Complexity of the realization of a linear function in the class of II-circuits

It is proved that the linear function gn(x₁,..., x_n) = x₁ + ... + x_nmod 2 is realized in the class of II-circuits with complexity L(g_n) n². Combination of this result with S. V. Yablonskii's upper bound yields L(g_n) n².Translated from Matematicheskie Zametki, Vol. 9, No. 1, pp. 35–40, January, 1971.     

Some Convergence Properties of Descent Methods

In this paper, we discuss the convergence properties of a class of descent algorithms for minimizing a continuously differentiable function f on R ⁿ without assuming that the sequence { x _k } of iterates is bounded. Under mild conditions, we prove that the limit infimum of is zero and that false convergence does not occur when f is convex. Furthermore, we discuss the convergence rate of { } and { f(x _k )} when { x _k } is unbounded and { f(x _k )} is bounded.     

Homomorphisms between Poisson JC*-Algebras

It is shown that every almost linear mapping of a unital Poisson JC*-algebra to a unital Poisson JC*-algebra is a Poisson JC*-algebra homomorphism when h(2 ⁿ uy) = h(2 ⁿ u) h(y), h(3 ⁿ u y) = h(3 ⁿ u) h(y) or h(q ⁿ u y) = h(q ⁿ u) h(y) for all , all unitary elements and n = 0, 1, 2, · · · , and that every almost linear almost multiplicative mapping is a Poisson JC*-algebra homomorphism when h(2x) = 2h(x), h(3x) = 3h(x) or h(qx) = qh(x) for all . Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings.Moreover, we prove the Cauchy–Rassias stability of Poisson JC*-algebra homomorphisms in Poisson JC*-algebras.*This work was supported by grant No. R05-2003-000-10006-0 from the Basic Research Program of the Korea Science & Engineering Foundation.     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Oscillatory criteria for nonlinearnth-order differential equations with quasiderivatives

Sufficient conditions are given for the existence of oscillatory proper solutions of a differential equation with quasiderivativesL _n y=f(t,L ₀ y, ..., L _n–1 y) under the validity of the sign conditionf(t,x ₁ ,...,x _n )x ₁0,f(t,0,x ₂ ,...,x _n )=0 on ₊ x ⁿ .     

Weak Variation of Gaussian Processes

Let X(t) (tR) be a real-valued centered Gaussian process with stationary increments. We assume that there exist positive constants ₀, C ₁, and c ₂ such that for any tR and hR with |h|₀ and for any 0r<min{|t|, ₀} where is regularly varying at zero of order (0 < < 1). Let be an inverse function of near zero such that (s)=(s) log log(1/s) is increasing near zero. We obtain exact estimates for the weak -variation of X(t) on [0,a].     

Iterative hyperidentities for theA n,m varieties of semigroups

Iterative hyperidentities are hyperidentities of the special formF ^a (x ₁,...,x _k =F ^a+b (x ₁,...,x _k ). This type of hyperidentity has been considered by Denecke and Pöschel, and by Schweigert. Here we consider iterative hyperidentities for the variety A_n,m of commutative semigroups satisfyingx ⁿ =x ^n+m ,n,m 1. We introduce two parameters(m, n) and(m) associated withn andm, and show thatA _nn,m satisfies the iterative hyperidentitiesF (x ₁,...,x _k =F ^+b (x ₁,...,x _k ) for every arityk. Moreover, the numbers and are minimal, making these hyperidentities irreducible in the sense of Schweigert. We also show how these hyperidentities for A_n,m may be used to prove that no non-trivial proper variety of commutative semigroups can have a finite hyperidentity basis.Presented by W. Taylor.Research supported by NSERC of Canada     

Radicals in pseudocomplemented lattices

For a pseudocomplemented latticeL, we prove that the filter D_n(L), 1n<, generated by then-strongly dense elements is contained in everyn-normal filter. Hence, D_n(L)=G_n(L)=Rad_n (L), where G_n(L) is the intersection of all n-normal filters, and Rad_n (L) is the intersection of alln-normal prime filters. Moreover, we prove that a prime filterP is n-normal iff D_n(L)=P. Consequently, for , we have D_n(L)=G_n(L)=Rad_n (L) and therefore iff Rad_n(L)={1} (or iff G_n(L)={1}).Considering the skeleton S(L) ofL, a complete clarification of the relationship between filters ofL and S(L) is given by studying th correspondence FFS(L).We state that D(L) (and that D₁(L), if is an irredundant intersection of maximal filters (resp. of *-maximal filters) iff S(L) is finite.Finally, for we state that the least *-congruence for which is that one generated by D_n(L).Presented by B. Jónsson.Research supported by the I.N.I:C, (Centro de Algebra da Universidade de Lisboa).     

Properties of a special class of doubly stochastic measures

Summary A measure on the unit squareI } I is doubly stochastic if(A } I) = (I } A) = the Lebesgue measure ofA for every Lebesgue measurable subsetA ofI = [0, 1]. By the hairpinL L ^–1, we mean the union of the graphs of an increasing homeomorphismL onI and its inverseL ^–1. By the latticework hairpin generated by a sequence {x _n :n Z} such thatx _n-1 < x_n (n Z), x _n = 0 and x _n = 1, we mean the hairpinL L ^–1 , whereL is linear on [x _n-1 ,x _n ] andL(n) =x _n-1 forn Z. In this note, a characterization of latticework hairpins which support doubly stochastic measures is given. This allows one to construct a variety of concrete examples of such measures. In particular, examples are given, disproving J. H. B. Kemperman's conjecture concerning a certain condition for the existence of doubly stochastic measures supported in hairpins.     

