Symmetry

Research

14 pages, 232 KiB

Open AccessArticle

On Certain Fourth-Order Linear Recursive Sequences

by Gül Karadeniz-Gözeri, Selin Sarı and Pınar Akgül

Symmetry 2025, 17(1), 41; https://doi.org/10.3390/sym17010041 - 28 Dec 2024

Cited by 1 | Viewed by 677

In this paper, we introduce a fourth-order linear recursive sequence that is related to the concept of subbalancing numbers. This sequence is constructed by using the third balancing number in the Diophantine equation of subbalancing numbers and is called the sequence of [...] Read more.

In this paper, we introduce a fourth-order linear recursive sequence that is related to the concept of subbalancing numbers. This sequence is constructed by using the third balancing number in the Diophantine equation of subbalancing numbers and is called the sequence of

B_{3}

-Lucas subbalancing numbers. Motivated by the results for

b_{3}

-Lucas subbalancing numbers, we obtain several algebraic identities regarding

B_{3}

-Lucas subbalancing numbers. Furthermore, we derive some algebraic relations between

B_{3}

-Lucas subbalancing numbers and some other integer sequences. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

21 pages, 1786 KiB

Open AccessArticle

Reliability Analysis for Degradation-Shock Processes with State-Varying Degradation Patterns Using Approximate Bayesian Computation (ABC) for Parameter Estimation

by Isyaku Muhammad, Mustapha Muhammad, Baohua Wang, Wang Chen, Badamasi Abba and Mustapha Mukhtar Usman

Symmetry 2024, 16(10), 1364; https://doi.org/10.3390/sym16101364 - 14 Oct 2024

Cited by 2 | Viewed by 3522

Abstract

The degradation of products is an integral part of their life-cycle, often following predictable trajectories. However, sudden, unexpected events, termed ’shocks’, can substantially alter these degradation paths. Shocks can significantly influence the pace of degradation, leading to accelerated system failure. Moreover, they may [...] Read more.

The degradation of products is an integral part of their life-cycle, often following predictable trajectories. However, sudden, unexpected events, termed ’shocks’, can substantially alter these degradation paths. Shocks can significantly influence the pace of degradation, leading to accelerated system failure. Moreover, they may initiate changes in degradation patterns, transitioning from linear to non-linear or random trajectories. To address this challenge, we present a novel multi-state reliability model for competing failure processes that account for degradation-shock dependencies by considering the state-varying degradation pattern. The degradation process is divided into s-states, with each state treated according to its pattern based on the time-transform Wiener process. The reliability function is derived based on soft failure caused by continuous degradation involving the s-states, the sudden increase in degradation caused by random shocks, and hard failure due to some shock processes. Additionally, we performed a sensitivity analysis to determine which parameters have the most significant impact on product reliability. Due to the complexity of the likelihood function, we adopted the ABC method to estimate the model parameters. A simulation study and a practical application with micro-electro-mechanical systems (MEMS) degradation results are used to demonstrate the efficiency and effectiveness of the proposed approach. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

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19 pages, 6053 KiB

Open AccessArticle

Assorted Spatial Optical Dynamics of a Generalized Fractional Quadruple Nematic Liquid Crystal System in Non-Local Media

by Mohammad A. Al Zubi, Kallekh Afef and Emad A. Az-Zo’bi

Symmetry 2024, 16(6), 778; https://doi.org/10.3390/sym16060778 - 20 Jun 2024

Viewed by 935

Abstract

Nematicons upgrade the recognition of light localization in the reorientation of non-local media. the current research employs a powerful integral scheme using a different procedure, namely, the modified simple equation method (MSEM), to analyze nematicons in liquid crystals from the controlling model. The [...] Read more.

