Related papers: String Dimension: VC Dimension for Infinite Shatte…
The study of strategic or adversarial manipulation of testing data to fool a classifier has attracted much recent attention. Most previous works have focused on two extreme situations where any testing data point either is completely…
Since some years, non-overlapping sets of strings (also called cross-bifix-free sets) have had an increasing interest in the frame of the researches about Theory of Codes. Recently some non-overlapping sets of strings with variable length…
Let $\mathcal{SN}$ be the $\sigma$-ideal of the strong measure zero sets of reals. We present general properties of forcing notions that allow to control of the additivity of $\mathcal{SN}$ after finite support iterations. This is applied…
High dimensional data and systems with many degrees of freedom are often characterized by covariance matrices. In this paper, we consider the problem of simultaneously estimating the dimension of the principal (dominant) subspace of these…
We prove three results on the dimension structure of complexity classes. 1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances…
We introduce the following variant of the VC-dimension. Given $S \subseteq \{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on every…
Using the concept of finite-size scaling, Monte Carlo calculations of various models have become a very useful tool for the study of critical phenomena, with the system linear dimension as a variable. As an example, several recent studies…
In a class of non-Abelian solitonic vortex strings supported in certain N=2 super-Yang-Mills theories we search for the vortex which can behave as a critical fundamental string. We use the Polchinski-Strominger criterion of the ultraviolet…
Fractals represent one of the fundamental manifestations of complexity, and fractal networks serve as tools for characterizing and investigating the fractal structures and properties of large-scale systems. Higher-order networks have…
We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse…
We discuss the definition and measurability questions of random fractals and find under certain conditions a formula for upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.
String theory requires additional degrees of freedom to maintain world-sheet reparameterisation invariance at the quantum level. These are often interpreted as extra dimensions, beyond the 4 space-time. I discuss a class of quasi-realistic…
In this work, we investigate the exclusive production of particles in scattering processes in the so-called saturation region. Within this scheme the phenomenon of geometric scaling takes place: cross sections are functions only of a…
This paper provides the first finite-dimensional characterization for the complete set of full-block, circle criterion multipliers. We consider the interconnection of a discrete-time, linear time-invariant system in feedback with a…
Estimating intrinsic dimensionality of data is a classic problem in pattern recognition and statistics. Principal Component Analysis (PCA) is a powerful tool in discovering dimensionality of data sets with a linear structure; it, however,…
Cicho\'n's diagram describes the connections between combinatorial notions related to measure, category, and compactness of sets of irrational numbers. In the second part of the 2010's, Goldstern, Kellner and Shelah constructed a forcing…
Four-particle tree-level scattering amplitudes in string theory are magically consistent with unitarity, reflected in the non-trivial fact that beneath the critical dimension, the residues of the amplitudes on massive poles can be expanded…
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…
Fractional superstrings are non-trivial generalizations of ordinary superstrings and heterotic strings, and have critical spacetime dimensions which are less than ten by virtue of a worldsheet fractional supersymmetry relating worldsheet…
The VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities. In this paper, we resolve two longstanding open problems on bounding the…