A Direct Proof of the Integral Formula for Arctangent

A direct proof is given for the fact that the integral formula for the XXX limit of the trace of the type I q-vertex operators satisfies the deformed Knizhnik-Zamolodchikov (dKZ) equation with level −4. We have also carried out one-time integration by taking the residue at infinity. As a corollary of these we can construct a family of the integral formulas for solutions to the dKZ equation. Another corollary is the integral formula for the correlation function of the inhomogeneous XXX model, whose number of integrals is less than that of the previously obtained correlator for the XXZ model.

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A Direct Proof of the Integral Formula for Arctangent

College Mathematics Journal ◽

10.2307/2686774 ◽

1989 ◽

Vol 20 (3) ◽

pp. 235 ◽

Cited By ~ 1

Author(s):

Arnold J. Insel

Keyword(s):

Integral Formula ◽

Direct Proof

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Integral Formula for Spectral Flow for -Summable Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-2017-030-2 ◽

2019 ◽

Vol 71 (2) ◽

pp. 337-379

Author(s):

Magdalena Cecilia Georgescu

Keyword(s):

Fredholm Operator ◽

Unbounded Operator ◽

Von Neumann Algebra ◽

Integral Formula ◽

Direct Proof ◽

Spectral Flow ◽

Fredholm Operators ◽

Von Neumann ◽

Bounded Perturbations ◽

The Ideal

AbstractFix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.

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Convergent-bceam diffraction studies on lead zirconate titanate Ceramics

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100143961 ◽

1986 ◽

Vol 44 ◽

pp. 482-483

Author(s):

M.L.A. Dass ◽

T.A. Bielicki ◽

G. Thomas ◽

T. Yamamoto ◽

K. Okazaki

Keyword(s):

Lead Zirconate Titanate ◽

Direct Proof ◽

Lead Zirconate ◽

Subsolidus Phase ◽

Pzt Ceramics ◽

Subsolidus Phase Diagram ◽

Zirconate Titanate ◽

Two Phases ◽

Cbd Method ◽

Titanate Ceramics

Lead zirconate titanate, Pb(Zr,Ti)O3 (PZT), ceramics are ferroelectrics formed as solid solutions between ferroelectric PbTiO3 and ant iferroelectric PbZrO3. The subsolidus phase diagram is shown in figure 1. PZT transforms between the Ti-rich tetragonal (T) and the Zr-rich rhombohedral (R) phases at a composition which is nearly independent of temperature. This phenomenon is called morphotropism, and the boundary between the two phases is known as the morphotropic phase boundary (MPB). The excellent piezoelectric and dielectric properties occurring at this composition are believed to.be due to the coexistence of T and R phases, which results in easy poling (i.e. orientation of individual grain polarizations in the direction of an applied electric field). However, there is little direct proof of the coexistence of the two phases at the MPB, possibly because of the difficulty of distinguishing between them. In this investigation a CBD method was found which would successfully differentiate between the phases, and this was applied to confirm the coexistence of the two phases.

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An integral formula for the even part of the texture function or « the apparition of the fπ and fω ghost distributions »

Journal de Physique ◽

10.1051/jphys:01982004302018900 ◽

1982 ◽

Vol 43 (2) ◽

pp. 189-195 ◽

Cited By ~ 6

Author(s):

Claude Esling ◽

Jacques Muller ◽

Hans-Joachim Bunge

Keyword(s):

Integral Formula

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THE HEXOSAMINE CONTENT OF THE ORBIT IN RELATION TO ANTI-THYROTROPHIN ANTIBODY TITRES IN SERA OF THYROTROPHIN-TREATED RABBITS WITH AND WITHOUT CORTISONE ADMINISTRATION

Acta Endocrinologica ◽

10.1530/acta.0.0410474 ◽

1962 ◽

Vol 41 (3) ◽

pp. 474-480 ◽

Cited By ~ 2

Author(s):

Otto Wegelius ◽

E. J. Jokinen

Keyword(s):

Connective Tissue ◽

Guinea Pigs ◽

Antibody Formation ◽

Direct Proof ◽

Significant Rise ◽

Extraocular Muscles ◽

Concomitant Treatment ◽

Antibody Titres ◽

Hexosamine Content ◽

Synergetic Action

ABSTRACT In all previous investigations on experimental exophthalmos, heterologous thyrotrophic pituitary extracts have been used. These protein hormones stimulate antihormone formation in the test animals. Cortisone has been reported to effectively block antibody formation. In addition, it has been shown to potentiate TSH-induced exophthalmos in guinea-pigs. With rabbits as test animals, the hexosamine content of the orbital tissues was determined and used as an index of exophthalmos development and at the same time the antibody titres in the sera were followed. TSH injections for six weeks led to a highly significant accumulation of hexosamine in the retrobulbar connective tissue and in the extraocular muscles, i. e. an increase of up to 400% as compared with the control animals. At the same time a significant rise in antihormonal titres was detectable in the sera. Concomitant treatment with cortisone brought about an equal or higher accumulation of hexosamine but significantly lower antibody titres. The known opposite peripheral actions of TSH and cortisone can be reconciled with the synergy in producing experimental exophthalmos by attributing the synergetic action of cortisone to the blocking of antihormone formation. If less antihormones are produced, the effect of TSH is enhanced. Our experiments do not provide direct proof for this hypothesis. High hexosamine values in the orbit and low antihormone titres in the serum are, however, concomitant phenomena.

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On terpenes. LXXXIX. Direct proof of the carbon skeleton of germacrone

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19582169 ◽

1958 ◽

Vol 23 (12) ◽

pp. 2169-2174 ◽

Cited By ~ 8

Author(s):

V. Herout ◽

M. Suchý

Keyword(s):

Direct Proof ◽

Carbon Skeleton

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ON p – q DUALITY AND EXPLICIT SOLUTIONS IN c ≤ 1 2D GRAVITY MODELS

International Journal of Modern Physics A ◽

10.1142/s0217751x95000577 ◽

1995 ◽

Vol 10 (08) ◽

pp. 1219-1236 ◽

Cited By ~ 17

Author(s):

S. KHARCHEV ◽

A. MARSHAKOV

Keyword(s):

String Theory ◽

Exact Solutions ◽

2D Gravity ◽

Integral Formula ◽

Integral Representations ◽

Gravity Models ◽

Explicit Solutions ◽

String Equation ◽

Duality Transformation

We study the role of integral representations in the description of nonperturbative solutions to c ≤ 1 string theory. A generic solution is determined by two functions, W(x) and Q(x), which behave at infinity like xp and xq respectively. The integral formula for arbitrary (p, q) models is derived, which explicitly realizes a duality transformation between (p, q) and (q, p) 2D gravity solutions. We also discuss the exact solutions to the string equation and reduction condition and present several explicit examples.

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