The importance of replication is often discussed and advocated -- not only in the domains of visualization and HCI, but in all scientific areas. When replicating a study, design decisions need to be made with regards which aspects of the original study will remain the same and which will be altered. We present a supporting multi-dimensional design space framework within which such decisions can be identified, categorized, compared and analyzed. The framework treats replication experimental design as a pairwise comparison problem, and represents the design by four practical dimensions defined by three comparison levels. The design space is therefore a framework that can be used for both retrospective characterization and prospective planning. We provide worked examples, and relate our framework to other attempts at describing the scope of replication studies.

Altermagnetism has emerged as a promising ingredient for realizing nontrivial Josephson phases, but so far explored in single Josephson junctions. In this work, we consider the coherent coupling of two Josephson junctions with spin-singlet $s$-wave superconductivity and demonstrate that $d$-wave altermagnetism gives rise to spin-polarized Andreev molecules due to the hybridization of Andreev bound states of each junction when the coupling is weak. Interestingly, these spin-polarized Andreev molecules induce an anomalous nonlocal Josephson effect, where the current flow across one Josephson junction due to phase changes across the other junction develops $0-π$ and $ϕ_{0}$ transitions originating from altermagnetism. Furthermore, the nonlocal Josephson current carried by spin-polarized Andreev molecules exhibits nonreciprocal critical currents, enabling a nonlocal Josephson diode effect whose polarity is tunable by the altermagnetic strength and right phase. Our findings put forward altermagnetism as a promising arena for designing nonlocal spin Josephson phenomena.

We test exact marginality of the deformation describing the resolution of a $\mathbb{Z}_2$ orbifold by analyzing the closed superstring equations of motion to third order in the size, including $α'$ corrections. We find that the third order correction is unobstructed for all deformation moduli. We are also able to reproduce the Eguchi-Hanson gravitational instanton up to the second order in the field theory limit with a suitable choice of moduli.

This document details the Fully Homomorphic Modified Rivest Scheme (FHMRS), a security issue in FHMRS, and a modification to FHMRS (mFHMRS) to mitigate the security issue.

Modern vehicles increasingly rely on advanced driver-assistance systems (ADAS), with radars playing a key role due to their cost-effectiveness and reliable performance. However, the growing number of radars operating in the same spectrum raises concerns about mutual interference, which could lead to system malfunctions and potential safety risks. This study focuses on a scenario in which all vehicles are equipped with frequency-modulated continuous-wave (FMCW) radars, and it assesses the impact of interference on radar functionality - expressed in terms of probability of failure - by considering both direct and reflected signals. The radars may employ one of the following proactive mitigation methods to reduce the impact of interference, all of which require no inter-vehicle coordination but differ in complexity: (i) random carrier-frequency hopping on a frame-by-frame basis, (ii) random carrier-frequency hopping on a chirp-by-chirp basis, and (iii) a directional, compass-based method specifically addressing interference from opposite directions, which can be combined with either of the two previous methods. In this work, we assume realistic simulated road traffic scenarios and develop a novel model that captures correlated interference and accounts for the main radar setting parameters. Results reveal that dense scenarios pose a high risk of radar malfunctions. Among the analyzed methods, chirp-by-chirp frequency hopping emerges as the most effective approach to mitigate interference and ensure system reliability, but only when combined with a sufficiently large bandwidth. The compass-based method, on the other hand, shows limited effectiveness and appears not worth the additional system complexity.

