The Strong Endomorphism Kernel Property in Ockham Algebras

An endomorphism on an algebra 𝒜 is said to be “strong” if it is compatible with every congruence on 𝒜; and 𝒜 is said to have the “strong endomorphism kernel property” if every congruence on 𝒜, different from the universal congruence, is the kernel of a strong endomorphism on 𝒜. Here we consider this property in the context of Ockham algebras. In particular, for those MS-algebras that have this property we describe the structure of their dual space in terms of 1-point compactifications of discrete spaces.     

On the Least Property of the Semilattice Congruences on PO-Semigroups

n on po-semigroups. We study the least property of (ordered) semilattice congruences, and prove: 1. N is the least ordered semilattice congruence on pr-semigroups (cf.[1]). 2. n is the least semilattice congruence on po-semigroups. 3. N is not the least semilattice congruence on po-semigroups in general. Thus, we give a complete solution to the problem posed by N. Kehayopulu in [1].     

Running uphill: energy needs and Naismith's Rule

In this note we make two points, from which follows a suggestion for running uphill:(1) an athlete's energy needs in running up a slope of gradient θ at a speed V may be considered to have three additive elements: that needed to stay alive; that needed to run at a speed V on the level; that needed to lift the body at a rate V sin θ.(2) corresponding to Naismith's rule of 1 to 8, in whole numbers, for walking or running in mountainous country (1 foot of ascent is equivalent to 8 feet on the level), the rule for running on a treadmill is 1 to 3 and the rule for running on roads is 1 to 4 or 5.This result suggests that, if possible, a runner should zigzag up slopes steeper than 1 in 8 on the fells and zigzag up slopes greater than 1 in 4 or 5 on roads.     

Fluid queue driven by aGI/GI/1 queue

Consider a model consisting of two phases: the GI/GI/1 queue and a buffer which is fed by a fluid arriving from a single-server queue. The fluid output from the GI/GI/1 queue is of the on/off type with on- and off-periods distributed as successive busy and idle periods in the GI/GI/1 queue. The fluid pours out of the buffer at a constant rate. The steady-state performance of this model is studied. We derive the Laplace-Stieltjes transform of the stationary distribution function of the buffer content in the case of the M/GI/1 queue in the first phase. It is shown that this distribution depends on the form of the service-time distribution. Therefore, the replacement of an M/GI/1 queue by an M/M/1 queue is not correct, in general. Continuity estimates are derived in the cast where the buffer is fed from the GI/GI/1 queue. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow Russia, 1996, Part II.     

Threshold model of a neural ensemble

Networks of threshold elements whose inputs are assigned positive and negative (inhibitory) weights and outputs take the values 0 and 1 are considered. A stationary ensemble is defined as a connected subnetwork of a threshold network for which the unit state (1, 1, …, 1) = 1 is stable. The transfer of an ensemble into the state 1 is called switching on. Necessary and sufficient conditions for a network to be an ensemble are given. It is shown that, in the proposed model, the switching on of one of two ensembles having common elements does not necessarily lead to the switching on of the other.     

The Endomorphism Kernel Property in Finite Distributive Lattices and de Morgan Algebras

Abstract

An algebra 𝒜 has the endomorphism kernel property if every congruence on 𝒜 different from the universal congruence is the kernel of an endomorphism on 𝒜. We first consider this property when 𝒜 is a finite distributive lattice, and show that it holds if and only if 𝒜 is a cartesian product of chains. We then consider the case where 𝒜 is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property.     

Application of wavelet bases in linear and nonlinear approximation to functions from Besov spaces

Linear and nonlinear approximations to functions from Besov spaces B _{p, q} ^σ ([0, 1]), σ > 0, 1 ≤ p, q ≤ ∞ in a wavelet basis are considered. It is shown that an optimal linear approximation by a D-dimensional subspace of basis wavelet functions has an error of order D ^{-min(σ, σ + 1/2 ? 1/p)} for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). An original scheme is proposed for optimal nonlinear approximation. It is shown how a D-dimensional subspace of basis wavelet functions is to be chosen depending on the approximated function so that the error is on the order of D ^?σ for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). The nonlinear approximation scheme proposed does not require any a priori information on the approximated function.     

Measures on phase space as solutions of the one-dimensional neutron transport equation

Summary The linear integral transport operator for slab geometry is formulated and studied as a mapping on the set of measures on the phase space of the underlying system, with the expected number of neutrons emergent from a collision represented by a measure on the space of outgoing velocities. Under appropriate assumptions it is shown that, if c represents the maximum number of secondary particles per collision, then there exists c ₁ ≥1 such that the system is subcritical for c≤c ₁ . An example shows that c ₁ ≥1 is sharp in general, but further assumptions are given under which one can deduce c ₁ >1. The idealized laws of elastic and inelastic scattering are shown to satisfy our assumptions. Entrata in Redazione il 27 ottobre 1975.     

Simple facts about algebras of unbounded operators

A topology, based essentially on order properties, is defined on 1-algebras of unbounded operators, and behaves properly with respect to the usual algebraic operations of $\text{C}$¹-algebras. It is shown that, for commutative 1-algebras, the Gelfand transformation is valid if and only if sufficiently analytic states exist in the algebra. The connection with the enveloping algebra of a Lie algebra is sketched.     

