A new fuzzy compactness defined by fuzzy nets

Let (Ω, $\text{A}$, μ) be a probability space and let $\text{B}$ be a subsigma algebra of $\text{A}$. Let A= L_∞Ω, $\text{A}$, μ , let A= L_∞Ω, $\text{B}$, μ, and let f?A. It is shown that best L_∞-approximations of f by elements of B comprise an interval in B; that is, there exists $\text{f}\text{,}\text{f}\text{?B}$ such that a function g?B is a best L_∞-approximation to f if and only if $\text{f}\text{? g ?}\text{f}$ a.e. on Ω. The difference, $\text{f}\text{? f}$, of $\text{f}$ and f is completely characterized in terms of special sets that have been developed in [2]. Then it is established that the best best L_∞-approximation, f_{$\text{B}$,∞}, to f by elements of B is the average of $\text{f}$ and $\text{f}$, where the function f_{$\text{B}$,∞} is defined by f_{$\text{B}$,∞}(ω) lim_{p → ∞}f_$\text{B}$,P(ξ) and f_$\text{B}$,P denotes the best L_p-approximation to f elements of L_p(Ω, $\text{B}$, μ).     

Some pathologies in the Mackey analysis for a certain nonseparable group

Suppose G is a separable locally compact group and N is a closed normal subgroup. If the dual N? is smooth and the orbit space $\text{N}\text{?}\text{G}$ is smooth for the natural action of G on $\text{N}\text{?}\text{(L}{}^{\mathrm{x}}\text{(n) = L(xnx}{}^{\mathrm{?1}}\text{))}$, the method of G. W. Mackey (Acta Math.99 (1958), 265–311) gives a fairly simple procedure for constructing the dual ?. In this paper we examine an example which shows that the nonseparable case is much more complicated. In the example, N is abelian, $\text{N}\text{?}\text{G}$ is finite and even when the stabilizer is N there are many irreducible representations of G associated with the same orbit.     

On the grading numbers of direct products of chains

For a finite lattice L, denote by $\text{l}{}^1\text{(L)}$ and $\text{l}{}_1\text{(L)}$ respectively the upper length and lower length of L. The grading number g(L) of L is defined as $\text{g(L) = l}{}^1\text{(}\text{Sub}\text{(L))-l}{}_1\text{(}\text{Sub}\text{(L))}$ where Sub(L) is the sublattice-lattice of L. We show that if K is a proper homomorphic image of a distributive lattice L, then $\text{l}{}_1\text{(}\text{Sub}\text{(K)) < l}{}_1\text{(}\text{Sub}\text{(L))}$; and derive from this result, formulae for $\text{l}{}_1\text{(}\text{Sub}\text{(L))}$ and g(L) where L is a product of chains.     

Regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions in Lp

The equation $\text{Lu = ?;(x, u)}\text{on}\text{B × (0, ∞), B}$ bounded, smooth domain in $\text{R}$ⁿ with nonlinear boundary conditions $\text{?u}\text{?v}\text{= g(x, u)}\text{on}\text{?B × (0, ∞)}$ is studied, L being the uniformly parabolic operator with time independent coefficients. Under suitable conditions on the nonlinearities (that do not involve monotonicity) global existence, uniqueness, compactness of the orbits and certain regularizing effects of the semigroup are established. In the case that L is in divergence form it is shown that under generic conditions orbits tend, as t → + ∞, to some equilibrium and that the stable equilibria attract essentially (Baire category) the whole space L²(B).     

Inverse matrix representation with one triangular array

We describe a technique that permits the representation of the inverse of a matrix A with only one additional triangular array. Let $\text{L1A = U}$, with L lower and U upper triangular arrays of order N. Algorithms are presented that use A and L to compute the matrix-vector products $\text{A}{}^{-1}\text{1b}$ and $\text{b}{}^{\mathrm{T}}\text{1A}{}^{-1}$ with N² multiplications and additions. The array L can be computed, with N³/3 multiplications, with a technique that avoids the computation of U. Standard Gaussian elimination simultaneously computes L and U as follows: start with $\text{I1A = A}$, where I is the identity matrix; perform identical linear combinations of rows on I and on the right hand side array A; gradually transform I into L and A into U. At an intermediate stage, where A has not yet been fully triangularized, we have $\text{L′1A = U′.L′}$ and Ú represent one of the pairs of arrays present before each linear combination of rows is performed. The key observation is that we only need two elements of Ú to compute each linear combination of rows of ?. Compute them with a scalar product of the appropriate rows of ? and columns of A. Instead of storing the arrays Ú, recompute their few needed elements whenever necessary.     

