Search results for "combinatoric"

Two-Dimensional Differential Systems with Asymmetric Principal Part

2013

We consider the Sturm–Liouville nonlinear boundary value problem $$\displaystyle\begin{array}{rcl} \left \{\begin{array}{l} x^{\prime} = f(t,y) + u(t,x,y),\\ y^{\prime} = -g(t, x) + v(t, x, y), \end{array} \right.& & {}\\ \begin{array}{l} x(0)\cos \alpha - y(0)\sin \alpha = 0,\\ x(1)\cos \beta - y(1)\sin \beta = 0, \end{array} & & {}\\ \end{array}$$ assuming that the limits \(\lim _{y\rightarrow \pm \infty }\frac{f(t,y)} {y} = f_{\pm }\), \(\lim _{x\rightarrow \pm \infty }\frac{g(t,x)} {x} = g_{\pm }\) exist. Nonlinearities u and v are bounded. The system includes various cases of asymmetric equations (such as the Fucik one). Two classes of multiplicity results are discussed. The first one …

Nonlinear anisotropic heat conduction in a transformer magnetic core

1996

In this chapter we deal with a quasilinear elliptic problem whose classical formulation reads: Find \( u \in {C^1}\left( {\bar \Omega } \right) \) such that u|Ω ∈ C 2(Ω) and $$ - div\left( {A\left( { \cdot ,u} \right)grad\;u} \right) = f\quad in\;\Omega $$ (9.1) $$ u = \bar u\quad on\;{\Gamma _1} $$ (9.2) $$ \alpha u + {n^T}A\left( { \cdot ,u} \right)grad\;u = g\quad on\;{\Gamma _2} $$ (9.3) where Ω ∈ L, n = (n 1, ..., n d ) T is the outward unit normal to ∂Ω, d ∈ {1, 2, ...,}, Γ1 and Γ2 are relatively open sets in the boundary ∂Ω, \({\overline \Gamma _1} \cup {\overline \Gamma _2} = \partial \Omega ,\,{\Gamma _1} \cap {\Gamma _2} = \phi\), \( A = \left( {{a_{ij}}} \right)_{i,j = 1}^d \) is…

Lusternik-Schnirelmann Critical Values and Bifurcation Problems

1987

We present a method to calculate bifurcation branches for nonlinear two point boundary value problems of the following type $$ \{ _{u(a) = u(b) = 0,}^{ - u'' = \lambda G'(u)} $$ (1.1) where G : R → R is a smooth mapping. This problem can be formulated equivalently as $$ g' \left(u \right)= \mu u, $$ (1.2) where $$ g \left(u \right)= \overset{b} {\underset{a} {\int}} G \left(u \left(t \right) \right) dt $$ (1.3) and μ = 1/λ. Solutions of this problem can be found by locating the critical points of the functional g : H → R on the spheres \(S_r= \lbrace x \in H \mid \;\parallel x \parallel =r \rbrace, r >0.\) (The Lagrange multiplier theorem.)

Entropy and Renormalization in Chaotic Visibility Graphs

2016

The Zero-Check for Eliminating Non-Significant Elements

1970

During continued matrix operations like the simplex method a lot of small non-significant elements, the actual value of which is zero,usually augment the working coefficient matrix. These elements are caused by round-off errors. They arise in the following manner in a computation of the type: $${\rm{d}}\, = \,{\rm{a}}\,{\rm{ - }}\,{\rm{b}}{\rm{.c}}$$ with e.g. the data (in FORTRAN notation) $${\rm{a}}\, = \,{\rm{2}}\,{\rm{ = }}\,{\rm{.20000000}}\,{\rm{E}}\,{\rm{01}}$$ $${\rm{b}}\, = \,{\rm{6}}\,{\rm{ = }}\,{\rm{.60000000}}\,{\rm{E}}\,{\rm{01}}$$ $${\rm{c}}\, = \,{\rm{1/3}}\,{\rm{ = }}\,{\rm{.33333333}}\,{\rm{E}}\,{\rm{00}}$$

Commutator Equations and the Equationally-Defined Commutator

2015

If \(\underline{s} = s_{1},\ldots,s_{m}\), and \(\underline{t} = t_{1},\ldots,t_{m}\) are sequences of terms (both sequences of the same length m) and X is a set of equations then $$\displaystyle{\underline{s} \approx \underline{ t} \in X}$$ abbreviates the fact that s i ≈ t i ∈ X for i = 1, …, m.)

