A cohomological description of massive fields

1981 ◽  
Vol 378 (1772) ◽  
pp. 141-154 ◽  
Keyword(s):  
Differential Equations ◽  
Cohomology Group ◽  
Rest Mass ◽  
Holomorphic Functions ◽  
Basic Properties ◽  
Free Fields ◽  
Time Required ◽  
Twistor Description ◽  
Positive Frequency ◽  
Massive Fields

In this paper we present a generalization of the theory of the P -transform that encompasses the n -twistor description of massive fields. Attention is devoted to the two-twistor description, for which it is shown that the cohomology group H 2 ( P + 3 x P + 3 , O m, s ( — ξ — 2, — ŋ — 2)) is naturally isomorphic to the space of positive-frequency free fields of mass m and spin s , provided s — 1/2 | ξ — ŋ | is a non-negative integer, and vanishes otherwise. The sheaf O m, s ( — ξ — 2, — ŋ — 2) is a subsheaf of the standard sheaf of twisted holomorphic functions O ( — ξ — 2, — ŋ — 2) on P + 3 x P + 3 and satisfies a pair of differential equations determining the mass and the spin. In establishing these results extensive use is made of a certain class of two-point fields on space-time, required to be of positive frequency and of zero rest mass in each variable separately, and also subject to a condition of definite total mass and total spin. Such fields are of considerable interest in their own right, for example in connection with the theory of twistor diagrams, and in this paper we formulate a number of their basic properties.

PENROSE TRANSFORM FOR INDEFINITE GRASSMANN MANIFOLDS

2011 ◽  
Vol 22 (01) ◽  
pp. 47-65
Author(s):  
HIDEKO SEKIGUCHI
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unitary Group ◽  
Cohomology Group ◽  
Grassmann Manifold ◽  
Symmetric Domain ◽  
Holomorphic Functions ◽  
Bounded Symmetric Domain ◽  
Dolbeault Cohomology ◽  
Penrose Transform

We construct the Radon–Penrose transform as an intertwining operator of the indefinite unitary group U(p,q), from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of positive k-planes to the space of holomorphic functions over the bounded symmetric domain. We prove that the Penrose transform is injective, and that its image is exactly the space of global holomorphic solutions to the system of partial differential equations of determinant type of size k + 1.

scholarly journals Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

2019 ◽  
Vol 52 (1) ◽  
pp. 482-489 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv ◽  
Liana Smolovyk
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Positive Continuous Function ◽  
Partial Differential ◽  
Partial Derivatives ◽  
Fixed Direction

AbstractIn the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}},where L : ℂn → ℝ+ is a positive continuous function, {\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.

scholarly journals Some Schemata for Applications of the Integral Transforms of Mathematical Physics

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 254 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Integral Transform ◽  
First Integral ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Basic Properties ◽  
Survey Article ◽  
Laplace Integral ◽  
Generating Operators

In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-type operational calculi that are also discussed in the article. The general schemata for applications of the integral transforms of mathematical physics are illustrated on an example of the Laplace integral transform. Finally, the Mellin integral transform and its basic properties and applications are briefly discussed.

Using Pontryagin's Maximum Principle to Solve One-Dimensional Optimization Problems, with and without Constraints, on an Iterative Analog Computer

SIMULATION ◽  
1965 ◽  
Vol 4 (6) ◽  
pp. 382-389 ◽  
Author(s):  
Hans L. Steinmetz
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Time Domain ◽  
Pontryagin’S Maximum Principle ◽  
Analog Computer ◽  
Solution Time ◽  
Pontryagin's Maximum Principle ◽  
Computer Technique ◽  
One Dimensional ◽  
Time Required

An analog computer technique is presented which enables application of Pontryagin's maximum prin ciple to the problem of optimizing control systems. The key problem in using Pontryagin's maximum principle is the extremization of the Hamiltonian function at every instant of time. Since the analog computer is an excellent differential equation solver, it is of advantage to convert this task into a dynamic problem. The technique used to do this is based upon the steepest ascent method. The method is applied to a one-dimensional control problem; higher-di mensional control problems can be treated using the same approach. The argument that an analog computer can solve differential equations with only one independent variable, corresponding to machine time, is true only in a technical sense. In practice it is feasible for cer tain types of problems to integrate one set of differ ential equations sufficiently fast enough so that, while integrating another set of differential equations at a much slower rate, the solution error associated with this approach remains within acceptable limits. When using the analog computer in this way, one time domain always corresponds to the solution time required for solving the differential equations de scribing the system; a second time domain corre sponds to the solution time required for solving an auxiliary set of differential equations which has no direct relationship with the system. Technological improvements and innovations made in the analog computer field during the recent past have contributed to the successful application of this approach.

scholarly journals CRITICAL TIMESCALES AND TIME INTERVALS FOR COUPLED LINEAR PROCESSES

2013 ◽  
Vol 54 (3) ◽  
pp. 127-142 ◽  
Author(s):  
MATTHEW J. SIMPSON ◽  
ADAM J. ELLERY ◽  
SCOTT W. MCCUE ◽  
RUTH E. BAKER
Keyword(s):  
Differential Equations ◽  
Applied Mathematics ◽  
Finite Measure ◽  
Single Species ◽  
Diffusion Reaction ◽  
Morphogen Gradient ◽  
Time Intervals ◽  
Linear Processes ◽  
Gradient Formation ◽  
Time Required

AbstractIn 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. Sample and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?”Biophys. J. 99(2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for example the analysis of models describing coupled chemical decay and cell differentiation processes.

scholarly journals ON THE ANNIHILATOR OF A DOLBEAULT GROUP

2010 ◽  
Vol 21 (03) ◽  
pp. 317-331
Author(s):  
IMRE PATYI
Keyword(s):  
Holomorphic Function ◽  
Open Subset ◽  
Cohomology Group ◽  
Holomorphic Functions ◽  
Dolbeault Cohomology ◽  
Infinite Dimensional ◽  
Continuous Character

We show that any Dolbeault cohomology group Hp,q(D), p ≥ 0, q ≥ 1, of an open subset D of a closed finite codimensional complex Hilbert submanifold of ℓ2 is either zero or infinite dimensional. We also show that any continuous character of the algebra of holomorphic functions of a closed complex Hilbert submanifold M of ℓ2 is induced by its evaluation at a point of M. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of ℓ2 admits a nowhere critical holomorphic function.

NEW TWISTOR DIAGRAMS FOR MASSLESS FREE FIELDS OF ARBITRARY SPIN

1993 ◽  
Vol 08 (14) ◽  
pp. 2437-2446 ◽  
Author(s):  
J. G. CARDOSO
Keyword(s):  
Rest Mass ◽  
Arbitrary Spin ◽  
Contour Integral ◽  
Integral Formulas ◽  
Free Fields

A method whereby the integral of a holomorphic projective twistor one-form may be re-expressed as the integral of an associated holomorphic projective twistor three-form is suggested. This method is used to represent the Penrose universal contour integral formulas for spinning zero-rest-mass free fields in terms of twistor diagrams.

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