Synthetic infrared spectra

We have designed, microfabricated, and characterized a diffractive optical element that reproduces the infrared spectrum of HF from 3600 to 4300 cm(-1) . The reflection-mode diffractive optic consists of 4096 lines, each 4.5mum wide, at 16 discrete depths relative to the substrate from 0 to 1.2 mum and was fabricated upon a silicon wafer by anisotropic reactive ion-beam etching in a four-mask-level process. We envisage the use of diffractive optical elements of this type as the basis for a new class of miniaturized, remote chemical sensor systems based on correlation spectroscopy.     

Enynones in organic synthesis. I. Spiroannulation by tandem oxy-cope rearrangement-electrocyclic ring closure

Bis-acetylenic alcohols of proper design undergo a facile oxy-Cope rearrangement to afford mixtures of E- and Z-enynones. These latter compounds afford methylenecyclopentenones upon enolization and electrocyclic ring closure.     

Enynones in organic synthesis. III. A novel synthesis of phenols

Enynones are converted to phenols by an acid catalyzed process which can be controlled to give either of two regioisomeric series of products.     

On the irregularity strength of trees

For any graph G, let n_i be the number of vertices of degree i, and . This is a general lower bound on the irregularity strength of graph G. All known facts suggest that for connected graphs, this is the actual irregularity strength up to an additive constant. In fact, this was conjectured to be the truth for regular graphs and for trees. Here we find an infinite sequence of trees with λ(T) = n₁ but strength converging to . © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 241–254, 2004     

Stability of exponential utility maximization with respect to market perturbations

We investigate the continuity of expected exponential utility maximization with respect to perturbation of the Sharpe ratio of markets. By focusing only on continuity, we impose weaker regularity conditions than those found in the literature. Specifically, we require, in addition to the V$V$-compactness hypothesis of Larsen and ?itkovi? (2007) [13], a local bmo$bmo$ hypothesis, a condition which is essentially implicit in the setting of [13]. For markets of the form S=M+∫λd〈M〉$S=M+\int \lambda d〈M〉$, these conditions are simultaneously implied by the existence of a uniform bound on the norm of λ⋅M$\lambda \cdot M$ in a suitable bmo$bmo$ space.     

The Hodge Structure on a Filtered Boolean Algebra

Let (B _n) be the order complex of the Boolean algebra and let B(n, k) be the part of (B _n) where all chains have a gap at most k between each set. We give an action of the symmetric group S _l on the l-chains that gives B(n, k) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The S _n action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary.We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods.Finally we present some data on the homology of B(n, k).     

Two counterexamples in normalization

In this paper, we offer a correction to Commutative Algebra by Oscar Zariski and Pierre Samuel by noting two erroneous theorems and offering counterexamples for each.

     

On an online random k‐SAT model

Given n Boolean variables x₁,…,x_n, a k‐clause is a disjunction of k literals, where a literal is a variable or its negation. Suppose random k‐clauses are generated one at a time and an online algorithm accepts or rejects each clause as it is generated. Our goal is to accept as many randomly generated k‐clauses as possible with the condition that it must be possible to satisfy every clause that is accepted. When cn random k‐clauses on n variables are given, a natural online algorithm known as Online‐Lazy accepts an expected (1 ? )cn + a_kn clauses for some constant a_k. If these clauses are given offline, it is possible to do much better, (1 ? )cn + Ω( )n can be accepted whp . The question of closing the gap between a_k and Ω( ) for the online version remained open. This article shows that for any k ≥ 1, any online algorithm will accept less than (1 ? )cn + (ln 2)n k‐clauses whp , closing the gap between the constant and Ω( ). Furthermore we show that this bound is asymptotically tight as k → ∞. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008