Multiple Gamma Functions Research Articles

Simple Moment Generating Function Optimisation Technique to Design Optimum Electronic Filter for Underwater Wireless Optical Communication Receiver

This paper introduces a new simple moment-generating function (MGF) design modelling method to conclude an optimum filter to maximize the Q-factor and increase the link communication span. This approach mitigates the pulse temporal dispersion, particularly the underwater wireless optical communication (UWOC) systems. Hence, some form of equalizing filter design is highly desirable. The model solution environment includes a Double Gamma Function (DGF) water channel impulse response, intersymbol interference (ISI), stochastic Poisson process, and additive Gaussian thermal noise (AGTN). The optimal filters exhibit temporal profiles comparable to those derived by published works based on complex Chernoff Bound (CB) and Modified Chernoff Bound (MCB) methods. The results show the impact of the optimum filter at a signal level and optical receiver level utilizing Eye-Diagrams and BER vs. Q-Factor, respectively. The computation involves four different UWOC propagation channel models for Coastal and Harbor waters. One of the main conclusions indicates that the optimum filter manages the temporal dispersion due to the ISI impairment correctly. Also, the proposed optimum filter reduces eye-opening and the corresponding Q-Factor by less than 15% for a five-times increase in pulse width for the same transmitted optical power level.

Subject-Specific Modeling of EEG-fNIRS Neurovascular Coupling by Task-Related Tensor Decomposition.

Neurovascular coupling (NVC) connects neural activity with hemodynamics and plays a vital role in sustaining brain function. Combining electroencephalography (EEG) and functional near-infrared spectroscopy (fNIRS) is a promising way to explore the NVC. However, the high-order property of EEG data and variability of hemodynamic response function (HRF) across subjects have not been well considered in existing NVC studies. In this study, we proposed a novel framework to enhance the subject-specific parametric modeling of NVC from simultaneous EEG-fNIRS measurement. Specifically, task-related tensor decomposition of high-order EEG data was performed to extract the underlying connections in the temporal-spectral-spatial structures of EEG activities and identify the most relevant temporal signature within multiple trials. Subject-specific HRFs were estimated by parameters optimization of a double gamma function model. A canonical motor task experiment was designed to induce neural activity and validate the effectiveness of the proposed framework. The results indicated that the proposed framework significantly improves the reproducibility of EEG components and the correlation between the predicted hemodynamic activities and the real fNIRS signal. Moreover, the estimated parameters characterized the NVC differences in the task with two speeds. Therefore, the proposed framework provides a feasible solution for the quantitative assessment of the NVC function.

Absolute Zeta fountions for Zeta functions of quantum cellular automata

Our previous work delt with the zeta function for the interacting particle system (IPS) including quantum cellular automaton (QCA) as a typical model in the study of ``IPS/Zeta Correspondence". On the other hand, the absolute zeta function is a zeta function over $\mathbb{F}_1$ defined by a function satisfying an absolute automorphy. This paper proves that a new zeta function given by QCA is an absolute automorphic form of weight depending on the size of the configuration space. As an example, we calculate an absolute zeta function for a tensor-type QCA, and show that it is expressed as the multiple gamma function. In addition, we obtain its functional equation by the multiple sine function.

Monte-Carlo based vertical underwater optical communication performance analysis with chlorophyll depth profiles

Although underwater wireless optical communication (UWOC) has the advantages of high speed, low latency, and high confidentiality, the transmission of light in water will be affected by the absorption and scattering of particles, which will lead to the aggravation of channel path loss as well as channel pulse spreading, finally causing false codes. Therefore, how to analyze the channel impulse response (CIR) effectively is a key task in channel modeling. In this paper, we consider a two-way underwater vertical line-of-sight (LOS) communication system model, based on the inherent optical property (IOP) model of chlorophyll, using the Kopelevich phase function containing water depth information, the CIR curves under different water types and transceiver configurations are plotted using the Monte-Carlo Simulation (MCS). The obtained simulation results are fitted with the double gamma function (DGF) model and the Gaussian model, respectively. The Gaussian model exhibits better properties than the DGF model in each water condition. Based on the closed-form expression of the CIR obtained from the Gaussian model, we solve for the bit error rate (BER) and 3-dB bandwidth of the system under different settings. The conclusions obtained can be used for the design and optimization of underwater vertical channels.

