Integer Degree Research Articles

Chaos and integrability of relativistic homogeneous potentials in curved space

Relativistic Hamiltonian systems of n degrees of freedom in static curved spaces are considered. The source of space-time curvature is a scalar potential V(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\varvec{q})$$\\end{document}. In the limit of weak potential 2V(q)/mc2≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2V(\\varvec{q})/mc^2\\ll 1$$\\end{document}, and small momentum |p|/mc≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\varvec{p} |/ mc\\ll 1$$\\end{document}, these systems transform into the corresponding non-relativistic flat Hamiltonian’s with the standard sum of kinetic energy plus potential V(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\varvec{q})$$\\end{document}. We compare the dynamics of the classical and the corresponding relativistic curved counterparts on examples of important physical models: the Hénon–Heiles system, the Armbruster–Guckenheimer–Kim galactic system and swinging Atwood’s machine. Our main results are formulated for relativistic Hamiltonian systems with homogeneous potentials of non-zero integer degree k of homogeneity. First, we show that the integrability of a non-relativistic flat Hamiltonian with a homogeneous potential is a necessary condition for the integrability of its relativistic counterpart in curved space-time with the same homogeneous potential or with a non-homogeneous potential that the lowest homogenous part coincides with this homogeneous potential. Moreover, we formulate necessary integrability conditions for relativistic Hamiltonian systems with homogeneous potentials in curved space-time. These conditions were obtained from analysis of the differential Galois group of variational equations along a particular straight-line solution defined by a non-zero vector d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{d}$$\\end{document} satisfying V′(d)=γd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V'(\\varvec{d})=\\gamma \\varvec{d}$$\\end{document} for a certain γ≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \ e 0$$\\end{document}. They are very strong: if the relativistic system is integrable in the Liouville sense, then either k=±2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=\\pm 2$$\\end{document}, or all non-trivial eigenvalues of the re-scaled Hessian γ-1V′′(d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{-1}V''(\\varvec{d})$$\\end{document} are either 0, or 1. Certain integrable relativistic systems are presented.

An integer degree for asymptotically conical self-expanders

An integer degree for asymptotically conical self-expanders

Power savings for counting solutions to polynomial-factorial equations

Let P be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions n≤N to n!=P(x) which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is o(N). The proof uses techniques of Diophantine and Padé approximation.

Integrability of Hamiltonian systems with gyroscopic term

We study the integrability of 2D Hamiltonian systems H_mu =frac{1}{2}(p_1^2+p_2^2) + omega (p_1q_2-p_2q_1) -tfrac{mu }{r}+ V(q_1,q_2), where r^2=q_1^2+q_2^2, and potential V(q_1,q_2) is a homogeneous rational function of integer degree k. The main result states that under very general assumptions: mu omega ne 0, |k|>2 and V(1,mathrm {i})ne 0 or V(-1,mathrm {i})ne 0, the system is not integrable. It was obtained by combining the Levi-Civita regularization, differential Galois methods, and the so-called coupling constant metamorphosis transformation. The proof that the regularized version of the problem is not integrable contains the most important theoretical results, which can be applied to study integrability of other problems.

On the representation of integers by binary forms defined by means of the relation (x + yi)n= Rn(x,y) + Jn(x,y)i

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d \geq 3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. In 2019 Stewart and Xiao proved that $R_F(Z) \sim C_FZ^{2/d}$ for some positive number $C_F$. We compute $C_{R_n}$ and $C_{J_n}$ for the binary forms $R_n(x, y)$ and $J_n(x, y)$ defined by means of the relation $ (x + yi)^n = R_n(x, y) + J_n(x, y)i$, where the variables $x$ and $y$ are real.

A uniqueness result for self-expanders with small entropy

In this short note, we prove a uniqueness result for small entropy self-expanders asymptotic to a fixed cone. This is a direct consequence of the mountain-pass theorem and the integer degree argument proved by J. Bernstein and L. Wang.

Topological Singularities in Periodic Media: Ginzburg\u2013Landau and Core-Radius Approaches

We describe the emergence of topological singularities in periodic media within the Ginzburg–Landau model and the core-radius approach. The energy functionals of both models are denoted by E_{varepsilon ,delta }, where varepsilon represent the coherence length (in the Ginzburg–Landau model) or the core-radius size (in the core-radius approach) and delta denotes the periodicity scale. We carry out the Gamma -convergence analysis of E_{varepsilon ,delta } as varepsilon rightarrow 0 and delta =delta _varepsilon rightarrow 0 in the |log varepsilon | scaling regime, showing that the Gamma -limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter λ=min{1,limε→0|logδε||logε|}\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\lambda =\\min \\Bigl \\{1,\\lim _{\\varepsilon \\rightarrow 0} {|\\log \\delta _\\varepsilon |\\over |\\log \\varepsilon |}\\Bigr \\} \\end{aligned}$$\\end{document}(upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller than varepsilon ^lambda we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than varepsilon ^lambda the concentration process takes place “after” homogenization.

