Closed-form Evaluation Research Articles

103.21 Closed form evaluation of a class of improper integrals

103.21 Closed form evaluation of a class of improper integrals

Simplified and accurate BER analysis of magnitude modulated M‐PSK signals

In this work, the authors consider phase shift keying (PSK)-type signals with very compacted spectrum and low envelope fluctuations obtained through the use of magnitude modulation (MM) techniques and stringent roll-off pulse shaping filtering. The analytical bit error rate (BER) of PSK-type signals combined with MM is assessed, for both flat fading and time-dispersive channel scenarios. The authors use a Gaussian model to approximate the statistical distribution of the distortion error introduced by MM to enable a closed-form evaluation of the BER. The proposed expression depends only on the constellation order and the Kullback–Leibler divergence of the MM factors' probability density function from the Gaussian one, and it was found to be very accurate.

Zeta determinant of the Laplacian on the real projective spaces

We give a closed form evaluation of the zeta determinant of the Laplace operator on spheres and projective spaces that clearly describes the arithmetic structure of this number. All the factors in the final formula for the determinant are easily computable.

A Bridge Between the Unit Square and Single Integrals for Real Functions of the Form <em>f</em>(<em>x · y</em>)

Sondow and co-workers have employed a key change of variables in order to evaluate double integrals over the unit square \([0,1] \times [0,1]\) in exact closed-form. Motivated by their results, I introduce here a change of variables which creates a ‘bridge’ between integrals of the form \(\,\int_0^1\!\!\int_0^1{f(x \cdot y)~dx \, dy}\,\) and single integrals of the form \(\int_0^1{f(p)\,\ln{p}~d p}\). This allows for prompt closed-form evaluations of several interesting integrals, including some of those investigated recently by Sampedro. I also show that the bridge holds when the intervals of integration are changed from \([0,1]\) to \([1,\infty)\). Finally, a generalization for higher dimensions is proved, which reveals an interesting link of those integrals to Mellin’s transform.

Buffer-Aided Serial Relaying for FSO Communications: Asymptotic Analysis and Impact of Relay Placement

In this paper, we analyze multi-hop free-space optical (FSO) communications in the context of decode-and-forward serial relaying, where the relays are equipped with finite-size buffers. Based on a Markov chain analysis, we derive closed-form asymptotic expressions for the system outage probability (OP) and average packet delay (APD) for an arbitrary number of relays N, and an arbitrary buffer size L. The closed-form evaluation links the system performance to the various network parameters in a simple and intuitive manner, and it is useful for offering clear insight on the impact of the relay placement and the selection of the buffer size for practical FSO systems. We prove that buffer-aided multi-hop systems can reap a diversity gain that ranges from [(N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</sub> /2)] + 1 to N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</sub> + 1 compared with multi-hop buffer-free systems while the asymptotic APD values can range from N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</sub> to (L - 1)N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</sub> for L ≥ 2. Our analysis also highlights the optimal solutions capable of concurrently minimizing the OP and APD.

Analytical evaluation of certain integrals occurring in studies of wireless communications systems using the Mellin-transform method

This paper illustrates the effectiveness of the Mellin-transform (MT) method for the exact calculation of certain one-dimensional definite integrals occurring in studies of wireless communications systems. The MT method, though it often requires laborious calculations, is a straightforward and extremely powerful technique for the closed-form evaluation of definite integrals. Only basic information is provided with regard to the MT and the closely related class of Mellin–Barnes (MB) integrals. On this basis, the MT method is applied to several examples of integrals occurring in studies of fading channels and wireless transmission systems. Closed-form expressions are obtained, which are either new or provided without proof in the literature. The new formulas are suitable for numerical evaluation.

Evaluation and regularization of phase-modulated Eisenstein series and application to double Schlömilch-type sums

In the study of periodic media, conditionally convergent series are frequently encountered and their regularization is crucial for applications. We derive an identity that regularizes two-dimensional phase-modulated Eisenstein series for all Bravais lattices, yielding physically meaningful values. We also obtain explicit forms for the phase-modulated series in terms of holomorphic Eisenstein series, enabling their closed-form evaluation for important high symmetry lattices. Results are then used to obtain representations for the related double Schlömilch-type sums, which are also given for all Bravais lattices. Finally, we treat displaced lattices of high symmetry, expressing them in terms of origin-centered lattices via geometric multi-set identities. These identities apply to all classes of two-dimensional sums, allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions.

A branching process approach to power markets

We propose and investigate a market model for power prices, including most basic features exhibited by previous models and taking into account self-exciting properties. The model proposed extends Hawkes-type models by introducing a twofold integral representation property. A Random Field approach was already exploited by Barndorff-Nielsen et al., who adopted the Ambit Field framework for describing the power price dynamics. The novelty contained in our approach consists of combining the basic features of both Branching Processes and Random Fields in order to get a realistic and parsimonious model setting. We shall provide some closed-form evaluation formulae for forward contracts. We discuss the risk premium behavior, by pointing out that in the present framework, a very realistic description arises. We outline a possible methodology for parameters estimation. We illustrate by graphical representation the main achievements of this approach.

