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This paper provides additive perturbation bounds for the generalized core inverse $ A^{d,\dag } $ A d , † of a complex matrix A. Given perturbation matrix E, we first present upper bounds for $ \frac {\|(A+E)^{d,\dag }-A^{d,\dag }\|_{2}}{\|A^{d,\dag }\|_{2}} $ ‖ (A + E) d , † − A d , † ‖ 2 ‖ A d , † ‖ 2 under the condition $ \mathcal {R}(E)\subseteq \mathcal {R}(A^k), \mathcal {N}(A^{k}A^{\dag })\subseteq \mathcal {N}(E), {\rm where}\ k=\mathrm {ind}(A) $ R (E) ⊆ R (A k) , N (A k A †) ⊆ N (E) , where k = ind (A) , and then under the condition $ (A+E)(A+E)^{d,\dag }=AA^{d,\dag } $ (A + E) (A + E) d , † = A A d , † . We then explore an integral formula for the generalized core inverse of a perturbed matrix associated with a semi-stable matrix. Furthermore, a perturbation bound for $ \|(A+E)^{d,\dag }-A^{d,\dag }\|_{F} $ ‖ (A + E) d , † − A d , † ‖ F is obtained without conditions. Sufficient conditions for the continuity of the generalized core inverse can be obtained as a result. Finally, numerical examples are presented to illustrate the derived bounds. [ABSTRACT FROM AUTHOR]

Three types of convex functions with respect to another function are introduced, and certain properties related to these functions are proved. Moreover, for each type of convexity, new inequalities of Hermite‐Hadamard type are derived. [ABSTRACT FROM AUTHOR]

The dynamics of quantum systems coupled to baths are typically studied using the Nakajima-Zwanzig memory kernel (K ) or the influence functions (I), particularly when memory effects are present. Despite their significance, formal connections between the two have not been explicitly known. We establish their connections by examining the system propagator for a N-level system linearly coupled to Gaussian baths with various types of system-bath coupling. For a certain class of problems, we devised a non-perturbative, diagrammatic approach to construct K from I for (driven) systems interacting with Gaussian baths, bypassing conventional projection-free dynamics inputs. Our work provides a way to interpret approximate path integral methods in terms of approximate memory kernels. Moreover, it offers a Hamiltonian learning procedure to extract the bath spectral density from reduced system trajectories, opening new avenues in quantum sensing and engineering. The insights we provide advance our understanding of non-Markovian dynamics and will serve as a stepping stone for future theoretical and experimental developments in this area. The Feynman-Vernon Path Integral and the Generalized Quantum Master Equation are the two main and oldest approaches to open quantum system dynamics modelling. Here, the authors discover a formal link between them, and use it to find a Hamiltonian learning method that can extract environmental spectral densities from the dynamics of the reduced system. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid load frequency control (LFC) scheme is proposed for multi-area interconnected power systems to decouple the intricate double control objectives, by dividing all subareas into the responsible areas and the free areas. The LFC in the responsible area has the function of regulating both the local frequency and the tie-line power, while the control objective of the LFC in the free area is thus simplified to regulate the local frequency only. Then, addressing the complex network coupling and uncertain dynamics, an integrated LFC controller is proposed for the free areas, which consists of two parts, namely, the coupling attenuation baseline controller and the disturbance compensation controller. The coupling attenuation baseline controller satisfying the predefined bounded L2-Gain condition is derived based on the solution to a multi-player zero-sum differential game. Additionally, a novel generalized integral observer is designed to estimate the system's integrated disturbance, and the corresponding disturbance compensation controller is derived. After that, the ultimately uniformly bounded (UUB) stability of the integrated LFC controller combining baseline controller and disturbance compensation controller is proven rigorously. Finally, the performance superiority of the proposed hybrid LFC scheme is validated by the simulations in challenging operating modes. [ABSTRACT FROM AUTHOR]

