Principal-value integrals – Revisited

2005 ◽  
Vol 83 (5) ◽  
pp. 565-575 ◽  
Author(s):  
Scott M Cohen ◽  
K TR Davies ◽  
R W Davies ◽  
M Howard Lee
Keyword(s):  
Path Integral ◽  
Complex Variable ◽  
Finite Limit ◽  
Specific Method ◽  
Basic Definition ◽  
First Order ◽  
Improper Integral ◽  
Principal Value ◽  
Definition Of ◽  
Finite Result

The principal-value (PV) integral has proved a useful tool in many fields of physics. The PV is a specific method for obtaining a finite result for an improper integral. When the integration passes through a simple pole, one speaks of a "first-order" PV. In this paper, we examine first-order PV integrals and analyze several of their important properties. First, we discuss how the PV agrees with one's naïve expectation about these integrals. Next, we show that the basic definition of the first-order PV gives a generalized formula for the complex-variable PV expression. Finally, we show the correspondence between the finite-limit PV integral of x–1 along the real axis and the path integral of z–1 (where z = x + iy) in the complex plane.PACS Nos.: 02.90.+p, 05.90.+m

scholarly journals On A. D. Sakharov’s Hypothesis of Cosmological Transitions with Changes in the Signature of the Metric

Universe ◽  
2021 ◽  
Vol 7 (5) ◽  
pp. 151
Author(s):  
Tatyana P. Shestakova
Keyword(s):  
Path Integral ◽  
Riemannian Manifolds ◽  
Integral Approach ◽  
Metric Tensor ◽  
Time Coordinate ◽  
Principal Value ◽  
Definition Of ◽  
The Universe ◽  
Philosophical Questions ◽  
Gauge Conditions

The paper discusses possible consequences of A. D. Sakharov’s hypothesis of cosmological transitions with changes in the signature of the metric, based on the path integral approach. This hypothesis raises a number of mathematical and philosophical questions. Mathematical questions concern the definition of the path integral to include integration over spacetime regions with different signatures of the metric. One possible way to describe the changes in the signature is to admit time and space coordinates to be purely imaginary. It may look like a generalization of what we have in the case of pseudo-Riemannian manifolds with a non-trivial topology. The signature in these regions can be fixed by special gauge conditions on components of the metric tensor. The problem is what boundary conditions should be imposed on the boundaries of these regions and how they should be taken into account in the definition of the path integral. The philosophical question is what distinguishes the time coordinate among other coordinates but the sign of the corresponding principal value of the metric tensor. In particular, there is an attempt in speculating how the existence of the regions with different signature can affect the evolution of the Universe.

scholarly journals DESCRIPTION OF NATURAL SEA STATES: REQUIREMENTS TO THE REPRODUCTION IN MODELS

1984 ◽  
Vol 1 (19) ◽  
pp. 34
Author(s):  
H. Lundgren ◽  
Elias Davidsen
Keyword(s):  
Numerical Models ◽  
Basic Definition ◽  
One Dimensional ◽  
Directional Spectrum ◽  
First Order ◽  
Definition Of ◽  
Order Components ◽  
Jonswap Spectrum

In order to attain a satisfactory reproduction in physical and numerical models of Nature's sea states, much development of the description of these sea states has to take place. This paper gives a discussion (i) of the problems that can be considered as solved today, and (ii) of the problem areas where much research is still needed. Firstly, the paper presents the basic definition of the directional spread of energy (for a given frequency) as derived from a field record. This definition is necessarily a statistical one and leads directly to the determination in practice of the directional spectrum. Secondly, the parameterization of spectra is discussed and it is proposed to express one-dimensional storm spectra by means of the four -parameter F.1 .-spectrum, which is more manageable than the JONSWAP-spectrum. Thirdly, it is concluded that the first order components of any spectrum have random phases.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

scholarly journals Solving puzzles in deformed JT gravity: phase transitions and non-perturbative effects

