Search results for "Combinatorics"
showing 10 items of 1770 documents
Nonlocal Heat Content
2019
The heat content of a Borel measurable set \(D \subset \mathbb {R}^N\) at time t is defined by M. van der Berg in [69] (see also [70]) as: $$\displaystyle \mathbb {H}_D(t) = \int _D T(t) {\chi }_D (x) dx, $$ with (T(t))t≥0 being the heat semigroup in \(L^2(\mathbb {R}^N)\). Therefore, the heat content represents the amount of heat in D at time t if in D the initial temperature is 1 and in \(\mathbb {R}^N \setminus D\) the initial temperature is 0.
Iterated Conditionals and Characterization of P-Entailment
2021
In this paper we deepen, in the setting of coherence, some results obtained in recent papers on the notion of p-entailment of Adams and its relationship with conjoined and iterated conditionals. We recall that conjoined and iterated conditionals are suitably defined in the framework of conditional random quantities. Given a family \(\mathcal {F}\) of n conditional events \(\{E_{1}|H_{1},\ldots , E_{n}|H_{n}\}\) we denote by \(\mathcal {C}(\mathcal {F})=(E_{1}|H_{1})\wedge \cdots \wedge (E_{n}|H_{n})\) the conjunction of the conditional events in \(\mathcal F\). We introduce the iterated conditional \(\mathcal {C}(\mathcal {F}_{2})|\mathcal {C}(\mathcal {F}_{1})\), where \(\mathcal {F}_{1}\)…
Locally Convex Quasi C*-Algebras and Their Structure
2020
Throughout this chapter \({{\mathfrak A}}_{\scriptscriptstyle 0}[\| \cdot \|{ }_{\scriptscriptstyle 0}]\) denotes a unital C*-algebra and τ a locally convex topology on \({{\mathfrak A}}_{\scriptscriptstyle 0}\). Let \(\widetilde {{{\mathfrak A}}_{\scriptscriptstyle 0}}[\tau ]\) denote the completion of \({{\mathfrak A}}_{\scriptscriptstyle 0}\) with respect to the topology τ. Under certain conditions on τ, a subspace \({\mathfrak A}\) of \(\widetilde {{{\mathfrak A}}_{\scriptscriptstyle 0}}[\tau ]\), containing \({{\mathfrak A}}_{\scriptscriptstyle 0}\), will form (together with \({{\mathfrak A}}_{\scriptscriptstyle 0}\)) a locally convex quasi *-algebra \(({\mathfrak A}[\tau ],{{\mathfrak…
Approximation Operators of Binomial Type
1999
Our objective is to present a unified theory of the approximation operators of binomial type by exploiting the main technique of the so- called “ umbral calculus” or “finite operator calculus” (see [18], [20]-[22]). Let us consider the basic sequence (bn)n≥0 associated to a certain delta operator Q. By supposing that b n (x) ≥ 0, x ∈ [0, ∞), our purpose is to put in evidence some approximation properties of the linear positive operators (L Q n ) n≥1 which are defined on C[0,1] by \( L_n^Qf = \sum\limits_{k = 0}^n {\beta _n^Q{,_k}f\left( {\frac{k}{n}} \right),\beta _{n{,_k}}^Q\left( x \right): = } \frac{1}{{{b_n}\left( n \right)}}\left( {\begin{array}{*{20}{c}} n \\ k \end{array}} \right){b_…
Action-Angle Variables
2001
In the following we will assume that the Hamiltonian does not depend explicitly on time; ∂H/∂t = 0. Then we know that the characteristic function W(q i , P i ) is the generator of a canonical transformation to new constant momenta P i , (all Q i , are ignorable), and the new Hamiltonian depends only on the P i ,: H = K = K(P i ). Besides, the following canonical equations are valid: $$ \dot Q_i = \frac{{\partial K}} {{\partial P_i }} = v_i = const. $$ (1) $$ \dot P_i = \frac{{\partial K}} {{\partial Q_i }} = 0. $$ (2)
Types of Motion in the Oblate Planet Problem
1985
We consider a mass point in the gravitational field of an oblate planet and in a meridianal plane. The Hamiltonian of the problem is: $$ \frac{1}{2}\left( {p_r^2 + \frac{{p_{\theta }^2}}{{{r^2}}}} \right) - \frac{1}{r} - \frac{\varepsilon }{{{r^3}}}\left( {1 - 3{{\sin }^2}\theta } \right) $$ .
Parameter Estimation for α-Fractional Bridges
2013
Let α, T > 0. We study the asymptotic properties of a least squares estimator for the parameter α of a fractional bridge defined as \(\mathrm{d}X_{t} = -\alpha \, \frac{X_{t}} {T-t}\,\mathrm{d}t + \mathrm{d}B_{t}\), 0 ≤ t \frac{1} {2}\). Depending on the value of α, we prove that we may have strong consistency or not as t → T. When we have consistency, we obtain the rate of this convergence as well. Also, we compare our results to the (known) case where B is replaced by a standard Brownian motion W.
Linear Oscillator with Time-Dependent Frequency
2001
Here is another important example of a path integral calculation, namely the time-dependent oscillator whose Lagrangian is given by $$\displaystyle{ L = \frac{m} {2} \dot{x}^{2} -\frac{m} {2} W(t)x^{2}\;. }$$ (21.1) Since L is quadratic, we again expand around a classical solution so that later on we will be dealing again with the calculation of the following path integral: $$\displaystyle{ \int _{x(t_{i})\,=\,0}^{x(t_{f})\,=\,0}[dx(t)]\text{exp}\left \{ \frac{\text{i}} {\hslash }\,\frac{m} {2} \int _{t_{i}}^{t_{f} }dt\left [\left (\frac{dx} {dt} \right )^{\!2} - W(t)x^{2}\right ]\right \}\;. }$$ (21.2) Using \(x(t_{i}) = 0 = x(t_{f}),\) we can integrate by parts and obtain $$\displaystyle{…
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