Search results for "Geometry"
showing 10 items of 4487 documents
Lie Algebras Generated by Extremal Elements
1999
We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.
Synthesis, characterization, and cytotoxic activity of copper(II) and platinum(II) complexes of 2-benzoylpyrrole and X-ray structure of bis[2-benzoyl…
2004
Copper(II) and platinum(II) complexes of 2-benzoylpyrrole (2-BZPH) were synthesized and characterized with IR, 1 H and 1 3 C NMR spectroscopies and coordination geometry with ligands arranged in transoid fashion. The crystal structure of [Cu I I (2-BZP) 2 ] was determined by X-ray diffraction. Death of complex treated Jurkat cells was measured by flow cytometry. The bis-chelate complexes [Cu I I (2-BZP) 2 ] and [Pt I I (2-BZP) 2 ] adopt square-planar coordination geometry with ligands, arranged in transoid fashion. Concentrations of 1-10 μM Platinum(II) complexes reduced cell survival from 100% to 20%, in contrast to the copper(II) complex which caused no cell death at a concentration of 10…
Preparation and structural characterization of organotin(IV) complexes with ligands containing a hetero {N} atom and a hydroxy group or hydroxy and c…
2005
AbstractTwenty-two n-butyltin(IV) and t-butyltin(IV) complexes of ligands containing an –OH (–C@O) group or –OH and –COOHgroups and an aromatic {N} donor atom were prepared by metathetical reactions. On the basis of the FT-IR and Mo¨ssbauer spec-troscopic data, molecular structures were assigned to these compounds. The binding sites of the ligands were identified by means ofFT-IR spectroscopic measurements, and it was found that in most cases the organotin(IV) moiety reacts with the phenolic form ofthese ligands. In the complexes with –OH and –COOH functions, the –COOH group is coordinated to the organotin(IV) centres in amonodentate manner. The 119 Sn Mo¨ssbauer and the FT-IR studies suppor…
The proof of Birman’s conjecture on singular braid monoids
2003
Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma_i^{+-1}) =_i^{+-1} and eta(tau_i) = sigma_i - sigma_i^{-1}, for 1 <= i <= n-1. The purpose of the present paper is to prove Birman's conjecture, namely, that the desingularization map eta is injective.
Automorphisms of 2–dimensional right-angled Artin groups
2007
We study the outer automorphism group of a right-angled Artin group AA in the case where the defining graph A is connected and triangle-free. We give an algebraic description of Out.AA/ in terms of maximal join subgraphs in A and prove that the Tits’ alternative holds for Out.AA/. We construct an analogue of outer space for Out.AA/ and prove that it is finite dimensional, contractible, and has a proper action of Out.AA/. We show that Out.AA/ has finite virtual cohomological dimension, give upper and lower bounds on this dimension and construct a spine for outer space realizing the most general upper bound. 20F36; 20F65, 20F28
Geometric rough paths on infinite dimensional spaces
2022
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $\alpha\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $\alpha$-rough paths by signatures of curves of bounded variation, given some tuning of the H\"older parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.
$\Omega$-symmetric measures and related singular integrals
2019
Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice continuously differentiable. Motivated by a question raised by Mattila and Preiss in [MP95], we prove the following: if a Radon measure $\mu$ has positive lower density and finte upper density almost everywhere, and the limit $$ \lim_{\epsilon \downarrow 0} \int_{\mathbb{C} \setminus B(x,\epsilon)} \frac{\Omega\left((x-y)/|x-y|\right)}{|x-y|} \, d\mu(y) $$ exists $\mu$-almost everywhere, then $\mu$ is $1$-rectifiable. To achieve this, we prove first that if …
Integrability of orthogonal projections, and applications to Furstenberg sets
2022
Let $\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of $\mathbb{R}^{d}$, and let $\pi_{V} \colon \mathbb{R}^{d} \to V$ be the orthogonal projection. We prove that if $\mu$ is a compactly supported Radon measure on $\mathbb{R}^{d}$ satisfying the $s$-dimensional Frostman condition $\mu(B(x,r)) \leq Cr^{s}$ for all $x \in \mathbb{R}^{d}$ and $r > 0$, then $$\int_{\mathcal{G}(d,n)} \|\pi_{V}\mu\|_{L^{p}(V)}^{p} \, d\gamma_{d,n}(V) \tfrac{1}{2}$ and $t \geq 1 + \epsilon$ for a small absolute constant $\epsilon > 0$. We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in $\mathbb{R}^{d}$. As another corollary of our method,…
Visible parts of fractal percolation
2009
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are 1-dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts.
Isometric embeddings of snowflakes into finite-dimensional Banach spaces
2016
We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.