An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form $(-\infty,t)$ is sufficient to determine uniquely the whole observable defined on quantum structures like $\sigma$-MV-algebras, $\sigma$-effect algebras, Boolean $\sigma$-algebras, monotone $\sigma$-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.

Homological quantum error correction uses tools of algebraic topology and homological algebra to derive Calderbank-Shor-Steane quantum error correcting codes from cellulations of topological spaces. This work is an exploration of the relevant topics, a journey from classical error correction, through homology theory, to CSS codes acting on qudit systems. Qudit codes have torsion in their logical spaces. This is interesting to study because it gives us extra logical qudits, of possibly different dimension. Apart from examples and comments on the topic, we prove an original result, the Structure Theorem for the Qudit Logical Space, an application of the Universal Coefficient Theorem from homological algebra, which gives us information about the logical space when torsion is involved, and that improves on a previous result in the literature. Furthermore, this work introduces our own abstracted and restricted version of the general notion of a cell complex, suited exactly to our needs.

We make an extensive empirical study of the market impact of large orders (metaorders) executed in the U.S. equity market between 2007 and 2009. We show that the square root market impact formula, which is widely used in the industry and supported by previous published research, provides a good fit only across about two orders of magnitude in order size. A logarithmic functional form fits the data better, providing a good fit across almost five orders of magnitude. We introduce the concept of an "impact surface" to model the impact as a function of both the duration and the participation rate of the metaorder, finding again a logarithmic dependence. We show that during the execution the price trajectory deviates from the market impact, a clear indication of non-VWAP executions. Surprisingly, we find that sometimes the price starts reverting well before the end of the execution. Finally we show that, although on average the impact relaxes to approximately 2/3 of the peak impact, the precise asymptotic value of the price depends on the participation rate and on the duration of the metaorder. We present evidence that this might be due to a herding phenomenon among metaorders.

Non-invasive prenatal testing or NIPT is currently among the top researched topic in obstetric care. While the performance of the current state-of-the-art NIPT solutions achieve high sensitivity and specificity, they still struggle with a considerable number of samples that cannot be concluded with certainty. Such uninformative results are often subject to repeated blood sampling and re-analysis, usually after two weeks, and this period may cause a stress to the future mothers as well as increase the overall cost of the test. We propose a supplementary method to traditional z-scores to reduce the number of such uninformative calls. The method is based on a novel analysis of the length profile of circulating cell free DNA which compares the change in such profiles when random-based and length-based elimination of some fragments is performed. The proposed method is not as accurate as the standard z-score; however, our results suggest that combination of these two independent methods correctly resolves a substantial portion of healthy samples with an uninformative result. Additionally, we discuss how the proposed method can be used to identify maternal aberrations, thus reducing the risk of false positive and false negative calls. Keywords: Next-generation sequencing, Cell-free DNA, Uninformative result, Method, Trisomy, Prenatal testing

We introduce a novel method that we call Single-Shot Cross-Spectroscopy (SSCS), for extracting the auto- and cross-power spectral densities of dephasing noise of a qubit pair. The method uses straightforward input, namely single-shot readouts from single-qubit Ramsey-type experiments, and is resilient against errors in state preparation and measurement. We apply it to experimental data from a semiconductor spin-qubit device and obtain noise spectra over five orders of magnitude in frequency (5 mHz--500 Hz). Compared to other techniques, SSCS enables access to noise correlations in the previously inaccessible intermediate-frequency range (1--500 Hz) for spin qubits, and can be further extended with faster readout. More broadly, the frequency range accessible with SSCS is limited only by the experiment repetition rate, and scales accordingly on other platforms.

Rule mining algorithms are one of the fundamental techniques in data mining for disclosing significant patterns in terms of linguistic rules expressed in natural language. In this paper, we revisit the concept of fuzzy implicative rule to provide a solid theoretical framework for any fuzzy rule mining algorithm interested in capturing patterns in terms of logical conditionals rather than the co-occurrence of antecedent and consequent. In particular, we study which properties should satisfy the fuzzy operators to ensure a coherent behavior of different quality measures. As a consequence of this study, we introduce a new property of fuzzy implication functions related to a monotone behavior of the generalized modus ponens for which we provide different valid solutions. Also, we prove that our modeling generalizes others if an adequate choice of the fuzzy implication function is made, so it can be seen as an unifying framework. Further, we provide an open-source implementation in Python for mining fuzzy implicative associative rules. We test the applicability and relevance of our framework for different real datasets and fuzzy operators.

We construct a set of criteria detecting genuine multipartite entanglement in arbitrary dimensional multipartite systems. These criteria are optimally suited for detecting multipartite entanglement in n-qubit Dicke states with m-excitations. In a detailed analysis we show that the criteria are also more robust to noise than any other criterion known so far, especially with increasing system size. Furthermore it is shown that the number of required local observables scales only polynomially with size, thus making the criteria experimentally feasible.

We present a complete description of subdirectly irreducible state BL-algebras as well as of subdirectly irreducible state-morphism BL-algebras. In addition, we present a general theory of state-morphism algebras, that is, algebras of general type with state-morphism which is an idempotent endomorphism. We define a diagonal state-morphism algebra and we show that every subdirectly irreducible state-morphism algebra can be embedded into a diagonal one. We describe generators of varieties of state-morphism algebras, in particular ones of state-morphism BL-algebras, state-morphism MTL-algebras, state-morphism non-associative BL-algebras, and state-morphism pseudo MV-algebras.