With the development of high repetition rate laser sources and advanced multi-particle correlation analyses such as covariance mapping, particle detection techniques such as velocity map imaging (VMI) are poised to offer unprecedented views into molecular phenomena. Taking full advantage of the high count rates in these experiments requires the development of detectors with sufficient spatial and temporal resolution that can process data in real time. The TimePix3 camera (TPX3CAM) is an event-based pixel detector capable of spatio-temporally localizing many simultaneous particle hits in an efficient manner. While the sparse nature of the data stream allows for compact representation of particle hits, it also presents algorithmic and computational challenges for clustering individual pixels into hits. Here we present the theory and application of a rapid data processing and centroiding algorithm for ion and electron hits collected in a VMI instrument. The array-based computations that comprise the algorithm take full advantage of the data sparsity of the TimePix3 data stream and localize particle hits on the microchannel plate (MCP) to better than a single pixel on the pixel detector. Centroiding can be parallelized on a commercially available graphics processing unit (GPU) for additional speed. Using these innovations, data processing occurs about 25 times faster than data acquisition, for a 1 kHz repetition rate instrument and tens of particles per shot. In addition to its speed, the TPX3CAM detector outperforms state-of-the-art delay line anode detectors at discriminating multiple simultaneous hits, enabling high-fidelity coincidence and covariance studies in the near future.

We present a data-driven investigation of the exhaustive ensemble of no-scale type IIB flux vacua constructed in \cite{Chauhan:2025rdj}. Using a combination of linear and non-linear dimensionality-reduction techniques, we analyse both flux and moduli spaces and demonstrate that the effective dimensionality of the underlying 12-dimensional flux space is substantially reduced. A central component of our study is a physics-informed autoencoder, which provides a non-linear compression of the flux and moduli data into a low-dimensional latent space. The learned latent representation organises vacua according to desired features and, in particular, isolates distinguished regions associated with small values of the flux superpotential $|W_0|$, revealing non-trivial correlations that are not captured by linear methods. In parallel, we apply tools from topological data analysis, specifically persistent homology, to probe the global structure of the vacuum distribution. This allows us to identify robust, long-lived topological features in both moduli and flux subspaces. This work is a necessary step for developing foundation models in string phenomenology.

In recent years, nonconvex minimax problems have attracted significant attention due to their broad applications in machine learning, including generative adversarial networks, robust optimization and adversarial training. Most existing algorithms for nonconvex stochastic minimax problems are developed under the standard Lipschitz smoothness assumption. In this paper, we study stochastic minimax problems under a generalized smoothness condition and propose an algorithm, NSGDA-M, which simultaneously updates the inner variable by stochastic gradient ascent and updates the outer variable by normalized stochastic gradient descent with momentum. When the objective function is nonconvex-strongly concave, we show that NSGDA-M finds an $ε$-stationary point of the primal function within $\mathcal{O}(ε^{-4})$ stochastic gradient evaluations in expectation, and $\mathcal{O}\left(ε^{-4}(\log(\frac{1}δ))^{3/2}\right)$ stochastic gradient evaluations in high probability, where $δ\in (0,1)$ is the failure probability. We verify the effectiveness of the proposed algorithm through numerical experiments on a distributionally robust optimization problem.

Despite many advances in query optimization, indexing techniques, and data storage, modern data platforms still face difficulties in delivering robust query performance under high concurrency and computationally intensive queries. This challenge is particularly pronounced in large-scale observability platforms handling high-volume, high-velocity data records. For instance, recurrent, expensive filtering queries at query time impose substantial computational and storage overheads in the analytical data plane. In this paper, we propose FluxSieve, a unified architecture that reconciles traditional pull-based query processing with push-based stream processing by embedding a lightweight in-stream precomputation and filtering layer directly into the data ingestion path. This avoids the complexity and operational burden of running queries in dedicated stream processing frameworks. Concretely, this work (i) introduces a foundational architecture that unifies streaming and analytical data planes via in-stream filtering and records enrichment, (ii) designs a scalable multi-pattern matching mechanism that supports concurrent evaluation and on-the-fly updates of filtering rules with minimal per-record overhead, (iii) demonstrates how to integrate this ingestion-time processing with two open-source analytical systems -- Apache Pinot as a Real-Time Online Analytical Processing (RTOLAP) engine and DuckDB as an embedded analytical database, and (iv) performs comprehensive experimental evaluation of our approach. Our evaluation across different systems, query types, and performance metrics shows up to orders-of-magnitude improvements in query performance at the cost of negligible additional storage and very low computational overhead.

