The Shuddering Pendulum

This paper treats the rich mathematical structure of the (dimensionless) equation of motion governing the behavior of an elastically restrained simple pendulum subject to a downward force of magnitude f(t) applied to its bob with $\dot{f}(t)>0$\dot{f}(t)>0 for all t>0 and f(t)→∞ as t→∞:

Gaussian upper bounds for fundamental solutions of a family of parabolic equations

We study the family of divergence-type second-order parabolic equations w_e(x)\frac?u?t=div(a(x)w_e(x) ?u), x ? \mathbbRⁿ{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and w_e=1{\omega_\varepsilon=1} for x _n  < 0 and w_e=e{\omega_\varepsilon=\varepsilon} for x _n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .     

On the existence and stability of periodic orbits in non ideal problems: General results

In this work, motivated by non-ideal mechanical systems, we investigate the following O.D.E. [(x)\dot] = f (x) + eg (x, t) + e²[^(g)] (x, t, e){\dot{x} = f (x) + \varepsilon g (x, t) + \varepsilon^{2}\widehat{g} (x, t, \varepsilon)} , where x ? W ì \mathbbRⁿ{x \in \Omega \subset \mathbb{R}^n} , g,[^(g)]{g,\widehat{g}} are T periodic functions of t and there is a ₀ ∈ Ω such that f ( a ₀) = 0 and f ′( a ₀) is a nilpotent matrix. When n = 3 and f (x) = (0, q (x ₃) , 0) we get results on existence and stability of periodic orbits. We apply these results in a non ideal mechanical system: the Centrifugal Vibrator. We make a stability analysis of this dynamical system and get a characterization of the Sommerfeld Effect as a bifurcation of periodic orbits.     

Existence of Positive Periodic Solutions for the Liénard Differential Equations with Weakly Repulsive Singularity

In this paper we discuss the existence of positive T-periodic solutions for the following second order differential equation

The stability of the elliptic equilibrium of planar quasi-periodic Hamiltonian systems

In this paper, we study the planar Hamiltonian system  = J (A(θ)x + ▽f(x, θ)), θ = ω, x ∈ R2 , θ∈ Td , where f is real analytic in x and θ, A(θ) is a 2 × 2 real analytic symmetric matrix, J = (1-1 ) and ω is a Diophantine vector. Under the assumption that the unperturbed system  = JA(θ)x, θ = ω is reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system.     

Multiple periodic solutions for nonautonomous Hamiltonian systems

In this paper, by the use of minimax method, we obtain some existence and multiplicity theorems for periodic solutions of nonautonomous Hamiltonian systems with bounded nonlinearity of the type:¶ J [(x)\dot] + ?H(t, x) + e(t) = 0. J \dot x + \nabla H(t, x) + e(t) = 0.     

Coincidence theorems and its applications to equilibrium problems

Given two maps h : X ×K ? \mathbbR{h : X \times K \rightarrow \mathbb{R}} and g : X → K such that, for all x ? X, h(x, g(x)) = 0{x \in X, h(x, g(x)) = 0} , we consider the equilibrium problem of finding [(x)\tilde] ? X{\tilde{x} \in X} such that h([(x)\tilde], g(x)) 3 0{h(\tilde{x}, g(x)) \geq 0} for every x ? X{x \in X} . This question is related to a coincidence problem.     

On certain functional equations related to Jordan triple (q, f){(\theta, \phi)}-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let q{\theta} and f{\phi} be automorphisms of R. Suppose T:R? R{T:R\rightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}, for all pairs x,y ? R{x,y\in R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=q\theta (x)+\phi (x)q}, for all x ? R{x\in R} and some fixed element q ? Q_s{q\in Q_{s}}.     

On the selfsimilar solutions of a diffusion convection equation

This paper is concerned with the equation¶¶ div(| ?u| ^p-2?u)+e| ?U| ^q+bx?U+aU=0, for  x ? \mathbbR^N div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.     

Optimal estimates on rotation number of almost periodic systems

In this paper, we will give some optimal estimates on the rotation number of the linear equation $\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x = 0,$\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x = 0, and that of the asymmetric equation: $\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x_{ + } + q(t)x_{ - } = 0,$\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x_{ + } + q(t)x_{ - } = 0, where p(t) and q(t) are almost periodic functions and x ₊ = max{ x,0} ,  x _- = min{ x,0} .x_{ + } = \max \{ x,0\} ,\;x_{ - } = \min \{ x,0\} . These estimates are obtained by introducing some kind of new norms in the spaces of almost periodic functions.     

On wave equations with supercritical nonlinearities

We prove that the solution operators e_t (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]¹ ?L_r+1) ×L₂(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]^s_q￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here e_t(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of {

Irregular Wavelet Frames and Gabor Frames

Given g { l^\fracn2 g( l_j x - kb ) }_jezjezⁿ ,where  l_j \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L ²(R ⁿ ) are given. For a class of functions g{ e^{zrib( j,x )} g( x - l_k ) }_{jezⁿ ,kez}\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame.     

