Conventional cardiovascular magnetic resonance (CMR) cine imaging relies on binning multiple heartbeats into a single cardiac cycle, which fails in arrhythmic patients where beat-to-beat variability causes motion artifacts and loss of functional information. Real-time 2D imaging captures individual beats but lacks volumetric coverage for mapping arrhythmic cardiac dynamics. We present a 3D real-time motion-field reconstruction method enabling continuous volumetric assessment in patients with premature ventricular contractions (PVCs) using a free-running CMR protocol. CMR-MOTUS was extended to jointly reconstruct real-time 3D motion fields and a motion-corrected reference image from continuous, ungated, non-breath-held data acquired with a variable-density Cartesian OPRA trajectory. Beat-to-beat ejection fraction (EF) was computed by propagating a single segmentation through all frames using the reconstructed motion fields. The method was validated on a cardiac motion phantom and tested in four healthy volunteers and four PVC patients. Phantom EF closely matched ground truth (22.1 +/- 0.6 percent vs. 21.9 percent). In healthy volunteers, EF values agreed with 2D references and showed narrow distributions reflecting physiological consistency. In PVC patients, EF distributions were bimodal, with the lower mode corresponding to PVC beats with markedly reduced EF. ECG confirmed alignment between EF irregularities and PVC episodes. These results show that 3D real-time motion-field reconstruction enables continuous beat-to-beat volumetric quantification in arrhythmia, revealing functional heterogeneity that conventional binning obscures. The bimodal EF distributions capture the true hemodynamic impact of PVCs and may provide clinically relevant metrics for monitoring and treatment evaluation.

In this note, we test the performance of six algorithms from the family of graph-based splitting methods [SIAM J. Optim., 34 (2024), pp. 1569-1594] specialized to normal cones of linear subspaces. To do this, we first implement some numerical experiments to determine the best relaxation parameter for each algorithm. Then, we compare the number of iterations each algorithm requires to reach a given stopping criterion, using the previously identified best relaxation parameter. The numerical results allow us to identify some relevant patterns and provide numerical evidence that may guide further theoretical analysis.

We investigate the impact of including a dynamical charm quark on the properties of light hadrons. Our study uses gauge ensembles generated with the tadpole-improved Symanzik gauge action, comparing 2+1+1 flavor (HISQ fermion) ensembles at four lattice spacings to 2+1 flavor (clover fermion) ensembles at six lattice spacings. For the light and strange flavor observables, we employ the same tadpole-improved clover fermion action. From numerical results for light and strange quark masses, pion and kaon decay constants, and $Ω$ baryon masses, we find that the values obtained after continuum, chiral, and infinite-volume extrapolations are consistent within uncertainties. Even though the mixed action setup can introduce additional discretization effects, our calculation shows evidences that those effects can cancel with the discretization error in the unitary setup, resulting in better convergence in the continuum extrapolation.

We study paths in the p-Bratteli diagram associated with hook partitions, where p is an odd prime. By comparing blocks along a path, we define inversions and descents. We prove that the sign balance derived from inversions vanishes at every vertex of the diagram. Using descents, we introduce the p^(k)-Fibonacci numbers and derive recurrence relations for them. For k=0, we recover the OEIS sequence A391520, while for k>=1 we obtain new families of Fibonacci-type sequences.

Machine learned interatomic potentials (MLIPs) have enabled atomistic simulations with ab initio accuracy for a fraction of the computational cost. However, many widely used MLIPs are short-ranged and do not accurately capture long-ranged electrostatic interactions. At interfaces with polar solvents, such as water, this deficiency can drive unphysical long-distance dipolar alignment far away from the interface. Here we reveal that neglecting long-ranged physics leads to spurious metallization of the water layer due to artificially large fluctuations of the total solvent dipole, similar to the electron rearrangement observed to prevent polar catastrophes at polar interfaces. This metallization is eliminated in MLIPs that explicitly include long-ranged electrostatics. Our results showcase a fundamental flaw of short-ranged MLIPs, highlighting that long-ranged electrostatics are essential for studying systems with a polar-liquid component, especially if one is interested in electronic properties.