Chemotaxis systems of Keller--Segel type constitute one of the central mathematical frameworks for understanding aggregation phenomena in biological and ecological systems. Over the past decades, the theory has evolved from the classical single-species model to increasingly sophisticated multi-species and multi-signal formulations that capture competition, cooperation, antagonistic chemotaxis, and interactions with fluid environments. This article provides a comprehensive exposition of multi-species Keller--Segel systems and their mathematical structure. We review fundamental analytical results concerning local and global well-posedness, mechanisms of finite-time blow-up, and the role of critical mass and dimensionality. Particular emphasis is placed on how cross-diffusion, antagonistic interactions, logistic effects, and nonlinear production terms alter the qualitative behavior of solutions. We further examine the mathematical mechanisms underlying pattern formation, including diffusion-driven instabilities, bifurcation phenomena, and the emergence of spatial and spatiotemporal structures. Connections between analytical thresholds and observed nonlinear dynamics are highlighted, and the interplay between reaction kinetics, chemotactic sensitivity, and diffusion is discussed from a unifying perspective. By synthesizing classical results with recent developments, this survey aims to clarify the structural principles governing multi-species chemotaxis systems, identify common analytical techniques, and outline open problems that remain central to the field. The exposition is intended to serve both specialists and researchers entering the area of nonlinear partial differential equations and mathematical biology.

Camera glasses create fundamental privacy tensions between wearers seeking recording functionality and bystanders concerned about unauthorized surveillance. We present a systematic multi-stakeholder evaluation of privacy mechanisms through surveys (N=525) and paired interviews (N=20) in China. Study 1 quantifies expectation-willingness gaps: bystanders consistently demand stronger information transparency and protective measures than wearers will provide, with disparities intensifying in sensitive contexts where 65-90% of bystanders would take defensive action. Study 2 evaluates twelve privacy-enhancing technologies, revealing four fundamental trade-offs that undermine current approaches: visibility versus disruption, empowerment versus burden, protection versus agency, and accountability versus exposure. These gaps reflect structural incompatibilities rather than inadequate goodwill, with context emerging as the primary determinant of privacy acceptability. We propose context-adaptive pathways that dynamically adjust protection strategies: minimal-friction visibility in public spaces, structured negotiation in semi-public environments, and automatic protection in sensitive contexts. Our findings contribute a diagnostic framework for evaluating privacy mechanisms and implications for context-aware design in ubiquitous sensing.

Let a Lie algebra $\mathfrak q$ be a linear sum of two complementary subalgebras $\mathfrak h$ and $\mathfrak r$. We continue our investigations initiated in (J. London Math. Soc. 103 (2021), 1577-1595), where compatible Poisson brackets associated with splitting $\mathfrak q=\mathfrak h\oplus\mathfrak r$ and related Poisson-commutative subalgebras of the symmetric algebra $\mathcal S(\mathfrak q)$ are studied. In this article, we further develop the general theory and study in more details splittings of the reductive Lie algebras such that both $\mathfrak h$ and $\mathfrak r$ are solvable horospherical subalgebras. We also derive some results of the Adler-Kostant-Symes theory using our approach.

