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A pervasive methodological error is the post-hoc interpretation of $p$-values. A $p$-value $p$ is the smallest significance level at which we would have rejected the null had we chosen level $p$. It is not the smallest significance level at which we reject the null. We introduce post-hoc $p$-values, that do admit such a post-hoc interpretation. We show that $p$ is a post-hoc $p$-value if and only if $1/p$ is an $e$-value, a recently introduced statistical object. The product of independent post-hoc $p$-values is a post-hoc $p$-value, making them easy to combine. Moreover, any post-hoc $p$-value can be trivially improved if we permit external randomization, but only (essentially) non-randomized post-hoc $p$-values can be arbitrarily merged through multiplication. In addition, we discuss what constitutes a `good' post-hoc $p$-value. Finally, we argue that post-hoc $p$-values eliminate the need of a pre-specified significance level, such as $\alpha = .05$ or $\alpha = .005$ (Benjamin et al., 2018). We believe this may take away incentives for $p$-hacking and contribute to solving the file-drawer problem, as both these issues arise from using a pre-specified significance level.

We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to $1/| \log h |^2$. This result holds in dimension $n=1$, in any dimension $n \geq 3$ if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension $n=3$ for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.

Given a graph $G$ and two independent sets of $G$, the independent set reconfiguration problem asks whether one independent set can be transformed into the other by moving a single vertex at a time, such that at each intermediate step we have an independent set of $G$. We study the complexity of this problem for $H$-free graphs under the token sliding and token jumping rule. Our contribution is twofold. First, we prove a reconfiguration analogue of Alekseev's theorem, showing that the problem is PSPACE-complete unless $H$ is a path or a subdivision of the claw. We then show that under the token sliding rule, the problem admits a polynomial-time algorithm if the input graph is fork-free.

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism from FG to Fn which maps G to the standard basis of Fn. Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space Fn, which does not assume an a priori group algebra structure on Fn. As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.

We study the problem of identifying a small set $k\sim n^\theta$, $0<\theta<1$, of infected individuals within a large population of size $n$ by testing groups of individuals simultaneously. All tests are conducted concurrently. The goal is to minimise the total number of tests required. In this paper we make the (realistic) assumption that tests are noisy, i.e.\ that a group that contains an infected individual may return a negative test result or one that does not contain an infected individual may return a positive test results with a certain probability. The noise need not be symmetric. We develop an algorithm called SPARC that correctly identifies the set of infected individuals up to $o(k)$ errors with high probability with the asymptotically minimum number of tests. Additionally, we develop an algorithm called SPEX that exactly identifies the set of infected individuals w.h.p. with a number of tests that matches the information-theoretic lower bound for the constant column design, a powerful and well-studied test design.

An $s{\operatorname{-}}t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices $s$ and $t$. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from $s$ to $t$. In this work we describe a quantum algorithm for the minimum $s{\operatorname{-}}t$ cut problem on undirected graphs. For an undirected graph with $n$ vertices, $m$ edges, and integral edge weights bounded by $W$, the algorithm computes with high probability the weight of a minimum $s{\operatorname{-}}t$ cut after $\widetilde O(\sqrt{m} n^{5/6} W^{1/3})$ queries to the adjacency list of $G$. For simple graphs this bound is always $\widetilde O(n^{11/6})$, even in the dense case when $m = \Omega(n^2)$. In contrast, a randomized algorithm must make $\Omega(m)$ queries to the adjacency list of a simple graph $G$ even to decide whether $s$ and $t$ are connected.

New low-order $H(\textrm{div})$-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the $(d+1)$-order normal-normal face bubble space. The reduced counterpart has only $d(d+1)^2$ degrees of freedom. Basis functions are explicitly given in terms of barycentric coordinates. Low-order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lam\'{e} coefficient $\lambda$, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.

We give an approximate Menger-type theorem for when a graph $G$ contains two $X-Y$ paths $P_1$ and $P_2$ such that $P_1 \cup P_2$ is an induced subgraph of $G$. More generally, we prove that there exists a function $f(d) \in O(d)$, such that for every graph $G$ and $X,Y \subseteq V(G)$, either there exist two $X-Y$ paths $P_1$ and $P_2$ such that the distance between $P_1$ and $P_2$ is at least $d$, or there exists $v \in V(G)$ such that the ball of radius $f(d)$ centered at $v$ intersects every $X-Y$ path.

The joint bidiagonalization (JBD) process iteratively reduces a matrix pair $\{A,L\}$ to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of $\{A,L\}$. The process has a nested inner-outer iteration structure, where the inner iteration usually can not be computed exactly. In this paper, we study the inaccurately computed inner iterations of JBD by first investigating influence of computational error of the inner iteration on the outer iteration, and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality of a part of Lanczos vectors. An error analysis of the rJBD is carried out to build up connections with Lanczos bidiagonalizations. The results are then used to investigate convergence and accuracy of the rJBD based GSVD computation. It is shown that the accuracy of computed GSVD components depend on the computing accuracy of inner iterations and condition number of $(A^T,L^T)^T$ while the convergence rate is not affected very much. For practical JBD based GSVD computations, our results can provide a guideline for choosing a proper computing accuracy of inner iterations in order to obtain approximate GSVD components with a desired accuracy. Numerical experiments are made to confirm our theoretical results.

The hull of a linear code $C$ is the intersection of $C$ with its dual code. We present and analyze the number of linear $q$-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension $k$ and length $n\ge 2k$ the number of all $[n,k]_q$ linear codes with hull dimension $l$ decreases as $l$ increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length $n$ and dimension $k$, comparing the obtained numbers with the number of all linear codes for the given $n$ and $k$.