Related papers: The 3x+1 problem: a lower bound hypothesis

We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set…

We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give…

In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear…

Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on…

The $3x+1$ problem, also called the Collatz conjecture, is a very interesting unsolved mathematical problem related to computer science. This paper generalized this problem by relaxing the constraints, i.e., generalizing this deterministic…

We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We…

We introduce new probabilistic arguments to derive optimal-order central moment bounds in planar directed last-passage percolation. Our technique is based on couplings with the increment-stationary variants of the model, and is presented in…

Under the 3x+1 problem, classified the number into four kind by mod 4. The four kind number can form a cycle base on 3x+b1 problem. Base on this cycle, if the number of kind number is zero the 3x+1 will be proofed.

We argue that proven exponential upper bounds on runtimes, an established area in classic algorithms, are interesting also in heuristic search and we prove several such results. We show that any of the algorithms randomized local search,…

The notion of (3+1)-avoidance has shown up in many places in enumerative combinatorics. The natural goal of enumeration of all (3+1)-avoiding posets remains open. In this paper, we enumerate graded (3+1)-avoiding posets for both reasonable…

Michel proved that for a one-parameter family of elliptic curves over Q(T) with non-constant j(T) that the second moment of the number of solutions modulo p is p^2 + O(p^{3/2}). We show this bound is sharp by studying y^2 = x^3 + Tx^2 + 1.…

Recently, Gilmer proved the first constant lower bound for the union-closed sets conjecture via an information-theoretic argument. The heart of the argument is an entropic inequality involving the OR function of two i.i.d.\ binary vectors,…

We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can…

This paper concerns the restricted 3-body problem. By applying topological methods we give a computer assisted proof of the existence of some classes of periodic orbits, the existence of symbolic dynamics and we give a rigorous lower…

There has been a resurgence of interest in lower bounds whose truth rests on the conjectured hardness of well known computational problems. These conditional lower bounds have become important and popular due to the painfully slow progress…

The clustering of integers with equal total stopping times has long been observed in the 3x + 1 Problem, and a number of elementary results about it have been used repeatedly in the literature. In this paper we introduce a simple…

It is often claimed that Bayesian methods, in particular Bayes factor methods for hypothesis testing, can deal with optional stopping. We first give an overview, using elementary probability theory, of three different mathematical meanings…

We axiomatize the first-order theories of exponential integer parts of real-closed exponential fields in a language with $2^x$, in a language with a predicate for powers of 2, and in the basic language of ordered rings. In particular, the…

We investigate the continuous non-monotone DR-submodular maximization problem subject to a down-closed convex solvable constraint. Our first contribution is to construct an example to demonstrate that (first-order) stationary points can…

Suppose $X$ is a uniformly distributed $n$-dimensional binary vector and $Y$ is obtained by passing $X$ through a binary symmetric channel with crossover probability $\alpha$. A recent conjecture by Courtade and Kumar postulates that…