Nematicons upgrade the recognition of light localization in the reorientation of non-local media. the current research employs a powerful integral scheme using a different procedure, namely, the modified simple equation method (MSEM), to analyze nematicons in liquid crystals from the controlling model. The expanded MSEM is investigated to enlarge the applicability of the standard one. The suggested expansion depends on merging the MSEM and the ansatz method. The new generalized nonlinear n-times quadruple power law is included. With the aid of the symbolic computational package Mathematica, new explicit complex hyperbolic, periodic, and more exact spatial soliton solutions are derived. Moreover, the related existence constraints are obtained. To show the dynamical properties of some of the obtained nematicons, three-dimensional profiles with corresponding contours are depicted with the choice of appropriate values of arbitrary parameters. The fractional impacts in various applicable senses are analyzed to investigate the generality of the considered model. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

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14 pages, 323 KiB

Open AccessArticle

Number of Volatility Regimes in the Muscat Securities Market Index in Oman Using Markov-Switching GARCH Models

by Brahim Benaid, Iman Al Hasani and Mhamed Eddahbi

Symmetry 2024, 16(5), 569; https://doi.org/10.3390/sym16050569 - 6 May 2024

Cited by 1 | Viewed by 1335

Abstract

The predominant approach for studying volatility is through various GARCH specifications, which are widely utilized in model-based analyses. This study focuses on assessing the predictive performance of specific GARCH models, particularly the Markov-Switching GARCH (MS-GARCH). The primary objective is to determine the optimal [...] Read more.

The predominant approach for studying volatility is through various GARCH specifications, which are widely utilized in model-based analyses. This study focuses on assessing the predictive performance of specific GARCH models, particularly the Markov-Switching GARCH (MS-GARCH). The primary objective is to determine the optimal number of regimes within the MS-GARCH framework that effectively captures the conditional variance of the Muscat Securities Market Index (MSMI). To achieve this, we employ the Akaike Information Criterion (AIC) to compare different MS-GARCH models, estimated via Maximum Likelihood Estimation (MLE). Our findings indicate that the chosen models consistently exhibit at least two regimes across various GARCH specifications. Furthermore, a validation using the Value at Risk (VaR) confirms the accuracy of volatility forecasts generated by the selected models. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

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15 pages, 287 KiB

Open AccessArticle

Two-Dimensional Quaternion Fourier Transform Method in Probability Modeling

by Nurwahidah Nurwahidah, Mawardi Bahri and Amran Rahim

Symmetry 2024, 16(3), 257; https://doi.org/10.3390/sym16030257 - 20 Feb 2024

Cited by 3 | Viewed by 1553

Abstract

The Fourier transform plays a crucial role in statistics, applied mathematics, and engineering sciences. In this study, we give a definition of the two-dimensional quaternion Fourier transform, which is an extension of the two-dimensional Fourier transform. We present a new convolution theorem including [...] Read more.

The Fourier transform plays a crucial role in statistics, applied mathematics, and engineering sciences. In this study, we give a definition of the two-dimensional quaternion Fourier transform, which is an extension of the two-dimensional Fourier transform. We present a new convolution theorem including this transformation. We study the characteristic function in the setting of quaternion algebra and obtain the essential properties. Based on this, we seek the expected value, variance, covariance, and their basic relations to the two-dimensional quaternion Fourier transform. We illustrate the results by giving examples to see how the obtained results differ from the classical case. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

22 pages, 349 KiB

Open AccessArticle

Actuarial Valuation and Hedging of Life Insurance Liabilities in the Presence of Stochastic Mortality Risk under the Locally Risk-Minimizing Hedging Approach

by Mohamed El Farissi, Mhamed Eddahbi and Ali Goumar

Symmetry 2024, 16(2), 165; https://doi.org/10.3390/sym16020165 - 31 Jan 2024

Viewed by 1555

Abstract

The paper examines the valuation and hedging of life insurance obligations in the presence of mortality risk using the local risk-minimizing hedging approach. Roughly speaking, it is assumed that the lifetime of policyholders in an insurance portfolio is modeled by a point process [...] Read more.

