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1.
By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term hh affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)u=b(x)g(u)+λh(x), u>0u>0 in ΩΩ, u|Ω=∞u|Ω=, where ΩΩ is a bounded domain with smooth boundary in RNRN, λ>0λ>0, g∈C1[0,∞)gC1[0,) is increasing on [0,∞)[0,), g(0)=0g(0)=0, gg is regularly varying at infinity with positive index ρρ, the weight bb, which is non-trivial and non-negative in ΩΩ, may be vanishing on the boundary, and the inhomogeneous term hh is non-negative in ΩΩ and may be singular on the boundary.  相似文献   

2.
In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L2L2 consisting of square integrable random vectors. We show that for the solution XX to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution xx for this inclusion that is a L2L2-continuous selection of XX. This result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions, as well as to consider the viability problem for stochastic differential inclusions.  相似文献   

3.
We prove that if for a continuous map ff on a compact metric space XX, the chain recurrent set, R(f)R(f) has more than one chain component, then ff does not satisfy the asymptotic average shadowing property. We also show that if a continuous map ff on a compact metric space XX has the asymptotic average shadowing property and if AA is an attractor for ff, then AA is the single attractor for ff and we have A=R(f)A=R(f). We also study diffeomorphisms with asymptotic average shadowing property and prove that if MM is a compact manifold which is not finite with dimM=2dimM=2, then the C1C1 interior of the set of all C1C1 diffeomorphisms with the asymptotic average shadowing property is characterized by the set of ΩΩ-stable diffeomorphisms.  相似文献   

4.
In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion XX in a bounded κκ-fat open set; if uu is a positive harmonic function with respect to XX in a bounded κκ-fat open set DD and hh is a positive harmonic function in DD vanishing on DcDc, then the non-tangential limit of u/hu/h exists almost everywhere with respect to the Martin-representing measure of hh.  相似文献   

5.
This paper is concerned with the Cauchy problem for the fast diffusion equation ut−Δum=αup1utΔum=αup1 in RNRN (N≥1N1), where m∈(0,1)m(0,1), p1>1p1>1 and α>0α>0. The initial condition u0u0 is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value u0u0 so that u(t,x)u(t,x) blows up in finite time, and we show how to get estimates on the profile of u(t,x)u(t,x) for small enough values of t>0t>0.  相似文献   

6.
This paper is devoted to construct a family of fifth degree cubature formulae for nn-cube with symmetric measure and nn-dimensional spherically symmetrical region. The formula fornn-cube contains at most n2+5n+3n2+5n+3 points and for nn-dimensional spherically symmetrical region contains only n2+3n+3n2+3n+3 points. Moreover, the numbers can be reduced to n2+3n+1n2+3n+1 and n2+n+1n2+n+1 if n=7n=7 respectively, the latter of which is minimal.  相似文献   

7.
In this paper, we consider Beta(2−α,α)(2α,α) (with 1<α<21<α<2) and related ΛΛ-coalescents. If T(n)T(n) denotes the length of a randomly chosen external branch of the nn-coalescent, we prove the convergence of nα−1T(n)nα1T(n) when nn tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ(n)σ(n) of collisions which occur in the nn-coalescent until the end of the chosen external branch, and for the block counting process associated with the nn-coalescent.  相似文献   

8.
Consider events of the form {Zs≥ζ(s),s∈S}{Zsζ(s),sS}, where ZZ is a continuous Gaussian process with stationary increments, ζζ is a function that belongs to the reproducing kernel Hilbert space RR of process ZZ, and S⊂RSR is compact. The main problem considered in this paper is identifying the function β∈RβR satisfying β(s)≥ζ(s)β(s)ζ(s) on SS and having minimal RR-norm. The smoothness (mean square differentiability) of ZZ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=sζ(s)=s for s∈[0,1]s[0,1] and ZZ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.  相似文献   

9.
In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error εε. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom dd. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error εε. The cost of the algorithm is polynomial in dd and ε−1ε1, while the number of qubits is polynomial in dd and logε−1logε1. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1)1δ,δ(0,1), with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error εε. The cost of the algorithm is polynomial in dd, ε−1ε1 and δ−1δ1, while the number of qubits is polynomial in dd, logε−1logε1 and logδ−1logδ1.  相似文献   

10.
Let kk be any field, GG be a finite group acting on the rational function field k(xg:g∈G)k(xg:gG) by h⋅xg=xhghxg=xhg for any h,g∈Gh,gG. Define k(G)=k(xg:g∈G)Gk(G)=k(xg:gG)G. Noether’s problem asks whether k(G)k(G) is rational (= purely transcendental) over kk. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if GG is a Frobenius group with abelian Frobenius kernel, then k(G)k(G) is retract kk-rational for any field kk satisfying some mild conditions. As an application, we show that, for any algebraic number field kk, for any Frobenius group GG with Frobenius complement isomorphic to SL2(F5)SL2(F5), there is a Galois extension field KK over kk whose Galois group is isomorphic to GG, i.e. the inverse Galois problem is valid for the pair (G,k)(G,k). The same result is true for any non-solvable Frobenius group if k(ζ8)k(ζ8) is a cyclic extension of kk.  相似文献   

