z)=E(z)B ^(r)(z), whereB ^(r)(z) is a right Blaschke-Potapov product andE(z) is a so called Arov singular matrix function. In factB ^(l)(z) may be chosen to obtain any Arov singular matrix functionE(z) in the second representation. This phenomenon and multiplicative representations of Arov singular functions are discussed.This paper was written while the author was a guest of the Department of Theoretical Mathematics of The Weizmann Institute of Science, Rehovot. The author would like to thank the Department for its support and hospitality.     

Uniform estimates for spherical functions on complex semisimple Lie groups

We prove new estimates for spherical functions and their derivatives on complex semisimple Lie groups, establishing uniform polynomial decay in the spectral parameter. This improves the customary estimate arising from Harish-Chandra's series expansion, which gives only a polynomial growth estimate in the spectral parameter. In particular, for arbitrary positive-definite spherical functions on higher rank complex simple groups, we establish estimates for which are of the form in the spectral parameter and have uniform exponential decay in regular directions in the group variable a _t . Here is an explicit constant depending on G, and may be singular, for instance.?The uniformity of the estimates is the crucial ingredient needed in order to apply classical spectral methods and Littlewood—Paley—Stein square functions to the analysis of singular integrals in this context. To illustrate their utility, we prove maximal inequalities in L ^p for singular sphere averages on complex semisimple Lie groups for all p in . We use these to establish singular differentiation theorems and pointwise ergodic theorems in L ^p for the corresponding singular spherical averages on locally symmetric spaces, as well as for more general measure preserving actions. Submitted: May 2000, Revised version: October 2000.     

On Multiple Singular Integrals along Polynomial Curves with Rough Kernels

This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integral operators as well as the corresponding maximal truncated singular integral operators for some fixed 1〈p〈 ∞,are given.     

Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities

In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix characteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t ⁻¹, were found. Conditions for the invertibility of the matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Submitted: June 15, 2007. Revised: October 25, 2007. Accepted: November 5, 2007.     

Analysis of Spectral Points of the Operators T [*] T and TT [*] in a Krein Space

Spectra and sets of regular and singular critical points of definitizable operators of the form T ^[*] T and TT ^[*] in a Krein space are compared. The relation between the Jordan chains of the above operators (corresponding to the same eigenvalue) is shown.      

An Off-DiagonalT1 Theorem and Applications

We give a general boundedness criterion, analogous to theT1 Theorem, for singular integrals mappingL^ptoL^q,q<p. As an easy corollary of this result, we deduce the Kato–Ponce “Leibniz Rule” for fractional order derivatives. We also study commutators of parabolic singular integrals and deduce pointwise a.e. convergence of truncations of a class of parabolic singular integrals which includes the caloric double-layer potential on the boundary of a non-cylindrical domain.     

Regularity of volume-minimizing flows on 3-manifolds

In Johnson and Smith (Indiana Univ Math J 44:45–85, 1995; Ann Global Anal Geometry 30:239–287, 2006; Proceedings of the VII International Colloquium on Differential Geometry, 1994, World Scientific, pp. 81–98), the authors characterized the singular set (discontinuities of the graph) of a volume-minimizing rectifiable section of a fiber bundle, showing that, except under certain circumstances, there exists a volume-minimizing rectifiable section with the singular set lying over a codimension-3 set in the base space. In particular, it was shown that for 2-sphere bundles over 3-manifolds, a minimizer exists with a discrete set of singular points. In this article, we show that for a 2-sphere bundle over a compact 3-manifold, such a singular point cannot exist. As a corollary, for any compact 3-manifold, there is a C ¹ volume-minimizing one-dimensional foliation. In addition, this same analysis is used to show that the examples, due to Pedersen (Trans Am Math Soc 336:69–78, 1993), of potentially volume-minimizing rectifiable sections (rectifiable foliations) of the unit tangent bundle to S ²ⁿ⁺¹ are not, in fact, volume minimizing.      

On Martos' and Chaehes-Coopeb's approach vis-a-vis

In this paper a linear fractional programming problem is studied in presence of “singular-points”. It is proved that “singular points”, if present, exist at an extreme point of S: = {x ? R ⁿ | Ax = b, x ≧0}

It is also shown that a “singular point” is adjacent to an optimal point of S and a characterization of a non-basic vector is obtained, whose entry into the optimal basis in Martos' approach yields the “singular point”.     

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C^∞, and let g_ij = δ_ij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W^2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C³ away from a certain singular set Σ and C^∞ away from a larger singular set Σ ∪ Σ₀. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ₀. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc.