scholarly journals The $Q_2$-Free Process in the Hypercube

10.37236/8864 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
J. Robert Johnson ◽  
Trevor Pinto
Keyword(s):  
Differential Equations ◽  
High Probability ◽  
Short Note ◽  
Heuristic Argument ◽  
Open Questions ◽  
Free Process ◽  
Random Subgraph ◽  
Analogous Process

The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the $Q_2$-free process in $Q_d$ and the random subgraph of $Q_d$ it generates. Our main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.

scholarly journals Semi-groups and differential equations

1969 ◽  
Vol 10 (1-2) ◽  
pp. 173-176
Author(s):  
J. D. Gray
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Short Note ◽  
Constant Coefficients

In this short note we shall apply the theory of semi-groups of operators, (cf: Hille and Phillips, [2]), to the problem of representing solutions of certain differential equations with non-constant coefficients. When the coefficients are constant, this representation reduces to the usual Laplace transform solution of the relevant equation.

scholarly journals PARAMETRIZING ELLIPTIC CURVES BY MODULAR UNITS

2015 ◽  
Vol 100 (1) ◽  
pp. 33-41 ◽  
Author(s):  
FRANÇOIS BRUNAULT
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Recent Work ◽  
Short Note ◽  
Torsion Subgroup ◽  
Modular Functions ◽  
Open Questions ◽  
Modular Units

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

Harmonic Spinors on Hyperbolic Space

1993 ◽  
Vol 36 (3) ◽  
pp. 257-262 ◽  
Author(s):  
Pierre-Yves Gaillard
Keyword(s):  
Differential Equations ◽  
Symmetric Space ◽  
Representation Theory ◽  
Hyperbolic Space ◽  
Harmonic Functions ◽  
Short Note ◽  
Harmonic Forms ◽  
Semisimple Symmetric Space ◽  
Invariant Differential ◽  
Harmonic Spinors

AbstractThe purpose for this short note is to describe the space of harmonic spinors on hyperbolicn-spaceHn. This is a natural continuation of the study of harmonic functions onHnin [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms onHnwere considered. The connection between invariant differential equations on a Riemannian semisimple symmetric spaceG/Kand homological aspects of the representation theory ofG, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

scholarly journals Parametrically Excited Oscillations of Second-Order Functional Differential Equations and Application to Duffing Equations with Time Delay Feedback

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Mervan Pašić
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Second Order ◽  
Oscillatory Behaviour ◽  
Quasilinear Equations ◽  
Open Questions ◽  
Control Gain ◽  
Main Equation ◽  
Functional Differential ◽  
Time Delayed Feedback

We study oscillatory behaviour of a large class of second-order functional differential equations with three freedom real nonnegative parameters. According to a new oscillation criterion, we show that if at least one of these three parameters is large enough, then the main equation must be oscillatory. As an application, we study a class of Duffing type quasilinear equations with nonlinear time delayed feedback and their oscillations excited by the control gain parameter or amplitude of forcing term. Finally, some open questions and comments are given for the purpose of further study on this topic.

scholarly journals Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Nathan Bush ◽  
Baasansuren Jadamba ◽  
Akhtar A. Khan ◽  
Fabio Raciti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Optimization Problem ◽  
Short Note ◽  
Variable Parameter ◽  
Partial Differential ◽  
Convergence Results ◽  
Equation Error

AbstractThe objective of this short note is to employ an equation error approach to identify a variable parameter in fourth-order partial differential equations. Existence and convergence results are given for the optimization problem emerging from the equation error formulation. Finite element based numerical experiments show the effectiveness of the proposed framework.

Some Open Questions in the Numerical Analysis of Singularly Perturbed Differential Equations

2015 ◽  
Vol 15 (4) ◽  
pp. 531-550 ◽  
Author(s):  
Hans-Görg Roos ◽  
Martin Stynes
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Singularly Perturbed ◽  
Defect Correction ◽  
Finite Difference Approximations ◽  
Difference Approximations ◽  
Open Questions ◽  
Convergence Results

AbstractSeveral open questions in the numerical analysis of singularly perturbed differential equations are discussed. These include whether certain convergence results in various norms are optimal, when supercloseness is obtained in finite element solutions, the validity of defect correction in finite difference approximations, and desirable adaptive mesh refinement results that remain to be proved or disproved.

scholarly journals On eigenvalues of a matrix arising in energy-preserving/dissipative continuous-stage Runge-Kutta methods

2021 ◽  
Vol 10 (1) ◽  
pp. 34-39
Author(s):  
Yusaku Yamamoto
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Collocation Method ◽  
Short Note ◽  
Complex Eigenvalues ◽  
Hilbert Matrix ◽  
Runge Kutta

Abstract In this short note, we define an s × s matrix Ks constructed from the Hilbert matrix H s = ( 1 i + j - 1 ) i , j = 1 s {H_s} = \left( {{1 \over {i + j - 1}}} \right)_{i,j = 1}^s and prove that it has at least one pair of complex eigenvalues when s ≥ 2. Ks is a matrix related to the AVF collocation method, which is an energy-preserving/dissipative numerical method for ordinary differential equations, and our result gives a matrix-theoretical proof that the method does not have large-grain parallelism when its order is larger than or equal to 4.

Quantum Mechanics and a Completely Integrable Dynamical System

1987 ◽  
Vol 42 (8) ◽  
pp. 819-824 ◽  
Author(s):  
W.-H. Steeb ◽  
A. J. van Tonder
Keyword(s):  
Differential Equations ◽  
Symmetric Matrix ◽  
Energy Levels ◽  
Equations Of Motion ◽  
Integrable Dynamical System ◽  
Finite Dimensional ◽  
Open Questions ◽  
The Matrix ◽  
Constants Of Motion ◽  
Energy Dependent

From the eigenvalue equation (H0 + λV) \ ψn(λ)) = En (λ) | ψn( λ ) ) one can derive an autonomous system of first order differential equations for the eigenvalues En (λ) and the matrix elements Vmn(X) = ( ψm(λ) \ V \ ψn(λ), where λ is the independent variable. If the initial values En (λ = 0) and ψn (λ = 0) are known the differential equations can be solved. Thus one finds the “motion” of the energy levels En(λ). Here we give two applications of this technique. Furthermore we describe the connection with the stationary state perturbation theory. We also derive the equations of motion for the extended case H = H0 + λ\ Vt + λ2 V2. Finally we investigate the case where the Hamiltonian is given by a finite dimensional symmetric matrix and derive the energy dependent constants of motion. Several open questions are also discussed.

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