On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the n-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length n to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.     

Schmidt representation of bilinear operators on Hilbert spaces

Current work defines Schmidt representation of a bilinear operator $T:H_1\times H_2\to K$, where $H_1,H_2$ and $K$ are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if $T$ is compact, and its singular values are ordered, then $T$ has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental.     

Generalized almost perfect nonlinear binomials and trinomials over fields of prime-square order

Let $p>3$ be a prime. We show that, for each integer d with $p\le d\le 2(p-1)$, there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over ${\mathbb{F}}_{p^2}$ of algebraic degree d. We start by deriving sufficient conditions for the function $G:{\mathbb{F}}_{p^2}\to {\mathbb{F}}_{p^2},X\mapsto X^{d_1}+u{X}^{d_2}$ to be GAPN in the case where one of the terms of G is GAPN. We then give explicit constructions of GAPN binomials over ${\mathbb{F}}_{p^2}$ of any odd algebraic degree between p and $2(p-1)$ and, in the case where p is not a Mersenne prime, also of any even algebraic degree in this range. To obtain GAPN functions of even algebraic degree also in the general case, we finally show how to construct GAPN trinomials over ${\mathbb{F}}_{p^2}$ of any even algebraic degree between p and $2(p-1)$ by applying a characterization of a special form of GAPN binomials by Özbudak and Sălăgean. Our constructed functions are the first GAPN functions of even algebraic degree over extension fields of odd characteristic reported so far.     

Forcing clique immersions through chromatic number

Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree $7t+7$ contains $K_t$ as an immersion and that every graph with chromatic number at least $3.54t+4$ contains $K_t$ as an immersion. We also show that every graph on $n$ vertices with no independent set of size three contains $K_{2\lfloor n∕5\rfloor }$ as an immersion.     

Standing waves for the NLS on the double-bridge graph and a rational–irrational dichotomy

We study standing waves of NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, $L_1$ and $L_2$. We study the solutions with possibly nontrivial components on the half-lines and a cnoidal component on the circle. The problem is equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter ω. After classifying the solutions with rational $L_1/L_2$, we turn to $L_1/L_2$ irrational showing that there exist standing waves only in correspondence to a countable set of negative frequencies ${\omega }_n$. Moreover we show that the frequency sequence admits cluster points and any negative real number can be a limit point of frequencies choosing a suitable irrational geometry $L_1/L_2$. These results depend on basic properties of diophantine approximation of real numbers.     

On the sum of two subnormal kernels

We show, by means of a class of examples, that if $K_1$ and $K_2$ are two positive definite kernels on the unit disc such that the multiplication by the coordinate function on the corresponding reproducing kernel Hilbert space is subnormal, then the multiplication operator on the Hilbert space determined by their sum $K_1+K_2$ need not be subnormal. This settles a recent conjecture of Gregory T. Adams, Nathan S. Feldman and Paul J. McGuire in the negative. We also discuss some cases for which the answer is affirmative.     

On three color Ramsey numbers R(C4,C4,K1,n)

For $k$ given graphs $G_1,G_2,\dots ,G_k$, $k\ge 2$, the $k$-color Ramsey number, denoted by $R(G_1,G_2,\dots ,G_k)$, is the smallest integer $N$ such that if we arbitrarily color the edges of a complete graph of order $N$ with $k$ colors, then it always contains a monochromatic copy of $G_i$ colored with $i$, for some $1\le i\le k$. Let $C_m$ be a cycle of length $m$ and $K_{1,n}$ a star of order $n+1$. In this paper, firstly we give a general upper bound of $R(C_4,C_4,\dots ,C_4,K_{1,n})$. In particular, for the 3-color case, we have $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+3$ and this bound is tight in some sense. Furthermore, we prove that $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+2$ for all $n={?}^2??$ and $?\ge 2$, and if $?$ is a prime power, then the equality holds.     

Higher Laplacians on pseudo-Hermitian symmetric spaces

One of the most fundamental operators studied in geometric analysis is the classical Laplace–Beltrami operator. On pseudo-Hermitian manifolds, higher Laplacians $L_m$ are defined for each positive integer m, where $L_1$ coincides with the Laplace–Beltrami operator. Despite their natural definition, these higher Laplacians have not yet been studied in detail. In this paper, we consider the setting of simple pseudo-Hermitian symmetric spaces, i.e., let $X=G/H$ be a symmetric space for a real simple Lie group G, equipped with a G-invariant complex structure. We show that the higher Laplacians $L_1,L_3,\dots ,L_{2r?1}$ form a set of algebraically independent generators for the algebra ${\mathbb{D}}_G(X)$ of G-invariant differential operators on X, where r denotes the rank of X. For higher rank, this is the first instance of a set of generators for ${\mathbb{D}}_G(X)$ defined explicitly in purely geometric terms, and confirms a conjecture of Engli? and Peetre, originally stated in 1996 for the class of Hermitian symmetric spaces.     

Graphs determined by their Aα-spectra

Let $G$ be a graph with $n$ vertices, and let $A\left(G\right)$ and $D\left(G\right)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1?\alpha )A\left(G\right)$for any real $\alpha \in [0,1]$. The collection of eigenvalues of $A_{\alpha }\left(G\right)$ together with multiplicities are called the $A_{\alpha }$-spectrum of $G$. A graph $G$ is said to be determined by its $A_{\alpha }$-spectrum if all graphs having the same $A_{\alpha }$-spectrum as $G$ are isomorphic to $G$. We first prove that some graphs are determined by their $A_{\alpha }$-spectra for $0\le \alpha <1$, including the complete graph $K_n$, the union of cycles, the complement of the union of cycles, the union of copies of $K_2$ and $K_1$, the complement of the union of copies of $K_2$ and $K_1$, the path $P_n$, and the complement of $P_n$. Setting $\alpha =0$ or $\frac{1}{2}$, those graphs are determined by $A$- or $Q$-spectra. Secondly, when $G$ is regular, we show that $G$ is determined by its $A_{\alpha }$-spectrum if and only if the join $G\vee K_m$ ($m\ge 2$) is determined by its $A_{\alpha }$-spectrum for $\frac{1}{2}<\alpha <1$. Furthermore, we also show that the join $K_m\vee P_n$ ($m,n\ge 2$) is determined by its $A_{\alpha }$-spectrum for $\frac{1}{2}<\alpha <1$. In the end, we pose some related open problems for future study.     

Spanning trees with at most 4 leaves in K1,5-free graphs

In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with ${\sigma }_4\left(G\right)\ge n?1$ contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw’s result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with ${\sigma }_5\left(G\right)\ge n?1$ contains a spanning tree with at most 4 leaves. Moreover, the degree sum condition “${\sigma }_5\left(G\right)\ge n?1$” is best possible.     

Multiplicative preprojective algebras of Dynkin quivers

For a commutative ring R and an ADE Dynkin quiver Q, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as R-algebras if and only if the bad primes for Q – 2 in type D, 2 and 3 for $Q=E_6$, $E_7$ and 2, 3 and 5 for $Q=E_8$ – are invertible in R. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2,1/3]$ for $Q=E_6$, $E_7$ and over $\mathbb{Z}[1/2,1/3,1/5]$ for $Q=E_8$. Conversely, if some bad prime is not invertible in R, we show that the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest.In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etgü–Lekili [5, Theorem 13] in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra, computed in Section 4, can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if $Q\ne A_1$, a departure from the additive preprojective algebra in characteristic 2 for $Q=D_{2n}$, $n\ge 2$ and $Q=E_7$, $E_8$.