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The Mean Integral

Description: The purpose of this paper is to examine properties of the mean integral. The mean integral is compared with the regular integral. If [a;b] is an interval, f is quasicontinuous on [a;b] and g has bounded variation on [a;b], then the man integral of f with respect to g exists on [a;b]. The following theorem is proved. If [a*;b*] and [a;b] each is an interval and h is a function from [a*;b*] into R, then the following two statements are equivalent: 1) If f is a function from [a;b] into [a*;b*], gi is a function from [a;b] into R with bounded variation and (m)∫^b_afdg exists then (m)∫^b_ah(f)dg exists. 2) h is continuous.
Date: December 1985
Creator: Spear, Donald W.
Partner: UNT Libraries

Advances in the theory of box integrals

Description: Box integrals - expectations <|{rvec r}|{sup s}> or <|{rvec r}-{rvec q}|{sup s}> over the unit n-cube (or n-box) - have over three decades been occasionally given closed forms for isolated n,s. By employing experimental mathematics together with a new, global analytic strategy, we prove that for n {le} 4 dimensions the box integrals are for any integer s hypergeometrically closed in a sense we clarify herein. For n = 5 dimensions, we show that a single unresolved integral we call K{sub 5} stands in the way of such hyperclosure proofs. We supply a compendium of exemplary closed forms that naturally arise algorithmically from this theory.
Date: June 25, 2009
Creator: Bailey, David H.; Borwein, J.M. & Crandall, R.E.
Partner: UNT Libraries Government Documents Department

Spaces of H-Integrable Functions

Description: In this thesis we consider integrals of a certain class of interval functions. Specifically we consider a nondegenerate number interval [a,b], a real valued function m, defined and nondecreasing on [a,b], and the set Hm, of real valued functions f, defined on [a,b] such that: 1) f(a)=0; 2) for each subinterval [p,q] of [a,b], if m(q)-m(p)=0, then f(q)-f(p)=0; and 3) the set of all sums of the form Σ(Δf)2/Δm for subdivisions D of [a,b] is bounded above.
Date: May 1968
Creator: Wittenmyer, Eugene L.
Partner: UNT Libraries

Determination of Band Curvatures by Angle-resolved Two-photonPhotoemission in thin films of C(60) on Ag(111)

Description: The thickness-dependent interfacial band structure wasdetermined for thin films of Co(6) on Ag(111) by angle-resolvedtwo-photon photoemission spectroscopy.
Date: March 1, 2005
Creator: Shipman, Steven T.; Szymanski, Paul; Garrett-Roe, Sean; Yang,Aram; Strader, Matthew L. & Harris, Charles B.
Partner: UNT Libraries Government Documents Department

Note on the Numerical Evaluation of Integrals of the Form [integral] [superscript infinity] [subscript infinity] f(x) [golden ratio constant] (x) dx, with Particular Reference to the Determiniation of the Expectation of a Function of a Normally Distributed Random Variable

Description: Abstract: A method is given for the rapid and accurate numerical integration of integrals of the form [integral] [superscript infinity] [subscript infinity] f(x) [golden ratio constant] (x) dx, where f(x) is "smooth."
Date: May 24, 1960
Partner: UNT Libraries Government Documents Department

Highly optimized fourth-order short-time approximation for pathintegrals

Description: We derive a fourth-order short-time approximation for use in imaginary-time path-integral simulations. The short-time approximation converges for all continuous and bounded from below potentials, attains quartic order of convergence for sufficiently smooth potentials, and utilizes statistically independent random variables for its construction. These properties recommend the approximation as a natural replacement of the trapezoidal Trotter-Suzuki approximation for physical systems with continuous distributions.
Date: October 1, 2006
Creator: Predescu, Cristian
Partner: UNT Libraries Government Documents Department

