M-estimators of structural parameters in pseudolinear models

Real valued M-estimators in a statistical model 1 with observations are replaced by -valued M-estimators in a new model with observations where are regressors, is a structural parameter and a structural function of the new model. Sufficient conditions for the consistency of are derived, motivated by the sufficiency conditions for the simpler parent estimator The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If F has a natural exponential density e^{x–b( x )} then our pseudolinear model with u = (g o )^–1 reduces to the well known generalized linear model, provided () = db()/d and g is the so-called link function of the generalized linear model. General results are illustrated for special pairs and leading to some classical M-estimators of mathematical statistics, as well as to a new class of generalized -quantile estimators.     

Anticipating Integral Equations

In this paper we deduce some estimates of the L ^p()-norm of the Skorohod and the forward integrals. These estimates allow us to study the existence of a unique solution to anticipating Volterra equations of the Skorohod and forward type. The coefficients F _i(t,s,x),ts, are -measurable and satisfy some differentiability conditions (in the sense of the stochastic calculus of variations).     

Cyclic Designs with Block Size 4 and Related Optimal Optical Orthogonal Codes

We prove the existence of a cyclic (4p, 4, 1)-BIBD—and hence, equivalently, that of a cyclic (4, 1)-GDD of type 4 ^p —for any prime such that (p–1)/6 has a prime factor q not greater than 19. This was known only for q=2, i.e., for . In this case an explicit construction was given for . Here, such an explicit construction is also realized for .We also give a strong indication about the existence of a cyclic (4p 4, 1)-BIBD for any prime , p>7. The existence is guaranteed for p>(2q ³–3q ²+1)²+3q ² where q is the least prime factor of (p–1)/6.Finally, we prove, giving explicit constructions, the existence of a cyclic (4, 1)-GDD of type 6 ^p for any prime p>5 and the existence of a cyclic (4, 1)-GDD of type 8 ^p for any prime . The result on GDD's with group size 6 was already known but our proof is new and very easy.All the above results may be translated in terms of optimal optical orthogonal codes of weight four with =1.     

A characterization of the generalized twisted field planes of characteristic ≥5

Let be a non-Desarguesian semifield plane of orderp ⁿ, p a prime number 5 andn3, and let denote the group induced by the autotopism groupG of on the line at infinity. We prove that is a generalized twisted field plane if, and only if, has an element of order (p ^k–1)((p ⁿ–1)/(p ^m–1)), for some integersk andm, wherek | m, m | n, andm.This work was supported in part by NSF grants RII-9014056, component IV of the EPSCoR of Puerto Rico grant and ARO grant for Cornell MSI     

Some relations among group-theoretic invariants of finite p-groups

In this paper there is given a formula for the number of solutions of the equation in an arbitrary finite p-group G (of exponent p ^l , 1 n 1) and a formula for the number of cyclic subgroups of G of any order. A connection is established among ¦G¦, p ^l , and the ranks of those subgroups of G of order greater than p^l; if G is regular, there are analogous relations among the orders of the characteristic subgroups , and the ranks of the subgroups of G of order greater than pⁿ. These results are precise; some of them strengthen the well-known classical theorems of Frobenius and Miller for p-groups.Translated from Matematicheskie Zametki, Vol. 17, No. 4, pp. 571–578, April, 1975.     

Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

Asymptotic bias and variance for a general class of varying bandwidth density estimators

Summary We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator , it is possible to iterate and compute a sequence of estimates , using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of is of orderh ^2km ,k=1, ...,t. Hereh ^m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<, it is possible to make asymptotically equivalent to the ideal estimator.     

Asymptotically Efficient Nonparametric Estimation of Nonlinear Spectral Functionals

The paper considers a problem of construction of asymptotically efficient estimators for functionals defined on a class of spectral densities. We define the concepts of H ₀- and IK-efficiency of estimators, based on the variants of Hájek–Ibragimov–Khas'minskii convolution theorem and Hájek–Le Cam local asymptotic minimax theorem, respectively. We prove that is a suitable sequence of T ^1/2-consistent estimators of unknown spectral density (), is H ₀- and IK-asymptotically efficient estimator for a nonlinear smooth functional ().     

A note on the diophantine equationx 2+ 4D=y p

LetD be a positive square free integer, and leth(–D) denote the class number of . Furthermore letp be an odd prime with . In this note we prove that ifp {5, 7} orp>3·10⁶, then the equation , has no positive integer solution (x, y).     

Some generalized theorems onp-hyponormal operators

A bounded linear operatorT is calledp-Hyponormal if (T ^*T)^p(TT ^*)^p, 0<p1. In Aluthge [1], we studied the properties of p-hyponormal operators using the operator . In this work we consider a more general operator , and generalize some properties of p-hyponormal operators obtained in [1].     