Heisenberg's uncertainty relation does not take into account that position and momentum are defined relative to a quantum reference frame (QRF). We introduce such a QRF as a covariant phase space observable to derive novel, frame-relative uncertainty relations and show that relative to such a frame, position and momentum appear compatible. We investigate the classical limit of the QRF, and demonstrate that in a Galilei-invariant setting, large frame mass corresponds to classical frame preparation. Finally, we give further conditions under which the standard uncertainty relations are recovered.

We study higher order quantum maps in the context of a *-autonomous category of affine subspaces. We show that types of higher order maps can be identified with certain Boolean functions that we call type functions. By an extension of this identification, the algebraic structure of Boolean functions is inherited by some sets of quantum objects including higher order maps. Using the Möbius transform, we assign to each type function a poset whose elements are labelled by subsets of indices of the involved spaces. We then show that the type function corresponds to a comb type if and only if the poset is a chain. We also devise a procedure for decomposition of the poset to a set of basic chains from which the type function is constructed by taking maxima and minima of concatenations of the basic chains in different orders. On the level of higher order maps, maxima and minima correspond to affine mixtures and intersections, respectively.

The matrix product structure is considered on a regular lattice in the hyperbolic plane. The phase transition of the Ising model is observed on the hyperbolic $(5, 4)$ lattice by means of the corner-transfer-matrix renormalization group (CTMRG) method. Calculated correlation length is always finite even at the transition temperature, where mean-field like behavior is observed. The entanglement entropy is also always finite.

Raman spectroscopy is a powerful experimental technique for characterizing molecules and materials that is used in many laboratories. First-principles theoretical calculations of Raman spectra are important because they elucidate the microscopic effects underlying Raman activity in these systems. These calculations are often performed using the canonical harmonic approximation which cannot capture certain thermal changes in the Raman response. Anharmonic vibrational effects were recently found to play crucial roles in several materials, which motivates theoretical treatments of the Raman effect beyond harmonic phonons. While Raman spectroscopy from molecular dynamics (MD-Raman) is a well-established approach that includes anharmonic vibrations and further relevant thermal effects, MD-Raman computations were long considered to be computationally too expensive for practical materials computations. In this perspective article, we highlight that recent advances in the context of machine learning have now dramatically accelerated the involved computational tasks without sacrificing accuracy or predictive power. These recent developments highlight the increasing importance of MD-Raman and related methods as versatile tools for theoretical prediction and characterization of molecules and materials.

We present results of comprehensive first-principles and kp-method studies of electronic, magnetic, and topological properties of graphene on a monolayer of CrI$_3$. First, we identify a twist angle between the graphene and CrI$_3$, that positions the graphene Dirac cones within the bandgap of CrI$_3$. Then, we derive the low-energy effective Hamiltonian describing electronic properties of graphene Dirac cones. Subsequently, we examine anomalous and valley Hall conductivity and discuss possible topological phase transition from a quantum anomalous Hall insulator to a trivial insulating state, concomitant a change in the magnetic ground state of CrI$_3$. These findings highlight the potential of strain engineering in two-dimensional van der Waals heterostructures for controlling topological and magnetic phases.

Rule-based models are essential for high-stakes decision-making due to their transparency and interpretability, but their discrete nature creates challenges for optimization and scalability. In this work, we present the Fuzzy Rule-based Reasoner (FRR), a novel gradient-based rule learning system that supports strict user constraints over rule-based complexity while achieving competitive performance. To maximize interpretability, the FRR uses semantically meaningful fuzzy logic partitions, unattainable with existing neuro-fuzzy approaches, and sufficient (single-rule) decision-making, which avoids the combinatorial complexity of additive rule ensembles. Through extensive evaluation across 40 datasets, FRR demonstrates: (1) superior performance to traditional rule-based methods (e.g., $5\%$ average accuracy over RIPPER); (2) comparable accuracy to tree-based models (e.g., CART) using rule bases $90\%$ more compact; and (3) achieves $96\%$ of the accuracy of state-of-the-art additive rule-based models while using only sufficient rules and requiring only $3\%$ of their rule base size.

Recently Jenei introduced a new structure called equality algebras which is inspired by ideas of BCK-algebras with meet. These algebras were generalized by Jenei and Kóródi to pseudo equality algebras which are aimed to find a connection with pseudo BCK-algebras with meet. We show that every pseudo equality algebra is an equality algebra. Therefore, we define a new type of pseudo equality algebras which more precisely reflects the relation to pseudo BCK-algebras with meet in the sense of Kabziński and Wroński. We describe congruences via normal closed deductive systems, and we show that the variety of pseudo equality algebras is subtractive, congruence distributive and congruence permutable.

Fuzzy implication functions are a key area of study in fuzzy logic, extending the classical logical conditional to handle truth degrees in the interval $[0,1]$. While existing literature often focuses on a limited number of families, in the last ten years many new families have been introduced, each defined by specific construction methods and having different key properties. This survey aims to provide a comprehensive and structured overview of the diverse families of fuzzy implication functions, emphasizing their motivations, properties, and potential applications. By organizing the information schematically, this document serves as a valuable resource for both theoretical researchers seeking to avoid redundancy and practitioners looking to select appropriate operators for specific applications.