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Quantization For The Mixtures Of Uniform Distributions On Connected And Disconnected Line Segments, Asha Barua, Gustavo Fernandez, Ashley Gomez, Olga Lopez, Mrinal Kanti Roychowdhury Sep 2025

Quantization For The Mixtures Of Uniform Distributions On Connected And Disconnected Line Segments, Asha Barua, Gustavo Fernandez, Ashley Gomez, Olga Lopez, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we have studied various mixed distributions generated by two uniform distributions: first, where the supports are two connected line segments, and second, where the supports are two disconnected line segments. For these mixed distributions, we have determined the optimal sets of n-means and the corresponding nth quantization errors for all positive integers n. The methods developed in this paper can be applied more generally to investigate optimal quantization for any mixed distribution P:=pP1+(1−p)P2, where P1 and P2 are arbitrary probability distributions supported on either connected or disconnected line segments, and (p,1−p) is any probability …

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How Many Symmetries Of The Regular N-Gon Are Even?, Thi Mai Khoi Ha Sep 2025

How Many Symmetries Of The Regular N-Gon Are Even?, Thi Mai Khoi Ha

Rose-Hulman Undergraduate Mathematics Journal

One of the simplest classes of finite groups used as a source of counterexamples in a first course of modern algebra is the class of finite dihedral groups. Among the subgroups of dihedral group, finding subgroups of index 2 is of interest in part because these subgroups are normal subgroups. In this article, we use the representations of the symmetries of the dihedral groups as permutations of the vertices and determine concretely all its subgroups of index 2. Under this representation or embedding, the article determines the intersection of the dihedral group with the corresponding alternating groups when they are …

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Incremental Increases Between Successive Integers When Raised To The Nth Power, Sutton J. Olesen Sep 2025

Incremental Increases Between Successive Integers When Raised To The Nth Power, Sutton J. Olesen

Rose-Hulman Undergraduate Mathematics Journal

For thousands of years, the beautiful field of number theory has captivated mathematicians with its elegant simplicity. Positive integers continue to reveal properties and relationships that are a joy to uncover, and in this paper, we investigate a pattern involving exponents and factorials while exploring some common notations in the field of number theory. Combinatorics, the field dealing with the mathematics of counting and arranging, also holds a presence in this paper. Pascal’s Triangle–the foundation of binomial expressions, also comes into play due to its tight relationship with combinatorics. Pascal’s Identity, the property that builds the triangle, becomes very useful …

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New Fiber And Graph Combinations Of Convex Bodies, Steven Hoehner, Sudan Xing Sep 2025

New Fiber And Graph Combinations Of Convex Bodies, Steven Hoehner, Sudan Xing

Mathematical Sciences Faculty Publications and Presentations

Three new combinations of convex bodies are introduced and studied: the fiber, chord, and graph combinations. These combinations are defined in terms of the fibers and graphs of pairs of convex bodies, and each operation generalizes the classical Steiner symmetral, albeit in different ways. For the fiber and chord combinations, we derive Brunn-Minkowski-type inequalities and the corresponding Minkowski's first inequalities. We also prove that the general affine surface areas are concave (respectively, convex) with respect to the graph sum, thereby generalizing fundamental results of Ye (Indiana Univ. Math. J. 14 (2014), 1-19) on the monotonicity of the general affine surface …

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Improved Berezin Radius Inequalities For Functional Hilbert Space Operators, Mojtaba Bakherad, Ulaş Yamanci, Mohammad W. Alomari Sep 2025

Improved Berezin Radius Inequalities For Functional Hilbert Space Operators, Mojtaba Bakherad, Ulaş Yamanci, Mohammad W. Alomari

Turkish Journal of Mathematics

The Berezin number of an operator S is defined by ber(S) = supλ∈Θ |Ŝ(λ)| = supλ∈Θ |⟨Sĸ̂λ, ĸ̂λ⟩|, where Ŝ(λ) = ⟨Sĸ̂λ, ĸ̂λ⟩ (λ ∈ Θ) is the Berezin transform of an operator S on the functional Hilbert space ℋ = ℋ(Θ) and ĸ̂λ = ĸλ / ‖ĸλ‖ is the normalized reproducing kernel of ℋ. In this paper, we show an improvement of the classical Cauchy–Schwarz inequality involving the Berezin number. Moreover, some Berezin numbers inequalities are proven for invertible operators.

