GENUINE-OPTIMAL CIRCULANT PRECONDITIONERS FOR WIENER-HOPF EQUATIONS

1. IntroductionWienerHopf equations are integral equations defined on the haif line:where rr > 0, a(.) C L1(ro and g(.) E L2(at). Here R = (--oo,oo) and ty [0,oo). Inou-r discussions, we assume that a(.) is colljugate symmetric, i.e. a(--t) = a(t). WienerHop f equations arise in a variety of practical aPplicatiolls in mathematics and ellgineering, forinstance, in the linear prediction problems fOr stationary stochastic processes [8, pp.145--146],diffuSion problems and scattering problems […     

Forced oscillation of second order differential equations with mixed nonlinearities

This article is concerned with the oscillation of the forced second order differential equation with mixed nonlinearities a(t) x ′ (t) γ′ + p 0 (t) x γ (g 0 (t)) + n i =1 p i (t) | x (g i (t)) | α i sgn x (g i (t)) = e(t), where γ is a quotient of odd positive integers, α i > 0, i = 1, 2, ··· , n, a, e, and p i ∈ C ([0, ∞ ) , R), a (t) > 0, gi : R → R are positive continuous functions on R with lim t →∞ g i (t) = ∞ , i = 0, 1, ··· , n. Our results generalize and improve the results in a recent article by Sun and Wong [29].     

A CLASS OF NONLOCAL BOUNDARY VALUE PROBLEMS OF NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

We consider the following boundary value problem ill the unbounded donain Liui = fi(x,u, Tu), i = 1, 2,' ! N,x E fl, (1) olLi "i0n Pi(x)t'i = gi(x,u), i = l, 2,',N,x E 0fl, (2) where x = (x i,', x.), u = (u1,' f uN), Th = (T1tti,', TNi'N) and [ n. 1 L, = -- I Z ajk(X)the i0j(X)C], Li,k=1' j=1 J] l Ltti = / K(x,y)ui(y)dy, x E n. jn K(x, y)ui(y)dy, x E n. Q denotes an unbounded dolllain in R", including the exterior of a boullded doinain and 0…     

THE CONVERGENCE OF APPROACH PENALTY FUNCTION METHOD FOR APPROXIMATE BILEVEL PROGRAMMING PROBLEM

' 1 IntroductionWe collsider the fOllowi11g bilevel programndng problen1:max f(x, y),(BP) s.t.x E X = {z E RnIAx = b,x 2 0}, (1)y e Y(x).whereY(x) = {argmaxdTyIDx Gy 5 g, y 2 0}, (2)and b E R", d, y E Rr, g E Rs, A, D.and G are m x n1 s x n aild 8 x r matrices respectively. If itis not very difficult to eva1uate f(and/or Vf) at all iteration points, there are many algorithmeavailable fOr solving problem (BP) (see [1,2,3etc1). However, in some problems (see [4]), f(x, y)is too com…     

非线性二阶阻尼微分方程的振动定理

Several oscillation criteria are given for the second order nonlinear differential equation with damped term of the form [α(t)(y'(t))σ]' p(t)(y'(t))σ q(t)f(y(t)) = 0, where α∈C(R, (0,∞)), p(t) and q(t) are allowed to change sign on [t0, ∞), and f∈C1 (R, R) such that xf(x) > 0 for x ≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.     

INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/（p^e）

Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence (a) over Z/(pe), there is a unique p-adic decomposition (a) = (a)0 (a)1·p … (a)e-1 ·pe-1, where each (a)i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and G' (f(x), pe) the set of all primitive sequences generated by f(x) over Z/(pe). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 deg(μ(x)),p- 1) = 1,set ψe-1 (x0, x1,…, xe-1) = xe-1·[ μ(xe-2) ηe-3 (x0, x1,…, xe-3)] ηe-2 (x0, x1,…, xe-2),which is a function of e variables over Z/(p). Then the compressing map ψe-1: G'(f(x),pe) → (Z/(p))∞,(a) (→)ψe-1((a)0, (a)1,… ,(a)e-1) is injective. That is, for (a), (b) ∈ G' (f(x), pe), (a) = (b) if and only if ψe - 1 ((a)0, (a)1,… , (a)e - 1) =ψe - 1 ((b)0,(b)1,… ,(b)e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions ψe-1 and ψe-1 over Z/(p) are both of the above form and satisfy ψe-1((a)0,(a)1,… ,(a)e-1) = ψe-1((b)0,(b)1,… ,(b)e-1) for (a),(b) ∈ G'(f(x),pe), the relations between (a) and (b), ψe-1 and ψe-1 are discussed.     

