Irregularity strength of digraphs

It is an elementary exercise to show that any non-trivial simple graph has two vertices with the same degree. This is not the case for digraphs and multigraphs. We consider generating irregular digraphs from arbitrary digraphs by adding multiple arcs. To this end, we define an irregular labeling of a digraph D to be an arc-labeling of the digraph such that the ordered pairs of the sums of the in-labels and out-labels at each vertex are all distinct. We define the strength of D to be the smallest of the maximum labels used across all irregular labelings. Similar definitions for graphs have been studied extensively and a different formulation of digraph irregularity was given in [H. Hackett, Irregularity strength of graphs and digraphs, Masters Thesis, University of Louisville, 1995]. Here we continue the study of irregular labelings of digraphs. We give a general lower bound on and determine exactly for tournaments, directed paths and cycles and the orientation of the path where all vertices have either in-degree 0 or out-degree 0. We also determine the irregularity strength of a union of directed cycles and a union of directed paths, the latter which requires a new result pertaining to finding circuits of given lengths containing prescribed vertices in the complete symmetric digraph with loops.     

A centralized approach to tandem mass spectrometry method development for high-throughput ADME screening

A centralized approach to acquisition and dissemination of tandem mass spectrometry (MS/MS) conditions within an ADME-screening bioanalytical mass spectrometry group has been developed. The method development process uses two automated software products (Autoscan and Automaton) specifically designed for mass spectrometers manufactured by MDS Sciex. Both provide the ability to quickly determine selected reaction monitoring (SRM) transitions for hundreds of compounds per day. In addition, Autoscan determines optimal polarity and collision energy (CE). Automaton also determines the optimal declustering potential (DP) as well as the CE. The resulting optimized conditions are loaded into a central database for access by LC/MS/MS bioanalysis workstations in the group. The effect of DP and CE on the sensitivity was investigated. Optimization of DP improved signal response about 27% on average. For approximately 10% of compounds, signal enhancement was greater than 50% compared to the generic setting. A generic setting of DP = 25 V can be used for the majority of ADME-screening applications. Optimization of CE can have a much larger impact on signal intensity and a minimum of three CE settings should be tested. We have determined that CE values of 1, 30 and 45 V provide adequate coverage for most small molecule drug discovery analytes.     

Synthesis of benzyl substituted naphthalenes from benzylidene tetralones

A convenient and efficient synthesis of novel highly substituted dimethoxybenzylnaphthalenes, which are precursors to several dihydroxynaphthoic acids, is described. The approach involves the use of aldol chemistry to provide a number of benzylidene tetralones, which are converted to the target naphthalenes in three steps, with good to excellent yields. Grignard reaction of intermediate benzyl tetralones provided 1-substituted benzyl naphthalenes. The reported synthesis is flexible and scalable and provides access to naphthalenes having a variety of substitution patterns. These benzyl substituted naphthalenes are being converted to naphthoic acids and the bioactivities of these compounds are currently being investigated.     

Synthesis and hydrolysis of a cis-chlorohydrin derived from a benzo[a]pyrene 7,8-diol 9,10-epoxide

(+/-)-7beta,8alpha-Dihydroxy-9beta,10beta-epoxy-7,8,9,10-tetrahydrobenzo[a]pyrene (DE-1) undergoes reaction with anhydrous HCl in dioxane to yield predominantly ( approximately 94%) a single chlorohydrin. This chlorohydrin was assigned structure 9, in which the chloro goup at C-10 is located cis to the C-9 hydroxyl group, on the basis of its (1)H NMR spectrum. This result is in contrast to the reaction of a diastereomeric benzo[a]pyrene 7,8-diol 9,10-epoxide (DE-2) with HCl, which yields only trans-chlorohydrin 8. The hydrolysis of cis-chlorohydrin 9 in 10:90 dioxane-water solutions yields the same ratio of tetrols ( approximately 89% cis/11% trans) as that formed by acid-catalyzed hydrolysis of DE-1. This result again contrasts with the hydrolysis of trans-chlorohydrin 8, which undergoes hydrolysis to give tetrols in a ratio different from that from acid-catalyzed hydrolysis of DE-2. A marked common ion rate depression in the hydrolysis of cis-chlorohydrin 9 is observed, which shows that hydrolysis proceeds via an intermediate carbocation that has a sufficient lifetime to be trapped by external chloride ion. The observation that DE-1 reacts with HCl to give mainly the cis-chlorohydrin is rationalized by quantum chemical calculations that suggest that the cis-chlorohydrin is more stable than the epimeric trans-chlorohydrin.     

The behavior of higher order mode shape derivatives in damaged,beam-like structures

This work considers the effect that damage has upon the higher order derivatives of mode shapes of structures having primarily beam-like vibration. Via numerical investigations, the sensitivity of various damage related parameters in inducing changes in these higher order modal derivatives is determined, leading to a more complete understanding of what factors make the most contribution to significant changes in these derivatives. It is concluded that higher order mode shape derivatives (e.g., modal curvature, third derivative, and fourth derivative) are better indicators of damage than the mode shapes. Three distinct types of response for the damage-induced higher order derivative discontinuities are identified as three key parameters (the mass loss, stiffness loss, and damage radius scale) vary. From this, formal approximations are obtained for the expected forms of the higher order derivative discontinuities based upon the underlying behavior predicted by a simple relation among these three parameters. These approximations are checked with numerical simulations, and an excellent level of agreement is observed under appropriate conditions. Finally, the potential of these higher order derivative changes for indicating the presence and location of damage in a global setting is examined.     

Synthesis of naphthalene and indene precursors to naphthoic and indenoic acids

As part of ongoing research to investigate structural requirements for lactate dehydrogenase inhibition by highly substituted naphthoic and indenoic acids, certain naphthalene and indene precursors to those types of compounds are required. Described here are efficient preparations of 1-naphthoic acid precursors 6-benzyl-2,3-dimethoxy-1-propylnaphthalenes, including compounds with substituted benzyl groups and 7-benzyl-2,3-dimethoxy-1-propylnaphthalene. Also described are the syntheses of indenoic acid precursors 2-benzyl-5,6-dimethoxy-7-propyl-1H-indenes, including compounds with substituted benzyl groups. These compounds were made from the key intermediates 6,7-dimethoxy-5-propyl-1-tetralone, 6,7-dimethoxy-8-propyl-1-tetralone, and 5,6-dimethoxy-4-propyl-1-indanone.     

A NEW FINITE ELEMENT APPROXIMATION OF A STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM

In this paper, we study numerical methods for an optimal control problem with pointwise state constraints. The traditional approaches often need to deal with the deltasingularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms （nonconforming finite element approximation） are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.