V. Levenshtein first proposed the sequence reconstruction problem in 2001. This problem studies the model where the same sequence from some set is transmitted over multiple channels, and the decoder receives the different outputs. Assume that the transmitted sequence is at distance $d$ from some code and there are at most $r$ errors in every channel. Then the sequence reconstruction problem is to find the minimum number of channels required to recover exactly the transmitted sequence that has to be greater than the maximum intersection between two metric balls of radius $r$, where the distance between their centers is at least $d$. In this paper, we study the sequence reconstruction problem of permutations under the Hamming distance. In this model we define a Cayley graph over the symmetric group, study its properties and find the exact value of the largest intersection of its two metric balls for $d=2r$. Moreover, we give a lower bound on the largest intersection of two metric balls for $d=2r-1$.
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We show how to reduce the computational time of the practical implementation of the Raviart-Thomas mixed method for second-order elliptic problems. The implementation takes advantage of a recent result which states that certain local subspaces of the vector unknown can be eliminated from the equations by transforming them into stabilization functions; see the paper published online in JJIAM on August 10, 2023. We describe in detail the new implementation (in MATLAB and a laptop with Intel(R) Core (TM) i7-8700 processor which has six cores and hyperthreading) and present numerical results showing 10 to 20% reduction in the computational time for the Raviart-Thomas method of index $k$, with $k$ ranging from 1 to 20, applied to a model problem.
We address the classical inverse problem of recovering the position and shape of obstacles immersed in a planar Stokes flow using boundary measurements. We prove that this problem can be transformed into a shape-from-moments problem to which ad hoc reconstruction methods can be applied. The effectiveness of this approach is confirmed by numerical tests that show significant improvements over those available in the literature to date.
This study examines, in the framework of variational regularization methods, a multi-penalty regularization approach which builds upon the Uniform PENalty (UPEN) method, previously proposed by the authors for Nuclear Magnetic Resonance (NMR) data processing. The paper introduces two iterative methods, UpenMM and GUpenMM, formulated within the Majorization-Minimization (MM) framework. These methods are designed to identify appropriate regularization parameters and solutions for linear inverse problems utilizing multi-penalty regularization. The paper demonstrates the convergence of these methods and illustrates their potential through numerical examples in one and two-dimensional scenarios, showing the practical utility of point-wise regularization terms in solving various inverse problems.
We present a short proof of a celebrated result of G\'acs and K\"orner giving sufficient and necessary condition on the joint distribution of two discrete random variables $X$ and $Y$ for the case when their mutual information matches the extractable (in the limit) common information. Our proof is based on the observation that the mere existence of certain random variables jointly distributed with $X$ and $Y$ can impose restriction on all random variables jointly distributed with $X$ and $Y$.
We construct a family of Markov decision processes for which the policy iteration algorithm needs an exponential number of improving switches with Dantzig's rule, with Bland's rule, and with the Largest Increase pivot rule. This immediately translates to a family of linear programs for which the simplex algorithm needs an exponential number of pivot steps with the same three pivot rules. Our results yield a unified construction that simultaneously reproduces well-known lower bounds for these classical pivot rules, and we are able to infer that any (deterministic or randomized) combination of them cannot avoid an exponential worst-case behavior. Regarding the policy iteration algorithm, pivot rules typically switch multiple edges simultaneously and our lower bound for Dantzig's rule and the Largest Increase rule, which perform only single switches, seem novel. Regarding the simplex algorithm, the individual lower bounds were previously obtained separately via deformed hypercube constructions. In contrast to previous bounds for the simplex algorithm via Markov decision processes, our rigorous analysis is reasonably concise.
Our research proposes a novel method for reducing the dimensionality of functional data, specifically for the case where the response is a scalar and the predictor is a random function. Our method utilizes distance covariance, and has several advantages over existing methods. Unlike current techniques which require restrictive assumptions such as linear conditional mean and constant covariance, our method has mild requirements on the predictor. Additionally, our method does not involve the use of the unbounded inverse of the covariance operator. The link function between the response and predictor can be arbitrary, and our proposed method maintains the advantage of being model-free, without the need to estimate the link function. Furthermore, our method is naturally suited for sparse longitudinal data. We utilize functional principal component analysis with truncation as a regularization mechanism in the development of our method. We provide justification for the validity of our proposed method, and establish statistical consistency of the estimator under certain regularization conditions. To demonstrate the effectiveness of our proposed method, we conduct simulation studies and real data analysis. The results show improved performance compared to existing methods.
The numerical solution of continuum damage mechanics (CDM) problems suffers from convergence-related challenges during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. In this work, we present a novel unified arc-length (UAL) method, and we derive the formulation of the analytical tangent matrix and governing system of equations for both local and non-local gradient damage problems. Unlike existing versions of arc-length solvers that monolithically scale the external force vector, the proposed method treats the latter as an independent variable and determines the position of the system on the equilibrium path based on all the nodal variations of the external force vector. This approach renders the proposed solver substantially more efficient and robust than existing solvers used in CDM problems. We demonstrate the considerable advantages of the proposed algorithm through several benchmark 1D problems with sharp snap-backs and 2D examples under various boundary conditions and loading scenarios. The proposed UAL approach exhibits a superior ability of overcoming critical increments along the equilibrium path. Moreover, the proposed UAL method is 1-2 orders of magnitude faster than force-controlled arc-length and monolithic Newton-Raphson solvers.
Natural revision seems so natural: it changes beliefs as little as possible to incorporate new information. Yet, some counterexamples show it wrong. It is so conservative that it never fully believes. It only believes in the current conditions. This is right in some cases and wrong in others. Which is which? The answer requires extending natural revision from simple formulae expressing universal truths (something holds) to conditionals expressing conditional truth (something holds in certain conditions). The extension is based on the basic principles natural revision follows, identified as minimal change, indifference and naivety: change beliefs as little as possible; equate the likeliness of scenarios by default; believe all until contradicted. The extension says that natural revision restricts changes to the current conditions. A comparison with an unrestricting revision shows what exactly the current conditions are. It is not what currently considered true if it contradicts the new information. It includes something more and more unlikely until the new information is at least possible.
Contract review is an essential step in construction projects to prevent potential losses. However, the current methods for reviewing construction contracts lack effectiveness and reliability, leading to time-consuming and error-prone processes. While large language models (LLMs) have shown promise in revolutionizing natural language processing (NLP) tasks, they struggle with domain-specific knowledge and addressing specialized issues. This paper presents a novel approach that leverages LLMs with construction contract knowledge to emulate the process of contract review by human experts. Our tuning-free approach incorporates construction contract domain knowledge to enhance language models for identifying construction contract risks. The use of a natural language when building the domain knowledge base facilitates practical implementation. We evaluated our method on real construction contracts and achieved solid performance. Additionally, we investigated how large language models employ logical thinking during the task and provide insights and recommendations for future research.
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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