Relativization makes contradictions harder for Resolution

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res(d)$\mathrm{Res}(d)$, as well as their tree-like versions, Res^?(d)${\mathrm{Res}}^{?}(d)$. The contradictions we use are natural combinatorial principles: the Least number principle  , _LNPn${\mathrm{LNP}}_n$ and an ordered variant thereof, the Induction principle  , _IPn${\mathrm{IP}}_n$.     

Homotopy of area decreasing maps by mean curvature flow

Let f:M→N$f:M\to N$ be a smooth area decreasing map between two Riemannian manifolds (M,_gM)$(M,{\mathrm{g}}_M)$ and (N,_gN)$(N,{\mathrm{g}}_N)$. Under weak and natural assumptions on the curvatures of (M,_gM)$(M,{\mathrm{g}}_M)$ and (N,_gN)$(N,{\mathrm{g}}_N)$, we prove that the mean curvature flow provides a smooth homotopy of f to a constant map.     

On the norming constants for normal maxima

Given n   independent standard normal random variables, it is well known that their maxima _Mn$M_n$ can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance _dn$d_n$ between the normalized _Mn$M_n$ and its associated limit distribution is less than 3/log?n$3/\log ?n$. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with _dn≤C(m)/log?n$d_n\le C(m)/\log ?n$ for n≥m≥5$n\ge m\ge 5$. Furthermore, the function C(m)$C(m)$ is computed explicitly, which satisfies C(m)≤1$C(m)\le 1$ and _limm→∞?C(m)=1/3${\lim }_{m\to \infty }?C(m)=1/3$. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.     

Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)$W(\mathbb{D})$ of analytic functions on the unit disc D$\mathbb{D}$ of the complex plane C$\mathbb{C}$. A complex number λ$\lambda$ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVA$\mathit{AV}=\lambda \mathit{VA}$. We prove that the set of all extended eigenvalues of V   is precisely the set C?{0}$\mathbb{C}?\{0\}$, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V$V$. The similar result for some weighted shift operator on _?p${?}_p$ spaces is also obtained.     

On additive-nilpotency of Jacobian matrices of polynomial maps

Let H:Cⁿ→Cⁿ$H:{\mathbb{C}}^n\to {\mathbb{C}}^n$ be a polynomial map, JH the Jacobian matrix of H  , and _VH$V_H$ the linear subspace of _Mn(C)$M_n(\mathbb{C})$ spanned by {JH(α)|α∈Cⁿ}$\{JH(\alpha )|\alpha \in {\mathbb{C}}^n\}$, the set of evaluated Jacobian matrices of H  . We describe the dimension of _VH$V_H$ through a power-linear Gorni–Zampieri mate of H, and give a numerical algorithm for computing this dimension and determining whether JH   is additive-nilpotent, i.e., whether _VH$V_H$ is a nilpotent subspace.