On operator-valued Fourier multipliers

We give a simpler proof of a result on operator-valued Fourier multipliers on Lp([0,2π]d;X) using an induction argument based on a known result when d=1.     

Harrington’s conservation theorem redone

Leo Harrington showed that the second-order theory of arithmetic WKL ₀ is -conservative over the theory RCA ₀. Harrington’s proof is model-theoretic, making use of a forcing argument. A purely proof-theoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cut-elimination argument.      

Global weak solutions of the wave map system to compact homogeneous spaces

We construct global weak solutions of the wave map problem in the class of maps with bounded energy, with values in an arbitrary compact homogeneous space, for arbitrary initial data inH _c ¹ . The proof proceeds by a ‘penalty approximation’ method, which generalizes J.Shatah's [5] argument for the case of maps with values in then-sphere. Supported in part by a grant from the National Science Foundation and Science Alliance.     

C1, 1 Regularity in semilinear elliptic problems

In this paper we give an astonishingly simple proof of C^{1, 1} regularity in elliptic theory. Our technique yields both new simple proofs of old results as well as new optical results. The setting we'll consider is the following. Let u be a solution to where B, is the unit ball in ?ⁿ, f(x, t) is a bounded Lipschitz function in x, and f_t′ is bounded from below. Then we prove that u ? C^{1, 1} (B_1/2). Our method is a simple corollary to a recent monotonicity argument due to Caffarelli, Jerison, and Kenig. © 2002 Wiley Periodicals, Inc.     

The strong partition relation on ω1 revisited

We give a new proof of the strong partition relation on ω₁, assuming the axiom of determinacy, which uses only a general argument not involving the complete analysis of a measure on ω₁. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients

This paper is devoted to the analysis of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on R⁺. We show global existence and Lipschitz continuity with respect to the model ingredients. In distinction to previous studies, where the L¹ norm was used, we apply the flat metric, similar to the Wasserstein W₁ distance. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Lipschitz continuous dependence with respect to the model coefficients and initial data and the uniqueness of the weak solutions are shown under the assumption on the Lipschitz continuity of the kinetic functions. The proof of this result is based on the duality formula and the Gronwall-type argument.     

On stationary flows of asymmetric fluids with heat convection

Existence, uniqueness and regularity of solutions of equations describing stationary flows of viscous incompressible isotropic fluids with an asymmetric stress tensor have been considered recently.⁵ In this paper we extend the results of Reference 5 to include heat convection in the hydrodynamic model. We show that the boundary value problem (1.1)–(1.6) below has solutions in appropriate Sobolev spaces, provided the viscosities v and c_a + c_d are sufficiently large. The proof is based on a fixed point argument. Moreover, we show that the solutions are unique if the heat conductivity κ is large enough.     

A proof of the Faber-Krahn inequality for the first eigenvalue of thep-Laplacian

Summary In this work we present an elementary proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian on bounded domains in ℝⁿ. Let λ₁ be the first eigenvalue and λ ₁ ^* be the first eigenvalue for the ball of the same volume. Then we show that λ₁=λ ₁ ^* iff the domain is a ball. Our proof makes considerable use of the corresponding Talenti's inequality and some well known properties of the first eigenfunction. Entrata in Redazione il 20 aprile 1998.     

Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second - Order Logic

We investigate the definability in monadic ∑₁¹ and monadic Π₁¹ of the problems REG_k, of whether there is a regular subgraph of degree k in some given graph, and XREG_k, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REG_k^k and XREG_k^k, respectively, for k ≥ 2 (all our graphs are undirected). Our motivation partly stems from the fact (which we prove here) that REG_k^k and XREG_k^k are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑₁¹ games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑₁¹ and monadic Π₁¹, and we compare the definability of these problems (in monadic ∑₁¹ and monadic Π₁¹ with their computational complexity (which varies from solvable using logspace to NP - complete).     