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form (M ¯ , g ¯) , where M ¯ = R × f M and g ¯ = ϵ d t 2 + f 2 (t) g M . We focus on the scalar curvature of these hypersurfaces, establishing upper and lower bounds, particularly in the case where (M ¯ , g ¯) is an Einstein manifold. These bounds facilitate the characterization of slices in GRW spacetimes. In addition, we use the vector field ∂ t and the so-called support function θ to derive generalized Minkowski-type integral formulas for compact Riemannian and spacelike hypersurfaces. These formulas are applied to establish, under certain conditions, results concerning the existence or non-existence of such compact hypersurfaces with scalar curvature, either bounded from above or below. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a new fractional extension of the interval Hermite–Hadamard ( H H ), H H –Fejér, and Pachpatte-type inequalities for left- and right-interval-valued harmonically convex mappings ( L R I V H convex mappings) with an exponential function in the kernel. We use fractional operators to develop several generalizations, capturing unique outcomes that are currently under investigation, while also introducing a new operator. Generally, we propose two methods that, in conjunction with more generalized fractional integral operators with an exponential function in the kernel, can address certain novel generalizations of increasing mappings under the assumption of L R I V convexity, yielding some noteworthy results. The results produced by applying the suggested scheme show that the computational effects are extremely accurate, flexible, efficient, and simple to implement in order to explore the path of upcoming intricate waveform and circuit theory research. [ABSTRACT FROM AUTHOR]

The study examines how generalized Ince–Gaussian beams (gIGBs) propagate through oceanic turbulence (OT) by decomposing them into a set of Laguerre–Gaussian beams. The Huygens–Fresnel diffraction integral is utilized to derive an analytical formula representing the average intensity distribution of these beams at the output surface of the OT medium. Detailed numerical results and discussions regarding both beam and medium parameters effects on gIGBs propagation in OT are provided. These findings showcase an examination of gIGBs for underwater wireless optical communication and imaging systems, employing laser light as conduit for transmitting signals. [ABSTRACT FROM AUTHOR]

In this work, novel Ostrowski-type inequalities for dissimilar function classes and generalized fractional integrals (FITs) are presented. We provide a useful identity for differentiable functions under FITs, which results in special expressions for functions whose derivatives have convex absolute values. A new condition for bounded variation functions is examined, as well as expansions to bounded and Lipschitzian derivatives. Our comprehension is improved by comparison with current findings, and recommendations for future study areas are given. [ABSTRACT FROM AUTHOR]

Hundreds of special functions have been employed in applied mathematics and computing sciences for many centuries due to their outstanding features and wide range of applications. When considering the relevance of these consequences in the evaluation of generalized integrals, applied physics, and many engineering areas, the illustration of image formulas involving one or more variable special functions is significant under various definite integrals. In this paper, it is devoted to study the various integral identities involving incomplete Fox-Wright functions and Srivastava’s polynomials. It is shown that the integrals of the Fox-Wright functions are also the Fox-Wright functions but of greater order. Due to the fact that our results are unified, a substantial number of new results can be constructed as special instances from our leading results. The results obtained in this work are general in nature and very useful in science, engineering and finance. [ABSTRACT FROM AUTHOR]

Due to the great success of hypergeometric functions, we provide the analytical solutions of certain definite logarithmic integrals and Nielsen-type integrals in terms of multi-variable Kampé de Fériet functions with suitable convergence conditions and higher harmonic sums by using series rearrangement technique and incomplete Gamma function. Further we also obtain the solution of other related logarithmic integrals in terms of generalized hypergeometric functions and Kummer’s confluent hypergeometric functions by using series rearrangement technique. The results presented in the paper and comparable outcomes are hoped to be supplied by the use of computer-aid programs, for example, Mathematica. [ABSTRACT FROM AUTHOR]

Motivated essentially by the fact that several interesting families of generalized elliptic-type integrals and many definite integrals of such families with respect to their modulus (or complementary modulus) are known to arise naturally in various physical contexts, we present a systematic account of the theory of these families of elliptic-type integrals in a unified and generalized manner. We also point out relevant connections of the general theory, which is discussed in this paper, with the results in numerous earlier publications on the theory and applications of the aforementioned families of elliptictype integrals. Several classes of hypergeometric functions of one, two and more variables are observed to emerge in our unified presentation of these families of elliptic-type integrals. [ABSTRACT FROM AUTHOR]

