Estimates for the state density for ordinary differential operators with white gaussian noise potential

1983 ◽  
Vol 95 (3-4) ◽  
pp. 263-274
Author(s):  
V. B. Moscatelli ◽  
M. Thompson
Keyword(s):  
Wiener Process ◽  
Lower Bounds ◽  
Gaussian Noise ◽  
Differential Operators ◽  
Asymptotic Properties ◽  
State Density ◽  
The State ◽  
Upper And Lower Bounds ◽  
Ordinary Differential Operators ◽  
Image Position

SynopsisThe present paper is concerned with developing the existence and asymptotic properties of the state density N(λ) associated with certain higher order random ordinary differential operators A of the formwhere Ao has homogeneous and ergodic coefficients with respect to the σ-algebra generated by the Wiener process q(ω, x). The analysis uses the Weyl min-max principle to determine rough upper and lower bounds for N(λ).

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

scholarly journals ON THE ORDER OF MAGNITUDE OF SUMS OF NEGATIVE POWERS OF INTEGRATED PROCESSES

2012 ◽  
Vol 29 (3) ◽  
pp. 642-658 ◽  
Author(s):  
Benedikt M. Pötscher
Keyword(s):  
Lower Bounds ◽  
Random Variables ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Integrated Process ◽  
Order Of Magnitude ◽  
Image Position ◽  
Integrated Processes

Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.

On the Zeros of Power Series with Exponential Logarithmic Coefficients

1981 ◽  
Vol 24 (3) ◽  
pp. 257-271 ◽  
Author(s):  
W. Gawronski ◽  
U. Stadtmüller
Keyword(s):  
Power Series ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Number Of Zeros ◽  
Logarithmic Coefficients ◽  
Image Position

In this paper we investigate the zeros of power series1for some functions of coefficients A. In particular, we derive upper and lower bounds for the number of zeros of f in its domain of analyticity.

Multifractal dimensions of product measures

1996 ◽  
Vol 120 (4) ◽  
pp. 709-734 ◽  
Author(s):  
L. Olsen
Keyword(s):  
Probability Measure ◽  
Lower Bounds ◽  
Borel Probability Measure ◽  
Multifractal Spectrum ◽  
Upper And Lower Bounds ◽  
Packing Measure ◽  
Multifractal Spectra ◽  
Product Measures ◽  
Borel Probability ◽  
Image Position

AbstractWe study the multifractal structure of product measures. for a Borel probability measure μ and q, t Є , let and denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [O11] Let μ be a Borel probability merasure on k and let v be a Borel probability measure on t. Fix q, s, t Є . We prove that there exists a number c > 0 such that for E ⊆k, F ⊆l and Hk+l provided that μ and ν satisfy the so-called Federer condition.Using these inequalities we give upper and lower bounds for the multifractal spectrum of μ × ν in terms of the multifractal spectra of μ and ν

An integral inequality with an application to ordinary differential operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 35-44 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Integral Inequality ◽  
Differential Operators ◽  
Linear Manifold ◽  
Integral Inequalities ◽  
Compact Interval ◽  
Ordinary Differential Operators ◽  
Image Position

SynopsisThis paper is concerned with integral inequalities of the formwhere p, q are real-valued coefficients, with p and w non-negative, on the compact interval [a, b] and D is a linear manifold of functions so chosen that all three integrals are absolutely convergent.

scholarly journals COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

Some inequalities for arithmetic and geometric means

1999 ◽  
Vol 129 (2) ◽  
pp. 221-228 ◽  
Author(s):  
Horst Alzer
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Geometric Means ◽  
Weighted Arithmetic ◽  
Image Position

Let An and Gn (respectively, A′n and G′n) be the weighted arithmetic and geometric means of x1, …, xn (respectively, 1 – x1, …, 1 – xn). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such thatholds for all xi ∈ [a, b] (i = 1, …, n; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author.

On Blasius's equation governing flow in the boundary layer on a flat plate

1973 ◽  
Vol 74 (1) ◽  
pp. 179-184 ◽  
Author(s):  
S. Richardson
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Power Series ◽  
Lower Bounds ◽  
Unknown Function ◽  
Asymptotic Properties ◽  
Inverse Function ◽  
Upper And Lower Bounds ◽  
Independent Variable ◽  
Original Approach

AbstractThe original approach of Blasius to the solution of the differential equation now associated with his name was to develop the unknown function as a power series. Unfortunately, this series has a limited radius of convergence, so that such a representation is not valid over the whole range of interest. It is shown here that, if we work instead with a particular inverse function, this can be expanded as a power series which converges for all relevant values of the independent variable. Moreover, the number associated with the solution which is of principal physical interest can be expressed in terms of the asymptotic properties of the coefficients of this series. Exploiting this relationship, we find upper and lower bounds for this number in terms of the zeros of two particular families of polynomials.

Limit Point Criteria for Differential Equations

1972 ◽  
Vol 24 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Differential Equations ◽  
Limit Point ◽  
Differential Operators ◽  
Central Importance ◽  
Linearly Independent ◽  
Ordinary Differential Operators ◽  
Image Position

For certain ordinary differential operators L of order 2n, this paper considers the problem of determining the number of linearly independent solutions of class L2[a, ∞) of the equation L(y) = λy. Of central importance is the operator0.1where the coefficients pi are real. For this L, classical results give that the number m of linearly independent L2[a, ∞) solutions of L(y) = λy is the same for all non-real λ, and is at least n [10, Chapter V]. When m = n, the operator L is said to be in the limit-point condition at infinity. We consider here conditions on the coefficients pi of L which imply m = n. These conditions are in the form of limitations on the growth of the coefficients.

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