On the spectrum of n quasigroups with given conjugate invariant subgroup

Let tⁿ be a vector of n positive integers that sum to 2n ? 1. Let u denote a vector of n or more positive integers that sum to n², and call u, n-universal if for every possible choice of t¹, t²,…, tⁿ, the components of the tⁱ can be arranged in the successive rows of an n-row matrix (with 0 in each unused cell) so that u is the vector of column sums.It is shown that (n,…, n)_{(n times)} is n-universal for every n. More generally, for odd n, any choice of t¹, t³,…, tⁿ can be placed in rows so that the column sums are (n, n?1,…, 2, 1); for even n, any choice of t², t⁴,…, tⁿ can be placed in rows so that the column sums are (n, n ?1,…, 2, 1). Hence, any u that can be obtained from the sum of two rows whose nonzero components are, respectively, n, n ?1,…, 2, 1 and n ?1, n ?2,…, 2, 1 (in any order, with 0's elsewhere) is n-universal.The problem examined is closely related to the graph conjecture that trees on 2, 3,…, n + 1 vertices can be superposed to yield the complete graph on n + 1 vertices.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

Remarks on the concepts of t-designs

We define the concept of t-design for real hyperbolic space ? ⁿ , as an analogue of the definition of Euclidean t-design. Then, we discuss the similarities between the concept of t-design on ? ⁿ or ? ⁿ , and the concept of relative t-design defined for association schemes by Delsarte: Pairs of vectors in the space of an association scheme (1977).     

Minimal free resolutions of monomial curves in P3k

Minimal free resolutions for prime ideals with generic zero (tn3,tn3?n10tn11,tn3?n20tn2, tn31), n1<n2<n3 positive integers, (n1,n2,n3)=1, are determined.     

Decay estimates for oscillatory integrals with polynomial phase in terms of p and p or in terms of p and p

We obtain estimates for certain oscillatory integrals with polynomial (degree n) phase, p(t). These estimates are stated in terms of differences between the roots, real or complex, of p⁽ⁿ⁻³⁾(t) and p⁽ⁿ⁻²⁾(t) or between p⁽ⁿ⁻²⁾(t) and p⁽ⁿ⁻¹⁾(t). The sharpness of these results is also explored. This result is a partial generalization of the results found in [J. Math. Anal. Appl. 280 (2003) 424].     

New classes of complete balanced Howell rotations

Complete balanced Howell rotations (CBHR) owe their origins to duplicate bridge tournaments but have since been shown to possess of deep combinatorial properties. They include many other combinatorial designs as special cases, such as: balanced Howell rotations, weak complete balanced Howell rotations, Room squares, Howell designs, and a class of balanced incomplete block designs.All known CBHR's are for n partnerships such that n = 2^t(p^r + 1), where p^r is an odd prime power and t a natural number. In most cases, p^r ≡ 3(mod 4) is also assumed. Berlekamp and Hwang gave constructions of CBHR's for each such n > 3 with t = 0; Schellenberg gave constructions for each such n with t = 1. In this paper, we construct CBHR for each such n with t arbitrary.     

Orthogonal designs II

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1,k) and ordern exist for everyk < n whenn = 2 ^t+2?3 andn = 2 ^t+2?5 (wheret is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n. Coupled with some results of earlier work, this means that theweighing matrix conjecture ‘For every ordern ≡ 0 (mod 4) there is, for eachk ?n, a square {0, 1, ? 1} matrixW = W(n, k) satisfyingWW ^t =kIn’ is resolved in the affirmative for all ordersn = 2^t+1?3,n = 2^t+1?5 (t a positive integer). The fact that the matrices we find are skew-symmetric for allk < n whenn ≡ 0 (mod 8) and because of other considerations we pose three other conjectures about weighing matrices having additional structure and resolve these conjectures affirmatively in a few cases. In an appendix we give a table of the known results for orders ? 64.     

Stationary distributions of the simplest dynamical systems in Rn perturbed by generalized Poisson process

Let z(t) ∈ Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) ∈ Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

On positive solutions of perturbed nonlinear differential equations

Some results are given concerning positive solutions of equations of the form x⁽ⁿ⁾ + P(t) G(x) = Q(t, x).Let class I (II) consist of all n-times differentiable functions x(t), such that x(t)>0 and x^{(n ? 1)}(t) ? 0 (x^{(n ? 1)}(t) ? 0) for all large t. Two theorems are given guaranteeing the nonexistence of solutions in class I and II, respectively, and three theorems ensure the convergence to zero of positive solutions. A recent result of Hammett concerning the second-order case is extended to the general case.     

Multi-variable weighted geometric means of positive definite matrices

We define a family of weighted geometric means {G(t;ω;A)}_t∈[0,1]ⁿ where ω and A vary over all positive probability vectors in Rⁿ and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0_n) and BMP (t=1_n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.     

On Dirichlet's approximation theorem

Though we cannot improve on the upper bound in Dirichlet's approximation theorem,Kaindl has shown that the upper bound can be lowered fromt ⁿ tot ⁿ ?t ^n?1?t ^n?2?...?t?1, if we admit equality. We show thatKaindl's upper bound is lowest possible in this case. The result is then generalized to linear forms.     

