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A new approach to analyzing intrinsic properties of the Josephus function, $J_{_k}$, is presented in this paper. The linear structure between extreme points of $J_{_k}$ is fully revealed, leading to the design of an efficient algorithm for evaluating $J_{_k}(n)$. Algebraic expressions that describe how recursively compute extreme points, including fixed points, are derived. The existence of consecutive extreme and also fixed points for all $k\geq 2$ is proven as a consequence, which generalizes Knuth result for $k=2$. Moreover, an extensive comparative numerical experiment is conducted to illustrate the performance of the proposed algorithm for evaluating the Josephus function compared to established algorithms. The results show that the proposed scheme is highly effective in computing $J_{_k}(n)$ for large inputs.

A universal partial cycle (or upcycle) for $\mathcal{A}^n$ is a cyclic sequence that covers each word of length $n$ over the alphabet $\mathcal{A}$ exactly once -- like a De Bruijn cycle, except that we also allow a wildcard symbol $\mathord{\diamond}$ that can represent any letter of $\mathcal{A}$. Chen et al. in 2017 and Goeckner et al. in 2018 showed that the existence and structure of upcycles are highly constrained, unlike those of De Bruijn cycles, which exist for any alphabet size and word length. Moreover, it was not known whether any upcycles existed for $n \ge 5$. We present several examples of upcycles over both binary and non-binary alphabets for $n = 8$. We generalize two graph-theoretic representations of De Bruijn cycles to upcycles. We then introduce novel approaches to constructing new upcycles from old ones. Notably, given any upcycle for an alphabet of size $a$, we show how to construct an upcycle for an alphabet of size $ak$ for any $k \in \mathbb{N}$, so each example generates an infinite family of upcycles. We also define folds and lifts of upcycles, which relate upcycles with differing densities of $\mathord{\diamond}$ characters. In particular, we show that every upcycle lifts to a De Bruijn cycle. Our constructions rely on a different generalization of De Bruijn cycles known as perfect necklaces, and we introduce several new examples of perfect necklaces. We extend the definitions of certain pseudorandomness properties to partial words and determine which are satisfied by all upcycles, then draw a conclusion about linear feedback shift registers. Finally, we prove new nonexistence results based on the word length $n$, alphabet size, and $\mathord{\diamond}$ density.

In this article, we study systems of $n \geq 1$, not necessarily linear, discrete differential equations (DDEs) of order $k \geq 1$ with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions of such equations. This part of the present article can be seen as a generalization of the pioneering work by Bousquet-M\'elou and Jehanne~(2006) who settled down the case $n=1$. Moreover, we obtain effective bounds for the algebraicity degrees of the solutions and provide an algorithm for computing annihilating polynomials of the algebraic series. Finally, we carry out a first analysis in the direction of effectivity for solving systems of DDEs in view of practical applications.

For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal H}$-subgraph-free graphs (for finite sets ${\cal H}$) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity ${\cal H}$-subgraph-free graphs is unknown. In this paper, we study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: $k$-Induced Disjoint Paths, $C_5$-Colouring, Hamilton Cycle and Star $3$-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and differs from problems that do satisfy all three conditions of the framework. Hence, we exhibit a rich complexity landscape among problems for ${\cal H}$-subgraph-free graph classes.

We derive general bounds on the probability that the empirical first-passage time $\overline{\tau}_n\equiv \sum_{i=1}^n\tau_i/n$ of a reversible ergodic Markov process inferred from a sample of $n$ independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct non-asymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Deep learning algorithms have been successfully applied to numerically solve linear Kolmogorov partial differential equations (PDEs). A recent research shows that the empirical risk minimization~(ERM) over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of linear Kolmogorov PDEs with bounded initial functions. However, the initial functions may be unbounded in many applications such as the Black Scholes PDEs in pricing call options. In this paper, we extend this result to the cases involving unbounded initial functions. We prove that for $d$-dimensional linear Kolmogorov PDEs with unbounded initial functions, under suitable assumptions, the number of training data and the size of the artificial neural network required to achieve an accuracy $\varepsilon$ for the ERM grow polynomially in both $d$ and $\varepsilon^{-1}$. Moreover, we verify that the required assumptions hold for Black-Scholes PDEs and heat equations which are two important cases of linear Kolmogorov PDEs.

