Showing papers on "Hadamard transform published in 2022"

Fractional SEIR Model and Data-Driven Predictions of COVID-19 Dynamics of Omicron Variant

Abstract: We study the dynamic evolution of COVID-19 caused by the Omicron variant via a fractional susceptible-exposed-infected-removed (SEIR) model. Preliminary data suggest that the symptoms of Omicron infection are not prominent and the transmission is, therefore, more concealed, which causes a relatively slow increase in the detected cases of the newly infected at the beginning of the pandemic. To characterize the specific dynamics, the Caputo-Hadamard fractional derivative is adopted to refine the classical SEIR model. Based on the reported data, we infer the fractional order and time-dependent parameters as well as unobserved dynamics of the fractional SEIR model via fractional physics-informed neural networks. Then, we make short-time predictions using the learned fractional SEIR model.

Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators

Abstract: Abstract In this article, the notion of interval-valued preinvex functions involving the Riemann–Liouville fractional integral is described. By applying this, some new refinements of the Hermite–Hadamard inequality for the fractional integral operator are presented. Some novel special cases of the presented results are discussed as well. Also, some examples are presented to validate our results. The established outcomes of our article may open another direction for different types of integral inequalities for fractional interval-valued functions, fuzzy interval-valued functions, and their associated optimization problems.

New Fractional Integral Inequalities for Convex Functions Pertaining to Caputo–Fabrizio Operator

Abstract: In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into account as an auxiliary result and with the assistance of Hölder, power-mean, Young, and Jensen inequality, some new estimations of the Hermite-Hadamard (H-H) type inequality as refinements are presented. Applications to special means and trapezoidal quadrature formula are presented to verify the accuracy of the results. Finally, a brief conclusion and future scopes are discussed.

Some New Versions of Hermite–Hadamard Integral Inequalities in Fuzzy Fractional Calculus for Generalized Pre-Invex Functions via Fuzzy-Interval-Valued Settings

Abstract: The purpose of this study is to prove the existence of fractional integral inclusions that are connected to the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for χ-pre-invex fuzzy-interval-valued functions. Some of the related fractional integral inequalities are also proved via Riemann–Liouville fractional integral operator, where integrands are fuzzy-interval-valued functions. To prove the validity of our main results, some of the nontrivial examples are also provided. As specific situations, our findings can provide a variety of new and well-known outcomes which can be viewed as applications of our main results. The results in this paper can be seen as refinements and improvements to previously published findings.

Hermite-Hadamard Inequalities in Fractional Calculus for Left and Right Harmonically Convex Functions via Interval-Valued Settings

Abstract: The purpose of this study is to define a new class of harmonically convex functions, which is known as left and right harmonically convex interval-valued function (LR-𝓗-convex IV-F), and to establish novel inclusions for a newly defined class of interval-valued functions (IV-Fs) linked to Hermite–Hadamard (H-H) and Hermite–Hadamard–Fejér (H-H-Fejér) type inequalities via interval-valued Riemann–Liouville fractional integrals (IV-RL-fractional integrals). We also attain some related inequalities for the product of two LR-𝓗-convex IV-Fs. These findings enable us to identify a new class of inclusions that may be seen as significant generalizations of results proved by Iscan and Chen. Some examples are included in our findings that may be used to determine the validity of the results. The findings in this work can be seen as a considerable advance over previously published findings.

Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions

Abstract: <abstract> <p>The inclusion relation and the order relation are two distinct ideas in interval analysis. Convexity and nonconvexity create a significant link with different sorts of inequalities under the inclusion relation. For many classes of convex and nonconvex functions, many works have been devoted to constructing and refining classical inequalities. However, it is generally known that log-convex functions play a significant role in convex theory since they allow us to deduce more precise inequalities than convex functions. Because the idea of log convexity is so important, we used fuzzy order relation $\left(\preceq \right)$ to establish various discrete Jensen and Schur, and Hermite-Hadamard (H-H) integral inequality for log convex fuzzy interval-valued functions (L-convex F-I-V-Fs). Some nontrivial instances are also offered to bolster our findings. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>

On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings

Abstract: In the present study, over a rectangle from the plane R2, we define and develop the conceptions of the interval-valued fractional double integrals having exponential kernels, from which we exploit Hermite–Hadamard, Fejér–Hermite–Hadamard, as well as Pachpatte type inclusion relations regarding the interval-valued co-ordinated convex mappings. These inclusion relations can be viewed as certain substantial generalizations of the previously reported findings. To identify the correctness of the inclusion relations constructed in this work, we also provide three examples regarding the interval-valued co-ordinated convex mappings.

