The Weibull function is frequently chosen to define psychometric functions. Tyler and Chen (Vis. Res. 40 (2000) 3121) criticised the high-threshold postulate implied by the Weibull function and argued that this function implies the assumption of multiplicative noise. It will be shown in this paper that in fact the Weibull function is compatible with the assumption of additive noise, and that th...

In transport logistics, routing is usually done by a central instance that is solving the optimization problem of finding the best solution to cover the current set of orders with the current set of vehicles under constraints such as punctuality, vehicle utilization etc. Approaches have been suggested recently which change this paradigm towards a distributed approach with autonomous entities de...

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...

We investigate the estimation of a multiplicative separable regression function from a bi-dimensional nonparametric regression model with random design. We present a general estimator for this problem and study its mean integrated squared error (MISE) properties. A wavelet version of this estimator is developed. In some situations, we prove that it attains the standard unidimensional rate of co...

Let X be a compact Hausdorff space; let τ : X → X be a topological involution; and let A ⊂ C(X, τ) be a real function algebra. Given an f ∈ A, the peripheral spectrum of f is the set σπ(f) of spectral values of f of maximum modulus. We demonstrate that if T1, T2 : A → B and S1, S2 : A → A are surjective mappings between real function algebras A ⊂ C(X, τ) and B ⊂ C(Y, φ) that satisfy σπ(T1(f)T2(...

We characterize the class of probability measures for which the multiplicative renormalization method can be applied for the function h(x) = 1 √ 1−x to obtain orthogonal polynomials. It turns out that this class consists of only uniform probability measures on intervals and probability measures being supported by one or two points. 1. Multiplicative Renormalization Method Let μ be a probability...

We introduce a general result relating “short averages” of a multiplicative function to “long averages” which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of μ(n) in almost all intervals of the form [x, x+ψ(x)] with ψ(x)→∞ arbitrarily slowly. This goes beyond what was previously known conditionally on t...

If E is an elliptic curve defined over Q and p is a prime of good reduction for E, let E(Fp) denote the set of points on the reduced curve modulo p. Define an arithmetic function ME(N) by setting ME(N) := #{p : #E(Fp) = N}. Recently, David and the third author studied the average of ME(N) over certain “boxes” of elliptic curves E. Assuming a plausible conjecture about primes in short intervals,...

we define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital banach algebra a and show that the δ-approximate spectrum σ_δ (a) of a is compact. the relation between the δ-approximate spectrum and the usual spectrum is investigated. also an analogue of the classical gleason-kahane-zelazko theorem is established: for each ε>0, there is δ>0 such that if ϕ is...

For a graph $G$ with edge set $E(G)$, the multiplicative sum Zagreb index of $G$ is defined as$Pi^*(G)=Pi_{uvin E(G)}[d_G(u)+d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$.In this paper, we first introduce some graph transformations that decreasethis index. In application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum Zagreb ...