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The study of the structure of translational tilings has captivated mathematicians, scientists, and the general public for centuries and continues to thrive at the crossroads of analysis, combinatorics, dynamics, logic, number theory, and…
Survey of several forms of updating, with a practical illustrative example. We study several updating (conditioning) schemes that emerge naturally from a common scenarion to provide some insights into their meaning. Updating is a subtle…
We introduce a new formal model -- based on the mathematical construct of sheaves -- for representing contradictory information in textual sources. This model has the advantage of letting us (a) identify the causes of the inconsistency; (b)…
A task of interest in machine learning (ML) is that of ascribing explanations to the predictions made by ML models. Furthermore, in domains deemed high risk, the rigor of explanations is paramount. Indeed, incorrect explanations can and…
A method to define the complex structure and separate the conformal mode is proposed for a surface constructed by two-dimensional dynamical triangulation. Applications are made for surfaces coupled to matter fields such as $n$ scalar fields…
Logical relations (LR) have been around for many years, and today they are used in many formal results. However, it can be difficult to LR beginners to find a good place to start to learn. Papers often use highly specialized LRs that use…
This work is about a new class of martingales: the vertical martingales. We construct the vertical martingale for smooth submersions and we develop a stochastic calculus for one. Furthermore, we gives a stochastic characterization for…
The main purpose of this note is to pose a couple of problems which are easily formulated thought some seem to be not yet solved. These problems are of general interest for discrete mathematics including a new twig of a bough of theory of…
We explore the possibility to derive basic calculus rules for some subdifferential constructions associated to set-valued maps between normed vector spaces. Then, we use these results in order to write optimality conditions for a special…
Skolem sequences and Skolem labeled graphs have been described and examined for several decades. This note explores weak Skolem labelling of cycle graphs, which we call Skolem circles. The relationship between Skolem sequences and Skolem…
A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…
We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a ``stack over a stack'' and the distinction between small stacks (which…
We introduce the notion of scale to generalize and compare different invariants of metric spaces and their measures. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They moreover are…
Dimensions are an integral part of many models we use every day. Without thinking about it, we frequently use the time dimension: many financial and accounting spreadsheets have columns representing months or years. Representing a second…
In mechanical engineering W\"ohler plots serve to measure the average number of load cycles before materials break, as a function of the maximal stress in each cycle. Although such plots are prevalent in engineering for more than 150 years,…
In order to apply nonstandard methods to modern algebraic geometry, as a first step in this paper we study the applications of nonstandard constructions to category theory. It turns out that many categorial properties are well behaved under…
Tackling the many open questions in particle physics will require the construction of new colliders. This short note includes a few considerations that seem to be brought up rarely.
The largely open problem of scaling laws in fully developed turbulence is discussed, with the stress put on similarities and differences with scaling in field theory. A soluble model of the passive advection is examined in more detail in…
We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems…
Lattice scalar field theories encounter a sign problem when the coupling constant is complex. This is a close cousin of the real-time sign problems that afflict the lattice Schwinger-Keldysh formalism, and a more distant relative of the…