We consider the equivalence between the two main categorical models for the type-theoretical operation of context comprehension, namely P. Dybjer's categories with families and B. Jacobs' comprehension categories, and generalise it to the non-discrete case. The classical equivalence can be summarised in the slogan: "terms as sections". By recognising "terms as coalgebras", we show how to use the structure-semantics adjunction to prove that a 2-category of comprehension categories is biequivalent to a 2-category of (non-discrete) categories with families. The biequivalence restricts to the classical one proved by Hofmann in the discrete case. It also provides a framework where to compare different morphisms of these structures that have appeared in the literature, varying on the degree of preservation of the relevant structure. We consider in particular morphisms defined by Claraimbault-Dybjer, Jacobs, Larrea, and Uemura.
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The problem of estimating a matrix based on a set of its observed entries is commonly referred to as the matrix completion problem. In this work, we specifically address the scenario of binary observations, often termed as 1-bit matrix completion. While numerous studies have explored Bayesian and frequentist methods for real-value matrix completion, there has been a lack of theoretical exploration regarding Bayesian approaches in 1-bit matrix completion. We tackle this gap by considering a general, non-uniform sampling scheme and providing theoretical assurances on the efficacy of the fractional posterior. Our contributions include obtaining concentration results for the fractional posterior and demonstrating its effectiveness in recovering the underlying parameter matrix. We accomplish this using two distinct types of prior distributions: low-rank factorization priors and a spectral scaled Student prior, with the latter requiring fewer assumptions. Importantly, our results exhibit an adaptive nature by not mandating prior knowledge of the rank of the parameter matrix. Our findings are comparable to those found in the frequentist literature, yet demand fewer restrictive assumptions.
Solving high-dimensional partial differential equations necessitates methods free of exponential scaling in the dimension of the problem. This work introduces a tensor network approach for the Kolmogorov backward equation via approximating directly the Markov operator. We show that the high-dimensional Markov operator can be obtained under a functional hierarchical tensor (FHT) ansatz with a hierarchical sketching algorithm. When the terminal condition admits an FHT ansatz, the proposed operator outputs an FHT ansatz for the PDE solution through an efficient functional tensor network contraction procedure. In addition, the proposed operator-based approach also provides an efficient way to solve the Kolmogorov forward equation when the initial distribution is in an FHT ansatz. We apply the proposed approach successfully to two challenging time-dependent Ginzburg-Landau models with hundreds of variables.
Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes
We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes $h_1,\ldots,h_t$ as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For t=4 our construction gives a new family of "almost-good" quantum LTCs -- with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.
Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure on $j$ eigenvalues and $k$ singular values that we coin $j,k$-point correlation measure. We fully derive all $j,k$-point correlation measures in the simplest cases for matrices of size $n=1$ and $n=2$. For $n>2$, we find a general formula for the $1,1$-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a P\'{o}lya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble.
We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.
This chapter explores the role of patent protection in algorithmic surveillance and whether ordre public exceptions from patentability should apply to such patents, due to their potential to enable human rights violations. It concludes that in most cases, it is undesirable to exclude algorithmic surveillance patents from patentability, as the patent system is ill-equipped to evaluate the impacts of the exploitation of such technologies. Furthermore, the disclosure of such patents has positive externalities from the societal perspective by opening the black box of surveillance for public scrutiny.
In this work we consider the Allen--Cahn equation, a prototypical model problem in nonlinear dynamics that exhibits bifurcations corresponding to variations of a deterministic bifurcation parameter. Going beyond the state-of-the-art, we introduce a random coefficient function in the linear reaction part of the equation, thereby accounting for random, spatially-heterogeneous effects. Importantly, we assume a spatially constant, deterministic mean value of the random coefficient. We show that this mean value is in fact a bifurcation parameter in the Allen--Cahn equation with random coefficients. Moreover, we show that the bifurcation points and bifurcation curves become random objects. We consider two distinct modelling situations: (i) for a spatially homogeneous coefficient we derive analytical expressions for the distribution of the bifurcation points and show that the bifurcation curves are random shifts of a fixed reference curve; (ii) for a spatially heterogeneous coefficient we employ a generalized polynomial chaos expansion to approximate the statistical properties of the random bifurcation points and bifurcation curves. We present numerical examples in 1D physical space, where we combine the popular software package Continuation Core and Toolboxes (CoCo) for numerical continuation and the Sparse Grids Matlab Kit for the polynomial chaos expansion. Our exposition addresses both, dynamical systems and uncertainty quantification, highlighting how analytical and numerical tools from both areas can be combined efficiently for the challenging uncertainty quantification analysis of bifurcations in random differential equations.
We explore the possibility of simulating the grade-two fluid model in a geometry related to a contraction rheometer, and we provide details on several key aspects of the computation. We show how the results can be used to determine the viscosity $\nu$ from experimental data. We also explore the identifiability of the grade-two parameters $\alpha_1$ and $\alpha_2$ from experimental data. In particular, as the flow rate varies, force data appears to be nearly the same for certain distinct pairs of values $\alpha_1$ and $\alpha_2$; however we determine a regime for $\alpha_1$ and $\alpha_2$ for which the parameters may be identifiable with a contraction rheometer.
The continental plates of Earth are known to drift over a geophysical timescale, and their interactions have lead to some of the most spectacular geoformations of our planet while also causing natural disasters such as earthquakes and volcanic activity. Understanding the dynamics of interacting continental plates is thus significant. In this work, we present a fluid mechanical investigation of the plate motion, interaction, and dynamics. Through numerical experiments, we examine the coupling between a convective fluid and plates floating on top of it. With physical modeling, we show the coupling is both mechanical and thermal, leading to the thermal blanket effect: the floating plate is not only transported by the fluid flow beneath, it also prevents the heat from leaving the fluid, leading to a convective flow that further affects the plate motion. By adding several plates to such a coupled fluid-structure interaction, we also investigate how floating plates interact with each other and show that, under proper conditions, small plates can converge into a supercontinent.
Functional data analysis is an important research field in statistics which treats data as random functions drawn from some infinite-dimensional functional space, and functional principal component analysis (FPCA) based on eigen-decomposition plays a central role for data reduction and representation. After nearly three decades of research, there remains a key problem unsolved, namely, the perturbation analysis of covariance operator for diverging number of eigencomponents obtained from noisy and discretely observed data. This is fundamental for studying models and methods based on FPCA, while there has not been substantial progress since Hall, M\"uller and Wang (2006)'s result for a fixed number of eigenfunction estimates. In this work, we aim to establish a unified theory for this problem, obtaining upper bounds for eigenfunctions with diverging indices in both the $\mathcal{L}^2$ and supremum norms, and deriving the asymptotic distributions of eigenvalues for a wide range of sampling schemes. Our results provide insight into the phenomenon when the $\mathcal{L}^{2}$ bound of eigenfunction estimates with diverging indices is minimax optimal as if the curves are fully observed, and reveal the transition of convergence rates from nonparametric to parametric regimes in connection to sparse or dense sampling. We also develop a double truncation technique to handle the uniform convergence of estimated covariance and eigenfunctions. The technical arguments in this work are useful for handling the perturbation series with noisy and discretely observed functional data and can be applied in models or those involving inverse problems based on FPCA as regularization, such as functional linear regression.
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