Inner functions in the unit ball of Cn

Let B be the open unit ball of $\text{C}$ⁿ, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have $\text{¦u¦ = 1}$ a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let $\text{¦X¦}$ be the norm closure, in L^∞ = L^∞(S), of X. Some results: set I is dense in the unit ball of H^∞(B) in the compact-open topology. On $\text{S,}\text{Q}\text{?}\text{Q}$ is weak¹-dense in $\text{L}{}^{\mathrm{\infty }}\text{, ¦}\text{Q}\text{?}\text{Q¦}$ does not contain $\text{H}{}^{\mathrm{\infty }}\text{, C(S) ?¦}\text{Q}\text{?}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ ¦}\text{H}\text{?}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ L}{}^{\mathrm{\infty }}$. (When $\text{n = 1, ¦Q¦ = H}{}^{\mathrm{\infty }}\text{and}\text{¦}\text{Q}\text{?}\text{Q¦ = L}{}^{\mathrm{\infty }}$.) Every unimodular $\text{?}\text{?}\text{L}{}^{\mathrm{\infty }}$ is a pointwise limit a.e. of products $\text{u}\text{v}\text{?}\text{, u}\text{?}\text{I, ν}\text{?}\text{I}$. The zeros of every $\text{? ? 0}$ in the ball algebra (but not of every H^∞-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if $\text{∥?∥}{}_{\mathrm{\infty }}\text{< 1}$. However, $\text{?}\text{u}$ cannot be bounded in B if u ? I is non-constant.     

On r-potent linear mappings

Let L(E) be the set of all linear mappings of a vector space E. Let Z₊ be the set of all positive integers. A nonzero element ? in L(E) is called an r-potent if $\text{?}{}^{\mathrm{r}}\text{=?}\text{and}\text{?}{}^{\mathrm{i}}\text{≠?}\text{for}\text{1<i<r (i,r∈Z}{}_{\mathrm{+}}\text{)}$. We prove that $\text{S(E)= {?∈L(E): ?}\text{is singular}\text{}}$ is a semigroup generated by the set of all r-potents in S(E), where r is a fixed positive integer with 2?r?n=dim(E).     

A constructive approach to the solution of a class of boundary value problems of mixed type

In this article we discuss the solution of boundary value problems which are described by the linear integrodifferential equation $\text{?xu}\text{?t}\text{(t, x) + u(t, x) ?}\text{1}\text{π}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{∝}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{exp}\text{(?y}{}^2\text{) u(t, y) dy = 0}$, where t∈J?$\text{R}$, x∈$\text{R}$. We interpret the equation in functional form as an ordinary differential equation for the mapping u:J→L₂(R,μ), where L₂(R,μ) is a weighted L₂-space. Emphasis is on the constructive aspects of the solution and on finding representations of the relevant isomorphisms.     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

.     

Reconstruction of a factor from measures on Takesaki's unitary equivalence relation

Let $\text{Ol}$?L^∞(S, μ) be a maximal abelian subalgebra of the factor $\text{F}$ on separable Hilbert space with modular involution J. ($\text{Ol}$∪ J$\text{Ol}$J)″ is represented naturally as L^∞(S × S, λ). If Takesaki's unitary equivalence relation $\text{R}$ ? S × S is not λ-null, it is a measure groupoid. If it is conull, and ($\text{Ol}$∪ J$\text{Ol}$J)″ is maximal abelian, $\text{F}$ and $\text{Ol}$ are reconstructed by the σ-left regular representation procedure. Examples show that these hypotheses are not always satisfied. An application shows that the L^∞ spectrum of a properly infinite ergodic transformation is null with respect to the L² spectrum.     

