Search results for "Mathematica"

Maslov Anomaly and the Morse Index Theorem

2001

Our starting point is again the phase space integral $$\displaystyle{ \text{e}^{\text{i}\hat{\varGamma }[\tilde{M}]} =\int \mathcal{D}\chi ^{a}\,\text{e}^{\text{i}S_{\text{fl}}[\chi,\tilde{M}]} }$$ (31.1) with periodic boundary conditions χ(0) = χ(T) and $$\displaystyle{ S_{\text{fl}}[\chi,\tilde{M}] = \frac{1} {2}\int _{0}^{T}dt\,\bar{\chi }_{ a}(t)\left [ \frac{\partial } {\partial t} -\tilde{M}(t)\right ]_{\phantom{a}b}^{a}\chi ^{b}(t)\;. }$$ (31.2) Here we have indicated that Sfl and \(\hat{\varGamma }\) depend on ηcl a and A i only through \(\tilde{M}_{\phantom{a}b}^{a}\): $$\displaystyle{ \tilde{M}(t)_{\phantom{a}b}^{a} =\omega ^{ac}\partial _{ c}\partial _{b}\mathcal{H}{\bigl (\eta _…

Orientation matters

2008

The optimal communication spanning tree (OCST) problem is a well known $\mathcal{NP}$-hard combinatorial optimization problem which seeks a spanning tree that satisfies all given communication requirements for minimal total costs. It has been shown that optimal solutions of OCST problems are biased towards the much simpler minimum spanning tree (MST) problem. Therefore, problem-specific representations for EAs like heuristic variants of edge-sets that are biased towards MSTs show high performance.In this paper, additional properties of optimal solutions for Euclidean variants of OCST problems are studied. Experimental results show that not only edges in optimal trees are biased towards low-…

Summing multi-norms defined by Orlicz spaces and symmetric sequence space

2016

We develop the notion of the \((X_1,X_2)\)-summing power-norm based on a~Banach space \(E\), where \(X_1\) and \(X_2\) are symmetric sequence spaces. We study the particular case when \(X_1\) and \(X_2\) are Orlicz spaces \(\ell_\Phi\) and \(\ell_\Psi\) respectively and analyze under which conditions the \((\Phi, \Psi)\)-summing power-norm becomes a~multinorm. In the case when \(E\) is also a~symmetric sequence space \(L\), we compute the precise value of \(\|(\delta_1,\cdots,\delta_n)\|_n^{(X_1,X_2)}\) where \((\delta_k)\) stands for the canonical basis of \(L\), extending known results for the \((p,q)\)-summing power-norm based on the space \(\ell_r\) which corresponds to \(X_1=\ell_p\), …

A non-linear Bishop–Phelps–BollobÁs type theorem

2018

The Neumann Problem for the Total Variation Flow

2004

This chapter is devoted to prove existence and uniqueness of solutions for the minimizing total variation flow with Neumann boundary conditions, namely $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = div\left( {\frac{{Du}} {{\left| {Du} \right|}}} \right) in Q = (0,\infty ) \times \Omega , \hfill \\ \frac{{\partial u}} {{\partial \eta }} = 0 on S = (0,\infty ) \times \partial \Omega , \hfill \\ u(0,x) = u_0 (x) in x \in \Omega , \hfill \\ \end{gathered} \right. $$ (2.1) where Ω is a bounded set in ℝ N with Lipschitz continuous boundary ∂ Ω and u0 ∈ L1(Ω). As we saw in the previous chapter, this partial differential equation appears when one uses the steepest descent method …

Hilbert Space Embeddings for Gelfand–Shilov and Pilipović Spaces

2017

We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, \(\mathcal{H}\), and its dual, \(\mathcal{H}^\prime\) . We prove that for such quasi-Banach space \(\mathcal{B}\), there are convenient Hilbert spaces, \(\mathcal{H}_{k}, k=1,2\), with normalized Hermite functions as orthonormal bases and such that \(\mathcal{B}\) lies between \(\mathcal{H}_1\; \mathrm{and}\;\mathcal{H}_2\), and the latter spaces lie between \(\mathcal{H}\; \mathrm{and}\;\mathcal{H}^\prime\).

h analogue of Newton's binomial formula

1998

In this letter, the $h$--analogue of Newton's binomial formula is obtained in the $h$--deformed quantum plane which does not have any $q$--analogue. For $h=0$, this is just the usual one as it should be. Furthermore, the binomial coefficients reduce to $\frac{n!}{(n-k)!}$ for $h=1$. \\ Some properties of the $h$--binomial coefficients are also given. \\ Finally, I hope that such results will contribute to an introduction of the $h$--analogue of the well--known functions, $h$--special functions and $h$--deformed analysis.

Explicit solutions of two-point boundary value operator problems

1988

Soit H un espace de Hilbert, complexe, separable et soit L(H) l'algebre de tous les operateurs lineaires bornes sur H. On etudie des conditions d'existence non triviales pour le probleme aux valeurs limites operateurs: t 2 X (2) +tA 1 X (1) +A 0 X=0; M 11 X(a)+N 11 X(b)+M 12 X (1) (a)+N 12 X (1) (b)=0, M 21 X(a)+N 21 X(b)+M 22 X (1) (a)+N 22 X (1) (b)=0, 0<a≤t≤b ou M ij , N ij , pour 1≤i, j≤2 et A 0 , A 1 sont des operateurs de L(H). Sous certaines hypotheses concernant l'existence des solutions d'une equation operateur algebrique X 2 +B 1 X+B 0 =0, on obtient des solutions explicites au probleme aux limites

Product Integration for Weakly Singular Integral Equations In ℝm

1985

In this note we discuss the numerical solution of the second kind Fredholm integral equation: $$ y(t) = f(t) + \lambda \int\limits_{\Omega } {{{\psi }_{\alpha }}(|t - s|)g(t,s)y(s)ds,\;t \in \bar{\Omega },} $$ (1) Where \( \lambda \in ;\not{ \subset }\backslash \{ 0\} \) , the functions f,g are given and continuous, |.| denotes the Euclidean norm, and φα, 0 \alpha > 0} \\ {\left\{ {\begin{array}{*{20}{c}} {\ln (r),} & {j = 0} \\ {{{r}^{{ - j}}}} & {j > 0} \\ \end{array} } \right\},\alpha = m} \\ \end{array} ,} \right. $$ with Cj not depending on r. Here Ω _ is the closure of a bounded domain Ω⊂ℝm.

Rigidity transition in two-dimensional random fiber networks

2000

Rigidity percolation is analyzed in two-dimensional random fibrous networks. The model consists of central forces between the adjacent crossing points of the fibers. Two strategies are used to incorporate rigidity: adding extra constraints between second-nearest crossing points with a probability p(sn), and "welding" individual crossing points by adding there four additional constraints with a probability p(weld), and thus fixing the angles between the fibers. These additional constraints will make the model rigid at a critical probability p(sn)=p(sn)(c) and p(weld)=p(weld)(c), respectively. Accurate estimates are given for the transition thresholds and for some of the associated critical e…