In this work, we presented the fuzzy generalized integral equation for population includes future surge in birthrates which is great interesting for future planning throughout the world. So, we formulated the fuzzy problem of birthrates to be investigated any surge in fuzzy birthrates. The fuzzy integral equation (FIE) included the fuzzy birthrates at time t and the fuzzy function of a girl living to age r2 and unclear role of a girl giving birth to a female kid at this age in an interval time dr2. This fuzzy integral contributed for fuzzy birthrate from woman in the suitable subinterval of the range of childbearing age r2. The homotopy analysis method (HAM) for explained the numerical fuzzy solution of periodicity in the surge of birthrate of generalized fuzzy dynamical birthrate of girl born at any as well as all the tables and figures are given in details also the behaviors of the birthrate which described the solution of fuzzy integral. [ABSTRACT FROM AUTHOR]

In the last decade, quadrotors have gained popularity among industry and academia due to their capabilities and the various applications in which they can be found. In addition to the above, because this is an underactuated system, researchers have found it a great challenge to control. Despite this, there is a wide variety of methodologies in the literature to control this type of system. Based on the above, this work proposed an alternative to trajectory tracking control for quadrotor unmanned aerial vehicles (UAV). The problem was divided into two main control loops: an outer control loop for the position coordinates, tackled through linear active disturbance rejection controllers (ADRC), and an inner control loop related to the orientation variables, addressed via robust proportional-integral-differential (PID) controllers. Furthermore, a generalized proportional integral observer (GPIO) was implemented to estimate the velocity and internal and external disturbances; therefore, the control strategy only depended on the attitude (position and orientation) quadrotor measurements. Then, the control performance was tested through numerical simulations and experimental tests, including wind disturbance inputs. [ABSTRACT FROM AUTHOR]

In this paper, fractional order integral operators are applied to the study of inequalities, which is also a research hotspot of fractional order integral operators in recent years. After introducing some basic concepts, we establish some new Volterra-Fredholm-type fractional integral inequalities. They can be used as an effective tool to study the properties of solutions of fractional differential equations and fractional integral equations. In addition, several examples are given to illustrate the effectiveness of the research results. [ABSTRACT FROM AUTHOR]

In 2015, Abdeljawad defined the conformable fractional derivative (Grunwald–Letnikov technique) to iterate the conformable fractional integral of order 0 < α ≤ 1 (Riemann approach), yielding Hadamard fractional integrals when α = 0 . The Gronwall type inequality for generalized operators unifying Riemann–Liouville and Hadamard fractional operators is obtained in this study. We use this inequality to show how the order and initial conditions affect the solution of differential equations with generalized fractional derivatives. More features for generalized fractional operators are established, as well as solutions to initial value problems in several new weighted spaces of functions. [ABSTRACT FROM AUTHOR]

This document is a summary of a special issue of the journal "Fractal & Fractional" titled "Mathematical Inequalities in Fractional Calculus and Applications." The special issue contains fifteen articles that explore mathematical inequalities involving fractional derivatives and fractional integral operators and their applications in various mathematical and related fields. The articles cover a range of topics, including convexity, quantum inequalities, fuzzy-interval-valued functions, integral operators, and symmetric derivatives. The authors of the summary express their gratitude to the contributing authors and reviewers for their valuable contributions to the special issue. [Extracted from the article]

This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j ∈ N) positively continuous and decaying functions in the finite interval a ≤ t ≤ x , the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results. [ABSTRACT FROM AUTHOR]

