This textbook is drawn from notes for a two-semester graduate course in quantum mechanics. It begins with the most constrained quantum system, and recovers the rest of the subject by relaxing those constraints one at a time. The starting point is a single qubit, the smallest nontrivial Hilbert space with the strongest possible restriction on its dynamics, made concrete by a Bloch cube whose six faces are the cardinal states of a spin-1/2 system. Tensor products admit many qubits; lattices give them a place to live; time evolution sets them in motion; the continuum limit produces wavefunctions; three-dimensional angular momentum, the hydrogen atom, and perturbation theory follow; Lorentz invariance promotes the lattice of spinors to the Dirac equation; and the renormalization group asks how theories at different scales relate. Each chapter loosens one feature of the qubit while keeping the others fixed, so that the standard apparatus of graduate quantum mechanics arrives as a sequence of controlled generalizations rather than as separate topics. Discrete-to-continuous transitions recur at four scales: in Hilbert-space dimension, in real space, in time, and in coupling. The book clos
We reveal a precise mathematical framework about a new family of generative models which we call Gradient Flow Drifting. With this framework, we prove an equivalence between the recently proposed Drifting Model and the Wasserstein gradient flow of the forward KL divergence under kernel density estimation (KDE) approximation. Specifically, we prove that the drifting field of drifting model (arXiv:2602.04770) equals, up to a bandwidth-squared scaling factor, the difference of KDE log-density gradients $ abla \log p_{\mathrm{kde}} - abla \log q_{\mathrm{kde}}$, which is exactly the particle velocity field of the Wasserstein-2 gradient flow of $KL(q\|p)$ with KDE-approximated densities. Besides that, this broad family of generative models can also include MMD-based generators, which arises as special cases of Wasserstein gradient flows of different divergences under KDE approximation. We provide a concise identifiability proof, and a theoretically grounded mixed-divergence strategy. We combine reverse KL and $χ^2$ divergence gradient flows to simultaneously avoid mode collapse and mode blurring, and extend this method onto Riemannian manifold which loosens the constraints on the kerne
Recent diffusion-based image editing methods commonly rely on text or high-level instructions to guide the generation process, offering intuitive but coarse control. In contrast, we focus on explicit, prompt-free editing, where the user directly specifies the modification by cropping and pasting an object or sub-object into a chosen location within an image. This operation affords precise spatial and visual control, yet it introduces a fundamental challenge: preserving the identity of the pasted object while harmonizing it with its new context. We observe that attention maps in diffusion-based editing models inherently govern whether image regions are preserved or adapted for coherence. Building on this insight, we introduce LooseRoPE, a saliency-guided modulation of rotational positional encoding (RoPE) that loosens the positional constraints to continuously control the attention field of view. By relaxing RoPE in this manner, our method smoothly steers the model's focus between faithful preservation of the input image and coherent harmonization of the inserted object, enabling a balanced trade-off between identity retention and contextual blending. Our approach provides a flexibl
Sparse attention accelerates Diffusion Transformers (DiTs) for video generation by computing only the important tokens while skipping the rest. The token selection strategy is key to balancing sparsity and accuracy. We formulate the token filtering process as a dual-goal optimization problem: maximizing sparsity and minimizing accuracy degradation. Existing algorithms cannot fulfill both objectives simultaneously. For example, Top-p only considers the accuracy constraint, while Top-k maintains a fixed computational budget but loosens the accuracy constraint. This paper demonstrates that maintaining a fixed recall rate is sufficient for ensuring accuracy, whereas a fixed threshold is suboptimal for reducing computational cost. Therefore, we propose a dynamic thresholding scheme to improve sparsity while maintaining the same level of accuracy. Furthermore, our algorithm is deeply integrated with Flash Attention (FA), eliminating the need for any additional masking computation overhead. Experimental results on Wan 2.2 validate that, compared to the BLASST algorithm which is also integrated with FA, our dynamic thresholding strategy enhances sparsity from 61.42\% to 82\% with a VBench
