A generalized moment problem

Let {λ _n} (n≧0) satisfy (1.1) we are considering the following problems: What are the necessary and sufficient conditions on a sequence {μ_n} (n≧0) in order that it should possess the representation (1.2) wherea(t) is of bounded variation or the representation (1.3) wheref(t) ∈L _M[0, 1] orf(t) is essentially bounded.     

Generalized partial dynamical symmetry in nuclei

We introduce the notion of a generalized partial dynamical-symmetry for which part of the eigenstates have part of the dynamical symmetry. This general concept is illustrated with the example of Hamiltonians with a partial dynamical O(6) symmetry in the framework of the interacting boson model. The resulting spectrum and electromagnetic transitions are compared with empirical data in 162Dy.     

Shape preserving widths of Sobolev-type classes ofs-monotone functions on a finite interval

LetI be a finite interval andr ∈ ℕ. Denote by △ ₊ ^s L _q the subset of all functionsy ∈L _q such that thes-difference △ _T ^s y(·) is nonnegative onI, ∀τ>0. Further, denote by △ ₊ ^s W _p ^r the class of functionsx onI with the seminorm ‖x ^(r) ‖L _p ≤1, such that △ _T ^s x≥0, τ > 0, τ>0. Fors=3,…,r+1, we obtain two-sided estimates of the shape preserving widths , whereM ⁿ is the set of all linear manifoldsM ⁿ inL _q , dimM ⁿ ≤n, such thatM ⁿ ⋂△ ₊ ^s L _q ≠ 0. Part of this work was done while the first author visited Tel Aviv University in 2001 and part of it while the second author was a member of the Industrial Mathematics Institute (IMI), University of South Carolina.     

Supersymmetric patterns in the pseudospin, spin, and coulomb limits of the dirac equation with scalar and vector potentials

We show that the Dirac equation in (3+1) dimensions gives rise to supersymmetric patterns when the scalar and vector potentials are (i). Coulombic with arbitrary strengths or (ii). when their sum or difference is a constant, leading to relativistic pseudospin and spin symmetries. The conserved quantities and the common intertwining relation responsible for such patterns are discussed.     

Stability and optimization of dispersion-managed soliton control

Dispersion-managed solitons with in-line all-optical regeneration are shown to be subject to amplitude and timing-jitter instabilities. We identify and discuss the different natures of these instabilities by means of a linear stability analysis, which is in good agreement with the full numerical solutions of the governing equations. Stable pulse propagation can be achieved through appropriate choices of dispersion map and pulse energy.     

Interpolatory Pointwise Estimates for Polynomial Approximation

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f∈ C[0,1] , by polynomials. For the sake of completeness, as well as in order to strengthen some existing results, we discuss briefly the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving affirmative results and by providing the necessary counterexamples in all other cases. November 16, 1998. Date revised: July 12, 1999. Date accepted: September 13, 1999.     

A chemical genetics approach reveals H,K-ATPase-mediated membrane voltage is required for planarian head regeneration

Biophysical signaling is required for both embryonic polarity and regenerative outgrowth. Exploiting endogenous ion transport for regenerative therapies will require direct regulation of membrane voltage. Here, we develop a pharmacological method to target ion transporters, uncovering a role for membrane voltage as a key regulator of anterior polarity in regenerating planaria. Utilizing the highly specific inhibitor, SCH-28080, our data reveal that H(+),K(+)-ATPase-mediated membrane depolarization is essential for anterior gene expression and brain induction. H(+),K(+)-ATPase-independent manipulation of membrane potential with ivermectin confirms that depolarization drives head formation, even at?posterior-facing wounds. Using this chemical genetics approach, we demonstrate that membrane voltage controls head-versus-tail identity during planarian regeneration. Our data suggest well-characterized drugs (already approved for human use) might be exploited to control adult stem cell-driven pattern formation during the regeneration of complex structures.     

Free-knot Splines Approximation of s-monotone Functions

Let I be a finite interval and r,sN. Given a set M, of functions defined on I, denote by ₊ ^s M the subset of all functions yM such that the s-difference ^s y() is nonnegative on I, >0. Further, denote by ₊ ^s W _p ^r , the class of functions x on I with the seminorm x ^(r)L _p 1, such that ^s x0, >0. Let M _n (h _k ):={ _i=1 ⁿ c _i h _k (w _i t– _i )c _i ,w _i , _i R, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions h _k (t)=t ₊ ^k , tR, kN ₀. We give two-sided estimates both of the best unconstrained approximation E( ₊ ^s W _p ^r ,M _n (h _k ))L _q , k=r–1,r, s=0,1,...,r+1, and of the best s-monotonicity preserving approximation E( ₊ ^s W _p ^r , ₊ ^s M _n (h _k ))L _q , k=r–1,r, s=0,1,...,r+1. The most significant results are contained in theorem 2.2.