Connections between semidefinite relaxations of the max-cut and stable set problems

We describe links between a recently introduced semidefinite relaxation for the max-cut problem and the well known semidefinite relaxation for the stable set problem underlying the Lovász’s theta function. It turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra. We also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations. This work was done while the author visited CWI, Amsterdam, The Netherlands. Svata Poljak untimely deceased in April 1995. We shall both miss Svata very much. Svata was an excellent colleague, from whom we learned a lot of mathematics and with whom working was always a very enjoyable experience; above all, Svata was a very nice person and a close friend of us. The research was partly done while the author visited CWI, Amsterdam, in October 1994 with a grant fom the Stieltjes Institute, whose support is gratefully acknowledged. Partially supported also by GACR 0519. Research support by Christian Doppler Laboratorium für Diskrete Optimierung.     

Comparison of Two Reformulation-Linearization Technique Based Linear Programming Relaxations for Polynomial Programming Problems

In this paper, we compare two strategies for constructing linear programmingrelaxations for polynomial programming problems using aReformulation-Linearization Technique (RLT). RLT involves an automaticreformulation of the problem via the addition of certain nonlinear impliedconstraints that are generated by using the products of the simple boundingrestrictions (among other products), and a subsequent linearization based onvariable redefinitions. We prove that applying RLT directly to the originalpolynomial program produces a bound that dominates in the sense of being atleast as tight as the value obtained when RLT is applied to the jointcollection of all equivalent quadratic problems that could be constructed byrecursively defining additional variables as suggested by Shor.     

Influence of moisture on the viscoelastic relaxations in long aliphatic chain contained semiaromatic polyamide, (PA9‐T) fiber

The effect of moisture on the mechanical relaxation processes of semiaromatic semicrystalline polyamides synthesized by a long‐chain aliphatic diamine and terephthalic acid was investigated by dynamic viscoelastic analysis (DVA) and differential scanning calorimetry (DSC). Moreover, the implication of moisture with the amorphous and crystalline domains was also examined by temperature‐dependent wide‐angle X‐ray diffraction and Fourier transform infrared spectra. The characteristics of the relaxations such as α, β, γ, and the pronounced peak shoulder appeared at 25–100 °C in DVA tan δ curves were found to be strongly susceptible to the presence of moisture. With moisture evaporation, the peak shoulder of 25–100 °C and the β‐relaxation disappeared. The former is anticipated to originate from to the side group motion of hydrogen‐bonded water, whereas the later one is from the motions of the amide–water complex units. With the disappearance of the β‐relaxation, the γ‐relaxation appeared simultaneously in much lower temperatures and ultimately coupled with the δ‐relaxation. The γ‐relaxation is attributed to be associated with the molecular motion of the amide group and δ‐relaxation with for the motion of the methylene units. The existence of two types of water was identified in the polymer, namely, tightly bound and loosely bound. The tightly bound water is believed to be directly connected by hydrogen bonding with the strong polar groups and the loosely bound water weakly links with those connected water making hydrogen bridges. The moisture acts as a plasticizer in the polymer matrix, which causes quite a large depression in its glass transition temperature (T_g). WAXD and FTIR studies corroborated the existence of water solely in amorphous regions, i.e., no rapport of water with the crystalline parts. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2878–2891, 2003     

Structural morphology of poly(styrene)-poly(butyl acrylate) polymer-polymer composites studied by dynamic mechanical measurements

The emulsion polymerization process allows production of polymer particles with different structural morphologies. Films obtained after coalescence keep some memory of this morphology, but large modifications can occur during coalescence. In the present case, one of the polymers, polystyrene (PS), exhibits a glass temperature (Tg) much higher than the filmification temperature (close to room temperature), while the other one, poly(butyl acrylate) (PBA), has a much lowerTg. Furthermore, it is well known that dynamic mechanical measurements can be very helpful in providing information on the morphology of polymer materials, i.e., on geometrical and topological arrangement of homopolymer domains. At first, this method was used for comparison of two types of films: i) the first one obtained from structured-core (PS)-shell (PBA) particles, ii) the second one obtained from a blend of homopolymer particles (PS and PBA). It appears that the expected core-shell particles lost their geometric structure in the second film. Second, comparison of the predicted dynamic modulus and experimental data shows that i) strong interactions exist between PS nodules unless their coalescence has occured, leading to an abnormally high modulus at room temperature, ii) after achieving their coalescence, PS forms a more or less continuous phase. Both phenomena strongly depend on the particle size and their respective volume fractions.     

