Bifurcation method for solving multiple positive solutions to Henon equation

Three algorithms based on the bifurcation method are applied to solving the D4 symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bi- furcation parameter, the D4-Σd(D4-Σ1, D4-Σ2) symmetry-breaking bifurcation points on the branch of the D4 symmetric positive solutions are found via the extended systems. Finally, Σd(Σ1, Σ2) sym- metric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.     

More Epsilonized Bounds of the Boas-Kac-Lukosz Type

In this paper we give a further investigation of the method introduced by the author in [1, Frequency-domain bounds for nonnegative unsharply band-limited functions] for proving bounds for functions with nonnegative Fourier transforms. We also dealt with the question of how large the supremum K^S of all numbers |f(u)| is with f the Fourier transform of a nonnegative integrable function F and f(0) = 1, |f(ku)| ≤ ε for k ∈ S. Here u > 0 and S ⊂ {2, 3, . . .}. This problem was related in [1] to finding the infimum M^S of all numbers M_h = max_ϑ [(1−h(ϑ))/(1− cos ϑ)] over all 2π-periodic even, smooth functions h whose Fourier cosine coefficients a_k vanish for k ∉ S, and it was proved and announced for several cases that M^S (1−K^S ) = 1. In this paper we prove the results announced in [1]. To that end we generalize the method given in [1] to include Fourier transforms f of probability measures on R and a certain generalized function h, and we show that the numbers K^S, M^S are assumed as |f(u)|, M_h for certain allowed f,h. Moreover, we establish a fundamental relation between finding the numbers K^S, M^S and the numbers K^T, M^T where T = {2, 3, . . .}\S. In particular, we show that M^T = 2K^S (2K^S − 1)−1,K^T = 1/2 M^S(M^S − 1)−1 and that M^T (1 − K^T) = 1,K^SK^T = 1/2 , whenever M^S (1 − K^S) = 1.     

On the exact non-null distribution of Wilks'L VC criterion and power studies

Letx ₁, x₂, ..., x_Nbep×1 random vectors distributed independently asN(u, Σ), Σ>0;u and Σ are unknown. In this paper, we derive the exact non-null distribution of Wilks' likelihood ratio criterion,L _VC, for testingH:∑=σ ²[(1−ρ)I+ρee′], σ>0 and ρ are unknown against the alternativeA≠H,e′=(1, 1, …, 1): 1×p. The distribution has been derived in three series forms: (1) a series of Meijer'sG-functions through Mellin transform, (2) an, alternate series using contour, intergration and (3) a series of chi square distributions. Powers have been computed based on these forms of the distribution forp=2 and 3.     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

On meromorphic univalent functions omitting a disc

The class Σ_b is defined to consist of meromorphic univalent functionsH omitting a disc with the radiusb:H(z)=z+ Σ ₀ ^∞ A _n z ⁻ⁿ ,z>1,H(b)>b ∈ (0, 1). By aid of FitzGerald inequalities the inverse coefficients of odd Σ_b-functions are maximized. The result extends the corresponding estimation, due to Netanyahu and Schober, fromb=0 to the whole interval (0, 1). The author wishes to express her gratitude to Professor O. Tammi for valuable discussions connected with the problem. This work was supported by a grant from the Finnish Ministry of Education.     

A C 0 Estimate for Nearly Umbilical Surfaces

Let Σ ⊂ R ³ be a smooth compact connected surface without boundary. Denote by A its second fundamental form and by Å the tensor A−(tr A/2)Id. In [4] we proved that, if ‖Å‖ _L ² (Σ) is small, then Σ is W ^2,2-close to a round sphere. In this note we show that, in addition, the metric of Σ is C ⁰–close to the standard metric of S ².The first author was supported by a grant of the Swiss National Science Foundation.     

A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

Moment convergence rates in the law of the logarithm for dependent sequences

Let {X _n ; n ≥ 1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set S _n = Σ _k=1 ⁿ X _k , M _n = max _k≤n |S _k |, n ≥ 1. Suppose σ ² = EX ₁² + 2Σ _k=2^∞ EX ₁ X _k (0 < σ < ∞). In this paper, the exact convergence rates of a kind of weighted infinite series of E{M _n −σɛ√n log n}₊ and E{|S _n | − σɛ√n log n}₊ as ɛ ↘ 0 and E{σɛ√π ² π/8logn − M _n }₊ as ɛ ↗ ∞ are obtained.     

