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1.
《Journal of Algebra》2002,247(2):509-540
Let Fm be a free group of a finite rank m ≥ 2 and let Xi, Yj be elements in Fm. A non-empty word w(x1,…,xn) is called a C-test word in n letters for Fm if, whenever (X1,…,Xn) = w(Y1,…,Yn) ≠ 1, the two n-typles (X1,…,Xn) and (Y1,…,Yn) are conjugate in Fm. In this paper we construct, for each n ≥ 2, a C-test word vn(x1,…,xn) with the additional property that vn(X1,…,Xn) = 1 if and only if the subgroup of Fm generated by X1,…,Xn is cyclic. Making use of such words vm(x1,…,xm) and vm + 1(x1,…,xm + 1), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u1, u2 ∈ Fm such that every endomorphism ψ of Fm with non-cyclic image is completely determined by ψ(u1), ψ(u2). 相似文献
2.
《Advances in Applied Mathematics》2002,28(3-4):360-394
A function which is homogeneous in x, y, z of degree n and satisfies Vxx + Vyy + Vzz = 0 is called a spherical harmonic. In polar coordinates, the spherical harmonics take the form rnfn, where fn is a spherical surface harmonic of degree n. On a sphere, fn satisfies ▵ fn + n(n + 1)fn = 0, where ▵ is the spherical Laplacian. Bounded spherical surface harmonics are well studied, but in certain instances, unbounded spherical surface harmonics may be of interest. For example, if X is a parameterization of a minimal surface and n is the corresponding unit normal, it is known that the support function, w = X · n, satisfies ▵w + 2w = 0 on a branched covering of a sphere with some points removed. While simple in form, the boundary value problem for the support function has a very rich solution set. We illustrate this by using spherical harmonics of degree one to construct a number of classical genus-zero minimal surfaces such as the catenoid, the helicoid, Enneper's surface, and Hennenberg's surface, and Riemann's family of singly periodic genus-one minimal surfaces. 相似文献
3.
《Advances in Applied Mathematics》2002,28(3-4):302-342
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P-matrices play an important role in the well-posedness of a linear complementarity problem (LCP). Similarly, the well-posedness of a horizontal linear complementarity problem (HLCP) is closely related to the column-W property of a matrix k-tuple.In this paper we first consider the problem of generating P-matrices from a given pair of matrices. Given a matrix pair (D, F) where D is a square matrix of order m and matrix F has m rows, “what are the conditions under which there exists a matrix G such that (D + FG) is a P-matrix?”. We obtain necessary and sufficient conditions for the special case when the column rank of F is m ? 1. A decision algorithm of complexity O(m2) to check whether the given pair of matrices (D, F) is P-matrisable is obtained. We also obtain a necessary and an independent sufficient condition for the general case when rank(F) is less than m ? 1.We then generalise the P-matrix generating problem to the generation of matrix k-tuples satisfying the column-W property from a given matrix (k + 1)-tuple. That is, given a matrix (k + 1)-tuple (D1, … ,Dk, F), where Djs are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D1 + FG1, … ,Dk + FGk) satisfies the column-W property. As in the case of P-matrices we obtain necessary and sufficient conditions for the case when rank(F) = m ? 1. Using these conditions a decision algorithm of complexity O(km2) to check whether the given matrix (k + 1)-tuple is column-W matrisable is obtained. Then for the case when rank(F) is less than m ? 1, we obtain a necessary and an independent sufficient condition.For a special sub-class of P-matrices we give a polynomial time decision algorithm for P-matrisability. Finally, we obtain a geometric characterisation of column-W property by generalising the well known separation theorem for P-matrices. 相似文献
6.
《Journal of Complexity》1996,12(2):167-174
LetKbe a closed basic set inRngiven by the polynomial inequalities φ1≥ 0, . . . , φm≥ 0 and let Σ be the semiring generated by the φkand the squares inR[x1, . . . ,xn]. Schmüdgen has shown that ifKis compact then any polynomial function strictly positive onKbelongs to Σ. Easy consequences are (1)f≥ 0 onKif and only iff∈R++ Σ (Positivstellensatz) and (2) iff≥ 0 onKbutf∈ Σ then asdtends to 0+, in any representation off + das an element of Σ in terms of the φk, the squares and semiring operations, the integerN(d) which is the minimum over all representations of the maximum degree of the summands must become arbitrarily large. A one-dimensional example is analyzed to obtain asymptotic lower and upper bounds of the formcd−1/2≤N(d) ≤Cd−1/2log (1/d). 相似文献
7.
