Some stochastic processes related to random density functions

If { _n } is an orthonormal system and {a _n} is a sequence of random variables such that _n (a _n )²=1 a.s. thenf(t)=| _n a _{n n} (t)|² produces a randomly selcted density function. We study the properties off under the assumptions that |a _n| is decreasing to zero at a geometric rate and { _n } is one of the following four function systems: trigonometric Jacobi, Hermite, or Laguerre. It is shown that, with probability one,f is an analytic function,f has at most a finite number of zeros in any finite interval, and the tail off goes to zero rapidly.     

Mappings that preserve cones in Lobachevskii space

Let ⁿ be n-dimensional Lobachevskii space, and {l_x:x ⁿ} be a family of lines, parallel to a linel ₀, 0ⁿ (in a given direction). Let {c_x:Xⁿ} be a family of circular cones in ⁿ of opening with axes l_X and vertex X. Then, iff:ⁿⁿ(n>2) is a bijective mapping andf(C_x)=C _f(x), it follows thatf is a motion in the space ⁿ.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 687–694, May, 1973.     

Line search by curve fitting in minimization algorithms

Summary Consider an unconstrained minimization of an uniformly convex functionf(z). Let be an algorithm that, for solving it constructs a sequence {z _i} withz _i+1 =z _i + ⁽ⁱ⁾ h _i ,h _i R ⁿ , ⁽ⁱ⁾ R ⁼ and –h _i ^T f(z _i ) > 0. For any algorithm that converges linearly and that uses parabolic or cubic interpolations for the line search, upper bounds on the number of function evaluations needed to approximate the minimum off(z) within a given accuracy, are calculated. The obtained results allow to compare the line search procedure under investigation.     

On convergence subsystems of orthonormal systems

It is proved that for any sequence {R _k} _k=1 of real numbers satisfyingR _kk (k1) andR _k=o(k log₂ k),k, there exists an orthonormal system {n _k(x)} _n=1 ,x (0;1), such that none of its subsystems {n _k(x)} _k=1 withn _kR_k (k1) is a convergence subsystem.     

Interpolating them-th power ofx at the zeros of then-th Chebyshev-polynomial yields an almost best Chebyshev-approximation

LetX={x ₁,x ₂,..., _n }I=[–1, 1] and . ForfC ₁(I) definef* byf–p _f =f*, wherep _f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT _n (x) and obtain x ^m –p _x ^m8eE _n–1(x ^m ), whereE _n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE _n (f) that omits theassumptionf ⁽ⁿ⁺¹⁾>0 in a similar estimate of Meinardus.     

Conjugation for polynomial mappings

We consider Keller's functions, namely polynomial functionsf:C ⁿ C ⁿ with detf(x)=1 at allx C ⁿ. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh , > 1, defined by power series in a ball with center at the origin, such thath(0)=I andh (f(x))=h (x). So eachh conjugates f to its linear part I in a ball where it is injective.We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s ^dg(x),g(x)ⁿ=0,xC ⁿ,sC the conjugationh for f is anentire function.     

Triple Positive Solutions for Multipoint Conjugate Boundary Value Problems

For the nth order nonlinear differential equation y ⁽ⁿ⁾(t)=f(y(t)), t [0,1], satisfying the multipoint conjugate boundary conditions, y ^(j)(a_i) = 0,1 i k, 0 j n _i - 1, 0 =a ₁ < a ₂ < < a _k = 1, and _i=1 ^k n _i =n, where f: [0, ) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone.     

Spectral properties of cosine operator functions

LetA be the generator of a cosine functionC _t ,t R in a Banach spaceX; we shall connect the existence and uniqueness of aT-periodic mild solution of the equationu = Au + f with the spectral property 1 (C _T ) and, in caseX is a Hilbert space, also with spectral properties ofA. This research was supported in part by DAAD, West Germany.     

A remark on generalized iterates

Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx Z, letf ^|n| denote then'th iterate off. Clearlyf ^|m|(f ^|n|(x))=f ^|m+n|(x) for allm,nZ and allxR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F ^||:G} on the real line with the properties (i)F ^||(F ^||(x))=F ^|+| (x) for allxR and (ii)F ^|n|(x)=f ^|n|(x) for allnZ andxR. That is,f can be embedded in a set ofG-generalized iterates. In particular,f can be embedded in a set of complex generalized iterates.Dedicated to Professor Janos Aczél on his 60th birthday     

Algebraic polynomial bases of space LP

Let (_n) be a system, close to the orthonormal complete system (x _n). An estimate is obtained for the deviation of the system {f_n}, obtained from {_n} by Schmidt's method, from the system {x_n}. This estimate is used to show that, in any L_P(–1,1), withp (1,4/3] [4,), and for any >e¦4 = i,13..., there exists an orthogonal algebraic system (P _n (x)) _n=0 , forming a basis in L_P and such that _n = degP _n (x) n for n>n_o(p,).Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 223–230, February, 1978.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .     

On rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces

Let R be a subring of the rationals with 1/2, 1/3R; let S _R ⁿ denote the R-local n-sphere and define _R ⁿ :=S _R ⁿ for n odd, _R ⁿ :=S _R ⁿ for n>0 even. An H-space (resp. a 1-conn. co-H-space) is decomposable over R, if it is homotopy equivalent to a weak product of spaces _R ⁿ (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M^*(E;R); here M^* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by _*(C) as a cogroup in the category of M-Lie algebras. For R = the functor [-,E] is also determined by the Lie algebra _*(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.     