Hemodynamic response function description in patients with glioma

IntroductionFunctional magnetic resonance imaging is a powerful tool that has provided many insights into cognitive sciences. Yet, as its analysis is mostly based on the knowledge of an a priori canonical hemodynamic response function (HRF), its reliability in patients’ applications has been questioned. There have been reports of neurovascular uncoupling in patients with glioma, but no specific description of the Hemodynamic Response Function (HRF) in glioma has been reported so far. The aim of this work is to describe the HRF in patients with glioma. MethodsForty patients were included. MR images were acquired on a 1.5T scanner. Activated clusters were identified using a fuzzy general linear model; HRFs were adjusted with a double-gamma function. Analyses were undertaken considering the tumor grade, age, sex, tumor location, and activated location. ResultsDifferences are found in the occipital, limbic, insular, and sub-lobar areas, but not in the frontal, temporal, and parietal lobes. The presence of a glioma slows the time-to-peak and onset times by 5.2 and 3.8 % respectively; high-grade gliomas present 8.1 % smaller HRF widths than low-grade gliomas. Discussion and conclusionThere is significant HRF variation due to the presence of glioma, but the magnitudes of the observed differences are small. Most processing pipelines should be robust enough for this magnitude of variation and little if any impact should be visible on functional maps. The differences that have been observed in the literature between functional mapping obtained with magnetic resonance vs. that obtained with direct electrostimulation during awake surgery are more probably due to the intrinsic difference in the mapping process: fMRI mapping detects all recruited areas while intra-surgical mapping indicates only the areas indispensable for the realization of a certain task. Surgical mapping might not be the gold standard to use when trying to validate the fMRI mapping process.

On the Barnes double gamma function

We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function . Second, we derive an algorithm for numerically computing the double gamma function and present its complete asymptotic expansion as . Third, we derive some new properties, including a new product identity and new representations of the gamma modular forms and .

Underwater VLC channel estimation based on Kalman filtering for direct current optical- and asymmetrically clipping optical- orthogonal frequency division multiplexing techniques

In this paper, we suppose an underwater optical wireless communication (UOWC) system, where the communication utilizes visible light communication for its advantages as wide spectrum, high data rate and high accuracy. The novelty of this paper is focused on improving the channel estimation between transmitter and receiver, where using Kalman Filter (KF) for channel estimation in UOWCs achieves the best results as compared to other traditional channel estimation methods. The scenario of this paper is summarized in transmitting data from transmitter to receiver via underwater harbor and coastal channels. Two channel models are utilized: weighted double gamma functions (WDGF) and a combination of exponential and arbitrary power function (CEAPF). The modulation technique used is optical orthogonal frequency division multiplexing with two kinds: direct current optical orthogonal frequency division multiplexing (DCO-OFDM) and asymmetrically clipping optical orthogonal frequency division multiplexing (ACO-OFDM). Three different techniques are used for channel estimation: Least Square (LS), minimum mean square error (MMSE), and KF. The simulation results reveal that the ACO-OFDM modulation technique with CEAPF channel modeling using KF achieves the lowest bit error rate (BER) compared to other channel estimation methods. The improvement percentage at BER = 10−1 is 13.3% for ACO-OFDM over DCO-OFDM with CEAPF in coastal water and is is 9.3% for WDGF. This indicates that CEAPF performs about 4% better than WDGF for ACO-OFDM than DCO-OFDM in terms of channel estimation.

Optimized deep learning/kalman filter-based underwater localization in VLC systems

Several applications depend on the localization technique in underwater visible light communication (UVLC) systems, as military, petroleum, and diving fields. Recent research aims to develop the localization system by different methods to obtain the optimum position of the receiver. In this paper, we use Kalman Filter (KF) algorithm with average Received Signal Strength (RSS) technique using optimization. Optimized Deep Learning Models (DLMs) are utilized to improve the system performance, including such as ResNet50V2, InceptionResNetV2, SSD, and RetinaNet. Two channel modeling Weighted Double Gamma Function (WDGF) with a Combination Exponential Arbitrary Power Function (CEAPF) are used for sea water to enhance the UVLC localization system. The obtained results show that using CEAPF channel modeling with ResNetV2 strategy achieves the best accuracy of the localization for different methods. Also, the ResNetV2 outperforms other strategies for using RSS average technique. The RSS with KF and DLM achieves a higher accuracy with ResNetV2 than InceptionResNetV2, RetinaNet and SSD. Using WDGF achieves accuracy less than that in CEAPF where for using KF with average RSS method. Applying the RSS with KF with CEAPF channel modeling improves the performance than using WDGF. We use an automatic hyper-parameter (HP) approach to the Bayesian optimization models ResNet50V2, InceptionResNetV2, SSD, and RetinaNet. The ResNet50V2 based on average RSS technique hybrid with KF in CEAPF channel model achieves 99.99% accuracy, 99.99% area under the curve (AUC), 99.98% precision, 99.89% F1-score, 0.099 RMSE and 0.43 s testing time.

