Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper: . Decomposer equations:

f(f^*(x)f(y))=f(y),f(f(x)f_*(y))=f(x)

.Strong decomposer equations:

f(f^*(x)y)=f(y),f(xf_*(y))=f(x)

.Canceler equations:

f(f(x)y)=f(xy),f(xf(y))=f(xy),f(xf(y)z)=f(xyz)

, where f^*(x) f(x) = f (x) f_* (x) = x. In this paper we solve them and introduce the general solution of the decomposer and strong decomposer equations in the sets with a binary operation and semigroups respectively and also associative equations in arbitrary groups. Moreover we state some equivalent equations to them and study the relations between the above equations. Finally we prove that the associative equations and the system of strong decomposer and canceler equations do not have any nontrivial solutions in the simple groups.     

4.

Outer approximation algorithm for nondifferentiable optimization problems     

D. Q. Mayne  E. Polak 《Journal of Optimization Theory and Applications》1984,42(1):19-30

It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ⁿ can be expressed as minimizing max{g(x, y)|y X}, whereg is a support function forf[f(x) g(x, y), for ally X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.This research was supported by the Science and Engineering Research Council, UK, and by the National Science Foundation under Grant No. ECS-79-13148.     

5.

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey     

Alexander Bendikov 《P-Adic Numbers, Ultrametric Analysis, and Applications》2018,10(1):1-11

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

6.

Relatively Undistorted Waves. New Examples     

A. P. Kiselev 《Journal of Mathematical Sciences》2003,117(2):3945-3946

New solutions of the wave equation with three space variables of the form u = g(x,y,z,t)f(), where the functions g and = (x,y,z,t) are some specified functions and f is an arbitrary function of one variable, are presented. Bibliography: 4 titles.     

7.

On a multiplicative functional transformation arising in nonlinear filtering theory     

M. H. A. Davis 《Probability Theory and Related Fields》1980,54(2):125-139

Summary This paper concerns the nonlinear filtering problem of calculating estimates E[f(x_t)¦y _s, st] where {x _t} is a Markov process with infinitesimal generator A and {y _t} is an observation process given by dy _t=h(x_t)dt +dw_twhere {w _t} is a Brownian motion. If h(x_t) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup T _s,t ^y obtained by a certain y-dependent multiplicative functional transformation of the signal process {x _t}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is A _t ^y f=e^y(t)h(A– 1/2h²)(e^–y(t)h f).Work partially supported by the U.S. Department of Energy (Contract ET-76-C-01-2295) at the Massachusetts Institute of Technology     

8.

Controllability of Nonlinear Evolution Systems with Preassigned Responses     

Bian  W. M. 《Journal of Optimization Theory and Applications》1999,100(2):265-285

In this paper, we study the approximate controllability with preassigned responses of the nonlinear delay systems x(t)=A(t)x(t)+f(t, x(t), x((t)), u(t)) and L(x(t), x(t))=A(t)x(t)+f(t, x(t), x((t)), u(t)). The controllability is not governed by an associated linear system, but by conditions on f or A involving the domain of A(t). No compactness assumptions are imposed in the main results.     

9.

Separation of spin synchronized signals     

T. Higuchi  G. K. Crawford  R. J. Strangeway  C. T. Russell 《Annals of the Institute of Statistical Mathematics》1994,46(3):405-428

In the measurements of VLF electric fields with the Pioneer Venus spacecraft in sunlight, spin synchronized signals often dominate over the naturally generated emissions. We present a method to separate natural emissions from the several possible sources of noise. Our major objective by this method is not to remove all spin modulation, but to effectively subtract the background noise caused by the identifiable noise sources. Examination of the data shows that the background spin synchronized noise is quite sensitive to (n), the angle between the sense axis and the solar direction. We model the observed data asy(n)=w(n)t(n)f((n))+x(n), wheref() represents the phase response of the background noise andx(n) is the estimated natural emissions.t(n) andw(n) are the long-term trend component and time- and phase-independent component of the intensity of the background noise, respectively. The method to decomposey(n) is based on the Bayesian approach which has been recently applied to various inversion problems such as nonstationary time series modeling and image reconstruction. In this procedure, the estimated parametersw(n),t(n),f(), andx(n) can be determined automatically. We will describe the Bayesian scheme and its application to the Pioneer Venus VLF electric field data.     

10.

