共查询到20条相似文献,搜索用时 46 毫秒
1.
We consider integrals of the form , where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? n , and θ ∈ ? k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.
相似文献
$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$
2.
Youri Davydov 《Lithuanian Mathematical Journal》2011,51(2):171-179
Let X i = {X i (t), t ∈ T} be i.i.d. copies of a centered Gaussian process X = {X(t), t ∈ T} with values in\( {\mathbb{R}^d} \) defined on a separable metric space T. It is supposed that X is bounded. We consider the asymptotic behavior of convex hullsand show that, with probability 1,(in the sense of Hausdorff distance), where the limit shape W is defined by the covariance structure of X: W = conv{K t , t ∈ T}, Kt being the concentration ellipsoid of X(t). We also study the asymptotic behavior of the mathematical expectations E f(W n ), where f is an homogeneous functional.
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$ {W_n} = {\text{conv}}\left\{ {{X_1}(t), \ldots, {X_n}(t),\,\,t \in T} \right\} $
$ \mathop {{\lim }}\limits_{n \to \infty } \frac{1}{{\sqrt {{2\ln n}} }}{W_n} = W $
3.
We consider some class of non-linear systems of the form where A(·) ∈ ? n × n , A i (·) ∈ ? n × n , b(·) ∈ ? n , whose coefficients are arbitrary uniformly bounded functionals.
$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$
A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation where ρ(·) ∈ ?1, m(·) ∈ ? n , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.
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$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$
4.
Óscar Ciaurri T. Alastair Gillespie Luz Roncal José L. Torrea Juan Luis Varona 《Journal d'Analyse Mathématique》2017,132(1):109-131
It is well known that the fundamental solution of with u(n, 0) = δ nm for every fixed m ∈ Z is given by u(n, t) = e ?2t I n?m (2t), where I k (t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series W t f(n) = Σ m∈Z e ?2t I n?m (2t)f(m). This formula allows us to analyze some operators associated with the discrete Laplacian using semigroup theory. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ? p (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. We also show that the Riesz transforms essentially coincide with the so-called discrete Hilbert transform defined by D. Hilbert at the beginning of the twentieth century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.
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$${u_t}\left( {n,t} \right) = u\left( {n + 1,t} \right) - 2u\left( {n,t} \right) + u\left( {n - 1,t} \right),n \in \mathbb{Z},$$
5.
O. P. Filatov 《Differential Equations》2008,44(3):349-363
We consider the system of differential inclusions , where F,G: D → Kυ (\(R^{m_1 } \)), Kυ (\(R^{m_2 } \)) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces \(R^{m_1 } \) and \(R^{m_2 } \), respectively, D = R + × \(R^{m_1 } \) × \(R^{m_2 } \) × [0, a], a > 0, and µ is a small parameter. The functions F and G and the right-hand side of the averaged problem \(\dot u\) ∈ µF 0(u), u(0) = x 0, F 0(u) ∈ Kυ (\(R^{m_1 } \)), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each ? > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ0] and any solution x µ(·), y µ(·) of the original problem, there exists a solution u µ(·) of the averaged problem such that ∥x µ(t) ? y µ(t) ∥ ≤ ? for t ∈ [0, 1/µ]. Furthermore, for each solution u µ(·)of the averaged problem, there exists a solution x µ(·), y µ(·) of the original problem with the same estimate.
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$$\dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$
6.
Let X be a real normed space and let f: ? → X be a continuous mapping. Let T f (t 0) be the contingent of the graph G(f) at a point (t 0, f(t 0)) and let S + ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 X ). We show that
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- 1.If dimX < ∞ and the dilation D(f, t 0) of f at t 0 is finite then T f (t 0) ∩ S + is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
- 2.If dimX = ∞, then, given any compact set F ? S +, there exists a Lipschitz mapping f: ? → X such that T f (t 0) ∩ S + = F.
- 3.But if a closed set F ? S + has cardinality greater than that of the continuum then the relation T f (t 0) ∩ S + = F does not hold for any Lipschitz f: ? → X.
7.
In L 2(?3;?3), we consider a self-adjoint operator ? ε , ε > 0, generated by the differential expression curl η(x/ε)?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? ε 1/2 ) and ? ε ?1/2 sin(τ? ε 1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?0)1/2) and (?0)?1/2 sin(τ(?0)1/2), respectively, in the norm of the operators acting from the Sobolev space H s (with a suitable s) to ?2. Here ?0 is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? τ 2 v ε = ?? ε v ε , div v ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε). 相似文献
8.
