It is well known that a smooth quartic curve has twenty-eight bitangent lines. For a reduced, possibly singular quartic curve, we introduce the notion of weak-bitangent line. This can be considered as a generalization of bitangent lines. In this article, we consider weak-bitangent lines for certain reduced quartic curves from the viewpoint of rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to deal with equations of weak-bitangent lines for certain reduced quartic curves. As a result, we can give new proofs for some classical results on singular quartic curves and their bitangent lines.

This paper presents a connection between Galois points and rational functions with small value sets over a finite field. This paper proves that a defining polynomial of any plane curve admitting two Galois points is an irreducible factor of a polynomial obtained from the equality of two rational functions in one variable for each. Under the assumption that Galois groups of two Galois points generate their semidirect product, a recent result of Bartoli, Borges, and Quoos indicates that one of these rational functions over a finite field has a very small value set. This paper shows that when two Galois points are external, the defining polynomial is an irreducible factor of the difference of two polynomials in one variable. This connects the study of Galois points to that of polynomials with small value sets.

Imanishi, Jinzenji and Kuwata provided a recipe for computing Euler number of Grassmann manifold $G\left(k,N\right)$ using physical model and its path-integral [S. Imanishi, M. Jinzenji and K. Kuwata, Journal of Geometry and Physics, Volume 180, October 2022, 104623]. They demonstrated that the cohomology ring of $G\left(k,N\right)$ is represented by fermionic variables. In this study, using only fermionic variables, we computed an integral of the Chern classes of the dual bundle of the tautological bundle on $G\left(k,N\right)$. In other words, the intersection number of the Schubert cycles is obtained using the fermion integral.

E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter [4] pointed out that by considering inner automorphism groups of quandles, one has a functor from the category of quandles with surjective homomorphisms to that of groups with surjective homomorphisms. In this paper, we focus on faithful quandles. As main results, we give a category equivalence between the category of faithful quandles with surjective quandle homomorphisms and that of pairs of groups and their conjugation-stable generators with suitable group homomorphisms. We are also interested in injective quandle homomorphisms. By defining suitable morphisms among pairs of groups and their conjugation-stable generators, we obtain a category which is equivalent to the category of faithful quandles with injective quandle homomorphisms.

The $L^p$ Sobolev-type inequality shows that the supremum of $\left|u\left(y\right)\right|$ defined on ${\mathbf{R}}^N$ is estimated from above by constant $C$ multiples of the $L^p$ norm of $\left(-\Delta +a^2\right)u\left(x\right)$. Among such constant $C$, the smallest constant is the best constant $C_0$. If we replace $C$ by $C_0$ in the $L^p$ Sobolev-type inequality, then the equality holds for the best function $U\left(x\right)$. The aim of this paper is to find $C_0$ and $U\left(x\right)$ of the $L^p$ Sobolev-type inequality. The Green function $G\left(x-y\right)$ of partial differential equation of elliptic type $\left(-\Delta +a^2\right)u\left(x\right)=f\left(x\right)$ defined on ${\mathbf{R}}^N$ is an important factor in this paper because $C_0$ and $U\left(x\right)$ consist of the Green function.

We study a general Lotka-Volterra competition-diffusion-advection system with general boundary conditions from river ecology. A complete classification on all possible long-time dynamical behaviors is established. Moreover, we investigate the joint effects of diffusion rates, advection rates, the inter-specific competition intensities and boundary conditions on global dynamics of the system. Finally, several numerical simulations are performed to verify the theoretical results. These results improve previously known ones by removing one condition and considering an interesting boundary condition where the species can be exposed to a net loss of individuals.