Related papers: A walk with Goodstein
In this note our aim is to give a proof of the Pontryagin maximum principle for a general optimal control problem with running state constraints and smooth dynamics. Our proof is based on the classical Ekeland variational principle. The…
This note is sketching a simple and natural mathematical construction for explaining the probabilistic nature of quantum mechanics. It employs nonstandard analysis and is based on Feynman's interpretation of the Heisenberg uncertainty…
Deterministic chaos, and even maximum computational complexity, have been discovered within Newtonian dynamics. Economists assume that prices and price changes can also obey abstract mathematical laws of motion. Sociologists and other…
We propose a new method to numerically calculate higher-order correlation functions of primordial fluctuations generated from any early-universe scenario. Our key-starting point is the realization that the tree-level In-In formalism is…
A graph $G$ is defined encapsulating the number theoretic notion of the Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach to the fundamental results on the coprimality of two natural numbers, through the use of…
The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their…
After a brief flirtation with logicism in 1917-1920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the…
We advance a famous principle - causality principle - but under a new view. This principle is a principium automatically leading to most fundamental laws of the nature. It is the inner origin of variation, rules evolutionary processes of…
The foundations of Statistical Mechanics can be recovered almost in their entirety from the Principle of Maximum Entropy. In this work we show that its non-equilibrium generalization, the Principle of Maximum Caliber (Jaynes, 1980), when…
This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical…
Current quantum theories of an elementary free particle assume unitary space inversion and anti-unitary time reversal operators. In so doing robust classes of possible theories are discarded. The present work shows that consistent theories…
The induction principle for natural numbers expresses that when a property holds for some natural number a and is hereditary, then it holds for all numbers greater than or equal to a. We present a similar principle for real numbers.
This paper investigates extreme value theory for processes obtained by applying transformations to stationary Gaussian processes, also called subordinated Gaussian processes. The main contributions are as follows. First, we refine the…
A new method of deriving comparative statics information using generalized compensated derivatives is presented which yields constraint-free semidefiniteness results for any differentiable, constrained optimization problem. More generally,…
We extend the Gibbs conditioning principle to an abstract setting combining infinitely many linear equality constraints and non-linear inequality constraints, which need not be convex. A conditional large large deviation principle (LDP) is…
Extending our own and others' earlier approaches to reasoning about termination of probabilistic programs, we propose and prove a new rule for termination with probability one, also known as "almost-certain termination". The rule uses both…
Motivated by a new formulation of the classical dividend problem, we show that Peskir's maximality principle can be transferred to singular stochastic control problems with 2-dimensional degenerate dynamics and absorption along the diagonal…
This paper shows that a wide class of effective learning rules -- those that improve a scalar performance measure over a given time window -- can be rewritten as natural gradient descent with respect to a suitably defined loss function and…
Statistical inference on biodiversity has a rich history going back to RA Fisher. An influential ecological theory suggests the existence of a fundamental biodiversity number, denoted $\alpha$, which coincides with the precision parameter…
We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences…