On a generalized Minkowski inequality and its relation to dominates for t-norms

Leth:?⁺ → ?⁺ be a continuous strictly increasing function withh(0) = 0. Such functionsh give rise to a generalization of the Minkowski inequality; namely, (1) $$h^{ - 1} (h(a + b) + h(c + d)) \leqq h^{ - 1} (h(a + c) + h(b + d))$$ wherea, b, c, andd are arbitrary non-negative real numbers. Theorem 1 shows that, ifh and logh′ (e ^x ) are both convex functions, thenh satisfies (1). Theorem 2, the major result, demonstrates that, if bothh ₁ andh ₂ satisfy the hypotheses of Theorem 1, then the composition ofh ₁ withh ₂ also satisfies the hypotheses of Theorem 1 and hence the inequality (1). The remainder of the paper shows how (1) and Theorems 1 and 2 impinge on the dominates relation for strict t-norms. In particular, Theorem 3 shows how (1) can be interpreted as equivalent to the dominates relation for two strict t-norms. Theorem 4 shows how to use Theorems 1 and 3 to construct a strict t-norm which dominates a given strict t-norm. And, Theorem 5 shows how Theorem 2 can be used to give a qualified answer of yes to the open question of whether or not the dominates relation is a transitive relation.     

The generalized analytic signal

This paper is an explication of the analytic signal in the generalized case, i.e., the analytic signal of a generalized function and of a generalized stochastic process. The contributions of the author are: (1) an L²-theory of distributions which, in the study of the analytic signal, has an advantage over the usual Schwartz-Itô-Gel'fand theory because the Cauchy representation is defined; (2) a proof (Theorem 2.5) that the Schwartz distributions δ, δ⁺, δ^? and ? may be extended to the L₂-case, expressions (Theorems 2.6 and 2.7) for their Hilbert and Fourier transforms in the L₂-case, and expressions (Section 2.1) for their analytic signals; (3) a proof (Theorem 3.3) that an orthogonal L₂-process, and therefore the Fourier transform of a second-order stationary stochastic process (Theorem 3.4), is strictly generalized; (4) a representation theorem (Theorem 3.5) which extends the Itô spectral representation theorem for stationary random distributions to the nonspectral, nonstationary, L₂-case; (5) expressions for the Cauchy representation (Theorem 3.6) and the analytic signal (Theorem 3.7) of an L₂-process; (6) an expression for and the covariance kernel of the analytic signal of white noise (Section 3.4). The word application in the text refers to the application of previously developed concepts.     

Quasilinear asymptotically periodic Schrödinger equations with subcritical growth

In this article the existence of one solution for a class of asymptotically periodic equations in the euclidean space is established. The basic tools employed here are the Mountain Pass Theorem and the Concentration-Compactness Principle. By using a change of variable, the quasilinear equation is reduced to a semilinear equation, whose respective associated functional is well defined in H¹(R^N) and satisfies the geometric hypotheses of the Mountain Pass Theorem.     

A noncommutative Brooks-Jewett Theorem

In classical measure theory the Brooks-Jewett Theorem provides a finitely-additive-analogue to the Vitali-Hahn-Saks Theorem. In this paper, it is studied whether the Brooks-Jewett Theorem allows for a noncommutative extension. It will be seen that, in general, a bona-fide extension is not valid. Indeed, it will be shown that a C^*-algebra A satisfies the Brooks-Jewett property if, and only if, it is Grothendieck, and every irreducible representation of A is finite-dimensional; and a von Neumann algebra satisfies the Brooks-Jewett property if, and only if, it is topologically equivalent to an abelian algebra.     

Essential and inessential points of local nonconvexity

LetS be a closed connected subset of a Hausdorff linear topological space,Q the points of local nonconvexity ofS, E the essential members ofQ, N the inessential. IfS～Q is connected, then the following are true: Theorem 1.If Q ?Øis countable, then S is planar. Theorem 2.If Q is finite and nonempty, then cardE≧cardN+1. Theorem 3.If SυR ² and N is infinite, then E is infinite.     

Fixed points of projectivities of prime order

This work begins with a review of the classical results for fixed points of projectivities in a projective plane over a general commutative field. The second section of this work features all the material necessary to prove the main result, which is presented in Theorem 2.8. It is shown that, in a finite projective plane of order q, there exists a projectivity g? of prime order p?>?3 if and only if p divides exactly one of the integers q ? 1, q, q?+?1, q ² + q + 1. Theorem 2.8 establishes a correspondence between the possible structures of points fixed by g?, as presented in Theorem 1.3, and the integer that is divisible by p. The special case of p = 2 is handled in Sect. 2.1, where it is shown that every involution is a harmonic homology for q odd and an elation for q even. The special case of p?=?3 is handled in Sect. 2.2, and Theorem 2.8 is adapted for p?=?3 and presented as Theorem 2.15. An application of Theorems 2.8 and 2.15 is determining the sizes of (n, r)-arcs that are stabilized by projectivities of prime order p in the finite projective plane of order q; in Sect. 3, this application is presented in Propositions 3.2 and 3.3.     