No cubic integer polynomial generates a Sidon sequence

AbstractIn this paper we show that for no integer n0 and no polynomial f with integer coefficients and degree at most 3 the sequence of values can be a Sidon sequence. This settles a corresponding conjecture of Ruzsa. To prove this result for each we construct infinitely many solutions of the Diophantine equation in pairwise distinct positive integers .

Periodic orbits of the spatial anisotropic Kepler problem with anisotropic perturbations

In this article, we study the periodic orbits of the spatial anisotropic Kepler problem with anisotropic perturbations on each negative energy surface, where the perturbations are homogeneous functions of arbitrary integer degree p. By choosing the different ranges of a parameter β, we show that there exist at least 6 periodic solutions for \(p>1\), while there exist at least 2 periodic solutions for \(p\le1\) on each negative energy surface. The proofs of main results are based on symplectic Delaunay coordinates, residue theorem, and averaging theory.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/63/abstr.html

Sine-Gordon on a wormhole

In an attempt to understand the soliton resolution conjecture, we consider the sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree n) there exists a unique linearly stable soliton, which we call the n-kink. We give numerical evidence that the n-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree n. When the radius of the wormhole throat a is large enough, the convergence to the n-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the one-kink using the Soffer–Weinstein weakly nonlinear perturbation theory.

Formal Schemes of Rational Degree

Formal schemes in algebraic geometry consist of power series with integer degree, this idea can be naturally carried over to power series with rational degree. In this paper formal schemes with rational degree are constructed. These schemes are non noetherian and thus require slight modification of the standard approach. The first part of the paper constructs non noetherian continuous valuation rings from discrete valuation rings, and these rings are reffered to as ‘eka’ (one in Hindi) rings. These new rings are designed to carry the properties of discrete valuation to continuous valuation faithfully. The second part of the paper constructs non noetherian formal schemes with rational degree and shows their admissibility. The corresponding flatness and coherence is proved. Finally line bundles of rational degree are constructed and their Čech cohomology computed.

SIMULATION OF KINETIC CURVES OF THERMO-OXIDATIVE DESTRUCTION OF PROLIMERS USING DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

In the paper it is proposed to consider fractal kinetics equations for process of thermo-oxidative destruction of melt polymers in the form of differential equations of fractional order, equaling to fraction of reactant groups of polymer, which are not subjected to destruction, with order of reaction n > 1. Small νd characterize the autoslowed-down type of reaction, and at big νd <1 - the autoaccelerated thermooxidation type with an decreasing speed of oxidation reaction, respectively much more smaller and close to an integer derivative on time of the first order. Comparative analysis of calculation results with corresponding results obtained by differential equations of integer order and degree of their matching with experimental data is given. Keywords: Fractal analysis, thermo-oxidative destruction, auto-slow and auto-accelerated type of thermal oxidation, differential equations of type and fractional orders.

Emergent Hierarchy Through Conductance-based Degree Constraints

The presence of hierarchy in many real-world networks is not yet fully understood. We observe that complex interaction networks are often coarse-grain models of vast modular networks, where tightly connected subgraphs are agglomerated into nodes for simplicity of representation and computational feasibility. The emergence of hierarchy in such growing complex networks may stem from one particular property of these ignored subgraphs: their graph conductance. Being a quantification of the main bottleneck of flow through the coarse-grain node, this scalar quantity implies a structural limitation and supports the consideration of heterogeneous degree constraints. The internal conductance values of the subgraphs are mapped onto integer degree restrictions, which we call internal bottlenecks, by using inverse cumulative density functions. This leads to a hidden variable model based on the rich-get-richer scheme. It is shown that imposing such restrictions generally leads to an increased measure of hierarchy and alters the tail of the degree distribution in a predictable way. Thus, a mechanism is provided whereby inherent limitations on network flow leads to hierarchical self-organization.