Interval Fejér-Type Inequalities for Left and Right-λ-Preinvex Functions in Interval-Valued Settings

Abstract: For left and right λ-preinvex interval-valued functions (left and right λ-preinvex IVFs) in interval-valued Riemann operator settings, we create Hermite–Hadamard (H-H) type inequalities in the current study. Additionally, we create Hermite–Hadamard–Fejér (H-H-Fejér)-type inequalities for preinvex functions of the left and right interval-valued type under some mild conditions. Moreover, some exceptional new and classical cases are also obtained. Some useful examples are also presented to prove the validity of the results.

Some results on the study of Caputo–Hadamard fractional stochastic differential equations

Abstract: In this article, we show the existence and uniqueness of solution of Caputo–Hadamardfractional stochastic differential equations (CHFSDE) via the Banach fixed point method (BFPM). We analyze the Ulam–Hyers stability (UHS) of CHFSDE by the generalized and the classical Gronwall inequalities. Two examples are presented to illustrate our results.

Hermite–Hadamard and Jensen-Type Inequalities for Harmonical (h1, h2)-Godunova–Levin Interval-Valued Functions

Abstract: There is no doubt that convex and non-convex functions have a significant impact on optimization. Due to its behavior, convexity also plays a crucial role in the discussion of inequalities. The principles of convexity and symmetry go hand-in-hand. With a growing connection between the two in recent years, we can learn from one and apply it to the other. There have been significant studies on the generalization of Godunova–Levin interval-valued functions in the last few decades, as it has tremendous applications in both pure and applied mathematics. In this paper, we introduce the notion of interval- valued harmonical (h1, h2)-Godunova–Levin functions. Using the new concept, we establish a new interval Hermite–Hadamard and Jensen-type inequalities that generalize the ones that exist in the literature. Additionally, we provide some examples to prove the validity of our main results.

Caputo-Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos.

Abstract: This study investigates Caputo-Hadamard fractional differential equations on time scales. The Hadamard fractional sum and difference are defined for the first time. A general logarithm function on time scales is used as a kernel function. New fractional difference equations and their equivalent fractional sum equations are presented by the use of fundamental theorems. Gronwall inequality, asymptotical stability conditions, and two discrete-time Mittag-Leffler functions of Hadamard type are obtained. Numerical schemes are provided and chaos in fractional discrete-time logistic equation and neural network equations are reported.

Generalized Hermite-Hadamard-Mercer type inequalities via majorization

Abstract: The Hermite-Hadamard inequality has been recognized as the most pivotal inequality which has grabbed the attention of several mathematicians. In recent years, load of results have been established for this inequality. The main theme of this article is to present generalized Hermite-Hadamard inequality via the Jensen-Mercer inequality and majorization concept. We establish a Hermite-Hadamard inequality of the Jensen-Mercer type for majorized tuples. With the aid of weighted generalized Mercer?s inequality, we also prove a weighted generalized Hermite-Hadamard inequality for certain tuples. The idea of obtaining the results of this paper, may explore a new way for derivation of several other results for Hermite-Hadamard inequality.

Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $ (h_1, h_2) $-Godunova-Levin functions

Abstract: Interval analysis distinguishes between inclusion relation and order relation. Under the inclusion relation, convexity and nonconvexity contribute to different kinds of inequalities. The construction and refinement of classical inequalities have received a great deal of attention for many classes of convex as well as nonconvex functions. Convex theory, however, is commonly known to rely on Godunova-Levin functions because their properties enable us to determine inequality terms more precisely than those obtained from convex functions. The purpose of this study was to introduce a ($ \subseteq $) relation to established Jensen-type and Hermite-Hadamard inequalities using $ (h_1, h_2) $-Godunova-Levin interval-valued functions. To strengthen the validity of our results, we provide several examples and obtain some new and previously unknown results.