Boundedness in the mean of orthonormalized polynomials

For the polynomials {p_n(t)} ₀ , orthonormalized on [–1, 1] with weightp(t) = (1–t) (1+t) _v=1 ^m , we obtain necessary and sufficient conditions for boundedness of the sequences of norms: 1) 2) and 3) with the conditions that on [–1, 1] and (H,)^–1 L²(0, 2), where(H,) is the modulus of continuity in C(–1, 1) of function H.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 759–770, May, 1973.     

Short Polynomial Representations for Square Roots Modulo p

Let p be an odd prime number and a a square modulo p. It is well known that the simple formula a mod p gives a square root of a when p 3 mod 4. Let us write p – 1 = 2 ⁿ s with s odd. A fast algorithm due to Shanks, with n steps, allows us to compute a square root of a modulo p. It will be shown that there exists a polynomial of at most 2 ^n–1 terms giving a square root of a. Moreover, if there exists a polynomial in a representing a square root of a modulo p, it will be proved that this polynomial would have at least 2 ^n–1 terms, except for a finite set _n of primes p depending on n.     

Weakly Cyclic Vectors with a Given Modulus

Let H^p be the Hardy space in the polydisk. Denote by the set of all holomorphic polynomials. A vector f ∈ H^p is called weakly cyclic if the product f is weakly dense in H^p, 0 < p < 1. We construct weakly cyclic vectors with a prescribed lower semi-continuous modulus of the boundary values. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 111–118.     

Primes dividing both 2 n –3 and 3 n –2 are rare

Let S(x) denote the number of primes p < x which divide both 2ⁿ–3 and 3ⁿ–2 for some We prove that

Looking for Difference Sets in Groups with Dihedral Images

We prove four theorems about groups with a dihedral (or cyclic) image containing a difference set. For the first two, suppose G, a group of order 2p with p an odd prime, contains a nontrivial (v, k, ) difference set D with order n = k – prime to p and self-conjugate modulo p. If G has an image of order p, then 0 2a + 2 for a unique choice of = ±1, and for a = (k – )/2p. If G has an image of order 2p, then and ( – 1)/( – 1). There are further constraints on n, a and . We give examples in which these theorems imply no difference set can exist in a group of a specified order, including filling in some entries in Smith's extension to nonabelian groups of Lander's tables. A similar theorem covers the case when p|n. Finally, we show that if G contains a nontrivial (v, k, ) difference set D and has a dihedral image D _2m with either (n, m) = 1 or m = p ^t for p an odd prime dividing n, then one of the C ₂ intersection numbers of D is divisible by m. Again, this gives some non-existence results.     

Inequalities of Putnam and Berger-Shaw forp-quasihyponormal operators

For an operatorT satisfying thatT ^*(T ^* T–TT ^*)T0, we shall show that and, moreover, tr itT isn-multicyclic.For an operatorT satisfying thatT ^* {(T ^* T) ^p –(TT ^*) ^p }T0 for somep (0, 1], we shall show that and, moreover, ifT isn-multicyclic.     

On the Approximability by Finite p-Groups of Free Products of Groups with Normal Amalgamation

A sufficient condition for the residual p-finiteness (approximability by the class of finite p-groups) of a free product G = (A * B; H) of groups A and B with a normal amalgamated subgroup H is obtained. This condition is used to prove that if A and B are extensions of residually -groups by -groups, where stands for the class of finitely generated torsion-free nilpotent groups, and if H is a normal p′-isolated polycyclic subgroup, then the group G is residually p-finite (i.e., residually -group), provided the quotient group G/H ^p H′ is residually p-finite.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 125–131.Original Russian Text Copyright © 2005 by E. V. Sokolov.     

Invariants Cohomologiques de Rost En Caractéristique PositiveRost''s Cohomological Invariants in Positive Characteristic

Let G/F be a semisimple algebraic group defined over a field F with characteristic . Let us denote by the Galois cohomology group introduced by Kato. If , we show that the p-primary part of Rost's invariant lifts in characteristic 0. This result allows to deduce properties of the Rost invariant in positive characteristic from known properties in characteristic 0. The case of Merkurjev–Suslin's invariant is specially interesting, i.e. if G/F=SL(D) for a central simple algebra D/F with degree p and class , one has and an element is a reduced norm if and only if the cup-product is trivial in ; one characterizes also in positive characteristic fields with p-dimension by the surjectivity of reduced norms.In a second part, we study Rost invariants when the base field is complete for a discrete valuation. As planned by Serre, invariants are then linked with Bruhat–Tits' theory, this yields a new proof of their nontriviality.