11.
We consider G=Γ×S1G=Γ×S1 with ΓΓ being a finite group, for which the complete Euler ring structure in U(G)U(G) is described. The multiplication tables for Γ=D6Γ=D6, S4S4 and A5A5 are provided in the Appendix. The equivariant degree for GG-orthogonal maps is constructed using the primary equivariant degree with one free parameter. We show that the GG-orthogonal degree extends the degree for GG-gradient maps (in the case of G=Γ×S1G=Γ×S1) introduced by G?ba in [K. G?ba, W. Krawcewicz, J. Wu, An equivariant degree with applications to symmetric bifurcation problems I: Construction of the degree, Bull. London. Math. Soc. 69 (1994) 377–398]. The computational results obtained are applied to a ΓΓ-symmetric autonomous Newtonian system for which we study the existence of 2π2π-periodic solutions. For some concrete cases, we present the symmetric classification of the solution set for the systems considered.  相似文献   

12.
We consider a multidimensional diffusion XX with drift coefficient b(Xt,α)b(Xt,α) and diffusion coefficient εa(Xt,β)εa(Xt,β) where αα and ββ are two unknown parameters, while εε is known. For a high frequency sample of observations of the diffusion at the time points k/nk/n, k=1,…,nk=1,,n, we propose a class of contrast functions and thus obtain estimators of (α,β)(α,β). The estimators are shown to be consistent and asymptotically normal when n→∞n and ε→0ε0 in such a way that ε−1n−ρε1nρ remains bounded for some ρ>0ρ>0. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.  相似文献   

13.
In many applications it has been observed that hybrid-Monte Carlo sequences perform better than Monte Carlo and quasi-Monte Carlo sequences, especially in difficult problems. For a mixed ss-dimensional sequence mm, whose elements are vectors obtained by concatenating dd-dimensional vectors from a low-discrepancy sequence qq with (s−d)(sd)-dimensional random vectors, probabilistic upper bounds for its star discrepancy have been provided. In a paper of G. Ökten, B. Tuffin and V. Burago [G. Ökten, B. Tuffin, V. Burago, J. Complexity 22 (2006), 435–458] it was shown that for arbitrary ε>0ε>0 the difference of the star discrepancies of the first NN points of mm and qq is bounded by εε with probability at least 1−2exp(−ε2N/2)12exp(ε2N/2) for NN sufficiently large. The authors did not study how large NN actually has to be and if and how this actually depends on the parameters ss and εε. In this note we derive a lower bound for NN, which significantly depends on ss and εε. Furthermore, we provide a probabilistic bound for the difference of the star discrepancies of the first NN points of mm and qq, which holds without any restrictions on NN. In this sense it improves on the bound of Ökten, Tuffin and Burago and is more helpful in practice, especially for small sample sizes NN. We compare this bound to other known bounds.  相似文献   

14.
We prove that if GG is a finite simple group which is the unit group of a ring, then GG is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2k−12k1 for some kk; or (c) a projective special linear group PSLn(F2)PSLn(F2) for some n≥3n3. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups GG with no non-trivial normal 2-subgroup.  相似文献   

15.
We consider a multidimensional diffusion XX with drift coefficient b(α,Xt)b(α,Xt) and diffusion coefficient ?σ(β,Xt)?σ(β,Xt). The diffusion sample path is discretely observed at times tk=kΔtk=kΔ for k=1…nk=1n on a fixed interval [0,T][0,T]. We study minimum contrast estimators derived from the Gaussian process approximating XX for small ??. We obtain consistent and asymptotically normal estimators of αα for fixed ΔΔ and ?→0?0 and of (α,β)(α,β) for Δ→0Δ0 and ?→0?0 without any condition linking ?? and ΔΔ. We compare the estimators obtained with various methods and for various magnitudes of ΔΔ and ?? based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework.  相似文献   

16.
Let x(s)x(s), s∈RdsRd be a Gaussian self-similar random process of index HH. We consider the problem of log-asymptotics for the probability pTpT that x(s)x(s), x(0)=0x(0)=0 does not exceed a fixed level in a star-shaped expanding domain T⋅ΔTΔ as T→∞T. We solve the problem of the existence of the limit, θ?lim(−logpT)/(logT)Dθ?lim(logpT)/(logT)D, T→∞T, for the fractional Brownian sheet x(s)x(s), s∈[0,T]2s[0,T]2 when D=2D=2, and we estimate θθ for the integrated fractional Brownian motion when D=1D=1.  相似文献   

17.
We consider NN independent stochastic processes (Xj(t),t∈[0,T])(Xj(t),t[0,T]), j=1,…,Nj=1,,N, defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable ?j?j and study the nonparametric estimation of the density of the random effect ?j?j in two kinds of mixed models. A multiplicative random effect and an additive random effect are successively considered. In each case, we build kernel and deconvolution estimators and study their L2L2-risk. Asymptotic properties are evaluated as NN tends to infinity for fixed TT or for T=T(N)T=T(N) tending to infinity with NN. For T(N)=N2T(N)=N2, adaptive estimators are built. Estimators are implemented on simulated data for several examples.  相似文献   

18.
19.
We show that if T:X→XT:XX is a continuous linear operator on an FF-space X≠{0}X{0}, then the set of frequently hypercyclic vectors of TT is of first category in XX, and this answers a question of A. Bonilla and K.-G. Grosse-Erdmann. We also show that if T:X→XT:XX is a bounded linear operator on a Banach space X≠{0}X{0} and if TT is frequently hypercyclic (or, more generally, syndetically transitive), then the TT-orbit of every non-zero element of XX is bounded away from 0, and in particular TT is not hypercyclic.  相似文献   

20.
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