Combinatorial Calabi flows on surfaces

For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow provides a new algorithm to find circle packings with prescribed curvatures. The proofs rely on careful analysis of combinatorial Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.Comment: 17 pages, 5 figure

A combinatorial analysis of Severi degrees

Based on results by Brugall\'e and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees $N^{d, \delta}$ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic version of a special function associated to long-edge graphs appeared in Fomin-Mikhalkin's formula, and conjectured it to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we consider a special multivariate function associated to long-edge graphs that generalizes their function. The main result of this paper is that the multivariate function we define is always linear. A special case of our result gives an independent proof of Block-Colley-Kennedy's conjecture. The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of $Q^{d, \delta}$ and a bound $\delta$ for the threshold of polynomiality of $N^{d, \delta}.$ Next, in joint work with Osserman, we apply the linearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the G\"ottsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series $B_1(q)$ and $B_2(q)$ appearing in the G\"ottsche-Yau-Zaslow formula. The proof of our linearity result is completely combinatorial. We define $\tau$-graphs which generalize long-edge graphs, and a closely related family of combinatorial objects we call $(\tau, n)$-words. By introducing height functions and a concept of irreducibility, we describe ways to decompose certain families of $(\tau, n)$-words into irreducible words, which leads to the desired results.Comment: 38 pages, 1 figure, 1 table. Major revision: generalized main results in previous version. The old results only applies to classical Severi degrees. The current version also applies to Severi degrees coming from special families of toric surface

Combinatorial Information Theory: I. Philosophical Basis of Cross-Entropy and Entropy

This study critically analyses the information-theoretic, axiomatic and combinatorial philosophical bases of the entropy and cross-entropy concepts. The combinatorial basis is shown to be the most fundamental (most primitive) of these three bases, since it gives (i) a derivation for the Kullback-Leibler cross-entropy and Shannon entropy functions, as simplified forms of the multinomial distribution subject to the Stirling approximation; (ii) an explanation for the need to maximize entropy (or minimize cross-entropy) to find the most probable realization; and (iii) new, generalized definitions of entropy and cross-entropy - supersets of the Boltzmann principle - applicable to non-multinomial systems. The combinatorial basis is therefore of much broader scope, with far greater power of application, than the information-theoretic and axiomatic bases. The generalized definitions underpin a new discipline of ``{\it combinatorial information theory}'', for the analysis of probabilistic systems of any type. Jaynes' generic formulation of statistical mechanics for multinomial systems is re-examined in light of the combinatorial approach. (abbreviated abstract)Comment: 45 pp; 1 figure; REVTex; updated version 5 (incremental changes

Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem

We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions (Dobiski-type formulas) are obtained for these quantities by both algebraic and combinatorial methods. By extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs. This link provides a combinatorial interpretation of the numbers arising in this normal ordering problem.Comment: 10 pages, 5 figure