Theorems on decomposition of operators in L 1 and their generalization to vector lattices

The Rosenthal theorem on the decomposition for operators in L ₁ is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in L ₁ is a projective component, which yields the known fact that a sum of narrow operators in L ₁ is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in L ₁, in particular the Liu decomposition. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 26–35, January, 2006.     

Bootstrap of the offspring mean in the critical process with a non-stationary immigration

In applications of branching processes, usually it is hard to obtain samples of a large size. Therefore, a bootstrap procedure allowing inference based on a small sample size is very useful. Unfortunately, in the critical branching process with stationary immigration the standard parametric bootstrap is invalid. In this paper, we consider a process with non-stationary immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively. We prove that 1+2α is the threshold for the validity of the bootstrap in this model. If β<1+2α, the standard bootstrap is valid and if β>1+2α it is invalid. In the case β=1+2α, the validity of the bootstrap depends on the slowly varying parts of the immigration mean and variance. These results allow us to develop statistical inferences about the parameters of the process in its early stages.     

Rates of convergence of a one-dimensional search based on interpolating polynomials

In this study, we derive the order of convergence of line search techniques based on fitting polynomials, using function values as well as information on the smoothness of the function. Specifically, it is shown that, if the interpolating polynomial is based on the values of the function and its firsts?1 derivatives atn+1 approximating points, the rate of convergence is equal to the unique positive rootr _n+1 of the polynomial $$D_{n + 1} (z) = z^{n + 1} - (s - 1)z^n - s\sum\limits_{j = 1}^n {z^{n - j} } .$$ For alln, r _n is bounded betweens ands+1, which in turn implies that the rate can be increased by as much as one wishes, provided sufficient information on the smoothness is incorporated.     

Polynomial approximation errors for functions of low-order continuity

Given a functionf defined on [-1, 1] we obtain, in terms of (n+1)st divided differences, expressions for the minimax errorE _n(f) and the errorS _n(f) obtained by truncating the Chebyshev series off aftern+1 terms. The advantage of using divided differences is thatf is required to have no more than a continuous second derivative on [-1, 1].     

Inverses of trees

LetT be a tree with a perfect matching. It is known that in this case the adjacency matrixA ofT is invertible and thatA ^?1 is a (0, 1, ?1)-matrix. We show that in factA ^?1 is diagonally similar to a (0, 1)-matrix, hence to the adjacency matrix of a graph. We use this to provide sharp bounds on the least positive eigenvalue ofA and some general information concerning the behaviour of this eigenvalue. Some open problems raised by this work and connections with Möbius inversion on partially ordered sets are also discussed.     

On Heisenberg-like Super Group Structures

The ring-like structures that can be defined on the ground supermanifolds \mathbb R^1|1{\mathbb R^{1\vert 1}} and \mathbb C^1|1{\mathbb C^{1\vert 1}} are classified up to equivalence in the category of smooth and complex Berezin-Kostant-Leites-Manin supermanifolds. It is proved that there are three different such equivalence classes in the real case, whereas there are two for the complex field. The corresponding module structures—defined componentwise on the product of k copies of \mathbb R^1|1{\mathbb R^{1\vert 1}} or \mathbb C^1|1{\mathbb C^{1\vert 1}}—are also classified up to equivalence. The notions of linearity and bilinearity are reviewed and used to define Heisenberg-like super group structures. It turns out that there are three non-isomorphic real such super groups, whereas only two over the complex field. The use of the appropriate exponential maps introduces the possibility of defining Heisenberg-like super group structures on the product of k copies of the ground supermanifold, with an appropriate super circle. The corresponding classification is also obtained.     

On the prime number theorem for the Selberg class

We show that the prime number theorem is equivalent with the non-vanishing on the 1-line, in the general setting of the Selberg class S \mathcal{S} of L \mathcal{L} -functions. The proof is based on a weak zero-density estimate near the 1-line and on a simple almost periodicity argument. We also give a conditional proof of the non-vanishing on the 1-line for every L \mathcal{L} -function in S \mathcal{S} , assuming a certain normality conjecture.     

A modulus of smoothness based on an algebraic addition

Summary The purpose of this paper is to present a new approach to smoothness of nonperiodic functions. We consider the space of continuous functions on [−1, 1] as well as the weighted L^p-space and introduce a modulus of smoothness that is based on an algebraic addition ⊕ defined on [−1, 1]. The present paper is mainly concerned with general properties and groundwork, whereas a second paper [4] is devoted to more complex properties, in particular to an equivalent K-functional and to the characterization of best algebraic approximation. Moreover the equivalence with the Butzer-Stens modulus will be shown there.     

Martingales and the Robbins-Monro procedure in D[0, 1]

The Robbins-Monro procedure for recursive estimation of a zero point of a regression function f is investigated for the case f defined on and with values in the space D[0, 1] of real-valued functions on [0, 1] that are right-continuous and have left-hand limits, endowed with Skorohod's J₁-topology. There are proved an a.s. convergence result and an invariance principle where the limit process is a Gaussian Markov process with paths in the space of continuous C[0, 1]-valued functions on [0, 1]. At first the case f(x) ≡ x, i.e., the case of a martingale in D[0, 1], is treated and by this then the general case. An application to an initial value problem with only empirically available function values is sketched.     

Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions

We characterize finite codimensional linear isometries on two spaces, C ⁽ⁿ⁾[0; 1] and Lip [0; 1], where C ⁽ⁿ⁾[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C ⁽ⁿ⁾[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.