144 results on '"fractal sets"'
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2. Generalized local fractional integral inequalities via generalized (h̃1,h̃2)-preinvexity on fractal sets.
- Author
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Al-Sa’di, Sa’ud, Bibi, Maria, Muddassar, Muhammad, and Budak, Hüseyin
- Subjects
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FRACTALS , *FRACTIONAL integrals , *INTEGRAL inequalities - Abstract
In this paper, we establish several general local fractional integral inequalities via generalized (h̃1,h̃2) preinvex mapping on fractal sets. By considering different parameter values, we develop particular applications of our result, such as midpoint-type inequality, generalized trapezoidal-type inequality, and generalized Simpson-type inequality. We present applications of the derived inequalities in numerical quadrature formulas, providing error estimates. [ABSTRACT FROM AUTHOR] more...
- Published
- 2025
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3. New Approaches to Fractal–Fractional Bullen's Inequalities Through Generalized Convexity.
- Author
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Saleh, Wedad, Boulares, Hamid, Moumen, Abdelkader, Albala, Hussien, and Meftah, Badreddine
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INTEGRAL calculus , *GENERALIZED integrals , *FRACTALS , *INTEGRALS , *FRACTIONAL integrals - Abstract
This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals and fractal calculus, but also offering a refinement of the well-known Bullen-type inequality. We further explore the connections between generalized convexity and fractal–fractional integrals, showing how the concept of generalized convexity enables the establishment of error bounds for fractal–fractional integrals involving lower-order derivatives, with an emphasis on their applications in various fields. The findings expand the current understanding of fractal–fractional inequalities and offer new insights into the use of local fractional derivatives for analyzing functions with fractional-order properties. [ABSTRACT FROM AUTHOR] more...
- Published
- 2025
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4. On Fractal–Fractional Simpson-Type Inequalities: New Insights and Refinements of Classical Results.
- Author
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Alsharari, Fahad, Fakhfakh, Raouf, and Lakhdari, Abdelghani
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GENERALIZED integrals , *FRACTALS , *SENSES , *LITERATURE - Abstract
In this paper, we introduce a novel fractal–fractional identity, from which we derive new Simpson-type inequalities for functions whose first-order local fractional derivative exhibits generalized s-convexity in the second sense. This work introduces an approach that uses the first-order local fractional derivative, enabling the treatment of functions with lower regularity requirements compared to earlier studies. Additionally, we present two distinct methodological frameworks, one of which achieves greater precision by refining classical outcomes in the existing literature. The paper concludes with several practical applications that demonstrate the utility of our results. [ABSTRACT FROM AUTHOR] more...
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- 2024
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5. METRIC RESULTS FOR THE EVENTUALLY ALWAYS HITTING POINTS AND LEVEL SETS IN SUBSHIFT WITH SPECIFICATION.
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WANG, BO and LI, BING
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SYMBOLIC dynamics , *FRACTALS , *HAUSDORFF measures , *POINT set theory ,FRACTAL dimensions - Abstract
We study the set of eventually always hitting points for symbolic dynamics with specification. The measure and Hausdorff dimension of such fractal set are obtained. Moreover, we establish the stronger metric results by introducing a new quantity L N (ω) which describes the maximal length of string of zeros of the prefix among the first N iterations of ω in symbolic space. The Hausdorff dimensions of the level sets for this quantity are also completely determined. [ABSTRACT FROM AUTHOR] more...
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- 2024
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6. MODIFIED SHRINKING TARGET PROBLEM FOR MATRIX TRANSFORMATIONS OF TORI.
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YUAN, NA and WANG, SHUAILING
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FRACTALS , *TORUS ,FRACTAL dimensions - Abstract
In this paper, we calculate the Hausdorff dimension of the fractal set x ∈ d : ∏ 1 ≤ i ≤ d | T β i n (x i) − x i | < ψ (n) for infinitely many n ∈ ℕ , where T β i is the standard β i -transformation with β i > 1 , ψ is a positive function on ℕ and | ⋅ | is the usual metric on the torus . Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let T be a d × d non-singular matrix with real coefficients. Then, T determines a self-map of the d -dimensional torus d : = ℝ d / ℤ d . For any 1 ≤ i ≤ d , let ψ i be a positive function on ℕ and Ψ (n) : = (ψ 1 (n) , ... , ψ d (n)) with n ∈ ℕ. We obtain the Hausdorff dimension of the fractal set { x ∈ d : T n (x) ∈ L (f n (x) , Ψ (n)) for infinitely many n ∈ ℕ } , where L (f n (x , Ψ (n))) is a hyperrectangle and { f n } n ≥ 1 is a sequence of Lipschitz vector-valued functions on d with a uniform Lipschitz constant. [ABSTRACT FROM AUTHOR] more...
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- 2024
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7. Synthesis of Multifractals by Brownian Dynamics of a Point in a Field of N Central Forces
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Grabar, Ivan G., Kubrak, Yuri O., Skiadas, Christos H., editor, and Dimotikalis, Yiannis, editor
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- 2024
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8. MILNE-TYPE INTEGRAL INEQUALITIES FOR MODIFIED pℎ, 𝑚q-CONVEX FUNCTIONS ON FRACTAL SETS
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J. E. Napoles, P. M. Guzman, and B. Bayraktar
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local fractional derivatives ,local fractional integrals ,fractal sets ,milne inequality ,convex modified functions of second type ,power mean inequality ,holder inequality ,Mathematics ,QA1-939 - Abstract
In the article, new versions of integral inequalities of Milne type are derived for pℎ, 𝑚q-convex modified functions of the second type on fractal sets. Based on a new generalized local fractional weighted integral operator, an identity is established as the foundation for subsequently obtained inequalities. Throughout our study, we obtained certain results known in the literature, which include particular cases of our findings. more...