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On Multiplicative Order Convergence And Multiplicative Order Continuous Operators, Hüma Gürkök, Bahri̇ Turan Sep 2025

On Multiplicative Order Convergence And Multiplicative Order Continuous Operators, Hüma Gürkök, Bahri̇ Turan

Turkish Journal of Mathematics

Let A and B be two f-algebras. This paper establishes the theoretical framework for multiplicative order convergence, clarifying its definition, properties, and relations to other convergence types. This includes a detailed discussion on the conditions under which multiplicative order convergence and order convergence, as well as unbounded order convergence, mutually imply each other. We establish an extension theorem for multiplicative order continuous operators, similar to Veksler’s theorem for order continuous operators. As a result of this theorem, we derive the order structure of the space of multiplicative order continuous operators. We show that the space Lmo(A, …

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The Action Of The Superposition Operator On Weighted Banach Sequence Spaces LP,Σ, Nermin Okičić, Fehim Dedagić Sep 2025

The Action Of The Superposition Operator On Weighted Banach Sequence Spaces LP,Σ, Nermin Okičić, Fehim Dedagić

Turkish Journal of Mathematics

We consider a nonlinear superposition operator on weighted Banach spaces lp,σ, where σ is the weighted function with the property that (∀n ∈ ℕ) σ(n) ≥ 1. We present the necessary and sufficient conditions for the action of the superposition operator from lp,σ to lq,τ, 1 ≤ p, q ≤ ∞, as well as the L-characteristic of the action of the considered operator.

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Nonoscillatory Solutions Of Third-Order Nonlinear Dynamic Equations: Existence And Nonexistence, Özkan Öztürk, Elvan Akin, Nesli̇han Nesli̇ye Pelen Sep 2025

Nonoscillatory Solutions Of Third-Order Nonlinear Dynamic Equations: Existence And Nonexistence, Özkan Öztürk, Elvan Akin, Nesli̇han Nesli̇ye Pelen

Turkish Journal of Mathematics

This paper investigates a third-order nonlinear dynamic equation on arbitrary time scales, a nonempty closed subset of the real numbers, unifying continuous and discrete analyses. We study the qualitative behavior of nonoscillatory solutions and their quasi-derivatives, focusing on their limiting behaviors. The existence of such solutions are established using improper integral criteria and Schauder’s and Knaster’s fixed point theorems. In addition, we establish the criteria for the nonexistence of nonoscillatory solutions. Furthermore, we prove the existence of Kneser-type solutions of the corresponding linear dynamic equation on isolated time scales, addressing an open problem in the literature. Several examples of theoretical …

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Majorization Properties For Analytic Functions With Positive Real Part, Oya Mert Sep 2025

Majorization Properties For Analytic Functions With Positive Real Part, Oya Mert

Turkish Journal of Mathematics

The majorization feature is important when investigating analytical functions’ geometric properties. The present work focuses on the majorization property for a general Ma-Minda type function class S*φ(γ, λ) of starlike functions of complex order, which is described with the Ruscheweyh differential operator. In the context of the main theorem, consequences are presented for various special functions φ having a positive real part. Furthermore, the Ruscheweyh differential operator extends a representation formula and investigates an upper-bound formula. Examples of upper bounds for analytic functions are also presented.

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Structural Stability For Damped Nonlinear Wave Equations, Sevda Isayeva, Varga Kalantarov Sep 2025

Structural Stability For Damped Nonlinear Wave Equations, Sevda Isayeva, Varga Kalantarov

Turkish Journal of Mathematics

We study the problem of continuous dependence on parameters for solutions to initial-boundary value problems for nonlinear strongly damped wave equations. Specifically, we consider three classes of equations: a strongly damped wave equation with a nonlinear damping term; a strongly damped wave equation incorporating both a nonlinear damping term and a nonlinear source term; and a strongly damped wave equation with a nonlinear source term and a linear weak damping term. For each class, we establish structural stability results, showing that the solutions depend continuously on the relevant parameters.

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Application On Convolution For Fuzzy Differential Subordination Of Meromorphic Functions, Kholood Mohamed Alsager, Fethi̇ye Müge Sakar, Alina Alb Lupas, Sheza Mohamed El-Deeb Sep 2025

Application On Convolution For Fuzzy Differential Subordination Of Meromorphic Functions, Kholood Mohamed Alsager, Fethi̇ye Müge Sakar, Alina Alb Lupas, Sheza Mohamed El-Deeb

Turkish Journal of Mathematics

This paper investigates fuzzy differential subordinations involving meromorphic functions defined in the punctured unit disk. A novel operator based on the Hadamard product (also known as convolution) is introduced to establish new results in the theory of fuzzy differential subordination. This operator provides a generalized framework that extends classical analytic methods to the fuzzy setting, offering new insights into the geometric behavior of meromorphic functions under fuzzy conditions. Several sufficient conditions for fuzzy differential subordination are derived, and their connections to existing results are discussed. The findings contribute to the growing body of literature at the intersection of classical geometric …

