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1.
The behavior of the sequence xn + 1 = xn(3N − xn2)/2N is studied for N > 0 and varying real x0. When 0 < x0 < (3N)1/2 the sequence converges quadratically to N1/2. When x0 > (5N)1/2 the sequence oscillates infinitely. There is an increasing sequence βr, with β−1 = (3N)1/2 which converges to (5N)1/2 and is such that when βr < x0 < βr + 1 the sequence {xn} converges to (−1)rN1/2. For x0 = 0, β−1, β0,… the sequence converges to 0. For x0 = (5N)1/2 the sequence oscillates: xn = (−1)n(5N)1/2. The behavior for negative x0 is obtained by symmetry. 相似文献
2.
Let X be a Banach space with closed unit ball B. Given k
, X is said to be k-β, respectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0 there exists δ, 0 < δ < 1, so that for every x B and every ε-separated sequence (xn) B there are indices (ni)ki = 1, respectively, (ni)k + 1i = 1, such that (1/(k + 1))||x + ∑ki = 1 xni|| ≤ 1 − δ, respectively, (1/(k + 1))||∑k + 1i = 1 xni|| ≤ 1 − δ. It is shown that a Banach space constructed by Schachermayer is 2-β, but is not isomorphic to any 2-NUC Banach space. Modifying this example, we also show that there is a 2-NUC Banach space which cannot be equivalently renormed to be 1-β. 相似文献
3.
Patrick S. Hagan 《Advances in Applied Mathematics》1981,2(4):400-416
We consider the general reaction-diffusion system At = F(A) + ε DM 2A + εg(→x, A), 0 < ε 1, where the small term εg(→x, A) represents the effects of localized impurities. We assume that the system At = F(A) has a stable time-periodic solution. Then we construct stable target pattern solutions of the full system. For typical initial conditions we find that these target patterns will arise only if g(→x, A) 0. Finally, we determine how target patterns interact and show that higher frequency target patterns eventually engulf neighboring lower frequency target patterns. 相似文献
4.
Gerhard Rosenberger 《Monatshefte für Mathematik》1978,85(3):211-233
x
1
2
+...+x
n
2
—ax
1...x
n
=b. First we describe a combinatorial presentation of a group of automorphisms of this equation, ifn=3, then we getPGL (2, ) as such a group of automorphisms of this equation. This gives analytical applications becausePGL (2, ) acts discontinously on the set {(x
1,x
2,x
3)0<x
1,x
2,x
3 andx
1
2
+x
2
2
+x
3
2
–x
1
x
2
x
3=b0}3. Further we ask for fundamental solutions of this equation. Finally, letx
1,x
2,x
3 withx
1
x
2
2
+x
3
2
––x
1
x
2
x
3=0 Then there areA, BSL(2, ) with trA=x
1, trB=x
2 and trA B=x
3, and the group (A, B) is a discrete free group of rank two. In analysis we are interested in the question whether there are evenA, BSL(2, ) with trA=x
1, trB=x
2 and trA B=x
3. We give necessary and sufficient conditions for that and remark that this question is connected with the ternary quadratic formk1p
2+k2q
2–r
2,k
1=x
1
2
,k
2=16(x
2
2
+x
1
2
+x
3
2
–x
1
x
2
x
3–4), which has some invariant properties. 相似文献
5.
Antonia W. Bluher 《Designs, Codes and Cryptography》2003,30(1):85-95
Let F be a field of characteristic 3 and 0 a F. We show that the 10 ways to factor x
6 + x + a into two cubics over the algebraic closure F are in natural Galois bijection with the 10 roots of x
10 + ax + 1. We use this to (1) prove the two polynomials have the same splitting field; (2) prove that a difference set constructed by Arasu and Player using the polynomial x
6 + x + a is isomorphic to a difference set constructed by Dillon using the polynomial x
10 + x + a; (3) obtain a natural realization for the accidental isomorphism between the alternating group A
6 and the special linear group PSL2(9); and (4) characterize how x
6 + x + a factors when F = GF (3m) with m odd. For example, x
6 + x + a is irreducible if and only if a can be written as – 36 + 4 with F
× and Tr(5) 0. 相似文献
6.
For any integersa
1,a
2,a
3,a
4 andc witha
1
a
2
a
3
a
4≢0(modp), this paper shows that there exists a solutionX=(x
1,x
2,x
3,x
4) ∈Z
4 of the congruencea
1
x
1
2
+a
2
x
2
2
+a
3
x
3
2
+a
4
x
4
2
≡c(modp) such that
Research of Zheng Zhiyong is supported by NNSF Grant of China. He would also like to thank the first author and the Mathematics
Department of Kansas, State University for their hospitality and support. 相似文献
7.
Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ?N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx even though Φ?1(y) is typically an (N ? m)—dimensional hyperplane; in addition, x is then equal to the element in Φ?1(y) of minimal ??1‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in Φ?1(y) with smallest ??2(w)‐norm. If x(n) is the solution at iteration step n, then the new weight w(n) is defined by w := [|x|2 + ε]?1/2, i = 1, …, N, for a decreasing sequence of adaptively defined εn; this updated weight is then used to obtain x(n + 1) and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x(n) converges for all y, regardless of whether Φ?1(y) contains a sparse vector. If there is a sparse vector in Φ?1(y), then the limit is this sparse vector, and when x(n) is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w = [|x|2 + ε]?1+τ/2, i = 1, …, N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc. 相似文献
8.