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- 2024
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9. New fractal–fractional Simpson estimates for twice differentiable functions with applications.
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Butt, Saad Ihsan, Khan, Ahmad, and Tipurić-Spužević, Sanja
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DIFFERENTIABLE functions , *INTEGRAL inequalities , *FRACTALS , *INTEGRAL operators , *RANDOM variables , *WAVE equation - Abstract
In this article, we establish a new auxiliary identity on fractal sets for twice local differentiable function involving extended fractal integral operators. Testing this identity together with generalized fractal Hölder’s and Power-mean integral inequalities, we develop some new fractal–fractional Simpson’s type inequalities. Furthermore, we use modified Yang Hölder’s and Power-mean inequality to create new fractal estimates. We also give comparison analysis of bounds and show how the modified form of Yang Hölder’s and Powermean integral inequalities can result in improved lower upper bounds. We also provide concrete examples to examine the validity of obtain results numerically and also justify them by 2D and 3D graphical analysis. As implementations, we operate our findings to get new applications in form of ζ-type special means, moment of random variables and wave equations. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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10. HARMONIC ANALYSIS ON THE SPACE OF M-POSITIVE VECTORS.
- Author
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Farkov, Yu. and Skopina, M.
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VECTOR spaces , *ALGEBRAIC spaces , *FOURIER transforms , *FRACTALS , *HARMONIC analysis (Mathematics) - Abstract
Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2, which is used in the Walsh analysis. The role of harmonics is played by some analogues of the classical Walsh functions. The concept of Fourier transform is introduced, and the Poisson summation formula, Plancherel theorem, Vilenkin-Chrestenson formulas and so on are proved. A kind of analogue of the Schwartz class is studied. This class consists of functions such that both the function itself and its Fourier transform have compact support. [ABSTRACT FROM AUTHOR] more...
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- 2024
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11. FRACTAL ORACLE NUMBERS.
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RATSABY, JOEL
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COMPLEX numbers , *RATIONAL numbers , *REAL numbers , *INFORMATION theory , *TURING machines , *KOLMOGOROV complexity , *FRACTALS - Abstract
Consider orbits (z , κ) of the fractal iterator f κ (z) : = z 2 + κ , κ ∈ ℂ , that start at initial points z ∈ K ̂ κ (m) ⊂ ℂ ̂ , where ℂ ̂ is the set of all rational complex numbers (their real and imaginary parts are rational) and K ̂ κ (m) consists of all such z whose complexity does not exceed some complexity parameter value m (the complexity of z is defined as the number of bits that suffice to describe the real and imaginary parts of z in lowest form). The set K ̂ κ (m) is a bounded-complexity approximation of the filled Julia set K κ . We present a new perspective on fractals based on an analogy with Chaitin's algorithmic information theory, where a rational complex number z is the analog of a program p , an iterator f κ is analogous to a universal Turing machine U which executes program p , and an unbounded orbit (z , κ) is analogous to an execution of a program p on U that halts. We define a real number Υ κ which resembles Chaitin's Ω number, where, instead of being based on all programs p whose execution on U halts, it is based on all rational complex numbers z whose orbits under f κ are unbounded. Hence, similar to Chaitin's Ω number, Υ κ acts as a theoretical limit or a "fractal oracle number" that provides an arbitrarily accurate complexity-based approximation of the filled Julia set K κ . We present a procedure that, when given m and κ , it uses Υ κ to generate K ̂ κ (m) . Several numerical examples of sets that estimate K ̂ κ (m) are presented. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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12. Study of Critical Behavior of Dynamic Systems.
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Sel'chenkova, N. I. and Uchaev, A. Ya.
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DYNAMICAL systems , *FRACTALS , *CONDENSED matter , *MULTIFRACTALS , *NONLINEAR systems , *HAMILTONIAN systems ,FRACTAL dimensions - Abstract
The phenomenon of dynamic chaos and self-organization is related to stochastic instability and critical behavior of nonlinear physical systems of various nature. These processes occur, e.g., in the dynamic failure phenomenon of condensed media and fully developed turbulence. In such nonlinear systems, a cascade of dynamic dissipative structure with a fractal structure arises. Multifractal properties are characterized by the spectral function f(dfi), determined by the behavior of the number of elements l required to cover the fractal sets with the similar probabilities Pi~ . In both cases, the transit of systems from one scale-time level to the next one via a cascade of bifurcation and quantitative process characteristics at developed stages do not depend on the Hamiltonian of interatomic interaction. The fractal organization of processes at all scale-time levels testifies to the process similarity and allows us to attribute these processes to a single class of universality. The evolution of the system as a whole is determined not by the interatomic interaction Hamiltonians, but by the incipient cascades of dissipative structures. The close values of fractal dimensions on all the studied scales, which characterize the failure structure, allow us to study the formation of micro-flaws of failure and macro-failure as the scale edges of the spectrum of a single process that has common order parameters. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
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13. Lossy compression of general random variables.
