This work is devoted to the study of a class of fractional boundary value problems using
(left) Caputo derivative, and with the particularity of being resonant, i.e., the associated homogeneous problem admits a nontrivial solution. Conditions to ensure the existence and uniqueness of solutions are presented. Using Mawhin’s coincidence degree, it is shown that the problem under consideration admits solutions and applying Banach contraction principle, sufficient conditions are obtained for which the solution is unique.

Machine-learning applications nowadays usually become a subject of data unavailability, complexity, and drift resulting from massive and rapid changes in data Volume, Velocity, and Variety (3V). Recent advances in deep learning have brought many improvements to the field providing, generative modeling, nonlinear abstractions, and adaptive learning to address these challenges respectively. In fact, deep learning aims to learn from representations that provide a consistent abstraction of the original feature space, which makes it more meaningful and less complex. However, data complexity related to different distortions such as higher levels of noise, for instance, remains difficult to overcome and challenging. In this context, recurrent expansion (RE) algorithms are recently unleashed to explore deeper representations than ordinary deep networks, providing further improvement in feature mapping. Unlike traditional deep learning, which extracts meaningful representations through inputs abstraction, RE allows entire deep networks to be merged into another one consecutively allowing exploration of Inputs, Maps, and estimated Targets (IMTs) as primary sources of learning; a total of three sources of information to provide additional information about their interaction in a deep network. Besides, RE makes it possible to study IMTs of several networks and learn significant features, improving its accuracy with each round. In this context, this paper presents a general overview of RE, its main learning rules, advantages, disadvantages, and its limits while going through an important state-of-the-art and some illustrative examples.

In this article we developed the idea of q-timescale calculus in quantum geometry. It introduces the q time scale-integral operator and differential operator. Its analysis some derived significant principles and findings which follow the calculus of q-time scale comparing with the Leibnitz-Newton usual calculus. The operator Δ_q- differential reduced method of transformation is proposed to transform the partial differential equation problems to time scale by utilizing the partial -differential equations. With easily computable coefficients the solution is calculated in the version of a power series which is convergent. It is also illustrated the performance and effectiveness of the proposed procedure and applying some important examples of partial differential equations. The newly obtained solution can be merges with usual calculus if the values of the parameter is set in the partial differential equation. The finding of the present work is that the Δ_q- differential transformation reduced method is convenient and efficient to solve partial differential equations such as heat equation , Laplace and Bernoulli equation.

Different sports have different fan base and in addition to this, there is a lot of crazes, enthusiasm, and zeal among people mainly youngsters. Cricket is one among them which has tremendous popularity not only among youngsters but also among all age groups. This will create a kind of pressure among the team members to perform well in every game as well as on the selectors to select the best players for the opening, middle orders, wicketkeeping, and bowling from the pool of players. As the game cannot be won by a single player or by openers or others as well, rather it is a collective effort of all the members of the team. So, it is necessary and one of the most important tasks to choose the players wisely so that they play well in their respective position. In this study, we try to formulate a model using MCDM (Multi criteria Decision Making Techniques) which evaluates not only the performance of the players but also the performance of different sets (i.e., openers, middle orders, wicket-keeper, and bowlers) and for this, we propose a novel ANP-DEA (Analytic Network Process – Data Envelopment Analysis) Technique and evaluate the best and worst performing set and their performance evaluation. A case study is done to properly visualize the proposed model.

This paper presents the effect of friction in the vibration behavior between two beams in relative motion according to Timoshenko's beam theory. The current system is composed of two cantilever beams screwed together. The nonlinear behavior can be divided into two phases: stick and slip. The differential equations of motion in the two phases are presented. In one hand, the viscous damping is added in the stick phase. In the other hand, the viscous damping and the frictional force applied are added in the slipping phase. The Galerkin method is used for the solution of differential equations to transfer the system of partial differential equations to the system of ordinary differential equations.

