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Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference --the logic of partial beliefs-- can be used to quantify the evidence that finite data provide in favor of a…
I discuss a seemingly unlikely confluence of topics in algebra, numerical computation, and computer vision. The motivating problem is that of solving multiples instances of a parametric family of systems of algebraic (polynomial or rational…
Fermat's Last Theorem is proved by using the philosophical and mathematical knowledge of 1637 when the French mathematician Pierre de Fermat claimed to have a truly marvelous proof of his conjecture. Our approach consists of setting three…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proven easy. We show the intractability of random instances of a…
We present the only proof of Pierre Fermat by descente infinie that is known to exist today. As the text of its Latin original requires active mathematical interpretation, it is more a proof sketch than a proper mathematical proof. We…
The problem of measuring similarity of graphs and their nodes is important in a range of practical problems. There is a number of proposed measures, some of them being based on iterative calculation of similarity between two graphs and the…
Probabilistic inferences distill knowledge from graphs to aid human make important decisions. Due to the inherent uncertainty in the model and the complexity of the knowledge, it is desirable to help the end-users understand the inference…
This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…
Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that…
In this paper we are concerned with various graph invariants (girth, diameter, expansion constants, eigenvalues of the Laplacian, tree number) and their analogs for weighted graphs -- weighing the graph changes a combinatorial problem to…
In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…
We introduce priors and algorithms to perform Bayesian inference in Gaussian models defined by acyclic directed mixed graphs. Such a class of graphs, composed of directed and bi-directed edges, is a representation of conditional…
To solve many problems on graphs, graph traversals are used, the usual variants of which are the depth-first search and the breadth-first search. Implementing a graph traversal we consequently reach all vertices of the graph that belong to…
Three new graph invariants are introduced which may be measured from a quantum graph state and form examples of a framework under which other graph invariants can be constructed. Each invariant is based on distinguishing a different number…
We identify a fragment of Presburger arithmetic enriched with free function symbols and cardinality constraints for interpreted sets, which is amenable to automated analysis. We establish decidability and complexity results for such a…
An algebraic framework in which to study infinite sums is proposed, complementing and augmenting the usual topological tools. The framework subsumes numerous examples in the literature. It is developed using many varied examples, with a…
We present an astonishingly simple and elegant proof of the celebrated Basel problem.
An original approach to solving rather difficult probabilistic problems arising in studying the readout of random discrete fields and having no exact analytical solutions at the moment is proposed. Several algorithms for direct, iterative,…
Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = \alpha^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this…