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AbstractIn a recent paper, Baker and Oliver Lorscheid showed that Descartes’s Rule of Signs and Newton’s Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebraic structures, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property. [ABSTRACT FROM AUTHOR]

Shame we are told is as intense as it is unrelenting. Shame we are told, further still, is as contagious as it is transmittable. By fastidiously attending to clinical reports and theoretical depictions of shame's totalizing character, I argue in this paper that we can notice something that has broadly gone unnoticed: shame assumes for itself absolute value. To bear this out, I contrast my view with that of the dominant one: that shame has an instrumental value. I show some limitations of the dominant view by reflecting on the role of shame in psychoanalytic training. The picture that emerges may seem quaintly commonsensical, and even commonplace, but it has, I think, vast clinical implications. I believe that it certainly has the potential to open some new horizons. The arguments for such a bold claim will occupy the concluding section of my paper. [ABSTRACT FROM AUTHOR]

The Capra operation is a graph theoretical operation that transforms the given graph into a structurally different and larger graph. It is commonly used in statistical analysis, data processing, and scientific research to compare and analyze data. The Capra operation involves subtracting the reference value from each numerical value, taking the absolute value of the resultand raising it to a specified power. This operation is used to create a new set of values that are more sensitive to differences between numerical values and can help to identify outliers and other anomalous data points. The Capra operation has numerous applications in fields such as finance, engineering, and biology, where it is used to analyze data and make predictions about future trends. Its ability to transform data and highlight differences between values makes it a valuable tool for researchers and analysts seeking to gain deeper insights into the patterns and structures within their data. In this paper, we studied the distance-based topological indices of some graphs under the Capra operation. [ABSTRACT FROM AUTHOR]

This work presents the controlling of the pitch orientation of a twin-rotor multi-input multi-output system (TRMS) using the sliding mode control (SMC) technique with variable gains. In this, the SMC is designed using variable gains where the pitch orientation error of the vertical dynamics of the TRMS system is intended to render the control law. The magnitude of the variable gains is related to the absolute value of the tracking error. Further, this technique minimised the chattering effect by ensuring the robustness property of the sliding modes. This design establishes a relationship between first and higher-order SMCs. The proposed controller is adapted from the twisting algorithm (TA) and conventional SMC. The control algorithm does not overestimate disturbances, resulting in a large reduction in control energy. Finally, a Lyapunov stability analysis is devised to demonstrate asymptotic stability. Simulations demonstrate the pitch orientation control and robustness of the proposed control scheme. [ABSTRACT FROM AUTHOR]

In the present work, a hybrid finite difference scheme is proposed and tested for the study of two-dimensional detonation problems. The hybrid scheme consists of a fifth-order weighted essentially nonoscillatory (WENO) method for discontinuous regions and a fourth or sixth-order robust conservative central finite difference scheme for the remaining domain. An arbitrary shock sensor is used to identify the discontinuities. The proposed shock sensor is based on the absolute value of the density gradient with an arbitrary threshold computed at each time step using a fast image segmentation technique. The sensor is complemented with a Ducros sensor to avoid the selection of vortices as discontinuities. The hybrid scheme is tested for several benchmark examples, including a two-dimensional transverse detonation wave, detonation wave diffraction, and a multiple obstacles test case. The obtained results show that the proposed sensor detects discontinuities adequately and the hybrid schemes give the same results as the fifth-order WENO in faster computational time. [ABSTRACT FROM AUTHOR]