Limiting distributions of homogeneous functions of sample spacings for distributions approaching the uniform distribution

Summary X ₁,...,X _n are independent random variables, identically distributed over the unit interval, with common probability density function 1 + r(x)/n for all sufficiently large n, where is a positive constant, and |r(x)| <D. V ₁, ..., V _n+1 are the sample spacings generated by X ₁,..., X _n . It is shown that in many cases, the asymptotic joint distribution of homogeneous functions of V ₁,..., V _n+1 can be found directly from the asymptotic joint distribution of homogeneous functions of independent exponential random variables.Research supported by NSF Grant GP 3783.     

Multiple Wiener-Ito integrals possessing a continuous extension

LetF(W) be a Wiener functional defined byF(W)=I _n(f) whereI _n(f) denotes the multiple Wiener-Ito integral of ordern of the symmetricL ²([0, 1] ⁿ ) kernelf. We show that a necessary and sufficient condition for the existence of a continuous extension ofF, i.e. the existence of a function ø(·) from the continuous functions on [0, 1] which are zero at zero to which is continuous in the supremum norms and for which ø(W)=F(W) a.s, is that there exists a multimeasure (dt ₁,...,dt _n ) on [0, 1] ⁿ such thatf(t ₁, ...,t _n ) = ((t ₁, 1]), ..., (t _n , 1]) a.e. Lebesgue on [0, 1] ⁿ . Recall that a multimeasure (A ₁,...,A _n ) is for every fixedi and every fixedA _i,...,A_i-1, A_i+1,...,A_n a signed measure inA _i and there exists multimeasures which are not measures. It is, furthermore, shown that iff(t ₁,t ₂, ...,t _n ) = ((t ₁, 1], ..., (t _n , 1]) then all the tracesf ^(k), off exist, eachf(^k) induces ann–2k multimeasure denoted by ^(k), the following relation holds

Spectral estimates of Cauchy's transform inL 2(Ω)

We prove that the singular numbers of the Cauchy transform onL ²(D) are asymptotically , whiles _n (C _| L _a ² (D))1/n (whereL _a ² (D) is the subspace of analytic functions inL ²(D)). Also, the singular numbers of the logarithmic potential onL ²(D) are asympoticallys _n (L)1/n, whiles _n(L _{|L a} ² (D))1/n ². Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform fromL _L ² () intoL ²() where and are rotation-invariant measures on .The author was partly supported by a grant from the national Science Foundation.     

Extremum Problem for Periodic Functions Supported in a Ball

We consider the Turan n-dimensional extremum problem of finding the value of A_n(hB ⁿ ) which is equal to the maximum zero Fourier coefficient of periodic functions f supported in the Euclidean ball hB ⁿ of radius h, having nonnegative Fourier coefficients, and satisfying the condition f(0)= 1. This problem originates from applications to number theory. The case of A₁([–h,h]) was studied by S. B. Stechkin. For A_n(hB ⁿ we obtain an asymptotic series as h 0 whose leading term is found by solving an n-dimensional extremum problem for entire functions of exponential type.     

On the convergence rate of imperfect minimization algorithms in Broyden'sβ-class

This paper presents a local convergence analysis of Broyden's class of rank-2 algorithms for solving unconstrained minimization problems, ,h C¹(R ⁿ ), assuming that the step-size ai in each iterationx _i+1 =x _i - _i H _i h(x _i ) is determined by approximate line searches only. Many of these methods including the ones most often used in practice, converge locally at least with R-order, .     

Generalized varieties of sums of powers

Let X ? P^N be an irreducible, non-degenerate variety. The generalized variety of sums of powers V S P_H^X(h) of X is the closure in the Hilbert scheme Hilb_h (X) of the locus parametrizing collections of points {x₁,..., x_h} such that the (h -1)-plane >x₁,..., x_h> passes through a fixed general point p ∈ P^N. When X = V_dⁿ is a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S P_H^X(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S P_H^X(h) are inherited from the birational geometry of variety X itself.     

On singular integrals along surfaces related to black spaces

Leth(t) be an arbitrary bounded radial function and let (x) be a real measurable and radial function defined onR ^n–1. Forx, yR ^n–1, we establish that the singular integral along surfacex (x, (x)):

Unique nontransitive additive conjoint measurement on finite sets

Nontransitive additive conjoint measurement for a binary relation>on a setX ₁ ×X ₂ ×...×X _n ofn-tuples(x ₁,...,x _n ), (y ₁,...,y _n ),... is concerned with the representation

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.     

On the product of the distances of a point from the vertices of a polytope

Letx ₁,...,x _m be points in the solid unit sphere ofE _n and letx belong to the convex hull ofx ₁,...,x _m. Then . This implies that all such products are bounded by (2/m) ^m (m −1) ^m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z ₀)| where andp′(z ₀)=0.