Optimizing reranking in advertising feeds is a constrained combinatorial problem, requiring simultaneous maximization of platform revenue and preservation of user experience. Recent generative ranking methods enable listwise optimization via autoregressive decoding, but their deployment is hindered by high inference latency and limited constraint handling. We propose a constraint-aware generative reranking framework that transforms constrained optimization into bounded neural decoding. Unlike prior approaches that separate generator and evaluator models, our framework unifies sequence generation and reward estimation into a single network. We further introduce constraint-aware reward pruning, integrating constraint satisfaction directly into decoding to efficiently generate optimal sequences. Experiments on large-scale industrial feeds and online A/B tests show that our method improves revenue and user engagement while meeting strict latency requirements, providing an efficient neural solution for constrained listwise optimization.

We study infinite-horizon Markov decision processes (MDPs) where the decision maker evaluates each of her strategies by aggregating the infinite stream of expected stage-rewards. The crucial feature of our approach is that the aggregation is performed by means of a given diffuse charge (a diffuse finitely additive probability measure) on the set of stages. The results of Neyman [2023] imply that in this setting, in every MDP with finite state and action spaces, the decision maker has a pure optimal strategy as long as the diffuse charge satisfies the time value of money principle. His result raises the question of existence of an optimal strategy without additional assumptions on the aggregation charge. We answer this question in the negative with a counterexample. With a delicately constructed aggregation charge, the MDP has no optimal strategy at all, neither pure nor randomized.

The widespread adoption of the Internet of Things (IoT) has positioned smart homes as paradigmatic examples of distributed automation systems, where reliability, efficiency, and interoperability depend critically on the underlying communication protocol. Among the low-power wireless technologies available for this scenario, Zigbee and Matter over Thread have emerged as leading contenders. While Zigbee represents a mature, non-IP mesh networking solution, Matter over Thread introduces an IP-based architecture designed to unify device interoperability across different ecosystems. However, despite extensive documentation of their design principles, there is a lack of empirical, comparative performance data under realistic network conditions. This paper presents a comprehensive experimental comparison between the two protocols, conducted on a testbed built from commercially available hardware. The proposed methodology focuses on different key performance dimensions, such as scalability, responsiveness, and fault tolerance. The results reveal that Zigbee achieves a lower baseline overhead and faster route recovery, making it more responsive in static small-scale deployments. Matter over Thread, conversely, exhibits superior scalability and robustness, maintaining stable throughput and predictable latency across multi-hop scenarios. Overall, we highlight that Zigbee and Matter over Thread embody distinct trade-offs between agility, efficiency, and scalability.

Motivated by recent experimental realizations of programmable spin models with long-range interactions, we investigate the non-equilibrium dynamics of the Power-of-Two (PWR2) model. This model consists of sparse long-range couplings between spin-$1/2$ objects separated by $d = 2^n$. In the absence of disorder, the system exhibits rapid scrambling and fast thermalization. We explore the impact of disorder in this system by analyzing the time evolution of the survival probability, half-chain entanglement entropy, and out-of-time-ordered correlators (OTOCs). We find that sufficiently strong disorder suppresses information spreading and induces localization. Remarkably, in the strong-disorder regime, the OTOCs display a non-monotonic spatial profile arising from the intrinsic nonlocality of the interactions, signaling qualitatively distinct scrambling dynamics compared to conventional long-range interacting systems. To characterize the localization transition, we extract the critical disorder strength $h_c$ from the spectral statistics and the eigenstate entanglement. We observe that $h_c$ increases with system size. Furthermore, at a fixed disorder strength, the eigenstate-averaged entanglement entropy increases with system size, while the inverse participation ratio decreases, indicating enhanced delocalization at larger sizes. These results collectively suggest that the PWR2 model remains ergodic in the thermodynamic limit for any finite disorder strength.

Arrow patterns were introduced by Berman and Tenner as a generalization of vincular patterns. They observed that arrow patterns have the potential to bridge the divide between a permutation's cycle notation and its one-line notation; in support of this, they used arrow avoidance to enumerate shallow and cyclic shallow permutations. More recently, $321$-avoiding cyclic permutations were recharacterized entirely in terms of arrow avoidance. Motivated by these results, we initiate a systematic study of arrow avoidance. In this paper, we prove structural results about arrow patterns, including defining arrow-Wilf equivalence, and enumerate several arrow avoidance classes. Finally, we consider the avoidance of pairs of arrow patterns, focusing on cases that prohibit fixed points in the underlying permutation.

