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1.
A new approach is established to show that the semigroup {S(t)}_(t≤0) generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L~q(?)(2 ≤q ∞) with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.  相似文献   

2.
The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n≤7. For dimension n 〉 7, the same result is established under the spin assumption.  相似文献   

3.
In this article, authors discuss the problem of uniform packing dimension of the image set of multiparameter stochastic processes without random uniform H(o)lder condition, and obtain the uniform packing dimension of multiparameter stable processes.If Z is a stable (N, d, α)-process and αN ≤ d, then the following holds with probability 1 Dim Z(E) = α DimE for any Borel setE ∈ B(R N),where Z(E) = {x: (E) t ∈ E, Z(t) = x}. Dim(E) denotes the packing dimension of E.  相似文献   

4.
Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.  相似文献   

5.
Let B1 ■ RNbe a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:-div(|▽u|p-2▽u) = |x|s|u|p*(s)-2u + λ|x|t|u|p-2u, x ∈ B1,u|■B1= 0,where t, s -p, 2 ≤ p N, p*(s) =(N+s)p N-pand λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N p(p- 1)t + p(p2- p + 1) and λ∈(0, λ1,t), where λ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤(ps+p) min{1,p+t p+s}+p2p-(p-1) min{1,p+t p+s}and λ 0 is small.  相似文献   

6.
In higher dimension, there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation. This article is concerned with the asymptotic behavior of solutions for the non–autonomous Chafee-Infante equation (?u)/(?t)-?u =λ(t)(u-u~3) in higher dimension, where λ(t) ∈ C~1[0, T ] and λ(t) is a positive, periodic function.We denote λ_1 as the first eigenvalue of-?? = λ?, x ∈ ?; ? = 0, x ∈ ??. For any spatial dimension N ≥ 1, we prove that if λ(t) ≤λ_1, then the nontrivial solutions converge to zero,namely, ■ u(x, t) = 0, x ∈ ?; if λ(t) λ_1 as t → +∞, then the positive solutions t→+∞are "attracted" by positive periodic solutions. Specially, if λ(t) is independent of t, then the positive solutions converge to positive solutions of-?U = λ(U-U~3). Furthermore,numerical simulations are presented to verify our results.  相似文献   

7.
ON A HYPER HILBERT TRANSFORM****   总被引:2,自引:0,他引:2  
The authors define the directional hyper Hilbert transform and give ita mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the Lp-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein.  相似文献   

8.
It is well known that a ring R is left hereditary iff every left ideal of R is projective, iff every submodule of a projective module is projective (ld R≤1), iff every quotient module of an injective module is injective (1cd R≤1), where 1d R and 1cd R means the left global dimension resp. codimension of the ring R. These rings may be generalized to those of weak left global dimension at most 1 (wld R≤1). The latter condition holds iff every left ideal of R is flat, iff every submodule  相似文献   

9.
In this article, we first present an equivalent formulation of the free boundary problem to 3-D incompressible Euler equations, then we announce our local wellposedness result concerning the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that (δ)p/(δ)n (ξ)|t=0 ≤ -2c0 < 0 with ξ being restricted to the initial surface.  相似文献   

10.
The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems. This approach is based on the monotonicity of several variational integrals, the Federer-Almgren dimension reduction and stratification theorems, and some simple PDE arguments.  相似文献   

11.
In this paper, we estimate the free entropy dimension of the group yon Neumann algebra(L)(Zt), which is less than 1/t,2 ≤ t ≤ ∞. This data is identical with the free dimension defined by Dykema.  相似文献   

12.
Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if where is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an with . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ0(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ0 is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II 1-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of . If M and N are strongly 1-bounded and MN is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II 1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded. Received: November 2005, Revision: March 2006, Accepted: March 2006  相似文献   

13.
In this paper we define a very simple invariant η(V^-) for a k-tuple V^-of unitaries in a finite factor von Neumann algebra, and we show how this invariant can replace free entropy in many of the important applications. We also introduce a notion of metric free entropy and some related concepts.We include proofs, using η, of the theorems of Liming Ge and of D. Voiculescu, respectively, on the primeness of and on the absence of Cartan snbalgebras in the noncommutative free group factors.  相似文献   

14.
We compute the exact value of Voiculescu's perturbation theoretic entropy of the boundary actions of free groups. This result is a partial answer of Voiculescu's question.