Deep learning/Kalman filter-based underwater localization in VLC systems

There is a huge importance for the localization system in underwater visible light communication (VLC) systems as in petroleum, military and diving fields. To enhance the localization system, we use the Kalman filter (KF) algorithm with average received signal strength (RSS) method to obtain the nearest estimated positions. In this paper, two channel modeling weighted double Gamma functions (WDGF) are applied and a combination exponential arbitrary power function (CEAPF) for enhancing localization in VLC underwater systems. Using the proposed KF enhances the localization by ~ 60% as compared to the than average RSS technique for WDGF channel modeling and ~ 78% for the CEAPF channel modeling. Based on the estimate of received signal strength (RSS) by deep learning models (DLMs), underwater localization utilizing VLC is introduced. Our proposed framework is categorized into two phases. First, data collection is collected based on MATLAB software. Second, the training and testing of DLMs, SSD, RetinaNet, ResNet50V2 and InceptionResNetV2 techniques are applied. The channel gains are the DLMs’ input data set, while the DLMs’ output is the RSS intensity technique coordinates for each detector. The DLMs are then developed and trained using Python software. The ResNet50V2 based on average RSS technique hybrid with KF in CEAPF channel model achieves 99.98% accuracy, 99.97% area under the curve, 98.99% precision, 98.88% F1-score, 0.101 RMSE and 0.32 s testing time.

New asymptotic expansions and Padé approximants related to the triple gamma function

In this work, our main focus is to establish asymptotic expansions for the triple gamma function in terms of the triple Bernoulli polynomials. As application, an asymptotic expansion for hyperfactorial function is also obtained. Furthermore, using these asymptotic expansions, Padé approximants related to the triple gamma function are derived as a consequence. The results obtained are new, and their importance is demonstrated by deducing several interesting remarks and corollaries.

Measurement Based Non-Line-of-Sight Vehicular Visible Light Communication Channel Characterization

Vehicular visible light communication (V-VLC) aims to provide secure complementary vehicle-to-everything-communications (V2X) to increase road safety and traffic efficiency. V-VLC provides directional transmissions, mainly enabling line-of-sight (LoS) communications. However, reflections due to nearby objects enable non-line-of-sight (NLoS) transmissions, extending the usage scenarios beyond LoS. In this paper, we propose wide-band measurement based NLoS channel characterization, and evaluate the performance of direct current biased optical orthogonal frequency division multiplexing (DCO-OFDM) V-VLC scheme for NLoS channel. We propose a distance based NLoS V-VLC channel path loss model considering reflection surface characteristics and NLoS V-VLC channel impulse response (CIR) incorporating the temporal broadening effect due to vehicle reflections through weighted double gamma function. The proposed path loss model yields higher accuracy up to 14 dB when compared to single order reflection model whereas CIR model estimates the full width at half maximum up to 2 ns accuracy. We further demonstrate that the target bit-error-rate of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-3}$</tex-math></inline-formula> can be achieved up to 7.86 m, 9.79 m, and 17.62 m distances for black, orange and white vehicle reflection induced measured NLoS V-VLC channels for DCO-OFDM transmissions.

On values of the higher derivatives of the Barnes zeta function at non-positive integers

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s \zeta_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbf Q(x)$, where $\zeta_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-positive integers for higher order derivatives of the Barnes zeta function in the case that $x$ is a positive rational number, involving the generalized Stieltjes constants and the values at positive integers of the Riemann zeta function. Our formulae also makes it possible to calculate an approximation in the case that $w_1, \ldots, w_N$ and $x$ are positive real numbers.

