arXiv subjects

Valerio De Angelis

Publications and source records attributed to Valerio De Angelis.

5 recordsLinked to original sources

arXiv2023

Estimating the $k$th coefficient of $(f(z))^n$ when $k$ is not too large

We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left( f\left( z\right) \right) ^{n}$ when $n\rightarrow \infty $ and $k$ is of order $n^{δ}$, where $0<δ<1,$ and $f\left( z\right) $ is a power series satisfying suitable positivity conditions and with $f\left( 0\right) \neq 0,$ $f^{\prime }\left( 0\right) =0.$ We also show that there is a positive number $\varepsilon <1$ (easily computed from the pattern of non-zero coefficients of $f\left( z\right) $) such that the same coefficient is positive for large $n$ and $\varepsilon <δ<1$, and admits an asymptotic expansion in inverse powers of $k$. We use the asymptotic estimates to prove that certain finite sums of exponential and trigonometric functions are non-negative, and illustrate the results with examples.

math.CA↗

arXiv2023

Wilf's Conjecture

In a Note in this Monthly, Klazar raised the question of whether the alternating sum of the Stirling numbers of the second kind $B^\pm(n)=\sum_{k=0}^n(-1)^kS(n,k)$ is ever zero for $n\neq 2$. In this article, we present an exposition of the history of this problem, and an economical account of a recent proof that there is at most one $n\neq 2$ for which $B^\pm(n)=0$.

math.NT↗

arXiv2018

Battery Life-Cycle Optimization and Runtime Control for Commercial Buildings Demand Side Management: A New York City Case Study

In metropolitan areas populated with commercial buildings, electric power supply is stringent especially during business hours. Demand side management using battery is a promising solution to mitigate peak demands, however long payback time creates barriers for large scale adoption. In this paper, we have developed a design phase battery life-cycle cost assessment tool and a runtime controller for the building owners, taking into account the degradation of battery. In the design phase, perfect knowledge on building load profile is assumed to estimate ideal payback time. In runtime, stochastic programming and load predictions are applied to address the uncertainties in loads for producing optimal battery operation. For validation, we have performed numerical experiments using the real-life tariff model serves New York City, Zn/MnO2 battery, and state-of-the-art building simulation tool. Experimental results shows a small gap between design phase assessment and runtime control. To further examine the proposed methods, we have applied the same tariff model and performed numerical experiments on nine weather zones and three types of commercial buildings. On contrary to the common practice of shallow discharging battery for preventing phenomenal degradation, experimental results show promising payback time achieved by optimally deep discharge a battery.

eess.SP↗

arXiv2015

The Stern diatomic sequence via generalized Chebyshev polynomials

Let a(n) be the Stern's diatomic sequence, and let x1,...,xr be the distances between successive 1's in the binary expansion of the (odd) positive integer n. We show that a(n) is obtained by evaluating generalized Chebyshev polynomials when the variables are given the values x1+1, ..., xr+1, and we derive a formula expressing the same polynomials in terms of sets of increasing integers of alternating parity. We also show that a(n) = Det(Ir + Mr), where Ir is the rxr identity matrix, and Mr is the rxr matrix that has x1,...,xr along the main diagonal, then all 1's just above and below the main diagonal, and all the other entries are 0.

math.NT↗

arXiv2013

The Zagier polynomials. Part II: Arithmetic properties of coefficients

The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D. Zagier in 1998 were recently extended to the polynomial case by replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Arithmetic properties of the coefficients of these polynomials are established. In particular, the 2-adic valuation of the modified Bernoulli numbers is determined. A variety of analytic, umbral, and asymptotic methods is used to analyze these polynomials.

math.NT↗