共查询到20条相似文献,搜索用时 46 毫秒
1.
Mervan Paši? 《Applied mathematics and computation》2011,218(5):2281-2293
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
2.
Donald Mills 《Journal of Number Theory》2002,92(1):87-98
Let Fq denote the finite field of q elements, q=pe odd, let χ1 denote the canonical additive character of Fq where χ1(c)=e2πiTr(c)/p for all c∈Fq, and let Tr represent the trace function from Fq to Fp. We are interested in evaluating Weil sums of the form S=S(a1, …, an)=∑x∈Fq χ1(D(x)) where D(x)=∑ni=1 aixpαi+pβi, αi?βi for each i, is known as a Dembowski-Ostrom polynomial (or as a D-O polynomial). Coulter has determined the value of S when D(x)=axpα+1; in this note we show how Coulter's methods can be generalized to determine the absolute value of S for any D-O polynomial. When e is even, we give a subclass of D-O polynomials whose Weil sums are real-valued, and in certain cases we are able to resolve the sign of S. We conclude by showing how Coulter's work for the monomial case can be used to determine a lower bound on the number of Flq-solutions to the diagonal-type equation ∑li=1 xpγ+1i+(xi+λ)pγ+1=0, where l is even, e/gcd(γ, e) is odd, and h (X)=λpe−γXpe−γ+λpγX is a permutation polynomial over Fq. 相似文献
3.
Let Un ⊂ Cn[a, b] be an extended Chebyshev space of dimension n + 1. Suppose that f0 ∈ Un is strictly positive and f1 ∈ Un has the property that f1/f0 is strictly increasing. We search for conditions ensuring the existence of points t0, …, tn ∈ [a, b] and positive coefficients α0, …, αn such that for all f ∈ C[a, b], the operator Bn:C[a, b] → Un defined by satisfies Bnf0 = f0 and Bnf1 = f1. Here it is assumed that pn,k, k = 0, …, n, is a Bernstein basis, defined by the property that each pn,k has a zero of order k at a and a zero of order n − k at b. 相似文献
4.
Hask?z Co?kun 《Journal of Mathematical Analysis and Applications》2002,276(2):833-844
We consider Hill's equation y″+(λ−q)y=0 where q∈L1[0,π]. We show that if ln—the length of the n-th instability interval—is of order O(n−(k+2)) then the real Fourier coefficients ank,bnk of q(k)—k-th derivative of q—are of order O(n−2), which implies that q(k) is absolutely continuous almost everywhere for k=0,1,2,…. 相似文献
5.
Yossi Moshe 《Journal of Number Theory》2007,123(1):224-240
Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind. 相似文献
6.
Irene Benedetti Luisa Malaguti Valentina Taddei 《Journal of Mathematical Analysis and Applications》2010,368(1):90-102
The paper deals with the multivalued initial value problem x′∈A(t,x)x+F(t,x) for a.a. t∈[a,b], x(a)=x0 in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x and the investigation includes the case when both A and F have a superlinear growth in x. We prove the existence of local and global classical solutions in the Sobolev space W1,p([a,b],E) with 1<p<∞. Introducing a suitably defined Lyapunov-like function, we are able to investigate the topological structure of the solution set. Our main tool is a continuation principle in Frechét spaces and we prove the required pushing condition in two different ways. Some examples complete the discussion. 相似文献
7.
Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains 总被引:1,自引:0,他引:1
Let a, n ? 1 be integers and S = {x1, … , xn} be a set of n distinct positive integers. The matrix having the ath power (xi, xj)a of the greatest common divisor of xi and xj as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (Sa). Similarly we can define the ath power LCM matrix [Sa]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S1 ∪ ? ∪ Sk, where k ? 1 is an integer and all Si (1 ? i ? k) are divisor chains such that (max(Si), max(Sj)) = gcd(S) for 1 ? i ≠ j ? k. In this paper, we first obtain formulae of determinants of power GCD matrices (Sa) and power LCM matrices [Sa] on the set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, we then show that det(Sa)∣det(Sb), det[Sa]∣det[Sb] and det(Sa)∣det[Sb] if a∣b and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime. 相似文献
8.
V.V. Bavula 《Journal of Pure and Applied Algebra》2008,212(10):2320-2337
Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra Pn?K[x1,…,xn], the ring of differential operators D(Pn) on Pn, D(Pn)⊗Pm, the n’th Weyl algebra An or An⊗Pm, the power series algebra K[[x1,…,xn]], the subalgebra Tk1,…,kn of D(Pn) generated by Pn and the higher derivations , 0≤j<pki, i=1,…,n (where k1,…,kn∈N), Tk1,…,kn⊗Pm or an arbitrary central simple (countably generated) algebra over an arbitrary field. 相似文献
9.
This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation ut(x, t) = uxx + qux(x, t) + p(t)u(x, t), with Dirichlet boundary conditions u(0, t) = ψ0, u(1, t) = ψ1. The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[·]:P→H1,2[0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only zero function, then the input-output mapping Φ[·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data ux(0, t) = f(t) the unknown parameter p(t) at (x, t) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[·]:P→H1,2[0,T] is given explicitly interms of the semigroup. 相似文献
10.