We develop a statistically robust framework for reconstructing metal--semiconductor contact regions using topological gradients. The inverse problem is formulated as the identification of an unknown contact region from boundary measurements governed by an elliptic model with piecewise coefficients. Deterministic stability of the topological gradient with respect to measurement noise is established, and the analysis is extended to a statistical setting with multiple independent observations. A central limit theorem in a separable Hilbert space is proved for the empirical topological gradient, yielding optimal $n^{-1/2}$ convergence and enabling the construction of confidence intervals and hypothesis tests for contact detection. To further refine the reconstruction, a shape optimization procedure is employed, where the free parameter $β$ in the CCBM formulation plays a crucial role in controlling interface sensitivity. While $β$ affects both topological and shape reconstructions, its influence is particularly pronounced in the shape optimization stage, allowing more accurate estimation of the size and geometry of the contact subregion. The proposed approach provides a rigorous criterion for distinguishing true structural features from noise-induced artifacts, and numerical experiments demonstrate the robustness, precision, and enhanced performance of the combined statistical, topological, and $β$-informed shape-based reconstruction.

We introduce a taxonomy of interaction types and show that graphs are focal hypergraphs: every graph is canonically a focal hypergraph via its closed neighbourhood structure, and every graph dynamical model is a special case of the general hypergraph dynamical model. The central distinction is between \emph{focal} interactions, in which the interaction domain is defined relative to a designated reference node, and \emph{non-focal} interactions, in which all participants stand in equivalent structural relationship. Closed graph neighbourhoods are precisely focal hyperedges, so hyperedges generalise graph neighbourhoods by removing the focal constraint. This yields a strict three-level hierarchy: graph models $\subsetneq$ focal hypergraph models $\subsetneq$ general hypergraph models. Moreover, graph models do encode genuinely higher-order (many-body) interactions, in the sense that each node's update function may depend jointly on all members of its closed neighbourhood, but they remain a strict special case of the hypergraph dynamical model, not equivalent to it. We further show that universal encodings such as bipartite factor graphs are neutral with respect to this hierarchy, and that the symmetry condition of the hypergraph dynamical model -- often treated as an additional constraint relative to the graph model -- is in fact the dynamical definition of a non-focal interaction. The taxonomy is grounded in concrete phenomena from physics, biology, ecology, and social systems, and yields a principle of representational alignment: the choice between graph and hypergraph models should be governed by the type of interaction, not by a blanket preference for one formalism over the other.

We investigate the relativistic scattering of spinning black holes using modern amplitude methods within a heavy-mass effective field theory formalism at third post-Minkowskian order. Using a systematic self-force expansion up to first order in the mass ratio, the gravitational amplitude and the associated eikonal-like phase are computed for a spin-aligned binary system comprising a heavy and a light black hole up to fifth order in the total spin and up to quadratic order in the spin of the light black hole. We also consider the resummation of the heavy black hole's spin in both the probe limit and the radiation-reaction sector, and verify that the resulting phase displays the characteristic ring singularity features associated with the Kerr metric.

Coherent control of motional degrees of freedom of ultracold atoms in optical lattices offers a promising route towards programmable quantum dynamics with massive particles. We propose and analyze a scheme for implementing coherent rearrangement of ultracold atoms, corresponding to arbitrary unitary transformations on single-particle motional states. Exploiting an analogy between dynamics in optical superlattices and discrete linear optics, we employ the Clements scheme to systematically construct any global $N$-dimensional single-particle unitary from tunneling and phase shifts in arrays of double wells. Tunneling is controlled globally, while local operations are achieved through site-resolved potential shifts. We numerically investigate the susceptibility of the scheme to intensity noise and addressing crosstalk. We identify key subroutines enabled by this unitary construction, including the Discrete Fourier Transform and the implementation of non-native Hamiltonians. Extending the scheme to two dimensions enables all-to-all atomic rearrangement with a circuit depth that scales sublinearly with the atom number, providing a high-density and highly scalable approach to atom rearrangement.

We consider a generalization of the Kuramoto model in which phase oscillators are represented by unit vectors coupled by a matrix of constant coefficients. We show that, when the oscillators are driven by an external periodic force, several resonances appear, giving rise to Arnold tongues that can be observed as the intensity and frequency of the external force are varied. Applying the Ott-Antonsen ansatz we obtain equations for the module and phase of order parameter. As these equations are explicitly time-dependent, we resort to extensive numerical simulations to uncover the resonant modes and their associated Arnold tongues and devil's staircases. These results contrast with the original forced Kuramoto model, where only $1:1$ resonance is possible.