A compound sliding mode controller for permanent magnet synchronous motor (PMSM) based on an adaptive terminal sliding mode reaching law with generalized proportional integral observer is proposed. First, an adaptive terminal reaching law is proposed to solve the defects of large chattering in commonly used sliding mode control, which consists of a piecewise function and a terminal term of state variables. The reaching law could adjust the gain according to the distance between the system state and the equilibrium point. On the premise of reducing chattering, a higher reaching speed is guaranteed. The convergence time has an upper bound and is not concerned with the initial value of the sliding mode variable. Based on the reaching law, an adaptive sliding mode controller is established. Second, to reduce the sliding mode gain and enhance the antidisturbance ability of the PMSM, a generalized proportional integral observer is established to evaluate the perturbations in PMSM and compensate them to the speed controller in real time. Simulations and experiments verify that the new compound control method has a faster-approaching speed, smaller chattering, and stronger antidisturbance ability than other methods. [ABSTRACT FROM AUTHOR]

In this paper, we conducted a detailed investigation on the evolution of a partially coherent generalized Hermite cosh-Gaussian beam (PCGHchGB) through a gradient-index medium. Analytical formula for a PCGHchGB propagating in the gradient-index medium is obtained based on ABCD transfer matrix and generalized diffraction integral formula. We discuss the periodic changes of the corresponding beam through the gradient-index medium by performing numerical examples. Our obtained results demonstrate that the gradient-index parameter of media and the initial beam parameters, such as the coherence width, beam waist width, decentered beam parameter, and beam orders, affect the evolution characteristics on the intensity distribution of the PCGHchGB in the gradient-index medium. The results of this work could be beneficial for applications in optical communication and nonlinear optics. [ABSTRACT FROM AUTHOR]

In this paper, specific Lp estimates for generalized Marcinkiewicz operators correlated to surfaces of revolution are proved. These estimates and the extrapolation procedure of Yano are employed to confirm the Lp boundedness of the above-mentioned integrals under weaker assumptions on the singular kernels. Our findings generalize and improve several known results. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms. [ABSTRACT FROM AUTHOR]

Modified Green–Lindsay generalized thermoelasticity theory was established by Yu et al. (Meccanica 53(10):2543–2554, 2018). On the basis of this theory, transient motions remove discontinuities in displacement fields. The goal of this article is to address a dynamical problem involving finite linear mode-I cracks in an isotropic and homogeneous elastic medium in a two-dimensional infinite space using the innovative framework of modified Green–Lindsay generalized thermoelasticity theory. There is a specified temperature and stress distribution on the crack's boundary. The integral transform techniques are used to obtain the numerical values of temperature, stress, displacement and stress intensity factor for copper material. These non-dimensional physical fields are explained graphically. Specifically, the present undertaking demonstrates its utility in addressing challenges related to fracture mechanics, geophysics and mining particularly in the context of coupling thermal and mechanical fields. This concerted effort proves valuable in exploring and resolving issues within these fields. [ABSTRACT FROM AUTHOR]

Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman's integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up and down ( U · D ) relations and over newly defined class U · D -ħ-Godunova–Levin convex fuzzy-number mappings. To demonstrate the unique properties of U · D -relations, recent findings have been developed using fuzzy Aumman's, as well as various other fuzzy partial order relations that have notable deficiencies outlined in the literature. Several compelling examples were constructed to validate the derived results, and multiple notes were provided to illustrate, depending on the configuration, that this type of integral operator generalizes several previously documented conclusions. This endeavor can potentially advance mathematical theory, computational techniques, and applications across various fields. [ABSTRACT FROM AUTHOR]

Chebyshev polynomials of the first kind are applied to construct the generalized Chebyshev–Poisson integral. The optimization problem for the generalized Chebyshev–Poisson operator as a functional of a function defined on an interval is solved, and its approximate properties on Holder classes H1 are analyzed. An exact equality is obtained for the deviation of Hölder class functions from the generalized Chebyshev–Poisson integral. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study to fractional calculus for class of Ψ function. The present study is designed to study generalized fractional derivatives and find their generalized transforms called Ψ-Laplace transform and Ψ-Sumudu transform. Moreover, find the analytical solutions of some applications in physics the form of generalized fractional derivatives by transform technique. [ABSTRACT FROM AUTHOR]