Differential Algebraicity of the Multiple Elliptic Gamma Function for a Rational Period

Differential Algebraicity of the Multiple Elliptic Gamma Function for a Rational Period

Ratios of Entire Functions and Generalized Stieltjes Functions

Monotonicity properties of the ratio $$\begin{aligned} \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, \end{aligned}$$where f is an entire function are investigated. Earlier results for Euler’s gamma function and other entire functions of genus 1 are generalised to entire functions of genus p with negative zeros. Derivatives of order comparable to p of the expression above are related to generalised Stieltjes functions of order \(p+1\). Our results are applied to the Barnes multiple gamma functions. We also show how recent results on the behaviour of Euler’s gamma function on vertical lines can be sharpened and generalised to functions of higher genus. Finally a connection to the so-called Prouhet-Tarry-Escott problem is described.

Autoencoder based underwater wireless optical communication with high data rate.

Underwater wireless optical communication (UWOC) has great potential to provide higher data rates and lower time delay communication compared to radio frequency and acoustic counterparts. However, UWOC systems with wide bandwidths are subject to photon absorption and scattering, which result in severe energy loss for optical beams and inter-symbol interference. To overcome these issues, this Letter interprets the UWOC system as an autoencoder (AE), named UWOC-AE, which takes advantage of the double Gamma function approximating channel impulse response of underwater optical links to learn the channel characteristics. Thus, within the AE framework, the encoder and decoder can be optimized jointly. Experiments indicate that the proposed UWOC-AE can achieve superior performance with high data rates compared to existing techniques.

Nielsen's beta function and some infinitely divisible distributions

AbstractWe show that a large collection of special functions, in particular Nielsen's beta function, are generalized Stieltjes functions of order 2, and therefore logarithmically completely monotonic. This includes the Laplace transform of functions of the form , where f is itself the Laplace transform of a sum of dilations and translations of periodic functions. Our methods are also applied to ratios of Gamma functions, and to the remainders in asymptotic expansions of the double Gamma function of Barnes.

A Quantized Riemann–Hilbert Problem in Donaldson–Thomas Theory

Abstract We introduce Riemann–Hilbert problems determined by the refined Donaldson–Thomas theory. They involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a quantum torus algebra. We study the simplest case in detail and use the Barnes double gamma function to construct a solution.

Inequalities Involving $q$-Analogue of Multiple Psi Functions

Logarithmic derivative of the multiple gamma function is known as the multiple psi function. In this work q-analogue of multiple psi functions of order n have been considered. Subadditive, superadditive and convexity properties of higher order derivatives of these functions are derived. Some related inequalities for these functions and their ratios are also obtained.

The Bayesian Cut.

An important task in the analysis of graphs is separating nodes into densely connected groups with little interaction between each other. Prominent methods here include flow based graph cutting procedures as well as statistical network modeling approaches. However, adequately accounting for this, the so-called community structure, in complex networks remains a major challenge. We present a novel generic Bayesian probabilistic model for graph cutting in which we derive an analytical solution to the marginalization of nuisance parameters under constraints enforcing community structure. As a part of the solution a large scale approximation for integrals involving multiple incomplete gamma functions is derived. Our multiple cluster solution presents a generic tool for Bayesian inference on Poisson weighted graphs across different domains. Applied on three real world social networks as well as three image segmentation problems our approach shows on par or better performance to existing spectral graph cutting and community detection methods, while learning the underlying parameter space. The developed procedure provides a principled statistical framework for graph cutting and the Bayesian Cut source code provided enables easy adoption of the procedure as an alternative to existing graph cutting methods.

On certain zeta integral: Transformation formula

We introduce an “L-function” L built up from the integral representation of the Barnes' multiple zeta function ζ. Unlike the latter, L is defined on a domain equipped with a non-trivial action of a group G. Although these two functions differ from each other, we can use L to study ζ. In fact, the transformation formula for L under G-transformations provides us with a new perspective on the special values of both ζ and its s-derivative.In particular, we obtain Kronecker limit formulas for ζ when restricted to points fixed by elements of G. As an illustration of this principle, we evaluate certain generalized Lambert series at roots of unity, establishing pertinent algebraicity results. Also, we express the Barnes' multiple gamma function at roots of unity as a certain infinite product.It should be mentioned that this work also considers twisted versions of ζ.