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(Α, Β)-Normal Differential Operators For First Order In Hilbert Spaces Of Vector-Functions, Zameddin I. Ismailov, Pembe İpek Al Sep 2025

(Α, Β)-Normal Differential Operators For First Order In Hilbert Spaces Of Vector-Functions, Zameddin I. Ismailov, Pembe İpek Al

Turkish Journal of Mathematics

In this article, the relationships between (α, β)-normality properties of the extensions of the minimal operator generated by the first-order differential operator expression in the Hilbert spaces of vector-functions in a finite interval and the variable operator coefficients of this differential operator expression are investigated. Then, the general form of all (α, β)-normal extensions of the minimal operator in these spaces is determined under the constraints obtained for the coefficient operators.

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On Certain Subclasses Of Starlike And Convex Functions With Respect To T-Symmetric Points, Osman Altintaş, Ni̇zami̇ Mustafa Sep 2025

On Certain Subclasses Of Starlike And Convex Functions With Respect To T-Symmetric Points, Osman Altintaş, Ni̇zami̇ Mustafa

Turkish Journal of Mathematics

In this study, we defined for the first time a new subclass of stalike and convex functions in the open unit disk of the complex plane and we examined coefficiend upper bound estimates for this class.

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The Generalization Of Rectifying Helices, Bülent Altunkaya Sep 2025

The Generalization Of Rectifying Helices, Bülent Altunkaya

Turkish Journal of Mathematics

This work establishes connections between different classes of arbitrary speed rectifying helices. We further analyze how the involute of a space curve can exhibit spherical helicity and how the evolute can inherit rectifying properties, thereby providing an alternative characterization of these helices. These relationships, along with their interplay with involutes and evolutes, are supported by illustrative examples that offer deeper insights into their geometric properties.

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Applications Of A New Linear Operator Involving Dziok-Srivastava Operator And Bessel Function Of The First Kind, Daniela Andrada Bardac-Vlada, Georgia Irina Oros Sep 2025

Applications Of A New Linear Operator Involving Dziok-Srivastava Operator And Bessel Function Of The First Kind, Daniela Andrada Bardac-Vlada, Georgia Irina Oros

Turkish Journal of Mathematics

This study introduces a new linear operator constructed from the well-known Dziok–Srivastava linear hypergeometric operator, the Bessel function of the first kind, and convolution. Applying the new operator, a class of functions with positive real part is defined and investigated concerning inclusion relations, the necessary and sufficient conditions for functions to belong to this class, as well as starlikeness and convexity conditions. The properties of the new class to be convex of a certain order, close-to-convex and starlike of a certain order are highlighted and examples are also included to prove that the new outcome is applicable. The methods employed …

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The Geometry Of Semislant Submersions Whose Total Manifolds Are Locally Conformal Kähler Manifolds, Çağrihan Çi̇men, Hakan Mete Taştan Sep 2025

The Geometry Of Semislant Submersions Whose Total Manifolds Are Locally Conformal Kähler Manifolds, Çağrihan Çi̇men, Hakan Mete Taştan

Turkish Journal of Mathematics

We study semislant submersions whose total manifolds are locally conformal Kähler manifolds. We provide conditions for the total geodesicity and integrability of the distributions involved in the definition. Next, we establish a condition under which such a submersion is a Clairaut submersion. Finally, we obtain some optimal inequalities for a semislant submersion whose total manifold is Hopf space form.

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Wave Intensity Analysis With Exercise Identifies Impairments In Pulmonary Hypertension, Christopher G. Lechuga, Farhan Raza, Mitchel J. Colebank, Claudia E. Korcarz, Jens C. Eickhoff, Naomi C. Chesler Sep 2025

Wave Intensity Analysis With Exercise Identifies Impairments In Pulmonary Hypertension, Christopher G. Lechuga, Farhan Raza, Mitchel J. Colebank, Claudia E. Korcarz, Jens C. Eickhoff, Naomi C. Chesler

Faculty Publications

Wave intensity analysis provides a novel approach to understanding the dynamic interactions between the right ventricle and pulmonary vasculature, particularly in pulmonary hypertension, a condition characterized by elevated pulmonary arterial pressures and vascular remodeling. This prospective study used wave intensity analysis to evaluate right ventricular and pulmonary vascular mechanics in 22 participants with pulmonary hypertension (including precapillary, isolated postcapillary, and combined pre/postcapillary pulmonary hypertension), and three without pulmonary hypertension. Forward and backward compression and decompression waves were quantified at rest and during incremental exercise (25, 50, and 75 W). Relationships between metrics of wave intensity analysis, hemodynamics, right ventricular function, …

