In this paper, we propose two algorithms for a hybrid construction of all $n\times n$ MDS and involutory MDS matrices over a finite field $\mathbb{F}_{p^m}$, respectively. The proposed algorithms effectively narrow down the search space to identify $(n-1) \times (n-1)$ MDS matrices, facilitating the generation of all $n \times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. To the best of our knowledge, existing literature lacks methods for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. In our approach, we introduce a representative matrix form for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. The determination of these representative MDS matrices involves searching through all $(n-1)\times (n-1)$ MDS matrices over $\mathbb{F}_{p^m}$. Our contributions extend to proving that the count of all $3\times 3$ MDS matrices over $\mathbb{F}_{2^m}$ is precisely $(2^m-1)^5(2^m-2)(2^m-3)(2^{2m}-9\cdot 2^m+21)$. Furthermore, we explicitly provide the count of all $4\times 4$ MDS and involutory MDS matrices over $\mathbb{F}_{2^m}$ for $m=2, 3, 4$.
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In this paper we propose a generalization of the Riemann curvature tensor on manifolds (of dimension two or higher) endowed with a Regge metric. Specifically, while all components of the metric tensor are assumed to be smooth within elements of a triangulation of the manifold, they need not be smooth across element interfaces, where only continuity of the tangential components are assumed. While linear derivatives of the metric can be generalized as Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization combining the classical angle defect and jumps of the second fundamental form across element interfaces, and rigorously prove correctness of this generalization. Specifically, if a piecewise smooth metric approximates a globally smooth metric, our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor arising from a globally smooth metric. Moreover, we show that if the metric approximation converges at some rate in a piecewise norm that scales like the $L^2$-norm, then the curvature approximation converges in the $H^{-2}$-norm at the same rate, under additional assumptions. By appropriate contractions of the generalized Riemann curvature tensor, this work also provides generalizations of scalar curvature, the Ricci curvature tensor, and the Einstein tensor in any dimension.
Markov chain Monte Carlo (MCMC) methods are one of the most common classes of algorithms to sample from a target probability distribution $\pi$. A rising trend in recent years consists in analyzing the convergence of MCMC algorithms using tools from the theory of large deviations. One such result is a large deviation principle for algorithms of Metropolis-Hastings (MH) type (Milinanni & Nyquist, 2024), which are a broad and popular sub-class of MCMC methods. A central object in large deviation theory is the rate function, through which we can characterize the speed of convergence of MCMC algorithms. In this paper we consider the large deviation rate function from (Milinanni & Nyquist, 2024), of which we prove an alternative representation. We also determine upper and lower bounds for the rate function, based on which we design schemes to tune algorithms of MH type.
We introduce the concept of an imprecise Markov semigroup $\mathbf{Q}$. It is a tool that allows to represent ambiguity around both the initial and the transition probabilities of a Markov process via a compact collection of plausible Markov semigroups, each associated with a (different, plausible) Markov process. We use techniques from geometry, functional analysis, and (high dimensional) probability to study the ergodic behavior of $\mathbf{Q}$. We show that, if the initial distribution of the Markov processes associated with the elements of $\mathbf{Q}$ is known and invariant, under some conditions that also involve the geometry of the state space, eventually the ambiguity around their transition probability fades. We call this property ergodicity of the imprecise Markov semigroup, and we relate it to the classical notion of ergodicity. We prove ergodicity both when the state space is Euclidean or a Riemannian manifold, and when it is an arbitrary measurable space. The importance of our findings for the fields of machine learning and computer vision is also discussed.
In this paper, our main aim is to investigate the strong convergence for a neutral McKean-Vlasov stochastic differential equation with super-linear delay driven by fractional Brownian motion with Hurst exponent $H\in(1/2, 1)$. After giving uniqueness and existence for the exact solution, we analyze the properties including boundedness of moment and propagation of chaos. Besides, we give the Euler-Maruyama (EM) scheme and show that the numerical solution converges strongly to the exact solution. Furthermore, a corresponding numerical example is given to illustrate the theory.
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 $\mathcal{X}=\{x_k\}_{k=1}^s$. Its condition number controls the stability of inversion, which is of great importance to super-resolution and nonuniform Fourier transforms. Under the assumption $m\geq 6s$ and without any restrictions on $\mathcal{X}$, the main theorems provide explicit lower bounds for the smallest singular value $\sigma_s(\Phi)$ in terms of distances between elements in $\mathcal{X}$. More specifically, distances exceeding an appropriate scale $\tau$ have modest influence on $\sigma_s(\Phi)$, while the product of distances that are less than $\tau$ dominates the behavior of $\sigma_s(\Phi)$. These estimates reveal how the multiscale structure of $\mathcal{X}$ affects the condition number of Fourier matrices. Theoretical and numerical comparisons indicate that the main theorems significantly improve upon classical bounds and recover the same rate for special cases but with relaxed assumptions.
