Arbitrary Real Function Research Articles

Nearly computable real numbers

We call a sequence ( a n ) n of elements of a metric space nearly computably Cauchy if for every increasing computable function r : N → N the sequence ( d ( a r ( n + 1 ) , a r ( n ) ) ) n converges computably to 0. We show that there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence ( a n ) n of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (Theory of Computing Systems 60 (2017) 396–420) and by Stephan and Wu (In New computational paradigms. First conference on computability in Europe, CiE 2005, Proceedings (2005) 461–469 Springer) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.

A General Method for Obtaining Degenerate Solutions to the Dirac and Weyl Equations and a Discussion on the Experimental Detection of Degenerate States

AbstractIn this work, a general method is described for obtaining degenerate solutions of the Dirac equation, corresponding to an infinite number of electromagnetic 4‐potentials and fields, which are explicitly calculated. More specifically, using four arbitrary real functions, one can automatically construct a spinor that satisfies the Dirac equation for an infinite number of electromagnetic 4‐potentials, defined by those functions. An interesting characteristic of these solutions is that, in the case of Dirac particles with nonzero mass, the degenerate spinors should be localized, both in space and time. The method is also extended to the cases of massless Dirac and Weyl particles, where the localization of the spinors is no longer required. Finally, two experimental methods are proposed for detecting the presence of degenerate states.

Euler top and freedom in supersymmetrization of one-dimensional mechanics

In this paper we formulate the Euler top as a system on the complex projective plane, playing the role of phase space, i.e., as a one-dimensional mechanical system. Then we suggest the supersymmetrization scheme of the generic one-dimensional systems with positive Hamiltonian which yields á priori integrable family of N=2k supersymmetric Hamiltonians parameterized by N/2 arbitrary real functions.

Generalised Fayet–Iliopoulos Terms in Supergravity

The U(1) vector multiplet theory with the Fayet-Iliopoulos (FI) term is one of the oldest and simplest models for spontaneously broken rigid supersymmetry. Lifting the FI term to supergravity requires gauged $R$-symmetry, as was first demonstrated in 1977 by Freedman within ${\cal N}=1$ supergravity. There exists an alternative to the standard FI mechanism, which is reviewed in this conference paper. It is obtained by replacing the FI model with a manifestly gauge-invariant action such that its functional form is determined by two arbitrary real functions of a single complex variable. One of these functions generates a superconformal kinetic term for the vector multiplet, while the other yields a generalised FI term. Coupling such a vector multiplet model to supergravity does not require gauging of the $R$-symmetry. These generalised FI terms are consistently defined for any off-shell formulation for ${\cal N}=1$ supergravity, and are compatible with a supersymmetric cosmological term.

Equivalence transformations and differential invariants of a generalized cubic–quintic nonlinear Schrödinger equation with variable coefficients

In this paper, a variable-coefficient cubic–quintic nonlinear Schrodinger equation involving five arbitrary real functions of space and time is analyzed from the point of view of symmetry analysis by using Lie’s invariance infinitesimal criterion. The infinitesimal generators of corresponding equivalence transformations are presented. The first-order differential invariants are constructed to identify when the equation can be mapped to a constant-coefficient cubic–quintic nonlinear Schrodinger equation. The constrained conditions on the variable coefficients we arrived here extend the cases discussed before and present more general results. Some brightlike and darklike solitary wave solutions for special potentials and cubic–quintic nonlinearities are obtained.

On the asymptotic behavior of some sums involving the number of prime divisors function

We consider sums of values of the composition of a real periodic arithmetic function and the number of prime divisors function over integers not exceeding a given bound. The prime divisors may be counted as with their multiplicity or without it, and we can restrict these divisors to the additional condition of belonging to some special set. This special set may be, for example, a sum of several arithmetic progressions with a given difference or imply an analog of prime number theorem with a power decrement in the remainder term. Moreover, instead of the number of prime divisors function we can consider an arbitrary real additive function that equals to one on primes. As an example of the periodic arithmetic function we can consider the Legendre symbol. In the paper we prove asymptotic formulae for such sums and investigate their behavior. The proof uses the decomposition of the periodic arithmetic function into additive characters of the residue group, so the problem reduces to special trigonometric sums with the number of prime divisors function in the exponent. In order to establish asymptotic formulae for such sums we consider the corresponding Dirichlet series, accomplish its analytic continuation and make use of the Perron formula and complex integration method in specially adapted form.

Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of the Lojasiewicz inequalities in the special case where the function is $C^1$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary real or complex analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Lojasiewicz inequalities when a function is $C^N$ and generalized Morse-Bott of order $N \geq 3$; we gave an elementary proof of the Lojasiewicz inequalities when a function is $C^2$ and Morse-Bott in arXiv:1708.09775v4 (finite-dimensional case) and arXiv:1706.09349 (infinite-dimensional case).