Presentations for groups acting on simply-connected complexes

New expansions for global semigroup theory are developed. Many expansions have a left and a right version, each with specific (dual) properties; e.g., the Rhodes expansions ?^$\text{L}$, resp. ?^$\text{R}$, have unambiguous $\text{L}$-resp. $\text{R}$-order. In applications one sometimes needs expansions having both properties simultaneously; these can be constructed by alternately applying the left and the right expansion (possibly infinitely often) while keeping the same set of generators. Thus one obtains an expansion which is invariant under application of the old two expansions and thus has the properties of both (e.g., one obtains -+ with

, and so -+ has unambiguous $\text{L}$-and $\text{R}$-order). It is proved that, in the case of the Rhodes expansion, the new expansion is ‘close’ to the original semigroup; in particular (and this is the main result of the paper), ?⁺_A is finite (resp. finite $\text{J}$-above) if S is finite (resp. finite$\text{J}$-above).     

On some results of Strichartz and of Rallis and Schiffman

Let (i, H, E) and (j, K, F) be abstract Wiener spaces and let α be a reasonable norm on E ? F. We are interested in the following problem: is ($\text{i ? j, H}\text{}\text{?}\text{bo}{}_2\text{K, E}\text{}\text{?}\text{bo}{}_{\mathrm{\alpha F}}$) an abstract Wiener space ? The first thing we do is to prove that the setting of the problem is meaningfull: namely, i ? j is always a continuous one to one map from $\text{H}\text{}\text{?}\text{bo}{}_2\text{K}$ into $\text{E}\text{}\text{?}\text{bo}{}_{\mathrm{\alpha }}\text{F}$. Then we exhibit an example which shows that the answer cannot be positive in full generality. Finally we prove that if F=L^p(X,$\text{X}$,λ) for some σ-finite measure λ ? 0 then $\text{(i?j, H}\text{?}{}_2\text{K,L}{}^{\mathrm{p}}$(X,$\text{X}$,λ) is an abstract Wiener space. By-products are some new results on γ-radonifying operators, and new examples of Banach spaces and cross norms for which the answer is affirmative (in particular α = π the projective norm, and F=L¹(X,$\text{X}$,λ)).     

Square roots of elliptic operators

Consider an elliptic sesquilinear form defined on $\text{V}$ × $\text{V}$ by $\text{J[u, v] = ∫}{}_{\mathrm{\Omega }}\text{a}{}_{\mathrm{jk}}\text{?u}\text{?x}{}_{\mathrm{k}}\text{}\text{?}\text{t6v}\text{?x}{}_{\mathrm{j}}\text{+ a}{}_{\mathrm{k}}\text{?u}\text{?x}{}_{\mathrm{k}}\text{v}\text{?}\text{+ α}{}_{\mathrm{j}}\text{u}\text{}\text{?}\text{t6v}\text{?x}{}_{\mathrm{j}}\text{+ au}\text{v}\text{?}\text{dx}$, where $\text{V}$ is a closed subspace of $\text{H}{}^1\text{(Ω)}$ which contains $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$, Ω is a bounded Lipschitz domain in $\text{R}$ⁿ, $\text{a}{}_{\mathrm{jk}}\text{, a}{}_{\mathrm{k}}\text{, α}{}_{\mathrm{j}}\text{, a ? L}{}_{\mathrm{\infty }}\text{(Ω),}\text{and Re}\text{a}{}_{\mathrm{jk}}\text{ζ}{}_{\mathrm{k}}\text{ζ}{}_{\mathrm{j}}\text{? κ > 0}$ for all ζ?$\text{C}$ⁿ with ¦ζ¦ = 1. Let L be the operator with largest domain satisfying J[u, v] = (Lu, v) for all υ∈$\text{V}$. Then L + λI is a maximal accretive operator in $\text{L}{}_2\text{(Ω)}$ for λ a sufficiently large real number. It is proved that $\text{(L + λI)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is a bounded operator from $\text{V}$ to $\text{L}{}_2\text{(Ω)}$ provided mild regularity of the coefficients is assumed. In addition it is shown that if the coefficients depend differentiably on a parameter t in an appropriate sense, then the corresponding square root operators also depend differentiably on t. The latter result is new even when the forms J are hermitian.     