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Room-Temperature Alignment-Free Magnetometry With Boron Vacancies In Hot-Pressed Hexagonal Boron Nitride, Raul Coto Cabrera Sep 2025

Room-Temperature Alignment-Free Magnetometry With Boron Vacancies In Hot-Pressed Hexagonal Boron Nitride, Raul Coto Cabrera

Mathematics Colloquium Series

Magnetic field sensing is essential for applications in communication, environmental monitoring, and biomedical diagnostics. Quantum magnetic field sensors based on solid-state spin defects, such as NV centers in diamond, typically require precise alignment between the external magnetic field and the defect’s spin quantization axis to achieve reliable and directional sensing. This alignment constraint imposes challenges for device integration, especially in scalable or fielddeployable applications. Here, we demonstrate room-temperature optically detected magnetic resonance (ODMR) from negatively charged boron vacancies (VB-) in commercially available hot-pressed polycrystalline hexagonal boron nitride (hBN). Owing to the random orientation of grains in the material, the system …

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Solitary Waves For Power Degenerate Non Linear Schro ̈Dinger Equation And Fourth Order Wave Equation., Vishnu Iyer Sep 2025

Solitary Waves For Power Degenerate Non Linear Schro ̈Dinger Equation And Fourth Order Wave Equation., Vishnu Iyer

All ETDs from UAB

In this dissertation we prove the existence and stability of solitary waves for the power degenerate non linear Schr ̈odinger equation (2.1.1) and a fourth order wave equation (3.1.1). Construction of the waves are variational in nature. For the NLS we consider a semilinear Schr ̈odinger equation, driven by the power degenerate second order differential operator ∇ · (|x|2a∇), a ∈ (0, 1). We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli- Kohn-Nirenberg’s inequality. Depending on the parameter a and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away …

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Calibrating A Holling-Tanner Model With Applications To S&P500 Forecasting, Jonathon Knowles Sep 2025

Calibrating A Holling-Tanner Model With Applications To S&P500 Forecasting, Jonathon Knowles

All ETDs from UAB

The primary goal of this thesis is to improve upon existing dynamical system models in economics. Existing models utilizing Lotka-Volterra dynamics display a tendency to suffer from unrealistic forecasts dominated by exponential growth and decay. Additionally, current inverse methods are only applicable in differential equation systems with linear coefficients. In this body of work, we propose novel methods for handling these challenges.

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Analytical And Numerical Approaches To Parameter Estimation In Damped Oscillatory Systems, Gracie Crooks, F. Ayça Çetinkaya Sep 2025

Analytical And Numerical Approaches To Parameter Estimation In Damped Oscillatory Systems, Gracie Crooks, F. Ayça Çetinkaya

CODEE Journal

We investigate the inverse problem of identifying damping and stiffness parameters in one-dimensional damped oscillatory systems governed by second-order differential equations. Focusing on mass–spring–damper models, we analyze the qualitative behavior of solutions across underdamped, critically damped, and overdamped regimes, and derive explicit conditions for parameter recovery based on time-domain observations such as equilibrium crossings and turnaround points. Two numerical estimation methods are developed and compared: a finite-difference least-squares approach based on central difference approximations, and a finite element formulation derived from a variational framework using piecewise linear basis functions. Computational experiments using synthetic data assess the accuracy, stability, and noise …

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How Euler Could Have Done It: Euler And A Direct Proof For The Functional Equation For The Riemann Zeta-Function, Alexander Aycock Sep 2025

How Euler Could Have Done It: Euler And A Direct Proof For The Functional Equation For The Riemann Zeta-Function, Alexander Aycock

Euleriana

We present a chain of argumentation for a direct proof of the functional equation for the Riemann ζ–function that Euler could have presented.

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Euler’S Original Derivation Of Elastica Equation, Shigeki Matsutani Sep 2025

Euler’S Original Derivation Of Elastica Equation, Shigeki Matsutani

Euleriana

Euler derived the differential equations of elastica by the variational method in 1744, but his original derivation has never been properly interpreted or explained in terms of modern mathematics. We elaborate Euler's original derivation of elastica and show that Euler used Noether's theorem concerning the translational symmetry of elastica, although Noether published her theorem in 1918. It is also shown that his equation is essentially the static modified KdV equation which is obtained by the isometric and isoenergy conditions, known as the Goldstein-Petrich scheme.

