Computational complexity of inner and outerj-radii of polytopes in finite-dimensional normed spaces

This paper studies the complexity of computing (or approximating, or bounding) the various inner and outer radii of ann-dimensional convex polytope in the space ⁿ equipped with an _p norm or a polytopal norm. The polytopeP is assumed to be presented as the convex hull of finitely many points with rational coordinates (V-presented) or as the intersection of finitely many closed halfspaces defined by linear inequalities with rational coefficients (-presented). The innerj-radius ofP is the radius of a largestj-ball contained inP; it isP's inradius whenj = n and half ofP's diameter whenj = 1. The outerj-radius measures how wellP can be approximated, in a minimax sense, by an (n — j)-flat; it isP's circumradius whenj = n and half ofP's width whenj = 1. The binary (Turing machine) model of computation is employed. The primary concern is not with finding optimal algorithms, but with establishing polynomial-time computability or NP-hardness. Special attention is paid to the case in whichP is centrally symmetric. When the dimensionn is permitted to vary, the situation is roughly as follows: (a) for general -presented polytopes in _p spaces with 1相似文献   

Inequalities for tails of adapted processes with an application to Wald's lemma

In this paper we introduce a new tail probability version of Wald's lemma for expectations of randomly stopped sums of independent random variables. We also make a connection between the works of Klass^{(18, 19)} and Gundy⁽¹¹⁾ on Wald's lemma. In making the connection, we develop new Lenglart and Good Lambda inequalities comparing the tails of various types of adapted processes. As a consequence of our Good Lambda inequalities we include the following result. Let {d _i }, {e _i } be two sequences of variables adapted to the same increasing sequence of σ-fields ? _n ↗?, (e.g., ? _n =σ({d _i } _i=1 ⁿ , {E _i } _i=1 ⁿ ), and letN?∞ be a stopping time adapted to {? _n }. Then for allp>0, there exists a constant 0<C _p <∞ depending onp only, such that $$\mathop {\overline {\lim } }\limits_\lambda \lambda ^p P\left( {\mathop {\sup }\limits_{1 \leqslant n \leqslant N} \left\| {\sum\limits_{i = 1}^n {d_i } } \right\| > \lambda } \right) \leqslant C_p \mathop {\overline {\lim } }\limits_\lambda \lambda ^p P\left( {\mathop {\sup }\limits_{1 \leqslant n \leqslant N} \left\| {\sum\limits_{i = 1}^n {e_i } } \right\| > \lambda } \right)$$ This result holds when the sequences are real, tangent, and either conditionally symmetric or nonnegative, or alternatively, if {d _i } is a sequence of independent random variables and {e _i } is an independent copy of {d _i }, withN a stopping time adapted to the filtration generated by {d _i } only. Other examples include Hilbert space valued differentially subordinate conditionally symmetric martingale differences. The result is true for more general operators applied to sequences as shown by an example comparing the square function of a conditionally symmetric sequence to the maximum of its absolute partial sums.     

The Role of Carnitine Acyltransferases in the Maintenance of Cell Function

Summary. Carnitine acyltransferases catalyse equilibria between acyl-CoA esters and the respective acylcarnitines. Therefore, they act not only as pathway enzymes, but also as modulators of acyl-CoA concentrations within individual sub-cellular compartments. Because acyl-CoA esters are potent biologically active metabolites, carnitine acyltransferase activities are potentially able to affect a diverse range of physiological processes, ranging from insulin secretion, to appetite control, and insulin sensitivity of tissues. The distinctive subcellular distributions of the different types of carnitine acyltransferases also enables them to participate in the transfer of acyl moieties across intracellular membranes, and of particular acylcarnitine esters across the plasma membrane and into the plasma. Pharmacological strategies that make use of these properties to improve cell function are discussed.     

Assembly and Dissociation of Α-Cyclodextrin [2]Pseudorotaxanes with α,ω-Bis(N-(N,N′-dimethylethylenediamine)alkane Threads

A series of novel bischelate bridging ligands, CH₃NH(CH₂)₂N(CH₃)(CH₂) _n N(CH₃)(CH₂)₂NHCH₃ (n = 9, 10, 11, and 12) were synthesized as hydrochloride salts and characterized by elemental analyses, electrospray mass spectrometry, and ¹H and ¹³C NMR spectroscopy. These ligands form [2]pseudorotaxanes with α-cyclodextrin (α-CD) and the stability constants have been determined from ¹H NMR titrations in D₂O. The kinetics and mechanism of the assembly and dissociation of a [2]pseudorotaxane in which α-CD has been threaded by the CH₃NH₂(CH₂)₂N(CH₃)(CH₂)₁₂N(CH₃)(CH₂)₂NH₂CH ₃ ²⁺ ligand were determined in aqueous solution using ¹H NMR spectroscopy. A weak inclusion of the dimethylethylenediamine end group precedes the passage of the α-CD onto the hydrophobic dodecamethylene chain.     

