Rm Sgn Research Articles

Extreme and moderate solutions of nonoscillatory second order half-linear differential equations

An existence and asymptotic theory is built for second order half-linear differential equations of the form $${\rm (A)}\qquad (p(t)|x^{\prime }|^\alpha {\rm sgn} \,x^{\prime })^{\prime } =q(t)|x|^\alpha {\rm sgn}\, x, \quad t \geq a \gt 0, $$ where $\alph

Edge States of a Periodic Chain with Four-Band Energy Spectrum

Tight-binding model on a finite chain is studied with four-fold alternated hopping parameters $t_{1,2,3,4}$. Imposing the open boundary conditions, the corresponding recursion is solved analytically with special attention paid to the occurrence of edge states. Corresponding results are strongly corroborated by numeric calculations. It is shown that in the system there exist four different edge phases if the number of sites is odd, and eight edges phases if the chain comprises even number of sites. Phases are labelled by $\sigma_1\equiv{\rm sgn}(t_1t_3-t_2t_4)$, $\sigma_2\equiv{\rm sgn}(t_1t_4-t_2t_3)$ and $\sigma_3\equiv{\rm sgn}(t_1t_2-t_3t_4)$. It is shown that these quantities represent gauge invariant topological indices emerging in the corresponding infinite chains.

Exponential estimates for solutions of half-linear differential equations

This paper is concerned with estimates, unimprovable in a certain sense, for positive solutions to the half-linear differential equation ( $${|y^{\prime}|^{\alpha - 1} {\rm sgn} y^{\prime})^{\prime} = p(t) |y|^{\alpha - 1} {\rm sgn} y}$$ , where p is a continuous nonnegative function on $${[0, \infty)}$$ and $${\alpha> 1}$$ . It is shown that any positive increasing solution y of the equation satisfies $${y(t) \geqq y(0) exp{h \int_{0}^{t} p^{\frac{1}{\alpha}}(s) {\rm d}s}}$$ , with $${h< (\alpha - 1)^{-\frac{1}{\alpha}}}$$ , for all t on the complement of a set of finite Lebesgue measure. Under an additional assumption, this estimate holds for all t. Further, a condition is established which guarantees that the equation has exponentially increasing solutions and exponentially decreasing solutions.

Spin representations of twisted central products of double covering finite groups and the case of permutation groups

Let $S$ be a finite group with a character, $\rm sgn$, of order 2, and $S'$ its central extension by a group $Z=\langle z\rangle$ of order 2. A representation $\pi$ of $S'$ is called {\it spin} if $\pi(z\sigma')=-\pi(\sigma')$ $(\sigma'\in S')$, and the set of all equivalence classes of spin irreducible representations (= IRs) of $S'$ is called the {\it spin dual} of $S'$. Take a finite number of such triplets $(S'_j,z_j,{\rm sgn}_j)$ $(1\le j\le m)$. We define twisted central product $S'=S'_1\hat{*}S'_2\hat{*}\cdots\hat{*}S'_m$ as a double covering of $S=S_1\times\cdots \times S_m$, $S_j=S'_j/\langle z_j\rangle$, and for spin IRs $\pi_j$ of $S'_j$, define twisted central product $\pi=\pi_1\hat{*}\pi_2\hat{*}\cdots\hat{*}\pi_m$ as a spin IR of $S'$. We study their characters and prove that the set of spin IRs $\pi$ of this type gives a complete set of representatives of the spin dual of $S'$. These results are applied to the case of representation groups $S'$ for $S={\mathfrak S}_n$ and ${\mathfrak A}_n$, and their (Frobenius-)Young type subgroups.

Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations

We provide lower bounds for the maximum number of limit cycles for the m-piecewise discontinuous polynomial differential equations $${\dot{x} = y+{\rm sgn}(g_m(x, y))F(x)}$$ , $${\dot{y} = -x}$$ , where the zero set of the function sgn(g m (x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane into sectors of angle 2π/m, and sgn(z) denotes the sign function.

The Intersection of Two Halfspaces Has High Threshold Degree

The threshold degree of a Boolean function $f\!\!\,: \{0,1\}^n \rightarrow \{-1, +1\}$ is the least degree of a real polynomial $p$ such that $f(x)\equiv{{\rm sgn}\,p(x)}.$ We construct two halfspaces on $\{0,1\}^n$ whose intersection has threshold degree $\Theta(\sqrt n),$ an exponential improvement on previous lower bounds. This solves an open problem due to Klivans [A Complexity-Theoretic Approach to Learning, Ph.D. thesis, MIT, Cambridge, MA, 2002] and rules out the use of perceptron-based techniques for PAC learning the intersection of two halfspaces, a central unresolved challenge in computational learning. We also prove that the intersection of two majority functions has threshold degree $\Omega({\rm log}\,n),$ which is tight and settles a conjecture of O'Donnell and Servedio [Proceedings of the $35$th Annual ACM Symposium on Theory of Computing (STOC), 2003, pp. 325--334]. Our proof consists of two parts. First, we show that for any nonconstant Boolean functions $f$ and $g,$ the intersection $f(x)\wedge g(y)$ has threshold degree $O(d)$ if and only if $\|f-F\|_\infty + \|g-G\|_\infty < 1$ for some rational functions $F,$ $G$ of degree $O(d).$ Second, we determine the least degree required for approximating a halfspace and a majority function to any given accuracy by rational functions. Our technique further allows us to obtain direct sum theorems for polynomial representations of composed Boolean functions. In particular, we give an improved lower bound on the approximate degree of the AND-OR tree.