Understanding the propagation of coronal mass ejections (CMEs) through interplanetary space is essential for space weather forecasting. Due to observational limitations, measurements of the photospheric polar magnetic fields remain highly uncertain, and their influence on CME propagation in the heliosphere is still poorly quantified. In this study, we systematically investigate how variations in the photospheric polar magnetic fields affect the Sun-Mars propagation of the 4 December 2021 CME using numerical simulations. The results show that stronger polar fields modify the background solar wind, producing higher plasma density, enhanced magnetic field strength, a flattened heliospheric current sheet, and weakened high-speed streams in the ecliptic plane. These changes markedly slow the CME's radial propagation and inhibit its lateral and radial expansion, leading to notably delayed arrivals at BepiColombo and MAVEN/Tianwen-1. Quantitatively, an enhancement of the polar magnetic fields with a peak value of 6 G at the pole decreases the mean propagation and expansion speeds by roughly 200 km s$^{-1}$ and halves the CME volume. Force analysis reveals that strengthening the polar fields produces only minor changes in the internal force balance of the CME, where the thermal pressure gradient force dominates over the Lorentz force, while it strongly affects the forces acting on the CME surface. At large heliocentric distances, the magnetic pressure of the background solar wind becomes comparable to or even exceeds the aerodynamic drag force, producing a strong confining effect that hinders the CME's motion.

Nonlinear Raman-Nath diffraction (NRND) and nonlinear Cherenkov radiation (NCR) are significant nonlinear diffraction phenomena in optics. Previous studies have primarily focused on NRND and NCR in uniaxial crystals, particularly in periodically poled lithium niobate (PPLN) crystals. However, research on these phenomena in biaxial crystals, such as periodically poled potassium titanyl phosphate (PPKTP), has been limited, and the study of NCR in PPKTP has not yet been undertaken.In this work, we experimentally investigated NRND and NCR phenomena in PPKTP crystals under varying incident angles, pump polarizations, poling periods, and crystal temperatures. Our findings indicate that PPKTP exhibits over ten times greater thermal stability compared to PPLN. This high thermal stability is promising for applications in parallel optical computing, as it helps reduce optical mode deviations and minimize bit error rates.

In-line digital holography (DIH) is a widely used lensless imaging technique, valued for its simplicity and capability to image samples at high throughput. However, capturing only intensity of the interference pattern during the recording process gives rise to some unwanted terms such as cross-term and twin-image. The cross-term can be suppressed by adjusting the intensity of reference wave, but the twin-image problem remains. The twin-image is a spectral artifact that superimposes a defocused conjugate wave onto the reconstructed object, severely degrading image quality. While deep learning has recently emerged as a powerful tool for phase retrieval, traditional Convolutional Neural Networks (CNNs) are limited by their local receptive fields, making them less effective at capturing the global diffraction patterns inherent in holography. In this study, we introduce HoloPASWIN, a physics-aware deep learning framework based on the Swin Transformer architecture. By leveraging hierarchical shifted-window attention, our model efficiently captures both local details and long-range dependencies essential for accurate holographic reconstruction. We propose a comprehensive loss function that integrates frequency-domain constraints with physical consistency via a differentiable angular spectrum propagator, ensuring high spectral fidelity. Validated on a large-scale synthetic dataset of 25,000 samples with diverse noise configurations (speckle, shot, read, and dark noise), HoloPASWIN demonstrates effective twin-image suppression and robust reconstruction quality.

Scientific knowledge increasingly depends on complex computational processes where both hardware and software layers can influence research outcomes. As computational complexity grows, classical-quantum integration provides a lens for examining how the scientific method adapts, particularly regarding a foundational principle of scientific validation - reproducibility. Building upon previous warnings of an ongoing reproducibility crisis in the computational context, this paper examines challenges across classical (HPC) and quantum computing. Despite its deterministic nature, HPC faces reproducibility threats from hardware dependencies, documentation inadequacies, disincentivizing research culture and infrastructure variation. Quantum computing, at low technological maturity, amplifies some challenges, while creating new ones through probabilistic outputs, hardware-specific noise, and tight software-hardware coupling. Classical-quantum integration reveals a telling pattern, where current reproducibility frameworks prove inadequate, as infrastructure blends with the results. Quantum integration serves as a catalyst exposing methodological limitations across the computational domain. We propose a workflow-centered path forward, pointing to the value of gradual cultural shift toward workflow-centered scientific practice. By developing meta-workflows that document both process abstractions and implementation contexts, we create a more robust foundation for scientific knowledge that acknowledges complexity without sacrificing rigor. The path forward involves embracing this evolution in understanding scientific knowledge rather than resisting it