Quantitative relation between noise sensitivity and influences

A Boolean function f: {0,1} ⁿ → {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm [3] showed that if the sum of squares of inuences of f is close to zero then f must be noise sensitive. We show a quantitative version of this result which does not depend on n, and prove that it is tight for certain parameters. Our results hold also for a general product measure µ _p on the discrete cube, as long as log1/p?logn. We note that in [3], a quantitative relation between the sum of squares of the inuences and the noise sensitivity was also shown, but only when the sum of squares is bounded by n ^?c for a constant c. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand’s lemma, which easily generalizes in various directions, including non-monotone functions.     

The slow wave solution to the FitzHugh-Nagumo equations

This paper gives a new existence proof for a travelling wave solution to the FitzHugh-Nagumo equations, u_t = u_xx +f(u)?w, w _t = ? (u?γw). The proof uses a contraction mapping argument, and also shows that the solution (u, c, w) to the travelling wave equations, where c is the wave speed, converges as ? → 0⁺ to the solution to the equations having ?=0, c=0, and w=0.     

Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus

Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δ_pu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rⁿ with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rⁿ (see [10], [3] and [11]).     

Epsilon substitution method for [Π0 1, Π0 1]-FIX

We formulate epsilon substitution method for a theory [Π⁰₁, Π⁰₁]-FIX for two steps non-monotonic Π⁰₁ inductive definitions. Then we give a termination proof of the H-processes based on Ackermann [1].     

An anti-Ramsey theorem

An edge-coloring of a graph is a partition of the set of edges into color-classes such that no two edges in the same class are adjacent. A subsetA of the vertex set isantihomogeneous if all edges in the subgraph induced byA have different colors. We study the maximum size of antihomogeneous subsets of the complete graphK _n with respect to edge-colorings. The greedy algorithm shows that this size is always at least (2n)^1/3. On the other hand, we prove thatK _n has an edgecoloring admitting no antihomogeneous subset of size 8 (nlnn)^1/3, answering a question of V. Rödl. The proof is not constructive; it employs a probabilistic argument. Related problems concerning Sidon sets in groups are mentioned.     

The quantum mechanical virial theorem and the absence of positive energy bound states of Schrödinger operators

For a class of potentialsq which may become as singular asr ^?2 at the origin we show that the eigenfunctionsu of the Friedrichs extensionT _Fof (?Δ+q)?C ₀ ^∞ (R ⁿ /{0}) satisfy the virial theorem 2(?Δu,u)=(u,rq,u). Two proofs of this relation are given. One extends the Fock-Weidmann method of scale transformations, the other makes an argument by Finkelstein rigorous and can be viewed of as a substitute of the “commutator proof” of (*) employed in the physical literature. The virial theorem provides a convenient tool for proving the nonexistence of eigenvalues embedded in the continuum or the total absence of eigenvalues ofT _F.     

Sharp converse theorem ofL p -polynomial approximation

For algebraic polynomial approximation on [?1,1] the analogue of the Zygmund-Timan type converse Marchaud inequality is proved. These are the exact converse estimates inL ^p spaces when 1<p<∞. The proof forp>2 uses Hirschman's multiplier theory, while forp相似文献   

On a theorem of Ganley

LetD be a relative difference with parameters (n, n, n, 1) in an abelian groupG of even ordern ². By a result of Ganley [3],n is necessarily a power of 2 andG is a direct sum of copies of ₄. We present a simple (and much shorter) alternative proof of this result, based on a geometric argument and a simple characterisation of the groups in question.     

A note on a strong uniqueness theorem of strauss

In this paper we present a correction of the proof of a strong uniqueness theorem given by H. Strauss^[1] in 1992 on approximation by reciprocals of functions of an n-dimensional space span (u₁, …, u_n) satisfying coefficient constraints.     

Grothendieck's theorem for noncommutative C1-algebras,with an Appendix on Grothendieck's constants

We study a conjecture of Grothendieck on bilinear forms on a C¹-algebra $\text{Ol}$. We prove that every “approximable” operator from $\text{Ol}$ into $\text{Ol}$¹ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C¹-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant k_G.