PAPER PLAINE

Graph coloring problems appear in scheduling, map coloring, and conflict resolution algorithms. Understanding when fair partitions are and aren't possible helps computer scientists know which real-world network problems have solutions and which don't, preventing wasted effort on impossible tasks.

Quantum control problems appear in emerging technologies like quantum computing and precision measurement devices. This work removes a major theoretical gap by extending control methods to realistic geometric boundaries—most real quantum systems don't sit in perfectly smooth domains. The finding that corners and edges are sufficient control points could simplify engineering of quantum devices by reducing where actuators need to be placed.

Operations research teams rely on optimization models to make million-dollar decisions about supply chains, scheduling, and resource allocation. If an AI formulates a model based on an incomplete or misunderstood problem statement, the resulting "solution" could be useless or actively harmful. Teaching AI systems to pause and ask clarifying questions before modeling—rather than guessing at missing details—is essential for making these tools trustworthy enough to use in real business contexts.

This work clarifies how disorder and randomness behave at extremes, which matters for fields ranging from statistical physics to information theory. The new framework—block-triangular majorisation—fills a conceptual gap between existing models and may help researchers better predict and control systems far from equilibrium, from biological networks to quantum computing.

Space missions to distant planets and asteroids often require trajectories that satisfy complex dynamic constraints—like staying in a certain orbital relationship to multiple bodies. Without these new transversality conditions, engineers can't reliably verify whether a proposed path is truly optimal or just locally good. This work gives mission planners the mathematical foundation to confidently design fuel-efficient routes through the solar system.

Many real control systems face uncertainty that changes with their own state or decisions — a robot's collision risk depends on where it is, a satellite's tumbling depends on its spin rate. Previous methods either couldn't handle this, required custom solvers that didn't scale up, or only worked for simple cases. This approach uses standard software and applies broadly, making it practical to design safer, more robust controllers for aerospace, robotics, and other safety-critical applications.

Leavitt path algebras appear in operator theory, symbolic dynamics, and quantum physics. Understanding when these algebras have predictable structural properties lets mathematicians and physicists apply them reliably in theory and computation, and simplifies the problem of determining when two seemingly different weighted graphs produce equivalent algebraic structures.

Entanglement of Formation is a central concept in quantum information theory, used in everything from quantum computing to quantum cryptography. If the mathematical framework physicists have been using doesn't actually work as promised, then calculations and proofs relying on that framework may be unreliable. This paper forces researchers to be more careful about when they can safely use these mathematical shortcuts and identifies which quantum states need special handling.

Quantum computers are being explored to solve logistics problems that classical computers struggle with at scale. But if they give invalid answers, they're useless. This work shows how to build in safeguards that force quantum solvers to respect real-world constraints—a necessary step before these machines can handle actual delivery networks.

Simpler control systems make planes safer and cheaper to design—they're less sensitive to errors in how engineers model the aircraft, which means they work better across different planes and changing conditions. Better control methods also let engineers more easily add safety features like automatic stall prevention or fault recovery without redesigning the entire system from scratch.

As neighborhoods add more solar panels and batteries, voltage instability threatens equipment damage and blackouts. This approach lets operators stabilize the grid automatically without real-time communication or constant adjustments—crucial for utilities managing thousands of unpredictable devices. The researchers also quantify how much protection is lost if regulations force operators to use weaker control settings, helping regulators decide whether current standards are safe enough.

Electric vertical takeoff aircraft (eVTOLs) could replace helicopters and short-haul flights, but their complexity makes them dangerous to pilot. A control system that reduces pilot workload while maintaining performance is essential for these aircraft to become practical and safe for commercial use—especially as companies race to deploy air taxis in cities.

Engineers designing electrical circuits and other dynamical systems increasingly want to use geometric algebra because it naturally represents rotations and multidimensional transformations. Without a properly defined Laplace transform in this framework, they have to convert between different mathematical systems, losing efficiency and clarity. This work removes that barrier, letting engineers stay within geometric algebra from problem setup through solution.