Geometric structure in stochastic approximation

1. IntroductionLet f: Re -- R be a differelltiable fUnction. f reaChes its extremes on the setJ = {x E R"lfx(x) = 0}, (1.1)where,x(X) = (V,..., V)". (1.2)If jx can be observed exactly at any x e R", then there are various numerical methods toconstruct {xh}, xk E Re such that the distance d(xk, J) between uk and J tends to zero ask -- co. However, in many application problems jx can only be observed with noise, i.e.,the observation at time k 1 isYk 1 = fi(~k) (k 1, (1'3)where xk is …     

A NOTE ON THE CONSISTENCY OF LS ESTIMATES IN LINEAR MODELS

Consider,the linear regression modelyi = x'iβ ei, 1≤i≤n, n≥1, (1)where x1, x2,' are known Hvectors, P is the unknown pdimensional vector of regressioncoefficiellts, e13 e2,' is a seqdence of iid. random errors, and y1, y2t' are known obser-vations of the dependellt variable. Denote by F the common distribution of e1, e2,' t andwrite9. = {F: j:xar=0, 0< j:lxl'aF< oo}, 1 5 r 5 2.The Least Squares (LS) estimate of P isn rs)n = Z SJ'x.ui, Sn = Zxix:.. i=1 i=1Here we tacitly assum…     

BOUNDEDNESS OF INFINITE DELAY DIFFERENCE SYSTEMS IN TERMS OF TWO MEASURES

1 PreliminariesLet R (R--), Z (Z--) denote the sets of non-negative (non-positive) realnumbers and nonnegative (nonpositive) integers, respectively, X= {of: { --r,'' 1--2, --1, 0} - Rk}, where r is a non-negative integer or r = oo. DenoteF == {h: Z X Rk - R , h(n, x) is continuous in x, and inf{h(n, x)} = 0},K = {a E C(R ,R ) t a(u) is strictly increasing in u and a(0) = 0},n LQ = {ry E C(R , R ): there are constants a, L 2 1 such that Z n(s) < a,s=n 1for all n E Z }, and in this …     

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

1. IntroductionConsider the large sparse system of linear equationsAx = b, (1.1)where, for a fixed positive integer cr, A e L(R") is a symmetric positive definite (SPD) matrir,having the bloCked formx,b E R" are the uDknwn and the known vectors, respectively, having the correspondingblocked formsni(ni S n, i = 1, 2,', a) are a given positthe integers, satisfying Z ni = n. This systemi= 1of linear equations often arises in sultable finite element discretizations of many secondorderseifad…     

带参数的一阶泛函差分方程的定号周期解

In this paper,the authors obtain the existence of one-signed periodic solutions of the first-order functional difference equation ?u(n) = a(n)u(n)-λb(n)f(u(n-τ(n))),n ∈ Z by using global bifurcation techniques,where a,b:Z → [0,∞) are T-periodic functions with ∑T n=1 a(n) 0,∑T n=1 b(n) 0;τ:Z → Z is T-periodic function,λ 0 is a parameter;f ∈ C(R,R) and there exist two constants s_2 0 s_1 such that f(s_2) = f(0) = f(s_1) = 0,f(s) 0 for s ∈(0,s_1) ∪(s_1,∞),and f(s) 0 for s ∈(-∞,s_2) ∪(s_2,0).     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

Mordell-Weil groups and Selmer groups of twin- prime elliptic curves

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.     