Preload loss in bolted joints results in alterations of the stiffness, damping, and nonlinearity of the structure, but existing monitoring techniques for rail-vehicle systems are often not capable of combining controlled shaker tests and sensing of nonlinear features. This paper proposes a method for detecting bolt loosening using a vibro-acoustic technique, where the structure is subjected to controlled shaker tests to sense the nonlinear features. A triaxial accelerometer was attached to the demonstrator, a microphone was placed in close proximity, and one of the bolts was tested under 0%, 20%, 40%, and 80% preload conditions. Single-tone and frequency-modulated (FM) signals close to the main natural frequency of 130 Hz, which was identified using sine sweep and narrow-band excitation, were applied to the demonstrator. When the structure was subjected to 130 Hz single-tone excitation, the loose state of the bolt exhibited several additional high-frequency spectral peaks. FM excitation between 125 and 135 Hz further distinguished between the states. Harmonic band power ratios, normalized to the carrier, distinguished between the loose state and the 80% preload state, where the dif
Ricci curvature and its associated flow offer powerful geometric methods for analyzing complex networks. While existing research heavily focuses on applications for undirected graphs such as community detection and core extraction, there have been relatively less attention on directed graphs. In this paper, we introduce a definition of Ricci curvature and an accompanying curvature flow for directed graphs. Crucially, for strongly connected directed graphs, this flow admits a unique global solution. We then apply this flow to detect strongly connected subgraphs from weakly connected directed graphs. (A weakly connected graph is connected overall but not necessarily strongly connected). Unlike prior work requiring graphs to be strongly connected, our method loosens this requirement. We transform a weakly connected graph into a strongly connected one by adding edges with very large artificial weights. This modification does not compromise our core subgraph detection. Due to their extreme weight, these added edges are automatically discarded during the final iteration of the Ricci curvature flow. For core evaluation, our approach consistently surpasses traditional methods, achieving be
Traditional k-means clustering underperforms on non-convex shapes and requires the number of clusters k to be specified in advance. We propose a simple geometric enhancement: after standard k-means, each cluster center is assigned a radius (the distance to its farthest assigned point), and clusters whose radii overlap are merged. This post-processing step loosens the requirement for exact k: as long as k is overestimated (but not excessively), the method can often reconstruct non-convex shapes through meaningful merges. We also show that this approach supports recursive partitioning: clustering can be performed independently on tiled regions of the feature space, then globally merged, making the method scalable and suitable for distributed systems. Implemented as a lightweight post-processing step atop scikit-learn's k-means, the algorithm performs well on benchmark datasets, achieving high accuracy with minimal additional computation.
We prove that the $5$-canonical map of every minimal projective $3$-fold $X$ with $K_X^3\geq 86$ is stably birational onto its image, which loosens previous requirements $K_X^3>4355^3$ and $K_X^3>12^3$ respectively given by Todorov and Chen. The essential technical ingredient of this paper is an efficient utilization of a moving divisor which grows from global $2$-forms by virtue of Chen-Jiang.
This paper presents tight upper and lower bounds for minimum number of samples (copies of a quantum state) required to attain a prescribed accuracy (measured by error variance) for scalar parameters estimation using unbiased estimators under quantum local differential privacy for qubits. Particularly, the best-case (optimal) scenario is considered by minimizing the sample complexity over all differentially-private channels; the worst-case channels can be arbitrarily uninformative and render the problem ill-defined. In the small privacy budget $ε$ regime, i.e., $ε\ll 1$, the sample complexity scales as $Θ(ε^{-2})$. This bound matches that of classical parameter estimation under local differential privacy. The lower bound however loosens in the large privacy budget regime, i.e., $ε\gg 1$. The upper bound for the minimum number of samples is generalized to qudits (with dimension $d$) resulting in sample complexity of $O(dε^{-2})$.
