Related papers: Fundamental sequences based on localization
In this paper, the first in a projected two-part series, we describe an organizing framework for the study of infinitary combinatorics. This framework is \v{C}ech cohomology. We show in particular that the \v{C}ech cohomology groups of the…
In these two related parts we present a set of methods, analytical and numerical, which can illuminate the behaviour of quantum system, especially in the complex systems. The key points demonstrating advantages of this approach are: (i)…
A central challenge in continual learning is forgetting, the loss of performance on previously learned tasks induced by sequential adaptation to new ones. While forgetting has been extensively studied empirically, rigorous theoretical…
We introduce an ordinal index which measures the complexity of a weakly null sequence, and show that a construction due to J. Schreier can be iterated to produce for each alpha < omega_1, a weakly null sequence (x^{alpha}_n)_n in…
In this note, by exploiting mean value properties of $s$-harmonic functions, we introduce some monotonicity formulas in the nonlocal setting. We take into account intrinsically nonlocal functionals mimicking those introduced by Alt,…
We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the help…
We study the sequential empirical process indexed by general function classes and its smoothed set-indexed analogue. Sufficient conditions for asymptotic equicontinuity are provided for nonstationary arrays of time series. This yields…
We present a new method for proving non-holonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an…
Sequential modelling with self-attention has achieved cutting edge performances in natural language processing. With advantages in model flexibility, computation complexity and interpretability, self-attention is gradually becoming a key…
How to best develop foundational models for time series forecasting remains an important open question. Tokenization is a crucial consideration in this effort: what is an effective discrete vocabulary for a real-valued sequential input? To…
The ordinal patterns of a fixed number of consecutive values in a time series is the spatial ordering of these values. Counting how often a specific ordinal pattern occurs in a time series provides important insights into the properties of…
We provide quantitative and abstract strong convergence results for sequences from a compact metric space satisfying a certain form of \emph{generalized Fej\'er monotonicity} where (1) the metric can be replaced by a much more general type…
We consider the problem of identifying universal low-dimensional features from high-dimensional data for inference tasks in settings involving learning. For such problems, we introduce natural notions of universality and we show a local…
We introduce cohomology and homology theories for small categories with general coefficient systems from simplex categories first studied by Thomason. These theories generalize at once Baues-Wirsching cohomology and homology and other more…
We prove a generalization of the fundamental theorem of algebraic K-theory for Verdier-localizing functors by extending the proof for algebraic K-theory of spaces to the realm of stable $\infty$-categories. The formula behaves much better…
We introduce a new dependence order, termed the conditional convex order, whose minimal and maximal elements characterize independence and perfect dependence. Moreover, it characterizes conditional independence, satisfies information…
We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, several concepts from analytical mechanics will be rediscovered from an entirely new perspective.…
We give a simple order-theoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order -- the extensional order -- and a bistable coherence, which captures…
This paper is devoted to a systematic study and characterizations of the fundamental notions of variational and strong variational convexity for lower semicontinuous functions. While these notions have been quite recently introduced by…
The purposes of this note are the following two; we first generalize Okada-Takeuti's well quasi ordinal diagram theory, utilizing the recent result of Dershowitz-Tzameret's version of tree embedding theorem with gap conditions. Second, we…