Dynamic investigation of triangles inscribed in a circle,which tend to an equilateral triangle

We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner.

First, using geometrical software, we investigate four theorems that represent interesting geometrical properties, after which we present formal proofs that rest on a combination between different fields of mathematics: trigonometry, algebra and geometry, and the use of the concept of standard deviation that is taken from statistics.     

Complexity analysis of an assignment problem with controllable assignment costs and its applications in scheduling

We extend the classical linear assignment problem to the case where the cost of assigning agent j to task i is a multiplication of task i’s cost parameter by a cost function of agent j. The cost function of agent j is a linear function of the amount of resource allocated to the agent. A solution for our assignment problem is defined by the assignment of agents to tasks and by a resource allocation to each agent. The quality of a solution is measured by two criteria. The first criterion is the total assignment cost and the second is the total weighted resource consumption. We address these criteria via four different problem variations. We prove that our assignment problem is NP-hard for three of the four variations, even if all the resource consumption weights are equal. However, and somewhat surprisingly, we find that the fourth variation is solvable in polynomial time. In addition, we find that our assignment problem is equivalent to a large set of important scheduling problems whose complexity has been an open question until now, for three of the four variations.     

Optical purification of a mixture of chiral forms by dimer formation

We introduce a readily executable method for the optical purification of "scalemic" (non 50%-50%) mixtures of chiral molecules of opposite handedness ("enantiomers"). The method relies on the formation of two types of dimers, (R-R or S-S) "homodimers" and (R-S) "heterodimers." The selectivity is linked to the difference in sign recently discovered by us to exist between certain transition-dipole matrix elements of opposite enantiomers. This sign difference results in differences in spectral propensity rules: In homodimers, transitions from the ground state can only take place to inversion symmetric excited states, while in the heterodimer the transitions are much more likely to proceed to antisymmetric excited states (although for heterodimers weak transitions to symmetric states might exist). These opposing propensity rules fully explain the observed large differences in the spectra of homodimers vs. heterodimers, which exist despite the almost identical energy levels positions. We illustrate the general concepts by computationally demonstrating the optically induced enantio-purification of scalemic mixtures of the hydropropionic C(3)H(6)O(3) (lactic) acid.     

Inversion of two-dimensional potentials from frequency-resolved spectroscopic data

We report the first successful reconstruction of two-dimensional potential energy surfaces (PES) using the magnitudes and positions of a set of frequency-resolved fluorescence (or absorption) lines. The inversion proceeds by first extracting the phases of the transition-dipole matrix elements, yielding, together with the (ground) PES to (from) which emission (absorption) occurs, a point by point reconstruction of the two-dimensional excited state PES. The inversion procedure is highly accurate even for PES with multiple minima and many missing lines, with typical RMS errors <0.002 cm(-1) in the classically allowed region and <0.018 cm(-1) in the classically forbidden region.     

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C^?-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.     

Thermal analysis characterization of isosorbide-containing thermosets

Sugar-based new monomers, polymers, and low molar mass additives have emerged as an exciting topic on green chemistry research, due to the worldwide focus on sustainable material. Isosorbide and its isomers, as ??Generally Recognized as Safe?? GRAS materials, possess unique stereochemistry and molecular geometry suitable for making cost-effective chemicals and polymers. With growing awareness of bisphenol A (BPA) as a xenoestrogen, isosorbide and its isomers holding the remarkable chemical properties and attractive price can be attached to glycidyl ether to make crosslinkable epoxy resin monomers with similar properties to BPA diglycidyl ether. By adding the hydrophobic functional group into the backbone of isosorbide epoxy or adjusting the amount and type of crosslinker, the mechanical properties and the water uptake ratios (from <1 to >50?wt%) of the isosorbide-derived epoxies could be optimized for different applications. The high water uptake epoxy with controllable biodegradation rate could be used as a drug delivery system or extracellular matrix for biomedical applications while the low water uptake epoxy with strong mechanical properties could be used for can coatings, bone cements, and other industrial additives and adhesives. The chemical structures and properties of the synthesized epoxy monomers and polymers were characterized by DSC, TG, and ¹H NMR.     

Cash flow optimization in delivery scheduling

The paper examines the inventory routing problem from the perspective of the present value of the cash flow associated with the distribution of a commodity such as propane. We analyze this problem for both deterministic and stochastic customer demands and validate our results on data from a real life distribution operation of propane. The analysis based on the present value of the cash flow indicates that optimization of propane deliveries based on efficiency/cost criteria alone will generate inferior solutions and it would be more advantageous for the company to set deliveries for a large percentage of the customers based on the present value of cash flow. In addition, in the case of stochastic demands, deliveries based on the cash flow consideration will tend to reduce the number of stockouts (i.e. improve both profit and service).     

Inverse Pattern Matching

Let a textstringTofnsymbols from some alphabet Σ and an integerm < nbe given. A patternPof lengthmover Σ is sought such thatPminimizes (alternatively, maximizes) the total number of pairwise character mismatches generated whenPis compared with allm-character substrings ofT. Two additional variants of the problem are obtained by adding the constraint thatPbe (respectively, not be) a substring ofT. Efficient sequential algorithms are proposed in this paper for the problem and its variants.     

A discrete-time model of failure and repair

In this paper the authors introduce and study a model of failures and repairs of units with discrete lifetimes. They suppose that a unit has a sequence of tasks to perform and that its lifetime is measured by the number of tasks performed before its final, fatal failure. Upon a failure the unit may be repaired (with some probability) and then it may attempt again to perform the current task. The unit dies when (with some probability) a repair cannot be completed. We derive some stochastic comparisons of pairs of such models. The stochastic comparisons are then applied for obtaining results regarding the inheritance of several aging properties by the repaired unit. Various examples illustrate the applicability of the model. Some variants of the model of this paper can be viewed as discrete analogues of the notion of imperfect repair.     

Maximal solutions of semilinear elliptic equations with locally integrable forcing term

We study the existence of a maximal solution of −Δu+g(u)=f(x) in a domain Ω ∈ ℝ ^N with compact boundary, assuming thatf ∈ (L _loc ¹ (Ω))₊ and thatg is nondecreasing,g(0)≥0 andg satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classicalC _1,2 Wiener criterion, then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition, we discuss the question of uniqueness of large solutions. This research was partially supported by an EC Grant through the RTN Program “Front-Singularities”, HPRN-CT-2002-00274.