This article introduces a measure of noncompactness in the generalized Morrey space. We study the applications of our new definition in investigating the conditions for the existence of solutions for systems of nonlinear integral equations. We can extend many useful theorems in L p (R N) for functions belonging to the generalized Morrey spaces. Compared to the L p (R N) spaces, the advantage of studying in the Morrey spaces is that we can research no compact support functions in our problems. Finally, significant examples are presented to show the efficiency of the main results. [ABSTRACT FROM AUTHOR]

Both theoretical and applied mathematics depend heavily on integral inequalities with generalized convexity. Because of its many applications, the theory of integral inequalities is currently one of the areas of mathematics that is evolving at the fastest pace. In this paper, based on fuzzy Aumann's integral theory, the Hermite–Hadamard's type inequalities are introduced for a newly defined class of nonconvex functions, which is known as  U · D preinvex fuzzy number-valued mappings ( U · D preinvex  F · N · V · M s) on coordinates. Some Pachpatte-type inequalities are also established for the product of two  U · D preinvex  F · N · V · M s, and some Hermite–Hadamard–Fejér-type inequalities are also acquired via fuzzy Aumann's integrals. Additionally, several new generalized inequalities are also obtained for the special situations of the parameters. Additionally, some of the interesting remarks are provided to acquire the classical and new exceptional cases that can be considered as applications of the main outcomes. Lastly, a few suggested uses for these inequalities in numerical integration are made. [ABSTRACT FROM AUTHOR]

The relationship between convexity and symmetry is widely recognized. In fuzzy theory, both concepts exhibit similar behavior. It is crucial to remember that real and interval-valued mappings are special instances of fuzzy-number-valued mappings ( F - N - V - M s ), as fuzzy theory relies on the unit interval, which is crucial to resolving problems with interval analysis and fuzzy number theory. In this paper, a new harmonic convexities class of fuzzy numbers has been introduced via up and down relation. We show several Hermite–Hadamard ( H ⋅ H ) and Fejér-type inequalities by the implementation of fuzzy Aumann integrals using the newly defined class of convexities. Some nontrivial examples are also presented to validate the main outcomes. [ABSTRACT FROM AUTHOR]

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus. [ABSTRACT FROM AUTHOR]

Based on the pseudo-order relation, we introduce the concept of left and right ℏ -preinvex interval-valued functions (LR- ℏ -PIVFs). Further, we establish the Hermite–Hadamard and Hermite–Hadamard–Fejér-type estimates for LR- ℏ -PIVFs using generalized fractional integrals. Finally, an example of interval-valued fractional integrals is provided to illustrate the validity of the results derived herein. Our results not only extend some existing inequalities for Hadamard, Riemann–Liouville, and Katugampola fractional integrals, but also provide new insights for future research on generalized convexity and IVFs, among others. [ABSTRACT FROM AUTHOR]

In this work, we establish some Parseval-Goldstein type identities and relations that include various new generalized integral transforms such as Lα,µ-transform and generalized Stieltjes transform. In addition, we evaluated improper integrals of some fundamental and special functions using our results. [ABSTRACT FROM AUTHOR]

A new fractional integral operator is used to present a generalized class of analytic functions in a complex domain. The method of definition is based on a Hadamard product of analytic function, which is called convolution product. Then we formulate a convolution integral operator acting on the subclass of normalized analytic functions. Consequently, we investigate the suggested convolution operator geometrically. Differential subordination inequalities, taking the starlike formula are given. Some consequences of well known results are illustrated. [ABSTRACT FROM AUTHOR]

In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some findings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals. [ABSTRACT FROM AUTHOR]

The current paper introduces a novel generalization of cone metric spaces called type I and type II composed cone metric spaces. Therefore, examples of a type I and type II composed cone metric space, which is not a cone metric space, are given. We establish some results of fixed point precisely about Hardy–Rogers type contraction on C2CMS and provide examples. Finally, we present an application of our results and how our results solve the Fredholm integral equation of generalizing several existing and unique fixed point theorems. [ABSTRACT FROM AUTHOR]