Influence of isotacticity and molecular weight on the properties of metallocenic isotactic polypropylene

The relationships between structure and properties have been established for isotactic polypropylene, iPP, homopolymers synthesized by metallocene catalyst systems. These iPPs exhibit different isotacticity degrees and molecular weights, and several thermal treatments during their processing have been applied. The most important factor affecting the structure and properties of these polymers is the isotacticity content. The thermal treatment, i.e., the rate of cooling from the melt, is also important and a clear molecular weight effect has been also found for the sample with lowest M_w. These factors affect the thermal properties, the degree of crystallinity and, therefore, the structural parameters and the viscoelastic behavior. A slow cooling from melt favors the formation of the γ phase instead of the α modification. The storage modulus, Young modulus and microhardness values increase as crystallinity does, independently of the origin of this increase: higher isotacticity or application of a slow crystallization from the melt.     

Raman spectra and structural features of ClOF2 + cation in a solution of anhydrous HF

The Raman spectra of ClOF₂ ⁺ cation in solutions of anhydrous HF were studied. In the ClOF₂ ⁺HF₂ ⁻ and ClOF₂ ⁺BF₄ ⁻−HF systems, this cation exists as a pyramidal structure (C _s symmetry), while in the ClOF₂ ⁺AuF₆ ⁻−HF system, it exists as a planar structure (C _2v symmetry). Based on nonempirical calculations by the Hartree-Fock-Roothaan method, an explanation for the dependence of the structure of the ClOF₂ ⁺ cation on the nature of the anion was proposed. For the Cl−O bond vibrations, the correlation functions of vibrational and rotational relaxations were calculated, and the characteristic times of these processes were determined. The main contribution to the formation of the band contours corresponding to the above-mentioned modes is made by the vibrational dephasing. Translated fromIzvestiya Akademii Nauk, Seriya Khimicheskaya, No. 3, pp. 432–437, March, 1998.     

On clique relaxation models in network analysis

Increasing interest in studying community structures, or clusters in complex networks arising in various applications has led to a large and diverse body of literature introducing numerous graph-theoretic models relaxing certain characteristics of the classical clique concept. This paper analyzes the elementary clique-defining properties implicitly exploited in the available clique relaxation models and proposes a taxonomic framework that not only allows to classify the existing models in a systematic fashion, but also yields new clique relaxations of potential practical interest. Some basic structural properties of several of the considered models are identified that may facilitate the choice of methods for solving the corresponding optimization problems. In addition, bounds describing the cohesiveness properties of different clique relaxation structures are established, and practical implications of choosing one model over another are discussed.     

Relaxations in thermosets. XIII. Effects of post-cure and aging on the sub-Tg relaxations of nonstoichiometric epoxide-based thermosets

The dielectric permittivity and loss of diglycidyl ether of bisphenol-A (DGEBA) cured with greater than and less than the stoichiometric amounts of diaminodiphenyl methane (DDM) have been measured over a temperature range 77–350 K prior to curing and gelation, after curing at about 340 K and further aging for a predetermined period. The height of the γ-relaxation peak monotonically decreases during the post-cure period and becomes masked by the contributions from the β-relaxation peak, whose height, in turn, first increases on postcuring to a same maximum value for both nonstoichiometric thermosets and then decreases. This decrease is attributed to physical aging effects. The β-relaxation peak shifts towards higher temperature on postcuring. Comparison between the changes in the dielectric properties of the saturated and starved thermosets show that while the γ-relaxation process may be attributed to the motion of the epoxide dipolar groups of the unreacted DGEBA, the β-relaxation process is not attributable entirely to the motion of ? OH groups and of the unreacted amines in the thermoset. Explanations involving the chain and network packing in the structure of a thermoset are necessary for the observed behavior of the β-relaxation process in amine saturated and starved thermosets.     

Fractal differential equations and fractal-time dynamical systems

Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate framework. A new calculus calledF ^α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF ^α-integral andF ^α-derivative respectively. TheF ^α-integral is suitable for integrating functions with fractal support of dimension α, while theF ^α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions ofF ^α-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems. We discuss construction and solutions of some fractal differential equations of the form

Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound

We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper, we specialize the algorithm to the box-constrained case and study its implementation, which is shown to be a state-of-the-art method for globally solving box-constrained nonconvex quadratic programs. S. Burer was supported in part by NSF Grants CCR-0203426 and CCF-0545514.