Classical Calculus Research Articles

On q-Hermite–Hadamard–Mercer and midpoint-Mercer inequalities for general convex functions with their computational analysis

This paper establishes some new inequalities of Hermite–Hadamard–Mercer type for [Formula: see text]-convex functions in the framework of [Formula: see text]-calculus and classical calculus. Some new [Formula: see text]-midpoint-Mercer type inequalities for the [Formula: see text]-differentiable [Formula: see text]-convex functions are also proved. Moreover, the computational analysis of newly established inequalities for convex and [Formula: see text]-convex functions are given to prove that the bounds of this paper are better than those of the existing ones. It is also shown with mathematical examples that the newly established inequalities are valid for [Formula: see text]-convex functions in [Formula: see text]-calculus.

Sheffer-Dunkl Sequences Via Umbral Calculus in the Dunkl Context

Umbral calculus refers to a series of techniques that can be used to prove some polynomial formulas. Nowadays, it mostly involves the study of Sheffer sequences. In this paper, we focus on a generalization of umbral calculus in a Dunkl context (that we call Dunkl-umbral calculus). Here, the operators of classical umbral calculus are naturally replaced by operators of the Dunkl theory on the real line. In this context we define for the first time Sheffer-Dunkl sequences, {sn,α(x)}n=0∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{s_{n,\\alpha }(x)\\}_{n=0}^\\infty $$\\end{document}, and provide some properties and examples. We also connect, via Dunkl-umbral calculus, properties of some polynomials in a Dunkl sense that have appeared in the literature in the recent years, like Bernoulli-Dunkl or Euler-Dunkl polynomials.

Brownian particle diffusion in generalized polynomial shear flows.

This study presents a mathematical framework for calculating the diffusion of Brownian particles in generalized shear flows. By solving the Langevin equationsusing stochastic instead of classical calculus, we propose a mathematical formulation that resolves the particle mean-square displacement (MSD) at all timescales for any two-dimensional parallel shear flow described by a polynomial velocity profile. We show that at long timescales, the polynomial order of time of the particle MSD is n+2, where n is the polynomial order of the transverse coordinate of the velocity profile. We generalize the method to resolve particle diffusion in any polynomial shear flow at all timescales, including the order of particle relaxation timescale, which is unresolved in current theories. Particle diffusion at all timescales is then studied for the cases of Couette and plane Poiseuille flows and a polynomial approximation of a hyperbolic tangent flow while neglecting the boundary effects. We observe three main stages of particle diffusion along the timeline for Couette and plane Poiseuille flows and four main stages for hyperbolic tangent flow. The particle MSD is distinctly different across these stages due to different dominant physical mechanisms. Thus, higher temporal and spatial resolution for diffusion processes in shear flows may be realized, suggesting a more accurate analytical approach for the diffusion of Brownian particles.

Sustaining quantitatively-grounded meanings for definite integrals in high school calculus

ABSTRACT Although the importance of developing students’ quantitatively-grounded meanings for integrals is widely acknowledged, there is limited research into how this can be accomplished in classroom settings. To address this limitation, I conducted a classroom design study in which I analyzed the collective development of classroom mathematical practices for establishing quantitatively-grounded meanings for integrals by a Year 12 calculus class. The findings suggest that the class established ways of approximating the value of definite integrals and integrals with a variable upper bound by progressing through the six layers of the Adding Up Pieces framework in ways that were largely similar to previous findings from studies conducted in clinical settings. The findings from this study also extend previous research by documenting the reasoning that students used to partition the domain and calculate target quantities using the trapezoidal rule, as well as the difficulty they encountered in developing the accumulation function.

Understanding Russia’s war against Ukraine: Political, eschatological and cataclysmic dimensions

ABSTRACT We apply Anatol Rapoport’s typology of war to reconstruct how Russian research communities explain war against Ukraine. We argue that the initial three-dimensional framework can be divided into two explanatory models: military-political and eschatological-cataclysmic. The first model explains the war with reference to classical calculus about military threats and national interests. The second model interprets war as an act of creating a new world order and as a struggle existential for Russia. These two models are not mutually exclusive, yet they illuminate evolution of meanings given for the war during the first two years of Russia’s large-scale invasion against Ukraine.

