The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup $C{l}_1(\mathbb{Z}P)$ of $S{K}_1(\mathbb{Z}P)$, in terms of a genetic basis of P. We also introduce a deflation map $C{l}_1(\mathbb{Z}P)\to C{l}_1(\mathbb{Z}(P/N))$, for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $S{K}_1(\mathbb{Z}P)$, when P is an elementary abelian p-group.     

GE2-rings and a graph of unimodular rows

For a commutative ring A we consider a related graph, $\mathrm{\Gamma }(A)$, whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that $\mathrm{\Gamma }(A)$ is path-connected if and only if A is a ${\mathrm{GE}}_2$-ring, in the terminology of P. M. Cohn. Furthermore, if $Y(A)$ denotes the clique complex of $\mathrm{\Gamma }(A)$, we prove that $Y(A)$ is simply connected if and only if A is universal for ${\mathrm{GE}}_2$. More precisely, our main theorem is that for any commutative ring A the fundamental group of $Y(A)$ is isomorphic to the group $K_2(2,A)$ modulo the subgroup generated by symbols.     

Arithmetic invariants from Sato–Tate moments

We give some arithmetic-geometric interpretations of the moments ${\mathrm{M}}_2[a_1]$, ${\mathrm{M}}_1[a_2]$, and ${\mathrm{M}}_1[s_2]$ of the Sato–Tate group of an abelian variety A defined over a number field by relating them to the ranks of the endomorphism ring and Néron–Severi group of A.     

Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem

$?\mathrm{\Delta }u=\lambda a(x)e^u?4\pi N{\delta }_0\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a smooth and bounded domain in ${\mathbb{R}}^2$ such that $0\in \mathrm{\Omega }$, $a(x)$ is a positive smooth function, N is a positive integer and $\lambda >0$ is a small parameter. Here ${\delta }_0$ defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution $u_{\lambda }$ blowing up at 0 and satisfying $\lambda {\int }_{\mathrm{\Omega }}a(x)e^{u_{\lambda }}\to 8\pi (N+1)$ as $\lambda \to 0^{+}$.     

A note on the Hambleton-Taylor-Williams conjecture

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in the spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group G and noetherian ring R. The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,{\mathbb{F}}_3)$ is also a counterexample to the conjectured HTW-decomposition. Nevertheless, we prove that for any finite group G the rank of $G_1(\mathbb{Z}G)$ does not exceed the rank of the expression in the HTW-decomposition. We also show that the HTW-decomposition predicts correct torsion for $G_1(\mathbb{Z}G)$ for any finite group G. Furthermore, we prove that for any degree other than $n=1$ the conjecture gives a correct prediction for the rank of $G_n(\mathbb{Z}G)$.     

Generalized rainbow Turán numbers of odd cycles

Given graphs F and H, the generalized rainbow Turán number $\mathrm{ex}(n,F,\mathrm{rainbow-}H)$ is the maximum number of copies of F in an n-vertex graph with a proper edge-coloring that contains no rainbow copy of H. B. Janzer determined the order of magnitude of $\mathrm{ex}(n,C_s,\mathrm{rainbow-}{C}_t)$ for all $s\ge 4$ and $t\ge 3$, and a recent result of O. Janzer implied that $\mathrm{ex}(n,C_3,\mathrm{rainbow-}{C}_{2k})=O(n^{1+1/k})$. We prove the corresponding upper bound for the remaining cases, showing that $\mathrm{ex}(n,C_3,\mathrm{rainbow-}{C}_{2k+1})=O(n^{1+1/k})$. This matches the known lower bound for k even and is conjectured to be tight for k odd.     

Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system

$\{\begin{array}{cc}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_1{v}_t=\mathrm{\Delta }v?v+w\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_2{w}_t=\mathrm{\Delta }w?w+u\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty )\hfill \end{array}$

under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain $\mathrm{\Omega }?{\mathbb{R}}^4$ with positive parameters ${\tau }_1,{\tau }_2,\chi >0$ and nonnegative smooth initial data $(u_0,v_0,w_0)$.Global existence and boundedness of solutions were shown if ${6u_{0}6}_{L^1(\mathrm{\Omega })}<{(8\pi )}^2/\chi$ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ${6u_{0}6}_{L^1(\mathrm{\Omega })}>{(8\pi )}^2/\chi$. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has $8\pi /\chi$-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in ${\mathbb{R}}^4$.