These bounds establish what's theoretically possible for cooling systems, giving engineers and designers a precise target for optimization. Knowing these limits helps distinguish between genuinely impossible designs and those that simply haven't been found yet—making research and development efforts more efficient by ruling out dead ends.

This finding catches a widespread testing flaw that makes optimization algorithms look better than they are. When companies or researchers evaluate AI solvers this way, they publish fake improvements and waste effort optimizing something that doesn't work. The authors provide a correction procedure and checklist so future evaluations don't repeat this error — and show where real gains actually hide.

This result applies to real systems where you're forced to choose between letting something run blind or paying the cost to reset and observe it again—from equipment maintenance to sensor-based control. Knowing that the optimal choice always has this threshold structure means engineers can search for good policies much more efficiently instead of exploring countless complex strategies, and can confidently build control systems that follow intuitive restart-based rules.

Synchronized networks appear everywhere: power grids must stay in sync to avoid blackouts, groups of robots coordinate through networked communication, and biological systems like fireflies flashing together rely on synchronization. By determining which network structures sync fastest and most robustly, engineers can design more stable and responsive systems across these applications.

As renewable energy expands, grid operators face a real problem: the physical properties that kept old power systems stable are disappearing. Planning stability measures upfront instead of as an afterthought cuts costs and steers investment toward technology that works harder—like batteries that stabilize while they store. This directly affects how expensive it is to build grids that can reliably handle high renewable penetration.

Knot invariants are central to understanding knot theory, with applications ranging from DNA topology to quantum physics. This work provides the first systematic method for computing one important invariant across all complexity levels, making it possible to distinguish and classify knots more efficiently. The graph-theoretic approach also opens doors to computational implementations that could handle larger or more complex knots than previous methods allowed.

Graph coloring problems appear throughout computer science, from scheduling tasks to assigning radio frequencies and optimizing network designs. Understanding which graph properties stay stable under these coloring rules and which algorithms solve them efficiently helps researchers solve real-world optimization problems faster. The finding that b*-coloring becomes tractable on certain structured graphs could improve how we handle large-scale network problems where computation speed matters.

Future 5G and 6G networks rely on these smart surfaces to reach more users with less power, but current optimization methods are too slow to work at scale. Faster algorithms mean networks can automatically adjust these surfaces throughout the day, adapting to changing demand and coverage gaps—directly translating to better phone signals, lower energy costs, and more reliable service in more locations.

Egyptian fractions aren't just historical curiosities—they appear in computer algorithms, cryptography, and number theory. This proof settles a foundational question about whether the simplest approximation method always works, which establishes new bounds on how large denominators can grow when repeatedly breaking down fractions. The work also introduces an optimal control framework that other researchers can now apply to similar problems in discrete mathematics.

Understanding growth rates of polynomial operations matters for computer algebra systems and symbolic computation, where knowing worst-case complexity helps optimize algorithms. This result provides a provable ceiling on how quickly these operations can spiral in complexity, which improves estimates for computational cost and feasibility.

Rotating detonation engines could be significantly more efficient than conventional combustors, but engineers first need to predict and control their unpredictable wave behavior. These models provide a practical tool to understand what's happening inside the engine in real time, which is essential for tuning operating conditions and preventing unwanted vibrations or instability that could damage hardware.

Understanding how materials partition into regions with specific junctions applies to phase transitions in alloys, domain formation in magnets, and interface patterns in composite materials. By proving these junction configurations are mathematically stable rather than temporary, the work provides theoretical confidence that engineers and scientists can predict and design materials where multiple phases meet at controlled points.

As populations age globally, healthcare systems are drowning in complexity. Better logistics alone won't solve the problem—you need systems that connect hospital decisions to what happens when patients go home, coordinate multiple medications safely, and adjust care timing to patients' actual biological rhythms. Without closing this gap, hospitals will keep optimizing the wrong things while seniors fall through the cracks between departments.

Reinforcement learning typically demands thousands of interactions with a system before learning to control it well—prohibitively expensive for physical equipment like robots or aircraft. By embedding physics equations directly into the learning process, this method cuts the number of required trials dramatically, making AI control practical for expensive real-world systems where trial-and-error is costly or dangerous.