Optimal Constant in Approximation by Bernstein Operators

We obtained that for any n N, C = 1 is the smallest constant for which the inequality ||B _n (f) - f|| C ₂(f, 1/n) holds on the class of continuous functions f, as well as on the class of bounded functions f, where B _n is the Bernstein operators of degree n, ₂ is the second order modulus and || || is the sup-norm.     

Training Multilayer Perceptrons Via Minimization of Sum of Ridge Functions

Motivated by the problem of training multilayer perceptrons in neural networks, we consider the problem of minimizing E(x)= _i=1 ⁿ f _i ( _i x), where _i R ^s , 1in, and each f _i ( _i x) is a ridge function. We show that when n is small the problem of minimizing E can be treated as one of minimizing univariate functions, and we use the gradient algorithms for minimizing E when n is moderately large. For large n, we present the online gradient algorithms and especially show the monotonicity and weak convergence of the algorithms.     

Two variable <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis> functions attached to eigenfamilies of positive slope

Consider a classical cusp eigenform f= _n=1 a _n (f)q ⁿ of weight k2 for ₀(N) with a Dirichlet character mod N, and let L _f (s,)= _n=1 (n)a _n (f)n ^-s denote the L-function of f twisted with an arbitrary Dirichlet character . For a prime number p5, consider a family of cusp eigenforms f _(k) of weight k , k {f _(k)= _n=1 a _n (f _(k))q ⁿ } containing f=f _(k), such that the Fourier coefficients a _n (f _(k)) are given by certain p-adic analytic functions k a _n (f _(k)). The purpose of this paper is to construct a two variable p-adic L function attached to Colemans family {f _(k)} of cusp eigenforms of a fixed positive slope =v _p ( _p )>0 where _p = _p (k ) is an eigenvalue (which depends on k ) of the Atkin operator U=U _p . Our p-adic L-function interpolates the special values L _f(k)(s,) at points (s,k ) with s=1,2,...,k -1. We give a construction using the Rankin-Selberg method and the theory of p-adic integration on a profinite group Y with values in an affinoid K-algebra A, where K is a fixed finite extension of Q _p . Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. In their turn, such measures come from Eisenstein distributions with values in certain Banach A-modules M =M (N;A) of families of overconvergent forms over A. To Robert Alexander Rankin in memoriam     

Homotopies for computation of fixed points on unbounded regions

Givenf: R ⁿ R ^n* with some conditions, our aim is to compute a fixed pointx f(x) off; hereR ⁿ isn-dimensional Euclidean space andR ^n* is the collection of nonempty subsets ofR ⁿ . A typical application of the algorithm can be motivated as follows: Beginning with the constant mapf ₀:R ⁿ {0} R ⁿ and its fixed pointx ₀ = 0, we deformf _t ast tof f and follow the pathx _t of fixed points off _t . Cluster points of thex _t 's ast are fixed points off. This research was supported in part by Army Research Office-Durham Contract DAHC-04-71-C-0041 and by National Science Foundation Grant GK-5695.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

Stationary self-similar extremal processes

Summary Let ( _k ) _k =– be a stationary sequence of random variables, and, forA, let where _n is an affine transformation of (has the forma _n·+b _n,a _n>0,b _n). ThenM _n is a random sup measure, that is, for arbitrary collections of open setsG . We show that the possible limiting random sup measures for such sequences (M _n) are those which are stationary (M(·+b)= _d M forb) and self-similar (M(a·)= _d ^loga (M) fora>0, where is an affine transformation of ). By applying simple transformations, we need only study stationaryM such thatM(a·)= _d aM fora>0. We show that these processes retain some but not all of the properties of the classical case. In particular, we display a nontrivial example such thattM (0,t] is continuous wp1. The classical planar point process representation of extremal processes is a special case of the present approach, but is not adequate for describing all possible limits.     

Future independent times and Markov chains

Summary A random timeT is a future independent time for a Markov chain (X _n ) ₀ ifT is independent of (X _T+n ) _n ^/ =0 and if (X _T+n ) _n ^/ =0 is a Markov chain with initial distribution and the same transition probabilities as (X _n ) ₀ . This concept is used (with the conditional stationary measure) to give a new and short proof of the basic limit theorem of Markov chains, improving somewhat the result in the null-recurrent case.This work was supported by the Swedish Natural Science Research Council and done while the author was visiting the Department of Statistics, Stanford University     

A Characterization of Minimal Legendrian Submanifolds in S2n+1

Let x: L ⁿ S²ⁿ⁺¹ R²ⁿ⁺² be a minimal submanifold in S²ⁿ⁺¹. In this note, we show that L is Legendrian if and only if for any A su(n + 1) the restriction to L of Ax, (–1)x satisfies f = 2(n + 1)f. In this case, 2(n + 1) is an eigenvalue of the Laplacian with multiplicity at least (n(n + 3)). Moreover if the multiplicity equals to ;(n(n + 3)), then L ⁿ is totally geodesic.