Surface runoff shapes planetary landscapes, but global hydrological models often lack the resolution and flexibility to simulate dynamic surface water bodies beyond Earth. Recent studies of Mars have revealed abundant geological and mineralogical evidence for past surface water, including valley networks, crater lakes, deltas and possible ocean margins dating from late Noachian to early Hesperian times. These features suggest that early Mars experienced periods allowing liquid water stability, runoff and sediment transport. To investigate where surface water could accumulate and how it may have been redistributed, we developed a global high-resolution (km-scale) surface hydrological model. The model uses a pre-computed hydrological database that maps topographic depressions, their spillover points, hierarchical connections between basins, and lake volume-area-elevation relationships. This database approach greatly accelerates simulations by avoiding repeated geomorphic processing. The model dynamically forms, grows, merges and dries lakes and putative seas without prescribing fixed coastlines, by transferring water volumes between depressions according to their storage capacities and overflow rules. We explore model behavior over the present-day Mars' topography measured by MOLA (Mars Orbiter Laser Altimeter) topography for a range of evaporation rates (from 0.1 m/yr to 10 m/yr) and total water inventories expressed as Global Equivalent Layer (from 1 mGEL to 1000 mGEL). 48 Simulations are iterated to reach the steady state. The model outputs the extent and depth of surface water bodies and identifies main drainage pathways using overflow fluxes as runoff indicators. Results show a transition toward a contiguous northern ocean between low (1-10 m) GEL values and increasing concentration of water in northern lowlands and major impact basins at higher GEL.

The classical Grunwald--Wang theorem asserts that, unless we are in the so-called special case, local cyclic Galois extensions at finitely many completions of a number field can be approximated by a global cyclic extension. In the special case the obstruction is measured by a group of order 2. It has been known for a long time that the Grunwald--Wang theorem extends to a very general context of valued fields. Therefore it is natural to ask whether in the special case the obstruction is always measured by a finite group and if so, is the order of this group bounded independently of the number of places under consideration. We show that the answer to both questions is negative in general, already for rational function fields and discrete valuations coming from points of the affine line. This has some interesting links to the arithmetic of function fields over Q or Q_p.

Collimation systems are essential in particle accelerators to safely and efficiently manage unavoidable beam losses during operation. These systems rely on collimators, which are specially designed movable jaws or absorbers positioned close to the beam envelope to intercept and localize beam losses. Their role is particularly critical in high-intensity hadron machines, where uncontrolled losses can lead to equipment damage or operational downtime. While the specific requirements vary across accelerator types, circular accelerators, especially present and future high-energy colliders, cannot operate safely without a well-optimized collimation system. This lecture offers an overview of the fundamental principles, design challenges and operational strategies of beam collimation, with emphasis on high-intensity hadron accelerators. The Large Hadron Collider, the most advanced example to date, will serve as the main reference for illustrating state-of-the-art collimation approaches and technologies.

Coupled beam optics can be geometrically described in terms of invariant eigenmode planes of a stable symplectic ``one-turn'' map $\mathcal M\in Sp(4)$. We show that the non-uniqueness of symplectically normalized bases within each eigenmode plane constitutes an in-plane gauge freedom $Sp(2)\times Sp(2)$, and that many coupled-optics parametrizations differ primarily by gauge choice. Building on this fact, we identify basis-independent descriptors of lattice and beam optics and introduce bounded, gauge-invariant coupling parameters or fractions $u_{k,\mathrm{inv}}$ computed from orthogonal projectors onto the eigenmode planes. To obtain smooth $s$-dependent optics functions and consistent mode labeling, we present a unifying and practical approach based on an $SO(2)$ continuity gauge (Procrustes alignment), together with diagnostics for stability and invariance. We further relate Edwards--Teng, Mais--Ripken, Lebedev--Bogacz, Wolski, and Sagan--Rubin parametrizations as gauge-equivalent representations within the respective $Sp(2)\times Sp(2)$ gauge freedom. Numerical examples of coupled lattices and beam optics illustrate the proposed invariants and show how representation-dependent scalar coupling parameters (e.g.\ in the Lebedev--Bogacz gauge) can leave their nominal bounds while $u_{k,\mathrm{inv}}$, defined here, remain bounded and physically interpretable.