- Author
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Riegler, Erwin, Koliander, Günther, and Bölcskei, Helmut
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FRACTALS , *RATE distortion theory , *IMAGE compression , *LOSSY data compression , *COMPRESSED sensing , *ERROR functions , *RANDOM variables , *GEOMETRIC quantization - Abstract
This paper is concerned with the lossy compression of general random variables, specifically with rate-distortion theory and quantization of random variables taking values in general measurable spaces such as, e.g. manifolds and fractal sets. Manifold structures are prevalent in data science, e.g. in compressed sensing, machine learning, image processing and handwritten digit recognition. Fractal sets find application in image compression and in the modeling of Ethernet traffic. Our main contributions are bounds on the rate-distortion function and the quantization error. These bounds are very general and essentially only require the existence of reference measures satisfying certain regularity conditions in terms of small ball probabilities. To illustrate the wide applicability of our results, we particularize them to random variables taking values in (i) manifolds, namely, hyperspheres and Grassmannians and (ii) self-similar sets characterized by iterated function systems satisfying the weak separation property. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
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14. On local fractional integral inequalities via generalized (h˜1,h˜2)\left({\tilde{h}}_{1},{\tilde{h}}_{2})-preinvexity involving local fractional integral operators with Mittag-Leffler kernel
- Author
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Vivas-Cortez Miguel, Bibi Maria, Muddassar Muhammad, and Al-Sa’di Sa’ud
- Subjects
generalized h̃1, h̃2-preinvex functions ,local fractional integrals ,generalized hermite-hadamard inequality ,fractal sets ,mittag-leffler kernel ,26a33 ,26a51 ,90c23 ,26d10 ,26d15 ,Mathematics ,QA1-939 - Abstract
Local fractional integral inequalities of Hermite-Hadamard type involving local fractional integral operators with Mittag-Leffler kernel have been previously studied for generalized convexities and preinvexities. In this article, we analyze Hermite-Hadamard-type local fractional integral inequalities via generalized (h˜1,h˜2)\left({\tilde{h}}_{1},{\tilde{h}}_{2})-preinvex function comprising local fractional integral operators and Mittag-Leffler kernel. In addition, two examples are discussed to ensure that the derived consequences are correct. As an application, we construct an inequality to establish central moments of a random variable. more...
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- 2023
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15. Some Remarks on Local Fractional Integral Inequalities Involving Mittag–Leffler Kernel Using Generalized (E , h)-Convexity.
- Author
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Saleh, Wedad, Lakhdari, Abdelghani, Almutairi, Ohud, and Kiliçman, Adem
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FRACTIONAL integrals , *FRACTALS , *FRACTIONAL calculus , *INTEGRAL operators , *INTEGRAL inequalities - Abstract
In the present work, we introduce two new local fractional integral operators involving Mittag–Leffler kernel on Yang's fractal sets. Then, we study the related generalized Hermite–Hadamard-type inequality using generalized (E , h) -convexity and obtain two identities pertaining to these operators, and the respective first- and second-order derivatives are given. In terms of applications, we provide some new generalized trapezoid-type inequalities for generalized ( E , h )-convex functions. Finally, some special cases are deduced for different values of δ , E, and h. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
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16. Study of Critical Behavior of Dynamic Systems
- Author
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Sel’chenkova, N. I. and Uchaev, A. Ya.
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- 2023
- Full Text
- View/download PDF
17. Some New Fractal Milne-Type Integral Inequalities via Generalized Convexity with Applications.
- Author
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Meftah, Badreddine, Lakhdari, Abdelghani, Saleh, Wedad, and Kiliçman, Adem
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INTEGRAL inequalities , *FRACTALS , *GENERALIZED integrals , *INTEGRAL functions , *CONVEX sets , *ARITHMETIC - Abstract
This study aims to construct some new Milne-type integral inequalities for functions whose modulus of the local fractional derivatives is convex on the fractal set. To that end, we develop a novel generalized integral identity involving first-order generalized derivatives. Finally, as applications, some error estimates for the Milne-type quadrature formula and new inequalities for the generalized arithmetic and p-Logarithmic means are derived. This paper's findings represent a significant improvement over previously published results. The paper's ideas and formidable tools may inspire and motivate further research in this worthy and fascinating field. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
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18. On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization.
- Author
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Elsadany, A. A., Aldurayhim, A., Agiza, H. N., and Elsonbaty, Amr
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FRACTAL analysis , *FRACTALS , *SYNCHRONIZATION , *COSINE function , *COMPUTER simulation - Abstract
In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map's fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map's parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
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19. Some Hermite‐Hadamard's type local fractional integral inequalities for generalized γ‐preinvex function with applications.
- Author
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Al‐Sa'di, Sa'ud, Bibi, Maria, and Muddassar, Muhammad
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FRACTIONAL integrals , *INTEGRAL inequalities , *GENERALIZED integrals , *FRACTALS , *RANDOM variables - Abstract
In this article, we define and investigate the concept of generalizedγ$$ \gamma $$‐preinvex function on the Yang's fractal set ℝξ(0<ξ≤1)$$ {\mathbb{R}}&#x0005E;{\xi}\kern3.0235pt \left(0&lt;\xi \le 1\right) $$. Based on the auxiliary definitions and involving local fractional integrals, we established several generalizations of Hermite‐Hadamard type inequalities under certain conditions. Additionally, we discuss some examples to test our outcomes and some applications in the form of bounds for generalized rth$$ {r}&#x0005E;{th} $$ moment of a continuous random variable. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
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20. Hybrid Approach to Combinatorial and Logic Graph Problems
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Kureichik, Vladimir, Zaruba, Daria, Kureichik, Vladimir, Jr., Kacprzyk, Janusz, Series Editor, Pal, Nikhil R., Advisory Editor, Bello Perez, Rafael, Advisory Editor, Corchado, Emilio S., Advisory Editor, Hagras, Hani, Advisory Editor, Kóczy, László T., Advisory Editor, Kreinovich, Vladik, Advisory Editor, Lin, Chin-Teng, Advisory Editor, Lu, Jie, Advisory Editor, Melin, Patricia, Advisory Editor, Nedjah, Nadia, Advisory Editor, Nguyen, Ngoc Thanh, Advisory Editor, Wang, Jun, Advisory Editor, Sharma, Harish, editor, Saraswat, Mukesh, editor, Yadav, Anupam, editor, Kim, Joong Hoon, editor, and Bansal, Jagdish Chand, editor more...