Manifolds accepting codimension-one- sphere-shape decompositions

A decomposition of a metric space is said to be CS^k-shape if each of its members is a compactum shape equivalent to a cohomology k-sphere. We will show that for m?2 every CS^m?1-shape decomposition of a closed m-manifold is upper semicontinuous (Theorem 3.1). Consequently, for m≠3, 4, 5, every connected closed m-manifold accepting an S^m?1-shape decomposition is homeomorphic to the total space of an (m?1)-sphere-fiber bundle over the circle (Theorem 4.2).     

Strongly ergodic Markov chains and rates of convergence using spectral conditions

For finite Markov chains the eigenvalues of P can be used to characterize the chain and also determine the geometric rate at which Pⁿ converges to Q in case P is ergodic. For infinite Markov chains the spectrum of P plays the analogous role. It follows from Theorem 3.1 that 6Pⁿ?Q6?Cβⁿ if and only if P is strongly ergodic. The best possible rate for β is the spectral radius of P?Q which in this case is the same as sup{|λ|: λ ? σ (P), λ ≠;1}. The question of when this best rate equals δ(P) is considered for both discrete and continous time chains. Two characterizations of strong ergodicity are given using spectral properties of P? Q (Theorem 3.5) and spectral properties of a submatrix of P (Theorem 3.16).     

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that homologically nontrivial generic smooth (2n−1)-parameter families of analytic discs in Cn, n?2, attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in C² as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in C² in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik-Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in Cn extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2).     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.     

A proof of the Borsuk Antipodal Theorem for Fredholm maps

With only the means of elementary analysis and the Smale-Sard lemma, a direct and self-contained proof of the Borsuk Antipodal Theorem for Fredholm maps is given. The discussion extends the arguments of J. C. Alexander and J. A. Yorke used in their analytical proof of the Antipodal Theorem in $\text{R}$ⁿ.     

A simple proof of the Prime Number Theorem

The Prime Number Theorem is proved using only properties of the Dirichlet series Σ_{n = 1}^∞n^?8 in its half plane of convergence, and simple facts of harmonic analysis.     

Sulla misurabilita' delle funzioni di piu' variabili

LetE be a Lebesgue measurable set in IR ^p+q andY a metric space. Iff:E→Y is such thatf(.,x) isL-measurable for almost allx andf(t,.) is continuous in each of theq variables separately for almost allt, thenf must beL-measurable (Theorem 1). By this result we deduce that a functionf:E→Y is almost-continuous iff it is almost-separately continuous. Finally, we give another characterization of the measurability of a functionf:IRp+q ^p+q→Y by means of properties of its sections (Theorem 2).     

Laurent expansion of an inverse of a function matrix

This paper suggests a general procedure based on the Taylor expansion of a function matrixF(z) for calculating the Laurent expansion ofF ^?1(z) around an isolated pole. It is shown that in order to compute thejth Laurent coefficient matrixB _j ofF ^?1(z), one needs in any case the Taylor coefficientsA ₀, A₁,..., A_2n+j ofF(z), wheren is the order of the pole. Theorem 1 helps to determine the order of the pole, while Theorem 2 shows also how the Laurent coefficients can be computed in the general case.     

A note on Hempel-McMillan coverings of 3-manifolds

Motivated by the concept of A-category of a manifold introduced by Clapp and Puppe, we give a different proof of a (slightly generalized) Theorem of Hempel and McMillan: If M is a closed 3-manifold that is a union of three open punctured balls then M is a connected sum of S³ and S²-bundles over S¹.     

The Kernel Theorem of Hilbert-Schmidt operators

All Hilbert-Schmidt operators acting on L²-sections of a vector bundle with fiber a separable Hilbert space H over a compact Riemannian manifold M, are characterized. This is achieved by defining the vector bundle of Hilbert-Schmidt operators on H, and then making use of a classical result known as the Kernel Theorem of Hilbert-Schmidt operators.     

Maximal regularity, the local inverse function theorem, and local well-posedness for the curve shortening flow

Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and L ²-maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented in the special situation only.     

Non-Liouville numbers and a Theorem of Hörmander

It is shown in this paper that Theorem 1 of [G. H. Meisters, “Translation-invariant linear forms and a formula for the Dirac measure,” J. Functional Analysis 8 (1971), 173–188] can be deduced from a very general result of Lars Hörmander, namely, Theorem 1 of “Generators for some rings of analytic functions” [Bull. Amer. Math. Soc.73 (1967), 943–949]. However, Hörmander's theorem is evidently not applicable in several other cases where Meisters'-type results have been obtained (e.g., Theorem 1 of G.H. Meisters and Wolfgang M. Schmidt, “Translation-invariant linear forms on L²(G) for compact abelian groups G,” J. Functional Analysis11 (1972), 407–424).     

Lipschitz Flow-box Theorem

A generalization of the Flow-box Theorem is proven. The assumption of a C¹ vector field f is relaxed to the condition that f be locally Lipschitz continuous. The theorem holds in any Banach space.     

Non-CLT groups of order pq 3

In this note we give a characterization of finite groups of order pq ³ (p, q primes) that fail to satisfy the Converse of Lagrange’s Theorem.