In this paper we study the orbit closure problem for a reductive group $G\subseteq GL(X)$ acting on a finite dimensional vector space $V$ over $\C$. We assume that the center of $GL(X)$ lies within $G$ and acts on $V$ through a fixed non-trivial character. We study points $y,z\in V$ where (i) $z$ is obtained as the leading term of the action of a 1-parameter subgroup $\lambda (t)\subseteq G$ on $y$, and (ii) $y$ and $z$ have large distinctive stabilizers $K,H \subseteq G$. Let $O(z)$ (resp. $O(y)$) denote the $G$-orbits of $z$ (resp. $y$), and $\overline{O(z)}$ (resp. $\overline{O(y)}$) their closures, then (i) implies that $z\in \overline{O(y)}$. We address the question: under what conditions can (i) and (ii) be simultaneously satisfied, i.e, there exists a 1-PS $\lambda \subseteq G$ for which $z$ is observed as a limit of $y$. Using $\lambda$, we develop a leading term analysis which applies to $V$ as well as to ${\cal G}= Lie(G)$ the Lie algebra of $G$ and its subalgebras ${\cal K}$ and ${\cal H}$, the Lie algebras of $K$ and $H$ respectively. Through this we construct the Lie algebra $\hat{\cal K} \subseteq {\cal H}$ which connects $y$ and $z$ through their Lie algebras. We develop the properties of $\hat{\cal K}$ and relate it to the action of ${\cal H}$ on $\overline{N}=V/T_z O(z)$, the normal slice to the orbit $O(z)$. We examine the case of {\em alignment} when a semisimple element belongs to both ${\cal H}$ and ${\cal K}$, and the conditions for the same. We illustrate some consequences of alignment. Next, we examine the possibility of {\em intermediate $G$-varieties} $W$ which lie between the orbit closures of $z$ and $y$, i.e. $\overline{O(z)} \subsetneq W \subsetneq O(y)$. These have a direct bearing on representation theoretic as well as geometric properties which connect $z$ and $y$.
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This paper focuses on representing the $L^{\infty}$-norm of finite-dimensional linear time-invariant systems with parameter-dependent coefficients. Previous studies tackled the problem in a non-parametric scenario by simplifying it to finding the maximum $y$-projection of real solutions $(x, y)$ of a system of the form $\Sigma=\{P=0, \, \partial P/\partial x=0\}$, where $P \in \Z[x, y]$. To solve this problem, standard computer algebra methods were employed and analyzed \cite{bouzidi2021computation}. In this paper, we extend our approach to address the parametric case. We aim to represent the "maximal" $y$-projection of real solutions of $\Sigma$ as a function of the given parameters. %a set of parameters $\alpha$. To accomplish this, we utilize cylindrical algebraic decomposition. This method allows us to determine the desired value as a function of the parameters within specific regions of parameter space.
This paper focuses on representing the $L^{\infty}$-norm of finite-dimensional linear time-invariant systems with parameter-dependent coefficients. Previous studies tackled the problem in a non-parametric scenario by simplifying it to finding the maximum $y$-projection of real solutions $(x, y)$ of a system of the form $\Sigma=\{P=0, \, \partial P/\partial x=0\}$, where $P \in \Z[x, y]$. To solve this problem, standard computer algebra methods were employed and analyzed \cite{bouzidi2021computation}. In this paper, we extend our approach to address the parametric case. We aim to represent the "maximal" $y$-projection of real solutions of $\Sigma$ as a function of the given parameters. %a set of parameters $\alpha$. To accomplish this, we utilize cylindrical algebraic decomposition. This method allows us to determine the desired value as a function of the parameters within specific regions of parameter space.
We define the notion of $k$-safe infinitary series over idempotent ordered totally generalized product $\omega $-valuation monoids that satisfy specific properties. For each element $k$ of the underlying structure (different from the neutral elements of the additive, and the multiplicative operation) we determine two syntactic fragments of the weighted $LTL$ with the property that the semantics of the formulas in these fragments are $k$ -safe infinitary series. For specific idempotent ordered totally generalized product $\omega $-valuation monoids we provide algorithms that given a weighted B\"{u}chi automaton and a weighted $LTL$ formula in these fragments, decide whether the behavior of the automaton coincides with the semantics of the formula.
A $3$-uniform hypergraph is a generalization of simple graphs where each hyperedge is a subset of vertices of size $3$. The degree of a vertex in a hypergraph is the number of hyperedges incident with it. The degree sequence of a hypergraph is the sequence of the degrees of its vertices. The degree sequence problem for $3$-uniform hypergraphs is to decide if a $3$-uniform hypergraph exists with a prescribed degree sequence. Such a hypergraph is called a realization. Recently, Deza \emph{et al.} proved that the degree sequence problem for $3$-uniform hypergraphs is NP-complete. Some special cases are easy; however, polynomial algorithms have been known so far only for some very restricted degree sequences. The main result of our research is the following. If all degrees are between $\frac{2n^2}{63}+O(n)$ and $\frac{5n^2}{63}-O(n)$ in a degree sequence $D$, further, the number of vertices is at least $45$, and the degree sum can be divided by $3$, then $D$ has a $3$-uniform hypergraph realization. Our proof is constructive and in fact, it constructs a hypergraph realization in polynomial time for any degree sequence satisfying the properties mentioned above. To our knowledge, this is the first polynomial running time algorithm to construct a $3$-uniform hypergraph realization of a highly irregular and dense degree sequence.
