On super-linear Emden–Fowler type differential equations

We study the second order Emden–Fowler type differential equation

(a(t)|x^′|^αsgnx^′)^′+b(t)|x|^βsgnx=0${(a(t){\left|x^{\prime }\right|}^{\alpha }\mathrm{sgn}{x}^{\prime })}^{\prime }+b(t){|x|}^{\beta }\mathrm{sgn}x=0$

in the super-linear case α<β$\alpha <\beta$. Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.     

Lipschitz gradients for global optimization in a one-point-based partitioning scheme

A global optimization problem is studied where the objective function f(x)$f\left(x\right)$ is a multidimensional black-box function and its gradient f^′(x)$f^{\prime }\left(x\right)$ satisfies the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant K$K$. Different methods for solving this problem by using an a priori given estimate of K$K$, its adaptive estimates, and adaptive estimates of local Lipschitz constants are known in the literature. Recently, the authors have proposed a one-dimensional algorithm working with multiple estimates of the Lipschitz constant for f^′(x)$f^{\prime }\left(x\right)$ (the existence of such an algorithm was a challenge for 15 years). In this paper, a new multidimensional geometric method evolving the ideas of this one-dimensional scheme and using an efficient one-point-based partitioning strategy is proposed. Numerical experiments executed on 800 multidimensional test functions demonstrate quite a promising performance in comparison with popular DIRECT-based methods.     

A characterization of quintic helices

A polynomial curve of degree 5, α,$\alpha ,$ is a helix if and only if both ∥α^′∥$\parallel {\alpha }^{\prime }\parallel$ and ∥α^′∧α^″∥$\parallel {\alpha }^{\prime }\wedge {\alpha }^{″}\parallel$ are polynomial functions.     

Splitting lemma in the infinity and a strong resonant problem with periodic nonlinearity

In this paper we study the periodic boundary problem: −u^″−u+g(u)=h(t),u(0)=u(2π),u^′(0)=u^′(2π)$-u^{″}-u+g\left(u\right)=h\left(t\right),u\left(0\right)=u\left(2\pi \right),u^{\prime }\left(0\right)=u^{\prime }\left(2\pi \right)$ with periodic nonlinearity g$g$. We proved that the problem has infinite solutions. The method we used is a variational reduction method of T. Bartsch and S.J. Li.     

Invariant tori and unboundedness for second order differential equations with asymmetric nonlinearities depending on the derivatives

In this paper, the boundedness and unboundedness of solutions of the oscillator

x^″+ax⁺−bx⁻+φ(x)ψ(x^′)+f(x)+g(x^′)=p(t)$x^{″}+a{x}^{+}-b{x}^{-}+\phi \left(x\right)\psi \left(x^{\prime }\right)+f\left(x\right)+g\left(x^{\prime }\right)=p\left(t\right)$

Estimating the ground state energy of the Schrödinger equation for convex potentials

In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε$\epsilon$. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d$d$. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$ and ε⁻¹${\epsilon }^{-1}$, while the number of qubits is polynomial in d$d$ and logε⁻¹$log{\epsilon }^{-1}$. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1)$1-\delta ,\delta \in (0,1)$, with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$, ε⁻¹${\epsilon }^{-1}$ and δ⁻¹${\delta }^{-1}$, while the number of qubits is polynomial in d$d$, logε⁻¹$log{\epsilon }^{-1}$ and logδ⁻¹$log{\delta }^{-1}$.     

Preserving closedness of operators under summation

We give a sufficient condition for the sum of two closed operators to be closed. In particular, we study the sum of two sectorial operators with the sum of their sectoriality angles greater than π  . We show that if one of the operators admits bounded H^∞$H^{\infty }$-calculus and the resolvent of the other operator satisfies a boundedness condition stronger than the standard sectoriality, but weaker than the bounded imaginary powers property in the case of UMD spaces, then the sum is closed. We apply the result to the abstract parabolic problem and give a sufficient condition for L^p$L^p$-maximal regularity.     

Müntz-type theorems on the half-line with weights

We consider the linear span S   of the functions t^_ak$t^{a_k}$ (with some _ak>0$a_k>0$) in weighted L²$L^2$ spaces, with rather general weights. We give one necessary and one sufficient condition for S to be dense. Some comparisons are also made between the new results and those that can be deduced from older ones in the literature.     

