New Farkas-Type Results for Vector-Valued Functions: A Non-abstract Approach

This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, called decision space. The contained and the container sets are described here by means of vector functions from the decision space to other two locally convex spaces which are equipped with the partial ordering associated with given convex cones. These new Farkas lemmas are obtained via the complete characterization of the conic epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the authors (Dinh et al. in J Optim Theory Appl 173:357–390, 2017), the aimed characterizations of the containment are expressed here in terms of the data.This research was supported by the National Foundation for Science & Technology Development (NAFOSTED) of Vietnam, Project 101.01-2015.27, Generalizations of Farkas lemma with applications to optimization, by the Ministry of Economy and Competitiveness of Spain and the European Regional Development Fund (ERDF) of the European Commission, Project MTM2014-59179-C2-1-P, and by the Australian Research Council, Project DP160100854

Tropical polyhedra are equivalent to mean payoff games

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.Comment: 28 pages, 5 figures; v2: updated references, added background materials and illustrations; v3: minor improvements, references update

Multi-Dimensional Sigma-Functions

In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by K.Weierstrass and F.Klein was aimed to extend to higher genera Weierstrass theory of elliptic functions based on the Weierstrass $\sigma$-functions. Our development was motivated by the recent achievements of mathematical physics and theory of integrable systems that were based of the results of classical theory of multi-dimensional theta functions. Both theta and sigma-functions are integer and quasi-periodic functions, but worth to remark the fundamental difference between them. While theta-function are defined in the terms of the Riemann period matrix, the sigma-function can be constructed by coefficients of polynomial defining the curve. Note that the relation between periods and coefficients of polynomials defining the curve is transcendental. Since the publication of our 1997-review a lot of new results in this area appeared (see below the list of Recent References), that promoted us to submit this draft to ArXiv without waiting publication a well-prepared book. We complemented the review by the list of articles that were published after 1997 year to develop the theory of $\sigma$-functions presented here. Although the main body of this review is devoted to hyperelliptic functions the method can be extended to an arbitrary algebraic curve and new material that we added in the cases when the opposite is not stated does not suppose hyperellipticity of the curve considered.Comment: 267 pages, 4 figure

A New Model of Wage Determination and Wage Inequality

This paper proposes a new model of wage determination and wage inequality. In this model, wage-setters set workers' wages; they do so either directly, as when individuals vote in a salary committee, or indirectly, as when political parties, via the myriad of social, economic, fiscal, and other policies, generate wages. The recommendations made by wage-setters (or arising from their policies) form a distribution, and all the wage-setter-specific distributions are combined into a single final wage distribution. There may be any number of wage-setters; some wage-setters count more than others; and the wage-setters may differ among themselves on both the wage distribution and the amounts recommended for particular workers. We use probability theory to derive initial results, including both distribution-independent and distribution-specific results. Fortuitously, elements of the model correspond to basic democratic principles. Thus, the model yields implications for the effects of democracy on wage inequality. These include: (1) The effects of the number of wage-setters and their power depend on the configuration of agreements and disagreements; (2) Independence of mind reduces wage inequality, and dissent does so even more; (3) When leaders of democratic nations seek to forge an economic consensus, they are unwittingly inducing greater economic inequality; (4) Arguments for independent thinking will be more vigorous in small societies than in large societies; (5) Given a fixed distributional form for wages and two political parties which either ignore or oppose each other's distributional ideas, the closer the party split to 50-50, the lower the wage inequality; and (6) Under certain conditions the wage distribution within wage-setting context will be normal, but the normality will be obscured, as cross-context mixtures will display a wide variety of shapes.shifted mirror-exponential distribution, shifted general Erlang distribution, wage-setter, power, consensus, independence of mind, dissent, form of government, probability distributions, shifted exponential distribution, Gini coefficient

Computational photochemistry of heteroaromatic biomolecules : photodynamic therapy and ultrafast relaxation

This thesis focuses on the photochemistry of heteroaromatic biomolecules. These molecular systems have a rich photochemistry and take part in photochemical reactions that have many very topical applications. Small heteroaromatics constitute important biological building blocks and are therefore a fundamental components of living organisms. Even though these compounds absorb light very efficiently, they also have ultrafast relaxation processes available to them. This means that they can remove the absorbed energy very fast and avoid harmfull photoproducts forming, which can lead to cell damage. Larger heteroaromatics have a similarly efficient absorption of electromagnetic light, and are present in compounds that are responsible for the harvesting of energy in nature, for example the chlorophyll molecule in green plants and bacteria. If large heteroaromatics are artificially presented to living cells however, the excess energy absorbed by these systems may also cause cell damage. This destructive force can however be utilised in therapy forms where there is a need to get rid of unwanted cells, such as in anti-cancer therapy. A form of therapy based on this principle is photodynamic therapy. The use of computational chemistry in the investigations of photochemical phenomena has increased following the improvements in the efficiency of computers and algorithms. Modern techniques have now reached a stage where ultrafast relaxation processes can be calculated for small heteroaromatics. As the experimental community has also reached a stage where these compounds can be probed using ultrafast laser experiments, there is a need for computational input to aid in the interpretation of the data of these phenomena. This thesis will present computational results concerning the relaxation dynamics of important small heteroaromatic biomolecules, and discuss them in terms of experimental data collected by collaborative groups. For the development of molecules to be used in photodynamic therapy, a lot of work is needed to ensure safety for use in human beings. With the computational chemistry community now being able to carry out absorption studies for large heteroaromatics, computational structure-absorption relationships can aid the development of this form of therapy. At the limits of modern photochemistry, methods are also appearing that can be used for studies of ultrafast relaxation in larger systems. These computations could contribute hugely to the understanding of the behaviour of these types of systems and aid their development. In a large component of this thesis, new structure-absorption relationships are presented for interesting heteroaromatics with potential for use in photodynamic therapy. One section is also devoted to exploratory work using methods that have not before been used in systems that are larger in size, and presents some promising results as well as current challenges in the field