The paper examines the valuation and hedging of life insurance obligations in the presence of mortality risk using the local risk-minimizing hedging approach. Roughly speaking, it is assumed that the lifetime of policyholders in an insurance portfolio is modeled by a point process whose stochastic intensity is controlled by a diffusion process. The stock price process is assumed to be a regime-switching Lévy process with non-zero regime-switching drift, where the parameters are assumed to depend on the economic states. Using the Föllmer–Schweizer decomposition, the main valuation and hedging results for a conditional payment process are determined. Some specific situations have been considered in which the local risk-minimizing strategies for a stream of liability payments or a unit-linked contract are presented. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

13 pages, 7975 KiB

Open AccessArticle

Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory

by Diego Caratelli, Pierpaolo Natalini and Paolo Emilio Ricci

Symmetry 2023, 15(10), 1900; https://doi.org/10.3390/sym15101900 - 10 Oct 2023

Cited by 4 | Viewed by 1516

Abstract

Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers [...] Read more.

Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

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15 pages, 4440 KiB

Open AccessArticle

Monetary Datum Fractal Analysis Using the Time Series Theory

by Maria-Alexandra Paun, Vladimir-Alexandru Paun and Viorel-Puiu Paun

Symmetry 2023, 15(10), 1896; https://doi.org/10.3390/sym15101896 - 10 Oct 2023

Cited by 3 | Viewed by 1565

Abstract

The paper will allow the interpretation of exchange rate fluctuations for several international currencies, the EUR—European currency and the CHF—Swiss Franc, respectively. The fractal dimension versus box-counting dimension, together with the fractal dimension versus log scale for CHF and EUR, respectively, are thoroughly [...] Read more.

The paper will allow the interpretation of exchange rate fluctuations for several international currencies, the EUR—European currency and the CHF—Swiss Franc, respectively. The fractal dimension versus box-counting dimension, together with the fractal dimension versus log scale for CHF and EUR, respectively, are thoroughly presented. The exchange rate time series for CHF and EUR during March–June 2022 were also analyzed. The Hurst exponent H was numerically evaluated. By the application of directional symmetry (DS) statistics, the model efficiency in predicting the direction modification of time series has been verified. Ultimately, the prognosis of EUR-RON and CHF-RON exchange rates of the time series fluctuations for the last months (August–December) of the year 2022 was performed. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

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17 pages, 321 KiB

Open AccessArticle

On Symmetric Additive Mappings and Their Applications

by Shakir Ali, Turki Alsuraiheed, Vaishali Varshney and Indah Emilia Wijayanti

Symmetry 2023, 15(10), 1874; https://doi.org/10.3390/sym15101874 - 6 Oct 2023

Viewed by 1868

Abstract

The key motive of this paper is to study symmetric additive mappings and discuss their applications. The study of these symmetric mappings makes it possible to characterize symmetric n-derivations and describe the structure of the quotient ring

S / P

, where [...] Read more.

The key motive of this paper is to study symmetric additive mappings and discuss their applications. The study of these symmetric mappings makes it possible to characterize symmetric n-derivations and describe the structure of the quotient ring

S / P

, where

S

is any ring and

P

is a prime ideal of

S

. The symmetricity of additive mappings allows us to transfer ring theory results to functional analyses, particularly to

C^{*}

-algebras. Precisely, we describe the structures of

C^{*}

-algebras via symmetric additive mappings. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

10 pages, 262 KiB

Open AccessArticle

New Oscillation Theorems for Second-Order Superlinear Neutral Differential Equations with Variable Damping Terms

by Munirah Aldiaiji, Belgees Qaraad, Loredana Florentina Iambor and Elmetwally M. Elabbasy

Symmetry 2023, 15(9), 1630; https://doi.org/10.3390/sym15091630 - 24 Aug 2023

Cited by 8 | Viewed by 1107

Abstract

In this paper, we investigate the oscillatory behavior of solutions of second-order differential equations with a superlinear neutral term and a damping term in noncanonical forms. We do not place any additional conditions on the damping term. Therefore, our results apply to new [...] Read more.

In this paper, we investigate the oscillatory behavior of solutions of second-order differential equations with a superlinear neutral term and a damping term in noncanonical forms. We do not place any additional conditions on the damping term. Therefore, our results apply to new classes of equations that were not covered by the previous criteria. According to symmetry between the negative and positive solutions of the studied equation, we present some new criteria that ensure the oscillation of all solutions by using the Riccati transformation and comparison method. We provide an example to illustrate our main results. Full article

(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)

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