Universal sums of three quadratic polynomials

Let a,b,c,d,e and f be integers with a≥ c≥ e> 0,b>-a and b≡a(mod 2),d>-c and d≡c(mod 2),f>-e and f≡e(mod 2).Suppose that b≥d if a=c,and d≥f if c=e.When b(a-b),d(c-d) and f(e-f) are not all zero,we prove that if each n∈N={0,1,2,...} can be written as x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 with x,y,z∈N then the tuple(a,b,c,d,e,f) must be on our list of 473 candidates,and show that 56 of them meet our purpose.When b∈[0,a),d∈[0,c) and f∈[0,e),we investigate the universal tuples(a,b,c,d,e,f) over Z for which any n∈N can be written as x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 with x,y,z∈Z,and show that there are totally 12,082 such candidates some of which are proved to be universal tuples over Z.For example,we show that any n∈N can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z∈Z,and conjecture that each n∈N can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z∈N.     

NEW PROOFS OF THE DECAY ESTIMATE WITH SHARP RATE OF THE GLOBAL WEAK SOLUTION OF THE n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u0∈ L1(Rn) ∩ L2(Rn) and the external force f ∈ L1(Rn× R+) ∩ L1(R+, L2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t 0, where the dimension n ≥ 2, C 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.     

NEUMANN PROBLEM FOR THE LANDAU-LIFSHITZ-MAXWELL SYSTEM IN TWO DIMENSIONS

91. IntroductionIn 1935, LandauLifshitz[1] proposed the fOllowing coupled system of the nonlinear evo-lution equationZr = --a,t x (2 x (b f H)) a,E x (b f A), (1.1)- 8E7 x H = -- aE, (1.2)0t- 0H 0ZV x E = ---- -- pfZ0t p7' (1'3)v. A gv. 2 = 0, v. E = 0, (l.4)where a1, a2, a, g are constants, cr1 2 0, a 2 0, Z(x,t) = (Z1(x,t), Z2(x,t), Z3(x,t))denotes the microscopic magnetization field, H = (H1 (x, f), H2(x, t), H3(x, t)) the magneticfield, E(x, t) = (E1(x, t), E2(x, t), E3(…     

LINEAR DILATATION OF QUASIREGULAR MAPPINGS IN SPACE

1 IntroductionIn this paPert we use the fOlowing notatioll for Euclidean n-space m and r = Rn U {oo}.Unit vectors in the directions of the rectangular coordinate axes in m are denoted by e1,', en.The open ball and sphere with center x and radius r > 0 are written as B"(x, r) and S"--'(x, r),respectively. We sometimes write Bn (r) = Bn (0, r), S"--'(r) = Sn-- 1 (0, r), Bn = B"(1), Sn-- 1 =sn-- '(1).Let f be a discrete and open maPping from a domain G of R" into R". Thenis called the l…     

A note on the existence of fractional f-factors in random graphs

Let G=Gn,p be a binomial random graph with n vertices and edge probability p=p(n),and f be a nonnegative integer-valued function defined on V(G) such that 0a≤f(x)≤bnp-2np ㏒n for every x ∈V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0,1] so that for each vertex x,we have dh G(x)=f(x),where dh G(x) = x∈e h(e) is the fractional degree of x in G. Set Eh = {e:e ∈E(G) and h(e)=0}.If Gh is a spanning subgraph of G such that E(Gh)=Eh,then Gh is called an fractional f-factor of G. In this paper,we prove that for any binomial random graph Gn,p with p≥n-23,almost surely Gn,p contains an fractional f-factor.     

OSCILLATION CRITERIA FOR SECOND ORDER QUASILINEAR DIFFERENTIAL EQUATIONS

1 IntroductionThis paper is concerned with the problem of oscillation of second-orderquasilinear differential equationsFOr equation (1), the following conditions, which are referred to (FO), arealways assumed to be valid(a) a > 0 is a constant;(b) F: [to, co) x R x R x R x R - R is a continuous function;(c) sgn F(t, x) u? v3 w) = sgn x for each t 2 to and x, u? v? w E R,(d) r(t) E C([to, co); (0, co)) and jco ddt = co.' (e) T: [to, co) - R is continuous, T(t) 5 t for t 2 to and ill7T(t)…     

Almansi decomposition for Dunkl operators

Let Ωbe a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ωwhich are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x) |x|2f1(x) … |x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.