Related papers: Singularity Theorems and Their Consequences
In singular models, the optimal set of parameters forms an analytic set with singularities and classical statistical inference cannot be applied to such models. This is significant for deep learning as neural networks are singular and thus…
This monograph discusses dualities in physics: what dualities are, their main examples--from quantum mechanics and electrodynamics to statistical mechanics, quantum field theory and string theory--and the philosophical questions they raise.…
The research process includes many decisions, e.g., how to entitle and where to publish the paper. In this paper, we introduce a general framework for investigating the effects of such decisions. The main difficulty in investigating the…
The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings,…
This review describes the history of discovery of violation of spatial parity P, charge conjugation parity C, combined parity CP. The hypothesis of existence of mirror particles was called upon by its authors to restore the symmetry between…
Our assumption that spacetime is a continuum leads to many challenges in mathematical physics. Singularities, divergent integrals and the like threaten many of our favorite theories, from Newtonian gravity to classical electrodynamics,…
We live at a time of contradictory messages about how successfully we understand gravity. General Relativity seems to work very well in the Earth's immediate neighborhood, but arguments abound that it needs modification at very small and/or…
General Relativity provides us with some solutions for rotating black holes. However, there are some problems associated with them: the appearance of singularities, the possibility of violations of the cosmic censorship conjecture, the…
The main aim of this paper is to make a remark about the relation between (i) dualities between theories, as `duality' is understood in physics and (ii) equivalence of theories, as `equivalence' is understood in logic and philosophy. The…
Interacting massive spin-1 fields have been widely used in cosmology and particle physics. We obtain a new condition on the validity of the classical limit of these theories related to the non-trivial constraints that exist for vector field…
The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce…
Self-similar models are important in general relativity and other fundamental theories. In this paper we shall discuss the ``similarity hypothesis'', which asserts that under a variety of physical circumstances solutions of these theories…
The paper has been withdrawn by the author due to a gap in Proof of Theorem 1.1.
Spurious correlations occur when a model learns unreliable features from the data and are a well-known drawback of data-driven learning. Although there are several algorithms proposed to mitigate it, we are yet to jointly derive the…
In this paper we review some general properties of probability distributions which exibit a singular behavior. After introducing the matter with several examples based on various models of statistical mechanics, we discuss, with the help of…
In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…
These notes are a slightly enlarged version of my habilitation thesis, where our research interest and main results in the past few years are summarized. Most of the discussion revolves around complex ordinary differential equations and…
The analysis of certain singularities in scalar-tensor gravity contained in a recent paper is completed, and situations are pointed out in which these singularities cannot occur.
This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and $\mathbb P^1_C\smallsetminus\{0,\infty\}$, where $C$ is…
In this paper we discuss singularity theorems in quantum gravity using effective field theory methods. To second order in curvature, this effective field theory contains two new degrees of freedom which have important implications for the…