- Published
- 2021
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21. A Novel Neighborhood Generation Method for Heuristics and Application to Traveling Salesman Problem
- Author
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Öztürk, Melike, Alabaş Uslu, Çiğdem, Kacprzyk, Janusz, Series Editor, Pal, Nikhil R., Advisory Editor, Bello Perez, Rafael, Advisory Editor, Corchado, Emilio S., Advisory Editor, Hagras, Hani, Advisory Editor, Kóczy, László T., Advisory Editor, Kreinovich, Vladik, Advisory Editor, Lin, Chin-Teng, Advisory Editor, Lu, Jie, Advisory Editor, Melin, Patricia, Advisory Editor, Nedjah, Nadia, Advisory Editor, Nguyen, Ngoc Thanh, Advisory Editor, Wang, Jun, Advisory Editor, Kahraman, Cengiz, editor, Cebi, Selcuk, editor, Cevik Onar, Sezi, editor, Oztaysi, Basar, editor, Tolga, A. Cagri, editor, and Sari, Irem Ucal, editor more...
- Published
- 2020
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22. Generalized Fractal Jensen–Mercer and Hermite–Mercer type inequalities via h-convex functions involving Mittag–Leffler kernel.
- Author
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Xu, Peng, Butt, Saad Ihsan, Yousaf, Saba, Aslam, Adnan, and Zia, Tariq Javed
- Subjects
FRACTIONAL integrals ,CONVEX functions ,FRACTALS ,INTEGRAL inequalities ,INTEGRAL operators ,FRACTIONAL calculus ,DIFFERENTIABLE functions - Abstract
In this paper, we present generalized Jensen-Mercer inequality for a generalized h -convex function on fractal sets. We proved Hermite-Hadamard-Mercer local fractional integral inequalities via integral operators pertaining Mittag-Leffler kernel. Also, we drive two new local fractional integral identities for differentiable functions. By employing these integral identities, we derive some new Hermite-Mercer type inequalities for generalized h -convex function in local fractional calculus settings. Finally, we give some examples to emphasize the applications of derived results. These results will be a significant addition to Jensen-type inequalities in the literature. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
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23. Qp 上分形多孔介质的流体动力学模型.
- Author
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吴波
- Subjects
FRACTALS ,PSEUDODIFFERENTIAL operators ,POROUS materials ,FLUID dynamics ,FLUIDS - Abstract
Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2022
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24. Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications
- Author
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Thabet Abdeljawad, Saima Rashid, Zakia Hammouch, İmdat İşcan, and Yu-Ming Chu
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Generalized convex function ,Generalized s-convex function ,Hermite–Hadamard inequality ,Simpson’s-like type inequality ,Generalized m-convex functions ,Fractal sets ,Mathematics ,QA1-939 - Abstract
Abstract The present article addresses the concept of p-convex functions on fractal sets. We are able to prove a novel auxiliary result. In the application aspect, the fidelity of the local fractional is used to establish the generalization of Simpson-type inequalities for the class of functions whose local fractional derivatives in absolute values at certain powers are p-convex. The method we present is an alternative in showing the classical variants associated with generalized p-convex functions. Some parts of our results cover the classical convex functions and classical harmonically convex functions. Some novel applications in random variables, cumulative distribution functions and generalized bivariate means are obtained to ensure the correctness of the present results. The present approach is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractals in computer graphics. more...
- Published
- 2020
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25. Some new local fractional inequalities associated with generalized ( s , m ) $(s,m)$ -convex functions and applications
- Author
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Thabet Abdeljawad, Saima Rashid, Zakia Hammouch, and Yu-Ming Chu
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Generalized convex function ,Generalized s-convex function ,Hermite–Hadamard inequality ,Simpson-type inequality ,Generalized m-convex functions ,Fractal sets ,Mathematics ,QA1-939 - Abstract
Abstract Fractal analysis is one of interesting research areas of computer science and engineering, which depicts a precise description of phenomena in modeling. Visual beauty and self-similarity has made it an attractive field of research. The fractal sets are the effective tools to describe the accuracy of the inequalities for convex functions. In this paper, we employ linear fractals R α $\mathbb{R}^{\alpha }$ to investigate the ( s , m ) $(s,m)$ -convexity and relate them to derive generalized Hermite–Hadamard (HH) type inequalities and several other associated variants depending on an auxiliary result. Under this novel approach, we aim at establishing an analog with the help of local fractional integration. Meanwhile, we establish generalized Simpson-type inequalities for ( s , m ) $(s,m)$ -convex functions. The results in the frame of local fractional showed that among all comparisons, we can only see the correlation between novel strategies and the earlier consequences in generalized s-convex, generalized m-convex, and generalized convex functions. We obtain application in probability density functions and generalized special means to confirm the relevance and computational effectiveness of the considered method. Similar results in this dynamic field can also be widely applied to other types of fractals and explored similarly to what has been done in this paper. more...
- Published
- 2020
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26. Some Hermite–Hadamard type inequalities for generalized h-preinvex function via local fractional integrals and their applications
- Author
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Wenbing Sun
- Subjects
Generalized h-preinvex function ,Hermite–Hadamard type inequalities ,Local fractional integrals ,Fractal sets ,Mathematics ,QA1-939 - Abstract
Abstract The concept of generalized h-preinvex function on real linear fractal sets R β $R^{\beta }$ ( 0 < β ≤ 1 $0 < \beta \le 1$ ) is introduced, which extends generalized preinvex, generalized s-preinvex, generalized Godunova–Levin preinvex, and generalized P-preinvex functions. In addition, some Hermite–Hadamard type inequalities for these classes of functions involving local fractional integrals are established. Lastly, the upper bounds for generalized expectation, generalized rth moment, and generalized variance of a continuous random variable are given to illustrate the applications of the obtained results. more...