Some Inequalities Using Generalized Convex Functions in Quantum Analysis

In the present work, the Hermite–Hadamard inequality is established in the setting of quantum calculus for a generalized class of convex functions depending on three parameters: a number in ( 0 , 1 ] and two arbitrary real functions defined on [ 0 , 1 ] . From the proven results, various inequalities of the same type are deduced for other types of generalized convex functions and the methodology used reveals, in a sense, a symmetric mathematical phenomenon. In addition, the definition of dominated convex functions with respect to the generalized class of convex functions aforementioned is introduced, and some integral inequalities are established.

Exact solutions of a (2+1)-dimensional extended shallow water wave equation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11671219 and 11871446).

We give the bilinear form and n-soliton solutions of a (2+1)-dimensional [(2+1)-D] extended shallow water wave (eSWW) equation associated with two functions v and r by using Hirota bilinear method. We provide solitons, breathers, and hybrid solutions of them. Four cases of a crucial , which is an arbitrary real continuous function appeared in f of bilinear form, are selected by using Jacobi elliptic functions, which yield a periodic solution and three kinds of doubly localized dormion-type solution. The first order Jacobi-type solution travels parallelly along the x axis with the velocity on (x, y)-plane. If , it is a periodic solution. If , it is a dormion-type-I solutions which has a maximum and a minimum . The width of the contour line is . If , we get a dormion-type-II solution (26) which has only one extreme value . The width of the contour line is . If , we get a dormion-type-III solution (21) which shows very strong doubly localized feature on (x,y) plane. Moreover, several interesting patterns of the mixture of periodic and localized solutions are also given in graphic way.

Curves, wavefronts and caustics determined by the intensity pattern of an adiffractional beam

The aim of the present work is threefold, first we show that the intensity pattern of a nondiffracting beam determines an arbitrary positive real function and a complete integral of both the eikonal and Laplace equations on the plane; second, by using this result we associate to the intensity pattern a two-parameter family of curves and a one-parameter family of caustics on the plane and third, we use the intensity pattern of the Bessel beam of order m to identify its maxima with the family of caustics. This result suggests to define a family of caustics as the maxima of a given intensity pattern.

Gravitino and Polonyi production in supergravity

We study production of gravitino and Polonyi particles in the minimal Starobinsky-Polonyi mathcal {N}=1 supergravity with inflaton belonging to a massive vector supermultiplet. Our model has only one free parameter given by the scale of spontaneous SUSY breaking triggered by Polonyi chiral superfield. The vector supermultiplet generically enters the action non-minimally, via an arbitrary real function. This function is chosen to generate the inflaton scalar potential of the Starobinsky model. Our supergravity model can be reformulated as an abelian supersymmetric gauge theory with the vector gauge superfield coupled to two (Higgs and Polonyi) chiral superfields interacting with supergravity, where the U(1) gauge symmetry is spontaneously broken. We find that Polonyi and gravitino particles are efficiently produced during inflation, and estimate their masses and the reheating temperature. After inflation, perturbative decay of inflaton also produces Polonyi particles that rapidly decay into gravitinos. As a result, a coherent picture of inflation and dark matter emerges, where the abundance of produced gravitinos after inflation fits the CMB constraints as a Super Heavy Dark Matter (SHDM) candidate. Our scenario avoids the notorous gravitino and Polonyi problems with the Big Bang Nucleosynthesis (BBN) and DM overproduction.

Spectral and scattering theory for differential and Hankel operators

We consider a class of Hankel operators H realized in the space L2(R+) as integral operators with kernels h(t+s) where h(t)=P(ln⁡t)t−1 and P(X)=Xn+pn−1Xn−1+⋯+p0 is an arbitrary real polynomial of degree n. This class contains the classical Carleman operator when n=0. We show that a Hankel operator H in this class can be reduced by an explicit unitary transformation (essentially by the Mellin transform) to a differential operator A=vQ(D)v in the space L2(R). Here Q(X)=Xn+qn−1Xn−1+⋯+q0 is a polynomial determined by P(X) and v(ξ)=π1/2(cosh⁡(πξ))−1/2 is the universal function. Then the operator A=vQ(D)v reduces by the generalized Liouville transform to the standard differential operator B=Dn+bn−1(x)Dn−1+⋯+b0(x) with the coefficients bm(x), m=0,…,n−1, decaying sufficiently rapidly as |x|→∞. This allows us to use the results of spectral theory of differential operators for the study of spectral properties of generalized Carleman operators. In particular, we show that the absolutely continuous spectrum of H is simple and coincides with R if n is odd, and it has multiplicity 2 and coincides with [0,∞) if n≥2 is even. The singular continuous spectrum of H is empty, and its eigenvalues may accumulate to the point 0 only. As a by-product of our considerations, we develop spectral theory of a new class of degenerate differential operators A=vQ(D)v where Q(X) is an arbitrary real polynomial and v(ξ) is an almost arbitrary real function decaying at infinity.

