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We study the second order Emden–Fowler type differential equation
(a(t)|x|αsgnx)+b(t)|x|βsgnx=0(a(t)|x|αsgnx)+b(t)|x|βsgnx=0
in the super-linear case α<βα<β. Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.  相似文献   

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A global optimization problem is studied where the objective function f(x)f(x) is a multidimensional black-box function and its gradient f(x)f(x) satisfies the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant KK. Different methods for solving this problem by using an a priori given estimate of KK, its adaptive estimates, and adaptive estimates of local Lipschitz constants are known in the literature. Recently, the authors have proposed a one-dimensional algorithm working with multiple estimates of the Lipschitz constant for f(x)f(x) (the existence of such an algorithm was a challenge for 15 years). In this paper, a new multidimensional geometric method evolving the ideas of this one-dimensional scheme and using an efficient one-point-based partitioning strategy is proposed. Numerical experiments executed on 800 multidimensional test functions demonstrate quite a promising performance in comparison with popular DIRECT-based methods.  相似文献   

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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   

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We give a sufficient condition for the sum of two closed operators to be closed. In particular, we study the sum of two sectorial operators with the sum of their sectoriality angles greater than π  . We show that if one of the operators admits bounded HH-calculus and the resolvent of the other operator satisfies a boundedness condition stronger than the standard sectoriality, but weaker than the bounded imaginary powers property in the case of UMD spaces, then the sum is closed. We apply the result to the abstract parabolic problem and give a sufficient condition for LpLp-maximal regularity.  相似文献   

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We consider the linear span S   of the functions taktak (with some ak>0ak>0) in weighted L2L2 spaces, with rather general weights. We give one necessary and one sufficient condition for S to be dense. Some comparisons are also made between the new results and those that can be deduced from older ones in the literature.  相似文献   

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We study smooth foliations on the solid torus S1×D2S1×D2 having S1×{0}S1×{0} and S1×∂D2S1×D2 as the only compact leaves and S1×{0}S1×{0} as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc.  相似文献   

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Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair {H,H}{H,H} of maximal accumulative operators has been proved. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H   and H=H+VH=H+V are maximal accumulative and V is trace class, we prove the existence of a summable   complex-valued spectral shift function. We also obtain trace formulas for pairs {H,H?}{H,H?}assuming only that H and  H?H?are resolvent comparable. In this case the determinant of the characteristic function of H is involved in trace formulas.  相似文献   

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It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem to be weakly tractable in the worst case. The complexity of linear tensor product problems in the worst case depends on the eigenvalues {λi}iN{λi}iN of a certain operator. It is known that if λ1=1λ1=1 and λ2∈(0,1)λ2(0,1) then λn=o((lnn)−2)λn=o((lnn)2), as n→∞n, is a necessary condition for a problem to be weakly tractable. We show that this is a sufficient condition as well.  相似文献   

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A four-stage Hermite–Birkhoff–Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form y=f(t,y)y=f(t,y) with initial conditions y(t0)=y0y(t0)=y0. Its formula uses yy, yy and y?y? as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived only once to obtain the values of Hermite–Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C ++, HBOQ(14)4 is superior to the Dormand–Prince Runge–Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/~remi.  相似文献   

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In this paper we give a weaker sufficient condition for the maximal monotonicity of the operator S+ATAS+ATA, where S:X?XS:X?X, T:Y?YT:Y?Y are two maximal monotone operators, A:X→YA:XY is a linear continuous mapping and X,YX,Y are reflexive Banach spaces. We prove that our condition is weaker than the generalized interior-point conditions given so far in the literature. This condition is formulated using the representative functions of the operators involved. In particular, we rediscover some sufficient conditions given in the past using the so-called Fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator, respectively.  相似文献   

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Using a generalization of Sturm’s comparison theorem, some new oscillation criteria are established for the matrix differential system with damping
(P(t)Y)+R(t)Y+Q(t)Y=0(P(t)Y)+R(t)Y+Q(t)Y=0
under the hypothesis: P(t)=P(t)>0P(t)=P(t)>0, Q(t)=Q(t)Q(t)=Q(t), Y(t)Y(t) are n×nn×n matrices of real valued continuous functions on the interval [t0,∞)[t0,), and R(t)=R(t)∈C1([t0,∞),Rn2)R(t)=R(t)C1([t0,),Rn2). Our results are sharper than some previous results.  相似文献   

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Let CC be an irreducible plane curve. A point PP in the projective plane is said to be Galois with respect to CC if the function field extension induced by the projection from PP is Galois. We denote by δ(C)δ(C) the number of Galois points contained in P2?CP2?C. In this article we will present two results with respect to determination of δ(C)δ(C) in characteristic two. First we determine δ(C)δ(C) for smooth plane curves of degree a power of two. In particular, we give a new characterization of the Klein quartic in terms of δ(C)δ(C). Second we determine δ(C)δ(C) for a generalization of the Klein quartic, which is related to an example of Artin–Schreier curves whose automorphism group exceeds the Hurwitz bound. This curve has many Galois points.  相似文献   

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This paper establishes a necessary and sufficient condition for the existence of a unique bounded solution to the classical Dirichlet problem in arbitrary open subset of RNRN (N≥3N3) with a non-compact boundary. The criterion is the exact analogue of Wiener’s test for the boundary regularity of harmonic functions and characterizes the “thinness” of a complementary set at infinity. The Kelvin transformation counterpart of the result reveals that the classical Wiener criterion for the boundary point is a necessary and sufficient condition for the unique solvability of the Dirichlet problem in a bounded open set within the class of harmonic functions having a “fundamental solution” kind of singularity at the fixed boundary point. Another important outcome is that the classical Wiener’s test at the boundary point presents a necessary and sufficient condition for the “fundamental solution” kinds of singularities of the solution to the Dirichlet problem to be removable.  相似文献   

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