- Published
- 2020
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27. Generalization of the Multiplicative and Additive Compounds of Square Matrices and Contraction Theory in the Hausdorff Dimension.
- Author
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Wu, Chengshuai, Pines, Raz, Margaliot, Michael, and Slotine, Jean-Jacques
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FOOD additives , *NONLINEAR dynamical systems , *MULTILINEAR algebra , *FRACTAL analysis , *FRACTALS ,FRACTAL dimensions - Abstract
The $k$ multiplicative and $k$ additive compounds of a matrix play an important role in geometry, multilinear algebra, the asymptotic analysis of nonlinear dynamical systems, and in bounding the Hausdorff dimension of fractal sets. These compounds are defined for the integer values of $k$. Here, we introduce generalizations called the $\alpha$ multiplicative and $\alpha$ additive compounds of a square matrix, with $\alpha$ real. We study the properties of these new compounds and demonstrate an application in the context of the Douady and Oesterlé theorem. Our results lead to a generalization of contracting systems to $\alpha$ -contracting systems, with $\alpha$ real. Roughly speaking, the dynamics of such systems contracts any set with the Hausdorff dimension larger than $\alpha$. For $\alpha =1$ , they reduce to standard contracting systems. We demonstrate our theoretical results by designing a state-feedback controller for a classical chaotic system, guaranteeing the well-ordered behavior of the closed-loop system. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
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28. Optimal control of the heat equation on a fractal set.
- Author
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Riane, Nizar and David, Claire
- Abstract
So far, controlling the solutions of PDE's on fractal sets has not been much explored. Whereas those structures bear interesting properties in terms of heat diffusion. We hereafter discuss the extension of classical results of control theory to self-similar sets, and apply them to the benchmark case of the Sierpiński Gasket. Our results show that this path will gain being developped. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
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29. NEW COMPUTATIONS OF OSTROWSKI-TYPE INEQUALITY PERTAINING TO FRACTAL STYLE WITH APPLICATIONS.
- Author
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AL QURASHI, MAYSAA, RASHID, SAIMA, KHALID, AASMA, KARACA, YELIZ, and CHU, YU-MING
- Subjects
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FRACTALS , *CONTINUOUS functions , *FRACTAL analysis , *CONCAVE functions , *CONVEX functions - Abstract
The purpose of this paper is to provide novel estimates of Ostrowski-type inequalities in a much simpler and shorter way of some recent significant results in the context of a fractal set ℝ α ̃ . By using our new approach, we established an auxiliary result that correlates with generalized convex () and concave functions for absolutely continuous functions with second-order local differentiable mappings. Moreover, we derived some companions of Ostrowski-type inequalities belonging to (2 α ̃) ∈ L ∞ [ s 1 , s 2 ] , (2 α ̃) ∈ L p [ s 1 , s 2 ] and (2 α ̃) ∈ L 1 [ s 1 , s 2 ] in local fractional sense. Our results generalize and offer better bounds than many known results in the existing literature associated with trapezoidal and midpoint formula. As an application perspective, we derived several estimation-type outcomes by the use of generalized α ̃ -type special means formula provided here to illustrate the usability of the obtained results. Our study contributes to a better understanding of fractal analysis and proves beneficial in exploring real-world phenomena. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
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30. NEW NEWTON'S TYPE ESTIMATES PERTAINING TO LOCAL FRACTIONAL INTEGRAL VIA GENERALIZED p-CONVEXITY WITH APPLICATIONS.
- Author
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LI, YONG-MIN, RASHID, SAIMA, HAMMOUCH, ZAKIA, BALEANU, DUMITRU, and CHU, YU-MING
- Subjects
- *
FRACTIONAL integrals , *GENERALIZED integrals , *FRACTALS , *CUMULATIVE distribution function , *THEORY of distributions (Functional analysis) , *CONVEX functions , *FRACTAL analysis - Abstract
This paper aims to investigate the notion of p -convex functions on fractal sets ℝ α ̂ (0 < α ̂ ≤ 1). Based on these novel ideas, we derived an auxiliary result depend on a three-step quadratic kernel by employing generalized p -convexity. Take into account the local fractal identity, we established novel Newton's type variants for the local differentiable functions. Several special cases are apprehended in the light of generalized convex functions and generalized harmonically convex functions. This novel strategy captures several existing results in the relative literature. Application is obtained in cumulative distribution function and generalized special weighted means to confirm the relevance and computational effectiveness of the considered method. Finally, we supposed that the consequences of this paper can stimulate those who are interested in fractal analysis. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
31. HERMITE–HADAMARD TYPE LOCAL FRACTIONAL INTEGRAL INEQUALITIES FOR GENERALIZED s-PREINVEX FUNCTIONS AND THEIR GENERALIZATION.
