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Motivated by modern machine learning applications where we only have access to empirical measures constructed from finite samples, we relax the marginal constraints of the classical Schrödinger bridge problem by penalizing the transport cost between the bridge's marginals and the prescribed marginals. We derive a duality formula for this transport-relaxed bridge and demonstrate that it reduces to a finite-dimensional concave optimization problem when the prescribed marginals are discrete and the reference distribution is absolutely continuous. We establish the existence and uniqueness of solutions for both the primal and dual problems. Moreover, as the penalty blows up, we characterize the limiting bridge as the solution to a discrete Schrödinger bridge problem and identify a leading-order logarithmic divergence. Finally, we propose gradient ascent and Sinkhorn-type algorithms to numerically solve the transport-relaxed Schrödinger bridge, establishing a linear convergence rate for both algorithms.

General-purpose vision-language-action models benefit from large vision-language priors, but effective manipulation also requires anticipating action-relevant scene changes. Existing world-action models often rely on large generative world models or dense future rollouts, which are expensive and spend capacity on visual details weakly coupled to control. We present Bridge-WA, a lightweight world-action framework that distills a frozen future-change teacher into three compact priors: future tokens for intended outcomes, change maps for intervention support, and motion-flow maps for local transition direction. A WorldBridge conditions the action transformer on these priors through multi-source attention memories and spatial-temporal biases, while the teacher model is removed at inference. Across VLABench, RoboTwin2.0, LIBERO-Plus and real-robot evaluations, Bridge-WA improves task success, progress, and robustness, with particularly clear gains under out-of-distribution visual shifts. By focusing action generation on where and how the scene will change, Bridge-WA suppresses nuisance appearance factors such as background, lighting, and distractors, leading to better generalization wit

The upcoming decade of observational cosmology will be shaped by large sky surveys, such as the ground-based LSST at the Vera C. Rubin Observatory and the space-based Euclid mission. While they promise an unprecedented view of the Universe across depth, resolution, and wavelength, their differences in observational modality, sky coverage, point-spread function, and scanning cadence make joint analysis beneficial, but also challenging. To facilitate joint analysis, we introduce A(stronomical)S(urvey)-Bridge, a bidirectional generative model that translates between ground- and space-based observations. AS-Bridge learns a diffusion model that employs a stochastic Brownian Bridge process between the LSST and Euclid observations. The two surveys have overlapping sky regions, where we can explicitly model the conditional probabilistic distribution between them. We show that this formulation enables new scientific capabilities beyond single-survey analysis, including faithful probabilistic predictions of missing survey observations and inter-survey detection of rare events. These results establish the feasibility of inter-survey generative modeling. AS-Bridge is therefore well-positioned

We investigate the martingale Schrödinger bridge, recently introduced by Nutz and Wiesel as a distinguished martingale transport plan between two probability measures in convex order. We show that this construction extends naturally to arbitrary dimension and admits several equivalent characterizations. In particular, we identify its continuous-time counterpart as the continuous martingale with prescribed marginals that minimizes a weighted quadratic energy measuring the deviation from Brownian motion. In the irreducible case, we prove that this continuous martingale Schrödinger bridge coincides with the Föllmer martingale, that is, with the Doob martingale associated to a suitable Föllmer process. More generally, we relate the martingale Schrödinger bridge to a variational problem over base measures and to the dual formulation of the corresponding weak optimal transport problem, thereby clarifying its connection with the classical Schrödinger bridge.

A hybrid digital twin framework is presented for bridge condition monitoring using existing traffic cameras and weather APIs, reducing reliance on dedicated sensor installations. The approach is demonstrated on the Peace Bridge (99 years in service) under high traffic demand and harsh winter exposure. The framework fuses three near-real-time streams: YOLOv8 computer vision from a bridge-deck camera estimates vehicle counts, traffic density, and load proxies; a Lighthill--Whitham--Richards (LWR) model propagates density $ρ(x,t)$ and detects deceleration-driven shockwaves linked to repetitive loading and fatigue accumulation; and weather APIs provide deterioration drivers including temperature cycling, freeze-thaw activity, precipitation-related corrosion potential, and wind effects. Monte Carlo simulation quantifies uncertainty across traffic-environment scenarios, while Random Forest models map fused features to fatigue indicators and maintenance classification. The framework demonstrates utilizing existing infrastructure for cost-effective predictive maintenance of aging, high-traffic bridges in harsh climates.

