Boundedness of planar jump discontinuities for homogeneous hyperbolic systems

2021 ◽  
Vol 18 (03) ◽  
pp. 511-538
Author(s):  
Jeffrey Rauch
Keyword(s):  
Source Function ◽  
Hyperbolic Systems ◽  
Partial Differential Operator ◽  
Real Analytic Function ◽  
Characteristic Variety ◽  
Jump Discontinuities ◽  
Conic Neighborhood ◽  
Conormal Variety ◽  
Real Analytic ◽  
Uniformly Bounded

Suppose that [Formula: see text] is a homogeneous constant coefficient strongly hyperbolic partial differential operator on [Formula: see text] and that [Formula: see text] is a characteristic hyperplane. Suppose that in a conic neighborhood of the conormal variety of [Formula: see text], the characteristic variety of [Formula: see text] is the graph of a real analytic function [Formula: see text] with [Formula: see text] identically equal to zero or the maximal possible value [Formula: see text]. Suppose that the source function [Formula: see text] is compactly supported in [Formula: see text] and piecewise smooth with singularities only on [Formula: see text]. Then the solution of [Formula: see text] with [Formula: see text] for [Formula: see text] is uniformly bounded on [Formula: see text]. Typically when [Formula: see text] on the conormal variety, the sup norm of the jump in the gradient of [Formula: see text] across [Formula: see text] grows linearly with [Formula: see text].

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

Inversion of the geomagnetic induction problem

1970 ◽  
Vol 315 (1521) ◽  
pp. 185-194 ◽  
Keyword(s):  
Frequency Spectrum ◽  
Real Data ◽  
Real Analytic Function ◽  
Numerical Application ◽  
Geomagnetic Induction ◽  
Induction Equation ◽  
Real Analytic ◽  
Principle Of Causality ◽  
Magnetic Potentials ◽  
Magnetic Induction Equation

An algorithm has been found for inverting the problem of geomagnetic induction in a con­centrically stratified Earth. It determines the (radial) conductivity distribution from the frequency spectrum of the ratio of internal to external magnetic potentials of any surface harmonic mode. The derivation combines the magnetic induction equation with the principle of causality in the form of an integral constraint on the frequency spectrum. This algorithm generates a single solution for the conductivity. This solution is here proved unique if the conductivity is a bounded, real analytic function with no zeros. Suggestions are made regarding the numerical application of the algorithm to real data.

On the convergence to equilibrium states for certain non-hyperbolic systems

1997 ◽  
Vol 17 (4) ◽  
pp. 977-1000 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Periodic Orbits ◽  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Markov Property ◽  
Standard Technique ◽  
Equilibrium States ◽  
Convergence To Equilibrium ◽  
Invariant Density ◽  
Frobenius Operator ◽  
Uniformly Bounded

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.

scholarly journals The Zero Set of a Real Analytic Function

2020 ◽  
Vol 107 (3-4) ◽  
pp. 529-530 ◽  
Author(s):  
B. S. Mityagin
Keyword(s):  
Analytic Function ◽  
Real Analytic Function ◽  
Zero Set ◽  
Real Analytic

scholarly journals A Construction of asymptotic solutions and the existence of smooth null-solutions for a class of non-Fuchsian partial differential operators

1997 ◽  
Vol 145 ◽  
pp. 125-142
Author(s):  
Takeshi Mandai
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Partial Differential Operator ◽  
Negative Integer ◽  
Asymptotic Solutions ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Real Analytic

Consider a partial differential operator(1.1) where K is a non-negative integer and aj,a are real-analytic in a neighborhood of (0, 0)

scholarly journals Topological invariants of germs of real analytic functions

1997 ◽  
Vol 39 (1) ◽  
pp. 85-89 ◽  
Author(s):  
Piotr Dudziński
Keyword(s):  
Real Analytic Function ◽  
Topological Invariant ◽  
Topological Invariants ◽  
Isolated Singularity ◽  
Simple Consequence ◽  
Milnor Fibre ◽  
Real Analytic Functions ◽  
Nonisolated Singularity ◽  
Real Analytic ◽  
Weighted Homogeneous Polynomial

Let f: (ℝn, 0)→ (ℝ,0) be a germ of a real analytic function. Let L and F(f) denote the link of f and the Milnor fibre of fc respectively, i. e., L = {x ∈ Sn−1 | f(x) = 0}, , where 0 ≤ ξ ≪ r ≪ 1, . In [2] Szafraniec introduced the notion of an -germ as a generalization of a germ defined by a weighted homogeneous polynomial satisfying some condition concerning the relation between its degree and weights (definition 1). He also proved that if f is an -germ (presumably with nonisolated singularity) then the number χ(F(f)/d mod 2 is a topological invariant of f, where χ(F(f)) is the Euler characterististic of F(f), and gave the formula for χ(L)/2 mod 2 (it is a well-known fact that F(L) is an even number). As a simple consequence he got the fact that χ(F(f)mod 2 is a topological invariant for any f, which is a generalization of Wall's result [3] (he considered only germs with an isolated singularity).

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