On the maximal number of solutions of a problem in linear inequalities

The problemy=Ax+c,x≧0,y≧0, (x, y)=0 is considered, where the square real matrixA and the real vectorc are the data and a solution is a pair of vectorsx, y. Under certain conditions on the matrixA there exists a solution for every vectorc, but it cannot be unique for everyc. We prove that under these conditions the maximal number of solutions is 2 ⁿ − 1.     

Bounded geometry for Kleinian groups

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations. Oblatum 31-VII-2000 & 9-V-2001?Published online: 20 July 2001     

Inducing efficiency in oligopolistic markets with increasing returns to scale

We consider a Cournot oligopoly market of firms possessing increasing returns to scale technologies (which may not be identical). It is shown that an external regulating agency can increase total social welfare without running a deficit by offering to subsidize one firm an amount which depends on the output level of that firm and the market price. The firms bid for this contract, the regulator collects the highest bid upfront and subsidizes the highest bidding firm. It is shown that there exists a subsidy schedule such that (i) the regulator breaks even, (ii) the subsidized firm obtains zero net profit and charges a price equal to its average cost, (iii) every other firm willingly exit the market and (iv) market price decreases, consumers are better off and total welfare improves.     

Colorful monochromatic connectivity

An edge-coloring of a connected graph is monochromatically-connecting if there is a monochromatic path joining any two vertices. How “colorful” can a monochromatically-connecting coloring be? Let mc(G) denote the maximum number of colors used in a monochromatically-connecting coloring of a graph G. We prove some nontrivial upper and lower bounds for mc(G) and relate it to other graph parameters such as the chromatic number, the connectivity, the maximum degree, and the diameter.     

Projected Subgradient Minimization Versus Superiorization

The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to regain feasibility. The latter poses a computational difficulty, and, therefore, the projected subgradient method is applicable only when the feasible region is “simple to project onto.” In contrast to this, in the superiorization methodology a feasibility-seeking algorithm leads the overall process, and objective function steps are interlaced into it. This makes a difference because the feasibility-seeking algorithm employs projections onto the individual constraints sets and not onto the entire feasible region. We present the two approaches side-by-side and demonstrate their performance on a problem of computerized tomography image reconstruction, posed as a constrained minimization problem aiming at finding a constraint-compatible solution that has a reduced value of the total variation of the reconstructed image.     

Functionalization of Poly[Ethylene Terephthalate) by Means of Glow-Discharge-Initiated Polymerization of Acrylic Acid

Glow-discharge-initiated polymerization of acrylic acid incorporated in poly(ethylene terephthalate) (PET) films was investigated. An increase in polymerization yield with plasma treatment duration and power was found. Polymerization was not confined to the film surface. At high power and long treatment time, polymerization in the bulk of the PET also took place. Water regain and contact angle of the PET-treated films were affected by the presence of poly(acrylic acid) (PAA). The carboxyl groups of the PAA chains incorporated in the PET matrix were utilized for further chemical modification of the PET film. Poly(ethylene glycol) (PEG) was grafted onto PAA by esterification. DSC studies showed the presence of both PAA and PEG in the PET matrix and shed light on the morphology of the multicomponent polymeric system. Free isocyanate groups were introduced into the PET matrix by reacting PAA carboxyl groups with hexamethylene diisocyanate.     

Block Polymers Derived from Poly(ethylene Oxide) and Carboxyl-Terminated Polybutadiene

The use of preformed poly(ethylene oxide) (PEO) and carboxyl-terminated polybutadiene (CTPB) for the preparation of block polymers is reported. Block polymerization was carried out by esterification and by coupling of equimolar amounts of these polymers with 2,4-toluene diisocyanate (TDI). When esterification was carried out, conversion of the two preformed polymers and block polymer composition varied with reaction temperature, catalyst used, and molecular weight of the PEO. Full conversions were not obtained. Better results were achieved when the preformed polymers were coupled with TDI. Tensile properties and water absorption capability of these block polymers were determined. Hydrogels with high water content up to 82% were obtained.     

Binomial Coefficients and Zero-Sum Ramsey Numbers

A counting argument is developed and divisibility properties of the binomial coefficients are combined to prove, among other results, that[formula]whereK_n, resp.K^k_n, is the complete, resp. completek-uniform, hypergaph andR(K_n, Z_p),R(K^k_n, Z₂) are the corresponding zero-sum Ramsey numbers.