In this paper we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations, namely for $\mathbf{H}(\mathbf{curl})$ problems. It is well known that convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement a near-kernel space made from the gradient of scalar functions. The new class of preconditioner is inspired by the idea of subspace decomposition, but based on spectral coarse spaces, and is specially designed for curl-conforming discretisations of Maxwell's equations in heterogeneous media on general domains which may have holes. Our approach has wider applicability and theoretical justification than the well-known Hiptmair-Xu auxiliary space preconditioner, with results extending to the variable coefficient case and non-convex domains at the expense of a larger coarse space.
Given an $m\times n$ binary matrix $M$ with $|M|=p\cdot mn$ (where $|M|$ denotes the number of 1 entries), define the discrepancy of $M$ as $\mbox{disc}(M)=\displaystyle\max_{X\subset [m], Y\subset [n]}\big||M[X\times Y]|-p|X|\cdot |Y|\big|$. Using semidefinite programming and spectral techniques, we prove that if $\mbox{rank}(M)\leq r$ and $p\leq 1/2$, then $$\mbox{disc}(M)\geq \Omega(mn)\cdot \min\left\{p,\frac{p^{1/2}}{\sqrt{r}}\right\}.$$ We use this result to obtain a modest improvement of Lovett's best known upper bound on the log-rank conjecture. We prove that any $m\times n$ binary matrix $M$ of rank at most $r$ contains an $(m\cdot 2^{-O(\sqrt{r})})\times (n\cdot 2^{-O(\sqrt{r})})$ sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank $r$ is at most $O(\sqrt{r})$.
An $(r, \delta)$-locally repairable code ($(r, \delta)$-LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, and has garnered significant interest among researchers. An $(r,\delta)$-LRC is called an optimal code if its parameters achieve the Singleton-like bound. In this paper, we construct three classes of $q$-ary optimal cyclic $(r,\delta)$-LRCs with new parameters by investigating the defining sets of cyclic codes. Our results generalize the related work of \cite{Chen2022,Qian2020}, and the obtained optimal cyclic $(r, \delta)$-LRCs have flexible parameters. A lot of numerical examples of optimal cyclic $(r, \delta)$-LRCs are given to show that our constructions are capable of generating new optimal cyclic $(r, \delta)$-LRCs.
In this paper we generalize the notion of $n$-equivalence relation introduced by Chen et al. in \cite{Chen2014} to classify constacyclic codes of length $n$ over a finite field $\mathbb{F}_q$, where $q=p^r$ is a prime power, to the case of skew constacyclic codes without derivation. We call this relation $(n,\sigma)$-equivalence relation, where $n$ is the length of the code and $ \sigma$ is an automorphism of the finite field. We compute the number of $(n,\sigma)$-equivalence classes, and we give conditions on $ \lambda$ and $\mu$ for which $(\sigma, \lambda)$-constacyclic codes and $(\sigma, \lambda)$-constacyclic codes are equivalent with respect to our $(n,\sigma)$-equivalence relation. Under some conditions on $n$ and $q$ we prove that skew constacyclic codes are equivalent to cyclic codes. We also prove that when $q$ is even and $\sigma$ is the Frobenius autmorphism, skew constacyclic codes of length $n$ are equivalent to cyclic codes when $\gcd(n,r)=1$. Finally we give some examples as applications of the theory developed here.
In this paper, we consider a problem which we call LTL$_f$ model checking on paths: given a DFA $\mathcal{A}$ and a formula $\phi$ in LTL on finite traces, does there exist a word $w$ such that every path starting in a state of $\mathcal{A}$ and labeled by $w$ satisfies $\phi$? The original motivation for this problem comes from the constrained parts orienting problem, introduced in [Petra Wolf, "Synchronization Under Dynamic Constraints", FSTTCS 2020], where the input constraints restrict the order in which certain states are visited for the first or the last time while reading a word $w$ which is also required to synchronize $\mathcal{A}$. We identify very general conditions under which LTL$_f$ model checking on paths is solvable in polynomial space. For the particular constraints in the parts orienting problem, we consider PSPACE-complete cases and one NP-complete case. The former provide very strong lower bound for LTL$_f$ model checking on paths. The latter is related to (classical) LTL$_f$ model checking for formulas with the until modality only and with no nesting of operators. We also consider LTL$_f$ model checking of the power-set automaton of a given DFA, and get similar results for this setting. For all our problems, we consider the case where the required word must also be synchronizing, and prove that if the problem does not become trivial, then this additional constraint does not change the complexity.
In this paper we prove that the $\ell_0$ isoperimetric coefficient for any axis-aligned cubes, $\psi_{\mathcal{C}}$, is $\Theta(n^{-1/2})$ and that the isoperimetric coefficient for any measurable body $K$, $\psi_K$, is of order $O(n^{-1/2})$. As a corollary we deduce that axis-aligned cubes essentially "maximize" the $\ell_0$ isoperimetric coefficient: There exists a positive constant $q > 0$ such that $\psi_K \leq q \cdot \psi_{\mathcal{C}}$, whenever $\mathcal{C}$ is an axis-aligned cube and $K$ is any measurable set. Lastly, we give immediate applications of our results to the mixing time of Coordinate-Hit-and-Run for sampling points uniformly from convex bodies.
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