相关论文: On The Interpretation Of The Master Equation

These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…

Beginning in the 1970s, statistician-cum-logician Per Martin-L\"of wrote a series of papers developing what became Martin-L\"of type theory, realizing a system where the distinction between mathematics and programming disappears. Inspired…

Large Language Models (LLMs) are widely used to evaluate natural language generation tasks as automated metrics. However, the likelihood, a measure of LLM's plausibility for a sentence, can vary due to superficial differences in sentences,…

We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it.…

We provide a semi-parametric analysis for the proportional likelihood ratio model, proposed by Luo & Tsai (2012). We study the tangent spaces for both the parameter of interest and the nuisance parameter, and obtain an explicit expression…

By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which…

We outline an unified introduction to the evolution equations of classical and quantum systems intended for a high school students audience. The attempt consists in circumventing the lack of mathematical knowledge with the use of simplified…

Partitions of the set of primes are introduced based on the Chebyshev polynomials at rationals. The prime densities of all such partitions are established. Euler's Criterion for $SL(2,\mathbb Q)$ is formulated, which is the bridge between…

These lecture notes are a concise introduction of recent techniques to prove local spectral universality for a large class of random matrices. The general strategy is presented following the recent book with H.T. Yau. We extend the scope of…

Nies and Scholz defined quantum Martin-L\"of randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum…

Modern applications require methods that are computationally feasible on large datasets but also preserve statistical efficiency. Frequently, these two concerns are seen as contradictory: approximation methods that enable computation are…

Automatically generating high-quality step-by-step solutions to math word problems has many applications in education. Recently, combining large language models (LLMs) with external tools to perform complex reasoning and calculation has…

The second-order equation in the (1/2,0)+(0,1/2) representation of the Lorentz group has been proposed by A. Barut in the 70s. It permits to explain the mass splitting of leptons (e,mu,tau). Recently, the interest has grown to this model…

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second order equation in divergence form with discontinuous coefficient. Our concern is to estimate the solutions with explicit constants,…

A short review is given of the simplified differential equations approach to Master Integrals, which was recently proposed by one of the authors. We show its applicability by calculating some non-trivial two-loop planar Master Integrals,…

Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The procedure of stochastization of one-step process was formulated. It allows…

Computing the probability of a formula given the probabilities or weights associated with other formulas is a natural extension of logical inference to the probabilistic setting. Surprisingly, this problem has received little attention in…

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on…

We propose a new representation for several quantum master equations in so-called quasithermodynamic form. This representation (when it exists) let one to write down dynamical equations both for diagonal and non-diagonal elements of density…

Reparameterization (RP) and likelihood ratio (LR) gradient estimators are used to estimate gradients of expectations throughout machine learning and reinforcement learning; however, they are usually explained as simple mathematical tricks,…