Journal of Progressive Research in Mathematics Journal of Progressive Research in Mathematics Scitech Research Organisation en-US Journal of Progressive Research in Mathematics 2395-0218 A Study on Sum Formulas for Generalized Tribonacci Numbers: Closed Forms of the Sum Formulas ∑_{k=0}ⁿkx^{k}W_{k}, ∑_{k=1}ⁿkx^{k}W_{-k} <p>In this paper, closed forms of the sum formulas ∑_{k=0}ⁿkx^{k}W_{k}, ∑_{k=1}ⁿkx^{k}W_{-k} for generalized Tribonacci numbers are presented. As special cases, we give summation formulas of Tribonacci, Tribonacci-Lucas, Padovan, Perrin, Narayana and some other third order linear recurrance sequences.</p> Yüksel Soykan ##submission.copyrightStatement## 2020-10-22 2020-10-22 17 1 1 40 The hydrodynamics of quantum spacetime - The minimal essentials of a new quantum theory <p>This is a somewhat long and extended abstract of a paper that presents a relatively short and concise review of a new quantum mechanics. This new theory is anchored in the hydrodynamical paradigm first introduced by L. Prandtl in his famous boundary layer theory.&nbsp; In addition the original ideas of L. Prandtl are expanded to encompass and combine with ideas from von Neumann-Connes’ pointless noncommutative geometry, Penrose-like fractal tiling cosmology, E-infinity Cantorian theory and the platonic golden mean number system based transfinite set theory.&nbsp; Proceeding in this way it is reasoned that while the pre-quantum particle and the pre-quantum wave may be best described as a multi dimensional zero set and empty set respectively in stringent mathematical terms, in physical terms however the new picture of a bluff body modelling the quantum particle and a surrounding Prandtl boundary layer modelling the quantum wave is virtually a quantum jump in our understanding of quantum physics in general and quantum wave collapse in particular.&nbsp; In that respect the work has some resemblance to that pilot wave theory of de Broglie and Bohm but is by no means more than that.&nbsp; The work is naturally connected to very specialized hydrodynamics related fields apart of Batchelor’s law and the important earthquake engineering subject of liquefaction which is of paramount importance for designing buildings with high resistance to earthquakes among other things.&nbsp; The concerted use of all these mathematical, experimental and number theoretical tools combine in the present paper to give a new synthesis for a deeper understanding of what we label the classical and quantum world predominantly for simplicity rather than logical, mandatory reasons.</p> Leila Marek-Crnjac Mohamed S. El Naschie ##submission.copyrightStatement## 2020-11-02 2020-11-02 17 1 41 53 A Study On Generalized (r,s,t,u,v,y)-Numbers <p>In this paper, we introduce the generalized (r,s,t,u,v,y) sequence and we deal with, in detail, three special cases which we call them (r,s,t,u,v,y), Lucas (r,s,t,u,v,y) and modified (r,s,t,u,v,y) sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.</p> Yüksel Soykan ##submission.copyrightStatement## 2020-11-27 2020-11-27 17 1 54 72 Homology Theory on Causal Random Groups <p>The objective of this paper is to analyze a new approach to homology theory that deals with Causal Random Groups, (CRG). It is shown that the evolution of (CRG) can be tracked down as these groups stay within the general random groups category. Random groups are algebraic groups that are not systematically the result of interaction of several random sub algebraic groups. Therefore, in general random groups are unstable. Evolution is change from a known state that can be traced back to that state. If change can not be traced back to its’ initial state, then this change results in a shift to an unknown zone. Change within the unknown zone is designated as evolution in complex domain. The method of tracing in complex domain is outlined and analyzed. It is proven that all change in complex domain stays within the limits defined by causal random algebraic groups. Eventually, (CRG) can reach stability (equilibrium) when repeated change in the complex domain finally leads to emergence from the unknown zone onto a state in real space. A rectangular slab is used to represent causal elements. The relationship between causal elements is depicted as a network of links and paths on each face of the slab, and the space between any two opposing faces. This space is refereed to as internal networks. Hyperplanes and hyperplane pencils are used to cut segments on the networks on the faces, and geodesics are used to explore zones in the internal networks. Zones on any face of the slab are considered to be in the real space, and zones in the internal network are in the complex space. If the boundary points of the zones are on tangent bundles to hyperplane pencils, then these points are non-singular, otherwise the zone contains singularity. Then the focus is shifted onto the internal networks, similar search is done this time with geodesics. Path on non-singular points connect stable, irreducible, cuasal elements, and any sub-group of (CRG) built on these points is complete and optimal.</p> M.M. Khoshyaran ##submission.copyrightStatement## 2020-12-15 2020-12-15 17 1 73 99 Variational Iteration Method for Partial Differential Equations with Piecewise Constant Arguments <p>In this paper, the variational iteration method is applied to solve the partial differential equations with piecewise constant arguments. This technique provides a sequence of functions which converges to the exact solutions of the problem and is based on the use of Lagrange<br>multipliers for identification of optimal value of a parameter in a functional. Employing this technique, we obtain the approximate solutions of the above mentioned equation in every interval [n, n + 1) (n = 0, 1, · · ·). Illustrative examples are given to show the efficiency of the<br>method.</p> Qi Wang ##submission.copyrightStatement## 2020-12-19 2020-12-19 17 1 100 108