Verifying Rollett’s Proviso on Active Devices Under Arbitrary Passive Embeddings

Some results implicit in Ohtomo’s original paper are reappraised and built upon to help high-frequency circuit designers avoid common misconceptions. An analysis of the small-signal stability of an intrinsic device is presented, based on a closed-form evaluation of the network determinant in admittance representation and on the relevant Nyquist plot. Since the intrinsic is studied as a three-port, the proposed method represents a rigorous tool for checking the fulfillment of the inherent-stability proviso under arbitrary passive embeddings. Conversely, the proviso is often simply assumed to hold, which can in principle lead to mistakenly predicted circuit stability, or even unconditional stability. Examples of the proposed graphical analysis are provided both for the intrinsic circuit of an actual device and for a modified version thereof, i.e., suitably tweaked for illustrative purposes. The latter example is then expanded to give the reader a full picture of the small-signal stability analysis process, both in a flawed and a correct form.

A family of polylog-trigonometric integrals

In this paper, we study the sequences $$\begin{aligned} I_n=\int _0^1\mathrm {Li}_n(\sin \pi x)\mathrm {d}x\quad \text{ and }\quad J_n=\int _0^1\mathrm {Li}_n(\cos \pi x)\mathrm {d}x, \end{aligned}$$ where \(\mathrm {Li}_n\) is the nth polylogarithm function. Among others, we determine their generating functions, asymptotic behaviour and their connection to the well-known log-sine integrals $$\begin{aligned} S_n=(-1)^n\int _0^1\log ^n(\sin \pi x)\mathrm {d}x. \end{aligned}$$ With the help of the explicit forms of \(I_n\) and \(J_n\), we deduce closed-form evaluations for a number of polylog-trigonometric definite integrals.

Measurement and optimization of responsiveness in supply chain networks with queueing structures

In this paper we consider supply chains with multiple stages of serial or network structure. The supply chains are endogenous in the sense that they involve queues because each order’s lead-time is dependent on the orders already in the system. We define supply chain responsiveness as the probability of fulfilling customer orders within a promised lead-time and study the problems of measuring and optimizing supply chain responsiveness using queueing network models. We first consider a single-server multi-stage serial supply chain and find a closed form expression for the fulfilment time distribution. For the multi-server multi-stage problem, the closed form evaluation of the fulfilment time distribution becomes intractable due to the dependency of the lead-times in different stages. We circumvent this difficulty by proposing a novel FCFS discipline which enables a closed-form analysis. For the multi-server multi-stage Jackson-type supply chain network, to enable analysis, we convert the system into an equivalent single server single stage system with state-dependent rates. For each case, we present detailed numerical examples for both measurement and the optimization of supply chain responsiveness.

Closed-form evaluation of integrals involving the gamma function

ABSTRACTIn this article, we obtain simplified as well as original closed-form expressions for integrals involving the gamma function. These explicit results are found neither in the literature nor using symbolic computation software. Subsequent results follow, giving rise to explicit expressions for integrals involving the error function, with application in order statistics. In particular, these results can be used within the framework of reliability theory.

An integral transform related to series involving alternating harmonic numbers

ABSTRACTWe introduce an integral transform which may be used to construct new closed-form formulas for series involving alternating harmonic numbers, such as the Ramanujan-like formula introduced in this article which elegantly relates Catalan's constant G with through an infinite summation involving Catalan numbers. Using this integral transform, we also obtain closed-form expressions for new series involving harmonic numbers of even index such as the new summation for Apéry's constant introduced in our article, as well as new proofs of known formulas for series containing harmonic numbers. The techniques introduced in this paper are also used to provide closed-form evaluations for some new definite integrals which state-of-the-art computer algebra systems cannot compute symbolically.

Evaluation of Euler-like sums via Hurwitz zeta values

In this paper we collect two generalizations of harmonic numbers (namelygeneralized harmonic numbers and hyperharmonic numbers) under one roof.Recursion relations, closed-form evaluations, and generating functions of thisunified extension are obtained. In light of this notion we evaluate someparticular values of Euler sums in terms of odd zeta values. We alsoconsider the noninteger property and some arithmetical aspects of this unifiedextension.

Closed form evaluation of Melham's reciprocal sums

Closed form evaluation of Melham's reciprocal sums

Closed-form evaluation of two-dimensional static lattice sums.

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S2 (for conductivity) and T2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S2 and T2 for different lattices are established. In particular, the value S2=π for the square and hexagonal arrays is discussed and T2=π/2 for the hexagonal is deduced.

Geometric polynomials: properties and applications to series with zeta values

We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.

On sums of squares of Fibonomial coefficients by q-calculus

We present some new kinds of sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. As proof method, we will follow the method given in [E. Kılıç and H. Prodinger, Closed form evaluation of sums containing squares of Fibonomial coefficients, accepted in Math. Slovaca]. For this, first we translate everything into [Formula: see text]-notation, and then to use generating functions and Rothe’s identity from classical [Formula: see text]-calculus.

Discrete analogues of Macdonald–Mehta integrals

We consider discretisations of the Macdonald–Mehta integrals from the theory of finite reflection groups. For the classical groups, Ar−1, Br and Dr, we provide closed-form evaluations in those cases for which the Weyl denominators featuring in the summands have exponents 1 and 2. Our proofs for the exponent-1 cases rely on identities for classical group characters, while most of the formulas for the exponent-2 cases are derived from a transformation formula for elliptic hypergeometric series for the root system BCr. As a byproduct of our results, we obtain closed-form product formulas for the (ordinary and signed) enumeration of orthogonal and symplectic tableaux contained in a box.

Closed form evaluation of sums containing squares of Fibonomial coefficients

Abstract We give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable q, and then to use generating functions and Rothe's identity from classical q-calculus.