Motivated by the question of rationality of cubic fourfolds, we show that a cubic X has an associated K3 surface in the sense of Hassett if and only if the variety F of lines on X is birational to a moduli space of sheaves on a K3 surface, but that having F birational to Hilb^2(K3) is more restrictive. We compare the loci in the moduli space of cubics where each condition is satisfied.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2016/0023/0001/MRL-2016-0023-0001-a001.pdf

On two rationality conjectures for cubic fourfolds

In a stochastic-flow network with budget constraint, the network reliability for level (d, b), i.e., R(d,b), where d is a given demand value and b is a budget limit, is the probability of transmitt...

Reliability evaluation of a stochastic-flow network in terms of minimal paths with budget constraint

System reliability of a multistate flow network can be computed in terms of all the lower boundary points, called d -minimal paths ( $d$ -MPs). Although several algorithms have been proposed in the literature for the $d$ -MP problem, there is still room for improvement upon its solution. Here, some new results are presented to improve the solution of the problem. A simple novel algorithm is improved to solve a Diophantine system that appeared in the $d$ -MP problem. Then, an improved algorithm is proposed for the $d$ -MP problem. It is also explained how the proposed algorithm can be used in order to assess the reliability of some smart grid communication networks. We provide the complexity results and show the main algorithm to be more efficient than the existing ones in terms of execution times through some benchmark networks and a general network example. Moreover, we compare the algorithms through one thousand randomly generated test problems using the Dolan and More's performance profile.

Finding all the Lower Boundary Points in a Multistate Two-Terminal Network

/pdf/a-note-on-the-diophantine-equation-a-x-b-y-c-z-2avmgryq5f.pdf

A note on the Diophantine equation $a^x + b^y = c^z$

An Introduction to Diophantine Equations

Preface.-Part 1: Diophantine Equations.-Elementary Methods for Solving Diophantine Equations.-The Decomposition Method.-Solving Diophantine Equations Using Inequalities.-The Parametric Method.-The Modular Arithmetic Method.-The Method of Mathematical Induction.-Fermat's Method of Infinite Descent (FMID).-Miscellaneous Diophantine Equations.-Some Classical Diophantine Equation.-Linear Diophantine Equation.-Pythagorean Triples and Related Problems.-Other Remarkable Equations.-Pell's-Type Equations.-Pell's Equation: History and Motivation.-Solving Pell's Equation by Elementary Methods.-The Equation ax^2-by^2=1.-The Negative Pell's Equation.-Part 2: Solutions to Exercises and Problems.-Elementary Methods for Solving Diophantine Equations.-The Decomposition Method.-Solving Diophantine Equations Using Inequalities.-The Parametric Method.-The Modular Arithmetic Method.-The Method of Mathematical Induction.-Fermat's Method of Infinite Descent (FMID).-Miscellaneous Diophantine Equations.-Some Classical Diophantine Equation.-Linear Diophantine Equation.-Pythagorean Triples and Related Problems.-Other Remarkable Equations.-Pell's-Type Equations.-Solving Pell's Equation by Elementary Methods.-The Equation ax^2-by^2=1.-The Negative Pell's Equation.-References.-Index.

An Introduction to Diophantine Equations: A Problem-Based Approach

An equation of the form 

$$a_1x_1 + \cdots + a_n x_n = c,$$

(2.1.1)

where a1, a2, …, a n , b are fixed integers, is called a linear Diophantine equation. We assume that n ≥ 1 and that coefficients a1, …, a n are all different from zero.

Some Classical Diophantine Equations

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An Introduction to Diophantine Equations

An Introduction to Diophantine Equations: A Problem-Based Approach

Some Classical Diophantine Equations