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In this paper we introduce a new methodology to determine an optimal coe- cient of penalized functional regression. We assume the dependent, independent variables and the regression coecients are functions of time and error dynamics follow a stochastic differential equation. First we construct our objective function as a time dependent residual sum of square and then minimize it with respect to regression coecients subject to dierent error dynamics such as LASSO, group LASSO, fused LASSO and cubic smoothing spline. Then we use Feynman-type path integral approach to determine a Schrodinger-type equation which have the entire information of the system. Using rst order conditions with respect to these coecients give us a closed form solution of them.

MSC 2010 subject classifications: Primary 60H05; Secondary 81Q30.

Keywords and phrases: Penalized regression coecients, Feynman-type path integrals, stochastic dierential equations.

We study random walks on the vertices of three non-isomorphic halfcubes obtained from a cube by a plane cut through its center. Starting from a particular vertex (called the origin), at each step a particle moves, independently of all previous moves, to one of the vertices adjacent to the current vertex with equal probability. We find the means and the standard deviations of the number of steps needed to: (1) return to origin, (2) visit all vertices, and (3) return to origin after visiting all vertices. We also find (4) the probability distribution of the last vertex visited.

Keywords: Absorbing State, Cover Time, Eventual Transition, One-step Transition, Recursive Relation, Transient State.
???????MSC Codes: 05C81, 97K60

The aim of this paper is to introduce the concepts of regularly continuous and fully continuous mappings as the mappings that have the preimages of regular open sets are regular open (respectively, open ). We investigate the connections between these classes and several well-known others of `generalized continuous mappings'. Several characterizations and decompositions of certain continuities are provided. In particular, a decomposition of complete continuity, which was first introduced by Arya and Gupta [1] , is also provided.

Keywords: Regular open set, Preopen set, t-set, Continuous mapping.
???????2020 AMS Subject Classification: 54C08, 54C05, 54C10.

At present, many achievements have been made in the research of Choquard equation. In this paper, some new conclusions about the Choqaurd type problems are obtained through the variational method. It is dierent from the other Choqaurd equations with critical exponent, in this study, we consider a kind of Choqaurd equations with doubly critical exponents and local perturbation at the same time. The key point of this paper is that Sobolev embedding from workspace to Lebesgue space is not compact. Therefore, it is almost impossible to use conventional methods to obtain the convergence of (PS)c sequences. In order to eliminate the diculty, We use the Pohozaev manifold to complete the proof of the existence of solutions for a class of Choquard equations. We also take some new tricks in this equation.
???????Keywords: Choquard equation; Ground state solution; local perturbation; Pohozaev manifold

2000 Mathematics Subject Classification. 34C37; 35A15; 37J45

In this paper, we present refinements of some probability inequal-ities. We begin with a brief introduction and mention the well-known Chebyshev's inequality which describes the fundamental relationship between mean and variance of a random variable. Here we first present a more general theorem which besides yielding several interesting re- sults, includes Chebyshev's inequality as a special case.Our next result also includes Chebyshev's inequality as a special case when  = 0. Besides, we use these results to prove a generalization as well as a re-finement of Chebyshev's inequality. We also present an example which shows that our result gives better bound than Chebyshev's inequal- ity . We also present two theorems, first of which includes Markov's inequality as a special case and the other provides a generalization as well as a refinement of Markov's inequality. Here we also present an example to show that our theorem gives better upper bound than Markov's inequality. We conclude this paper by proving two more results, the first of which provides a generalization as well as a re- finement of Chernoff's inequality and the second includes Chernoff's inequality as a special case.

Key words: Probability Inequalities: Chebyshev's inequality, Markov's in- equality and Cherno's inequality. Mean, Variance.

Einstein Equations aren't hyperbolic because they are invariant under an invertible change of 4-dimensional variables. A possible solution of this problem is to consider a particular set of this 4-dimensional variables in order to reduce the number of the unknowns appearing in the metric tensor. The choice of these variables depends on the particular physical situation where we are working. In fact, in the right hand side of Einstein Equations there is the energy-momentum tensor of the sources; if this is all the matter contained in the Universe, then the problem becomes too complicated to deal with. An approximation can be used in particular situations. For example here the situation is considered of a polyatomic gas generating its own gravity field and sufficiently far from the other matter, so as not to be aected by its in uence on the metric tensor. The isotropy of the Universe is imposed by using the Representation Theorems jointly with another change of 4-dimensional variables so as to reduce the unknowns appearing in the 10 components of the metric tensor to only 2 scalar functions. In this way hyperbolic is achieved.

Keywords: Higher-order upper radial sets, Higher-order upper radial derivatives, Setvalued vector optimization, Weak eciency, Higher-order optimality conditions.

In this paper, we present the new particle swarm optimization algorithm by defining the inertia weight which depends on linearly the iteration step for solving the unsaturated soil water problem. By comparing with the exist particle swarm optimizations such as such as PSO algorithm, FPSO algorithm, P-PSO-SA algorithm and IDWPSO algorithm, numerical experiments verify that our proposed algorithm is more efficient and accurate to reach the optimal solution in the multimodal function extremum problem. Combining the characteristic difference method, our algorithm is applied to solve the unsaturated soil water problem.

Keywords: Particle swarm optimization; Unsaturated soil water; Inertia weight; Characteristic difference; Parameter identification.

Lot of mathematical models including differential equations play important role in healthcare and biotechnology. One of them is Malthus model. This model was developed by Thomas Malthus, in his essay on world population growth and resource supply. Another interesting equation is Advection diffusion equation and Predator prey model. We use a integral transform called as Soham transform to obtain the solutions of these models which are important in biotechnology and health sciences.
???????Key Words: Mathematical Models in health care; Soham transform; Integral transform; Malthus model; Predator ­Prey model; Logistic model; System of differential equations.