Related papers: The fair soup division and approximating numbers
Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum…
In this note we study a problem of fair division in the absence of full information. We give an algorithm which solves the following problem: n $\ge$ 2 persons want to cut a cake into n shares so that each person will get at least 1/n of…
We explore an algorithm for approximating roots of integers, discuss its motivation and derivation, and analyze its convergence rates with varying parameters and inputs. We also perform comparisons with established methods for approximating…
This paper is an overview of the classical level crossing problem which is studied extensively in the literature and is fundamental in many branches of applied probability. We discuss a number of approximations with an emphasis on their…
We analyze solvability of a special form of distributed order fractional differential equations within the space of tempered distributions supported by the positive half-line.
In this paper, we investigate the Mill's ratio estimation problem and get two new inequalities. Compared to the well known results obtained by Gordon, they becomes tighter. Furthermore, we also discuss the inverse Q-function approximation…
Combining several (sample approximations of) distributions, which we term sub-posteriors, into a single distribution proportional to their product, is a common challenge. Occurring, for instance, in distributed 'big data' problems, or when…
We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that…
For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…
We give a possible explanation for the mystery of a missing number in the statement of a problem that asks for the non-negative integers to be partitioned into three subsets. We interpret the missing number as one of the clues that can lead…
The division of goods in the online realm poses opportunities and challenges. While innovative mechanisms can be developed, uncertainty about the future may hinder effective solutions. This project aims to explore fair distribution models…
We introduce a very natural generalization of the well-known problem of simultaneous congruences. Instead of searching for a positive integer $s$ that is specified by $n$ fixed remainders modulo integer divisors $a_1,\dots,a_n$ we consider…
In this brief note we critically examine the process of partial and of total differentiation, showing some of the problems that arise when we relate both concepts. A way to solve all the problems is proposed.
In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due…
We consider the problem of fairly dividing a two dimensional heterogeneous good among multiple players. Applications include division of land as well as ad space in print and electronic media. Classical cake cutting protocols primarily…
We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.
The design of fair voting rules has been addressed quite often in the literature. Still, the so-called inverse problem is not entirely resolved. We summarize some achievements in this direction and formulate explicit open questions and…
Fairness in multiwinner elections is studied in varying contexts. For instance, diversity of candidates and representation of voters are both separately termed as being fair. A common denominator to ensure fairness across all such contexts…
We give a simple inequality for the sum of independent bounded random variables. This inequality improves on the celebrated result of Hoeffding in a special case. It is optimal in the limit where the sum tends to a Poisson random variable.
It is proved that one cannot approximate stably the first derivative of a smooth function given noisy values of this function and a bound on this function and its first derivative. Such an approximation is shown to be possible if an a…