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Clifford V. Johnson ◽  
Felipe Rosso
Keyword(s):  
Phase Structure ◽  
Scalar Potential ◽  
String Equation ◽  
Two Dimensional ◽  
Leading Order ◽  
Classical Analysis ◽  
First Order ◽  
First Order Phase Transition ◽  
Definition Of ◽  
First Order Phase

Abstract Recent work has shown that certain deformations of the scalar potential in Jackiw-Teitelboim gravity can be written as double-scaled matrix models. However, some of the deformations exhibit an apparent breakdown of unitarity in the form of a negative spectral density at disc order. We show here that the source of the problem is the presence of a multi-valued solution of the leading order matrix model string equation. While for a class of deformations we fix the problem by identifying a first order phase transition, for others we show that the theory is both perturbatively and non-perturbatively inconsistent. Aspects of the phase structure of the deformations are mapped out, using methods known to supply a non-perturbative definition of undeformed JT gravity. Some features are in qualitative agreement with a semi-classical analysis of the phase structure of two-dimensional black holes in these deformed theories.

scholarly journals The Use of Terms and Definitions in the Study of Be Stars (Review Paper)

1987 ◽  
Vol 92 ◽  
pp. 3-21
Author(s):  
George W. Collins
Keyword(s):  
Review Paper ◽  
Be Stars ◽  
Basic Definition ◽  
The Subject ◽  
Definition Of ◽  
Incorrect Usage

AbstractIn this paper I shall examine the use and misuse of some astronomical terminology as it is commonly found in the literature. The incorrect usage of common terms, and sometimes the terms themselves, can lead to confusion by the reader and may well indicate misconceptions by the authors. A basic definition of the Be phenomena is suggested and other stellar characteristics whose interpretation may change when used for non-spherical stars, is discussed. Special attention is paid to a number of terms whose semantic nature is misleading when applied to the phenomena they are intended to represent. The use of model-dependent terms is discussed and some comments are offered which are intended to improve the clarity of communication within the subject.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

scholarly journals Macroscopic degeneracy and order in the 3D plaquette Ising model

2015 ◽  
Vol 29 (20) ◽  
pp. 1550109 ◽  
Author(s):  
Desmond A. Johnston ◽  
Marco Mueller ◽  
Wolfhard Janke
Keyword(s):  
Low Temperature ◽  
Magnetic Order ◽  
Finite Size ◽  
Temperature Phase ◽  
Finite Size Scaling ◽  
First Order ◽  
Magnetic Order Parameter ◽  
First Order Transition ◽  
Definition Of ◽  
Low Temperature Phase

The purely plaquette 3D Ising Hamiltonian with the spins living at the vertices of a cubic lattice displays several interesting features. The symmetries of the model lead to a macroscopic degeneracy of the low-temperature phase and prevent the definition of a standard magnetic order parameter. Consideration of the strongly anisotropic limit of the model suggests that a layered, “fuki-nuke” order still exists and we confirm this with multi-canonical simulations. The macroscopic degeneracy of the low-temperature phase also changes the finite-size scaling corrections at the first-order transition in the model and we see this must be taken into account when analyzing our measurements.

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

Modelling spatial reasoning systems with shape algebras and formal logic

1997 ◽  
Vol 11 (4) ◽  
pp. 273-285 ◽  
Author(s):  
Scott C. Chase
Keyword(s):  
Spatial Reasoning ◽  
Predicate Logic ◽  
Spatial Relations ◽  
Large Set ◽  
First Order ◽  
Reasoning Systems ◽  
Geometric Objects ◽  
Small Set ◽  
Definition Of ◽  
High Level

AbstractThe combination of the paradigms of shape algebras and predicate logic representations, used in a new method for describing designs, is presented. First-order predicate logic provides a natural, intuitive way of representing shapes and spatial relations in the development of complete computer systems for reasoning about designs. Shape algebraic formalisms have advantages over more traditional representations of geometric objects. Here we illustrate the definition of a large set of high-level design relations from a small set of simple structures and spatial relations, with examples from the domains of geographic information systems and architecture.

Sign in / Sign up

Export Citation Format

Share Document