We present some conceptual reflections on time in the context of thermodynamics with internal variables or with internal fluxes with relaxation times of several different orders of magnitude. The combination between reversible and irreversible contributions to the dy- namics depends on the observation timescale, and opens new aspects to the usual discussion on the contrast between reversibility and irreversibility. As an illustration of such situations, two different kinds of hierarchies of internal variables are presented. In one of them, all the variables are vectors following linear relaxational evolution equations, with their respective relaxation times following a potential law. In the other one, all the variables have a different tensorial order, in such a way that the variable of order i is the flux of the precedent variable of order i - 1, but all of them have the same relaxation time. Some mathematical and physical features of both hierarchies are examined.

Among Celestino Galiani’s papers, some handwritten documents concerning Galiani’s activity as a consultant to the papal throne have been found, all concerning issues that arose between the end of the seventeenth century and the beginning of the eighteenth century, when the various states began to play a significant role in the management of the lotto game and discussions began regarding its ethical legality and the actual forms of management. The problem was particularly felt in the Papal State and the reflections of Galiani, who had assumed several roles and functions of primary importance within the Church, played an important role in the decisions taken by the papal court. In this paper, we present Galiani’s mathematical education and an initial analysis, based on manuscript documents, of Galiani’s activity and thought regarding the Lotto game.

In this article we discuss some controversial orthographic rules for the Romanian language regarding the letters 1 and a, both of which represent a single sound, similar to the one corresponding to the letter i in the English word pencil, but with a slightly different pronunciation nuance. The rules, established by the orthographic reforms of the Romanian Academy in 1904 and 1993, refer to the positions of the twin letters 1 and a in Romanian words that imply the corresponding sound. We show that these rules are illogical and sometimes lead to writing difficulties for Romanian speakers. Therefore, we strongly propose the use of the letter 1 everywhere, except for the words roman, Romania and their derivatives, which recall the Latin origin of the Romanian language. Let us note that in 1953 the same Academy had proposed the use of the letter 1 everywhere, including in the words romln, Romlnia and their derivatives, so our proposal is not completely new. But our logical analysis of the orthographic rules of 1904 and 1993 legitimizes our proposal, which guarantees clarity and simplicity. Please note that the full article that follows is written in Romanian, as it is of interest mainly to Romanian speakers.

The paper provides the existence of a positive weak solution and a priori estimates for a class of parametric Dirichlet problems with unbounded variable exponents and exhibiting exponential growth.The approach relies on a special sub-supersolution method that we develop in our general setting.The bound of the admissible values of the parameter is explicitly determined.Applications to the regularity properties and asymptotic behavior with respect to the parameter are also given.Examples demonstrate the applicability of the stated results.

The purpose of our paper is to investigate the geometric behavior of complete noncompact and stochastically complete quasi k-Yamabe gradient solitons under appropriate conditions in order to obtain new triviality and nonexistence results. For this, we derive a suitable Bochner type formula and we apply it jointly with integrability criteria, Liouville type results and several maximum principles dealing, in particular, with the notion of convergence to zero at infinity and the concept of polynomial and exponential volume growth.

First-passage problems are considered for a Wright-Fisher diffusion process {X(t), t > 0}, which is important in population genetics and mathematical finance. Let t(x) be the first time that X(t), starting from X(0) = x, is equal to 0 or to 1. We compute the expected area covered by the process in the interval [0, t (x)]. We also compute the probability that the process will take on a value smaller than or equal to 0 before a value greater than or equal to 1 in the case when there are uniform jumps according to a Poisson process. Finally, stochastic control problems called homing problems are solved explicitly in particular cases for a controlled version of the process {X (t), t > 0}.

We present new Zabczyk-Rolewicz type methods of exploring the uniform exponential stability of nonautonomous dynamics.First, we give a characterization of Zabczyk type for uniform exponential stability of discrete nonautonomous systems.Next, we obtain a new criterion of Zabczyk-Rolewicz type for uniform exponential stability, using general nonlinear trajectories.In addition, we discuss consequences for uniformly bounded discrete nonautonomous systems.Furthermore, we apply our results to deduce the famous stability theorem of Rolewicz for continuous-time dynamics.

Conversations about weather, environment, health, cuisine, and even politics all involve the word ”temperature”. It was an attempt to understand the working of the heat engines that gave the temperature a clear definition. In this paper, we put the equilibrium thermodynamics definition into a larger context of multiscale thermodynamics, the multiscale rate thermodynamics, and nonphysical environments.

Let B2t be a bipartite planar graph with an even number of regions. We are able to find bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to the graph. We will also investigate the minimal vertex covers and the maximum matchings related to such a graph.