《Advances in Applied Mathematics》2002,28(3-4):580-591
For fixed positive integer k, let En denote the set of lattice paths using the steps (1, 1), (1, − 1), and (k, 0) and running from (0, 0) to (n, 0) while remaining strictly above the x-axis elsewhere. We first prove bijectively that the total area of the regions bounded by the paths of En and the x-axis satisfies a four-term recurrence depending only on k. We then give both a bijective and a generating function argument proving that the total area under the paths of En equals the total number of lattice points on the x-axis hit by the unrestricted paths running from (0, 0) to (n − 2, 0) and using the same step set as above. 相似文献
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《Journal of Complexity》1998,14(4):448-453
LetP⊂[0, 1]dbe ann-point set and letw: P→[0, ∞) be a weight function withw(P)=∑z∈P w(z)=1. TheL2-discrepancy of the weighted set (P, w) is defined as theL2-average ofD(x)=vol(Bx)−w(P∩Bx) overx∈[0, 1]d, where vol(Bx) is the volume of thed-dimensional intervalBx=∏dk=1 [0, xk). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd⩾1 and allε>0 there exists a weighted set of at mostKε−ppoints in [0, 1]dwithL2-discrepancy at mostε, whereK=K(p) is a suitable number independent ofεandd. Wasilkowski and Woźniakowski proved thatp*⩽1.4779, by combining known bounds for the error of numerical integration and using their relation toL2-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*⩾1.04882, and by a slight modification of the proof we getp*⩾1.0669. Determiningp* exactly seems to be quite a difficult problem. 相似文献
9.
《Linear algebra and its applications》2006,412(2-3):396-407
The support of an [n, k] linear code C over a finite field Fq is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position. The rth generalized Hamming weight dr(C), 1 ⩽ r ⩽ k, of C is defined as the minimum of the cardinalities of the supports of all [n, r] subcodes of C. The sequence (d1(C), d2(C), … , dk(C)) is called the Hamming weight hierarchy (HWH) of C. The HWH, dr(C) = n − k + r; r = 1, 2 , …, k, characterizes maximum distance separable (MDS) codes. Therefore the matrix characterization of MDS codes is also the characterization of codes with the HWH dr(C) = n − k + r; r = 1, 2, … , k. A linear code C with systematic check matrix [I∣P], where I is the (n − k) × (n − k) identity matrix and P is a (n − k) × k matrix, is MDS iff every square submatrix of P is nonsingular. In this paper we extend this characterization to linear codes with arbitrary HWH. Using this result, we characterize Near-MDS codes, Near-Near-MDS (N2-MDS) codes and Aμ-MDS codes. The MDS-rank of C is the smallest integer η such that dη+1 = n − k + η + 1 and the defect vector of C with MDS-rank η is defined as the ordered set {μ1(C), μ2(C), μ3(C), … , μη(C), μη+1(C)}, where μi(C) = n − k + i − di(C). We call C a dually defective code if the defect vector of the code and its dual are the same. We also discuss matrix characterization of dually defective codes. Further, the codes meeting the generalized Greismer bound are characterized in terms of their generator matrix. The HWH of dually defective codes meeting the generalized Greismer bound are also reported. 相似文献
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In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c1(τ) and using a limiting argument for c = c1(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?2f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?2f(0, 0) = 0. 相似文献
11.
Fayequa B. Majid Arjuna I. Ranasinghe 《Communications in Nonlinear Science & Numerical Simulation》2009,14(5):1861-1867
The initial-boundary-value problem on the semi-infinite interval and on a finite interval for the Burgers equation ut = uxx + 2uxu is solved using a stream function ? and a linearizing transformation w = e?. The transformation reduces the equation to a heat equation with appropriate initial and homogeneous time-dependent linear boundary conditions. One advantage of this method is that we never need to find an explicit expression for ? in our computations. 相似文献
12.
《European Journal of Operational Research》2006,171(2):693-707
We are concerned with a variation of the knapsack problem as well as of the knapsack sharing problem, where we are given a set of n items and a knapsack of a fixed capacity. As usual, each item is associated with its profit and weight, and the problem is to determine the subset of items to be packed into the knapsack. However, in the problem there are s players and the items are divided into s + 1 disjoint groups, Nk (k = 0, 1, … , s). The player k is concerned only with the items in N0 ∪ Nk, where N0 is the set of ‘common’ items, while Nk represents the set of his own items. The problem is to maximize the minimum of the profits of all the players. An algorithm is developed to solve this problem to optimality, and through a series of computational experiments, we evaluate the performance of the developed algorithm. 相似文献
13.