V$_2$O$_5$ is a promising battery electrode material that can intercalate not only Li, but also more abundant alkaline metals such as Na and K, and even multivalent ions such as Al, Ca, Cu, Mg, and Zn. V$_2$O$_5$ exhibits several different polymorphs, and phase transitions between the polymorphs can occur depending on intercalant or external conditions. At least 8 different layered polymorphs have been observed. However, detailed information about the energetics and structural properties of each polymorph is still lacking. To obtain a reliable computational reference, we use hybrid density functional theory calculations to investigate the properties of layered V$_2$O$_5$ polymorphs. We benchmarked several methods to include van der Waals interactions in combination with hybrid functionals, and found that the Grimme D3 method is most accurate. We obtain detailed information on the electronic properties and structures of the various unintercalated polymorphs and show that the main electronic effect of intercalants is a filling of the lowest conduction bands, as the intercalant contributions are well above the conduction-band minimum. Despite the structural differences between the unintercalated polymorphs, we find that they have very similar band gaps and band structures, with the exception of the high temperature and pressure phase $β$.

We prove a converse Lyapunov theorem for boundedness of reachability sets for a general class of control systems whose flow is Lipschitz continuous on compact intervals with respect to trajectory-dominated inputs. We show that this condition is satisfied by many semi-linear evolution equations. For ordinary differential equations, as a consequence of our results, we obtain a converse Lyapunov theorem for forward completeness, without a priori restrictions on the magnitude of inputs.

This work presents a general framework for the operationally driven optimal siting and sizing of battery energy storage systems in power transmission networks, aimed at enhancing their resource adequacy. The approach considers multi-period planning horizons, enforces network constraints at high temporal resolution, and targets large-scale meshed systems. The resulting computationally complex mixed-integer non-linear programming problem is reformulated as a mixed-integer second-order cone programming problem and solved via Generalized Benders Decomposition, with feasibility cuts enabling congestion management and voltage regulation under binding network limits. A tailored heuristic recovers an alternating-current power-flow-feasible operating point from the relaxed solution. The proposed formulation is parallelizable, yielding excellent computational performance, while featuring rigorous guarantees of convergence.

Epistemic uncertainty arises in lack of complete knowledge about the state of a system. There are multiple mathematical frameworks for measuring such uncertainty quantitatively, often referred to as imprecise probability theories. Inspired by work of Opdan, we introduce a general category theoretic definition of epistemic calculi, which we use as a foundation for modelling and studying contradictions and synergies between several philosophical epistemological concepts. We further develop an enriched category theoretic process for changing calculi, and use this to study relationships between existing examples, like possibility theory and certainty factors. Finally, we introduce a general categorical form of belief updating based on change of enrichment, and prove that Bayesian updating and possibilistic conditioning arise as examples.

In this paper, we study evolutive Hamilton Jacobi equations with Hamiltonians that are discontinuous in time, posed on a simple network consisting of two edges on the real line connected at a single junction. We introduce a notion of (flux-limited) viscosity solution for Hamiltonians H=H(t,x,p) that are assumed to be only measurable in time t. The flux limiter, A=A(t), acting at the junction, is not required to be continuous but only in L infinity. In the case of convex Hamiltonians, we prove a comparison principle and establish an existence result via the construction of an optimal control problem. Generalisations to the nonconvex case and to more general networks are also discussed.

Context graphs are essential for modern AI applications including question answering, pattern discovery, and data analysis. Building accurate context graphs from structured databases requires inferring join relationships between entities. Invalid joins introduce ambiguity and duplicate records, compromising graph quality. We present a scalable join inference approach combining statistical pruning with Large Language Model (LLM) reasoning. Unlike purely statistics-based methods, our hybrid approach mimics human semantic understanding while mitigating LLM hallucination through data-driven inference. We first identify primary key candidates and use LLMs for adjudication, then detect inclusion dependencies with the same two-stage process. This statistics-LLM combination scales to large schemas while maintaining accuracy and minimizing false positives. We further leverage the database query history to refine the join inferences over time as the query workloads evolve. Our evaluation on TPC-DS, TPC-H, BIRD-Dev, and production workloads demonstrates that the approach achieves high precision (78-100%) on well-structured schemas, while highlighting the inherent difficulty of join discovery in poorly normalized settings.

A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of tournaments, it is well known that every tournament contains a transitive subgraph of size $\log n$, and that this is best possible up to a constant factor. A restricted family of tournaments that has been extensively studied is the family of $k$-majority tournaments. They are obtained by taking $2k-1$ linear orders of a set $X$, and defining a tournament on $X$ which has an edge from $u$ to $v$ if $u$ precedes $v$ in at least $k$ of these orders. Milans, Schreiber, and West proved that such tournaments indeed have significantly larger transitive tournaments. More precisely, they proved that every $k$-majority tournament contains a transitive tournament of size $n^{2^{-Θ(k)}}$. Our main goal in this paper is to give an exponential improvement in the dependence of the exponent on $k$ by showing that every $k$-majority tournament contains a transitive set of size $n^{Ω(1/k)}$. Finally, we highlight several open problems and conjectural directions related to random $k$-majority tournaments.