The minimal bound of the thermodynamic uncertainty relation (TUR) is modulated from that of the classical counterpart ($\mathcal{Q}_{\rm min}=2$) when a quantumness is present in the dynamical process far from equilibrium. A recent study on a dissipative two-level system (TLS) subject to an external field indicates that quantum coherence can suppress the fluctuations of the irreversible current and loosens the TUR bound to $\mathcal{Q}_{\rm min}^{\rm TLS}\approx 1.25$. Here, we extend on the field-driven single TLS % in a photonic bath to a quantum-mechanically coupled two-qubit system (TQS), and explore how the quantum coupling between the two qubits, an additional complexity introduced to the probem of TLS, affects the photon current, fluctuations, and the TUR bound. We find that the TUR bound of TQS depends on the strength of coupling, such that $\mathcal{Q}_{\rm min}^{\rm TQS}=\mathcal{Q}_{\rm min}^{\rm TLS}\approx 1.25$ when the two qubits are effectively decoupled under weak coupling, whereas another loose bound $\mathcal{Q}_{\rm min}^{\rm TQS}\approx 1.36$ is identified for two strongly coupled qubits under strong fields. By contrasting the TQS against two coupled noisy osci
Semantic interpretability in Reinforcement Learning (RL) enables transparency and verifiability of decision-making. Achieving semantic interpretability in reinforcement learning requires (1) a feature space composed of human-understandable concepts and (2) a policy that is interpretable and verifiable. However, constructing such a feature space has traditionally relied on manual human specification, which often fails to generalize to unseen environments. Moreover, even when interpretable features are available, most reinforcement learning algorithms employ black-box models as policies, thereby hindering transparency. We introduce interpretable Tree-based Reinforcement learning via Automated Concept Extraction (iTRACE), an automated framework that leverages pre-trained vision-language models (VLM) for semantic feature extraction and train a interpretable tree-based model via RL. To address the impracticality of running VLMs in RL loops, we distill their outputs into a lightweight model. By leveraging Vision-Language Models (VLMs) to automate tree-based reinforcement learning, iTRACE loosens the reliance the need for human annotation that is traditionally required by interpretable mo
We present an experimental test of Kubo formula performed on a nonlinear quantum conductor, a Superconductor-Insulator-Superconductor tunnel junction, driven far from equilibrium by a DC voltage bias. We implement the proposal of Lesovik and Loosen [1] and demonstrate experimentally that it is possible to extract both the emission and absorption noise of the conductor by measuring the power it exchanges with a linear detection circuit whose occupation is tuned close to vacuum levels. We then compare their difference to the real part of the admittance which is independently measured by coherent reflectometry, finding that Kubo formula holds within experimental accuracy. Last, we show theoretically that the spectral density of power exchanged between a quantum conductor and its linear detection circuit follows a Lesovik and Loosen like formula, even in the presence of strong detection back-action. This result applies as long as the conductor acts as a current source for the detection circuit and the detection circuit is not singular.
In this paper we extend the Aw-Rascle-Zhang (ARZ) non-equilibrium traffic flow model to take into account the look-ahead capability of connected and autonomous vehicles (CAVs), and the mixed flow dynamics of human driven and autonomous vehicles. The look-ahead effect of CAVs is captured by a non-local averaged density within a certain distance (the look-ahead distance). We show, using wave perturbation analysis, that increased look-ahead distance loosens the stability criteria. Our numerical experiments, however, showed that a longer look-ahead distance does not necessarily lead to faster convergence to equilibrium states. We also examined the impact of spatial distributions and market penetrations of CAVs and showed that increased market penetration helps stabilizing mixed traffic while the spatial distribution of CAVs have less effect on stability. The results revealed the potential of using CAVs to stabilize traffic, and may provide qualitative insights on speed control in the mixed autonomy environment.
