Related papers: Tree Codes and a Conjecture on Exponential Sums
We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…
This is an expository paper aiming to introduce Zilber's Exponential Closedness conjecture to a general audience. Exponential Closedness predicts when (systems of) equations involving addition, multiplication, and exponentiation have…
A new synthesis scheme is proposed to effectively generate a random vector with prescribed joint density that induces a (latent) Gaussian tree structure. The quality of synthesis is measured by total variation distance between the…
An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over…
In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit…
We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…
We describe a new Pr\"ufer code which works also for infinite trees.
Existing ordinal trees and random forests typically use scores that are assigned to the ordered categories, which implies that a higher scale level is used. Versions of ordinal trees are proposed that take the scale level seriously and…
The class of $\ell$-maximum distance separable ($\ell$-MDS) codes {is a} generalization of maximum distance separable (MDS) codes {that} has attracted a lot of attention due to its applications in several areas such as secret sharing…
Linear-time computational techniques have been developed for combining evidence which is available on a number of contending hypotheses. They offer a means of making the computation-intensive calculations involved more efficient in certain…
Recursive neural networks (RvNN) have been shown useful for learning sentence representations and helped achieve competitive performance on several natural language inference tasks. However, recent RvNN-based models fail to learn simple…
Entailment trees have been proposed to simulate the human reasoning process of explanation generation in the context of open--domain textual question answering. However, in practice, manually constructing these explanation trees proves a…
In this paper, we define an intuitionistic version of Computation Tree Logic. After explaining the semantic features of intuitionistic logic, we examine how these characteristics can be interesting for formal verification purposes.…
We give an algorithm to enumerate the results on trees of monadic second-order (MSO) queries represented by nondeterministic tree automata. After linear time preprocessing (in the input tree), we can enumerate answers with linear delay (in…
We show that for a wide range of channels and code ensembles with pairwise-independent codewords, with probability tending to 1 with the code length, expurgating an arbitrarily small fraction of codewords from a randomly selected code…
We present an explicit connected spanning structure that appears in a random graph just above the connectivity threshold with high probability.
The first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove that the first-order theory of a string automatic…
Imagine being able to ask questions to a black box model such as "Which adversarial examples exist?", "Does a specific attribute have a disproportionate effect on the model's prediction?" or "What kind of predictions could possibly be made…
Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However, they are not always robust and tend to overfit the data. Additionally, if…
We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle \frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short…