It is shown that the maximum size of a set $S$ of vectors of a $k$-dimensional vector space over ${𝔽}_q$, with the property that every subset of size $k$ is a basis, is at most $q+1$, if $k\le p$, and at most $q+k-p$, if $q\ge k\ge p+1\ge 4$, where $q=p^k$ and $p$ is prime. Moreover, for $k\le p$, the sets $S$ of maximum size are classified, generalising Beniamino Segre’s “arc is a conic” theorem. These results have various implications. One such implication is that a $k\times (p+2)$ matrix, with $k\le p$ and entries from ${𝔽}_p$, has $k$ columns which are linearly dependent....

For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl${}_c$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...

The symbol ${\left(X_I\right)}_{\kappa }$ (with κ ≥ ω) denotes the space $X_I:={\prod }_{i\in I}{X}_i$ with the κ-box topology; this has as base all sets of the form $U={\prod }_{i\in I}{U}_i$ with $U_i$ open in $X_i$ and with $|i\in I:U_i\ne X_i|<\kappa$. The symbols w, d and S denote respectively the weight, density character and Suslin number. Generalizing familiar classical results, the authors show inter alia: Theorem 3.1.10(b). If κ ≤ α⁺, |I| = α and each $X_i$ contains the discrete space 0,1 and satisfies $w\left(X_i\right)\le \alpha$, then $w\left(X_{\kappa }\right)={\alpha }^{<\kappa }$. Theorem 4.3.2. If $\omega \le \kappa \le \left|I\right|\le 2^{\alpha }$ and $X={\left(D\left(\alpha \right)\right)}^I$ with D(α) discrete, |D(α)| = α, then $d\left({\left(X_I\right)}_{\kappa }\right)={\alpha }^{<\kappa }$. Corollaries 5.2.32(a)...

Let $G:=\mathrm{SO}{(n,1)}^{\circ }$ and $\Gamma (n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma$ in a homogeneous space $H\setminus G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{ℬ}_T\subset H\setminus G\}$ of compact subsets, there exists $\eta >0$ such that $\#\left[e\right]\Gamma \cap {ℬ}_T=ℳ\left({ℬ}_T\right)+O\left(ℳ{\left({ℬ}_T\right)}^{1-\eta }\right)$ for an explicit measure $ℳ$ on $H\setminus G$ which depends on $\Gamma$. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma$ in arithmetic...

A real form $G$ of a complex semi-simple Lie group $G^{ℂ}$ has only finitely many orbits in any given $G^{ℂ}$-flag manifold $Z=G^{ℂ}/Q$. The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle...

Given an integral scheme $X$ over a non-archimedean valued field $k$, we construct a universal closed embedding of $X$ into a $k$-scheme equipped with a model over the field with one element ${𝔽}_1$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^{\mathrm{an}}$. Moreover, using the scheme-theoretic...

Given a locale $L$ and a join semilattice $J$ with bottom element $0_J$, a new concept $(\sigma ,J)$ called $L$-slice is defined,where $\sigma$ is as an action of the locale $L$ on the join semilattice $J$. The $L$-slice $(\sigma ,J)$ adopts topological properties of the locale $L$ through the action $\sigma$. It is shown that for each $a\in L$, ${\sigma }_a$ is an interior operator on $(\sigma ,J)$.The collection $M=\{{\sigma }_a;a\in L\}$ is a Priestly space and a subslice of $L$-$Hom(J,J)$. If the locale $L$ is spatial we establish an isomorphism between the $L$-slices $(\sigma ,L)$ and $(\delta ,M)$. We have shown that the fixed...

In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ ${𝒫}_{\mathrm{lf},\mathrm{c}}$ (Every locally finite connected graph has a maximal independent set). $\circ$ ${𝒫}_{\mathrm{lc},\mathrm{c}}$ (Every locally countable connected graph has a maximal independent set). $\circ$ CAC${}_1^{{\aleph }_{\alpha }}$ (If in a partially ordered set all antichains are finite and all chains have size ${\aleph }_{\alpha }$, then the set has size ${\aleph }_{\alpha }$) if ${\aleph }_{\alpha }$ is regular. $\circ$ CWF (Every partially ordered set has a...

A metacyclic group $H$ can be presented as $\langle \alpha ,\beta :{\alpha }^n=1$, ${\beta }^m={\alpha }^t$, $\beta \alpha {\beta }^{-1}={\alpha }^r\rangle$ for some $n$, $m$, $t$, $r$. Each endomorphism $\sigma$ of $H$ is determined by $\sigma \left(\alpha \right)={\alpha }^{x_1}{\beta }^{y_1}$, $\sigma \left(\beta \right)={\alpha }^{x_2}{\beta }^{y_2}$ for some integers $x_1$, $x_2$, $y_1$, $y_2$. We give sufficient and necessary conditions on $x_1$, $x_2$, $y_1$, $y_2$ for $\sigma$ to be an automorphism.

Let $G$ be the semidirect product $V⋊K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$. Let $\pi$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$. We construct an invariant symbolic calculus for $\pi$, under some technical hypothesis. We give some examples including the Poincaré group. ...

Let $X$ be the Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff non-single point spaces $X_{\alpha }$. Let $p\in X^{*}$ be a point in the closure of some $G\subset X$ whose weak Lindelöf number is strictly less than the cofinality of $\tau$. Then we show that $\beta X\setminus \left\{p\right\}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $\beta X$. In particular, this is true if either $X={\omega }^{\tau }$ or $X=R^{\tau }$ and $\tau$ is infinite and not countably cofinal.

Given a Tychonoff space $X$ and an infinite cardinal $\kappa$, we prove that exponential $\kappa$-domination in $X$ is equivalent to exponential $\kappa$-cofinality of $C_p\left(X\right)$. On the other hand, exponential $\kappa$-cofinality of $X$ is equivalent to exponential $\kappa$-domination in $C_p\left(X\right)$. We show that every exponentially $\kappa$-cofinal space $X$ has a ${\kappa }^{+}$-small diagonal; besides, if $X$ is $\kappa$-stable, then $nw\left(X\right)\le \kappa$. In particular, any compact exponentially $\kappa$-cofinal space has weight not exceeding $\kappa$. We also establish that any exponentially $\kappa$-cofinal...

Let $k$ be a field of characteristic $p\&gt;0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...

Let ${𝕄}_{n,m}$ be the set of all $n\times m$ real or complex matrices. For $A,B\in {𝕄}_{n,m}$, we say that $A$ is row-sum majorized by $B$ (written as $A{\prec }^{\mathrm{rs}}B$) if $R\left(A\right)\prec R\left(B\right)$, where $R\left(A\right)$ is the row sum vector of $A$ and $\prec$ is the classical majorization on ${ℝ}^n$. In the present paper, the structure of all linear operators $T:{𝕄}_{n,m}\to {𝕄}_{n,m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on ${ℝ}^n$ and then find the linear preservers of row-sum majorization of these relations on ${𝕄}_{n,m}$. ...

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...