5D BPS quivers and KK towers

We explore BPS quivers for D = 5 theories, compactified on a circle and geometrically engineered over local Calabi-Yau 3-folds, for which many of known machineries leading to (refined) indices fail due to the fine-tuning of the superpotential. For Abelian quivers, the counting reduces to a geometric one, but the technically challenging L2 cohomology proved to be essential for sensible BPS spectra. We offer a mathematical theorem to remedy the difficulty, but for non-Abelian quivers, the cohomology approach itself fails because the relevant wavefunctions are inherently gauge-theoretical. For the Cartan part of gauge multiplets, which suffers no wall-crossing, we resort to the D0 picture and reconstruct entire KK towers. We also perform numerical checks using a multi-center Coulombic routine, with a simple hypothesis on the quiver invariants, and extend this to electric BPS states in the weak coupling chamber. We close with a comment on known Donaldson-Thomas invariants and on how L2 index might be read off from these.

An extension of Shannon’s entropy to explain taxa diversity and human diseases

In this study, with the use of the information theory, we have proposed and proved a mathematical theorem by which we argue the reason for the existence of human diseases. To introduce our theoretical frame of reference, first, we put forward a modification of Shannon’s entropy, computed for all available proteomes, as a tool to compare systems complexity and distinguish between the several levels of biological organizations. We establish a new approach to differentiate between several taxa and corroborate our findings through the latest tree of life. Furthermore, we found that human proteins with higher mutual information, derived from our theorem, are more prone to be involved in human diseases. We further discuss the dynamics of protein network stability and offer probable scenarios for the existence of human diseases and their varying occurrence rates. Moreover, we account for the reasoning behind our mathematical theorem and its biological inferences.

Be grateful when a solution exists, because of Brouwer’s Fixed Point Theorem

“Be grateful when solutions exist, because of Brouwer’s Fixed Point Theorem” offers a introduction to a mathematical theorem asserting the existence of solutions to problems in engineering, medicine, economics, and other fields, as long as certain criteria are satisfied. The discussion is illustrated with numerous hand-drawn sketches. While this theorem assures the existence of a solution—a “fixed point”—it does not provide insight on how to find it. Still, it can be reassuring to know that a solution exists. For example, economist John von Newmann used this theorem in his 1937 economic model establishing that there exist prices at which supply equals demand. Mathematics students and enthusiasts are encouraged to be grateful for knowledge that a solution exists in mathematical and life pursuits, even when the solution itself remains elusive. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Does Geometric Algebra Provide a Loophole to Bell’s Theorem?

In 2007, and in a series of later papers, Joy Christian claimed to refute Bell’s theorem, presenting an alleged local realistic model of the singlet correlations using techniques from geometric algebra (GA). Several authors published papers refuting his claims, and Christian’s ideas did not gain acceptance. However, he recently succeeded in publishing yet more ambitious and complex versions of his theory in fairly mainstream journals. How could this be? The mathematics and logic of Bell’s theorem is simple and transparent and has been intensely studied and debated for over 50 years. Christian claims to have a mathematical counterexample to a purely mathematical theorem. Each new version of Christian’s model used new devices to circumvent Bell’s theorem or depended on a new way to misunderstand Bell’s work. These devices and misinterpretations are in common use by other Bell critics, so it useful to identify and name them. I hope that this paper can serve as a useful resource to those who need to evaluate new “disproofs of Bell’s theorem”. Christian’s fundamental idea is simple and quite original: he gives a probabilistic interpretation of the fundamental GA equation a · b = ( a b + b a ) / 2 . After that, ambiguous notation and technical complexity allows sign errors to be hidden from sight, and new mathematical errors can be introduced.

Small Steps and Great Leaps in Thought

We are justified in employing the rule of inference Modus Ponens (or one much like it) as basic in our reasoning. By contrast, we are not justified in employing a rule of inference that permits inferring to some difficult mathematical theorem from the relevant axioms in a single step. Such an inferential step is intuitively “too large” to count as justified. What accounts for this difference? This chapter canvasses several possible explanations. It argues that the most promising approach is to appeal to features like usefulness or indispensability to important or required cognitive projects. On the resulting view, whether an inferential step counts as large or small depends on the importance of the relevant rule of inference in our thought.

CALCULATION OF RATINGS AND PROGRAMMING LANGUAGES

Determining the rating is a significant and necessary factor in any assessment and decision. Rating calculation is based on mathematical theorem and very difficult calculations. Development of technology and programming languages simplified the rating process and made it available for everyone.

Two Flavours of Mathematical Explanation

A proof of a mathematical theorem tells us that the theorem is true (or should be accepted), but some proofs go further and tell us why the theorem is true (or should be accepted). That is, some, but not all, proofs are explanatory. Call this intra-mathematical explanation and it is to be contrasted with extra-mathematical explanation, where mathematics explains things external to mathematics. This chapter focuses on the intra-mathematical case. The authors consider a couple of examples of explanatory proofs from contemporary mathematics. They determine whether these proofs share some common feature that may account for their explanatoriness. The authors conclude with two plausible, but competing, accounts of mathematical explanation and suggest that there might be more than one kind of explanation at work in mathematics.

Teorema matematika dalam perancangan desain

World industry flourished and grown rapidly in all fields, as a piece of design, scattered in the discipline of interior design, graphic design, fashion design, product design, craft industry. But behind the glorious and legends in the design is not much known about the Chronicle and its struggle would be a mathematical theorem. Through this small scale study tries to analyze flashbacks oversize presence behind the creation of the design. With the approach of a mathematical theorem how do stories, it was created and in what form is present in the design?Keywords: mathematical theorems, applications in industrial design

Shoelace Formula: Connecting the Area of a Polygon and the Vector Cross Product

Understanding how one representation connects to another and how the essential ideas in that relationship are generalized can result in a mathematical theorem or a formula. In this article, we demonstrate this process by connecting a vector cross product in algebraic form to a geometric representation and applying a key mathematical idea from the relationship to prove the Shoelace theorem.