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We consider the following coupled system wave/Wentzell :

u_tt-Δu+μ₁g₁(u_t)+μ₂g₁(u_t(t-τ))=0, in Ω×(0,∞),

v_tt+∂_νu-Δ_Tv+μ₁′g₂(v_t)+μ₂′g₂(v_t(t-τ))=0, on Γ₁×(0,∞),

u=v, on Γ×(0,∞),

u=0, on Γ₀×(0,∞),

where Ω is a bounded domain in Rⁿ(n≥2), with smooth boundary Γ divided into two closed and disjoint subsets Γ₀ and Γ₁. We denote by ∇_T the tangential gradient on Γ, by Δ_T the tangential Laplacian on Γ and by ∂_ν the normal derivative where ν represents the unit outward normal to Γ.
μ₁, μ₂, μ₁′ and μ₂′ are positive real numbers, the two functions g₁(u_t(t-τ)) and g₂(v_t(t-τ)) describe the delays on the nonlinear frictional dissipations g₁(u_t) and g₂(v_t), on Ω and Γ₁, respectively, τ>0 is a time delay.

we will prove that the above problem is well-posed by proving the existence and uniqueness of a solution using the Faedo-Galerkin method.

The immune response includes the cascade of reactions which are aimed at destroying the pathogen and eliminating the consequences of the pathogen invasion. The understanding of the mechanism of the immune response will help to improve existing methods of treatment of infectious diseases.

Immune response includes the innate and the adaptive components. The innate immune response develops right after the pathogen invasion however it cannot always stop the infection progression. The adaptive immune response develops later during further infection progression. It acts to eliminate specific pathogens or pathogen-infected cells. The interaction of these two parts of the immune response provides the effective counteraction of the human organism to viral infection.

In this work, we construct a series of hierarchical models in order to investigate the influence of the immune response to the virus infection progression. These models are based on the local and non-local reaction-diffusion equations in one space dimension. The one-dimensional formulation allows the investigation of these models analytically, namely, evaluating the total viral load and the speed of the reaction-diffusion wave propagation. For the respiratory viral infections, the total viral load corresponds to the infectivity of the virus, i.e., the rate of infection transmission between individuals, and the wave speed corresponds to the virulence of the virus, i.e., the severity of the disease. The dependence of these characteristics on the parameters of the immune response has been investigated, and it is shown that the stronger immune response results in reduced both infectivity and virulence. The character of this dependence on the innate (interferon) and adaptive (antibodies and cytotoxic T-lymphocytes) immune response is evaluated.

In this work, we consider a generalized Mittag-Leffler type function and establish several integral formulas involving Jacobi and related transforms. We also establish a new composition of generalized fractional calculus operators associated with the generalized Mittag-Leffler type function. Additionally, certain special cases of generalized Mittag-Leffler function have been corollarily presented.

As s generalization of semi-invariant Riemannian maps, B. Sahin first introduce the idea of "Generic Riemannian Maps" in the article "Generic Riemannian Maps". He considered the total space as a Kaehler manifold. We extend the notion of generic Riemannian map to the case when the total manifold is nearly Kaehler manifold. We study the integrability conditions for the horizontal distribution while the vertical distribution is always integrable. We also study the geometry of foliation of two distributions and obtain the necessary and sufficient condition for generic Riemannian maps to be totally geodesic. Additionally, we study the generic Riemannian map with umbilical fibers.

The use of a multi-valued logic in ethics is not a new idea, but, to the best of our knowledge, there is not any integrated proposal for its application reported in the literature until now. In this work soft and neutrosophic sets are used as tools for introducing a multi-valued logic in ethics. Our target is not to add, in the already existing long catalogue, another theory about ethics, but instead to create a new basis enabling a modern approach to the subject. The paper starts by examining the role of human logic and statistical thinking to the creation and evolution of ethics. A brief historical account of the development of ethics follows with emphasis to the moral dilemmas, the existence of which motivates the application of a multi-valued logic to ethics. The basic information about fuzzy sets, fuzzy logic soft sets and neutrosophic sets, needed for the understanding of the rest of the paper, is also presented, before using the soft and neutrosophic sets for a mathematical representation of the ethical rules, and in extension of the moral theories. The final conclusions of all this discussion are summarized at the end of the paper.