With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph. In particular, the set of satisfiable formulae in k-conjunctive normal form with ≤ n variables has the homotopy type of Θ(Cube(n, n− k)), where Cube(n, n− k) is a hypergraph associated to the (n− k)-skeleton of an n-cube. We make...

A new sequence of eigenfunctions is developed and studied in depth. These theta polynomials are derived from a recent analytic solution of the canonical Cauchy problem for parabolic equations, namely, the inverse heat conduction problem. By appealing to the methods of the operator calculus, it is possible to categorize the new functions as polynomials of binomial and Sheffer types. The connecti...

In this paper, we discuss generating theorems of polyhedral quadrangulations closed surfaces. We prove that the set eight reductional operations {R1, …, R8} defined for is finitizable any surface F2, is, there exist finitely many minimal F2 using such R1, R7 and R8. Furthermore, show proper subset not torus.

Abstract We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I series, we showed that, for many pairs infinite cardinals $\theta < \kappa $ , the existence a coloring $c:[\kappa ]^2 \rightarrow \theta is theorem $\textsf{ZFC}$ . Adding requirement subadditivity to significant strengthening, though, and here see that in cases independent conn...

Real-world applications now deal with a massive amount of data, and information about the world is inaccurate, incomplete, or uncertain. Therefore, we present in our paper proposed model for solving problems. This based on class locally generalized closed sets, namely, simply* alpha closed* sets closed** (briefly, -sets -sets), open set. We also introduce various concepts their properties relat...

We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal T}_{\rm max}(E)$, min}(E)$ and ort}(E)$. show that these are not isolated examples; in any given system of at least one will occur as a subprojection. This characterizes those which have is the first general structural result for such systems.

We study the convergence and rates of a multi-block proximal alternating direction method multipliers (PADMM) for solving linearly constrained separable nonconvex nonsmooth optimization problems. This algorithm is an important variant (ADMM) which includes term in each subproblem, to cancel out complicated terms applications where subproblems are not easy solve or do admit simple closed form so...