Generalising the concept of a complete permutation polynomial over a finite field, we define completness to level $k$ for $k\ge1$ in fields of odd characteristic. We construct two families of polynomials that satisfy the condition of high level completeness for all finite fields, and two more families complete to the maximum level a possible for large collection of finite fields. Under the binary operation of composition of functions one family of polynomials is an abelian group isomorphic to the additive group, while the other is isomorphic to the multiplicative group.

For an undirected unweighted graph $G=(V,E)$ with $n$ vertices and $m$ edges, let $d(u,v)$ denote the distance from $u\in V$ to $v\in V$ in $G$. An $(\alpha,\beta)$-stretch approximate distance oracle (ADO) for $G$ is a data structure that given $u,v\in V$ returns in constant (or near constant) time a value $\hat d (u,v)$ such that $d(u,v) \le \hat d (u,v) \le \alpha\cdot d(u,v) + \beta$, for some reals $\alpha >1, \beta$. If $\beta = 0$, we say that the ADO has stretch $\alpha$. Thorup and Zwick~\cite{thorup2005approximate} showed that one cannot beat stretch 3 with subquadratic space (in terms of $n$) for general graphs. P\v{a}tra\c{s}cu and Roditty~\cite{patrascu2010distance} showed that one can obtain stretch 2 using $O(m^{1/3}n^{4/3})$ space, and so if $m$ is subquadratic in $n$ then the space usage is also subquadratic. Moreover, P\v{a}tra\c{s}cu and Roditty~\cite{patrascu2010distance} showed that one cannot beat stretch 2 with subquadratic space even for graphs where $m=\tilde{O}(n)$, based on the set-intersection hypothesis. In this paper we explore the conditions for which an ADO can be stored using subquadratic space while supporting a sub-2 stretch. In particular, we show that if the maximum degree in $G$ is $\Delta_G \leq O(n^{1/2-\varepsilon})$ for some $0<\varepsilon \leq 1/2$, then there exists an ADO for $G$ that uses $\tilde{O}(n^{2-\frac {2\varepsilon}{3}})$ space and has a sub-2 stretch. Moreover, we prove a conditional lower bound, based on the set intersection hypothesis, which states that for any positive integer $k \leq \log n$, obtaining a sub-$\frac{k+2}{k}$ stretch for graphs with maximum degree $\Theta(n^{1/k})$ requires quadratic space. Thus, for graphs with maximum degree $\Theta(n^{1/2})$, obtaining a sub-2 stretch requires quadratic space.

A 2-packing set for an undirected graph $G=(V,E)$ is a subset $\mathcal{S} \subset V$ such that any two vertices $v_1,v_2 \in \mathcal{S}$ have no common neighbors. Finding a 2-packing set of maximum cardinality is a NP-hard problem. We develop a new approach to solve this problem on arbitrary graphs using its close relation to the independent set problem. Thereby, our algorithm red2pack uses new data reduction rules specific to the 2-packing set problem as well as a graph transformation. Our experiments show that we outperform the state-of-the-art for arbitrary graphs with respect to solution quality and also are able to compute solutions multiple orders of magnitude faster than previously possible. For example, we are able to solve 63% of our graphs to optimality in less than a second while the competitor for arbitrary graphs can only solve 5% of the graphs in the data set to optimality even with a 10 hour time limit. Moreover, our approach can solve a wide range of large instances that have previously been unsolved.

We generalized a modified exponentialized estimator by pushing the robust-optimal (RO) index $\lambda$ to $-\infty$ for achieving robustness to outliers by optimizing a quasi-Minimin function. The robustness is realized and controlled adaptively by the RO index without any predefined threshold. Optimality is guaranteed by expansion of the convexity region in the Hessian matrix to largely avoid local optima. Detailed quantitative analysis on both robustness and optimality are provided. The results of proposed experiments on fitting tasks for three noisy non-convex functions and the digits recognition task on the MNIST dataset consolidate the conclusions.