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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]