The present work builds on previous investigations of the authors (and their collaborators) regarding bridges, a certain type of morphisms between encryption schemes, making a step forward in developing a (category theory) language for studying relations between encryption schemes. Here we analyse the conditions under which bridges can be performed sequentially, formalizing the notion of composability. One of our results gives a sufficient condition for a pair of bridges to be composable. We illustrate that composing two bridges, each independently satisfying a previously established IND-CPA security definition, can actually lead to an insecure bridge. Our main result gives a sufficient condition that a pair of secure composable bridges should satisfy in order for their composition to be a secure bridge. We also introduce the concept of a complete bridge and show that it is connected to the notion of Fully composable Homomorphic Encryption (FcHE), recently considered by Micciancio. Moreover, we show that a result of Micciancio which gives a construction of FcHE schemes can be phrased in the language of complete bridges, where his insights can be formalised in a greater generality.

Bitcoin's limited scripting capabilities and lack of native interoperability mechanisms have constrained its integration into the broader blockchain ecosystem, especially decentralized finance (DeFi) and multi-chain applications. This paper presents a comprehensive taxonomy of Bitcoin cross-chain bridge protocols, systematically analyzing their trust assumptions, performance characteristics, and applicability to the Artificial Intelligence of Things (AIoT) scenarios. We categorize bridge designs into three main types: naive token swapping, pegged-asset bridges, and arbitrary-message bridges. Each category is evaluated across key metrics such as trust model, latency, capital efficiency, and DeFi composability. Emerging innovations like BitVM and recursive sidechains are highlighted for their potential to enable secure, scalable, and programmable Bitcoin interoperability. Furthermore, we explore practical use cases of cross-chain bridges in AIoT applications, including decentralized energy trading, healthcare data integration, and supply chain automation. This taxonomy provides a foundational framework for researchers and practitioners seeking to design secure and efficient cross-cha

We introduce a new class of stochastic processes called fractional Wiener-Weierstrass bridges. They arise by applying the convolution from the construction of the classical, fractal Weierstrass functions to an underlying fractional Brownian bridge. By analyzing the $p$-th variation of the fractional Wiener-Weierstrass bridge along the sequence of $b$-adic partitions, we identify two regimes in which the processes exhibit distinct sample path properties. We also analyze the critical case between those two regimes for Wiener-Weierstrass bridges that are based on standard Brownian bridge. We furthermore prove that fractional Wiener-Weierstrass bridges are never semimartingales, and we show that their covariance functions are typically fractal functions. Some of our results are extended to Weierstrass bridges based on bridges derived from a general continuous Gaussian martingale.

This is the second of three papers motivated by the author's desire to understand and explain "algebraically" one aspect of Dmitriy Zhuk's proof of the CSP Dichotomy Theorem. In this paper we extend Zhuk's "bridge" construction to arbitrary meet-irreducible congruences of finite algebras in locally finite varieties with a Taylor term. We then connect bridges to centrality and similarity. In particular, we prove that Zhuk's bridges and our "similarity bridges" (defined in our first paper) convey the same information in locally finite Taylor varieties.

Inverse protein folding is a fundamental task in computational protein design, which aims to design protein sequences that fold into the desired backbone structures. While the development of machine learning algorithms for this task has seen significant success, the prevailing approaches, which predominantly employ a discriminative formulation, frequently encounter the error accumulation issue and often fail to capture the extensive variety of plausible sequences. To fill these gaps, we propose Bridge-IF, a generative diffusion bridge model for inverse folding, which is designed to learn the probabilistic dependency between the distributions of backbone structures and protein sequences. Specifically, we harness an expressive structure encoder to propose a discrete, informative prior derived from structures, and establish a Markov bridge to connect this prior with native sequences. During the inference stage, Bridge-IF progressively refines the prior sequence, culminating in a more plausible design. Moreover, we introduce a reparameterization perspective on Markov bridge models, from which we derive a simplified loss function that facilitates more effective training. We also modulat

This paper employs various computational techniques to determine the bridge numbers of both classical and virtual knots. For classical knots, there is no ambiguity of what the bridge number means. For virtual knots, there are multiple natural definitions of bridge number, and we demonstrate that the difference can be arbitrarily far apart. We then acquired two datasets, one for classical and one for virtual knots, each comprising over one million labeled data points. With the data, we conduct experiments to evaluate the effectiveness of common machine learning models in classifying knots based on their bridge numbers.

