Related papers: Solving the 100 Swiss Francs Problem

We study collusion in a second-price auction with two bidders in a dynamic environment. One bidder can make a take-it-or-leave-it collusion proposal, which consists of both an offer and a request of bribes, to the opponent. We show that…

This study actually draws from and builds on an earlier paper (Kumar and Bhattacharya, 2002). Here we have basically added a neutrosophic dimension to the problem of determining the conditional probability that a financial fraud has been…

In this paper two conjectures are proposed based on which we can prove the first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1 (\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first case of FLT can be…

Spectral learning recently generated lots of excitement in machine learning, largely because it is the first known method to produce consistent estimates (under suitable conditions) for several latent variable models. In contrast, maximum…

We use state-of-art lattice algorithms to improve the upper bound on the lowest counterexample to the Mertens conjecture to $\approx \exp(1.96 \times 10^{19})$, which is significantly below the conjectured value of $\approx \exp(5.15 \times…

Walks in the plane taking unit-length steps north and east from $(0,0)$ to $(n,n)$ never dropping below $y=x$ and parking cars subject to preferences are two intriguing ingredients in a formula conjectured in 2005, now famously known as the…

We develop a maximum penalized quasi-likelihood estimator for estimating in a nonparametric way the diffusion function of a diffusion process, as an alternative to more traditional kernel-based estimators. After developing a numerical…

The goal of this article is to give a positive answer to Rockafellar's maximality of the sum conjecture in the linear multi-valued operator case.

Logistic regression models for binomial responses are routinely used in statistical practice. However, the maximum likelihood estimate may not exist due to data separability. We address this issue by considering a conjugate prior penalty…

We determine an explicit formula for the Laplace transform of the price of an option on a maximal interest rate when the instantaneous rate satisfies Cox-Ingersoll-Ross's model. This generalizes considerably one result of Leblanc-Scaillet.

Randomstrasse101 is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in…

Large language models (LLMs) play a vital role in almost every domain in today's organizations. In the context of this work, we highlight the use of LLMs for sentiment analysis (SA) and explainability. Specifically, we contribute a novel…

We re-visit the classical problem of optimal payment of dividends and determine the degree to which the diffusion approximation serves as a valid approximation of the classical risk model for this problem. Our results parallel some of those…

Let $\mu_1 \ge \dotsc \ge \mu_n > 0$ and $\mu_1 + \dotsm + \mu_n = 1$. Let $X_1, \dotsc, X_n$ be independent non-negative random variables with $EX_1 = \dotsc = EX_n = 1$, and let $Z = \sum_{i=1}^n \mu_i X_i$. Let $M = \max_{1 \le i \le n}…

We study multivariate Gaussian models that are described by linear conditions on the concentration matrix. We compute the maximum likelihood (ML) degrees of these models. That is, we count the critical points of the likelihood function over…

We present SGDPLL(T), an algorithm that solves (among many other problems) probabilistic inference modulo theories, that is, inference problems over probabilistic models defined via a logic theory provided as a parameter (currently,…

Numerous capability and safety techniques of Large Language Models (LLMs), including RLHF, automated red-teaming, prompt engineering, and infilling, can be cast as sampling from an unnormalized target distribution defined by a given reward…

In this paper, we present a new approachment for Sturm-Liouville problem having special potentials. We acquire the representations of solutions and asymptotic formulas for solutions with regard to initial conditions. Also, a few…

The union-closed sets conjecture, attributed to P\'eter Frankl from 1979, states that for any non-empty finite union-closed family of finite sets not consisting of only the empty set, there is an element that is in at least half of the sets…

The Polignac's Conjecture, first formulated by Alphonse de Polignac in 1849, asserts that, for any even number M, there exist infinitely many couples of prime numbers P, P+M. When M = 2, this reduces to the Twin Primes Conjecture. Despite…