Long-time behavior of solutions to Hamilton–Jacobi equations with quadratic gradient term

We study a rate of convergence appearing in the long-time behavior of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation

Simultaneous Abel equations

Let S{\mathcal{S}} be a set of homeomorphisms of an open interval such that the group generated by S{\mathcal{S}} is disjoint, i.e., the graphs of any two distinct functions in it do not intersect. We give necessary and sufficient conditions for the system of Abel equations

Semiconcavity estimates for viscous Hamilton–Jacobi equations

We present sharp Hessian estimates of the form D² S^e(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation

On the Mean Value of the Index of Composition of an Integer

For each integer n l(n)=[(log n)/(log g(n))]\lambda(n)={{\rm log}\, n\over{\rm log}\, \gamma(n)} be the index of composition of n, where g(n)=?_p|np\gamma(n)=\prod_{p\vert n}p . For convenience, we write ?_{x ￡ n ￡ x+?x}l(n)\sum_{x\le n\le x+\sqrt{x}}\lambda(n) and ?_{n ￡ x}l(n)\sum_{n\le x}\lambda(n) , as well as for ?_{x ￡ n ￡ x+?x}1/l(n)\sum_{x\le n\le x+\sqrt{x}}1/\lambda(n) and ?_{n ￡ x}1/l(n)\sum_{n\le x}1/\lambda(n) . Finally we study the sum of running over shifted primes.     

Hamiltonization of Higher-Order Nonlinear Ordinary Differential Equations and the Jacobi Last Multiplier

It is known that Jacobi’s last multiplier is directly connected to the deduction of a Lagrangian via Rao’s formula (Madhava Rao in Proc. Benaras Math. Soc. (N.S.) 2:53–59, 1940). In this paper we explicitly demonstrate that it also plays an important role in Hamiltonian theory. In particular, we apply the recent results obtained by Torres del Castillo (J. Phys. A Math. Theor. 43:265202, 2009) and deduce the Hamiltonian of a second-order ODE of the Lienard type, namely, [(x)\ddot]+f(x)[(x)\dot]²+g(x)=0\ddot{x}+f(x){\dot{x}}^{2}+g(x)=0. In addition, we consider cases where the coefficient functions may also depend on the independent variable t. We illustrate our construction with various examples taken from astrophysics, cosmology and the Painlevé-Gambier class of differential equations. Finally we discuss the Hamiltonization of third-order equations using Nambu-Hamiltonian mechanics.     

Nonoscillation theorems for certain second order nonlinear differential equations

Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation (r(t)y(x(t))|x^￠(t)|^p-1x^￠(t))^￠+c(t)f(x(t))=0,    t 3 t₀,(r(t)\psi(x(t))\vert x^{\prime}(t)\vert^{p-1}x^{\prime}(t))^{\prime}+c(t)f(x(t))=0,\quad t\ge t_0,     

On some Dirichlet series related to the Riemann zeta function,I

On the assumption of the truth of the Riemann hypothesis for the Riemann zeta function we construct a class of modified von-Mangoldt functions with slightly better mean value properties than the well known function L\Lambda . For every e ? (0,1/2)\varepsilon \in (0,1/2) there is a [(L)\tilde] : \Bbb N ? \Bbb C\tilde {\Lambda} : \Bbb N \to \Bbb C such that¶ i) [(L)\tilde] (n) = L (n) (1 + O(n^-1/4  logn))\tilde {\Lambda} (n) = \Lambda (n) (1 + O(n^{-1/4\,} \log n)) and¶ii) ?_{n \leqq x} [(L)\tilde] (n) (1- [(n)/(x)]) = [(x)/2] + O(x^1/4+e) (x \geqq 2).\sum \limits_{n \leqq x} \tilde {\Lambda} (n) \left(1- {{n}\over{x}}\right) = {{x}\over{2}} + O(x^{1/4+\varepsilon }) (x \geqq 2).¶Unfortunately, this does not lead to an improved error term estimation for the unweighted sum ?_{n \leqq x} [(L)\tilde] (n)\sum \limits_{n \leqq x} \tilde {\Lambda} (n), which would be of importance for the distance between consecutive primes.     

The gradient and heavy ball with friction dynamical systems: the quasiconvex case

We consider the gradient system $\dot x(t)+\nabla\phi(x(t))=0$ and the so-called heavy ball with friction dynamical system $\ddot x(t) +\lambda\dot x(t)+\nabla\phi(x(t))=0$ , as well as an implicit discrete (proximal) version of it, and study the asymptotic behavior of their solutions in the case of a smooth and quasiconvex objective function Φ. Minimization properties of trajectories are obtained under various additional assumptions. We finally show a minimizing property of the heavy ball method which is not shared by the gradient method: the genericity of the convergence of each trajectory, at least when Φ is a Morse function, towards local minimum of Φ.