Cameron–Liebler line classes are sets of lines in PG(3, q) that contain a fixed number x of lines of every spread. Cameron and Liebler classified Cameron–Liebler line classes for x ∈ {0, 1, 2, q2 ? 1, q2, q2 + 1} and conjectured that no others exist. This conjecture was disproven by Drudge for q = 3 [8] and his counterexample was generalized to a counterexample for any odd q by Bruen and Drudge [4]. A counterexample for q even was found by Govaerts and Penttila [9]. Non‐existence results on Cameron–Liebler line classes were found for different values of x. In this article, we improve the non‐existence results on Cameron–Liebler line classes of Govaerts and Storme [11], for q not a prime. We prove the non‐existence of Cameron–Liebler line classes for 3 ≤ x < q/2. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 342–349, 2008 相似文献
9.
The linear equation Δ2u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?2 has the particularly interesting nonlinear generalization Δg2u = 1, where Δg = e?2u Δ is the Laplace‐Beltrami operator for the metric g = e2ug0, with g0 the standard Euclidean metric on ?2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+Kg(x) exp(2u(x)) = 0 and Δ Kg(x) + exp(2u(x)) = 0, with x ∈ ?2, describing a conformally flat surface with a Gauss curvature function Kg that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ Kg exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K+ ∈ L1(B1(y)), uniformly w.r.t. y ∈ ?2. We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc. 相似文献
10.
We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line {(x1,x2)|x10,x2=0} before time n. It is proved that for aperiodic random walk with mean zero and finite 2+(>2)-th absolute moment, this probability times n1/4 converges to some positive constant c* as
. We show that c* is expressed by using the characteristic function of the increment of the random walk. For the simple random walk, this expression gives
Mathematics Subject Classification (2000):60G50, 60E10 相似文献
11.
Carl M. Bender 《Advances in Mathematics》1978,30(3):250-267
For many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by−″ + (x2/4+εx4/4−E)y=0, y(±∞)=0,the perturbation coefficients An in the expansion for the ground-state energysimplify dramatically as n → ∞:.We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λ4 model for summing the set of all graphs as the number of vertices → ∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron. 相似文献
12.
We provide additional methods for the evaluation of the integral
N0,4(a;m) : = ò0¥ \fracdx( x4 + 2ax2 + 1 )m+1,N_{0,4}(a;m) := \int_{0}^{\infty} \frac{dx}{( x^{4} + 2ax^{2} + 1 )^{m+1}}, 相似文献
13.
Liao Hua-kui 《Archiv der Mathematik》2001,76(2):149-160
We classify surfaces with affine Gauss-Kronecker curvature zero in locally and globally. If x:M ? A3x:M \rightarrow A^3 is Euclidean complete and its affine conormal surface U:M ? \Bbb R3U:M \rightarrow \Bbb R^3 is complete, thenx(M) is an elliptic paraboloid or unimodular affine equivalent to a surface determined by equation x21-x22+C2 x^2_1-x^2_2+C^2\,cos h2x3=0\,\hbox {h}^2x_3=0 where x2 \leqq Cx_2 \leqq C and C is a positive constant. 相似文献
14.
Consider the Hill's operator Q = ?d2/dx2 + q(x) in which q(x), 0 ≤ x ≤ 1, is a white noise. Denote by f(μ) the probability density function of ?λ0(q), the negative of the ground state eigenvalue, at μ. We prove the detailed asymptotics as μ → + ∞. This result is based on a precise Laplace analysis of a functional integral representation for f(μ) established by S. Cambronero and H. P. McKean in 5 . © 2005 Wiley Periodicals, Inc. 相似文献
15.
R. Dietmann 《Archiv der Mathematik》2000,75(3):195-197
We give a somewhat improved estimate for the smallest non-trivial solution of a cubic congruence of the form a1 x13 + ?+ as xs3 o 0 (mod m)a_1 x_1^3 + \cdots + a_s x_s^3 \equiv 0 \,(\hbox {mod}\,\, m). This also yields new results about small fractional parts of additive cubic forms, where we restrict ourselves to the case of five variables. 相似文献
16.
S. I. El‐Zanati G. F. Seelinger P. A. Sissokho L. E. Spence C. Vanden Eynden 《组合设计杂志》2008,16(4):329-341
Let Vn(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of Vn(q) is a partition of Vn(q) if every nonzero element of Vn(q) is contained in exactly one element of . Suppose there exists a partition of Vn(q) into xi subspaces of dimension ni, 1 ≤ i ≤ k. Then x1, …, xk satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of Vn(q). In this article, we show that there exists a partition of Vn(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2n ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of Vn(q) induce uniformly resolvable designs on qn points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008 相似文献
17.
Wojciech M. Zaja̧czkowski 《Mathematical Methods in the Applied Sciences》2011,34(2):191-197
Let f∈L2, ? µ(?3), where where x = (x1, x2, x3) is the Cartesian system in ?3, x′ = (x1, x2), , µ∈?+\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?3 Given f∈L2, ? µ(?3) we show the existence of u∈H(?3) such that where Since f, u, g are defined in ?3 we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
18.
O. M. Fomenko 《Journal of Mathematical Sciences》2011,178(2):227-233
New results on the distribution of integral points on the cones
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