On the kth root partition function

For any positive integer [Formula: see text], let [Formula: see text] be the number of solutions of the equation [Formula: see text] with integers [Formula: see text], where [Formula: see text] is the integral part of real number [Formula: see text]. Recently, Luca and Ralaivaosaona gave an asymptotic formula for [Formula: see text]. In this paper, we give an asymptotic development of [Formula: see text] for all [Formula: see text]. Moreover, we prove that the number of such partitions is even (respectively, odd) infinitely often.

Estimation of Thermal Resistance Field in Layered Materials by Analytical Asymptotic Method

In this paper, the problem of the quantitative characterization of thermal resistance fields in a multilayer sample is addressed by using the classical front face flash method as the thermal excitation and infrared thermography (IRT) as the monitoring sensor. In this challenging problem, the complete inverse processing of a multilayer analytical model is difficult due to the lack of sensitivity of some parameters (layer thickness, depth of thermal resistance, etc.) and the expansive computational iterative processing. For these reasons, the proposed strategy is to use a simple multilayer problem where only one resistive layer is estimated. Moreover, to simplify the inverse processing often based on iterative methods, an asymptotic development method is proposed here. Regarding the thermal signal reconstruction (TSR) methods, the drawback of these methods is the inability to be quantitative. To overcome this problem, the method incorporates a calibration process originating from the complete analytical quadrupole solution to the thermal problem. This analytical knowledge allows self-calibration of the asymptotic method. From this calibration, the quantitative thermal resistance field of a sample can be retrieved with a reasonable accuracy lower than 5%.

“What Kind of a Science is Sustainability Science?” An Evidence-Based Reexamination

Sustainability science (SS), rooted in multiple disciplines, has been developing rapidly during the last two decades and become a well-recognized new field of study. However, the “identity” of SS remains unclear. Therefore, this study was intended to help synthesize the key characteristics of SS by revisiting the question raised by the leading sustainability scientist, Robert Kates (2011): “What kind of a science is sustainability science?” Specifically, we reviewed the literature in SS, and developed a synthesis of definitions and core research questions of SS, using multiple methods including change-point detection, word cloud visualization, and content and thematic analyses. Our study has produced several main findings: (1) the development of SS exhibited an S-shaped growth pattern, with an exponential growth phase through to 2012, and a asymptotic development phase afterwards; (2) ten key elements from the existing definitions of SS were identified, of which understanding “human–environment interactions” and “use-inspired” were most prominent; and (3) sixteen core questions in SS were derived from the literature. We further proposed an eight-theme framework of SS to help understand how the sixteen questions are related to each other. We argue that SS is coming of age, but more integrative and concerted efforts are still needed to further consolidate its identity by developing a coherent and rigorous scientific core.

Axial–transversal coupling in the free nonlinear vibrations of Timoshenko beams with arbitrary slenderness and axial boundary conditions

The nonlinear free oscillations of a straight planar Timoshenko beam are investigated analytically by means of the asymptotic development method. Attention is focused for the first time, to the best of our knowledge, on the nonlinear coupling between the axial and the transversal oscillations of the beam, which are decoupled in the linear regime. The existence of coupled and uncoupled motion is discussed. Furthermore, the softening versus hardening nature of the backbone curves is investigated in depth. The results are summarized by means of behaviour charts that illustrate the different possible classes of motion in the parameter space. New, and partially unexpected, phenomena, such as the changing of the nonlinear behaviour from softening to hardening by adding/removing the axial vibrations, are highlighted.