Fractional Degree Stochastic Dominance

We develop a continuum of stochastic dominance rules for expected utility maximizers. The new rules encompass the traditional integer-degree stochastic dominance; between adjacent integer degrees, they formulate the consensus of individuals whose absolute risk aversion at the corresponding integer degree has a negative lower bound. By extending the concept of “uniform risk aversion” previously proposed in the literature to high-order risk preferences, we interpret the fractionalized degree parameter as a benchmark individual relative to whom all considered individuals are uniformly no less risk averse in the lottery choices. The equivalent distribution conditions for the new rules are provided, and the fractional degree “increase in risk” is defined. We generalize the previously defined notion of “risk apportionment” and demonstrate its usefulness in characterizing comparative statics of risk changes in fractional degrees. This paper was accepted by David Simchi-Levi, decision analysis.

On the representation of $k$-free integers by binary forms

Let F be a binary form with integer coefficients, non-zero discriminant and degree d with d at least 3 and let r denote the largest degree of an irreducible factor of F over the rationals. Let k be an integer with k \geq 2 and suppose that there is no prime p such that p^k divides F(a,b) for all pairs of integers (a,b) . Let R_{F,k}(Z) denote the number of k -free integers of absolute value at most Z which are represented by F . We prove that there is a positive number C_{F,k} such that R_{F,k}(Z) is asymptotic to C_{F,k} Z^{2/d} provided that k exceeds {7r}/{18} or (k,r) is (2,6) or (3,8) .

THE DISTRIBUTION OF DIVISORS OF POLYNOMIALS

Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. We determine the order of magnitude of $H_F(x, y, z)$ uniformly for $y+y/\log^C y < z\le y^2$ and $y\le x^{1-\delta}$, showing that the order is the same as the order of $H(x,y,z)$, the number of positive integers $n\le x$ with a divisor in $(y,z]$. Here $C$ is an arbitrarily large constant and $\delta>0$ is arbitrarily small.

Asymptotics of the generalized Gegenbauer functions of fractional degree

The generalized Gegenbauer functions of fractional degree (GGF-Fs), denoted by rGν(λ)(x) (right GGF-Fs) and lGν(λ)(x) (left GGF-Fs) with x∈(−1,1),λ>−1∕2 and real ν≥0, are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGF-Fs become indispensable for optimal error estimates of polynomial approximation to singular functions, and have intimate relations with several families of nonstandard basis functions recently introduced for solving fractional differential equations. However, some properties of GGF-Fs, which are important pieces for the analysis and applications, are unknown or under explored. The purposes of this paper are twofold. The first is to show that for λ>−1∕2 and x=cosθ with θ∈(0,π),(sinθ)λrGν(λ)(cosθ)=2λΓ(λ+1∕2)π(ν+λ)λcos((ν+λ)θ−λπ∕2)+Rν(λ)(θ),and derive the precise expression of the “residual” term Rν(λ)(θ) for all real ν≥ν0 (with some ν0>0). With this at our disposal, we obtain the bounds of GGF-Fs uniform in ν. Under an appropriate weight function, the bounds are uniform for θ∈[0,π] as well. Moreover, we can study the asymptotics of GGF-Fs with large fractional degree ν. The second is to present miscellaneous properties of GGF-Fs for better understanding of this family of useful special functions.

Completely monotonic integer degrees for a class of special functions

Let $f_{n}(x)$ $\left(n = 0, 1, \cdots\right)$ be the remainders for the asymptotic formula of $\ln\Gamma (x)$ and $R_{n}(x) = \left(-1\right)^{n}f_{n}(x)$. This paper introduced the concept of completely monotonic integer degree and discussed the ones for the functions $\left(-1\right)^{m}R_{n}^{(m)}(x)$, then demonstrated the correctness of the existing conjectures by using a elementary simple method. Finally, we propose some operational conjectures which involve the completely monotonic integer degrees for the functions $\left(-1\right) ^{m}R_{n}^{(m)}(x)$ for $m = 0, 1, 2, \cdots$.

Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems

A Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems is presented via an auxiliary integer degree polynomial. The presented Routh test is a classical Routh table test on the auxiliary integer degree polynomial derived from and for the commensurate fractional degree polynomial. The theoretical proof of this proposed approach is provided by utilizing Argument principle and Cauchy index. Illustrative examples show efficiency of the presented approach for stability test of commensurate fractional degree polynomials and commensurate fractional order systems. So far, only one Routh-type test approach [1] is available for the commensurate fractional degree polynomials in the literature. Thus, this classical Routh-type test approach and the one in [1] both can be applied to stability analysis and design for the fractional order systems, while the one presented in this paper is easy for peoples, who are familiar with the classical Routh table test, to use.

On the representation of integers by binary forms

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_F$ such that $R_F(Z)$ is asymptotic to $C_F Z^{\frac{2}{d}}$.