On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses

Abstract: Hadamard fractional calculus is one of the most important fractional calculus theories. Compared with a single Hadamard fractional order equation, Hadamard fractional differential equations have a more complex structure and a wide range of applications. It is difficult and challenging to study the dynamic behavior of Hadamard fractional differential equations. This manuscript mainly deals with the boundary value problem (BVP) of a nonlinear coupled Hadamard fractional system involving fractional derivative impulses. By applying nonlinear alternative of Leray-Schauder, we find some new conditions for the existence of solutions to this nonlinear coupled Hadamard fractional system. Our findings reveal that the impulsive function and its impulsive point have a great influence on the existence of the solution. As an application, we discuss an interesting example to verify the correctness and validity of our results.

Weighted Fractional Calculus: A General Class of Operators

Abstract: We conduct a formal study of a particular class of fractional operators, namely weighted fractional calculus, and its extension to the more general class known as weighted fractional calculus with respect to functions. We emphasise the importance of the conjugation relationships with the classical Riemann–Liouville fractional calculus, and use them to prove many fundamental properties of these operators. As examples, we consider special cases such as tempered, Hadamard-type, and Erdélyi–Kober operators. We also define appropriate modifications of the Laplace transform and convolution operations, and solve some ordinary differential equations in the setting of these general classes of operators.

Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator

Abstract: We present new Mercer variants of Hermite-Hadamard (HH) type inequalities via Atangana-Baleanu (AB) fractional integral operators pertaining non-local and non-singular kernels. We establish trapezoidal type identities for fractional operator involving non-singular kernel and give Jensen-Mercer (JM) variants of Hermite-Hadamard type inequalities for differentiable mapping $ \Upsilon $ possessing convex absolute derivatives. We establish connections of our results with several renowned results in the literature and also give applications to special functions.

Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator

Abstract: <abstract><p>We present new Mercer variants of Hermite-Hadamard (HH) type inequalities via Atangana-Baleanu (AB) fractional integral operators pertaining non-local and non-singular kernels. We establish trapezoidal type identities for fractional operator involving non-singular kernel and give Jensen-Mercer (JM) variants of Hermite-Hadamard type inequalities for differentiable mapping $ \Upsilon $ possessing convex absolute derivatives. We establish connections of our results with several renowned results in the literature and also give applications to special functions.</p></abstract>

Revisiting Fejer-Hermite-Hadamard Type Inequalities in Fractal Domain and Applications

Abstract: In this paper, some new fractal versions of Fejér–Hermite–Hadamard (FHH) type variants for generalized Raina [Formula: see text]-convex mappings are established benefiting from Raina’s function and fractal set [Formula: see text]. By means of three integral identities coupled with Raina’s function and local differentiation, we established some bounds for the difference between the left and central parts and also the difference between the center and right parts in FHH inequality. Besides that, some illustrative examples and noted special cases are apprehended. Additionally, we developed various generalizations for random variables, cumulative distribution functions, and special function theory as applications of local fractional integrals. The consequences established can provide contribution to inequality theory, fractional calculus and probability theory from the viewpoint of application to establish the other associated classes of functions. With the aid of these methodologies, it is promising to comprise further bounds of other type of variants which involve local fractional techniques.

Hadamard fractional calculus on time scales

Abstract: This study defines a Hadamard fractional sum by use of the time-scale theory. Then a [Formula: see text]-fractional difference is given and fundamental theorems are proved. Initial value problems of fractional difference equations are presented and their equivalent fractional sum equations are provided. The discrete Mittag-Leffler function solutions of linear fractional difference equations are obtained. It can be concluded that the new discrete fractional calculus of Hadamard type is well defined.

Some Fuzzy Inequalities for Harmonically s-Convex Fuzzy Number Valued Functions in the Second Sense Integral

Abstract: Many fields of mathematics rely on convexity and nonconvexity, especially when studying optimization issues, where it stands out for a variety of practical aspects. Owing to the behavior of its definition, the idea of convexity also contributes significantly to the discussion of inequalities. The concepts of symmetry and convexity are related and we can apply this because of the close link that has grown between the two in recent years. In this study, harmonic convexity, also known as harmonic s-convexity for fuzzy number valued functions (F-NV-Fs), is defined in a more thorough manner. In this paper, we extend harmonically convex F-NV-Fs and demonstrate Hermite–Hadamard (H.H) and Hermite–Hadamard Fejér (H.H. Fejér) inequalities. The findings presented here are summaries of a variety of previously published studies.