Singular Nonlinear (n − 1,1) Conjugate Boundary Value Problems     

Eloe  Paul W.  Henderson  Johnny 《Georgian Mathematical Journal》1997,4(5):401-412

Solutions are obtained for the boundary value problem, y ⁽ⁿ⁾ + f(x,y) = 0, y ⁽ⁱ⁾(0) = y(1) = 0, 0 i n – 2, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

11.

On theL 1 exact penalty function with locally lipschitz functions     

Liansheng Zhang  Zhijian Huang 《应用数学学报(英文版)》1988,4(2):145-153

In this paper, we discuss the following inequality constrained optimization problem (P) minf(x) subject tog(x)0,g(x)=(g ₁(x), ...,g _r (x)) , wheref(x),g _j (x)(j=1, ...,r) are locally Lipschitz functions. TheL ₁ exact penalty function of the problem (P) is (PC) minf(x)+cp(x) subject tox R ⁿ , wherep(x)=max {0,g ₁(x), ...,g _r (x)},c>0. We will discuss the relationships between (P) and (PC). In particular, we will prove that under some (mild) conditions a local minimum of (PC) is also a local minimum of (P).     

12.

Existence-uniqueness results for a class of singular boundary value problems-II     

R.K. Pandey  A.K. Verma 《Journal of Mathematical Analysis and Applications》2008,338(2):1387-1396

In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem −(p(x)y^′′(x))=q(x)f(x,y) for 0<x?b and y(0)=a, α₁y(b)+β₁y^′(b)=γ₁, where p(x) satisfies (i) p(x)>0 in (0,b), (ii) p(x)∈C¹(0,r), and for some r>b, (iii) is analytic in and q(x) satisfies (i) q(x)>0 in (0,b), (ii) q(x)∈L¹(0,b) and for some r>b, (iii) is analytic in with quite general conditions on f(x,y). Region for multiple solutions have also been determined.     

13.

On some composite functional inequalities     

Włodzimierz Fechner 《Aequationes Mathematicae》2010,79(3):307-314

The aim of the paper is to deal with the following composite functional inequalities

Outer approximation algorithm for nondifferentiable optimization problems

It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ⁿ can be expressed as minimizing max{g(x, y)|y X}, whereg is a support function forf[f(x) g(x, y), for ally X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.This research was supported by the Science and Engineering Research Council, UK, and by the National Science Foundation under Grant No. ECS-79-13148.     

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

Relatively Undistorted Waves. New Examples

New solutions of the wave equation with three space variables of the form u = g(x,y,z,t)f(), where the functions g and = (x,y,z,t) are some specified functions and f is an arbitrary function of one variable, are presented. Bibliography: 4 titles.     

On a multiplicative functional transformation arising in nonlinear filtering theory

Summary This paper concerns the nonlinear filtering problem of calculating estimates E[f(x_t)¦y _s, st] where {x _t} is a Markov process with infinitesimal generator A and {y _t} is an observation process given by dy _t=h(x_t)dt +dw_twhere {w _t} is a Brownian motion. If h(x_t) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup T _s,t ^y obtained by a certain y-dependent multiplicative functional transformation of the signal process {x _t}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is A _t ^y f=e^y(t)h(A– 1/2h²)(e^–y(t)h f).Work partially supported by the U.S. Department of Energy (Contract ET-76-C-01-2295) at the Massachusetts Institute of Technology     

Controllability of Nonlinear Evolution Systems with Preassigned Responses

In this paper, we study the approximate controllability with preassigned responses of the nonlinear delay systems x(t)=A(t)x(t)+f(t, x(t), x((t)), u(t)) and L(x(t), x(t))=A(t)x(t)+f(t, x(t), x((t)), u(t)). The controllability is not governed by an associated linear system, but by conditions on f or A involving the domain of A(t). No compactness assumptions are imposed in the main results.     

Separation of spin synchronized signals

In the measurements of VLF electric fields with the Pioneer Venus spacecraft in sunlight, spin synchronized signals often dominate over the naturally generated emissions. We present a method to separate natural emissions from the several possible sources of noise. Our major objective by this method is not to remove all spin modulation, but to effectively subtract the background noise caused by the identifiable noise sources. Examination of the data shows that the background spin synchronized noise is quite sensitive to (n), the angle between the sense axis and the solar direction. We model the observed data asy(n)=w(n)t(n)f((n))+x(n), wheref() represents the phase response of the background noise andx(n) is the estimated natural emissions.t(n) andw(n) are the long-term trend component and time- and phase-independent component of the intensity of the background noise, respectively. The method to decomposey(n) is based on the Bayesian approach which has been recently applied to various inversion problems such as nonstationary time series modeling and image reconstruction. In this procedure, the estimated parametersw(n),t(n),f(), andx(n) can be determined automatically. We will describe the Bayesian scheme and its application to the Pioneer Venus VLF electric field data.     