Studies of QCD phase transition signals are often conducted under spatially uniform temperature conditions. However, the influence of spatial temperature gradients on the signals emerging at the phase interface in the fireball generated by heavy-ion collisions has not yet been fully explored. Based on an Ising-like effective potential, we study the locally equilibrated systems with temperature gradients. In a 2D disk geometry, the low-energy fluctuation spectrum is explicitly resolved into radial and angular momentum modes. The nonlocal correaltions of singular eigen-mode exhibits strong anisotropy, which are long-ranged along isotherms but suppressed radially due to the thermal geometry of the system. Unlike homogeneous systems where the zero-momentum mode dominates, correlations in such inhomogeneous system result from the superposition of a series of zero and non-zero angular momentum modes with comparable contributions. We extract the singular angular momentum modes and establish their connection to experimentally observable anisotropic flow. We find azimuthally sensitive observables may offer a previously unexplored avenue for detecting the QCD phase transition.

The density matrix is a positive semidefinite operator of trace 1 characterizing the state of a quantum system. We consider the inverse problem to reconstruct such density matrices from indirect measurements, also known as quantum state tomography. To solve such inverse problems in high or infinite dimensional settings, we study variational regularization using the quantum relative entropy as penalty functional. Quantum relative entropy is an analog of the well-known maximum entropy functional with compositions of functions replaced by the spectral functional calculus. The main aim of this paper is to establish the regularizing property of this scheme. As a crucial intermediate step, we establish lower semi-compactness of the penalty functional with respect to the weak-$*$-topology. Moreover, we compute the subgradient, proximal operator, and conjugate functional of the quantum relative entropy on finite dimensional spaces. This enables us to apply iterative algorithms from convex optimization to solve the regularized problems numerically. To show the validity and practical value of our results, we apply our theory to the examples of Photon-Induced Near-field Electron Microscopy (PINEM) and to optical homodyne tomography.

Aims: We investigate long-term radial velocity (RV) variability in the K-dwarf star GJ 1137 (HD 93083, HIP52521), a known Saturn-mass exoplanet host, and assess the role of stellar activity in shaping the observed signals. Methods: We analyse 13 years of archival high-precision spectroscopic observations obtained with the High Accuracy Radial velocity Planet Searcher spectrograph (HARPS). We performed an extensive spectroscopic analysis of the stellar activity indicators and applied an RV modelling approach, incorporating Keplerian fits, Gaussian process regression as a proxy for stellar activity, and other stellar activity diagnostics. Furthermore, we refined the orbital parameters and the minimum mass of the known exoplanet GJ 1137 b and searched for additional planetary candidates in the system. Results: We detect a long-period RV signal that, if interpreted as planetary, would suggest the presence of a Jovian analogue companion. However, our spectroscopic activity analysis provides strong evidence that this variability is induced by the star's long-term magnetic cycle ( Pcyc = 5870+(480)-(350) days) rather than by an orbiting planet. The signal is detected in both full width at half maximum (FWHM) of the crosscorrelation function and the chromospheric activity index log R'Hk. We measure the stellar rotation period to Prot = 32.3+(1.2)-(1.3) d and identify a significant short-period RV signal, which we attribute to a Super Earth with a period of 9.6412+(12)-(11) d and a minimum mass of 5.12+(0.70)-(0.69) Earth masses, making GJ 1137 a multiple-planet system.

