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Counting ideals in abelian number fields
No full textAlready Dedekind and Weber considered the problem of counting integral ideals of norm at most x in a given number field K. Here we improve on the existing results in case K/Q is abelian and has degree at least four. For these fields, we obtain as a consequence an improvement of the available results on counting pairs of coprime ideals each having norm at most x
Formal multiple Eisenstein series and their derivations
Full text linkWe introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl-Kaneko-Zagier as a hybrid of classical Eisenstein series and multiple zeta values. In depth one, we obtain formal versions of the Eisenstein series satisfying the same algebraic relations as the classical Eisenstein series. In particular, they generate an algebra whose elements we call formal quasimodular forms. We show that the algebra of formal multiple Eisenstein series is an sl2-algebra by formalizing the usual derivations for quasimodular forms and extending them naturally to the whole algebra. Additionally, we introduce some families of derivations for general quasi-shuffle algebras, providing a broader context for these derivations. Further, we prove that a quotient of this algebra is isomorphic to the algebra of formal multiple zeta values. This gives a novel and purely formal approach to classical (quasi)modular forms and builds a new link between (formal) multiple zeta values and modular forms
Multi-plateau high-harmonic generation in liquids driven by off-site recombination
Full text linkNon-perturbative high-harmonic generation has recently been observed in the liquid phase, and the underlying mechanism was shown to be different from that in gases and solids. Liquid-phase high-harmonic generation is currently understood in terms of a recollision mechanism with electron trajectories limited by electron scattering. The cut-off energy and its independence of the driving laser parameters are reproduced by this mechanism. However, when the driving laser intensity is increased, no extension of the cut-off energy is observed, which contrasts with the general expectations from most nonlinear media. Here we observe the appearance of a second plateau in high-harmonic generation from multiple liquids (water, heavy water, propanol and ethanol) and explore its origin. From the combined analysis of experimental, computational and theoretical results, we find that electrons recombining at neighbouring molecular sites instead of the ionization site are responsible and verify this feature through the characteristic dependence of the second-plateau yield on the ellipticity of the driving field. We find that the second plateau is dominated by electrons recombining at the first or second solvation shell, relying on hole delocalization. Theoretical results predict the appearance of yet higher plateaus, indicating a general trend. Our work establishes a previously unexplored physical phenomenon in the highly nonlinear optical response of liquid
Generalization of Optimal Control Saturation Pulse Design for Robust and High CEST Contrast
No full textPurpose: Optimal control (OC) chemical exchange saturation transfer (CEST) pulse design for singular pulses that can be used flexibly and robustly with high saturation at different duty cycles, saturation durations, and magnetic field strengths.Theory and methods: An OC framework was developed to design a single-pulse shape that can be flexibly applied for arbitrary pulse train parameters and outperform typically used CEST saturation pulse shapes. The pulse design was developed primarily with a continuous-wave (CW) spectrum as the optimization target, but can be easily adapted to specific scenarios. The generalized OC pulse was evaluated through simulations, a phantom, and in vivo measurements on a 3 T clinical scanner. Performance was assessed in terms of contrast, robustness to field inhomogeneities, and resilience against artifacts such as Rabi oscillations and sidebands, compared to established saturation techniques.Results: Investigations showed that the generalized OC pulse achieved a contrast matching CW saturation and also functioned well under field inhomogeneities. Low-pass filtering of the optimized pulse shape effectively suppressed artifacts outside the initial optimization frequency range, enabling generalization across different field strengths. Phantom experiments consistently showed higher contrast than Gaussian, Fermi, and adiabatic spin-lock (aSL) pulses for various CEST agents covering most clinically relevant regimes. In vivo imaging demonstrated substantially enhanced CEST contrast for both creatine/phosphocreatine in muscle and amide proton transfer (APT) in the brain compared to Gaussian saturation.Conclusion: The generalized OC pulse provides a robust and flexible alternative to conventional CEST saturation strategies. Its integration into the open-source Pulseq-CEST framework supports simple reproducibility and a vendor-independent implementation
Supramolecular engineering of high-efficiency nanozymes via chain-length-directed crystallization of cellulose oligomers
No full textNanozymes are nanomaterials designed to mimic the catalytic functions of natural enzymes, offering advantages such as enhanced stability, tunability, and scalability. Although precise control over the spatial arrangement of catalytic centers is essential for maximizing nanozyme activity, it remains a fundamental challenge in nanozyme design. Here, we present a supramolecular strategy to achieve molecular-level engineering of catalytic centers by grafting hemin onto monodisperse cellulose oligomers (MCOs). The crystallization-driven self-assembly of MCOs directs the spatial organization of hemin while preventing its detrimental aggregation. Systematically tuning of the cellulose chain length reveals that a degree of polymerization (DP) of 6 optimally balances supramolecular packing and increased availability of active sites, whereas shorter (DP1) and longer (DP20) chains compromise catalytic performance due to aggregation or decreased substrate binding affinity. Structural analyses reveal that chain-length-directed crystallization governs nanozyme morphology, aggregation behavior, and catalytic performance. Through this approach, we achieve an approximately 500-fold enhancement in catalytic efficiency over free hemin, while structural analyses elucidate the role of chain-length-dependent crystallization in governing nanozyme morphology and performance. This study establishes transferable cellulose-based supramolecular strategy for engineering high-performance nanozymes with broad applicability across diverse catalytic systems
Long-range input to cortical microcircuits shapes EEG-BOLD correlation
Full text linkAlthough generated by different mechanisms, electroen-cephalography (EEG) rhythms and blood-oxygen-level-dependent (BOLD) activity have been shown to be correlated. The level of correlation varies between EEG frequency bands, brain regions and experimental paradigms, but the underpinning mechanisms of this correlation remain poorly understood. Here we create a mathematical, data-informed model of a cortical microcircuit that encompasses all major neuron types across cortical layers, and use it to generate EEG and BOLD under various external input conditions. The model exhibits noise-driven fluctuations mimicking EEG rhythms, with external inputs modulating EEG spectral characteristics. Consistent with experimental results, we observe negative alpha-BOLD correlation and positive gamma-BOLD correlation across different input configurations. Temporal variability of the input is found to increase EEG-BOLD correlation and to improve the correspondence with experimental results. This study provides a mathematical framework to theoretically study the correlation of EEG and BOLD features in a comprehensive way