Factories, power plants, and infrastructure systems lose money both when machines fail unexpectedly and when they replace equipment too often. These algorithms let operators automatically tune maintenance schedules to their specific equipment based on what actually happens in the field, rather than guessing or using generic rules—potentially cutting total maintenance costs by 10–20% depending on the equipment and failure patterns.

Drones carrying sensors or packages need to visit specific locations efficiently — think inspecting power lines, surveying disaster zones, or making deliveries. Current planning methods ignore physics, creating routes that drones can't actually fly. This work bridges that gap, producing real trajectories that drones can follow, which matters for any mission where time and fuel are tight.

Weather forecasting, climate modeling, and other complex systems often measure things indirectly: a satellite might infer temperature from radiation, or a sensor might measure a combination of quantities rather than one thing directly. Standard filtering techniques fail in these cases. This method makes it feasible to improve forecasts in situations where current tools break down, potentially extending accurate prediction windows for weather and other dynamical systems that rely on difficult-to-interpret observations.

Optimization algorithms power everything from training neural networks to portfolio design, but engineers have long worked without knowing if their chosen method is efficient or wasteful. These bounds tell practitioners when stochastic approximation methods will work well and when the error rates are guaranteed tight—eliminating guesswork about algorithm quality.

These results resolve long-standing questions about equidistribution—how objects distribute evenly—in algebraic spaces that appear throughout modern number theory and representation theory. The exact error formulas allow mathematicians to move beyond just proving things converge 'eventually' and instead predict precisely how fast and where deviations occur, enabling more refined analysis of dynamical systems on these structures.

Trucking is a $800+ billion U.S. industry where millisecond-by-millisecond acceptance decisions directly affect profits and service reliability. A public benchmark lets companies and researchers test new bidding strategies without exposing proprietary fleet data, accelerating improvements that could reduce empty miles, lower shipping costs, and improve on-time delivery across the supply chain.

Cloud seeding and weather modification remain experimental, but this work shows a computationally practical way to steer real weather models toward specific precipitation outcomes. If scaled to operational forecasts, such methods could eventually help mitigate flooding or drought in water-stressed regions, though implementation would require careful environmental and policy frameworks before deployment.

Control systems steer everything from robots to aircraft, and understanding their information complexity helps engineers design more efficient controllers. This result corrects a long-standing misconception about which mathematical tools actually measure the same thing, preventing researchers from accidentally using interchangeable definitions that aren't interchangeable. It also reveals a new source of complexity that arises not from the system spreading apart dynamically, but from the geometry of the constraints themselves.

As grids add more wind and solar, they rely increasingly on these uncertain sources for emergency backup power. Setting reliability rules too high wastes money; too low risks blackouts. This work provides a concrete method to find the actual optimal threshold instead of guessing — potentially saving millions in grid costs across Europe's renewable-heavy systems while maintaining the same safety margin.

Eigenvalue optimization underlies real engineering problems: designing drums or membranes that vibrate at specific frequencies, tuning acoustic properties of rooms, and optimizing quantum systems. This work provides a practical tool for engineers and physicists to systematically find the best configurations without trial and error, while also proving that solutions actually exist — a guarantee that wasn't previously certain in all cases.

Many scientific and engineering problems require searching through billions of possible solutions—from designing power grids to optimizing industrial processes. Until now, researchers had to completely rewrite their code to move from testing on a personal computer to running on university supercomputers, wasting months on infrastructure work instead of actual problem-solving. This framework eliminates that friction, letting a researcher test an idea on their laptop Monday and scale to a supercomputer Thursday without touching the core code.

Many real optimization problems can only be queried through comparisons—ranking models, A/B testing, or noisy systems where you can compare outcomes but not measure them precisely. This algorithm makes it practical to find good solutions in those scenarios. The quantum version hints at how quantum computers might eventually offer speedups for real-world optimization beyond brute-force advantage.

Many real machine-learning algorithms pick or filter their data based on what they see, not randomly in advance. Without reliable guarantees for this setting, practitioners can't know whether their statistical conclusions are trustworthy. This work closes that gap, providing theoretical backing for algorithms that adaptively select subsets of data while maintaining provable recovery guarantees—particularly relevant for clustering tasks where picking the right groups is inherently data-dependent.

