Related papers: Self-embeddings of models of arithmetic; fixed poi…

Higher-dimensional theories provide a promising framework for unified extensions of the supersymmetric standard model. Compactifications to four dimensions often lead to U(1) symmetries beyond the standard model gauge group, whose breaking…

Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of…

Consider the semialgebraic structure over the real field. More generally, let an ominimal structure be over a real closed field. We show that a definable metric space X with a definable metric d is embedded into a Euclidean space so that…

We prove that every continuous function $f:E\to Y$ depends on countably many coordinates, if $E$ is an $(\aleph_1,\aleph_0)$-invariant pseudo-$\aleph_1$-compact subspace of a product of topological spaces and $Y$ is a space with a regular…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{N}\trianglelefteq\mathbb{E}$ of locally compact groups and a twisted action…

Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpi\'nski and Knaster-Urbanik theorems are presented. Embeddable properties of countable metric spaces are generalized onto uncountable metric…

Let $M$ be a separable metric space. We say that $f=(f_n):M\to c_0$ is a good-$\lambda$-embedding if, whenever $x,y\in M$, $x\ne y$ implies $d(x,y)\le\Vert f(x)-f(y)\Vert$ and, for each $n$, $Lip(f_n)<\lambda$, where $Lip(f_n)$ denotes the…

A foundational question in the theory of linear compartmental models is how to assess whether a model is structurally identifiable -- that is, whether parameter values can be inferred from noiseless data -- directly from the combinatorics…

When proving the correctness of a method for slicing probabilistic programs, it was previously discovered by the authors that for a fixed point iteration to work one needs a non-standard starting point for the iteration. This paper presents…

We show that if $X$ is a complete metric space with uniform relative normal structure and $G$ is a subgroup of the isometry group of $X$ with bounded orbits, then there is a point in $X$ fixed by every isometry in $G$. As a corollary, we…

Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This…

In this paper, we explore various ways in which a factor $\sigma$-algebra $\mathscr{B}$ can sit in a dynamical system $\mathbf{X} :=(X, \mathscr{A}, \mu, T)$, i.e. we study some possible structures of the extension $\mathscr{A} \rightarrow…

Let f be a continuous map of a complete separable metric space E onto the irrationals. We show that if a complete separable metric space M contains isometric copies of every closed relatively discrete set in E, then M contains also an…

For a finite and positive measure space $(\Omega,\Sigma,\mu)$ and any weakly compact convex subset of $L\sp\infty(\Omega,\Sigma,mu)$, a fixed point theorem for a class of nonexpansive self-mappings is proved. An analogous result is obtained…

Cosine similarity has become a standard metric for comparing embeddings in modern machine learning. Its scale-invariance and alignment with model training objectives have contributed to its widespread adoption. However, recent studies have…

We construct an indecomposable continuum with exactly one strong non-cut point. The method is an adaptation of Bellamy $[1]$. We start with an $\omega_1$-chain of indecomposable metric continua and retractions. The inverse limit is an…

Simpson showed that every countable model $\mathcal{M} \models \mathsf{PA}$ has an expansion $(\mathcal{M}, X) \models \mathsf{PA}^*$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a…

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is…

Let $R$ be a constant. Let $\mathcal{M}^R_\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\Omega^n$ ($n\ge 3$) with smooth boundary $\Sigma $ such that $g$ has constant scalar curvature $R$ and $g|_{\Sigma}$ is a…