Binary codes of length $n$ may be viewed as subsets of vertices of the Boolean hypercube $\{0,1\}^n$. The ability of a linear error-correcting code to recover erasures is connected to influences of particular monotone Boolean functions. These functions provide insight into the role that particular coordinates play in a code's erasure repair capability. In this paper, we consider directly the influences of coordinates of a code. We describe a family of codes, called codes with minimum disjoint support, for which all influences may be determined. As a consequence, we find influences of repetition codes and certain distinct weight codes. Computing influences is typically circumvented by appealing to the transitivity of the automorphism group of the code. Some of the codes considered here fail to meet the transitivity conditions requires for these standard approaches, yet we can compute them directly.
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Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry, gave suboptimal conditions for existence of solutions and proposed an $H^2$-conformal finite element method to approximate them. In this paper, the existence of Miura surfaces is studied using a mixed formulation. It is also proved that the constraints propagate from the boundary to the interior of the domain for well-chosen boundary conditions. Then, a numerical method based on a least-squares formulation, Taylor--Hood finite elements and a Newton method is introduced to approximate Miura surfaces. The numerical method is proved to converge at order one in space and numerical tests are performed to demonstrate its robustness.
We analyze the conforming approximation of the time-harmonic Maxwell's equations using N\'ed\'elec (edge) finite elements. We prove that the approximation is asymptotically optimal, i.e., the approximation error in the energy norm is bounded by the best-approximation error times a constant that tends to one as the mesh is refined and/or the polynomial degree is increased. Moreover, under the same conditions on the mesh and/or the polynomial degree, we establish discrete inf-sup stability with a constant that corresponds to the continuous constant up to a factor of two at most. Our proofs apply under minimal regularity assumptions on the exact solution, so that general domains, material coefficients, and right-hand sides are allowed.
We identify a family of $O(|E(G)|^2)$ nontrivial facets of the connected matching polytope of a graph $G$, that is, the convex hull of incidence vectors of matchings in $G$ whose covered vertices induce a connected subgraph. Accompanying software to further inspect the polytope of an input graph is available.
Let $(M,g)$ be a Riemannian manifold. If $\mu$ is a probability measure on $M$ given by a continuous density function, one would expect the Fr\'{e}chet means of data-samples $Q=(q_1,q_2,\dots, q_N)\in M^N$, with respect to $\mu$, to behave ``generically''; e.g. the probability that the Fr\'{e}chet mean set $\mbox{FM}(Q)$ has any elements that lie in a given, positive-codimension submanifold, should be zero for any $N\geq 1$. Even this simplest instance of genericity does not seem to have been proven in the literature, except in special cases. The main result of this paper is a general, and stronger, genericity property: given i.i.d. absolutely continuous $M$-valued random variables $X_1,\dots, X_N$, and a subset $A\subset M$ of volume-measure zero, $\mbox{Pr}\left\{\mbox{FM}(\{X_1,\dots,X_N\})\subset M\backslash A\right\}=1.$ We also establish a companion theorem for equivariant Fr\'{e}chet means, defined when $(M,g)$ arises as the quotient of a Riemannian manifold $(\widetilde{M},\tilde{g})$ by a free, isometric action of a finite group. The equivariant Fr\'{e}chet means lie in $\widetilde{M}$, but, as we show, project down to the ordinary Fr\'{e}chet sample means, and enjoy a similar genericity property. Both these theorems are proven as consequences of a purely geometric (and quite general) result that constitutes the core mathematics in this paper: If $A\subset M$ has volume zero in $M$ , then the set $\{Q\in M^N : \mbox{FM}(Q) \cap A\neq\emptyset\}$ has volume zero in $M^N$. We conclude the paper with an application to partial scaling-rotation means, a type of mean for symmetric positive-definite matrices.
We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions, demonstrate the potential increase in the smallest singular values, and represent a qualitative model for the increase in the small singular values after a matrix has been downcast to a lower arithmetic precision. Numerical experiments confirm the qualitative validity of this model and its ability to predict singular values changes in the presence of decreased arithmetic precision.