Singular Nonlinear (n − 1,1) Conjugate Boundary Value Problems

Solutions are obtained for the boundary value problem, y ⁽ⁿ⁾ + f(x,y) = 0, y ⁽ⁱ⁾(0) = y(1) = 0, 0 i n – 2, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

On theL 1 exact penalty function with locally lipschitz functions

In this paper, we discuss the following inequality constrained optimization problem (P) minf(x) subject tog(x)0,g(x)=(g ₁(x), ...,g _r (x)) , wheref(x),g _j (x)(j=1, ...,r) are locally Lipschitz functions. TheL ₁ exact penalty function of the problem (P) is (PC) minf(x)+cp(x) subject tox R ⁿ , wherep(x)=max {0,g ₁(x), ...,g _r (x)},c>0. We will discuss the relationships between (P) and (PC). In particular, we will prove that under some (mild) conditions a local minimum of (PC) is also a local minimum of (P).     

Existence-uniqueness results for a class of singular boundary value problems-II

In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem −(p(x)y^′′(x))=q(x)f(x,y) for 0<x?b and y(0)=a, α₁y(b)+β₁y^′(b)=γ₁, where p(x) satisfies (i) p(x)>0 in (0,b), (ii) p(x)∈C¹(0,r), and for some r>b, (iii) is analytic in and q(x) satisfies (i) q(x)>0 in (0,b), (ii) q(x)∈L¹(0,b) and for some r>b, (iii) is analytic in with quite general conditions on f(x,y). Region for multiple solutions have also been determined.     

On some composite functional inequalities

The aim of the paper is to deal with the following composite functional inequalities

Existence of unique solutions for perturbed autonomous differential equations

We consider the existence of unique absolutely continuous solutionsfor x' = p(t)f(x) + p(t)h(t), t 0, x(0) = 0, where p, f, andh are positive almost everywhere, but none of them needs becontinuous or monotone. Moreover, p and f can be unbounded aroundzero. Our uniqueness results are not based on assumptions onthe differences f(x) – f(y), as it is usual in most uniquenessresults, and they are new even when p, f, and h are continuous.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)

Oscillation criteria for second order nonlinear neutral differential equations

By refining the standard integral averaging technique, we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form

(r(t)(x(t)+p(t)x(t-τ))^′)^′+q(t)f(x(t),x(σ(t)))=0.     

Oscillation theorem for superlinear second order damped differential equations

In this paper, we are concerned with the oscillation of second order superlinear differential equations of the form

(a(t)y^′(t))^′+p(t)y^′(t)+q(t)f(y(t))=0.     

Boundary-value problems for shallow elastic membrane caps

Consider the general nonlinear boundary-value problem (p(t)y' (t))' = p(t)q(t) f (t, y(t), y' (t)), t 1, g(y(1), y' (1))= 0, where the function f may be singular at the point y(1)= 0 and p(1) 0. We obtain conditions which guarantee existenceof positive and bounded solutions of the above problem. As anapplication we prove existence and uniqueness of rotationallysymmetric solutions to a nonlinear boundary-value problem, representingthe elastic deformation of a spherical cap.     

Linear Discrepancy and Bandwidth

The linear discrepancy of a partially ordered set P=(X,), denoted by ld(P), is the least integer k for which there exists an injection f:XZ satisfying (i) if xy, then f(x)<f(y); and (ii) if x and y are incomparable, then |f(x)–f(y)|k. There is an apparent connection between ld(P) and the bandwidth of the incomparability graph of P. We prove that, in fact, these two quantities are always equal.     

Existence Theorems for Certain Elliptic and Parabolic Semilinear Equations

Existence theorems for the nonlinear parabolic differential equation −∂u/∂t + Δu + |u|^p + f(x, t) = 0 in ⁿ × [0, ∞) with zero initial value are established given explicit conditions on the nonhomogeneous termf(x, t). An existence theorem is also demonstrated for the corresponding elliptic equation.     

A comparative study of functional equations characterising sine and cosine

A comparative study of the functional equationsf(x+y)f(x–y)=f ²(x)–f ²(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated.     

Quotient mean,its invariance with respect to a quasi-arithmetic mean-type mapping,and some applications

Under some conditions on the functions f and g defined in a real interval I the function