Random version of Dvoretzky’s theorem in ℓpn

We study the dependence on $\epsilon$ in the critical dimension $k(n,p,\epsilon )$ for which one can find random sections of the ${?}_p^n$-ball which are $(1+\epsilon )$-spherical. We give lower (and upper) estimates for $k(n,p,\epsilon )$ for all eligible values $p$ and $\epsilon$ as $n\to \infty$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\infty$. Toward this end, we provide tight bounds for the Gaussian concentration of the ${?}_p$-norm.     

The odd primary homology of SU(n), Sp(n) and Spin(n)-gauge groups

For an odd prime $p$, we calculate the mod-$p$ homology of $SU\left(n\right)$-gauge groups over a simply-connected, closed 4-manifold for all $n\ge 2$. Similar calculations are obtained for the structure groups $Sp\left(n\right)$ when $n\ge 1$ and $Spin\left(n\right)$ for $n\ge 3$ (except for some cases when $n$ is even and $p=3$).     

Some applications of differential subordination

The authors introduce a two-parameter function $G(p,\lambda ;z)$ which is shown to play an important rôle in a unified presentation of many interesting subclasses of analytic and $p$-valent functions in the open unit disk $\mathbb{U}$. Making use of the familiar principle of differential subordination, several properties and relationships involving the function $G(p,\lambda ;z)$ are derived. Some sufficient conditions for $p$-valent starlikeness are also obtained.     

Lehmer's totient problem over Fq[x]

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in {\mathbb{F}}_q[x]$ and $\phi (q,p(x))$ be the Euler's totient function of $p(x)$ over ${\mathbb{F}}_q[x]$, where ${\mathbb{F}}_q$ is a finite field with q elements. We prove that $\phi (q,p(x))|(q^{\mathrm{deg}(p(x))}?1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3$, $p(x)$ is the product of any 2 non-associate irreducibles of degree 1; or (iii) $q=2$, $p(x)$ is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.     

The adversary degree-associated reconstruction number of double-brooms

A vertex-deleted subgraph of a graph G is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree-associated reconstruction number $\mathrm{adrn}(G)$ is the least k such that every set of k dacards determines G. We determine $\mathrm{adrn}(D_{m,n,p})$, where the double-broom $D_{m,n,p}$ with $p\ge 2$ is the tree with $m+n+p$ vertices obtained from a path with p vertices by appending m leaves at one end and n leaves at the other end. We determine $\mathrm{adrn}(D_{m,n,p})$ for all $m,n,p$. For $2\le m\le n$, usually $\mathrm{adrn}(D_{m,n,p})=m+2$, except $\mathrm{adrn}(D_{m,m+1,p})=m+1$ and $\mathrm{adrn}(D_{m,m+2,p})=m+3$. There are exceptions when $(m,n)=(2,3)$ or $p=4$. For $m=1$ the usual value is 4, with exceptions when $p\in \{2,3\}$ or $n=2$.     

Existence and multiplicity of solutions for a discontinuous problems with critical Sobolev exponents

In this paper, we consider the equation$?{\mathrm{\Delta }}_{p}u=\lambda {|u|}^{p^{?}?2}u+f(x,u)\text{in}{\mathbf{R}}^N,$ with discontinuous nonlinearity, where $1<p<N$, $\lambda >0$ is a real parameter and $p^{?}=\frac{Np}{N?p}$ is the critical Sobolev exponent. Under proper conditions on f, applying the nonsmooth critical point theory for locally Lipschitz functionals, we obtain at least one nontrivial nonnegative solution provided that $\lambda <{\lambda }_0$ and for any $k\in \mathbf{N}$, it has k pairs of nontrivial solutions if $\lambda <{\lambda }_k$, where ${\lambda }_0$ and ${\lambda }_k$ are positive numbers. In particular, we obtain the existence results for f is discontinuous in just one point.