This paper proposes a novel adaptive Koopman Model Predictive Control (MPC) framework, termed HPC-AK-MPC, designed to address the dual challenges of time-varying dynamics and safe operation in complex industrial processes. The framework integrates two core strategies: online learning and historically-informed safety constraints. To contend with process time-variance, a Recursive Extended Dynamic Mode Decomposition (rEDMDc) technique is employed to construct an adaptive Koopman model capable of updating its parameters from real-time data, endowing the controller with the ability to continuously learn and track dynamic changes. To tackle the critical issue of safe operation under model uncertainty, we introduce a novel Historical Process Constraint (HPC) mechanism. This mechanism mines successful operational experiences from a historical database and, by coupling them with the confidence level of the online model, generates a dynamic "safety corridor" for the MPC optimization problem. This approach transforms implicit expert knowledge into explicit, adaptive constraints, establishing a dynamic balance between pursuing optimal performance and ensuring robust safety. The proposed HPC-A
Every year more than 2.3 million joint replacement is performed worldwide. Around 10% of these replacements fail those results in revisions at a cost of $8 billion per year. In particular patients younger than 55 years of age face higher risks of failure due to greater demand on their joints. The long-term failure of joint replacement such as implant loosening significantly decreases the life expectancy of replacement. One of the main challenges in understanding and treatment of implant loosening is lack of a low-cost screening device that can detect or predict loosening at very early stages. In this work we are proposing a novel method of screening implant condition via ultrasonic signals. In this method we are applying ultrasonic signals to the joint via several piezoresistive discs while reading signals with several other piezoresistive sensors. We are introducing a new approachin interpreting ultrasonic signals and we prove in a finite element environment that our method can be used to assess replacement condition. We show how our new concept can detect and distinguish between different implant fixation failure types sizes and even locate the position of the failure. We believe
The safety and integrity of engineered structures are critically dependent on maintaining sufficient preload in their bolted joints. This preload can be dynamically lost due to sustained vibrations or sudden shock that are large enough to induce slip in the threads. While high-fidelity finite element simulations and analytical methods can accurately model the loss of preload for a single, their prohibitive computational expense and complexity render them unfeasible for analyzing large-scale structures with many bolts. This creates a critical need for reduced-order models that capture the essential physics of loosening while remaining computationally efficient. This paper introduces a reduced-order modeling methodology for predicting the loosening of bolted lap joints subjected to transverse shock excitation. The core idea is to treat the bolt tension as a dynamic degree-of-freedom that governs the effective properties of the joint through tension-dependent stiffness and damping that couple the components together. The methodology is applied to a pair of oscillators coupled by with a single lap joint with a strain-sensing bolt. Three different sets of experimental measurements are u
Recent results from DESI combined with cosmic microwave background data give the tightest constraints on the sum of neutrino masses to date. However, these analyses approximate the neutrino mass hierarchy by three degenerate-mass (DM) neutrinos, instead of the normal (NH) and inverted hierarchies (IH) informed by terrestrial neutrino oscillation experiments. Given the stringency of the upper limits from DESI data, we test explicitly whether the inferred neutrino constraints are robust to the choice of neutrino mass ordering using both Bayesian and frequentist methods. For Planck data alone, we find that the DM hierarchy presents a good approximation to the physically motivated hierarchies while showing a strong dependence on the assumed lower bound of the prior, confirming previous studies. For the combined Planck and DESI baryon acoustic oscillation data, we find that assuming NH ($M_\mathrm{tot} < 0.13\,\mathrm{eV}$) or IH ($M_\mathrm{tot} < 0.16\,\mathrm{eV}$) loosens the Bayesian upper limits compared to the DM approximation ($M_\mathrm{tot} < 0.086\,\mathrm{eV}$). The frequentist analysis shows that the different neutrino models fit the data equally well and the loose
Weak-value-amplification permits small effects to be measured as observable changes at the sacrifice of power due to post-selection. The power recycling scheme has been proven to eliminate this inefficiency of the rare post-selection, thus surpassing the limit of the shot noise and improving the precision of the measurement. However, the improvement is strictly limited by the system setup, especially the system loss. Here we introduce a dual recycling model based on the interferometric weak-value-based deflection measurement. Two mirrors, the power-recycling mirror and signal-recycling mirror, are placed at the bright and dark port of the interferometer respectively, creating a composite resonator. The results show that both the power and the signal-to-noise ratio (SNR) are greatly enhanced in a wider range of experimental parameters compared to the power-recycling scheme. This work considerably loosens the constraint of the system setup and further explores the real advantage of weak measurement over traditional schemes.
Reinforcement learning (RL) has achieved promising results on most robotic control tasks. Safety of learning-based controllers is an essential notion of ensuring the effectiveness of the controllers. Current methods adopt whole consistency constraints during the training, thus resulting in inefficient exploration in the early stage. In this paper, we propose an algorithm named Constrained Policy Optimization with Extra Safety Budget (ESB-CPO) to strike a balance between the exploration efficiency and the constraints satisfaction. In the early stage, our method loosens the practical constraints of unsafe transitions (adding extra safety budget) with the aid of a new metric we propose. With the training process, the constraints in our optimization problem become tighter. Meanwhile, theoretical analysis and practical experiments demonstrate that our method gradually meets the cost limit's demand in the final training stage. When evaluated on Safety-Gym and Bullet-Safety-Gym benchmarks, our method has shown its advantages over baseline algorithms in terms of safety and optimality. Remarkably, our method gains remarkable performance improvement under the same cost limit compared with ba