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A unified geometric approach for the stability analysis of traveling pulse solutions for reaction diffusion equations with skew-gradient structure has been established in a previous paper (see Paul Cornwell [Indiana Univ. Math. J. 68 (2019), pp. 1801-1832]), but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equations. [ABSTRACT FROM AUTHOR]

We propose a "geometric Chevalley-Warning" conjecture, that is, a motivic extension of the Chevalley-Warning theorem in number theory. Its statement is equivalent to a recent question raised by F. Brown and O. Schnetz. In this paper we show that the conjecture is true for linear hyperplane arrangements, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in P3. The last section is devoted to verifying the conjecture for certain special kinds of hypersurfaces of any dimension. As a by-product, we obtain information on the Grothendieck classes of the affine "Potts model" hypersurfaces considered by Aluffi and Marcolli. [ABSTRACT FROM AUTHOR]

It has been proved in Janssens, Jespers, and Temmerman [Proc. Amer. Math. Soc. 145 (2017), pp. 2771–2783] that if h is an element of prime order p in a finite nilpotent group G and u=h+(h-1)g\widehat {h}\in \mathbb {Z}G, u\not \in G, then \langle u^*,u\rangle \approx C_p\ast C_p. We offer a simple geometric approach to generalize this result. [ABSTRACT FROM AUTHOR]

In this article, we take a geometric approach to extend Newton's well-known inequalities. A family of inequalities, which are invariant under affine transformations, is proposed for parallelotopes. We prove some of them and show the connections they have with matrix inequalities, convex geometry, and geometric inequalities. [ABSTRACT FROM AUTHOR]

We give a geometric characterization of the sets E ⊂ R for which every quasisymmetric embedding f : E → Rn extends to a quasisymmetric embedding f : R → RN for some N ≥ n. [ABSTRACT FROM AUTHOR]