相关论文: On the grasshopper problem with signed jumps

Let an $n\times n$ array $\left( a_{ij}\right) $ of lights be given, each either on (when $a_{ij}=1$) or off (when $a_{ij}=-1$). For each row and each column there is a switch so that if the switch is pulled ($x_{i}=-1$ for row $i$ and…

The pole placement problem belongs to the classical problems of linear systems theory. It is often assumed that the ground field is the real numbers R or the complex numbers C. The major result over the complex numbers derived in 1981 by…

We use SMT technology to address a class of problems involving uninterpreted functions and nonlinear real arithmetic. In particular, we focus on problems commonly found in mathematical competitions, such as the International Mathematical…

Motivated by the recent 10 million dollar AIMO challenge, this paper targets the problem of finding all functions conforming to a given specification. This is a popular problem at mathematical competitions and it brings about a number of…

Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this decomposition to construct a two-player game. Given a fixed integer $n$ and an initial decomposition of $n=n…

We investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability {\cal L}_N(t) that the leftmost particle remains the leftmost as a…

The Erd\H{o}s--Ginzburg--Ziv Problem is a classical extremal problem in discrete geometry. Given $m$ and $n$, the problem asks about the smallest number $s$ such that among any $s$ points in the integer lattice $\mathbb{Z}^n$ one can find…

Non-Stop University CRYPTO is the International Olympiad in Cryptography that was held for the eight time in 2021. Hundreds of university and school students, professionals from 33 countries worked on mathematical problems in cryptography…

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which…

For a given set of intervals on the real line, we consider the problem of ordering the intervals with the goal of minimizing an objective function that depends on the exposed interval pieces (that is, the pieces that are not covered by…

In this work, we establish near-linear and strong convergence for a natural first-order iterative algorithm that simulates Von Neumann's Alternating Projections method in zero-sum games. First, we provide a precise analysis of Optimistic…

Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that: (i) all prefix sums of the ordering are nonnegative, and (ii) the maximum value of a prefix sum is minimized. Kellerer…

A good range of problems on trees can be described by the following general setting: Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$ and a vector $s\in\mathbb R^d$, we need to estimate the largest possible absolute value…

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L_1 and L_2 the goal is to decide whether there exists an orthogonal linear transformation mapping L_1 to L_2. Our main result is an algorithm for this problem…

Let $\mathcal{O}_n$ be the set of all maximal outerplanar graphs of order $n$. Let $ar(\mathcal{O}_n,F)$ denote the maximum positive integer $k$ such that $T\in \mathcal{O}_n$ has no rainbow subgraph $F$ under a $k$-edge-coloring of $T$.…

We deal with the Neumann boundary value problem \begin{equation*} \begin{cases} \, u" + \bigl{(} \lambda a^{+}(t)-\mu a^{-}(t) \bigr{)}g(u) = 0, \\ \, 0 < u(t) < 1, \quad \forall\, t\in\mathopen{[}0,T\mathclose{]},\\ \, u'(0) = u'(T) = 0,…

The usefulness of parameterized algorithmics has often depended on what Niedermeier has called, "the art of problem parameterization". In this paper we introduce and explore a novel but general form of parameterization: the number of…

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a…

These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…

We concern the problem of modifying the edge lengths of a tree in minimum total cost so that the prespecified $p$ vertices become the $p$-maxian with respect to the new edge lengths. This problem is called the inverse $p$-maxian problem on…