Semi-classical Pseudo-differential Operators on ħZn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar \\mathbb {Z}^n$$\\end{document} and Applications

In this paper we consider the semiclassical version of pseudo-differential operators on the lattice space ħZn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar {{\\mathbb {Z}}^{n}}$$\\end{document}. The current work is an extension of the previous work (Botchway et al. in J Funct Anal 278(11):108473, 33, 2020) and agrees with it in the limit of the parameter ħ→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar \\rightarrow 1$$\\end{document}. The various representations of the operators will be studied as well as the composition, transpose, adjoint and the link between ellipticity and parametrix of operators. We also give the conditions for the ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^p$$\\end{document}, weighted ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2$$\\end{document} boundedness and ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^p$$\\end{document} compactness of operators. We investigate the relation between the classical and semi-classical quantization in the spirit of Ruzhansky and Turunen (Pseudo-differential operators and symmetries. Pseudo-differential operators, vol 2. Theory and Applications, Birkhäuser, Basel, 2010; J Fourier Anal Appl 16(6):943–982, 2010) RTspsJFAA and employ its applications to Schatten–von Neumann classes on ℓ2(ħZn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2( \\hbar \\mathbb {Z}^n)$$\\end{document}. We establish Gårding and sharp Gårding inequalities, with an application to the well-posedness of parabolic equations on the lattice ħZn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar \\mathbb {Z}^n$$\\end{document}. Finally we verify that in the limiting case where ħ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar \\rightarrow 0$$\\end{document} the semi-classical calculus of pseudo-differential operators recovers the classical Euclidean calculus, but with a twist.

Erroneous Applications of Fractional Calculus: The Catenary as a Prototype

In this work, we study the equation of the catenary curve in the context of the Caputo derivative. We solve this equation and compare the solution with real physical models. From the experiments, we find that the best approximation is achieved in the classical case. Therefore, introducing a fractional parameter arbitrarily can be detrimental. However, we observe that, when adding a certain weight to the chain, fractional calculus produces better results than classical calculus for modeling the minimum height.

General fractional calculus with respect to exponential function used in fundamental viscoelastic models

A type of general fractional calculus with respect to another function is introduced in this paper. The kernel function within the fractional derivative and integral is specified as the Exponential function in a typical way. Based on the excellent property of the newly defined general fractional calculus, we proposed two viscoelastic models in the fractional manner compared with the classical calculus.

Notion of quadratic variation in Banach spaces

This article introduces the notion of global quadratic variation which is based on a notion weaker than the usual tensor topology appearing in the literature of stochastic processes in Banach separable spaces. It is based on convergence in the weak-star topology of approximating sequences. This calculus pursues in the infinite dimensional space the stochastic calculus via regularization introduced by Russo and Vallois for real valued processes and developed in Banach space by Di Girolami and Russo. This article in particular focuses on the relation with classical calculus in order to compare our global quadratic variation with the classical existing notions of quadratic variations in the literature.

Sequent Calculi for First-order ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document}

Strict-Tolerant Logic (ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document}) underpins naïve theories of truth and vagueness (respectively including a fully disquotational truth predicate and an unrestricted tolerance principle) without jettisoning any classically valid laws. The classical sequent calculus without Cut is sometimes advocated as an appropriate proof-theoretic presentation of ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document}. Unfortunately, there is only a partial correspondence between its derivability relation and the relation of local metainferential ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document}-validity – these relations coincide only upon the addition of elimination rules and only within the propositional fragment of the calculus, due to the non-invertibility of the quantifier rules. In this paper, we present two calculi for first-order ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document} with an eye to recapturing this correspondence in full. The first calculus is close in spirit to the Epsilon calculus. The other calculus includes rules for the discharge of sequent-assumptions; moreover, it is normalisable and admits interpolation.

Integrating classical and fractional calculus rheological models in developing hydroxyapatite-enhanced hydrogels

This study presents a novel method for comprehending the rheological behavior of biomaterials utilized in bone regeneration. The focus is on gelatin, alginate, and hydroxyapatite nanoparticle composites to enhance their mechanical properties and osteoconductive potential. Traditional rheological models are insufficient for accurately characterizing the behavior of these composites due to their complexity and heterogeneity. To address this issue, we utilized fractional calculus rheological models, such as the Scott-Blair, Fractional Kelvin-Voigt, Fractional Maxwell, and Fractional Kelvin-Zener models, to accurately represent the viscoelastic properties of the hydrogels. Our findings demonstrate that the fractional calculus approach is superior to classical models in describing the intricate, time-dependent behaviors of the hydrogel-hydroxyapatite composites. Furthermore, the addition of hydroxyapatite not only improves the mechanical strength of hydrogels but also enhances their bioactivity. These findings demonstrate the potential of these composites in bone tissue engineering applications. The study highlights the usefulness of fractional calculus in biomaterials science, providing new insights into the design and optimization of hydrogel-based scaffolds for regenerative medicine.