- Author
-
SUN, WENBING
- Subjects
- *
GENERALIZED integrals , *FRACTALS , *NUMERICAL integration , *FRACTIONAL integrals , *ABSOLUTE value , *INTEGRAL inequalities - Abstract
In this paper, the definition of generalized s-preinvex function on Yang's fractal sets ℝ γ (0 < γ ≤ 1) is proposed, and the generalized Hermite–Hadamard's inequality for this class of functions is established. By using this convexity, some generalized Hermite–Hadamard type integral inequalities with parameters are established. For these inequalities, the absolute values of twice local fractional order derivative of the functions are generalized s-preinvex functions. Some special integral inequalities can be obtained by assigning special values to the obtained inequalities, and two examples are given to illustrate our results. Finally, we propose the applications of the results in numerical integration and error estimation. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
32. On new generalized unified bounds via generalized exponentially harmonically s-convex functions on fractal sets.
- Author
-
Chu, Yu-Ming, Rashid, Saima, Abdeljawad, Thabet, Khalid, Aasma, and Kalsoom, Humaira
- Subjects
- *
FRACTALS , *SET functions , *FRACTAL analysis , *CONVEX functions - Abstract
The visual beauty reflects the practicability and superiority of design dependent on the fractal theory. Based on the applicability in practice, it shows that it is the completely feasible, self-comparability and multifaceted nature of fractal sets that made it an appealing field of research. There is a strong correlation between fractal sets and convexity due to its intriguing nature in the mathematical sciences. This paper investigates the notions of generalized exponentially harmonically (G E H ) convex and G E H s-convex functions on a real linear fractal sets R α (0 < α ≤ 1 ). Based on these novel ideas, we derive the generalized Hermite–Hadamard inequality, generalized Fejér–Hermite–Hadamard type inequality and Pachpatte type inequalities for G E H s-convex functions. Taking into account the local fractal identity; we establish a certain generalized Hermite–Hadamard type inequalities for local differentiable G E H s-convex functions. Meanwhile, another auxiliary result is employed to obtain the generalized Ostrowski type inequalities for the proposed techniques. Several special cases of the proposed concept are presented in the light of generalized exponentially harmonically convex, generalized harmonically convex and generalized harmonically s-convex. Meanwhile, an illustrative example and some novel applications in generalized special means are obtained to ensure the correctness of the present results. This novel strategy captures several existing results in the corresponding literature. Finally, we suppose that the consequences of this paper can stimulate those who are interested in fractal analysis. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
33. Measurement Brownian Dimension of Von Koch Curve
- Author
-
Mahasin Younis
- Subjects
brownian dimension ,brownian motion ,fractal sets ,Mathematics ,QA1-939 ,Electronic computers. Computer science ,QA75.5-76.95 - Abstract
The aim of this paper, it's calculate Brownian dimension of fractal pattern has self similarity (Von Koch Curve). This method is Random Middle Third Displacement in [0,1] has Gaussian distribution. Random processes are the main focus of research by analyzing dynamical systems to determine these ‘chaotic’ systems function. One such dynamical system is Brownian Motion. Basis of current definitions of physical phenomena. more...
- Published
- 2018
- Full Text
- View/download PDF
34. ON MAX–MIN MEAN VALUE FORMULAS ON THE SIERPINSKI GASKET.
- Author
-
NAVARRO, JOSE CARLOS and ROSSI, JULIO D.
- Subjects
- *
HARMONIC functions , *FRACTALS , *INFINITY (Mathematics) , *TRIANGLES , *MEAN value theorems - Abstract
In this paper, we study solutions to the max–min mean value problem 1 2 max q ∈ V m , p { f (q) } + 1 2 min q ∈ V m , p { f (q) } = f (p) in the Sierpinski Gasket with a prescribed Dirichlet datum at the three vertices of the first triangle. In the previous mean value, formula p is a vertex of one triangle at one stage in the construction of the Sierpinski Gasket and V m , p is the set of vertices that are adjacent to p at that stage. For this problem, it was known that there are existence and uniqueness of a continuous solution, a comparison principle holds, and, moreover, solutions are Lipschitz continuous. Here we continue the analysis of this problem and prove that the solution is piecewise linear on the segments of the Sierpinski Gasket. Moreover, we also show for which values at the three vertices of the first triangle solutions to this mean value formula coincide with infinity harmonic functions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
35. On the symmetric intersection of Rauzy fractals associated with the k-bonacci substitution.
- Author
-
Ammar, Hamdi, Cassaigne, Julien, and Sellami, Tarek
- Subjects
- *
FRACTALS , *ALGORITHMS , *SUBSTITUTIONS (Mathematics) - Abstract
In this article, we study the intersection of Rauzy fractals associated with the k-bonacci substitution and its reversed substitution. Applying the balanced pair algorithm to these two substitutions, we characterise all minimal balanced pairs, and we obtain a general formula for the associated intersection substitution. This substitution is defined over letters. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
36. Some new local fractional inequalities associated with generalized (s,m)-convex functions and applications.
- Author
-
Abdeljawad, Thabet, Rashid, Saima, Hammouch, Zakia, and Chu, Yu-Ming
- Subjects
FRACTAL analysis ,FRACTALS ,INTEGRAL inequalities ,PROBABILITY density function ,CONVEX functions ,COMPUTER engineering - Abstract
Fractal analysis is one of interesting research areas of computer science and engineering, which depicts a precise description of phenomena in modeling. Visual beauty and self-similarity has made it an attractive field of research. The fractal sets are the effective tools to describe the accuracy of the inequalities for convex functions. In this paper, we employ linear fractals R α to investigate the (s , m) -convexity and relate them to derive generalized Hermite–Hadamard (HH) type inequalities and several other associated variants depending on an auxiliary result. Under this novel approach, we aim at establishing an analog with the help of local fractional integration. Meanwhile, we establish generalized Simpson-type inequalities for (s , m) -convex functions. The results in the frame of local fractional showed that among all comparisons, we can only see the correlation between novel strategies and the earlier consequences in generalized s-convex, generalized m-convex, and generalized convex functions. We obtain application in probability density functions and generalized special means to confirm the relevance and computational effectiveness of the considered method. Similar results in this dynamic field can also be widely applied to other types of fractals and explored similarly to what has been done in this paper. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
