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1.
We establish the existence of at least three positive solutions to the second-order three-point boundary value problem, u″ + f(tu) = 0, u(0) = 0, αu(η) = u(1), where η: 0 lt; η < 1, 0 < α < 1/η, and f: [0, 1] × [0, ∞) → [0, ∞) is continuous. We accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works.  相似文献   

2.
In this paper we study the critical exponents of the Cauchy problem in Rn of the quasilinear singular parabolic equations: ut = div(|∇u|m − 1u) + ts|x|σup, with non-negative initial data. Here s ≥ 0, (n − 1)/(n + 1) < m < 1, p > 1 and σ > n(1 − m) − (1 + m + 2s). We prove that pc ≡ m + (1 + m + 2s + σ)/n > 1 is the critical exponent. That is, if 1 < p ≤ pc then every non-trivial solution blows up in finite time, but for p > pc, a small positive global solution exists.  相似文献   

3.
We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form − Δu = λh(x)H(u − a)uq + u2* − 1 in RN, where λ, a > 0 are parameters, h(x) is both nonnegative and integrable on RN, H is the Heaviside function, 2* is the critical Sobolev exponent, and 0 ≤ q < 2* − 1. We obtain existence, multiplicity and regularity of solutions by distinguishing the cases 0 ≤ q ≤ 1 and 1 < q < 2* − 1.  相似文献   

4.
In this paper, we consider the problem of finding u = u(xyt) and p = p(t) which satisfy ut = uxx + uyy + p(t)u + ? in R × [0, T], u(xy, 0) = f(xy), (xy) ∈ R = [0, 1] × [0, 1], u is known on the boundary of R and u(xyt) = E(t), 0 < t ? T, where E(t) is known and (xy) is a given point of R. Through a function transformation, the nonlinear two-dimensional diffusion problem is transformed into a linear problem, and a backward Euler scheme is constructed. It is proved by the maximum principle that the scheme is uniquely solvable, unconditionally stable and convergent in L norm. The convergence orders of u and p are of O(τ + h2). The impact of initial data errors on the numerical solution is also considered. Numerical experiments are presented to illustrate the validity of the theoretical results.  相似文献   

5.
In [J.-M. Chang, J.-S. Yang. Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2008) 760-767] the authors claim that every alternating group graph AGn is (n − 4)-fault-tolerant edge 4-pancyclic. Which means that if the number of faults ∣F∣ ? n − 4, then every edge in AGn − F is contained in a cycle of length ?, for every 4 ? ? ? n!/2 − ∣F∣. They also claim that AGn is (n − 3)-fault-tolerant vertex pancyclic. Which means that if ∣F∣ ? n − 3, then every vertex in AGn − F is contained in a cycle of length ?, for every 3 ? ? ? n!/2 − ∣F∣. Their proofs are not complete. They left a few important things unexplained. In this paper we fulfill these gaps and present another proofs that AGn is (n − 4)-fault-tolerant edge 4-pancyclic and (n − 3)-fault-tolerant vertex pancyclic.  相似文献   

6.
This paper presents a generalized Gaussian quadrature method for numerical integration over triangular, parallelogram and quadrilateral elements with linear sides. In order to derive the quadrature rule, a general transformation of the regions, R1 = {(xy)∣a ? x ? bg(x) ? y ? h(x)} and R2 = {(xy)∣a ? y ? bg(y) ? x ? h(y)}, where g(x), h(x), g(y) and h(y) are linear functions, is given from (xy) space to a square in (ξη) space, S: {(ξη)∣0 ? ξ ? 1, 0 ? η ? 1}. Generlized Gaussian quadrature nodes and weights introduced by Ma et.al. in 1997 are used in the product formula presented in this paper to evaluate the integral over S, as it is proved to give more accurate results than the classical Gauss Legendre nodes and weights. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear sides. The performance of the method is illustrated for different functions over different two-dimensional regions with numerical examples.  相似文献   

7.
The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices Aj and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ?f(A1) + f(A2) + ? + f(Am)? ? ? f(A1 + A2 + ? + Am)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A1) + f(A2) + ? + f(Am)? ? f(?A1 + A2 + ? + Am?), where f is a non-negative concave function on [0, ∞) and ? · ? is normalized.  相似文献   

8.
Let b = b(A) be the Boolean rank of an n × n primitive Boolean matrix A and exp(A) be the exponent of A. Then exp(A) ? (b − 1)2 + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101-116]. In this paper, we show that for each 3 ? b ? n − 1, there are n × n primitive Boolean matrices A with b(A) = b such that exp(A) = (b − 1)2 + 1, and we explicitly describe all such matrices.  相似文献   

9.
A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p ? 1, A ∈ p − QH, if A(∣A2p − ∣A2p)A ? 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A−1(0) ⊆ A∗-1(0), A ∈ p − QH, a necessary and sufficient condition for the adjoint of a pure p − QH operator to be supercyclic is proved. Operators in p − QH satisfy Bishop’s property (β). Each A ∈ p − QH has the finite ascent property and the quasi-nilpotent part H0(A − λI) of A equals (A − λI)-1(0) for all complex numbers λ; hence f(A) satisfies Weyl’s theorem, and f(A) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ(A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p − QH.  相似文献   

10.
We consider the boundary value problem (?p(u′))′ + λF(tu) = 0, with p > 1, t ∈ (0, 1), u(0) = u(1) = 0, and with λ > 0. The value of λ is chosen so that the boundary value problem has a positive solution. In addition, we derive an explicit interval for λ such that, for any λ in this interval, the existence of a positive solution to the boundary value problem is guaranteed. In addition, the existence of two positive solutions for λ in an appropriate interval is also discussed.  相似文献   

