We present a method for finding large fixed-size primes of the form $X^2+c$. We study the density of primes on the sets $E_c = \{N(X,c)=X^2+c,\ X \in (2\mathbb{Z}+(c-1))\}$, $c \in \mathbb{N}^*$. We describe an algorithm for generating values of $c$ such that a given prime $p$ is the minimum of the union of prime divisors of all elements in $E_c$. We also present quadratic forms generating divisors of Ec and study the prime divisors of its terms. This paper uses the results of Dirichlet's arithmetic progression theorem [1] and the article [6] to rewrite a conjecture of Shanks [2] on the density of primes in $E_c$. Finally, based on these results, we discuss the heuristics of large primes occurrences in the research set of our algorithm.
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We develop the theory of illfounded and cyclic proof systems in the context of the modal $\mu$-calculus. A fine analysis of provability and admissibility bridges the finitary, cyclic and illfounded notions of proof for this logic and re-enforces the subtlety of two important normal form theorems: guardedness and disjunctiveness.
We explore new interactions between finite model theory and a number of classical streams of universal algebra and semigroup theory. A key result is an example of a finite algebra whose variety is not finitely axiomatisable in first order logic, but which has first order definable finite membership problem. This algebra witnesses the simultaneous failure of the {\L}os-Tarski Theorem, the SP-preservation theorem and Birkhoff's HSP-preservation theorem at the finite level as well as providing a negative solution to a first order formulation of the long-standing Eilenberg Sch\"utzenberger problem. The example also shows that a pseudovariety without any finite pseudo-identity basis may be finitely axiomatisable in first order logic. Other results include the undecidability of deciding first order definability of the pseudovariety of a finite algebra and a mapping from any fixed template constraint satisfaction problem to a first order equivalent variety membership problem, thereby providing examples of variety membership problems complete in each of the classes $\texttt{L}$, $\texttt{NL}$, $\texttt{Mod}_p(\texttt{L})$, $\texttt{P}$, and infinitely many others (depending on complexity-theoretic assumptions).
Fix a positive integer $n$, a real number $p\in (0,1]$, and a (perhaps random) hypergraph $\mathcal{H}$ on $[n]$. We introduce and investigate the following random multigraph model, which we denote $\mathbb{G}(n,p\, ; \,\mathcal{H})$: begin with an empty graph on $n$ vertices, which are labelled by the set $[n]$. For every $H\in \mathcal{H}$ choose, independently from previous choices, a doubleton from $H$, say $D = \{i,j\} \subset H$, uniformly at random and then introduce an edge between the vertices $i$ and $j$ in the graph with probability $p$, where each edge is introduced independently of all other edges.
We consider the gradient descent flow widely used for the minimization of the $\mathcal{L}^2$ cost function in Deep Learning networks, and introduce two modified versions; one adapted for the overparametrized setting, and the other for the underparametrized setting. Both have a clear and natural invariant geometric meaning, taking into account the pullback vector bundle structure in the overparametrized, and the pushforward vector bundle structure in the underparametrized setting. In the overparametrized case, we prove that, provided that a rank condition holds, all orbits of the modified gradient descent drive the $\mathcal{L}^2$ cost to its global minimum at a uniform exponential convergence rate; one thereby obtains an a priori stopping time for any prescribed proximity to the global minimum. We point out relations of the latter to sub-Riemannian geometry.
We continue to investigate the $k$ nearest neighbour learning rule in separable metric spaces. Thanks to the results of C\'erou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Gy\"{o}rfi, Krzy\.{z}ak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (that is, those of Nagata dimension zero). Combining the theorem of C\'erou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the $k$-NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the $k$-NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Kor\'anyi and Reimann (1995) and Sawyer and Wheeden (1992).
We propose a novel test procedure for comparing mean functions across two groups within the reproducing kernel Hilbert space (RKHS) framework. Our proposed method is adept at handling sparsely and irregularly sampled functional data when observation times are random for each subject. Conventional approaches, which are built upon functional principal components analysis, usually assume a homogeneous covariance structure across groups. Nonetheless, justifying this assumption in real-world scenarios can be challenging. To eliminate the need for a homogeneous covariance structure, we first develop the functional Bahadur representation for the mean estimator under the RKHS framework; this representation naturally leads to the desirable pointwise limiting distributions. Moreover, we establish weak convergence for the mean estimator, allowing us to construct a test statistic for the mean difference. Our method is easily implementable and outperforms some conventional tests in controlling type I errors across various settings. We demonstrate the finite sample performance of our approach through extensive simulations and two real-world applications.
For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.
The minimum set cover (MSC) problem admits two classic algorithms: a greedy $\ln n$-approximation and a primal-dual $f$-approximation, where $n$ is the universe size and $f$ is the maximum frequency of an element. Both algorithms are simple and efficient, and remarkably -- one cannot improve these approximations under hardness results by more than a factor of $(1+\epsilon)$, for any constant $\epsilon > 0$. In their pioneering work, Gupta et al. [STOC'17] showed that the greedy algorithm can be dynamized to achieve $O(\log n)$-approximation with update time $O(f \log n)$. Building on this result, Hjuler et al. [STACS'18] dynamized the greedy minimum dominating set (MDS) algorithm, achieving a similar approximation with update time $O(\Delta \log n)$ (the analog of $O(f \log n)$), albeit for unweighted instances. The approximations of both algorithms, which are the state-of-the-art, exceed the static $\ln n$-approximation by a rather large constant factor. In sharp contrast, the current best dynamic primal-dual MSC algorithms achieve fast update times together with an approximation that exceeds the static $f$-approximation by a factor of (at most) $1+\epsilon$, for any $\epsilon > 0$. This paper aims to bridge the gap between the best approximation factor of the dynamic greedy MSC and MDS algorithms and the static $\ln n$ bound. We present dynamic algorithms for weighted greedy MSC and MDS with approximation $(1+\epsilon)\ln n$ for any $\epsilon > 0$, while achieving the same update time (ignoring dependencies on $\epsilon$) of the best previous algorithms (with approximation significantly larger than $\ln n$). Moreover, [...]
We propose a new representation of functions in Sobolev spaces on an $N$-dimensional hyper-rectangle, expressing such functions in terms of their admissible derivatives, evaluated along lower-boundaries of the domain. These boundary values are either finite-dimensional or exist in the space $L_{2}$ of square-integrable functions -- free of the continuity constraints inherent to Sobolev space. Moreover, we show that the map from this space of boundary values to the Sobolev space is given by an integral operator with polynomial kernel, and we prove that this map is invertible. Using this result, we propose a method for polynomial approximation of functions in Sobolev space, reconstructing such an approximation from polynomial projections of the boundary values. We prove that this approximation is optimal with respect to a discrete-continuous Sobolev norm, and show through numerical examples that it exhibits better convergence behavior than direct projection of the function. Finally, we show that this approach may also be adapted to use a basis of step functions, to construct accurate piecewise polynomial approximations that do not suffer from e.g. Gibbs phenomenon.
We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation, and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.
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