Riemann’s theorems on the rearrangement of absolutely convergent and conditionally convergent series of real numbers are analysed within Bishop-style constructive mathematics. The constructive proof that every rearrangement of an absolutely convergent series has the same sum is relatively straightforward; but the proof that a conditionally convergent series can be rearranged to converge to what...

ing from the multiplication of 8\ by S2 . Special example : T* • T.* is absolutely convergent, but 1\ and T* are each divergent when r < § and * < §. In the divergent series 8/ the terms increase without limit in numerical value, as v increases without limit. The same is true of S2 . Herein lies the difference between this pair of divergent series yielding an absolutely convergent product, and ...

In the Fictitious Domain Method with Lagrange multiplier (FDM) the physical domain is embedded into a simpler but larger domain called the fictitious domain. The partial differential equation is extended to the fictitious domain using a Lagrange multiplier to enforce the prescribed boundary conditions on the physical domain while all the other data are extended to the fictitious domain. This le...

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new approximate displacement, stress, and rotation using two-scale discretizations. first scale level setting consists approximating traction variable (Lagrange multiplier) in discontinuous polynomial spaces, computing elementwise rigid body modes. In...

In this paper, we propose an MSD (most significant digit) multiplier based on an MSB (most significant bit) multiplier over GF(2). The proposed multiplier is based on connecting D (digit size)-bit bit-operations in series. In each digit operation in our proposed multiplier, the “left shift and reduction operation” is serially performed for each of D bits. Because registers for storing intermedi...

Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers (1 − |ξ| − |η|)+ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from L × L into L with minimal smoothness, i.e., any δ > 0, and we obtain estimates for other pairs of spaces for...

The construction of asymptotically distribution free time series models specification tests based on estimated residual autocorrelations is considered from a general view point. Test statistics are weighted sums of the estimated residual autocorrelations, and have an asymptotic standard normal distribution when the specification is correct, despite of the estimated parameters effect. The weight...

The notions of permutable and weak-permutable convergence of a series ∑∞ n=1 an of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s prin...

In this paper we prove several new characterizations of weakly unconditionally Cauchy series in Banach spaces and in the dual space of a normed space. For a given series ζ, we consider the spaces S(ζ), Sw(ζ) and S0(ζ) of bounded sequences of real numbers (ai)i such that the series P i aixi is convergent, weakly convergent or ∗-weakly convergent, respectively. By means of these spaces we charact...