Related papers: An elementary proof that P =/= NP

The following paper follows on from work by Kamae, and gives a rigorous proof of the Ergodic Theorem, using nonstandard analysis.

It is proved that $P(\omega)/\triangle_d$, where $\triangle_d$ is the ideal of sets of asymptotic density zero, is universal in the sense of embeddings.

Given an algebraic differential equation of order greater than one, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already…

We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for any such $p$-semi-regular type, "domination…

We develop non-asymptotically justified methods for hypothesis testing about the $p-$dimensional coefficients $\theta^{*}$ in (possibly nonlinear) regression models. Given a function $h:\,\mathbb{R}^{p}\mapsto\mathbb{R}^{m}$, we consider…

A general structural equation model is fitted on a panel data set that consists of $I$ correlated samples. The correlated samples could be data from correlated populations or correlated observations from occasions of panel data. We consider…

Symmetry tests provide an important probe for the structure of elementary particle interactions and for the validity of the standard model. However, it is pointed out that in the interpretation of such experiments one must keep in mind that…

We give a simple new proof that regular languages defined by first-order sentences with no quantifier alteration can be defined by such sentences in which only regular atomic formulas appear. Earlier proofs of this fact relied on arguments…

Let X be a d dimensional vector of covariates and Y be the response variable. Under the nonparametric model Y = m(X) + {\sigma}(X) \in we develop an ANOVA-type test for the null hypothesis that a particular coordinate of X has no influence…

The preferential attachment (PA) model is a popular way of modeling dynamic social networks, such as collaboration networks. Assuming that the PA function takes a parametric form, we propose and study the maximum likelihood estimator of the…

As suggested by the title, it has recently become clear that theorems of Nonstandard Analysis (NSA) give rise to theorems in computability theory (no longer involving NSA). Now, the aforementioned discipline divides into classical and…

We prove an impossibility result for conditional Probably Approximately Correct (PAC)-efficient reasoning in large language models. While recent work has established marginal PAC efficiency guarantees for composite models that switch…

We consider polynomials in R[x] which map the set of nonnegative (element-wise) matrices of a given order into itself. Let n be a positive integer and define P(n)= {p in R[x] : p(A) is nonnegative (element-wise), for all A, A an n-by-n…

In this paper we prove uniqueness results for renormalized solutions to a class of nonlinear parabolic problems.

In this note we prove that the parametric fundamental equation of information is stable in the sense of Hyers and Ulam provided that the parameter is nonpositive. We also prove, as a corollary, that the system of equations that defines the…

We render a thorough, physicist's account of the formulation of the Standard Model (SM) of particle physics within the framework of noncommutative differential geometry (NCG). We work in Minkowski spacetime rather than in Euclidean space.…

We construct a class of noncommutative spectra and give the basic properties of the class of noncommutative spectra.

A prefixed polynomial equation is an equation of the form $P(t_1,\ldots,t_n) = 0$, where $P$ is a polynomial whose variables $t_1,\ldots,t_n$ range over the natural numbers, preceded by quantifiers over some, or all, of its variables. Here,…

We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.

Let $p$ be a prime integer and $\mathbb{Z}_p$ be the ring of $p$-adic integers. By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group ${\rm…