A. M. Bikchentaev 《Mathematical Notes》2016,100(3-4):515-525
Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let Lp(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ Lp(M, τ), then A2 ∈ Lp(M, τ). Let n be a positive integer, n > 2, and A = An ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μt(A) belong to the set {0} ∪ [‖An?2‖?1, ‖A‖] for all t > 0; 2) either μt(A) ≥ 1 for all t > 0 or there is a t0 > 0 such that μt(A) = 0 for all t > t0. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z2 = 0, ZP = 0, and PZ = Z. Here, if Q ∈ Lp(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L1(M, τ) and if A = A3 and A ? A2 ∈ M, then τ(A) ∈ R. 相似文献
9.
Adel Mahmoud Gomaa 《Czechoslovak Mathematical Journal》2017,67(2):339-365
We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation where {?(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([?d, 0],E) be the Banach space of continuous functions from [?d, 0] into E and f d : [a, b] × C([?d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ t x(s) = x(t + s) for each s ∈ [?d, 0]. We prove that, under certain conditions, the differential equation with delay has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.
相似文献
$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$
(P)
$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$
(Q)
10.
A. V. Faminskii 《Mathematical Notes》2008,83(1-2):107-115
In the strip П = (?1, 0) × ?, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u t + u xxx + uu x = 0 with initial condition either 1) u(?1, x) = ?xθ(x), or 2) u(?1, x) = ?xθ(?x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (?1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity. 相似文献
11.
For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, T_u) = P{inf (t∈[0,S]_(s∈[t,t+T_u])) sup R_u(s) 0}, S, T_u 0.For X being a general Gaussian process we derive approximations of P_S(u, T_u) as u →∞. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval. 相似文献
12.
Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators. 相似文献
13.
F. E. Lomovtsev 《Differential Equations》2008,44(6):866-871
We prove the well-posed solvability in the strong sense of the boundary value Problems where the unbounded operators A s (t), s > 0, in a Hilbert space H have domains D(A s (t)) depending on t, are subordinate to the powers A 1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A 0(t) and the symmetric operators A s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that , where the smoothing operators are defined by .
相似文献
$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$
$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$
$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$
14.
V. M. Badkov 《Proceedings of the Steklov Institute of Mathematics》2009,266(1):64-77
Two-sided pointwise estimates are established for polynomials that are orthogonal on the circle |z| = 1 with respect to the weight ?(τ): = h(τ)|sin(τ/2)|?1 g(|sin(τ/2)|) (τ ∈ ?), where g(t) is a concave modulus of continuity slowly changing at zero such that t ?1 g(t) ∈ L 1[0, 1] and h(τ) is a positive function from the class C 2π with a modulus of continuity satisfying the integral Dini condition. The obtained estimates are applied to find the order of the distance from the point t = 1 to the greatest zero of a polynomial orthogonal on the segment [?1, 1]. 相似文献
15.
In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u0(x) = u(x, 0) and the boundary data g0(t) = u(0, t), g1(t) = ux(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.
相似文献
$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$
16.
Let u =(uh, u3) be a smooth solution of the 3-D Navier-Stokes equations in R3× [0, T). It was proved that if u3 ∈ L∞(0, T;˙B-1+3/p p,q(R3)) for 3 p, q ∞ and uh∈ L∞(0, T; BMO-1(R3)) with uh(T) ∈ VMO-1(R3), then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al.(2016), which requires u ∈ L∞(0, T;˙B-1+3/pp,q(R3)). Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest. 相似文献
17.
We investigate the equiconvergence on TN = [?π, π)N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ Lp(TN) and g ∈ Lp(RN), p > 1, N ≥ 3, g(x) = f(x) on TN, in the case where the “partial sums” of these expansions, i.e., Sn(x; f) and Jα(x; g), respectively, have “numbers” n ∈ ZN and α ∈ RN (nj = [αj], j = 1,..., N, [t] is the integral part of t ∈ R1) containing N ? 1 components which are elements of “lacunary sequences.” 相似文献
18.
This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation on a time scale \(\mathbb{T}\) where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on \(\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}\) and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty \). We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.
相似文献
$$(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$$
19.
Basudeb Dhara 《Czechoslovak Mathematical Journal》2018,68(1):95-119
Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x1,..., x n ) a multilinear polynomial over C which is not central valued on R. If for all r = (r1,..., r n ) ∈ I n , then one of the following conditions holds:
相似文献
$$F(f(r))G(f(r)) = H(f(r)^2 )$$
- (1)there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R
- (2)there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C
- (3)there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R
- (4)f(x1,..., x n )2 is central valued on R and one of the following conditions holds
- (a)there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’
- (b)there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.
- (a)
20.
In the space L 2(?2), we consider the operator . We study the spectrum of H and, for V ∈ C 0 2 (?2), prove the trace formula where c 0 = π ?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ k (i) are the eigenvalues of H.
相似文献
$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$
$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $