On arithmetic and asymptotic properties of up-down numbers

Let σ=(σ₁,…,σ_N), where σ_i=±1, and let C(σ) denote the number of permutations π of 1,2,…,N+1, whose up-down signature sign(π(i+1)-π(i))=σ_i, for i=1,…,N. We prove that the set of all up-down numbers C(σ) can be expressed by a single universal polynomial Φ, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that Φ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers C(σ), for fixed N. We prove a concise upper bound for C(σ), which describes the asymptotic behaviour of the up-down function C(σ) in the limit C(σ)?(N+1)!.     

Stability of Friedrichs's scheme in the maximum norm for hyperbolic systems in one space dimension

We are concerned with Friedrichs's scheme for an initial value problem u_t(t, x) = A(t, x)u_x(t, x), u(0, x) = u₀(x), where u₀(x) belongs to L_∞, not to L₂. We show that Friedrichs's scheme is stable in the maximum norm ·_L∞, provided that the system is regularly hyperbolic and that the eigenvalues d_i(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λd_i(t, x)?0 (i = 1,2,..., N), where λ is a mesh ratio.     

Limit theorems for semigroups with perturbed generators,with applications to multiscaled random evolutions

For parameters η, let {B(η)} denote infinitesimal operators of strongly continuous semigroups, with resolvents R(λ; B(η)) satisfying λR(λ; B(η)) = P(η) + λV(η) + o(λ). For parameters α, let {A(α)} denote possibly unbounded, linear operators for which {A(α) + B(η)} are infinitesimal operators of strongly continuous semigroups {U_α·η(t)}. For α, η converging simultaneously, we show strong convergence of the semigroups U_α·η(t) to a strongly continuous semigroup U(t), with limiting infinitesimal operator characterized by lim_α·η ∑_jP(η) A(α) × (V(η) A(α))ⁱf. We give applications of the abstract perturbation theorems to limit theorems of random evolutions and associated abstract Cauchy problems, in which multiscaling occurs in the convergence.     

An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U _f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U _f -algebra. For any U-algebraAsetU _A =U _i εI({i}×(A ^K _i—domf _i ^A )), where (K _i) _i εI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U _f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ?N such that there exists a bijective mapping σ:U _{F(N, U)} →N ?M satisfying σ((i, α)) ? α(K _i ) for all (i, α) ∈U _{F (N, U)}, and, for the structureg=(g _i)_iεI defined by ,g _i : =f _i ^{F(N, U)} ∪ {(α, σ((i, α))) | (i, α ∈U _{F(N, U)}} id _M induces an isomorphism betweenF(M, U _f ), and (F(N, U)g).     

Homothetic interval orders

We give a characterization of the non-empty binary relations ? on a N^*-set A such that there exist two morphisms of N^*-sets u₁,u₂:A→R₊ verifying u₁?u₂ and x?y⇔u₁(x)>u₂(y). They are called homothetic interval orders. If ? is a homothetic interval order, we also give a representation of ? in terms of one morphism of N^*-sets u:A→R₊ and a map such that x?y⇔σ(x,y)u(x)>u(y). The pairs (u₁,u₂) and (u,σ) are “uniquely” determined by ?, which allows us to recover one from each other. We prove that ? is a semiorder (resp. a weak order) if and only if σ is a constant map (resp. σ=1). If moreover A is endowed with a structure of commutative semigroup, we give a characterization of the homothetic interval orders ? represented by a pair (u,σ) so that u is a morphism of semigroups.     

Zero density of L-functions related to Maass forms

Let f(z) be a Hecke-Maass cusp form for SL ₂(?), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T) be the number of zeros ρ = β +iγ of L(s, f) with |γ| ? T, β ? σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4 ? σ ? 1 ? ?, there exists a constant C = C(?) such that N(σ,T) ? T ^2(1?σ)/σ(logT) ^C , which improves the previous results.     

On the existence of search designs with continuous factors

Consider the search linear model defined as follows. Lety(N×1) be a vector ofN observations such that (1) $$E(y) = A_1 \xi _1 + A_2 \xi _2 ,V(y) = \sigma ^2 I_N$$ whereσ ² may or may not be known,A ₁(N × υ ₁) andA ₂(N ×υ ₂) are known matrices, ξ₁(υ ₁ × 1) is unknown and ξ₂(υ ₂ × 1) is partly known in the following sense. We known that at mostk elements of ξ₂ are non zero but we do not know particularly which these nonzero elements are. The problem is to make inferences about the elements of ξ₁ and, furthermore, to search the nonzero elements of ξ₁ and make inferences about them. We wanty to be such that the above problem can be resolved with certainty whenσ ²=0; the underlying design corresponding toy is then called a search design. It has been shown in earlier work that for a search design, we must haveN≧υ ₁+2k. In this paper, we consider the special case of search linear models, when the object of the experiment is to fit an appropriate response surface. We establish a basic result, namely, that when the true response surface is representable by a polynomial, then search designs exist for whichN=υ ₁+2k, irrespective of the value ofυ ₂.     

Permutation-operator inequalities on certain symmetry classes of tensors

If A∈T(m, N), the real-valued N-linear functions on E_m, and σ∈S_N, the symmetric group on {…,N}, then we define the permutation operator P_σ: T(m, N) → T(m, N) such that P_σ(A)(x₁,x₂,…,x_N = A(x_σ(1),x_σ(2),…, x_σ(N)). Suppose Σ^q_i=1n_i = N, where the n_i are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈P_σ(A₁?A₂???A_q),A₁?A₂?? ? A_q〉 ? 0 for A_i∈S(m, n_i), where S(m, n_i) denotes the subspace of T(m, n_i) consisting of all the fully symmetric members of T(m, n_i). Also we present a broad generalization of the Neuberger identity which is sometimes useful in answering questions of the type described below. Suppose G and H are subgroups of S_N. We let T_G(m, N) denote all A∈T(m, N) such that P_σ(A) = A for all σ∈G. We define the symmetrizer $\text{S}$_G: T(m, N)→T_G(m,N) such that $\text{S}{}_{\mathrm{G}}\text{(A) = 1/|G|Σ}{}_{\mathrm{\sigma \in G}}\text{P}{}_{\mathrm{\sigma }}\text{(A)}$. Suppose H is a subgroup of G and A∈T_H(m, N). Clearly $\text{6}\text{S}{}_{\mathrm{<mtext>G</mtext>}}\text{6(A) 6? 6A6.}$ We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of T_H(m,N), then can we explicitly present a constant C>0 such that $\text{6}\text{S}{}_{\mathrm{G}}\text{(A)6?C6A6}$ for all A∈D?     

Unbiased invariant minimum norm estimation in generalized growth curve model

This paper considers the generalized growth curve model subject to R(X_m)⊆R(X_m-1)⊆?⊆R(X₁), where B_i are the matrices of unknown regression coefficients, X_i,Z_i and U are known covariate matrices, i=1,2,…,m, and E splits into a number of independently and identically distributed subvectors with mean zero and unknown covariance matrix Σ. An unbiased invariant minimum norm quadratic estimator (MINQE(U,I)) of tr(CΣ) is derived and the conditions for its optimality under the minimum variance criterion are investigated. The necessary and sufficient conditions for MINQE(U,I) of tr(CΣ) to be a uniformly minimum variance invariant quadratic unbiased estimator (UMVIQUE) are obtained. An unbiased invariant minimum norm quadratic plus linear estimator (MINQLE(U,I)) of is also given. To compare with the existing maximum likelihood estimator (MLE) of tr(CΣ), we conduct some simulation studies which show that our proposed estimator performs very well.     

Multiple solutions of coupled Sturm-Liouville systems

This paper deals with the coupled Sturm-Liouville system ? (pu′)′ + Pu + rv = λ₁u + λ₁N₁₁(u, v) + λ₂N₂₁(u, v), ? (qv′)′ + Qv + ru = λ₂v + λ₁N₁₂(u, v) + λ₂N₂₂(u, v), α₁₁u(0) + α₁₂u′(0) = 0 = α₂₁v(0) + α₂₂v′(0), β₁₁u(1) + β₁₂u′(1) = 0 = β₂₁v(1) + β₂₂v′(1). The functions p, P, q, Q, r are smooth; λ₁ and λ₂ are eigenparameters; N_ij(u, v) is analytic and of higher order. The linearized problem, all $\text{N}{}_{\mathrm{ij}}\text{\&z.tbnd; 0}$, is shown to have eigenvalues (λ₁, λ₂) which are continuously distributed along a sequence of monotonically decreasing curves in the λ₁λ₂-plane. A generalized Lyapunov-Schmidt method establishes that if (λ₁, λ₂) is near a simple eigenvalue of the linearized problem, then the number of small solutions of the nonlinear problem corresponds to the number of real roots of a certain polynomial.     

Stability and bifurcation of equilibrium solutions of reaction-diffusion equations on an infinite space interval

Under the condition that f(x, y, z, α) and its partial derivatives decay sufficiently fast as $\text{¦x¦ → ∞}$ we will study the (linear) stability and bifurcation of equilibrium solutions of the scalar problem u_t = u_xx + f(x, u, u_x, α), u_x(?∞, t) = u_x(∞, t) = 0 (1) where α is a real bifurcation parameter. After introducing appropriate function spaces X and Y the problem (1) can be rewritten $\text{d}\text{dt}\text{u = G(u, α)}$, (7) where G:X×R → Y is given by G(u, α)(x) = u″(x) + f(x, u(x), u′(x), α). It will be shown, for each (u, α)?X × R, that the Fréchet derivative G_u(u,a): X → Y is not a Fredholm operator. This difficulty is due to the fact that the domain of the space variable x, is infinite and cannot be eliminated by making another choice of X and Y. Since G_u(u, α) is not Fredholm, the hypotheses of most of the general stability and bifurcation results are not satisfied. If (u₀, α₀?S = {(u, α): G(u, α) = 0}, (i.e., (u₀,α₀) is an equilibrium solution of (7)), a necessary condition on the spectrum of G_u(u₀, α₀) for a change in the stability of points in S to occur at G_u(u₀, α₀) will be given. When this condition is met, the principle of exchange of stability which means, in a neighborhood of (u₀, α₀), that adjacent equilibrium solutions for the same α have opposite stability properties in a weakened sense will be established. Also, when G_u or its first order partial derivatives, evaluated at (u₀, α₀), are not too degenerate, the shape of S in a neighborhood of (u₀, α₀) will be described and a strenghtened form of the principle of exchange of stability will be obtained.     

An identity with generalized derivations on lie ideals, right ideals and Banach algebras

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u ε L, then one of the following holds:

1. there exists α ε C such that F(x) = α x for all x ε R
2. R satisfies the standard identity s ₄ and there exist a ε U and α ε C such that F(x) = ax + xa + αx for all x ε R.

We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.     

Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u_t - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q_T =]0,T[×Ω, Ω ⊂ R^N, with an initial condition u(0) = u₀, where u₀ is not bounded, |H(t,x, u, ξ)⩽ β|ξ|^p + f(t,x) + βe^λ₁^|u|f, |g|^p/(p-1) ∈ L^r(Q_T) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator.     

Stability of solutions bifurcating from multiple eigenvalues

Let X and Y be real Banach spaces and G:X × $\text{R}$ be a twice continuously differentiate function which is not necessarily linear. Suppose G(u₀, α₀) = 0 and the dimension of the null space of G_u(u₀, α₀) is m, where 1 ? m < ∞. Usually, S = {(u, α):G(u, α) = 0}, in a neighborhood of (u₀, α₀), consists of a finite number of curves emanating from (u₀, α₀). We will determine the stability of points, (u, α), in S (i.e., the maximum of the real parts of the spectrum of G_u(u, α) for each (u, α) ∈ S) using a general perturbation theorem of Kato. Our results contain as a special case the stability theorems of Crandall and Rabinowitz for the case m = 1. We will also tie our stability theorems together with some bifurcation results of Decker and Keller. Finally we apply our results to systems of reaction diffusion equations.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Economic planning based on social preference functions

We argue that the most desirable social welfare functions for practical use (here sometimes called social preference functions) are those determined by Σ_ilog(u_i(x)?α) where u_i(x) is the utility of alternative x to individual i and α is the minimum standard of living decided upon by the society.     

L∞ bounds for solutions of parametrized elliptic equations

This generalizes earlier results (T. I. Seidman, Indiana Univ. Math. J.30 (1981), 305–311) for ?Δu = λf(u). For the family of equations (su1) Au = g(u, λ) with appropriate boundary conditions the object is to construct from g and the boundary conditions a function η(λ, r) such that a bound y(λ) on ∥u∥_∞ can be obtained by solving the ODE: y′(λ) = η(λ, y) with y(λ₀) = B(λ₀) = bound at λ = λ₀.     

A convergent two-time method for periodic differential equations

Two timing, an ad hoc method for studying periodic evolution equations, can be given a rigorous justification when the problem is in standard form, u = ?f(t, u). First solve $\text{dw}\text{dσ}\text{= ?(I ? M) f(σ, w)}\text{for}\text{w(σ, v)}$, where M is the mean value operator and v is any initial value. Then w(σ, v) is periodic in σ but does not satisfy the original equation. Now, force a solution u(t), using nonlinear variation of constants, in the form w(σ, v(τ)), where σ = t is the fast time and τ = ?t is the slow time. With the resulting differential equation for v, one reads off from its nonconstant solutions thè approximate transient behavior of u(t) for times of order ?^?1. On the other hand, the equilibrium points (constant solutions) v₀ correspond to steady state (periodic solutions) of the original system. Interesting applications, such as to one-dimensional wave equations with cubic damping, can be given.     

Some results on V-ary asymmetric tries

Tries (radix search tries) find many applications in computer science and telecommunications. It is assumed that a trie is built over an alphabet U = {σ₁,…,σ_v} (V-ary trie), and n (possible infinite) strings of elements from U (i.e., keys) are stored in external nodes of the trie. The occurrence of an element σ_i in a key is represented by a probability p_i (asymmetric trie). Our main interest is to compute all moments of the depth of a leaf (external node) in a random family of tries. By solving a system of recurrences we find an exact formula for all factorial moments of the depth, and—using the Mellin transform technique—we derive asymptotic approximations for them. We prove that the m th factorial moment of the depth of a leaf in a trie with n keys is equal to αln^mn + βln^m−1n + O(ln^m−2n), where the constants α and β are functions of the probabilities, p_i, i = 1,…,V. In particular, we show that for symmetric tries the variance of the depth is O(1), while for asymmetric tries it is αlnn + O(1), and we determine explicitly the constant O(1). These results extend previous analyses by Knuth [12], Flajolet and Sedgewick [6], Jacquet and Regnier [10], and Kirschenhofer and Prodinger [11].     

Triple Solutions for Second-Order Three-Point Boundary Value Problems

We establish the existence of at least three positive solutions to the second-order three-point boundary value problem, u″ + f(t, u) = 0, u(0) = 0, αu(η) = u(1), where η: 0 lt; η < 1, 0 < α < 1/η, and f: [0, 1] × [0, ∞) → [0, ∞) is continuous. We accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works.