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A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring N of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring B, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over B by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over B and left query algorithms over N. In general, there are properties that admit a left query algorithm over N but not over B. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over B if and only if it admits a left query algorithm over N. In other words and rather surprisingly, homomorphism counts over N do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over B and a right query algorithm over B.

This work presents a study of Kolmogorov complexity for general quantum states from the perspective of deterministic-control quantum Turing Machines (dcq-TM). We extend the dcq-TM model to incorporate mixed state inputs and outputs, and define dcq-computable states as those that can be approximated by a dcq-TM. Moreover, we introduce (conditional) Kolmogorov complexity of quantum states and use it to study three particular aspects of the algorithmic information contained in a quantum state: a comparison of the information in a quantum state with that of its classical representation as an array of real numbers, an exploration of the limits of quantum state copying in the context of algorithmic complexity, and study of the complexity of correlations in quantum systems, resulting in a correlation-aware definition for algorithmic mutual information that satisfies symmetry of information property.

We study the relationship between certain Groebner bases for zero dimensional ideals, and the interpolation condition functionals of ideal interpolation. Ideal interpolation is defined by a linear idempotent projector whose kernel is a polynomial ideal. In this paper, we propose the notion of "reverse" complete reduced basis. Based on the notion, we present a fast algorithm to compute the reduced Groebner basis for the kernel of ideal projector under an arbitrary compatible ordering. As an application, we show that knowing the affine variety makes available information concerning the reduced Groebner basis.

This paper evaluates AIcon2abs (Queiroz et al., 2021), a recently proposed method that enables awareness among the general public on machine learning. Such is possible due to the use of WiSARD, an easily understandable machine learning mechanism, thus requiring little effort and no technical background from the target users. WiSARD is adherent to digital computing; training consists of writing to RAM-type memories, and classification consists of reading from these memories. The model enables easy visualization and understanding of training and classification tasks' internal realization through ludic activities. Furthermore, the WiSARD model does not require an Internet connection for training and classification, and it can learn from a few or one example. This feature makes it easier to observe the machine, increasing its accuracy on a particular task with each new example used. WiSARD can also create "mental images" of what it has learned so far, evidencing key features pertaining to a given class. The assessment of the AIcon2abs method's effectiveness was conducted through the evaluation of a remote course with a workload of approximately 6 hours. It was completed by thirty-four Brazilian subjects: 5 children between 8 and 11 years old; 5 adolescents between 12 and 17 years old; and 24 adults between 21 and 72 years old. Data analysis adopted a hybrid approach. AIcon2abs was well-rated by almost 100% of the research subjects, and the data collected revealed quite satisfactory results concerning the intended outcomes. This research has been approved by the CEP/HUCFF/FM/UFRJ Human Research Ethics Committee.

This article presents an in-depth educational overview of the latest mathematical developments in coupled cluster (CC) theory, beginning with Schneider's seminal work from 2009 that introduced the first local analysis of CC theory. We offer a tutorial review of second quantization and the CC ansatz, laying the groundwork for understanding the mathematical basis of the theory. This is followed by a detailed exploration of the most recent mathematical advancements in CC theory.Our review starts with an in-depth look at the local analysis pioneered by Schneider which has since been applied to analyze various CC methods. We then move on to discuss the graph-based framework for CC methods developed by Csirik and Laestadius. This framework provides a comprehensive platform for comparing different CC methods, including multireference approaches. Next, we delve into the latest numerical analysis results analyzing the single reference CC method developed by Hassan, Maday, and Wang. This very general approach is based on the invertibility of the CC function's Fr\'echet derivative. We conclude the article with a discussion on the recent incorporation of algebraic geometry into CC theory, highlighting how this novel and fundamentally different mathematical perspective has furthered our understanding and provides exciting pathways to new computational approaches.

E-commerce, a type of trading that occurs at a high frequency on the Internet, requires guaranteeing the integrity, authentication and non-repudiation of messages through long distance. As current e-commerce schemes are vulnerable to computational attacks, quantum cryptography, ensuring information-theoretic security against adversary's repudiation and forgery, provides a solution to this problem. However, quantum solutions generally have much lower performance compared to classical ones. Besides, when considering imperfect devices, the performance of quantum schemes exhibits a significant decline. Here, for the first time, we demonstrate the whole e-commerce process of involving the signing of a contract and payment among three parties by proposing a quantum e-commerce scheme, which shows resistance of attacks from imperfect devices. Results show that with a maximum attenuation of 25 dB among participants, our scheme can achieve a signature rate of 0.82 times per second for an agreement size of approximately 0.428 megabit. This proposed scheme presents a promising solution for providing information-theoretic security for e-commerce.

1. Temporal trends in species distributions are necessary for monitoring changes in biodiversity, which aids policymakers and conservationists in making informed decisions. Dynamic species distribution models are often fitted to ecological time series data using Markov Chain Monte Carlo algorithms to produce these temporal trends. However, the fitted models can be time-consuming to produce and run, making it inefficient to refit them as new observations become available. 2. We propose an algorithm that updates model parameters and the latent state distribution (e.g. true occupancy) using the saved information from a previously fitted model. This algorithm capitalises on the strength of importance sampling to generate new posterior samples of interest by updating the model output. The algorithm was validated with simulation studies on linear Gaussian state space models and occupancy models, and we applied the framework to Crested Tits in Switzerland and Yellow Meadow Ants in the UK. 3. We found that models updated with the proposed algorithm captured the true model parameters and latent state values as good as the models refitted to the expanded dataset. Moreover, the updated models were much faster to run and preserved the trajectory of the derived quantities. 4. The proposed approach serves as an alternative to conventional methods for updating state-space models (SSMs), and it is most beneficial when the fitted SSMs have a long run time. Overall, we provide a Monte Carlo algorithm to efficiently update complex models, a key issue in developing biodiversity models and indicators.

Characterizing the solution sets in a problem by closedness under operations is recognized as one of the key aspects of algorithm development, especially in constraint satisfaction. An example from the Boolean satisfiability problem is that the solution set of a Horn conjunctive normal form (CNF) is closed under the minimum operation, and this property implies that minimizing a nonnegative linear function over a Horn CNF can be done in polynomial time. In this paper, we focus on the set of integer points (vectors) in a polyhedron, and study the relation between these sets and closedness under operations from the viewpoint of 2-decomposability. By adding further conditions to the 2-decomposable polyhedra, we show that important classes of sets of integer vectors in polyhedra are characterized by 2-decomposability and closedness under certain operations, and in some classes, by closedness under operations alone. The most prominent result we show is that the set of integer vectors in a unit-two-variable-per-inequality polyhedron can be characterized by closedness under the median and directed discrete midpoint operations, each of these operations was independently considered in constraint satisfaction and discrete convex analysis.

In this article, we develop comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point pattern and its tapered counterpart. Under second-order stationarity, we show that both the DFTs and the tapered DFTs are asymptotically jointly independent Gaussian even when the DFTs share the same limiting frequencies. Based on these results, we establish an $\alpha$-mixing central limit theorem for a statistic formulated as a quadratic form of the tapered DFT. As applications, we derive the asymptotic distribution of the kernel spectral density estimator and establish a frequency domain inferential method for parametric stationary point processes. For the latter, the resulting model parameter estimator is computationally tractable and yields meaningful interpretations even in the case of model misspecification. We investigate the finite sample performance of our estimator through simulations, considering scenarios of both correctly specified and misspecified models. Furthermore, we extend our proposed DFT-based frequency domain methods to a class of non-stationary spatial point processes.

When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.