The aim of this work is to study by numerical simulation a mathematical modelling technique describing charge trapping during initial charge injection in an insulator submitted to electron beam irradiation. A two-fluxes method described by a set of two stationary transport equations is used to split the electron current je(z) into coupled forward je+(z) and backward je?(z) currents and such that je(z) = je+(z) ? je?(z). The sparse algebraic linear system, resulting from the vertex-centered finite-volume discretization scheme is solved by an iterative decoupled fixed point method which involves the direct inversion of a bi-diagonal matrix.The sensitivity of the initial secondary electron emission yield with respect to the energy of incident primary electrons beam, that is penetration depth of the incident beam, or electron cross sections (absorption and diffusion) is investigated by numerical simulations. 相似文献
14.
《International Journal of Approximate Reasoning》2006,41(3):331-342
In many real-life situations, we know the probability distribution of two random variables x1 and x2, but we have no information about the correlation between x1 and x2; what are the possible probability distributions for the sum x1 + x2? This question was originally raised by A.N. Kolmogorov. Algorithms exist that provide best-possible bounds for the distribution of x1 + x2; these algorithms have been implemented as a part of the efficient software for handling probabilistic uncertainty. A natural question is: what if we have several (n > 2) variables with known distribution, we have no information about their correlation, and we are interested in possible probability distribution for the sum y = x1 + ⋯ + xn? Known formulas for the case n = 2 can be (and have been) extended to this case. However, as we prove in this paper, not only are these formulas not best-possible anymore, but in general, computing the best-possible bounds for arbitrary n is an NP-hard (computationally intractable) problem. 相似文献
15.
《Journal of Algebra》1999,211(2):562-577
LetRbe a Krull ring with quotient fieldKanda1,…,aninR. If and only if theaiare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb1,…,bninRan interpolating integer-valued polynomial, i.e., anf ∈ K[x] withf(ai) = biandf(R) ⊆ R.IfSis an infinite subring of a discrete valuation ringRvwith quotient fieldKanda1,…,aninSare pairwise incongruent mod allMkv ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of theaiamong residue classes of the idealsMkv ∩ S) such that for allb1,…,bn ∈ Rvthere exists a polynomialf ∈ K[x] of degree at mostdwithf(ai) = biandf(S) ⊆ Rv. 相似文献
16.
《Linear algebra and its applications》2006,412(2-3):154-160
In this paper we consider the positive definite solutions of nonlinear matrix equation X + A⋆X−δA = Q, where δ ∈ (0, 1], which appears for the first time in [S.M. El-Sayed, A.C.M. Ran, On an iteration methods for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001) 632–645]. The necessary and sufficient conditions for the existence of a solution are derived. An iterative algorithm for obtaining the positive definite solutions of the equation is discussed. The error estimations are found. 相似文献
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In this paper, we estimate the reliability of some parallel and series multi-component stress–strength models. We determine the reliability of a system composed of k dependent components subjected to n dependent stresses. We study the cases, when the components are either arranged in series or in parallel. The components strengths are assumed to have (k + 1)-parameter multivariate Marshall–Olkin exponential distribution, while the stresses are (n + 1)-parameter multivariate Marshall–Olkin exponentially distributed. 相似文献
19.
《Journal of the Egyptian Mathematical Society》2014,22(1):11-13
Let n ⩾ 1 be a fixed integer and let R be an (n + 1)!-torsion free 1-ring with identity element e. If F, d:R → R are two additive mappings satisfying F(xn+1) = F(x)(x1)n + xd(x)(x1)n−1 + x2d(x)(x1)n−2+ ⋯ +xnd(x) for all x ∈ R, then d is a Jordan 1-derivation and F is a generalized Jordan 1-derivation on R. 相似文献
20.
Somayeh Rashedi Ghodrat Ebadi Anjan Biswas 《Journal of the Egyptian Mathematical Society》2013,21(3):175-183
In this paper, we establish the formulas of the extermal ranks of the quaternion matrix expression f(X1, X2) = C7 ? A4X1B4 ? A5X2B5 where X1, X2 are variant quaternion matrices subject to quaternion matrix equations A1X1 = C1, A2X1 = C2, A3X1 = C3, X2B1 = C4, X2B2 = C5, X2B3 = C6. As applications, we give a new necessary and sufficient condition for the existence of solutions to some systems of quaternion matrix equations. Some results can be viewed as special cases of the results of this paper. 相似文献