Invariance principle for integral type functionals of square-integrable martingales

Let X_n = {X_n(t): 0 ⩽ t ⩽1}, n ⩾ 0, be a sequence of square-integrable martingales. The main aim of this paper is to give sufficient conditions under which ∫^·₀f_n (A_n(t), X_n(t)) dX_n(t) converges weakly in D[0, 1] to ∫^·₀f₀(A₀(t), X₀(t)) dX₀ (t) as n → ∞, where {A_n, n ⩾ 0} is some sequence of increasing processes corresponding to the sequence {X_n, n ⩾ 0}.     

On the Fourier transforms of retarded Lorentz-invariant functions

In this article we evaluate the Fourier transforms of retarded Lorentz-invariant functions (and distributions) as limits of Laplace transforms. Our method works generally for any retarded Lorentz-invariant functions φ(t) (t?Rⁿ) which is, besides, a continuous function of slow growth. We give, among others, the Fourier transform of G_R(t, α, m², n) and G_A(t, α, m², n), which, in the particular case α = 1, are the characteristic functions of the volume bounded by the forward and the backward sheets of the hyperboloid u = m² and by putting α = ?k are the derivatives of k-order of the retarded and the advanced-delta on the hyperboloid u = m². We also obtain the Fourier transform of the function W(t, α, m², n) introduced by M. Riesz (Comm. Sem. Mat. Univ. Lund4 (1939)). We finish by evaluating the Fourier transforms of the distributional functions G_R(t, α, m², n), G_A(t, α, m², n) and W(t, α, m², n) in their singular points.     

An inequality for a sum of quadratic forms with applications to probability theory

Let {x_t} be a sequence of p-component vectors, and let A_t=∑^T_s=1x_tx′_t with A_n nonsingular for some n?p. It is shown that ∑^t_t=q+1x′_tA^-2_tx_t?trA^-1_q for n?q<T. An application of this proposition is the convergence of a certain martingale with probability 1. A matrix version of Kronecker's lemma then leads to strong consistency of least-squares estimates under a certain condition.     

Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the n-dimensional hypercube so only two different in-degrees a and b occur. We show that this can be done, for two specified in-degrees, if and only if obvious necessary conditions hold. Namely, we need 0?a,b?n and also there exist non-negative integers s and t so that s+t=n² and as+bt=n²n−1. This is connected to a question arising from constructing a strategy for a “hat puzzle”.     

Slow entropy for some smooth flows on surfaces

We study slow entropy in some classes of smooth mixing flows on surfaces. The flows we study can be represented as special flows over irrational rotations and under roof functions which are C² everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol’d flows), we show that in the scale a_n(t) = n(log n)^t slow entropy equals 1 (the speed of orbit growth is n log n) for a.e. irrational α. If the singularity is of power type (x^?γ, γ ∈ (0, 1)) (Kochergin flows), we show that in the scale a_n(t) = n^t slow entropy equals 1 + γ for a.e. α.We show moreover that for local rank one flows, slow entropy equals 0 in the n(log n)^t scale and is at most 1 for scale n^t. As a consequence we get that a.e. Arnol’d and a.e Kochergin flow is never of local rank one.     

A poisson convergence theorem for a particle system with dependent constant velocities

Consider an infinite collection of particles travelling in d-dimensional Euclidean space and let X_n denote the initial position of the n^th particle. Assume that the n^th particle has through all time the random velocity V_n and that {V_n} is a sequence of dependent random variables. Let X_n(t) = X_n + V_nt denote the position of the n^th particle at time t. Conditions are obtained for the convergence of {X_n(t)} to a Poisson process as t→∞. Essentially they require that the dependence in the V_n-sequence decrease with increasing distance between the initial positions and that the conditional distribution of V_n given the initial positions of all the particles and V_nk≠n be absolutely continuous with respect to Lebesgue measure.     

Oscillation theorems for nth-order differential equations with deviating arguments

This paper is concerned with the study of the oscillatory behavior of solutions of the nth-order nonlinear differential equation (a(t)x^{(n ? v)}(t))^(v) + q(t)f(x[g(t)]) = 0, where n is even and 1 ? v ? n ? 1. A systematic study is attempted which extends and correlates a number of existing results.     

Some soluble cases of the discrete logarithm problem

Letg be a primitive λ-root modn. Then the powersg ^t mod,n, fort=1, 2, ..., λ(n) represent a (cyclic) subgroupC _λ(n) (of order λ(n)) ofM _n , the group of order ?(n), representingall primitive residue classes modn. To computet backwards fromg ^t modn is calledthe discrete logarithm problem inC _λ(n), also expressible by $$a \equiv g^t \bmod n \Leftrightarrow t \equiv \log _g a\bmod \lambda \left( n \right).$$ The purpose of this paper is to point out some cases, in which this problem can be solved by astraight-forward formula only, with no element of guessing or collecting factorizations or the like involved at all, taking timeO(log³ n) or less to compute (orO(log^2+ε n), if fast multiplication of multiple precision integers is used).—One very simple case is given by the modulusn=10 ^s , fors≥4. To give just one instance of this case: Forn=10⁸, if the primitive λ-rootg=3¹⁷ ≡ 29140163 modn is chosen, anda=g ^t modn, then $$a \equiv 4962873 \cdot \frac{{a^{250000} - 1}}{{5000000}}\bmod 5000000.$$