Asymptotic expansions for the distributions of latent roots ofS h S e −1 and of certain test statistics in MANOVA

S _e andS _n are independent central and noncentral Wishart matrices having Wishart distributionsW _p (n _e , Σ) andW _p (n _h , Σ; Ω) respectively. Asymptotic expansions are given for the distributions of latent roots ofS _h S _e ⁻¹ and of certain test statistics in MANOVA under the assumption thatn=n _e +n _h becomes large with a fixed ration _e ∶n _h =e∶h(e+h=1,e>0,h>0) andΩ=O(n).     

Decomposition of ranges of vector measures

The following conditions on a zonoidZ, i.e., a range of a non-atomic vector measure, are equivalent: (i) the extreme set containing 0 in its relative interior is a parallelepiped; (ii) the zonoidZ determines them-range of any non-atomic vector measure with rangeZ, where them-range of a vector measure μ is the set ofm-tuples (μ(S ₁), …, μ(S _m), whereS ₁, …S _m are disjoint measurable sets and (iii) there is avector measure space (X, Σ, μ) such that any finite factorization ofZ, Z =ΣZ _i , in the class of zonoids could be achieved by decomposing (X, Σ). In the case of ranges of non-atomic probability measures (i) is automatically satisfied, so (ii) and (iii) hold. Partially supported by NSF grant MCS-79-06634     

Evolution equations with a free boundary condition

In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂₁,u −Δu = Γ(u)(∇u, ∇u) [tT T_{u u}N, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥_u T_u(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = u_o(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝⁿ, in which the nonlinear equation reduces to ∂_tu-Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than K_N <-0 and Σ is totally geodesic in N.     

On calculation of moments of ladder heights

Let X,X ₁,X ₂, … be independent identically distributed random variables, F(x) = P{X < x}, S ₀ = 0, and S _n =Σ _i=1 ⁿ X _i . We consider the random variables, ladder heights Z ₊ and Z ₋ that are respectively the first positive sum and the first negative sum in the random walk {S _n }, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z ₊ and Z ₋ in the qualitatively different cases EX > 0, EX < 0, and EX = 0. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006.     

Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ ¹ or the circle S ¹, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h ⁻¹ for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ _f the set of critical points of f, D(M,Σ _f ) the subgroup of D(M) preserving Σ _f , and S(f), S (f,Σ _f ), O(f), and O(f,Σ _f ) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ _f ). In fact S(f) = S(f,Σ _f ).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ _f ). It is proved that except for few cases the connected components of S(f) and O(f,Σ _f ) are contractible, π _k O(f) = π _k M for k ≥ 3, π₂ O(f) = 0, and π₁ O(f) is an extension of π₁ D(M) ⊕ Z ^k (for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ _f ) are tame Fréchet manifolds. Communicated by Peter Michor Vienna Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45.     

UNIFORM CONVERGENCE OF CESARO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that iff ∈ C([-π,π]2) and the function f is bounded partial p-variation for some p ∈ [1, ∞), then the double trigonometric Fourier series of a function f is uniformly (C;-α,-β) summable (α β＜ 1/p,α,β＞ 0) in the sense of Pringsheim. If α β≥ 1/p, then there exists a continuous function f0 of bounded partial double trigonometric Fourier series of fo diverge over cubes.     

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S _j for j=1,…,K at unknown spot locations x _j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S _j and x _j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S _j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.      

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

The <Emphasis Type="Italic">f</Emphasis>-Vector of the Descent Polytope

For a positive integer n and a subset S⊆[n−1], the descent polytope DP  _S is the set of points (x ₁,…,x _n ) in the n-dimensional unit cube [0,1] ⁿ such that x _i ≥x _i+1 if i∈S and x _i ≤x _i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F _S (t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.     

On the range of random walk

Let {S _n , n=0, 1, 2, …} be a random walk (S _n being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE _d , thed-dimensional integer lattice. Letf _n =Prob {S ₁ ≠ 0, …,S _n −1 ≠ 0,S _n =0 |S ₀=0}. The random walk is said to be transient if and strongly transient if . LetR _n =cardinality of the set {S ₀,S ₁, …,S _n }. It is shown that for a strongly transient random walk with p<1, the distribution of [R _n −np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S ₀, …,S _n }. For a finite setA inE _d , let C(A=Σ _x∈A ) Prob {S _n ∉A, n≧1 |S ₀=x} be the capacity ofA. A strong law forC{S ₀, …,S _n } is proved for a transient random walk, and some related questions are also considered. This research was partially supported by the National Science Foundation.