37. Arithmetic on self-similar sets.
- Author
-
Zhao, Bing, Ren, Xiaomin, Zhu, Jiali, and Jiang, Kan
- Abstract
Let K 1 and K 2 be two one-dimensional homogeneous self-similar sets with the same ratio of contractions. Let f be a continuous function defined on an open set U ⊂ R 2 . Denote the continuous image of f by f U (K 1 , K 2) = { f (x , y) : (x , y) ∈ (K 1 × K 2) ∩ U }. In this paper we give a sufficient condition which guarantees that f U (K 1 , K 2) contains some interiors. Our result is different from Simon and Taylor's (2020, Proposition 2.9) as we do not need the condition that the product of the thickness of K 1 and K 2 is strictly greater than 1. As a consequence, we give an application to the univoque sets in the setting of q -expansions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
38. GENERALIZED h-CONVEXITY ON FRACTAL SETS AND SOME GENERALIZED HADAMARD-TYPE INEQUALITIES.
- Author
-
SUN, WENBING
- Subjects
- *
CONVEX functions , *DEFINITIONS , *MATHEMATICAL equivalence - Abstract
In this paper, we introduce the α -type concept of generalized h -convex function on real linear fractal sets ℝ α (0 < α ≤ 1) , from which the known definitions of generalized convex functions and generalized s -convex functions are derived, and from this, we obtain generalized Godunova–Levin functions and generalized P -functions. Some properties of generalized h -convex functions are discussed. Lastly, some generalized Hadamard-type inequalities of these classes functions are given. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
39. Embeddings of weighted generalized Morrey spaces into Lebesgue spaces on fractal sets.
- Author
-
Samko, Natasha
- Subjects
- *
GENERALIZED spaces , *QUASI-metric spaces , *EMBEDDINGS (Mathematics) , *SPACE , *FRACTALS - Abstract
We study embeddings of weighted local and consequently global generalized Morrey spaces defined on a quasi-metric measure set (X, d, μ) of general nature which may be unbounded, into Lebesgue spaces Ls(X), 1 ≤ s ≤ p < ∞. The main motivation for obtaining such an embedding is to have an embedding of non-separable Morrey space into a separable space. In the general setting of quasi-metric measure spaces and arbitrary weights we give a sufficient condition for such an embedding. In the case of radial weights related to the center of local Morrey space, we obtain an effective sufficient condition in terms of (fractional in general) upper Ahlfors dimensions of the set X. In the case of radial weights we also obtain necessary conditions for such embeddings of local and global Morrey spaces, with the use of (fractional in general) lower and upper Ahlfors dimensions. In the case of power-logarithmic-type weights we obtain a criterion for such embeddings when these dimensions coincide. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
40. 分形空间中的广义预不变凸函数与相关的 Hermite-Hadamard 型积分不等式.
- Author
-
孙文兵
- Abstract
Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2019
- Full Text
- View/download PDF
41. SOME LOCAL FRACTIONAL INTEGRAL INEQUALITIES FOR GENERALIZED PREINVEX FUNCTIONS AND APPLICATIONS TO NUMERICAL QUADRATURE.
- Author
-
SUN, WENBING
- Subjects
- *
NUMERICAL functions , *FRACTIONAL integrals , *GENERALIZED integrals , *INTEGRAL inequalities , *NUMERICAL integration , *FRACTALS - Abstract
In this paper, a new identity with parameters involving local fractional integrals is derived. Using this identity, some general local fractional integral inequalities for generalized preinvex functions are established. A parallel development is deduced for generalized preconcave functions. Taking special values for the parameters, some generalized midpoint inequalities, trapezoidal inequalities and Simpson inequalities are obtained. Finally, as some applications, error estimates of numerical integration for local fractional integrals are given. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
42. Spectral triples for the variants of the Sierpiński gasket.
- Author
-
Arauza Rivera, Andrea
- Subjects
- *
FRACTALS , *GASKETS , *HAUSDORFF measures , *TOPOLOGICAL spaces , *FRACTAL analysis , *GEOMETRY ,FRACTAL dimensions - Abstract
Fractal geometry is the study of sets which exhibit the same pattern at multiple scales. Developing tools to study these sets is of great interest. One step towards developing some of these tools is recognizing the duality between topological spaces and commutative C -algebras. When one lifts the commutativity axiom, one gets what are called noncommutative spaces and the study of noncommutative geometry. The tools built to study noncommutative spaces can in fact be used to study fractal sets. In what follows we will use the spectral triples of noncommutative geometry to describe various notions from fractal geometry. We focus on the fractal sets known as the harmonic Sierpiński gasket and the stretched Sierpiński gasket, and show that the spectral triples constructed in [7] and [23] can recover the standard self-affine measure in the case of the harmonic Sierpiński gasket and the Hausdorff dimension, geodesic metric, and Hausdorff measure in the case of the stretched Sierpiński gasket. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
43. NONLINEAR MEAN-VALUE FORMULAS ON FRACTAL SETS.
- Author
-
NAVARRO, J. C. and ROSSI, J. D.
- Subjects
- *
TRIANGLES , *GEOMETRIC vertices , *FRACTALS , *MEAN value theorems , *LIPSCHITZ spaces - Abstract
In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem 1 2 max q ∈ V m , p { f (q) } + 1 2 min q ∈ V m , p { f (q) } − f (p) = 0 in the Sierpiński gasket with prescribed values f (p 1) , f (p 2) and f (p 3) at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
- Full Text
- View/download PDF
44. Hermite-Hadamard type inequalities for generalized convex functions on fractal sets style
- Author
-
Muharrem Tomar, Praveen Agarwal, and Junesang Choi
- Subjects
Fractal sets ,Generalized convex functions ,Generalized Hermite--Hadamard inequality ,Generalized Holder inequality ,Mathematics ,QA1-939 - Abstract
We aim to establish certain generalized Hermite-Hadamard's inequalities for generalized convex functions via local fractional integral. As special cases of some of the results presented here, certain interesting inequalities involving generalized arithmetic and logarithmic means are obtained. more...
