Logarithmic Integrals Research Articles

Tackling carbon peak and carbon neutrality challenges: A method with long-range energy alternatives planning system and Logarithmic Mean Divisia Index Integration

China aims to carbon peak emissions before 2030 and achieve carbon neutrality by 2060 in response to climate change challenges. Studying the feasibility of this goal, identifying potential gaps, and formulating corresponding emission reduction pathways are urgent issues that need to be addressed. This study creatively integrates Logarithmic Mean Divisia Index into Long-range Energy Alternatives Planning System to assess energy consumption and carbon emissions in Gansu Province from 2001 to 2060, historical and scenario analysis was conducted from a novel perspective. Results indicate that the economic development effect is the most significant driving factor for the emission growth of 131.54 million tons from 2001 to 2020, and its contribution rate is 176.31 %. The leading negative driving factor is the energy intensity effect, contributing at a rate of −62.00 %. Additionally, the energy structure and industrial structure effects also play minor negative roles. Scenario analysis suggests achieving a carbon peak by 2030 requires a 29.53 % reduction in energy intensity from the 2020 baseline, with electricity consumption must reach 46.37 % and clean energy generating 64 %. Carbon peak emissions are projected to be 220.37 million tons, with reduction of carbon emissions by 123.84 million tons by 2060.

Nonlinear step responses and how to find them

AbstractThe unit step response is a standard tool for experimental identification; its shape is equivalent to a solution of an appropriate governing equation. In this study, we investigate two variants of nonlinear first-order governing equation, $${y'=k(1-y^\beta )/\beta }$$ y ′ = k ( 1 - y β ) / β , and mainly its rescaled version $${y'=k\left( 1-y^\beta \right) }$$ y ′ = k 1 - y β . We explore a continuum of response shapes, ranging from constant zero to constant one, including standard shapes such as $$1-\exp (-t)$$ 1 - exp ( - t ) and $${\tanh (t)}$$ tanh ( t ) , but also uncommon shapes such as a part of the logarithmic integral. Two methods for estimating the parameter $$\beta $$ β are proposed: one based on analyzing the curvature of a single response and the other based on analyzing initial slopes for multiple responses, while varying the amplitude of the step. Our investigation is conducted in the context of gas permeation through a homogeneous porous medium, where nonlinear response shapes have been observed. This context allows for a meaningful interpretation of the results, consistent with expectations for specific thermodynamic processes. However, neither method is constrained by this context, and either might be considered for any first-order system that fails to have amplitude-scale invariance or for which the shapes seem to exhibit an effect of saturation.

An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuable. These methods also offer avenues for number-theoretic exploration through analytical manipulation. In this work, we introduce a novel approximation function, ϕ(n), which adds to the existing repertoire of approximation methods and provides a fresh perspective for number-theoretic studies. Deeper analytical investigation of ϕ(n) reveals modified representations of the Chebyshev function, prime number theorem, and Riemann zeta function. Computational studies indicate that the difference between ϕ(n) and the logarithmic integral at magnitudes greater than 10100 is less than 1%.

An Algorithmic Evaluation of a Family of Logarithmic Integrals and Associated Euler Sums

Numerous logarithmic integrals have been extensively documented in the literature. This paper presents an algorithmic evaluation of a specific class of these integrals. Our systematic approach, rooted in logarithmic principles, enables us to extend our findings to other cases within this family of integrals. Furthermore, we explore special cases derived from our main results, thereby enhancing the applicability and significance of our work for a wider audience of researchers.

Prime Orbit Theorems for expanding Thurston maps: Genericity of strong non-integrability condition

In the second paper [16] of this series, we obtained an analog of the prime number theorem for a class of branched covering maps on the 2-sphere S2 called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumption. More precisely, the number of primitive periodic orbits, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued Hölder continuous function on S2 satisfying the α-strong non-integrability condition, is asymptotically the same as the well-known logarithmic integral, with an exponential error bound. In this third and last paper of the series, we show that the α-strong non-integrability condition is generic in the class of α-Hölder continuous functions.