This paper studies the extreme singular values of non-harmonic Fourier matrices. Such a matrix of size $m\times s$ can be written as $\Phi=[ e^{-2\pi i j x_k}]_{j=0,1,\dots,m-1, k=1,2,\dots,s}$ for some set $\mathcal{X}=\{x_k\}_{k=1}^s$. The main results provide explicit lower bounds for the smallest singular value of $\Phi$ under the assumption $m\geq 6s$ and without any restrictions on $\mathcal{X}$. They show that for an appropriate scale $\tau$ determined by a density criteria, interactions between elements in $\mathcal{X}$ at scales smaller than $\tau$ are most significant and depends on the multiscale structure of $\mathcal{X}$ at fine scales, while distances larger than $\tau$ are less important and only depend on the local sparsity of the far away points. Theoretical and numerical comparisons show that the main results significantly improve upon classical bounds and achieve the same rate that was previously discovered for more restrictive settings.
We study the maximum-average submatrix problem, in which given an $N \times N$ matrix $J$ one needs to find the $k \times k$ submatrix with the largest average of entries. We study the problem for random matrices $J$ whose entries are i.i.d. random variables by mapping it to a variant of the Sherrington-Kirkpatrick spin-glass model at fixed magnetization. We characterize analytically the phase diagram of the model as a function of the submatrix average and the size of the submatrix $k$ in the limit $N\to\infty$. We consider submatrices of size $k = m N$ with $0 < m < 1$. We find a rich phase diagram, including dynamical, static one-step replica symmetry breaking and full-step replica symmetry breaking. In the limit of $m \to 0$, we find a simpler phase diagram featuring a frozen 1-RSB phase, where the Gibbs measure is composed of exponentially many pure states each with zero entropy. We discover an interesting phenomenon, reminiscent of the phenomenology of the binary perceptron: there exist efficient algorithms that provably work in the frozen 1-RSB phase.
We present a novel stabilized isogeometric formulation for the Stokes problem, where the geometry of interest is obtained via overlapping NURBS (non-uniform rational B-spline) patches, i.e., one patch on top of another in an arbitrary but predefined hierarchical order. All the visible regions constitute the computational domain, whereas independent patches are coupled through visible interfaces using Nitsche's formulation. Such a geometric representation inevitably involves trimming, which may yield trimmed elements of extremely small measures (referred to as bad elements) and thus lead to the instability issue. Motivated by the minimal stabilization method that rigorously guarantees stability for trimmed geometries [1], in this work we generalize it to the Stokes problem on overlapping patches. Central to our method is the distinct treatments for the pressure and velocity spaces: Stabilization for velocity is carried out for the flux terms on interfaces, whereas pressure is stabilized in all the bad elements. We provide a priori error estimates with a comprehensive theoretical study. Through a suite of numerical tests, we first show that optimal convergence rates are achieved, which consistently agrees with our theoretical findings. Second, we show that the accuracy of pressure is significantly improved by several orders using the proposed stabilization method, compared to the results without stabilization. Finally, we also demonstrate the flexibility and efficiency of the proposed method in capturing local features in the solution field.
A set of vertices of a graph $G$ is said to be decycling if its removal leaves an acyclic subgraph. The size of a smallest decycling set is the decycling number of $G$. Generally, at least $\lceil(n+2)/4\rceil$ vertices have to be removed in order to decycle a cubic graph on $n$ vertices. In 1979, Payan and Sakarovitch proved that the decycling number of a cyclically $4$-edge-connected cubic graph of order $n$ equals $\lceil (n+2)/4\rceil$. In addition, they characterised the structure of minimum decycling sets and their complements. If $n\equiv 2\pmod4$, then $G$ has a decycling set which is independent and its complement induces a tree. If $n\equiv 0\pmod4$, then one of two possibilities occurs: either $G$ has an independent decycling set whose complement induces a forest of two trees, or the decycling set is near-independent (which means that it induces a single edge) and its complement induces a tree. In this paper we strengthen the result of Payan and Sakarovitch by proving that the latter possibility (a near-independent set and a tree) can always be guaranteed. Moreover, we relax the assumption of cyclic $4$-edge-connectivity to a significantly weaker condition expressed through the canonical decomposition of 3-connected cubic graphs into cyclically $4$-edge-connected ones. Our methods substantially use a surprising and seemingly distant relationship between the decycling number and the maximum genus of a cubic graph.
We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.
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