- Published
- 2018
- Full Text
- View/download PDF
45. A higher-dimensional Bourgain–Dyatlov fractal uncertainty principle
- Author
-
Rui Han and Wilhelm Schlag
- Subjects
Cartan estimates ,Pure mathematics ,Uncertainty principle ,32U15 ,Zhàng ,01 natural sciences ,Set (abstract data type) ,symbols.namesake ,Mathematics - Analysis of PDEs ,uncertainty principle ,Fractal ,Dimension (vector space) ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Fractal set ,Beurling–Malliavin theorem ,42B30 ,0101 mathematics ,subharmonic functions ,Mathematics ,Numerical Analysis ,Applied Mathematics ,010102 general mathematics ,fractal sets ,16. Peace & justice ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,symbols ,010307 mathematical physics ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of $\delta$-regular sets in one dimension, but relative to arbitrary axes. Our results remain true if $Y$ is distorted by diffeomorphisms. Our method combines the original approach by Bourgain and Dyatlov, in the more quantitative 2017 rendition by Jin and Zhang, with Cartan set techniques., Comment: 44 pages, arguments simplified and 2 figures added more...
- Published
- 2020
- Full Text
- View/download PDF
46. Scaling Limit for the Incipient Spanning Clusters
- Author
-
Aizenman, Michael, Friedman, Avner, editor, Gulliver, Robert, editor, Golden, Kenneth M., editor, Grimmett, Geoffrey R., editor, James, Richard D., editor, Milton, Graeme W., editor, and Sen, Pabitra N., editor more...
- Published
- 1998
- Full Text
- View/download PDF
47. Cellular outline segmentation using fractal estimators
- Author
-
Adrián Salvatelli, José Caropresi, Claudio Delrieux, María F. Izaguirre, and Víctor Casco
- Subjects
image processing ,segmentation ,imunofluorescence microscopy images ,fractal sets ,Computer engineering. Computer hardware ,TK7885-7895 ,Electronic computers. Computer science ,QA75.5-76.95 - Abstract
Segmentation in biological images is essential for the determination of biological parameters that allow the construction of models of several biological problems. This helps to establish clear relationships between those models and the parameter estimation, and for elaboration of key experiments that give support to biological theories. Segmentation is the process of qualitative or quantitative information extraction (shape, texture, physical and geometric properties, among others). These quantities are needed to compute the biological descriptors for further classification (v.g., cell counting, development stage assessment, and many others). This process is almost always supervised (i.e., human assisted), since the quality of the images that are produced with classic microscopy technologies have defects that in general disallow the application of unsupervised segmentation techniques. In this paper we investigate the use of the a local fractal dimension estimation as an image descriptor for microscopy images. This local descriptor appears to be robust enough to perform unsupervised or semisupervised segmentations, specifically in our study. We applied this technique on microscopy images of amphibian embryos' skin in which, using immunofluorescence techniques, we have labeled the cell adhesion molecule E-Cadherin. This molecule is one of the key factors of the Ca2+- dependent cell-cell adhesion. Segmentation of the cellular outlines was performed using a processing workflow, which can be repeatedly applied to a set of similar images, from which information is extracted for characterization and eventual quantification purposes. more...
- Published
- 2007
48. New Hermite-Hadamard and Jensen Type Inequalities for h−Convex Functions on Fractal Sets.
- Author
-
VIVAS, MIGUEL, HERNÁNDEZ, JORGE, and MERENTES, NELSON
- Subjects
- *
CONVEX functions , *FRACTALS , *JENSEN'S inequality , *FRACTIONAL calculus , *BANACH spaces - Abstract
In this paper, some new Jensen and Hermite-Hadamard inequalities for h-convex functions on fractal sets are obtained. Results proved in this paper may stimulate further research in this area. [ABSTRACT FROM AUTHOR] more...
- Published
- 2016
49. On Max–Min Mean Value Formulas on the Sierpinski Gasket
- Author
-
Universidad de Alicante. Departamento de Matemáticas, Navarro Climent, José Carlos, Rossi, Julio D., Universidad de Alicante. Departamento de Matemáticas, Navarro Climent, José Carlos, and Rossi, Julio D. more...
- Abstract
In this paper, we study solutions to the max–min mean value problem ½ max q∈Vm,p {f(q)} + ½ min q∈Vm,p {f(q)} = f(p) in the Sierpinski Gasket with a prescribed Dirichlet datum at the three vertices of the first triangle. In the previous mean value, formula p is a vertex of one triangle at one stage in the construction of the Sierpinski Gasket and Vm,p is the set of vertices that are adjacent to p at that stage. For this problem, it was known that there are existence and uniqueness of a continuous solution, a comparison principle holds, and, moreover, solutions are Lipschitz continuous. Here we continue the analysis of this problem and prove that the solution is piecewise linear on the segments of the Sierpinski Gasket. Moreover, we also show for which values at the three vertices of the first triangle solutions to this mean value formula coincide with infinity harmonic functions. more...
- Published
- 2021
50. HARNACK INEQUALITIES FOR ENERGY FORMS ON FRACTALS SETS.
- Author
-
VIVALDI, MARIA AGOSTINA
- Subjects
MATHEMATICAL inequalities ,FRACTALS ,ELLIPTIC equations ,DIVERGENCE theorem ,RADON measures - Published
- 2009
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