arXiv subjects
Publications and source records attributed to Alexandre Laugier.
In this article, based on ideas and results by J. S\'andor (2001, 2004), we define $k$-multiplicatively $e$-perfect numbers and $k$-multiplicatively $e$-superperfect numbers and prove some results on them. We also characterize the $k$-$T_0T^\ast$-perfect numbers defined by Das and Saikia (2013) in details.
In this paper we propose a definition of a recurrence relation homomorphism and illustrate our definition with a few examples. We then define the period of a k-th order of linear recurrence relation and deduce certain preliminary results associated with them.
Some basic properties of the ring of integers $\mathbb{Z}$ are extended to entire rings. In particular, arithmetic in entire principal rings is very similar than arithmetic in the ring of integers $\mathbb{Z}$. These arithmetic properties are derived from a $\star$-ring extension of the considered entire ring (ring extension with conjugation) equipped with a real function which is a multiplicative structure-preserving map between two algebras. The algebra of this ring extension is studied in detail. Some examples of such ring extension are given.
We give a new proof of Lucas' Theorem in elementary number theory.
In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for $F_p$, where $p$ is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, $m$ and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.
We give a few remarks on the periodic sequence $a_n=\binom{n}{x}~(mod~m)$ where $x,m,n\in \mathbb{N}$, which is periodic with minimal length of the period being $$\ell(m,x)={\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor+b_i}_i}=m{\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor}_i}$$ where $m=\prod^w_{i=1}p^{b_i}_i$. We prove certain interesting properties of $\ell(m,x)$ and derive a few other results and congruences.
We complete a proof of a theorem that was inspired by an Indian Olympiad problem, which gives an interesting characterization of a prime number $p$ with respect to the binomial coefficients ${n\choose p}$. We also derive a related result which generalizes the theorem in one direction.
Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, we considered the definition of orthonormal basis in Minkowski space, the structure of metric tensor relative to orthonormal basis, procedure of orthogonalization. Linear transformation of Minkowski space mapping at least one orthonormal basis into orthonormal basis is called motion. The set of motions of Minkowski space V generates not complete group SO(V) which acts single transitive on the basis manifold. Passive transformation of Minkowski space mapping at least one orthonormal basis into orthonormal basis is called quasimotion of Minkowski space. The set of passive transformations of Minkowski space generates passive representation of not complete group SO(V) on basis manifold. Since twin representations (active and passive) of not complete group SO(V) on basis manifold are single transitive, then we may consider definition of geometric object.
Given a bidirected ring with capacities and a demand graph, we present an approximation algorithm to the problem of finding the minimum $\alpha$ such that there exists a feasible unsplittable routing of the demands after multiplying each capacity by $\alpha$. We also give an approximation scheme to the problem.
We introduce a new class of optimization problems called integer Minkowski programs. The formulation of such problems involves finitely many integer variables and nonlinear constraints involving functionals defined on families of discrete or polyhedral sets. We show that, under certain assumptions, it is possible to reformulate them as integer linear programs, by making use of integral generating sets. We then apply this technique to the network design problem for fractional and integral flows subject to survivability constraints.
In solid phase the pressure correlates to the elastic related volume change while the temperature to the thermal related volume change. These volume changes are not compatible with the exception of constant volume condition when the expanded volume converts completely compressed volume. Separating the thermal and elastic related volume changes the work functions for each of the thermodynamic conditions are derived. Based on theoretical consideration it is suggested that the thermal related volume change do not result mechanical work. Homogeneous model, both the system and the surrounding have the same phase, can completely explain the lack of the thermal related work and provide a self-consistent thermodynamic description for the elastic solids.
Using the contemporary thermodynamic equations of elastic solids leads to contradictions with the fundamental statements of thermodynamics. Two examples are presented to expose the inconsistencies. In example one the internal energy between the initial and final states shows path dependency while in example two changing the temperature of a system at constant volume produces mechanical work. These results are contradictory with the fundamentals of thermodynamics and indicate that the contemporary description of elastic solids needs to be revisited and revised.