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Prior-data fitted networks (PFNs) have achieved exceptional performance on tabular classification tasks. However, like other classifiers, their performance can suffer under the effect of class imbalance, resulting in poor performance for…
Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important…
We derive properties of powers of a function satisfying a second-order linear differential equation. In particular we prove that the n-th power of the function satisfies an (n+1)-th order differential equation and give a simple method for…
Completely multiplicative functions whose sum is zero ($CMO$).The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits\_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all…
Equivariant neural networks provide a principled framework for incorporating symmetry into learning architectures and have been extensively analyzed through the lens of their separation power, that is, the ability to distinguish inputs…
In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$, where we restrict the class of weight functions to strongly…
Let $p$ be an odd prime, $k,\ell$ be positive integers, $q=p^k, Q=p^{\ell}$. In this paper we characterise planar functions of the form $f_{\underline{c}}(X)=c_0X^{qQ+q}+c_1X^{qQ+1}+c_2X^{Q+q}+c_3X^{Q+1}$ over $\mathbb{F}_{q^2}$ for any…
The study of finite projective planes involves planar functions, namely, functions f : F_q --> F_q such that, for each nonzero a in F_q, the function c --> f(c+a) - f(c) is a bijection on F_q. Planar functions are also used in the…
Graph Neural Networks (GNNs) has been widely used in a variety of fields because of their great potential in representing graph-structured data. However, lacking of rigorous uncertainty estimations limits their application in high-stakes.…
We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word…
We characterize the computational power of neural networks that follow the graph neural network (GNN) architecture, not restricted to aggregate-combine GNNs or other particular types. We establish an exact correspondence between the…
A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject…
We introduce and study a new class of differential fields in positive characteristic. We call them separably differentially closed fields and demonstrate that they are the differential analogue of separably closed fields. We prove several…
The partition function $ p_{[1^c11^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{11}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{11 n})^d}.\] In this paper, we prove infinite families of…
Probabilistic circuits (PCs) have gained prominence in recent years as a versatile framework for discussing probabilistic models that support tractable queries and are yet expressive enough to model complex probability distributions.…
A new class of regular quaternionic functions, defined by power series in a natural fashion, has been introduced in recent years. Several results of the theory recall the classical complex analysis, whereas other results reflect the…
We show that neural networks with access to randomness can outperform deterministic networks by using amplification. We call such networks Coin-Flipping Neural Networks, or CFNNs. We show that a CFNN can approximate the indicator of a…
Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers-Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under…
Recent studies on binomials of the form $F_r(x) = x^r(1 + \chi(x))$ over $\mathbb{F}_{p^n}$ have shown that these functions can exhibit very low boomerang uniformity. In this paper, we focus on the specific behavior of such binomials in…
We introduce a new point of view towards Glaeser's theorem on composite $C^\infty$ functions [Ann. of Math. 1963], with respect to which we can formulate a ``$C^k$ composite function property" that is satisfied by all semiproper real…