Primitives of continuous functions via continuous piecewise linear functions

In classical calculus textbooks, the existence of primitive functions of continuous functions is proved by using Riemann integrals. Recently, Patrik Lundström gave a proof via polynomials, based on the Weierstrass approximation theorem. In this note, it is shown that the proof will be easy by using continuous piecewise linear functions.

James reduced product schemes and double quasisymmetric functions

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that QSym manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product JCP∞. In recent work, we used this viewpoint to develop topologically-motivated bases of QSym and initiate a Schubert calculus for JCP∞ in both cohomology and K-theory.Here, we study the torus-equivariant cohomology of JCP∞. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson-type rule for the structure coefficients of this basis.Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex (quasi-)projective variety also carries the structure of a (quasi-)projective variety.

New error bounds for Newton’s formula associated with tempered fractional integrals

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson’s second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature.

A New Generalized Definition of Fractal–Fractional Derivative with Some Applications

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function’s derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow’s rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition.

About Logically Probable Sentences

The starting point of this paper is the empirically determined ability to reason in natural language by employing probable sentences. A sentence is understood to be logically probable if its schema, expressed as a formula in the language of classical propositional calculus, takes the logical value of truth for the majority of Boolean valuations, i.e., as a logically probable formula. Then, the formal system P is developed to encode the set of these logically probable formulas. Based on natural semantics, a strong completeness theorem for P is proved. Alternative notions of consequence for logically probable sentences are also considered.

Attack principles in sequent-based argumentation theory

Abstract Attack principles have been introduced in semi-abstract argumentation frameworks and, in the present work, we interpret them in sequent-based argumentation frameworks. Thus, we investigate the role of minimality and consistency of the support set of an argument. Through the notion of preservation of strength, we introduce a formal criterion to sort out the attack principles; isolate the more “acceptable” ones, i.e. those easier to justify; and recover a new argumentative semantics for the non-classical logic that arises from dropping the rules $(\neg , r)$, $(\land , r)$ and $(\supset , l)$ from Gentzen’s classical sequent calculus for classical logic $\textsf{LK}$.

Class Satisfaction with Non-Face-To-Face Calculus Classes Centered on Video Lectures and Real-Time Video Lectures

This study attempted to derive implications related to class operation to improve the quality of non-face-to-face online classes in the post-COVID-19 era by analyzing college students' class satisfaction with non-face-to-face online Calculus classes. On this purpose, a study was conducted based on the results of lecture evaluation of Calculus opened in the first semester of 2020 and the first semester of 2021. In the first semester of 2020, video lectures were conducted, and in the first semester of 2021, real-time lectures were conducted using Zoom. The method of the study was compared and analyzed using the results of lecture evaluation conducted in each semester. In 2020, because it was a video lecture, I could hardly communicate and give feedback, so the lecture evaluation was worse than other items, but in 2021, it was conducted real time non-face-to-face, but the evaluation of the same item improved a lot. Since it is mathematics, not any other subject, it was found that communication and feedback with students can give students quality of class and good learning results.

Are Ancient Logics Explosive?

The twentieth-century logical mainstream, derived from works by Łukasiewicz and Scholz, pictures the history of logic for the most part as the prehistory of Boolean–Fregean mathematical logic. Particularly, with respect to classical propositional calculus, the Stoic logic has been pictured as an early stage of it and Aristotle's or the Peripatetics' logic as a theory that assumes it. Although it was not emphasised, it follows that the ancient logics contain the principle of explosion. In the endmost quarter of the twentieth century, a competitive view began to spread to the effect that the ancient systems of logic were paraconsistent or relevantistic. In the twenty-first century, the latter view prevails and has every chance of becoming a new orthodoxy. It is claimed that although in Łukasiewicz's argument for the classicality of ancient logics, there are gaps, it may be demonstrated that the ancient logics contain the principle of explosion.

Student experience using synchronous and asynchronous instruction in mathematics classes

This paper aims to study the effects that asynchronous and synchronous instructional methods have on student perceptions and attitudes towards online education. We analyze both qualitative and quantitative survey responses from 496 students in three large (greater than 200 enrolled students) first-year calculus classes at the University of Guelph. The survey results show significant differences among the three online instructional formats in students’ attitudes toward asking questions, the similarity of each instructional modality to face-to-face delivery, technology issues encountered, students’ perceived ability to keep up with course content, how collaborative the course felt, and student preferences for mode of delivery. There were no significant differences reported in relation to time-management skills or communication with instructors. Qualitative analysis of student comments also revealed perceived strengths and weaknesses of each mode of instruction. The results are used to make suggestions for improving student experience in online, post-secondary mathematical teaching.