Experimental computation with oscillatory integrals

Description: A previous study by one of the present authors, together with D. Borwein and I. Leonard [8], studied the asymptotic behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x and along the way looked at closed forms for integer values of p. In this study we address these integrals with the tools of experimental mathematics, namely by computing their numerical values to high precision, both as a challenge in itself, and also in an attempt to recognize the numerical values as closed-form constants. With this approach, we are able to reproduce several of the results of [8] and to find new results, both numeric and analytic, that go beyond the previous study.
Date: June 26, 2009
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department

The Evaluation of Definite Integrals, and a Quasi-Monte-Carlo Method Based on the Properties of Algebraic Numbers

Description: A formula is given for the approximate evaluation of multiple definite integrals using the ergodic property of a certain transformation of the unit cube into itself. Estimates of the rate of convergence are made for sufficiently smooth integrand. The work was motivated by a belief, that appeared at one time justified, that a substantial improvement of the accuracy of Monte Carlo method would result from use of the principles described herein. Although that belief proved groundless in numerical tests, it is deemed worthwhile to give this report of the work. Part of the theoretical development, the work of L.G. Peck, will be reported independently. (auth)
Date: October 13, 1951
Creator: Richtmyer, R. D.
Partner: UNT Libraries Government Documents Department

SPAN-2--AN IMB-704 CODE TO CALCULATE UNCOLLIDED FLUX OUTSIDE A CIRCULAR CYLINDER

Description: The SPAN-2 code calculates the uncollided gamma flux at a point outside a right circular cylinder which is surrounded by cyllndrical shell shields and above which are plane slab shields. The cylinder is assured to contain a source of gamma radiation which varies in the radial and axial directions only. Field points may be located in a plane through the axis of the cylinder. The method of integration used is three-dimensional Gaussian quadrature. Typical computing and editing time for a 20 field point problem, in which there are 10 side and 10 top shields, is four minutes per energy level. A 32,768 word IBM-704 is required for the code. (auth)
Date: August 1, 1959
Creator: Gillis, P.A.; Lawton, T.J. & Brand, K.W.
Partner: UNT Libraries Government Documents Department

Box Integrals

Description: By a "box integral" we mean here an expectation $\langle|\vec r - \vec q|^s \rangle$ where $\vec r$runs over the unit $n$-cube,with $\vec q$ and $s$ fixed, explicitly:\begin eqnarray*&&\int_01 \cdots \int_01 \left((r_1 - q_1)2 + \dots+(r_n-q_n)2\right)^ s/2 \ dr_1 \cdots dr_n.\end eqnarray* The study ofbox integrals leads one naturally into several disparate fields ofanalysis. While previous studies have focused upon symbolic evaluationand asymptotic analysis of special cases (notably $s = 1$), we workherein more generally--in interdisciplinary fashion--developing resultssuch as: (1) analytic continuation (in complex $s$), (2) relevantcombinatorial identities, (3) rapidly converging series, (4) statisticalinferences, (5) connections to mathematical physics, and (6)extreme-precision quadrature techniques appropriate for these integrals.These intuitions and results open up avenues of experimental mathematics,with a view to new conjectures and theorems on integrals of thistype.
Date: June 1, 2006
Creator: Bailey, David H.; Borwein, Jonathan M. & Crandall, Richard E.
Partner: UNT Libraries Government Documents Department

Haar Measure on the Cantor Ternary Set

Description: The purpose of this thesis is to examine certain questions concerning the Cantor ternary set. The second chapter deals with proving that the Cantor ternary set is equivalent to the middle thirds set of [0,1], closed, compact, and has Lebesgue measure zero. Further a proof that the Cantor ternary set is a locally compact, Hausdorff topological group is given. The third chapter is concerned with establishing the existence of a Haar integral on certain topological groups. In particular if G is a locally compact and Hausdorff topological group, then there is a non-zero translation invariant positive linear form on G. The fourth chapter deals with proving that for any Haar integral I on G there exists a unique Haar measure on G that represents I.
Date: August 1990
Creator: Naughton, Gerard P. (Gerard Peter)
Partner: UNT Libraries