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We provide a characterization of those log-concave distributions in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^n$$\\end{document} that are contoured distributions, through the Kp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_p$$\\end{document}-bodies of the distribution, defined by Ball. Our method uses the logarithmic integral for the solution of a Bernstein type approximation problem. In the second part of the paper we state a question for contoured distributions that would provide an alternative approach to the isotropic constant problem.

Certain logarithmic integrals and associated Euler sums

We evaluate the following family of logarithmic integrals in closed form:12π∫02π|1+reit|2nlogm⁡|1+reit|dt for n∈Z⩾0 and m=0,1,2,3,4, r∈[−1,1]. In 1995, Shen conducted evaluations for specific instances where r equals −1 with n equal to 0, as well as when m assumes the values 1, 2, 3, and 4. Moreover, by employing these integrals, we derive generating functions for certain sequences that incorporate harmonic numbers and binomial coefficients. Furthermore, we assess a selection of series that encompass squared central binomial coefficients and harmonic numbers.

Memristive tonotopic mapping with volatile resistive switching memory devices

To reach the energy efficiency and the computing capability of biological neural networks, novel hardware systems and paradigms are required where the information needs to be processed in both spatial and temporal domains. Resistive switching memory (RRAM) devices appear as key enablers for the implementation of large-scale neuromorphic computing systems with high energy efficiency and extended scalability. Demonstrating a full set of spatiotemporal primitives with RRAM-based circuits remains an open challenge. By taking inspiration from the neurobiological processes in the human auditory systems, we develop neuromorphic circuits for memristive tonotopic mapping via volatile RRAM devices. Based on a generalized stochastic device-level approach, we demonstrate the main features of signal processing of cochlea, namely logarithmic integration and tonotopic mapping of signals. We also show that our tonotopic classification is suitable for speech recognition. These results support memristive devices for physical processing of temporal signals, thus paving the way for energy efficient, high density neuromorphic systems.

Logarithmic Learning Differential Convolutional Neural Network

Convolutional Neural Networks (CNNs) have revolutionized image classification through their innovative design and training methodologies in computer vision. Differential convolutional neural network with simultaneous multidimensional filter realization improved the performance of the convolutional neural network with calculation cost drawback. This paper introduces logarithmic learning integration into the differential Convolutional neural network to overcome the drawback by supplying faster error minimization and convergence. This task is done by incorporating LogRelu activation, a Logarithmic Cost Function, and unique logarithmic learning method. The effectiveness of the proposed approaches are evaluated by using various datasets and SGD/Adam optimizers. The first step is the adaptation of LogRelu activation function to convolutional and differential convolutional neural networks. The experiment results show that LogRelu integration to convolutional neural network and differential convolutional neural network yields performance improvements ranging from 1.61% to 5.44%. The same integration on ResNet-18, ResNet-34, and ResNet-50 enhances top-1 accuracy in the range of 3.07% and 9.96%. In addition to LogRelu activation function, a Logarithmic Cost Function with logarithmic learning method is also proposed and adapted to differential convolutional neural network. These improvements lead to a new differential convolutional neural network named as Logarithmic Differential Convolutional Neural Network (LDiffCNN), It consistently outperforms standard CNN by increasing the accuracy up to 3.02%. Notably, Logarithmic Differential Convolutional Neural Network demonstrates reduced training iterations up to 38% with faster convergence. The experimental results proved the efficiency of the proposed approach.

Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker Wκ,μx Function II

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function Mκ,μx. In this second part, first derivatives with respect to the parameters of the Whittaker function Wκ,μx are calculated. Using the confluent hypergeometric function, these derivatives can be expressed as infinite sums of quotients of the digamma and gamma functions. Furthermore, it is possible to obtain these parameter derivatives in terms of infinite integrals, with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions), from the integral representation of Wκ,μx. These infinite sums and integrals can be expressed in closed form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function wiκ,μx and its derivative with respect to κ, as well as some reduction formulas for the integral Whittaker functions Wiκ,μx and wiκ,μx, are calculated.

Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker Mκ,μ(x) Function I

In this paper, first derivatives of the Whittaker function Mκ,μx are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of Mκ,μx it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions). These infinite sums and integrals can be expressed in closed form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions Mκ,μx and integral Whittaker functions Miκ,μx and miκ,μx are calculated.

Binet's second formula, Hermite's generalization, and two related identities

Abstract Legendre was the first to evaluate two well-known integrals involving sines and exponentials. One of these integrals can be used to prove Binet’s second formula for the logarithm of the gamma function. Here, we show that the other integral leads to a specific case of Hermite’s generalization of Binet’s formula. From the analogs of Legendre’s integrals, with sines replaced by cosines, we obtain two integration identities involving logarithms and trigonometric functions. Using these identities, we then subsequently derive generalizations of Binet’s and Hermite’s formulas involving the integral of a complex logarithm.

Logarithmic integrals with applications to BBP and Euler-type sums

Logarithmic integrals with applications to BBP and Euler-type sums

On the average value of

AbstractWe prove that the Riemann hypothesis is equivalent to the condition $\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$ for all $x>2$ . Here, $\pi (t)$ is the prime-counting function and $\operatorname {\textrm {li}}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function $\theta (t)$ and discuss the extent to which one can make related claims unconditionally.

On the extension of Hermite–Hadamard type inequalities for coordinated convex mappings

In this paper, we obtain an important inequalities for coordinated convex functions and as a result of these inequalities we give the extension of Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral and logarithmic integral. The inequalities obtained in this study provide generalizations of some result given in earlier works.

Another Alternating Analogue of Euler’s Constant

In 2005 Jonathan Sondow found several interesting analogies between Euler’s constant γ and revealing that the latter may be regarded as the alternating analogue of Euler’s constant. In this article, we show that the constant , where stands for the logarithmic integral, also shares several properties in common with Euler’s constant and, therefore, may also be regarded as an alternating analogue of Euler’s constant.

De Branges canonical systems with finite logarithmic integral

Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We further develop this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. This result can be viewed as a spectral version of the classical Szego theorem in the theory of polynomials orthogonal on the unit circle. It extends Krein-Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.

Definite Integral of Power and Algebraic Functions in terms of the Lerch Function

Bierens de haan (1867) evaluated a definite integral involving the cotangent function and this result was also listed in Gradshteyn and Ryzhik (2007). The objective of this present note is to use this integral along with Cauchy's integral formula to derive a definite logarithmic integral in terms of the Lerch function. We will use this integral formula to produce a table of known and new results in terms of special functions and thereby expanding the list of definite integrals in both text books.

DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION

A family of generalized definite logarithmic integrals given by $$\int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i m} (\log (a)-i \log (x))^k\right)}{(x+1)^2}dx$$built over the Lerch function has its analytic properties and special values listed in explicit detail. We use the general method as given in [6] to derive this integral. We then give a number of examples that can be derived from the general integral in terms of well known functions.

Approximation of Cauchy-type singular integrals with high frequency Fourier kernel

Two types of splitting algorithms are proposed for approximation of Cauchy type singular integrals having high frequency Fourier kernel. To evaluate non-singular integrals, modified Levin collocation methods with multiquadric radial basis function and Chebyshev polynomials are proposed. In the scenario of interval splitting, a multi-resolution quadrature is used to tackle the singularity ridden kernel. While in the case of integrand splitting, the singular part integral is evaluated analytically. Logarithmic singular integrals with oscillatory kernels are transformed to Cauchy principal value integrals and computed with the new algorithms. Error analysis of the component algorithms, as well as the individual methods, is performed theoretically. Validation of accuracy and error estimates of the methods are performed numerically as well.