We study how to allocate resources to participants who can strategically misrepresent their deservingness at a cost. A principal assigns item(s) (or money) among multiple agents on the basis of their costly signals. Each agent's signal reflects their private type in the absence of misrepresentation but can be inflated above their true type at a cost. The principal is a social planner who aims to maximize the weighted average of matching efficiency and a utilitarian objective. Strategic misrepresentation introduces novel incentive-compatibility constraints, under which we characterize the optimal mechanism. We apply our characterization to two kinds of markets, distinguished by resource scarcity, and show that the principal strictly benefits from randomizing the allocations based on costly signals when the population of participants is large enough. Interestingly, in large markets with scarce resources, the format of the optimal mechanism converges to a winner-takes-all contest; however, there is a non-diminishing value in randomizing allocations to middle types as the population of participants grows.

One of the three common experimental channels searching for charged lepton flavor violation (CLFV) via $μ-e$ conversion involves first capturing the muon on a nuclear target. This channel provides a unique opportunity to constrain four-point quark-lepton operators that may contribute to CLFV, in addition to the electromagnetic operators that can also be accessed in the other channels. We investigate the leading Lorentz- and CPT-violating contributions for nuclear $μ-e$ conversion experiments within the Standard-Model Extension and obtain the first bounds on the relevant quark-lepton operators using the results of the SINDRUM II experiment; we also consider possibilities for improved constraints from upcoming experiments.

The increasingly complex Web3 ecosystem and decentralized finance (DeFi) landscape demand ever higher levels of technical expertise and financial literacy from participants. The Intent-Centric paradigm in DeFi has thus emerged in response, which allows users to focus on their trading intents rather than the underlying execution details. However, existing approaches, including Typed-intent design and LLM-driven solver, trade off expressiveness, trust, privacy, and composability. We present OMNIINTENT, a language-runtime co-design that reconciles these requirements. OMNIINTENT introduces ICL, a domain-specific Intent-Centric Language for precise yet flexible specification of triggers, actions, and runtime constraints; a Trusted Execution Environment (TEE)-based compiler that compiles intents into signed, state-bound transactions inside an enclave; and an execution optimizer that constructs transaction dependency graphs for safe parallel batch submission and a mempool-aware feasibility checker that predicts execution outcomes. Our full-stack prototype processes diverse DeFi scenarios, achieving 89.6% intent coverage, up to 7.3x throughput speedup via parallel execution, and feasibility-prediction accuracy up to 99.2% with low latency.

One of the emerging problems in algebraic geometry is to characterize the affine $n$-space $\mathbb{A}^n$ among smooth affine schemes up to $\mathbb{A}^1$-contractibility. Recent efforts show that this characterization holds in dimensions $n<3$ over certain fields. In this thesis, we extend this observation to "reasonably" arbitrary base schemes in relative dimensions $d<3$, exploiting the Zariski local triviality and the triviality of the sheaf of relative differentials. From dimensions $n\geq 3$, the existence of smooth "exotic" affine schemes - those that are $\mathbb{A}^1$-contractible but not isomorphic to the affine $n$-space - has already been established. A well-studied family constitutes the Koras-Russell threefolds $\mathcal{K}$ and their higher-dimensional prototypes $\mathcal{X}_n$, whose $\mathbb{A}^1$-contractibility has been so far proven over fields of characteristic zero. Here, we extend the relative $\mathbb{A}^1$-contractibility of $\mathcal{K}$ and $\mathcal{X}_n$ over a Noetherian base scheme in arbitrary dimensions. Then, using these prototypes, we study the existence of "exotic spheres" - $n$-dimensional smooth schemes that are $\mathbb{A}^1$-homotopic, but not isomorphic to $\mathbb{A}^n \backslash \{0\}$ - in motivic homotopy theory. This result can be seen as the "compact" analog of the study of exotic affine schemes. Our main result shows that in all dimensions $n\geq 4$, the quasi-affine varieties $\mathcal{X}_n \backslash \{\bullet\}$ give a model for the exotic motivic spheres over infinite perfect fields. The novelty is that these constitute the first family of examples of smooth motivic spheres of dimension $n$, which are not isomorphic to $\mathbb{A}^n \backslash \{0\}$.