Functions of perturbed operators

We prove that if 0<\a<1 and $f$ is in the H\"older class \L_\a(\R), then for arbitrary self-adjoint operators $A$ and $B$ with bounded $A-B$, the operator $f(A)-f(B)$ is bounded and \|f(A)-f(B)\|\le\const\|A-B\|^\a. We prove a similar result for functions $f$ of the Zygmund class \L_1(\R): \|f(A+K)-2f(A)+f(A-K)\|\le\const\|K\|, where $A$ and $K$ are self-adjoint operators. Similar results also hold for all H\"older-Zygmund classes \L_\a(\R), \a>0. We also study properties of the operators $f(A)-f(B)$ for f\in\L_\a(\R) and self-adjoint operators $A$ and $B$ such that $A-B$ belongs to the Schatten--von Neumann class \bS_p. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions.Comment: 6 page

Computer program for calculating water and steam properties

Computer subprogram, WASP, accepts any two of pressure, temperature, and density as input conditions. Pressure and either entropy or enthalpy are also allowable input variables. This flexibility is especially useful in cycle analysis. Metastable calculations can also be made using WASP

Computer program for calculating thermodynamic and transport properties of fluids

Computer code has been developed to provide thermodynamic and transport properties of liquid argon, carbon dioxide, carbon monoxide, fluorine, helium, methane, neon, nitrogen, oxygen, and parahydrogen. Equation of state and transport coefficients are updated and other fluids added as new material becomes available

An Interesting Class of Operators with unusual Schatten-von Neumann behavior

We consider the class of integral operators Q_\f on $L^2(\R_+)$ of the form (Q_\f f)(x)=\int_0^\be\f (\max\{x,y\})f(y)dy. We discuss necessary and sufficient conditions on $\phi$ to insure that $Q_{\phi}$ is bounded, compact, or in the Schatten-von Neumann class \bS_p, $1<p<\infty$. We also give necessary and sufficient conditions for $Q_{\phi}$ to be a finite rank operator. However, there is a kind of cut-off at $p=1$, and for membership in \bS_{p}, $0<p\leq1$, the situation is more complicated. Although we give various necessary conditions and sufficient conditions relating to Q_{\phi}\in\bS_{p} in that range, we do not have necessary and sufficient conditions. In the most important case $p=1$, we have a necessary condition and a sufficient condition, using $L^1$ and $L^2$ modulus of continuity, respectively, with a rather small gap in between. A second cut-off occurs at $p=1/2$: if \f is sufficiently smooth and decays reasonably fast, then \qf belongs to the weak Schatten-von Neumann class \wS{1/2}, but never to \bS_{1/2} unless \f=0. We also obtain results for related families of operators acting on $L^2(\R)$ and $\ell^2(\Z)$. We further study operations acting on bounded linear operators on $L^{2}(\R^{+})$ related to the class of operators Q_\f. In particular we study Schur multipliers given by functions of the form $\phi(\max\{x,y\})$ and we study properties of the averaging projection (Hilbert-Schmidt projection) onto the operators of the form Q_\f.Comment: 87 page

Defective Gut Function in \u3cem\u3eDrop-Dead\u3c/em\u3e Mutant \u3cem\u3eDrosophila\u3c/em\u3e

Mutation of the gene drop-dead (drd ) causes adult Drosophila to die within 2 weeks of eclosion and is associated with reduced rates of defecation and increased volumes of crop contents. In the current study, we demonstrate that flies carrying the strong allele drdlwf display a reduction in the transfer of ingested food from the crop to the midgut, as measured both as a change in the steady-state distribution of food within the gut and also in the rates of crop emptying and midgut filling following a single meal. Mutant flies have abnormal triglyceride (TG) and glycogen stores over the first 4 days post-eclosion, consistent with their inability to move food into the midgut for digestion and nutrient absorption. However, the lifespan of mutants was dependent upon food presence and quality, suggesting that at least some individual flies were able to digest some food. Finally, spontaneous motility of the crop was abnormal in drdlwf flies, with the crops of mutant flies contracting significantly more rapidly than those of heterozygous controls. We therefore hypothesize that mutation of drd causes a structural or regulatory defect that inhibits the entry of food into the midgut

Functions of operators under perturbations of class \bS_p

This is a continuation of our paper \cite{AP2}. We prove that for functions $f$ in the H\"older class \L_\a(\R) and 1, the operator $f(A)-f(B)$ belongs to \bS_{p/\a}, whenever $A$ and $B$ are self-adjoint operators with A-B\in\bS_p. We also obtain sharp estimates for the Schatten--von Neumann norms \big\|f(A)-f(B)\big\|_{\bS_{p/\a}} in terms of \|A-B\|_{\bS_p} and establish similar results for other operator ideals. We also estimate Schatten--von Neumann norms of higher order differences \sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big). We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on $f$ for $f(A)-f(B)$ to belong to \bS_q under the assumption that A-B\in\bS_p. We also obtain Schatten--von Neumann estimates for quasicommutators $f(A)Q-Qf(B)$, and introduce a spectral shift function and find a trace formula for operators of the form $f(A-K)-2f(A)+f(A+K)$.Comment: 49 page