We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary binary morphisms. Such morphisms map binary letters 0 and 1 to complementary words. We prove that every prefix of the famous Thue-Morse word $\mathbf{t}$ gives a complementary morphism that is $3^+$-free and then $\alpha$-free for any real $\alpha>3$. We also describe the lengths of all prefixes of $\mathbf{t}$ that give cubefree complementary morphisms by a 4-state binary finite automaton. Next we show that cubefree complementary morphisms of length $k$ exist for all $k\ne\{3,6\}$. Moreover, if $k$ is not representable as $3\cdot2^n$, then the images of letters can be chosen to be factors of $\mathbf{t}$. In addition to more traditional techniques of combinatorics on words, we also rely on the Walnut theorem-prover. Its use and limitations are discussed.
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Traditional manual detection for solder joint defect is no longer applied during industrial production due to low efficiency, inconsistent evaluation, high cost and lack of real-time data. A new approach has been proposed to address the issues of low accuracy, high false detection rates and computational cost of solder joint defect detection in surface mount technology of industrial scenarios. The proposed solution is a hybrid attention mechanism designed specifically for the solder joint defect detection algorithm to improve quality control in the manufacturing process by increasing the accuracy while reducing the computational cost. The hybrid attention mechanism comprises a proposed enhanced multi-head self-attention and coordinate attention mechanisms increase the ability of attention networks to perceive contextual information and enhances the utilization range of network features. The coordinate attention mechanism enhances the connection between different channels and reduces location information loss. The hybrid attention mechanism enhances the capability of the network to perceive long-distance position information and learn local features. The improved algorithm model has good detection ability for solder joint defect detection, with mAP reaching 91.5%, 4.3% higher than the You Only Look Once version 5 algorithm and better than other comparative algorithms. Compared to other versions, mean Average Precision, Precision, Recall, and Frame per Seconds indicators have also improved. The improvement of detection accuracy can be achieved while meeting real-time detection requirements.
The causal inference literature frequently focuses on estimating the mean of the potential outcome, whereas the quantiles of the potential outcome may carry important additional information. We propose a universal approach, based on the inverse estimating equations, to generalize a wide class of causal inference solutions from estimating the mean of the potential outcome to its quantiles. We assume that an identifying moment function is available to identify the mean of the threshold-transformed potential outcome, based on which a convenient construction of the estimating equation of quantiles of potential outcome is proposed. In addition, we also give a general construction of the efficient influence functions of the mean and quantiles of potential outcomes, and identify their connection. We motivate estimators for the quantile estimands with the efficient influence function, and develop their asymptotic properties when either parametric models or data-adaptive machine learners are used to estimate the nuisance functions. A broad implication of our results is that one can rework the existing result for mean causal estimands to facilitate causal inference on quantiles, rather than starting from scratch. Our results are illustrated by several examples.
We explore new interactions between finite model theory and a number of classical streams of universal algebra and semigroup theory. A key result is an example of a finite algebra whose variety is not finitely axiomatisable in first order logic, but which has first order definable finite membership problem. This algebra witnesses the simultaneous failure of the {\L}os-Tarski Theorem, the SP-preservation theorem and Birkhoff's HSP-preservation theorem at the finite level as well as providing a negative solution to a first order formulation of the long-standing Eilenberg Sch\"utzenberger problem. The example also shows that a pseudovariety without any finite pseudo-identity basis may be finitely axiomatisable in first order logic. Other results include the undecidability of deciding first order definability of the pseudovariety of a finite algebra and a mapping from any fixed template constraint satisfaction problem to a first order equivalent variety membership problem, thereby providing examples of variety membership problems complete in each of the classes $\texttt{L}$, $\texttt{NL}$, $\texttt{Mod}_p(\texttt{L})$, $\texttt{P}$, and infinitely many others (depending on complexity-theoretic assumptions).
Quantum kernel methods are a candidate for quantum speed-ups in supervised machine learning. The number of quantum measurements N required for a reasonable kernel estimate is a critical resource, both from complexity considerations and because of the constraints of near-term quantum hardware. We emphasize that for classification tasks, the aim is reliable classification and not precise kernel evaluation, and demonstrate that the former is far more resource efficient. Furthermore, it is shown that the accuracy of classification is not a suitable performance metric in the presence of noise and we motivate a new metric that characterizes the reliability of classification. We then obtain a bound for N which ensures, with high probability, that classification errors over a dataset are bounded by the margin errors of an idealized quantum kernel classifier. Using chance constraint programming and the subgaussian bounds of quantum kernel distributions, we derive several Shot-frugal and Robust (ShofaR) programs starting from the primal formulation of the Support Vector Machine. This significantly reduces the number of quantum measurements needed and is robust to noise by construction. Our strategy is applicable to uncertainty in quantum kernels arising from any source of unbiased noise.
We consider the gradient descent flow widely used for the minimization of the $\mathcal{L}^2$ cost function in Deep Learning networks, and introduce two modified versions; one adapted for the overparametrized setting, and the other for the underparametrized setting. Both have a clear and natural invariant geometric meaning, taking into account the pullback vector bundle structure in the overparametrized, and the pushforward vector bundle structure in the underparametrized setting. In the overparametrized case, we prove that, provided that a rank condition holds, all orbits of the modified gradient descent drive the $\mathcal{L}^2$ cost to its global minimum at a uniform exponential convergence rate; one thereby obtains an a priori stopping time for any prescribed proximity to the global minimum. We point out relations of the latter to sub-Riemannian geometry.
Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree $n$ converge at the root-exponential rate $O(\exp(-2\rho\sqrt{n}))$ with $\rho>0$ when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at $z=-\rho^2$. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.
For two real symmetric matrices, their eigenvalue configuration is the arrangement of their eigenvalues on the real line. We study the problem of determining a quantifier-free necessary and sufficient condition for two real symmetric matrices to realize a given eigenvalue configuration as a generalization of Descartes' rule of signs. We exploit the combinatorial properties of our definition for eigenvalue configuration to reduce a two-polynomial root counting problem into several single-polynomial root counting problems of symmetric polynomials. We then leverage the fundamental theorem of symmetric polynomials to derive a final quantifier-free necessary and sufficient condition for two real symmetric matrices to realize a given eigenvalue configuration.
For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.
We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation, and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.
Detecting differences in gene expression is an important part of single-cell RNA sequencing experiments, and many statistical methods have been developed for this aim. Most differential expression analyses focus on comparing expression between two groups (e.g., treatment vs. control). But there is increasing interest in multi-condition differential expression analyses in which expression is measured in many conditions, and the aim is to accurately detect and estimate expression differences in all conditions. We show that directly modeling single-cell RNA-seq counts in all conditions simultaneously, while also inferring how expression differences are shared across conditions, leads to greatly improved performance for detecting and estimating expression differences compared to existing methods. We illustrate the potential of this new approach by analyzing data from a single-cell experiment studying the effects of cytokine stimulation on gene expression. We call our new method "Poisson multivariate adaptive shrinkage", and it is implemented in an R package available online at //github.com/stephenslab/poisson.mash.alpha.
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