Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

The de Rham Witt complex,cohomological kernels and pm-extensions in characteristic p

In earlier work the authors determined the Brauer kernel of extensions of degree p in characteristic $p>2$ where the Galois group is a semidirect product of order ps for $s|(p?1)$. This result is extended here and tools are developed to compute the cohomological kernels $H_{p^m}^{n+1}(E_m/F)$ for all $n\ge 0$ where $[E_m:F]=p^m$ and the Galois closure is a semidirect product of cyclic groups order $p^m$ and s where $s|(p?1)$. A six-term exact sequence describing the K-theory and cohomology of the extension is obtained. As an application it is shown that any F-division p-algebra of index $p^m$ split in $E_m$ is cyclic; a characteristic p analogue of a result of Vishne. The proofs use the de Rham Witt complex and Izhboldin groups, extending techniques developed earlier for the study of degree 4 extensions in characteristic two. The paper also provides background on the de Rham Witt Complex and Izhboldin groups difficult to track down in the literature.     

On the Schur multiplier of special p-groups

Let G be a special p-group minimally generated by $d\ge 3$ elements and having derived subgroup of order $p^{\frac{1}{2}d(d?1)}$. Berkovich asked to find the Schur multiplier and covering groups of such groups G Berkovich and Janko (2011) [1]. We try to give an answer to this question in this article.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

Cubic non-normal Cayley graphs of order 4p2

A Cayley graph $\mathrm{Cay}(G,S)$ on a group $G$ is said to be normal if the right regular representation $R\left(G\right)$ of $G$ is normal in the full automorphism group of $\mathrm{Cay}(G,S)$. In this paper all connected cubic non-normal Cayley graphs of order $4p^2$ are constructed explicitly for each odd prime $p$. It is shown that there are three infinite families of cubic non-normal Cayley graphs of order $4p^2$ with $p$ odd prime. Note that a complete classification of cubic non-Cayley vertex-transitive graphs of order $4p^2$ was given in [K. Kutnar, D. Marus?ic?, C. Zhang, On cubic non-Cayley vertex-transitive graphs, J. Graph Theory 69 (2012) 77–95]. As a result, a classification of cubic vertex-transitive graphs of order $4p^2$ can be deduced.     

On exceptional groups of order p5

A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient.The smallest examples of exceptional p-groups have order $p^5$. For an odd prime p, we classify all pairs $(G,Q)$ where G has order $p^5$ and Q is a distinguished quotient. (The case $p=2$ has already been treated by Easdown and Praeger.) We establish the striking asymptotic result that as p increases, the proportion of groups of order $p^5$ with at least one exceptional quotient tends to 1/2.     

Partial cohomology of groups and extensions of semilattices of abelian groups

We extend the notion of a partial cohomology group $H^n(G,A)$ to the case of non-unital A and find interpretations of $H^1(G,A)$ and $H^2(G,A)$ in the theory of extensions of semilattices of abelian groups by groups.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

On the Lp boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

Let $H=?\mathrm{\Delta }+V$ be a Schrödinger operator on $L^2({\mathbb{R}}^2)$ with real-valued potential V, and let $H_0=?\mathrm{\Delta }$. If V has sufficient pointwise decay, the wave operators $W_{\pm }=s?{\lim }_{t\to \pm \infty }?e^{itH}{e}^{?it{H}_0}$ are known to be bounded on $L^p({\mathbb{R}}^2)$ for all $1<p<\infty$ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on $L^p({\mathbb{R}}^2)$ for $1<p<\infty$. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents p.     

Quasilinear elliptic problems with critical exponents and Hardy terms

Let $\Omega \ni 0$ be an open bounded domain, $\Omega \subset R^N(N>p^2)$. We are concerned with the multiplicity of positive solutions of $-{\mathrm{\Delta }}_{p}u-\mu \frac{|u{|}^{p-2}u}{|x{|}^p}=\lambda |u{|}^{p-2}u+Q(x)|u{|}^{p^{*}-2}u,u\in W_0^{1,p}(\Omega )\text{,}$where $-{\mathrm{\Delta }}_{p}u=-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u),1<p<N,p^{*}=\frac{\mathit{Np}}{N-p},0<\mu <{\left(\frac{N-p}{p}\right)}^p,\lambda >0$and $Q(x)$ is a nonnegative function on $\overline{\Omega }$. By investigating the effect of the coefficient of the critical nonlinearity, we, by means of variational method, prove the existence of multiple positive solutions.     

On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O\left(k^5\right)}?n^9$. In this article we improve this running time to $2^{O\left(k^2\right)}?n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O\left(k^5\right)$ to $O\left(k^2\right)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p?1$ for $p\ge 3$, we speed up the running time to $2^{O(k?k^{\frac{1}{p?1}})}?n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p\ge 3$, Weighted Independent Set cannot be solved in time $2^{o\left(k\right)}?n^{O\left(1\right)}$ in the class of $\{\text{bull},C_4,\dots ,C_{2p?1}\}$-free graphs unless the ETH fails.     

Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Let $\Omega ?{\mathbb{R}}^N$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)$\mathrm{?}\zeta =G\zeta ,\zeta {|}_{\Gamma }=0,$ vanishes if $G\in {\mathsf{L}}^1(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ and $\zeta \in {\mathsf{W}}^{1,1}(\Omega ;{\mathbb{R}}^N)$. In particular, square-integrable solutions ζ of (1) with $G\in {\mathsf{L}}^1\cap {\mathsf{L}}^2(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ vanish. As a consequence, we prove that:$???:{\mathsf{C}}_{°}^{\infty }(\Omega ,\Gamma ;{\mathbb{R}}^3)\to [0,\infty ),u?6\mathrm{sym}\left(\mathrm{?}u{P}^{?1}\right)6_{{\mathsf{L}}^2(\Omega )}$ is a norm if $P\in {\mathsf{L}}^{\infty }(\Omega ;{\mathbb{R}}^{3\times 3})$ with $\mathrm{Curl}P\in {\mathsf{L}}^p(\Omega ;{\mathbb{R}}^{3\times 3})$, $\mathrm{Curl}{P}^{?1}\in {\mathsf{L}}^q(\Omega ;{\mathbb{R}}^{3\times 3})$ for some $p,q>1$ with $1/p+1/q=1$ as well as $\det P?c^{+}>0$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $\Phi \in {\mathsf{H}}^1(\Omega ;{\mathbb{R}}^3)$, $\Omega ?{\mathbb{R}}^3$, satisfy $\mathrm{sym}(\mathrm{?}{\Phi }^{?}\mathrm{?}\Psi )=0$ for some $\Psi \in {\mathsf{W}}^{1,\infty }(\Omega ;{\mathbb{R}}^3)\cap {\mathsf{H}}^2(\Omega ;{\mathbb{R}}^3)$ with $\det \mathrm{?}\Psi ?c^{+}>0$. Then there exists a constant translation vector $a\in {\mathbb{R}}^3$ and a constant skew-symmetric matrix $A\in \mathfrak{so}(3)$, such that $\Phi =A\Psi +a$.