310,495 research outputs found
A novel approach to fractional calculus: utilizing fractional integrals and derivatives of the Dirac delta function
While the definition of a fractional integral may be codified by Riemann and
Liouville, an agreed-upon fractional derivative has eluded discovery for many
years. This is likely a result of integral definitions including numerous
constants of integration in their results. An elimination of constants of
integration opens the door to an operator that reconciles all known fractional
derivatives and shows surprising results in areas unobserved before, including
the appearance of the Riemann Zeta Function and fractional Laplace and Fourier
Transforms. A new class of functions, known as Zero Functions and closely
related to the Dirac Delta Function, are necessary for one to perform
elementary operations of functions without using constants. The operator also
allows for a generalization of the Volterra integral equation, and provides a
method of solving for Riemann's "complimentary" function introduced during his
research on fractional derivatives
New anisotropic models from isotropic solutions
We establish an algorithm that produces a new solution to the Einstein field
equations, with an anisotropic matter distribution, from a given seed isotropic
solution. The new solution is expressed in terms of integrals of known
functions, and the integration can be completed in principle. The applicability
of this technique is demonstrated by generating anisotropic isothermal spheres
and anisotropic constant density Schwarzschild spheres. Both of these solutions
are expressed in closed form in terms of elementary functions, and this
facilitates physical analysis.Comment: 23 pages, To appear in Math. Meth. Appl. Sc
Hyperfinite stochastic integration for LĂ©vy processes with finite-variation jump part
This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite LĂ©vy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump LĂ©vy processes with finite-variation jump part. Since the hyperfinite ItĂ´ integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to LĂ©vy jump-diffusions with finite-variation jump part. As an application, we provide a short and direct nonstandard proof of the generalized ItĂ´ formula for stochastic differentials of smooth functions of LĂ©vy jump-diffusions whose jumps are bounded from below in norm.LĂ©vy processes, stochastic integration, nonstandard analysis, ItĂ´ formula
Composite structural motifs of binding sites for delineating biological functions of proteins
Most biological processes are described as a series of interactions between
proteins and other molecules, and interactions are in turn described in terms
of atomic structures. To annotate protein functions as sets of interaction
states at atomic resolution, and thereby to better understand the relation
between protein interactions and biological functions, we conducted exhaustive
all-against-all atomic structure comparisons of all known binding sites for
ligands including small molecules, proteins and nucleic acids, and identified
recurring elementary motifs. By integrating the elementary motifs associated
with each subunit, we defined composite motifs which represent
context-dependent combinations of elementary motifs. It is demonstrated that
function similarity can be better inferred from composite motif similarity
compared to the similarity of protein sequences or of individual binding sites.
By integrating the composite motifs associated with each protein function, we
define meta-composite motifs each of which is regarded as a time-independent
diagrammatic representation of a biological process. It is shown that
meta-composite motifs provide richer annotations of biological processes than
sequence clusters. The present results serve as a basis for bridging atomic
structures to higher-order biological phenomena by classification and
integration of binding site structures.Comment: 34 pages, 7 figure
Elementary And Integral-elementary Functions
By an integral-elementary function we mean any real function that can be obtained from the constants, sin x, e(x), log x, and arcsin x (defined on (-1, 1)) using the basic algebraic operations, composition and integration. The rank of an integral-elementary function f is the depth of the formula defining f. The integral-elementary Functions of rank less than or equal to n are real-analytic and satisfy a common algebraic differential equation P-n(f, f',..., f((k))) = 0 with integer coefficients. We prove that every continuous function f: R --> R can be approximated uniformly by integral-elementary functions of bounded rank. Consequently, there exists an algebraic differential equation with integer coefficients such that its everywhere analytic solutions approximate every continuous function uniformly. This solves a problem posed by L. A. Rubel. Using the same basic functions as above, but allowing only the basic algebraic operations and compositions, we obtain the class of elementary functions. We show that every differentiable function with a derivative not exceeding an iterated exponential can be uniformly approximated by elementary functions of bounded rank. If we include the function arcsin x defined on [-1, 1], then the resulting class of naive-elementary functions will approximate every continuous function uniformly. We also show that every sequence can be uniformly approximated by elementary functions, and that every integer sequence can be represented in the form f(n), where f is naive-elementary
- …