Quadratically constrained minimum cross-entropy analysis

Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the strong duality theorem and derive a dual-to-primal conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find an-optimal solution for the quadratically constrained minimum cross-entropy analysis.     

On consistency of bounding operations in deterministic global optimization

This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence _q in Lemma 2.1, and a derivation of Proposition 2.1 is presented that includes a new and simple consistency proof of the classical bounding by convex envelopes used in many branch-and-bound procedures.     

Metastable patterns in solutions of ut = ϵ2uxx − f(u)

We consider the equation in question on the interval 0 ≦ x ≦ 1 having Neumann boundary conditions, with f(u) = F(u), where F is a double well energy density with equal minima at u = ±1. The only stable states of the system are patternless constant solutions. But given two-phase initial data, a pattern of interfacial layers typically forms far out of equilibrium. The ensuing nonlinear relaxation process is extremely slow: patterns persist for exponentially long times proportional to exp{A_±l/?, where A = F(±1) and l is the minimum distance between layers. Physically, a tiny potential jump across a layer drives its motion. We prove the existence and persistence of these metastable patterns, and characterise accurately the equations governing their motion. The point of view is reminiscent of center manifold theory: a manifold parametrising slowly evolving states is introduced, a neighbourhood is shown to be normally attracting, and the parallel flow is characterised to high relative accuracy. Proofs involve a detailed study of the Dirichlet problem, spectral gap analysis, and energy estimates.     

Crystal structures of dimethylsuccinate and dimethyloxalate: Carbonyl group orientation for C13 chemical shielding tensor studies

The crystal structures of dimethylsuccinate (DMS) and dimethyloxalate (DMO) have been determined to facilitate the determination of the C-13 chemical shielding tensors of the carbonyl carbon in esters. Crystals of DMS are monoclinic, space groupC2/c,Z=4,a=13.154(4),b=6.156(1),c=9.363(4)Å,=98.53(3)°. The structure was solved by direct methods and refined by leastsquares procedures to giveR=0.071 for 932 observed data. Crystals of DMO are monoclinic space group,P2₁/n,Z=2, witha=3.891(1),b=11.879(2),c=6.213(2) Å,=103.32(2)°. The structure is the same (within experimental error) as that reported by Dougill and Jeffrey (1953) and refined to giveR=0.074 for 395 observed data.     

Windmill polynomials over fields of characteristic two

Chambers andSmeets [3] have designed a windmill arrangement of linear feedback shift registers (LFSRs) to generate pn-sequences overGF(2) with high speed. When the windmill hasv vanes, the associated minimal feedback polynomial (having degreen, relatively prime tov) can be taken to have the shapef ₁(x ^v )+x ⁿ f ₂(x ^–v ), where the polynomialsf ₁ andf ₂ have degree [n/v]. Their numerical evidence, whenv is divisible by 4, suggests that, surprisingly, there areno such windmill polynomials which are irreducible ifn±3 (mod 8), while about twice as many irreducible and primitive windmill polynomials as they expected occur ifn±1 (mod 8). A discussion of this behaviour is presented here with proofs. The brief explanation is that the Galois group of the underlying generic windmill polynomial overGF (4) is equal to the alternating groupA _n .     

Anticancer agent development: X-ray crystal structure and keto-enol tautomerism of dimethyl 1-hydroxy-6,7-methylenedioxy-4-(3′,4′,5′-trimethoxyphenyl-trans-3,4-dihydronaphthalene-2,3-dicarboxylate

Structural and conformational information obtained from the crystal structure and solution¹H nmr investigations of the title compound are compared. The 4-aryltetralone, C₂₄H₂₄O₁₀, crystallizes as a chloroform solvate in the monoclinic space group, P2₁/n, witha=12.519(4),b=17.938(6),c=12.534(9)Å,=111.90(5)°, and D_calc=1.51 g cm^–3 forZ=4. The data for this compound were collected at –150°C. Least-squares refinement of 2796 observed [F _o5(F _o)] reflections led to the final agreement index ofR=0.062. A threefold static disorder was observed for one of the carboxyl groups. The second carboxyl group participates in an intramolecular hydrogen bond and is thus ordered. The¹H nmr spectrum revealed the title compound to exist as a keto-enol tautomeric mixture in solution. Vicinal hydrogen coupling constant analysis proved reliable in ascertaining B-ring stereochemistry of 2,3-disubstituted-4-aryltetralones.     

Comparison theorems forL-monosplines of minimal norm

Summary LetLM _N be the set of allL-monosplines withN free knots, prescribed by a pair (x;E) of pointsx = {x _i } ₁ ⁿ ,a <x ₁ < ... <x _n <b and an incidence matrixE = (e _ij ) _i=1 ⁿ , _r-1 ^j=0 with Denote byLM _N ^O the subset ofLM _N consisting of theL-monosplines withN simple knots (n=N). We prove that theL-monosplines of minimalL _p-norms inLM _N belong toLM _N ^O .The results are reformulated as comparison theorems for quadrature formulae.     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

Circles through two points that always enclose many points

This paper proves that any set of n points in the plane contains two points such that any circle through those two points encloses at least points of the set. The main ingredients used in the proof of this result are edge counting formulas for k-order Voronoi diagrams and a lower bound on the minimum number of semispaces of size at most k.Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under Grant CCR-8714565, by the second author has been partially supported by the Digital Equipment Corporation, by the fourth author has been partially supported by the Office of Naval Research under Grant N00014-86K-0416.     

Algorithms for flows with parametric capacities

We consider the problem of finding maximal flows with respect to capacities which are linear functions of a parametert [0,T]. Since this problem is a special case of a parametric linear program the classichorizontal approach can be applied in which optimal solutions are computed for successive subintervals of [0,T]. We discuss an alternative algorithm which approximates in each iteration the optimal solution for allt [0,T]. Thisvertical algorithm is a labeling type algorithm where the flow variables are piecewise linear functions. Flow augmentations are done alongconditional flow augmenting paths which can be found by modified path algorithms. The vertical algorithm can be used to solve the parametric flow problem optimally as well as to compute a good approximation for allt if the computation of the optimal solution turns out to be too time consuming.Partially supported by NSF Grants ECS-8412926 and INT-8521433, and NATO Grant RG 85/0240.