11.
For a prescribed real number s ∈ [1, 2), we give some sufficient conditions on the coefficients p(x) and q(x) such that every solution y = y(x), y ∈ C2((0, T]) of the linear differential equation (p(x)y′)′ + q(x)y = 0 on (0, T], is bounded and fractal oscillatory near x = 0 with the fractal dimension equal to s. This means that y oscillates near x = 0 and the fractal (box-counting) dimension of the graph Γ(y) of y is equal to s as well as the s dimensional upper Minkowski content (generalized length) of Γ(y) is finite and strictly positive. It verifies that y admits similar kind of the fractal geometric asymptotic behaviour near x = 0 like the chirp function ych(x) = a(x)S(φ(x)), which often occurs in the time-frequency analysis and its various applications. Furthermore, this kind of oscillations is established for the Bessel, chirp and other types of damped linear differential equations given in the form y″ + (μ/x)y′ + g(x)y = 0, x ∈ (0, T]. In order to prove the main results, we state a new criterion for fractal oscillations near x = 0 of real continuous functions which essentially improves related one presented in [1].  相似文献   

12.
Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑k = 0nakxk ∈ K[x], an ≠ 0. For p ∈ K[x]\F[x], define DF(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : akF}. For p ∈ F[x], define DF(p) = n. Let p(x) = ∑k = 0nakxk and let q(x) = ∑j = 0mbjxj, with an ≠ 0, bm ≠ 0, anbm ∈ F, bjF for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and DF(p ° q) = DF(q). With only the assumption that anbm ∈ F, we prove the inequality DF(p ° q) ≥ DF(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(xy)), where p is a polynomial in one variable.  相似文献   

13.
In this paper we consider a problem of preemptive scheduling of multiprocessor tasks on dedicated processors in order to minimize the sum of completion times. Using a standard notation, our problem can be denoted as P ∣ fixj, pmtn ∣ ∑Cj. We give a polynomial-time algorithm to solve P ∣ fixj, G = {P4, dart}-free, pmtn ∣ ∑Cj problem. This result generalizes the following problems: P2 ∣ fixj, pmtn ∣ ∑Cj, P ∣ ∣fixj∣ ∈ {1, m}, pmtn ∣ ∑Cj and P4 ∣ fixj = 2, pmtn ∣ ∑Cj.  相似文献   

14.
We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Qij = 1 for 0 ? j − i ? N − 1 and Qij = 0 otherwise. We prove that det(QQ) is the minimum of det(RR) over all complex matrices R with the same dimensions as Q satisfying ∣Rij∣ ? 1 whenever Qij = 1 and Rij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ.  相似文献   

15.
Nonlinear matrix equation Xs + AXtA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.  相似文献   

16.
For real A, B such that −1 ? B < A ? 1, we denote by R(A,B) the class of functions, such that f′ ? (1 + Az)/(1 + Bz). Sharp distortion results for functions from RA,B are obtained.  相似文献   

17.
We study commutative algebras which are generalizations of Jordan algebras. The associator is defined as usual by (xyz) = (x y)z − x(y z). The Jordan identity is (x2yx) = 0. In the three generalizations given below, t, β, and γare scalars. ((x x)y)x + t((x x)x)y = 0, ((x x)x)(y x) − (((x x)x)y)x = 0, β((x x)y)x + γ((x x)x)y − (β + γ)((y x)x)x = 0. We show that with the exception of a few values of the parameters, the first implies both the second and the third. The first is equivalent to the combination of ((x x)x)x = 0 and the third. We give examples to show that our results are in some reasonable sense, the best possible.  相似文献   

18.
Let G = (VE) be a connected graph. The distance between two vertices u, v ∈ V, denoted by d(uv), is the length of a shortest u − v path in G. The distance between a vertex v ∈ V and a subset P ⊂ V is defined as , and it is denoted by d(vP). An ordered partition {P1P2, … , Pt} of vertices of a graph G, is a resolving partition of G, if all the distance vectors (d(vP1), d(vP2), … , d(vPt)) are different. The partition dimension of G, denoted by pd(G), is the minimum number of sets in any resolving partition of G. In this article we study the partition dimension of Cartesian product graphs. More precisely, we show that for all pairs of connected graphs G, H, pd(G × H) ? pd(G) + pd(H) and pd(G × H) ? pd(G) + dim(H), where dim(H) denotes the metric dimension of H. Consequently, we show that pd(G × H) ? dim(G) + dim(H) + 1.  相似文献   

19.
The Barnes’ G-function G(x) = 1/Γ2, satisfies the functional equation logG(x + 1) − logG(x) = logΓ(x). We complement W. Krull’s work in Bemerkungen zur Differenzengleichung g(x + 1) − g(x) = φ(x), Math. Nachrichten 1 (1948), 365-376 with additional results that yield a different characterization of the function G, new expansions and sharp bounds for G on x > 0 in terms of Gamma and Digamma functions, a new expansion for the Gamma function and summation formulae with Polygamma functions.  相似文献   

20.
Suppose that p(XY) = A − BX − X(∗)B(∗) − CYC(∗) and q(XY) = A − BX + X(∗)B(∗) − CYC(∗) are quaternion matrix expressions, where A is persymmetric or perskew-symmetric. We in this paper derive the minimal rank formula of p(XY) with respect to pair of matrices X and Y = Y(∗), and the minimal rank formula of q(XY) with respect to pair of matrices X and Y = −Y(∗). As applications, we establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.  相似文献   

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