We present Roomify, a spatially-grounded transformation system that generates themed virtual environments anchored to users' physical rooms while maintaining spatial structure and functional semantics. Current VR approaches face a fundamental trade-off: full immersion sacrifices spatial awareness, while passthrough solutions break presence. Roomify addresses this through spatially-grounded transformation - treating physical spaces as "spatial containers" that preserve key functional and geometric properties of furniture while enabling radical stylistic changes. Our pipeline combines in-situ 3D scene understanding, AI-driven spatial reasoning, and style-aware generation to create personalized virtual environments grounded in physical reality. We introduce a cross-reality authoring tool enabling fine-grained user control through MR editing and VR preview workflows. Two user studies validate our approach: one with 18 VR users demonstrates a 63% improvement in presence over passthrough and 26% over fully virtual baselines while maintaining spatial awareness; another with 8 design professionals confirms the system's creative expressiveness (scene quality: 5.95/7; creativity support: 6.08/7) and professional workflow value across diverse environments.

Determining the physically accessible unitary dynamics of a quantum system under finite Hamiltonian resources is a central problem in quantum control and Hamiltonian engineering. Dynamical Lie algebras (DLAs) provide the fundamental link between available control Hamiltonians and the resulting quantum dynamics. While the structural classification of DLAs is well-established, how to systematically engineer and reshape these algebraic structures under realistic physical constraints remains largely unexplored. In this work, building upon recent results on direct sums of identical DLAs, we develop a unified framework for engineering Hamiltonian-driven quantum dynamics based on DLAs: (i) constructing qubit-efficient direct-sum Hamiltonian structures via spectral decomposition of Hermitian operators, enabling parallel simulation of multiple quantum subsystems; (ii) identifying Hamiltonian modifications that preserve full controllability, including the $\mathfrak{su}(2^N)$ algebra, even when additional physically motivated control terms are introduced; and (iii) engineering restricted Hamiltonian sets that confine quantum dynamics to target subalgebras through irreducible Lie-algebra decompositions, providing a principled approach to symmetry-based dynamical reduction. By bridging these Lie-algebraic insights with practical control objectives, our framework provides a systematic pathway for engineering expressive and resource-efficient unitary evolutions, thus unlocking greater structural flexibility of Hamiltonian-driven quantum systems.

An implementation of an electron temperature-dependent interaction potential for copper in a two-temperature model-molecular dynamics framework is presented. An algorithm for enforcing energy conservation when using such an interaction is provided that is needed due to the changing interaction strength with the degree of excitation. Furthermore, focus is put on how to treat the pressure differences due to an electron temperature gradient following laser irradiation. The influence of various extensions is investigated in large-scale two-temperature model molecular dynamics simulations and compared to existing approaches.

This paper presents a method to derate the dependency on the decimation factor, $M$, of the pass-band droop inherent to $N$-th ordered comb decimators. It is achieved by cascading a symmetric 3-tap FIR filter in the integral stage of the corresponding comb decimator and choosing the coefficients only as a function of order $N$. The proposed derating method derived from the conventional comb decimator can be readily applied to any recently developed comb decimator and droop-compensation filter design method.

Physical reservoir computing (PRC) is a promising brain-inspired computing architecture for overcoming the von Neumann bottleneck by utilizing the intrinsic dynamics of physical systems. However, a major obstacle to its real-world implementation lies in the tension between extracting sufficient information for high computational performance and maintaining a hardware-feasible, high-speed architecture. Here, we report spectral dynamics reservoir computing (SDRC), a broadly applicable framework based on analogue filtering and envelope detection that bridges this gap. SDRC effectively exploits the fast spectral dynamics embedded in short-time, coarse spectra of material responses to attain strong computational capability while maintaining high-speed processing and minimal hardware overhead. This approach circumvents the need for implementation-intensive, precision-sensitive integrated circuits required in high-speed time-multiplexing measurements, while enabling real-time use of the material's spectral manifold as a high-dimensional computational resource. We implement and experimentally demonstrate SDRC applied to spin waves that achieves state-of-the-art-level performance with only 56 nodes on benchmark tasks of parity-check and second-order nonlinear autoregressive moving average, as well as high accuracy of 98.0% on a real-world problem of speech recognition.