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Early Theories On Fluid Resistance And Translation Of Euler’S “Dilucidationes De Resistentia Fluidorum”, Sylvio R. Bistafa Sep 2025

Early Theories On Fluid Resistance And Translation Of Euler’S “Dilucidationes De Resistentia Fluidorum”, Sylvio R. Bistafa

Euleriana

In 1763, Euler published Dilucidationes de resistentia fluidorum (Explanations on the resistance of fluids), a memoir that challenges the fluid resistance theories proposed by Isaac Newton and d’Alembert. Euler's work explores the resistance experienced by solid bodies moving through fluids, critiquing both Newton's "common rule" and d’Alembert's paradox, which predicted zero resistance for non-viscous fluids. Euler's treatise is divided into two parts: the first focuses on the mathematical modeling of fluid flow patterns, while the second addresses the calculation of fluid resistance on surfaces. Despite significant advancements, Euler's work remains constrained by the limitations of non-viscous fluid assumptions, ultimately grappling …

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Assistance For The Calculation Of Sines (Translation Of E246), Julian Schennach Sep 2025

Assistance For The Calculation Of Sines (Translation Of E246), Julian Schennach

Euleriana

Paralleling his famous relation eiφ = cos(φ) + i sin(φ), Euler establishes the equality (cos φ + i sin φ)n = (cos nφ + i sin nφ). He uses it to comprehensively derive trigonometric identities that convert arbitrary powers of sines and cosines of an angle (and products thereof) into sums of sines and cosines of multiples of that angle. Some negative and fractional powers are shown to yield infinite series. Euler further describes a general method to evaluate various infinite series involving weighted trigonometric functions. These results foreshadow Fourier series. As Euler points out, the scope of …

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On Amicable Numbers, Jonathan David Evans Sep 2025

On Amicable Numbers, Jonathan David Evans

Euleriana

This is an English translation of Euler's 1750 paper "De numeris amicabilibus" (E152), the most substantial of his three works with this name. In it, he expounds at great length the ad hoc methods he has developed to search for pairs of amicable numbers, concluding with a list of around 60 new pairs.

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A Reader's Guide To Daniel Bernoulli's "Recurrent Series", Stacy Langton Sep 2025

A Reader's Guide To Daniel Bernoulli's "Recurrent Series", Stacy Langton

Euleriana

This is a commentary and reader’s guide to Daniel Bernoulli’s article “Observations concerning recurrent series”. It is intended to help the modern reader understand what Bernoulli is doing in that article.

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Daniel Bernoulli's "Observations Concerning Recurrent Series", Stacy Langton Sep 2025

Daniel Bernoulli's "Observations Concerning Recurrent Series", Stacy Langton

Euleriana

Daniel Bernoulli’s article "Observations concerning Recurrrent Series" is one of the gems of the mathematical literature. It describes a numerical method for solving polynomial equations. This method, now known as “Bernoulli’s method”, is still in use today. The basis of the method is the study of certain sequences of numbers that satisfy what we now call linear recurrence relations. Bernoulli’s article shows how to find solutions of these recurrence relations. A nice application is a now well-known formula for the Fibonacci numbers. Bernoulli also uses his theory of recurrence relations to give a proof of an early form of De …

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Comprehensive Conversations, Christopher Goff, Erik Tou Sep 2025

Comprehensive Conversations, Christopher Goff, Erik Tou

Euleriana

An introduction to the contents in Volume 5 (Issue 1) of Euleriana.

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Generalized Transversality Conditions For Fuzzy Quantum-Symmetric Variational Problems Via Granular Approach, Martin Bohner, Ewa Girejko, Agnieszka B. Malinowska, Linh Nguyen, Baruch Schneider, Tri Truong Sep 2025

Generalized Transversality Conditions For Fuzzy Quantum-Symmetric Variational Problems Via Granular Approach, Martin Bohner, Ewa Girejko, Agnieszka B. Malinowska, Linh Nguyen, Baruch Schneider, Tri Truong

Mathematics and Statistics Faculty Research & Creative Works

This paper investigates fuzzy q-symmetric variational problems with natural boundary conditions. Based on the relative distance measure fuzzy arithmetic and horizontal membership functions (HMFs), we propose novel concepts of differentiability and integrability for fuzzy functions on quantum geometric subsets of real numbers. Then, fundamental foundations of q-symmetric calculus of variations based on HMFs are provided. With the help of HMFs and granular q-symmetric differentiability, we derive necessary optimality conditions for fuzzy q-symmetric variational problems that depend on free endpoints. Moreover, sufficient conditions for minimizers of q-symmetric variational problems are obtained. Some numerical examples illustrating the proposed approach are presented.

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