Bifurcation of Soliton Families from Linear Modes in Non‐‐Symmetric Complex Potentials

Continuous families of solitons in the nonlinear Schrödinger equation with non‐‐symmetric complex potentials and general forms of nonlinearity are studied analytically. Under a weak assumption, it is shown that stationary equations for solitons admit a constant of motion if and only if the complex potential is of a special form , where is an arbitrary real function. Using this constant of motion, the second‐order complex soliton equation is reduced to a new second‐order real equation for the amplitude of the soliton. From this real soliton equation, a novel perturbation technique is employed to show that continuous families of solitons bifurcate out from linear discrete modes in these non‐‐symmetric complex potentials. All analytical results are corroborated by numerical examples.

Quadruple symmetric real signals spectral even and odd decomposition

The spectral properties of quadruple symmetric real signals are analyzed in the study. Six number theorems are formulated and proofed analytically in a capacity of central results of the research. Lasted theorem could be used to construct complex Fourier spectrum for arbitrary real function by even - odd decomposition. The theorem is illustrated numerically. The initial signal with length N (analogous values length interval or number of discrete samples) in the time domain is Fourier transformed through two spectral - real and imaginary parts with length N in the frequency domain. The real and imaginary parts of the complex Fourier spectrum of the initial signal, could be obtained by procedure, described in the paper. Spectral parts could be calculated by equivalent functions-signals. Even left and odd right equivalent functions-signals contain N/2 nonzero analogous values or discrete samples. This strategy allows constructing complex Fourier spectrum of the initial signal with length N in the time domain based on equivalent real and imaginary spectral parts with the length N/2 in the frequency domain. The study is an extension and resume of AMC 221(2013) pp. 344-350.

Existence and classification of three-dimensional Lorentzian manifolds with prescribed distinct Ricci eigenvalues

We prove that there exists at least one three-dimensional Lorentzian manifold with any prescribed distinct Ricci eigenvalues, which are given as arbitrary real analytic functions. Moreover, we prove that such Lorentzian manifolds depend locally on three arbitrary functions of two variables.

On completely monotone of an arbitrary real parameter function involving the gamma function

In this article, we present some completely monotone properties for an arbitrary real parameter function involving the Gamma function. As an application of these results, some meaningful and useful inequalities are obtained.

Rings of real functions in pointfree topology

This paper deals with the algebra F ( L ) of real functions on a frame L and its subclasses LSC ( L ) and USC ( L ) of, respectively, lower and upper semicontinuous real functions. It is well known that F ( L ) is a lattice-ordered ring; this paper presents explicit formulas for its algebraic operations which allow to conclude about their behaviour in LSC ( L ) and USC ( L ) . As applications, idempotent functions are characterized and previous pointfree results about strict insertion of functions are significantly improved: general pointfree formulations that correspond exactly to the classical strict insertion results of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces are derived. The paper ends with a brief discussion concerning the frames in which every arbitrary real function on the α-dissolution of the frame is continuous.

Generalized versus Selected Descriptions of Quantum <I>LC</I>-Circuits

Proofs are given that the quantum-mechanical description of the LC -circuit with a time dependent external source can be readily established by starting from a more general discretization rule of the electric charge. For this purpose one resorts to an arbitrary but integer-dependent real function F(n) instead of n. This results in a nontrivial generalization of the discrete time dependent Schrodinger-equation established before via F(n)=n, as well as to modified charge conservation laws. However, selected descriptions can also be done by looking for a unique derivation of the effective inductance. This leads to site independent inductances, but site dependent ones get implied by accounting for periodic solutions to F(n) in terms of Jacobian elliptic functions. Many-charge generalizations of quantum circuits, including the modified continuity equation for total charge and current densities, have also been discussed.

Characterizing Polynomial Time Computability of Rational and Real Functions

Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real functions. This article is divided into two main parts. The first part investigates polynomial time computability of rational functions and the role of continuity in such computation. On the one hand this is interesting for its own sake. On the other hand it provides insights into polynomial time computability of real functions for the latter, in the sense of recursive analysis, is modeled as approximations of rational computations. The main conclusion of this part is that continuity does not play any role in the efficiency of computing rational functions. The second part defines polynomial time computability of arbitrary real functions, characterizes it, and compares it with the corresponding notion over rational functions. Assuming continuity, the main conclusion is that there is a conceptual difference between polynomial time computation over the rationals and the reals manifested by the fact that there are polynomial time computable rational functions whose extensions to the reals are not polynomial time computable and vice versa.

The differential properties of one class of surfaces in Euclidean space

We consider n-dimensional smooth surfaces of class ℰ1 in the Euclidean space of dimension n + m satisfying the following condition. Given two distinct points of the surface, the surface normals at these points either are disjoint or meet at the distance from both of these points bounded below by some fixed positive constant. We establish that every surface of this type carries in a neighborhood of each point a parametrization with bounded second order generalized derivatives in the sense of Sobolev. The proof is based on using geometric properties of the surfaces of this form and on the proposition that establishes sufficient conditions for the existence of bounded second order generalized derivatives of an arbitrary real function. In the Appendix we prove an analog of this lemma in the case of derivatives of arbitrary order.