Hypersurfaces in hyperbolic space with support function

Based on [19], we develop a global correspondence between immersed hypersurfaces ?:Mⁿ→Hⁿ⁺¹$?:{\mathrm{M}}^n\to {\mathbb{H}}^{n+1}$ satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e^2ρ_gSn$e^{2\rho }{g}_{{\mathbb{S}}^n}$ on domains Ω in the boundary Sⁿ${\mathbb{S}}^n$ at infinity of Hⁿ⁺¹${\mathbb{H}}^{n+1}$, where ρ   is the horospherical support function, _∂∞?(Mⁿ)=∂Ω${\partial }_{\infty }?({\mathrm{M}}^n)=\partial \mathrm{\Omega }$, and Ω is the image of the Gauss map G:Mⁿ→Sⁿ$G:{\mathrm{M}}^n\to {\mathbb{S}}^n$. To do so we first establish results on when the Gauss map G:Mⁿ→Sⁿ$G:{\mathrm{M}}^n\to {\mathbb{S}}^n$ is injective. We also discuss when an immersed horospherically concave hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed hypersurfaces in Hⁿ⁺¹${\mathbb{H}}^{n+1}$ and conformal metrics on domains in Sⁿ${\mathbb{S}}^n$. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in Hⁿ⁺¹${\mathbb{H}}^{n+1}$ of constant mean curvature.     

On singular foliations on the solid torus

We study smooth foliations on the solid torus S¹×D²$S^1\times D^2$ having S¹×{0}$S^1\times \{\mathbf{0}\}$ and S¹×∂D²$S^1\times \partial D^2$ as the only compact leaves and S¹×{0}$S^1\times \{\mathbf{0}\}$ as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc.     

Trace formulas for additive and non-additive perturbations

Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair {H^′,H}$\{H^{\prime },H\}$ of maximal accumulative operators has been proved. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H   and H^′=H+V$H^{\prime }=H+V$ are maximal accumulative and V is trace class, we prove the existence of a summable   complex-valued spectral shift function. We also obtain trace formulas for pairs {H,H^?}$\{H,H^{?}\}$assuming only that H and  H^?$H^{?}$are resolvent comparable. In this case the determinant of the characteristic function of H is involved in trace formulas.     

On the tractability of linear tensor product problems in the worst case

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem to be weakly tractable in the worst case. The complexity of linear tensor product problems in the worst case depends on the eigenvalues _{λi}i∈N${\left\{{\lambda }_i\right\}}_{i\in \mathbb{N}}$ of a certain operator. It is known that if _λ1=1${\lambda }_1=1$ and _λ2∈(0,1)${\lambda }_2\in (0,1)$ then _λn=o((lnn)⁻²)${\lambda }_n=o\left({(lnn)}^{-2}\right)$, as n→∞$n\to \infty$, is a necessary condition for a problem to be weakly tractable. We show that this is a sufficient condition as well.     

Hermite–Birkhoff–Obrechkoff four-stage four-step ODE solver of order 14 with quantized step size

A four-stage Hermite–Birkhoff–Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form y^′=f(t,y)$y^{\prime }=f(t,y)$ with initial conditions y(_t0)=_y0$y\left(t_0\right)=y_0$. Its formula uses y^′$y^{\prime }$, y^″$y^{″}$ and y^?$y^{?}$ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived only once to obtain the values of Hermite–Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C ++, HBOQ(14)4 is superior to the Dormand–Prince Runge–Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/~remi.     

A new condition for maximal monotonicity via representative functions

In this paper we give a weaker sufficient condition for the maximal monotonicity of the operator S+A^∗TA$S+A^{\ast }TA$, where S:X?X^∗$S:X?X^{\ast }$, T:Y?Y^∗$T:Y?Y^{\ast }$ are two maximal monotone operators, A:X→Y$A:X\to Y$ is a linear continuous mapping and X,Y$X,Y$ are reflexive Banach spaces. We prove that our condition is weaker than the generalized interior-point conditions given so far in the literature. This condition is formulated using the representative functions of the operators involved. In particular, we rediscover some sufficient conditions given in the past using the so-called Fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator, respectively.