Singular perturbation problems for linear parabolic differential operators of arbitrary order

We consider the first initial-boundary value problem for $\text{(}\text{?u}\text{?t}\text{) + ?L}{}_1\text{u + L}{}_0\text{u = f(L}{}_0$ and L₁ are linear elliptic partial differential operators) and investigate the properties of u(x, t, ?) as ? ↓ 0 in the maximum norm. Special attention is paid to approximations obtained by the boundary layer method. We use a priori estimates.     

A new proof for the first-fit decreasing bin-packing algorithm

This paper provides a new proof of a classical result of bin-packing, namely the $\text{11}\text{9}$ performance bound for the first-fit decreasing algorithm. In bin-packing, a list of real numbers in (0,1] is to be packed into a minimal number of bins, each of which holds a total of at most 1. The first-fit decreasing (FFD) algorithm packs each number in order of nonincreasing size into the first bin in which it fits. In his doctoral dissertation, D. S. Johnson (“Near-Optimal Bin Packing Algorithms,” Doctoral thesis, MIT, Cambridge, Mass., 1973) proved that for every list L, $\text{FFD}\text{(L) ?}\text{11}\text{9}\text{OPT}\text{(L) + 4}$, where FFD(L) and OPT(L) denote the number of bins used by FFD and an optimal packing, respectively. Unfortunately, his proof required more than 100 pages! This paper contains a much shorter and simpler proof that $\text{FFD}\text{(L) ?}\text{11}\text{9}\text{OPT}\text{(L) + 3}$.     

Regularizing effects for ut + Aϑ(u) = 0 in L1

Various initial-boundary value problems and Cauchy problems can be written in the form $\text{du}\text{dt}\text{+ A?(u) = 0}$, where ?:R→$\text{R}$ is nondecreasing and A is the linear generator of strongly continuous nonexpansive semigroup e^?tA in an L¹ space. For example, if A = ?Δ (subject, perhaps, to suitable boundary conditions) we obtain equations arising in flow in a porous medium or plasma physics (depending on the choice of ?) while if $\text{A =}\text{?}\text{?x}$ acting in L¹($\text{R}$) we have a scalar conservation law. In this paper we show that if M, m > 0 and m?′² ? ν??′' ? M?′², where ν ? {1,?1}, then (roughly speaking), the norm of $\text{t}\text{du}\text{dt}$ may be estimated in terms of the initial data u₀ in L¹. Such estimates give information about the regularity of solutions, asymptotic behaviour, etc., in applications. Side issues, such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator A?, are also dealt with. These considerations are of independent interest.     

On a valence problem in extremal graph theory

Let L ≠ Kinp be a p-chromatic graph and e be an edge of L such that L ? e is (p?1)-chromatic If Gⁿ is a graph of n vertices without containing L but containing K_p, then the minimum valence of Gⁿ is

$\text{?n}\text{1?}\text{1}\text{p?}\text{3}\text{2}\text{+}\text{O}\text{(1).}$

     

Invariant subspaces of analytic multiparticle Hamiltonians

The quantum mechanics of n particles interacting through analytic two-body interactions can be formulated as a problem of functional analysis on a Hilbert space $\text{G}$ consisting of analytic functions. On $\text{G}$, there is an Hamiltonian H with resolvent R(λ). These quantities are associated with families of operators H(?) and R(λ, ?) on $\text{L}$, the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) consists of possible isolated points, plus a number of half-lines starting at the thresholds of scattering channels and making an angle 2? with the real axis.Assuming that the two-body interactions are in the Schmidt class on the two-particle space $\text{G}$, this paper studies the resolvent R(λ, ?) in the case ? ≠ 0. It is shown that a well known Fredholm equation for R(λ, ?) can be solved by the Neumann series whenever ¦λ¦ is sufficiently large and λ is not on a singular half-line. Owing to this, R(λ, ?) can be integrated around the various half-lines to yield bounded idempotent operators P_p(?) (p = 1, 2,…) on $\text{L}$. The range of P_p(?) is an invariant subspace of H(?). As ? varies, the family of operators P_p(?) generates a bounded idempotent operator P_p on a space $\text{G}$. The range of this is an invariant subspace of H. The relevance of this result to the problem of asymptotic completeness is indicated.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.