We report the absence of orbital Hall magnetoresistance (OMR) in nonmagnet/ferromagnet bilayers, challenging the general assumption that orbital transport mimics spin transport. Despite the observation of giant orbital torques, confirming the generation of orbital currents, thickness-dependent magnetoresistance measurements reveal that the signal is dominated by the intrinsic magnetoresistance of the ferromagnet and current shunting, with no discernible OMR contribution. We attribute this contradiction to the distinct transport properties of orbital compared with spin. Orbital currents undergo isotropic bulk absorption in the ferromagnet rather than anisotropic interfacial reflection required for OMR. Furthermore, we find that texture-induced magnetoresistance and self-torques in Ni-based bilayers can generate misleading signals, suggesting that caution is required when employing Ni in orbitronic studies. These findings clarify the distinct physical rules governing orbital transport and provide a simple method to distinguish spin and orbital currents.

In this paper, we construct higher Chow cycles of type $(2, 1)$ on a family of surfaces related to a product of curves, which are certain degree $N$ abelian covers of $\mathbb{P}^1$ branched over $n+2$ points. We prove that for a very general member, these cycles generate a subgroup of the indecomposable part of $\operatorname{rank} \ge n\cdot φ(N)$, where $φ(N)$ is Euler's totient function, by computing their images under the transcendental regulator map.

Internal-external field separation is crucial for many aspects of geomagnetism, aiming at distinguishing contributions of the magnetic field generated within the Earth (or any other planet) from those produced in the exterior. When data is available on a full spherical observation surface, this separation is a standard, stable, and widely used procedure dating back to Gauss. However, when data is only available in a subdomain of the observation surface (as is the case for aeromagnetic and ground-based surveys), the situation changes. Here we show that, without prior assumptions, an internal-external field separation is not uniquely possible. Given the geophysically reasonable assumption that the exterior sources (e.g., ionospheric current systems) are located above a source-free spherical shell, we show that a unique separation becomes possible but that it is highly unstable. The results are based on a Hardy-Hodge decomposition of spherical vector fields and provide an explanation of the intrinsic difficulties of regional data-based internal-external field separation.

Quantum weight reduction procedures ease the implementation of quantum codes by sparsifying them, resulting in low-weight checks and low-degree qubits. However, to date, only few quantum weight reduction methods have been explored. In this work we introduce a simple and general procedure for quantum weight reduction that achieves check weight 6 and total qubit degree 6, lower than existing procedures at the cost of a potentially larger qubit overhead. Our quantum weight reduction procedure replaces each qubit and check in an arbitrary Calderbank-Shor-Steane code with an ample patch of surface code, these patches are then joined together to form a geometrically nonlocal Layer Code. This is a quantum analog of the simple classical weight reduction procedure where each bit and check is replaced by a repetition code. Due to the simplicity of our weight reduction procedure, bounds on the weight and degree of the resulting code follow directly from the Layer Code construction and hence are easily verified by inspection. Our procedure is well suited for implementation in modular architectures that consist of surface code patches networked via long-range interconnects.

We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set $\mathcal{D}\subseteq [0,T]\times\mathbb{R}^d$, a diffusion $X$ in $\mathbb{R}^d$ must be linearly controlled in order to keep the time-space process $(t,X_t)$ inside the set $\mathcal{C}:=([0,T]\times\mathbb{R}^d)\setminus\mathcal{D}$, while at the same time minimising an expected cost that depends on the state $(t,X_t)$ and is quadratic in the speed of the control exerted. We find a probabilistic representation for the value function and an optimal control under a set of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of the set $\mathcal{D}$. The optimally controlled dynamics is in strong form, in the sense that it is adapted to the filtration generated by the driving Brownian motion. Fully explicit formulae are presented in some relevant examples.

We study the relationship between the frequency of a ternary digit in a number and the asymptotic mean value of the digits. The conditions for the existence of the asymptotic mean of digits in a ternary number are established. We indicate an infinite everywhere dense set of numbers without frequency of digits but with the asymptotic mean of the digits.