Three new approaches to the solution of the Bratu problem are presented. The first approach realizes the idea of successive differentiating the initial equation of this problem with expansion of the sought for function at the symmetry point of a space. The second approach is associated with the additional integration of the differential Bratu equation, and it represents a hybrid integral method. The third approach is based on the combined application of the successive differentiation of the Bratu equation, the hybrid integral method, the expansion of the sought for function at two points of the space, and an additional integral relation. The indicated approaches are characterized by the simplicity of the calculations required for their realization, and they reduce the solution of the Bratu problem to the solution of a system of linear algebraic equations. The numerical results of solving this problem demonstrate the high efficiency of the approaches proposed, providing the obtaining of the solutions whose accuracy exceeds the accuracy of the analogous solutions, obtained on the basis of the known numerical and numerical-analytical methods, by three to five orders of magnitude. This is especially true for the third approach that allows one to obtain two classical solutions of the Bratu problem fairly simply and with a very high accuracy. It is shown that the obtaining of an approximate solution of the Bratu problem with this approach calls for a small number of series terms, and the solutions obtained converge quickly and approximate the problem highly exactly. [ABSTRACT FROM AUTHOR]

In this paper, we study pursuit and evasion differential game problems of one pursuer/one evader and many pursuers/one evader, respectively, in the space R n . In both problems, we obtain sufficient conditions that guarantee the completion of a pursuit and an evasion. We construct the players' optimal strategies in both problems, and we estimate the possible distance that an evader can preserve from pursuers. Lastly, we illustrate our results via some numerical examples. [ABSTRACT FROM AUTHOR]

In this work, we characterize the boundedness of t-generalized composition operators from F (p ,   q ,   s) spaces to iterated weighted-type Banach space. We also provide estimates of the norm and the essential norm of t-generalized composition operators from F (p ,   q ,   s) spaces to iterated weighted-type Banach space. As corollaries, we obtain approximations of the essential norm of integral operators and generalized composition operators from F (p ,   q ,   s) spaces to iterated weighted-type Banach space. Moreover, we conclude our work by discussing the t-generalized composition operators and the special cases of F (p ,   q ,   s) . [ABSTRACT FROM AUTHOR]

In this paper, based on Yang's fractal theory, the Hermite–Hadamard's inequalities for generalized h -preinvex function are proved. Then, using the local fractional integral identity proposed by Sun [Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals 27(5) (2019) 1950071] as auxiliary function, some parameterized local fractional integral inequalities for generalized h -preinvex functions are established. For the special cases of the parameters, some generalized Simpson-type, midpoint-type and trapezoidal inequalities are established. Finally, some applications of these inequalities in numerical integration are proposed. [ABSTRACT FROM AUTHOR]

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $\mathrm{AC}_{\mathbb{R}}$ and $\mathrm{NC}_{\mathbb{R}}^{}$ classes for this setting. We give a theorem in the style of Immerman's theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the $\mathrm{AC}_{R}$ and $\mathrm{NC}_R^{}$ hierarchy. Those generalizations apply to the Boolean $\mathrm{AC}$ and $\mathrm{NC}$ hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains. [ABSTRACT FROM AUTHOR]

Numerous recently introduced chaotic systems exhibit straightforward algebraic representations. In this study, we explore the potential for identifying a global analytic first integral in a generalized 3-dimensional chaotic system (2). Our work involves detailing the model of a new 3-D chaotic system characterized by three Lyapunov exponents—positive, zero, and negative. We depict the phase trajectories, illustrate bifurcation patterns, and visualize Lyapunov exponent graphs. The investigation encompasses both local and global analytic first integrals for the system, providing results on the existence and non-existence of these integrals for different parameter values. Our findings reveal that the system lacks a global first integral, and the presence or absence of analytic first integrals is contingent upon specific parameter values. Additionally, we present a formal series for the system, demonstrating 3D and 2D projections of the system (2) for a given set of initial conditions achieved by selecting alternative values for parameters a, b, c, d, r and l. [ABSTRACT FROM AUTHOR]

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F ˜ λ , we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F ˜ λ. [ABSTRACT FROM AUTHOR]