Many real-world optimization problems—from tuning industrial processes to training AI models with limited data—can't measure gradients directly and must contend with noisy measurements. This method makes high-dimensional optimization faster and more stable when noise is severe, without requiring expensive matrix calculations or gradient estimation that doesn't work reliably anyway.

Spacecraft currently rely on hand-coded control systems or policies developed through labor-intensive manual research. This framework could compress that development cycle while building in built-in verification that results are trustworthy—crucial for safety-critical aerospace applications where false confidence in a control system could end in collision or mission failure.

Robot-controlled labs waste time and resources repositioning equipment between every tiny parameter adjustment. CLUSTER reduces this waste by being smarter about which parameters to change together, cutting experiment time significantly. For labs running hundreds of optimization experiments—from drug discovery to materials science—this 50% speedup translates directly to faster results and lower costs.

Many optimization and engineering problems require finding a single matrix that bounds multiple others, but unlike ordering regular numbers, matrices often have no unique smallest upper bound. This algorithm provides a guaranteed way to find one, enabling faster and more precise solutions in quantum information processing, control systems design, and numerical computations that rely on comparing matrices in this specific way.

Hydrogen systems are being built now as part of the shift to renewable energy, but operators don't yet know how to run them efficiently. This framework provides a practical toolkit for predicting when to make hydrogen and when to sell it back to the grid, potentially reducing waste and improving revenue. The finding that machine learning uncovers real dynamics hidden from traditional statistics means energy operators need both approaches working together to actually optimize these systems.

Control engineers routinely work with nonlinear systems (robots, aircraft, power grids) that are hard to analyze and control. If a system can be Koopman linearized, standard linear control techniques become available, making design faster and more reliable. These geometric conditions let engineers quickly determine whether linearization will work for their specific system, avoiding wasted effort on impossible transformations.

Discovering the equations behind real-world systems — from weather patterns to turbulent flows to chemical reactions — often requires costly experiments or simulations. This approach could make equation discovery practical in fields where data collection is expensive or slow, allowing engineers and scientists to understand complex behavior with far fewer measurements. For systems where each experiment costs time or money, needing 5 measurements instead of 50 makes the difference between feasible and infeasible research.

Sparse signal recovery underpins everything from MRI scanners to compressed sensing applications where you need to reconstruct images or signals from far fewer measurements than classical theory says possible. By automating parameter tuning, this method gets to accurate answers faster without requiring engineers to hand-tune settings for each new problem—cutting computational time while maintaining accuracy in both clean and noisy data.

Sparse recovery is fundamental to medical imaging, radar, and data compression. Faster, more reliable algorithms mean clearer MRI scans with less radiation, better quality images from fewer measurements, and quicker processing of real-time signals in communications and sensing systems.

Ensemble Kalman Inversion is used across science and engineering to solve inverse problems—inferring unknown causes from observed effects—in fields like medical imaging, materials science, and climate modeling. By fixing its tendency to fail on certain problems, this improved version makes the method more reliable without requiring derivatives, which are often expensive or impossible to compute in real applications.

Policy makers constantly struggle to design incentives—carbon taxes, congestion pricing, welfare programs—without knowing exactly how people will respond or what constraints they face. This framework provides a principled way to adjust incentives over time as you learn, ensuring you don't waste resources on poorly-tuned policies while fumbling in the dark. It trades short-term exploration (slightly suboptimal incentives that reveal preferences) for long-term efficiency, with proven mathematical guarantees on how much welfare you'll recover.

Matrix nearness problems appear in signal processing, computer vision, and control systems—anywhere engineers need to find the closest match to data while respecting real-world constraints. This work makes it practical to solve versions of these problems that were previously unsolvable, expanding what's computationally feasible in applications from image compression to robotic control.

This work bridges two competing models for organizing complex algebraic objects, letting mathematicians working in different corners of the field understand they're studying the same underlying landscape. By revealing these hidden connections, it provides a unified foundation for higher-dimensional algebra—a framework that increasingly underpins applications from representation theory to mathematical physics.