Comparative Studies of Cathodically-Promoted and Base-Catalyzed Michael Addition Reactions of Levoglucosenone

Regioselective Michael addition of nitro and heterocyclic compounds to levoglucosenone, 1, is effectively catalyzed by amines and also by cathodic electrolysis. In comparison to the base-catalyzed reaction, it was found that under electrochemical conditions the reaction proceeds under milder conditions and with higher yields. Cathodically-initiated Michael addition of thiols to levoglucosenone using small currents produces the previously unknown threo addition product in several instances. The normal erythro isomer, identified as the kinetic product, tends to be formed when large currents are used. In contrast, slow, low current electrolyses promote equilibration of the two forms so that erythro can be converted to threo by the retro reaction and readdition. Addition of 2-naphthalenethiol to (R)-(+)-apoverbenone is also reported.     

Torsional isomerization of biologically active bicyclic molecules

Rotational isomerization of bipyridines C₅H₄N? C₅H₄N was studied by CNDO /2, PPP /CI , and CNDO /CI methods. It is shown that CNDO /2 overestimates the angle of rotation ? between the pyridine rings ca. two times. The angle ? was determined for 2,2′-bipyridine by means of correlation of the theoretical (CNDO /2) and experimental dipole moment. It was also found from the correlation between the theoretical and experimental UV spectra. It is shown that there is an explicit dependence of the results upon the distance between heteroatoms (PPP /CI ). It has been found that the CNDO /CI method correctly predicts the value of the rotational angles and their sequence in bipyridines.     

Comparison of the electrophilic and free-radical addition of halogens with hexafluoro-1,3-butadiene and 1,3-butadiene

Ionic and photochemical reaction of chlorine (Cl₂), bromine (Br₂) and iodine monochloride (ICl) to hexafluoro-1,3-butadiene (1) and 1,3-butadiene (2) were carried out under conditions that would provide product distributions under controlled ionic or free-radical conditions. Product distributions for ionic reaction of Cl₂ and Br₂ with 1 are similar and suggest a weakly-bridged halonium ion species. Theoretical calculations support weakly-bridged chloronium and bromonium ions for both dienes 1 and 2. There are more of the 1,4-dihalo-2-butene products from ionic halogenation of 1 than 2 which correlates with the greater charge density on carbon-4 of halonium ions from 1. Ionic and free-radical reactions of ICl with 1 give 8 and 2% of 3-chloro-4-iodohexafluoro-1-butene and 4-chloro-3-iodohexafluoro-1-butene, respectively. The minor cis-1,4-dihalo-2-butene products from 1 and 2 are reported when formed.     

Toeplitz operators in n-dimensions

The interplay between the theory of Toeplitz operators on the circle and the theory of pseudodifferential operators on the line (i. e. Wiener-Hopf operators) is by now well-known and well-understood. In this article we show that there is a parallel situation in higher dimensions. To begin with, by using pseudodifferential multipliers, one can simplify the composition rules for Toeplitz operators, (§ 3), and describe precisely how Toeplitz operators of Bergmann type are related to Toeplitz operators of Szegö type (§ 9). Furthermore, it turns out that the ring of pseudodifferential operators on a compact manifold, M, is isomorphic with the ring of Toeplitz operators on an appropriate Grauert tube about M (§ § 4–6), and the ring of Weyl operators on ⁿ is isomorphic with the ring of Toeplitz operators on the complex ball in ⁿ (§ § 7–10).     

On the origin of dynamic instability of molecular systems

The vibronic origin of dynamic instability of molecular systems considered earlier, is here given a more complete and rigorous treatment. It is shown that the nonvibronic contribution to the curvature of the adiabatic potential arising due to nuclear displacements under fixed electronic density distribution, is always positive, and hence the only reason for dynamic instability is the pseudo Jahn-Teller effect. For some examples of special interest (planar equilateral NH₃, planar square CH₄ and linear H ₃ ⁺ ) the molecular excited states, responsible for the instability of the ground state, are revealed by means of ab initio calculations.