For a finite measure λ, let L0(λ) denote the space of λ-measurable functions equipped with the topology of convergence in measure. We prove that a series in L0(λ) is subseries (or unconditionally) convergent provided each of its lacunary subseries converges. A series in a topological vector space is said to be subseries convergent if each of its subseries converges. As is well known, a subserie...

In 1870 Cantor proved that representation of a function of one variable by a trigonometric series can be done in only one way. In 1996 Bourgain proved the same thing for spherical convergence and multiple trigonometric series. His proof involves injecting a lot of new ideas into the theory of uniqueness. We give here an exposition of Bourgain’s proof, specialized to the case of dimension 2. Our...

A CMOS current-mode companding multiplier/divider composed of three compact logarithm circuits and one compact exponential circuit is proposed. Approaches for constructing the logarithm circuits and the exponential circuit are based on second-order Taylor series approximations. By gathering the three logarithm results and passing through the exponential circuit, we can obtain the result of zp(x...

T H E present writer has given examples in which an absolutely convergent series is obtained as the result of multiplying two conditionally convergent series together, or of multiplying one conditionally convergent series by a divergent series.* He has also given an example of two divergent series whose product is absolutely convergent.f Pringsheim has treated this subject from a more general p...

Let φ(n) be any function which grows more slowly than exponentially in n, i.e., limsup n→∞ φ(n)1/n ≤ 1. There is a double trigonometric series whose coefficients grow like φ(n), and which is everywhere convergent in the square, restricted rectangular, and one-way iterative senses. Given any preassigned rate, there is a one dimensional trigonometric series whose coefficients grow at that rate, b...

We show how to convert divergent series, which typically occur in many applications in physics, into rapidly convergent inverse factorial series. This can be interpreted physically as a novel resummation of perturbative series. Being convergent, these new series allow rigorous extrapolation from an asymptotic region with a large parameter, to the opposite region where the parameter is small. We...

In time series analysis remaining autocorrelation in the errors of a model implies that it is failing to properly capture the structure of time-dependence of the series under study. This can be used as a diagnostic checking tool and as an indicator of the adequacy of the model. Through the study of the errors of the model in the Lagrange Multiplier testing framework, in this paper we derive (an...

Cell voltage equalizers are necessary to ensure years of operation and maximize the chargeable/dischargeable energy of seriesconnected supercapacitors (SCs). A two-switch voltage equalizer using a series-resonant voltage multiplier operating in frequency-multiplied discontinuous conduction mode (DCM) is proposed for series-connected SCs in this paper. The frequency-multiplied mode virtually inc...