We consider the linear nonautonomous system of difference equations xn+1−xn+P(n)xn−k=0, n=0,1,2,… , where k∈Z, P(n)∈Rrxr. We obtain sufficient conditions for the system to be oscillatory. The conditions based on the eigenvalues of the matrix coefficients of the system. 相似文献
11.
Kazushi Yoshitomi 《Indagationes Mathematicae》2005,16(2):289-299
Let q ∈ {2, 3} and let 0 = s0 < s1 < … < sq = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set lj = sj − sj−1 for j ∈ 1, q. Given (p1,, pq) ∈ Rq, let b: Z → R be a periodic function of period T such that b(·) = pj on sj−1 + 1, sj for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l2(Z). By [λ2j , λ2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that pm ≠ pn for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1. 相似文献
12.
In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ0, u(1,t)=ψ1. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C1[0,T], Ψ[⋅]:K→C1[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k′(0) and k′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C1[0,T], Ψ[⋅]:K→C1[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1. 相似文献
13.
Let k ? k′ be a field extension. We give relations between the kernels of higher derivations on k[X] and k′[X], where k[X]:= k[x 1,…, x n ] denotes the polynomial ring in n variables over the field k. More precisely, let D = {D n } n=0 ∞ a higher k-derivation on k[X] and D′ = {D′ n } n=0 ∞ a higher k′-derivation on k′[X] such that D′ m (x i ) = D m (x i ) for all m ? 0 and i = 1, 2,…, n. Then (1) k[X] D = k if and only if k′[X] D′ = k′; (2) k[X] D is a finitely generated k-algebra if and only if k′[X] D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[X] D of a higher derivation D of k[X] can be generated by a set of closed polynomials. 相似文献
14.
Shaofang Hong 《Linear algebra and its applications》2006,416(1):124-134
Let S = {x1, … , xn} be a set of n distinct positive integers and f be an arithmetical function. Let [f(xi, xj)] denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its i, j-entry and (f[xi, xj]) denote the n × n matrix having f evaluated at the least common multiple [xi, xj] of xi and xj as its i, j-entry. The set S is said to be lcm-closed if [xi, xj] ∈ S for all 1 ? i, j ? n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x1, … , xn} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ? p) for any prime p, then the matrix [f(xi, xj)] (resp. (f[xi, xj])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1-14], we also obtain reduced formulas for det(f(xi, xj)) and det(f[xi, xj]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices. 相似文献
15.
《代数通讯》2013,41(3):937-951
ABSTRACT Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I 2(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M 2 m (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 n?2 ? 2 and n ≥ 5, we show that q σ ∈ I n (K), where q σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms. 相似文献
16.
Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space 总被引:1,自引:0,他引:1
E.U. Ofoedu 《Journal of Mathematical Analysis and Applications》2006,321(2):722-728
Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n?0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}n?0⊂[0,1] be such that ∑n?0αn=∞, and ∑n?0αn(kn−1)<∞. Suppose {xn}n?0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n?0, and suppose there exists a strictly increasing continuous function , ?(0)=0 such that 〈Tnx−x∗,j(x−x∗)〉?kn‖x−x∗‖2−?(‖x−x∗‖), ∀x∈K. It is proved that {xn}n?0 converges strongly to x∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n?0 (where {un}n?0 is a bounded sequence in K and {an}n?0, {bn}n?0, {cn}n?0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T. 相似文献
17.
R.S. Singh 《Journal of multivariate analysis》1976,6(2):338-342
Let Xj = (X1j ,…, Xpj), j = 1,…, n be n independent random vectors. For x = (x1 ,…, xp) in Rp and for α in [0, 1], let Fj(x) = αI(X1j < x1 ,…, Xpj < xp) + (1 ? α) I(X1j ≤ x1 ,…, Xpj ≤ xp), where I(A) is the indicator random variable of the event A. Let Fj(x) = E(Fj(x)) and Dn = supx, α max1 ≤ N ≤ n |Σ0n(Fj(x) ? Fj(x))|. It is shown that P[Dn ≥ L] < 4pL exp{?2(L2n?1 ? 1)} for each positive integer n and for all L2 ≥ n; and, as n → ∞, with probability one. 相似文献
18.
A smoothing self-adaptive Levenberg-Marquardt algorithm for solving system of nonlinear inequalities
In this paper, we consider the smoothing self-adaptive Levenberg-Marquardt algorithm for the system of nonlinear inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations H(x) = 0. A smoothing self-adaptive Levenberg-Marquardt algorithm is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The Levenberg-Marquardt parameter μk is chosen as the product of μk = ∥Hk∥δ with δ ∈ (0, 2] being a positive constant. We will show that if ∥Hk∥δ provides a local error bound, which is weaker than the non-singularity, the proposed method converges superlinearly to the solution for δ ∈ (0, 1), while quadratically for δ ∈ [1, 2]. Numerical results show that the new method performs very well for system of inequalities. 相似文献
19.
Y. Stein 《Israel Journal of Mathematics》1995,89(1-3):301-319
LetP=x n +P n?1(y)x n?1+…+P 0(y),Q=x m +Q m?2(y)x m?2+…+Q 0(y) belong toK[x, y], whereK is a field of characteristic zero. The main result of this paper is the following: Assume thatP x Q y ?P y Q x =1. Then:*
- K[Q m?2(y), …,Q 0(y)]=K[y],
- K[P, Q]=K[x, y] ifQ=x m +Q k (y)x k +Q r (y)x r