Canonical Adiabatic Theory

2001

In the present chapter we are concerned with systems, the change of which—with the exception of a single degree of freedom—should proceed slowly. (Compare the pertinent remarks about \(\varepsilon\) as slow parameter in Chap. 7) Accordingly, the Hamiltonian reads: $$\displaystyle{ H = H_{0}{\bigl (J,\varepsilon p_{i},\varepsilon q_{i};\varepsilon t\bigr )} +\varepsilon H_{1}{\bigl (J,\theta,\varepsilon p_{i},\varepsilon q_{i};\varepsilon t\bigr )}\;. }$$ (12.1) Here, \((J,\theta )\) designates the “fast” action-angle variables for the unperturbed, solved problem \(H_{0}(\varepsilon = 0),\) and the (p i , q i ) represent the remaining “slow” canonical variables, which do not necessarily have…

Particle in Harmonic E-Field E(t) = Esinω 0 t; Schwinger–Fock Proper-Time Method

2017

Since the Green’s function of a Dirac particle in an external field, which is described by a potential A μ (x), is given by $$\displaystyle{ \left [\gamma \cdot \left (\frac{1} {i} \partial - eA\right ) + m\right ]G(x,x^{{\prime}}\vert A) =\delta (x - x^{{\prime}}) }$$ (37.1) the Green operator G+[A] is defined by $$\displaystyle{ \left (\gamma \Pi + m\right )G_{+} = 1\,,\quad \Pi _{\mu } = p_{\mu } - eA_{\mu } }$$ or $$\displaystyle\begin{array}{rcl} G_{+}& =& \frac{1} {\gamma \Pi + m - i\epsilon }\,,\quad \epsilon > 0 {}\\ & =& \frac{\gamma \Pi - m} {\left (\gamma \Pi \right )^{2} - m^{2} + i\epsilon } = \frac{-\gamma \Pi + m} {m^{2} -\left (\gamma \Pi \right )^{2} - i\epsilon } {}\\ & =&…

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

2017

Let \begin{document}$A∈{\rm{Sym}}(n× n)$\end{document} be an elliptic 2-tensor. Consider the anisotropic fractional Schrodinger operator \begin{document}$\mathscr{L}_A^s+q$\end{document} , where \begin{document}$\mathscr{L}_A^s: = (-\nabla·(A(x)\nabla))^s$\end{document} , \begin{document}$s∈ (0, 1)$\end{document} and \begin{document}$q∈ L^∞$\end{document} . We are concerned with the simultaneous recovery of \begin{document}$q$\end{document} and possibly embedded soft or hard obstacles inside \begin{document}$q$\end{document} by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain \begin{document}$Ω$\end{document} associated with \begin{document}$\mathscr{L}_A^s+q$\end{docume…

Glassy dynamics in confinement: planar and bulk limits of the mode-coupling theory.

2014

We demonstrate how the matrix-valued mode-coupling theory of the glass transition and glassy dynamics in planar confinement converges to the corresponding theory for two-dimensional (2D) planar and the three-dimensional bulk liquid, provided the wall potential satisfies certain conditions. Since the mode-coupling theory relies on the static properties as input, the emergence of a homogeneous limit for the matrix-valued intermediate scattering functions is directly connected to the convergence of the corresponding static quantities to their conventional counterparts. We show that the 2D limit is more subtle than the bulk limit, in particular, the in-planar dynamics decouples from the motion …