Based on an extension of Riemann sums, Moore and Yang have defined an integral notion for the context of continuous inclusion monotonic interval functions in which the limits of integration are real numbers. This integral notion generalizes the usual one for real-valued functions based on Riemann sums. In this paper we extend this approach by considering intervals as limits of integration and abolishing the inclusion monotonic restriction of the integrable interval functions. Also, such a new integration notion is used to define interval probability density functions and use it in interval probability distribution functions. [ABSTRACT FROM AUTHOR]

We introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity results, while Euler calculus conveys topological information and allows for compatibility with operations on constructible functions. We conduct a systematic study of such transforms and introduce two new ones: the Euler–Fourier and Euler–Laplace transforms. We show that the first has a left inverse and that the second provides a satisfactory generalization of Govc and Hepworth's persistent magnitude to constructible sheaves, in particular to multi-parameter persistent modules. Finally, we prove index-theoretic formulae expressing a wide class of hybrid transforms as generalized Euler integral transforms. This yields expectation formulae for transforms of constructible functions associated with (sub)level-sets persistence of random Gaussian filtrations. [ABSTRACT FROM AUTHOR]

This work presents a novel investigation that utilizes the integral operator I p , λ n in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula for a generalized integral operator. Additionally, certain sandwich theorems were discovered. [ABSTRACT FROM AUTHOR]

We study generating functions of 1 4 -BPS states in N = 4 super Yang-Mills at finite N by attempting to generalize the Harish-Chandra-Itzykson-Zuber integral to multiple commuting matrices. This allows us to compute the overlaps of two or more generating functions; such calculations arise in the computation of two-point correlators in the free-field limit. We discuss the four-matrix HCIZ integral in the U(2) context and lay out a prescription for finding a more general formula for N > 2. We then discuss its connections with the restricted Schur polynomial operator basis. Our results generalize readily to arbitrary numbers of matrices, opening up the opportunity to study more generic BPS operators. [ABSTRACT FROM AUTHOR]

Some questions concerning the inversion of the classical and generalized integral Radon transforms are discussed. The main issue is to determine information about the integrand if the values of some integrals are known. A feature of this work is that a function is integrated over hyperplanes in a finite-dimensional Euclidean space and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. The independent variables describing the known integrals are fewer than those in the unknown integrand. We consider discontinuous integrands defined on specifically introduced pseudoconvex sets. A Stefan-type problem of finding discontinuity surfaces of the integrand is posed. Formulas for solving the problem under study are derived by applying special integro-differential operators to known data. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new generalized integral transform on Dzherbashian-Nersesian fractional operator and give some special cases. We make a generalization of the application of integral transformations to different fractional operators, where several previous results can be invoked from a single relation. We also use the new results obtained to solve some fractional differential equations involving the recent revival of Dzherbashian-Nersesian fractional operators. [ABSTRACT FROM AUTHOR]

Beech wood pyrolysis is capable of providing fuel as well as invaluable materials of industrial importance besides being CO2-neutral. Mechanistic insights revealed by the kinetics and thermodynamics of beech wood pyrolysis may not only lead to controlling the process but also to optimize its efficiency. However, the literature lacks in some reliable and physically meaningful mechanistic insights into beech wood pyrolysis process. Moreover, beech wood pyrolysis thermodynamics has not yet been reported. Therefore, the present paper puts forward a detailed kinetic and thermodynamic study on the beech wood pyrolysis. The beech wood pyrolysis is firstly deconvoluted into three isolated pseudo-hemicellulose, pseudo-cellulose and pseudo-lignin thermal degradation processes by an effective deconvolution function. Afterwards, generalized integral isoconversional method is applied. It suggests that the three processes follow single-step kinetics. Advanced reaction model determination methodology reveals that the thermal degradation processes of pseudo-hemicellulose, pseudo-cellulose and pseudo-lignin go to completion by respectively following, second order (RO), two-dimensional (2D) nucleation/growth and complicated diffusion mechanisms. The thermodynamics of beech wood pyrolysis puts forth interesting and important information regarding the endothermicity of the processes involved and arrangement/orientation of the activated complexes in transition state. The practical valuation of the present research is also pointed out and discussed. [ABSTRACT FROM AUTHOR]