In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to an elliptic partial differential equation (PDE) with a lognormally distributed coefficient and a normally distributed source term. There is extensive previous work on using QMC to compute expected values in UQ, which have proven very successful in tackling a range of different PDE problems. However, the use of QMC for density estimation applied to UQ problems will be explored here for the first time. Density estimation presents a more difficult challenge compared to computing the expected value due to discontinuities present in the integral formulations of both the distribution and density. Our strategy is to use preintegration to eliminate the discontinuity by integrating out a carefully selected random parameter, so that QMC can be used to approximate the remaining integral. First, we establish regularity results for the PDE quantity of interest that are required for smoothing by preintegration to be effective. We then show that an $N$-point lattice rule can be constructed for the integrands corresponding to the distribution and density, such that after preintegration the QMC error is of order $\mathcal{O}(N^{-1+\epsilon})$ for arbitrarily small $\epsilon>0$. This is the same rate achieved for computing the expected value of the quantity of interest. Numerical results are presented to reaffirm our theory.
This paper surveys innovative protocols that enhance the programming functionality of the Bitcoin blockchain, a key part of the "Bitcoin Ecosystem." Bitcoin utilizes the Unspent Transaction Output (UTXO) model and a stack-based script language for efficient peer-to-peer payments, but it faces limitations in programming capability and throughput. The 2021 Taproot upgrade introduced the Schnorr signature algorithm and P2TR transaction type, significantly improving Bitcoin's privacy and programming capabilities. This upgrade has led to the development of protocols like Ordinals, Atomicals, and BitVM, which enhance Bitcoin's programming functionality and enrich its ecosystem. We explore the technical aspects of the Taproot upgrade and examine Bitcoin Layer 1 protocols that leverage Taproot's features to program non-fungible tokens (NFTs) into transactions, including Ordinals and Atomicals, along with the fungible token standards BRC-20 and ARC-20. Additionally, we categorize certain Bitcoin ecosystem protocols as Layer 2 solutions similar to Ethereum's, analyzing their impact on Bitcoin's performance. By analyzing data from the Bitcoin blockchain, we gather metrics on block capacity, miner fees, and the growth of Taproot transactions. Our findings confirm the positive effects of these protocols on Bitcoin's mainnet, bridging gaps in the literature regarding Bitcoin's programming capabilities and ecosystem protocols and providing valuable insights for practitioners and researchers.
The general theory of greedy approximation with respect to arbitrary dictionaries is well developed in the case of real Banach spaces. Recently, some of results proved for the Weak Chebyshev Greedy Algorithm (WCGA) in the case of real Banach spaces were extended to the case of complex Banach spaces. In this paper we extend some of known in the real case results for other than WCGA greedy algorithms to the case of complex Banach spaces.
In this paper, we study a class of special linear codes involving their parameters, weight distributions, self-orthogonal properties, deep holes, and the existence of error-correcting pairs. We prove that such codes must be maximum distance separable (MDS) codes or near MDS codes and completely determine their weight distributions with the help of the solutions to some subset sum problems. Based on the Schur method, we show that such codes are not equivalent to generalized Reed-Solomon (GRS) codes. A sufficient and necessary condition for such codes to be self-orthogonal is also characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two classes of almost self-dual codes. Additionally, we find a class of deep holes of such codes and determine the existence of their error-correcting pairs in most cases, which also reveal more connections between such codes and GRS codes.
In this work, we develop Crank-Nicolson-type iterative decoupled algorithms for a three-field formulation of Biot's consolidation model using total pressure. We begin by constructing an equivalent fully implicit coupled algorithm using the standard Crank-Nicolson method for the three-field formulation of Biot's model. Employing an iterative decoupled scheme to decompose the resulting coupled system, we derive two distinctive forms of Crank-Nicolson-type iterative decoupled algorithms based on the order of temporal computation and iteration: a time-stepping iterative decoupled algorithm and a global-in-time iterative decoupled algorithm. Notably, the proposed global-in-time algorithm supports a partially parallel-in-time feature. Capitalizing on the convergence properties of the iterative decoupled scheme, both algorithms exhibit second-order time accuracy and unconditional stability. Through numerical experiments, we validate theoretical predictions and demonstrate the effectiveness and efficiency of these novel approaches.
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