The Bridge graph is a special type of graph which are constructed by connecting identical connected graphs with path graphs. We discuss different types of bridge graphs $B_{n\times l}^{m\times k}$ in this paper. In particular, we discuss the following: complete-type bridge graphs, star-type bridge graphs, and full binary tree bridge graphs. We also bound the second eigenvalues of the graph Laplacian of these graphs using methods from Spectral Graph Theory. In general, we prove that for general bridge graphs, $B_{n\times l}^2$, the second eigenvalue of the graph Laplacian should be between $0$ and $2$, inclusive. In the end, we talk about future work on infinite bridge graphs. We created definitions and found the related theorems to support our future work about infinite bridge graphs.

The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set. We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realising the inverse design of materials from new invariant values.

A bridge in a graph is an edge whose removal disconnects the graph and increases the number of connected components. We calculate the fraction of bridges in a wide range of real-world networks and their randomized counterparts. We find that real networks typically have more bridges than their completely randomized counterparts, but very similar fraction of bridges as their degree-preserving randomizations. We define a new edge centrality measure, called bridgeness, to quantify the importance of a bridge in damaging a network. We find that certain real networks have very large average and variance of bridgeness compared to their degree-preserving randomizations and other real networks. Finally, we offer an analytical framework to calculate the bridge fraction , the average and variance of bridgeness for uncorrelated random networks with arbitrary degree distributions.

Suppose a knot in a $3$-manifold is in $n$-bridge position. We consider a reduction of the knot along a bridge disk $D$ and show that the result is an $(n-1)$-bridge position if and only if there is a bridge disk $E$ such that $(D, E)$ is a cancelling pair. We apply this to an unknot $K$, in $n$-bridge position with respect to a bridge sphere $S$ in the $3$-sphere, to consider the relationship between a bridge disk $D$ and a disk in the $3$-sphere that $K$ bounds. We show that if a reduction of $K$ along $D$ yields an $(n-1)$-bridge position, then $K$ bounds a disk that contains $D$ as a subdisk and intersects $S$ in $n$ arcs.

NASA's Science Mission Directorate (SMD) has initiated a program to enhance the participation of historically underrepresented institutions and communities in NASA's mission. Currently known as the NASA SMD Bridge Program, its goal is to establish enduring partnerships among these institutions, research-intensive universities, and NASA centers. There are concerns about using "Bridge" in the program's name, with stakeholders suggesting that it might stigmatize students and mislead applicants about its focus. In this white paper, we address these concerns and conclude that a name change that better reflects the mission of this SMD effort is necessary to address these concerns.

In line with the Astro2020 Decadal Report State of the Profession findings and the NASA core value of Inclusion, the NASA Science Mission Directorate (SMD) Bridge Program was created to provide financial and programmatic support to efforts that work to increase the representation and inclusion of students from under-represented minorities in the STEM fields. To ensure an effective program, particularly for those who are often left out of these conversations, the NASA SMD Bridge Program Workshop was developed as a way to gather feedback from a diverse group of people about their unique needs and interests. The Early Career Perspectives Working Group was tasked with examining the current state of bridge programs, academia in general, and its effect on students and early career professionals. The working group, comprised of 10 early career and student members, analyzed the discussions and responses from workshop breakout sessions and two surveys, as well as their own experiences, to develop specific recommendations and metrics for implementing a successful and supportive bridge program. In this white paper, we will discuss the key themes that arose through our work, and highlight sele

Vector graphics, known for their scalability and user-friendliness, provide a unique approach to visual content compared to traditional pixel-based images. Animation of these graphics, driven by the motion of their elements, offers enhanced comprehensibility and controllability but often requires substantial manual effort. To automate this process, we propose a novel method that integrates implicit neural representations with text-to-video diffusion models for vector graphic animation. Our approach employs layered implicit neural representations to reconstruct vector graphics, preserving their inherent properties such as infinite resolution and precise color and shape constraints, which effectively bridges the large domain gap between vector graphics and diffusion models. The neural representations are then optimized using video score distillation sampling, which leverages motion priors from pretrained text-to-video diffusion models. Finally, the vector graphics are warped to match the representations resulting in smooth animation. Experimental results validate the effectiveness of our method in generating vivid and natural vector graphic animations, demonstrating significant impro