COUPLING OF MULTIDIMENSIONAL PARABOLIC AND HYPERBOLIC EQUATIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 53-80 ◽  
Author(s):  
GLORIA AGUILAR ◽  
LAURENT LÉVI ◽  
MONIQUE MADAUNE-TORT
Keyword(s):  
Hyperbolic Equations ◽  
Type Equation ◽  
Entropy Inequality ◽  
Order Operator ◽  
Viscosity Method ◽  
Vanishing Viscosity ◽  
Reaction Type ◽  
First Order ◽  
Definition Of ◽  
Uniqueness Proof

This paper deals with the mathematical analysis of a quasilinear parabolic-hyperbolic problem in a multidimensional bounded domain Ω. In a region Ωp a diffusion-advection-reaction type equation is set, while in the complementary Ωh ≡ Ω\Ωp, only advection-reaction terms are taken into account. To begin we provide a definition of a weak solution through an entropy inequality on the whole domain. Since the interface ∂Ωp ∩ ∂Ωh contains outward characteristics for the first-order operator in Ωh, the uniqueness proof starts by considering first the hyperbolic zone and then the parabolic one. The existence property uses the vanishing viscosity method and to pass to the limit on the hyperbolic zone, we refer to the notion of process solution.

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.

On the Strong Stability of First-Order Operator-Differential Equations

2004 ◽  
Vol 40 (5) ◽  
pp. 703-710 ◽  
Author(s):  
D. R. Bojovic ◽  
B. S. Jovanovic ◽  
P. P. Matus
Keyword(s):  
Differential Equations ◽  
Strong Stability ◽  
Order Operator ◽  
First Order

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 NEW L1-GRADIENT TYPE ESTIMATES OF SOLUTIONS TO ONE-DIMENSIONAL QUASILINEAR PARABOLIC SYSTEMS

2010 ◽  
Vol 12 (01) ◽  
pp. 85-106 ◽  
Author(s):  
S. N. ANTONTSEV ◽  
J. I. DÍAZ
Keyword(s):  
Type Equation ◽  
Parabolic Type ◽  
Parabolic Systems ◽  
Diffusion Term ◽  
One Dimensional ◽  
First Order ◽  
Estimates Of Solutions ◽  
Gradient Type ◽  
Degenerate Type ◽  
Parabolic Type Equation

We consider a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of the degenerate type. We derive some new L1-gradient type estimates for its solutions which are uniform in the sense that they do not depend on the coefficients nor on the size of the spatial domain. We also give some applications of such estimates to gas dynamics, filtration problems, a p-Laplacian parabolic type equation and some first order systems of Hamilton–Jacobi or conservation laws type.

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.

scholarly journals Formulae of partial reduction for linear systems of first order operator equations

2010 ◽  
Vol 23 (11) ◽  
pp. 1367-1371 ◽  
Author(s):  
Branko Malešević ◽  
Dragana Todorić ◽  
Ivana Jovović ◽  
Sonja Telebaković
Keyword(s):  
Linear Systems ◽  
Partial Reduction ◽  
Order Operator ◽  
Operator Equations ◽  
First Order

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.

Sign in / Sign up

Export Citation Format

Share Document