Block Walsh–Hadamard Transform-based Binary Layers in Deep Neural Networks

Abstract: Convolution has been the core operation of modern deep neural networks. It is well known that convolutions can be implemented in the Fourier Transform domain. In this article, we propose to use binary block Walsh–Hadamard transform (WHT) instead of the Fourier transform. We use WHT-based binary layers to replace some of the regular convolution layers in deep neural networks. We utilize both one-dimensional (1D) and 2D binary WHTs in this article. In both 1D and 2D layers, we compute the binary WHT of the input feature map and denoise the WHT domain coefficients using a nonlinearity that is obtained by combining soft-thresholding with the tanh function. After denoising, we compute the inverse WHT. We use 1D-WHT to replace the 1 × 1 convolutional layers, and 2D-WHT layers can replace the 3 × 3 convolution layers and Squeeze-and-Excite layers. 2D-WHT layers with trainable weights can be also inserted before the Global Average Pooling layers to assist the dense layers. In this way, we can reduce the number of trainable parameters significantly with a slight decrease in trainable parameters. In this article, we implement the WHT layers into MobileNet-V2, MobileNet-V3-Large, and ResNet to reduce the number of parameters significantly with negligible accuracy loss. Moreover, according to our speed test, the 2D-FWHT layer runs about 24 times as fast as the regular 3 × 3 convolution with 19.51% less RAM usage in an NVIDIA Jetson Nano experiment.

The New Versions of Hermite–Hadamard Inequalities for Pre-invex Fuzzy-Interval-Valued Mappings via Fuzzy Riemann Integrals

Abstract: Abstract In this study, we use the fuzzy order relation to show some novel variants of Hermite–Hadamard inequalities for pre-invex fuzzy-interval-valued mappings ( F-I∙V-Ms ), which we term fuzzy-interval Hermite–Hadamard inequalities and fuzzy-interval Hermite–Hadamard–Fejér inequalities. This fuzzy order relation is defined as the level of the fuzzy-interval space by the Kulisch–Miranker order relation. There are also some new exceptional instances mentioned. The theory proposed in this research is shown with practical examples that demonstrate its usefulness. This paper's approaches and methodologies might serve as a springboard for future study in this field.

Hermite–Hadamard and Pachpatte Type Inequalities for Coordinated Preinvex Fuzzy-Interval-Valued Functions Pertaining to a Fuzzy-Interval Double Integral Operator

Abstract: Many authors have recently examined the relationship between symmetry and generalized convexity. Generalized convexity and symmetry have become a new area of study in the field of inequalities as a result of this close relationship. In this article, we introduce the idea of preinvex fuzzy-interval-valued functions (preinvex F∙I-V∙F) on coordinates in a rectangle drawn on a plane and show that these functions have Hermite–Hadamard-type inclusions. We also develop Hermite–Hadamard-type inclusions for the combination of two coordinated preinvex functions with interval values. The weighted Hermite–Hadamard-type inclusions for products of coordinated convex interval-valued functions discussed in a recent publication by Khan et al. in 2022 served as the inspiration for our conclusions. Our proven results expand and generalize several previous findings made in the body of literature. Additionally, we offer appropriate examples to corroborate our theoretical main findings.

Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation

Abstract: Abstract It is a well-known fact that inclusion and pseudo-order relations are two different concepts which are defined on the interval spaces, and we can define different types of convexities with the help of both relations. By means of pseudo-order relation, the present article deals with the new notions of convex functions which are known as left and right log- s s -convex interval-valued functions (IVFs) in the second sense. The main motivation of this study is to present new inequalities for left and right log- s s -convex-IVFs. Therefore, we establish some new Jensen-type, Hermite-Hadamard (HH)-type, and Hermite-Hadamard-Fejér (HH-Fejér)-type inequalities for this kind of IVF, which generalize some known results. To strengthen our main results, we provide nontrivial examples of left and right log- s s -convex IVFs.

Hermite-Hadamard-Type Fractional Inclusions for Interval-Valued Preinvex Functions

Abstract: We introduce a new class of interval-valued preinvex functions termed as harmonically h-preinvex interval-valued functions. We establish new inclusion of Hermite–Hadamard for harmonically h-preinvex interval-valued function via interval-valued Riemann–Liouville fractional integrals. Further, we prove fractional Hermite–Hadamard-type inclusions for the product of two harmonically h-preinvex interval-valued functions. In this way, these findings include several well-known results and newly obtained results of the existing literature as special cases. Moreover, applications of the main results are demonstrated by presenting some examples.