The Tricomi problem of a quasi-linear Lavrentiev–Bitsadze mixed type equation

In this paper, we consider the Tricomi problem of a quasi-linear Lavrentiev–Bitsadze mixed type equation $$\begin{array}{lll}({\rm sgn}\,u_y) {\frac{\partial ^2 u}{\partial x^2}} + {\frac{\partial ^2 u}{\partial y^2}}-1=0,\end{array}$$ whose coefficients depend on the first-order derivative of the unknown function. We prove the existence of solution to this problem by using the hodograph transformation. The method can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.

Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Nous considérons la solution d’une équation différentielle stochastique, dirigée par un mouvement brownien linéaire standard, dont le terme de dérive varie avec le temps $\rho\operatorname{sgn}(x)|x|^{\alpha}/t^{\beta}$. Ce processus peut être vu comme un mouvement brownien évoluant dans un potentiel dépendant du temps, éventuellement singulier. Nous montrons des résultats d’existence et d’unicité et nous étudions le comportement asymptotique de la solution. Les propriétés de récurrence ou de transience de cette diffusion sont décrites en fonction des paramètres $\rho$, $\alpha$ et $\beta$, et nous donnons les vitesses de transience et d’explosion. Des résultats de convergence en loi et des lois de type logarithme itéré sont également obtenus.

Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation

Sufficient conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the second-order nonlinear differential equation $$x^{\prime\prime}(t)+q(t)|x(t)|^{\gamma}\,{\rm sgn}\, x(t)=0, \quad \quad (A)$$where γ is a positive constant different from 1 and q : [a, ∞) → (0, ∞) is a continuous integrable function. We show how an application of the theory of regular variation gives the possibility of determining the precise asymptotic behavior of solutions of both superlinear and sublinear equation (A).

A role of the coefficient of the differential term in qualitative theory of half-linear equations

The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation $(r(t)\Phi (y^{\Delta}))^{\Delta} +p(t)\Phi (y^{\sigma})=0$, where $\Phi (u)=|u|^{\alpha -1}\mathop{\rm sgn}u$ with $\alpha >1$. We discuss sign and smoothness conditions posed on $r$, (non)availability of some transformations, and mainly we show how the behavior of $r$, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati type technique, which are supplemented by some new observations.

On the asymptotic behavior at infinity of solutions to quasi-linear differential equations

Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation $$y^{(n)}+\sum _{j=0}^{n-1}a_j(x)y^{(j)}+p(x)|y|^k \mathop {\rm sgn} y =0$$ with $ n\ge 1$, real (not necessarily natural) $k>1$, and continuous functions $p(x)$ and $a_j(x)$ defined in a neighborhood of $+\infty $. For this equation with positive potential $p(x)$ a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. \endgraf Sufficient conditions are obtained for existence of solution to this equation which is equivalent to a polynomial.

Bose-Einstein condensation and symmetry breaking

Adding a gauge symmetry breaking field -\nu\sqrt{V}(a_0+a_0^*) to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis \lim_{V\to\infty}< a_0>/\sqrt{V}={\rm sgn}\nu \lim_{V\to\infty}\sqrt{< a_0^*a_0>/V} where the averages are taken in the ground state or in thermal equilibrium states. Letting \nu tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode (k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.

Periodic solutions of non-smooth friction oscillators

We study the differential equation $x''+g(x')+\mu(x)\,{\rm sgn}\, x'+f(x)=\varphi(t)$ with T-periodic right-hand side, which models e.g. a mechanical system with one degree of freedom subjected to dry friction and periodic external force. If, in particular, the damping term g is present and acts, up to a bounded difference, like a linear damping, we get existence of a T-periodic solution.¶In the more difficult case g = 0, we concentrate on the model equation $x''+\mu(x)\,{\rm sgn}\,x'+x=\varphi(t)$ and obtain sufficient conditions for the existence of a T-periodic solution by application of Brouwer's fixed point theorem. For this purpose we show that a certain associated autonomous differential equation admits a periodic orbit such that the surrounded set (minus some neighborhood of the equilibria) is forward invariant for the equation above. Under additional assumptions on μ we prove boundedness of all solutions.¶Finally, we provide a principle of linearized stability for periodic solutions without deadzones, where the “linearized” differential equation is an impulsive Hill equation.

An Autocorrelation Criterion for the Time-Diversity Reception of Speech Over Burst-Error Channels

This paper proposes a new approach to signal selection in time-diversity systems. Specifically, we consider the problem of digital speech transmission over a burst-error channel using two-channel time-diversity reception. Let every speech segment (of length W) be transmitted twice so that at least one of the transmissions escapes an error burst, with a certain useful probability. Let the received speech segments be Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> . We propose an autocorrelation-maximizing signal selection procedure of the following form. Select Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> (or Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ) as the “cleaner” speech segment according as C(Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , W) ≧ (or <) C(Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , W), where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$ C(Y_u,W)=\sum_{r=2}^W({\rm sgn}\ Y_{ur}\cdot{\rm sgn}\ Y_{u(r-1)})/(W-1);u=1,2.$</tex> Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uτ</inf> is the speech amplitude at sample τ in Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</inf> , W is a computational window that is typically a few milliseconds long, and sgn Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uτ</inf> is a polarity function that is assumed to have zero mean and unit variance. The use of sgn Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uτ</inf> instead of Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uτ</inf> leads to a simply implemented selection procedure, and computer simulations have demonstrated its practical utility. For example, in one study of three-bit dpcm coding, autocorrelation-based burst-error detection proved to be more useful than a procedure where dpcm samples were error-protected on a bit-by-bit basis, rather than in blocks.