Many real problems—from compressing high-dimensional data to fitting machine learning models—require finding the best solution subject to multiple geometric constraints that intersect in complex ways. This work removes a major computational barrier: instead of struggling with coupled constraints, practitioners can now use a straightforward algorithm with guaranteed convergence. This opens the door to faster, more reliable solutions in fields like signal processing, dimensionality reduction, and scientific computing.

As data centers consume more electricity, grids face real capacity limits. This approach lets data centers become grid helpers instead of problems—they can absorb power at off-peak times and reduce demand during crunch hours. For grid operators, that means deferring expensive infrastructure upgrades. For data center operators, it means lower bills and new revenue from selling grid services. Under tight grid conditions, the value compounds: the same battery system becomes twice as valuable simply because computation and storage work as a team.

Networks with these symmetries appear throughout engineering, physics, and computer science — from electrical grids to molecular structures to recommendation systems. Being able to verify optimal configurations algebraically instead of numerically means engineers can confidently design these systems without the computational bottlenecks and rounding errors that plague existing methods.

This could improve medical imaging (like ultrasound or tomography) where doctors need to identify internal boundaries or detect cavities without full access to the object. It also applies to materials testing and nondestructive inspection, where engineers need to locate internal flaws or structural features by measuring only from the surface. The constrained optimization approach makes the method more robust when real-world measurements contain errors.

This type of nested optimization appears in machine learning applications like training robust AI models that resist adversarial attacks, game-playing systems, and fairness-aware machine learning. Faster algorithms mean these systems can be trained in hours instead of days, making it practical to deploy protective techniques that were previously too slow to be useful in real applications.

Real-world decision systems need both speed and accountability. Energy traders need to know how their storage decisions respond to price swings; pension fund managers need to measure exposure across dozens of risk factors; pharmaceutical manufacturers must document how process changes affect product quality for regulators. SNAPO delivers these sensitivities during training rather than afterward, cutting computation time by orders of magnitude — sensitivity analysis that took hours now takes milliseconds — while keeping the same training budget. This makes AI-driven optimization practical for industries where understanding risk isn't optional.

Linear systems theory relies on these realizations to describe how systems behave, and the new algebraic proof works in characteristic 2 fields, which appear in coding theory and digital systems where all arithmetic happens modulo 2. The simpler method makes it practical to verify whether a given system has a valid mathematical representation without running complex algorithms, and also reveals new connections between realizability and field extensions that could inform future designs.

Standard portfolio advice assumes smooth, memoryless markets — assumptions that fail during crashes and volatility clusters. This work provides investors and fund managers with mathematically rigorous strategies tailored to real market behavior, potentially improving returns and risk management when applied to multi-asset portfolios.

Most classical optimization methods assume clean data and ideal conditions, but real-world problems involve measurement errors, uncertainty, and unexpected disturbances. This framework solves that problem by building robustness directly into the method, allowing engineers and scientists to find good solutions reliably in noisy, uncertain environments—from robotics to power systems to machine learning.

This theorem simplifies proofs across multiple network structures by eliminating the need to separately verify bounds under different sparsity conditions. Graph theorists and computer scientists studying network properties, coloring algorithms, and combinatorial optimization can now transfer known results between maximum-degree and average-degree settings, reducing redundant work and expanding what we know about when large independent sets must exist in sparse networks.

This completes a research program started years ago on matching problems in increasingly complex graphs. The methods used here — breaking down the problem by identifying which small matchings drive the minimum — create a template for solving similar extremal problems on other graph families, potentially accelerating progress on open questions in combinatorics.

This result helps engineers and mathematicians understand the boundaries between minimal rigidity and redundancy. In applications like robot design, mechanical linkages, and structural analysis, knowing exactly when a structure must be completely symmetric versus when it can be sparser tells engineers how much flexibility they have in their designs without sacrificing stability.

Error-correcting codes are fundamental to reliable data transmission—from cloud storage to wireless communications. These tighter bounds help engineers understand what's theoretically possible and avoid wasting resources searching for codes that don't exist, while the new optimization methods could improve the design of more efficient communication systems.