  相似文献   


15.
熵是度量复杂系统无序性的重要物理量,而且现实中的大多数网络都呈现出无标度网络的特性.在网络的节点熵和结构熵概念的基础上,给出了BA模型的网络结构熵演化的解析结论和数值模拟.从解析结论和数值模拟可以得到,网络结构熵随网络大小以对数的速度增长;但在同样规模下,无标度网络的结构熵小于随机网络的结构熵.  相似文献   

16.
By means of random matrix approximation procedure, we re-prove Biane and Voiculescus free analog of Talagrands transportation cost inequality for measures on R in a more general setup. Furthermore, we prove the free transportation cost inequality for measures on T as well by extending the method to special unitary random matrices.Supported in part by Grant-in-Aid for Scientific Research (C)14540198 and by the program R&D support scheme for funding selected IT proposals of the Ministry of Public Management, Home Affairs, Posts and Telecommunications.Supported in part by MTA-JSPS project (Quantum Probability and Information Theory) and by OTKA T032662.Supported in part by Grant-in-Aid for Young Scientists (B)14740118.Mathematics Subject Classification (2000): Primary: 46L54, 46L53; Secondary: 60F10, 15A52, 94A17.  相似文献   

17.
In this paper, we introduce the topological entropy of a free semigroup action generated by proper maps, which extends the notions of the topological entropy of the free semigroup actions defined by Bufetov in 1999 and topological entropy of the proper maps defined by Patrão in 2010. We then give some properties of these notions and discuss the relations between them. We also give a partial variational principle for locally compact separable metric spaces. Moreover, the relationship between topological entropy of the free semigroup generated by proper maps and topological entropy of a skew-product transformation is given. These results extend the results obtained by Patrão, Bufetov and Lin, Ma and Wang in 2018.  相似文献   

18.
We propose a new way of defining entropy of a system, whichgives a general form that is non-extensive like Tsallis entropy,but is linearly dependent on component entropies, like Renyientropy, which is extensive. This entropy has a conceptuallynovel but simple origin and is mathematically easy to defineby a very simple expression involving a derivative. It leadsto a probability distribution function involving the Lambertfunction resulting from optimizing the entropy, which has hithertonever appeared in this context, and is somewhat more complexthan the Shannon or Boltzmann form, but is nevertheless mathematicallyquite tractable. We have compared it numerically with the Tsallisand Shannon entropies. We have also considered constraints onthe energy spectra imposed by the properties of the Lambertfunction, which are absent in the Shannon form. It may turnout to be a more appropriate candidate in a physical situationwhere the probability distribution does not suit any of thepreviously defined forms, especially when the probability densityfunction sought is expected to be stiffer than that resultingfrom maximizing the other entropies. We consider the problemof defining free energy and other thermodynamic functions whenthe entropy is given as a general function of the probabilitydistribution, including that for non-extensive forms. We thenfind that the free energy, which is central to the determinationof all other quantities of interest in a thermodynamic context,can be obtained uniquely, at least numerically, even when itis the root of a transcendental equation. In particular, weexamine the cases of the Tsallis form and the new form proposedby us. We compare the free energy, the internal energy and thespecific heat of a simple system of two energy states for eachof these forms and find significant departures for some quantities,while some others are less sensitive to the parametrization.  相似文献   

19.
20.
Emergence is analyzed by the principle of least action. The supreme law of nature describes diverse systems as actions that evolve from one state to another by consuming free energy in least time. As the system will either gain or lose quanta at the step of evolution, the systemic characteristics after the change of state cannot be reduced to those before the change. The change will invariably entail also a change in inter‐actions, which are flows of energy on paths that integrate ingredient systems to a synergistic system. New qualities will emerge along with opening interactions just as old ones will disappear along with ceasing interactions. The analysis of emergence as a natural process reveals that its irreducible and unpredictable nature does not result from complexity as such but because the natural process itself is molding the surroundings where it is evolving. © 2012 Wiley Periodicals, Inc. Complexity, 2012  相似文献   

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