In this paper, we consider the Cauchy problem for the rod equation in the line. By constructing an explicit smooth initial data, we present a new method to prove that this problem is ill-posed in $H^s(\R)$ with $1< s<3/2$ in the sense of {\it norm inflation}, i.e., an initial data is smooth and arbitrarily small in $H^s(\R)$ with $1< s<3/2$, but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.

The efficient execution of fault-tolerant quantum algorithms is fundamentally limited by the production rate of magic states required for non-Clifford operations. While circuit optimization typically targets T-depth, static T-depth does not reliably predict executable performance under bounded T-state delivery. We introduce a model that captures demand-supply imbalance using two key quantities: slack ratio, a structural indicator of scheduling flexibility, and Delta_max, a measure of cumulative demand surplus. We show that Delta_max is a strong schedule-level indicator of execution slowdown and yields a provable lower bound on executable makespan for a fixed schedule. Empirical evaluation on constructed directed acyclic graph (DAG) families, with arithmetic circuits and exact quantum Fourier transform (QFT) traces providing additional grounding, shows that slack ratio is a stronger structural predictor than T-depth for stall and inversion risk, while Delta_max is the strongest predictor of slowdown. Across 4,904 instances, the lower bound shows zero violations, with 88.9% of cases within one cycle. These results highlight the importance of explicitly modeling delivery constraints in fault-tolerant quantum compilation.

Let $\mathfrak{g}$ be a simple Lie algebra. Frenkel and Reshetikhin introduced the deformed $W$-algebra $\mathbf{W}_{qt}(\mathfrak{g})$. In this work, we propose a formal reformulation of this definition in a different context. In this framework, we reformulate and prove the well-definedness of an algorithm (arxiv:2103.15247, arxiv:2205.08312) inspired by the Frenkel-Mukhin algorithm (arXiv:math/9911112) which, starting from a given dominant monomial $m$ satisfying some degree conditions, produces elements of the deformed $W$-algebra. Then, we apply this algorithm to construct explicitly some specific elements of $\mathbf{W}_{q,t}(\mathfrak{g})$. In particular, we apply this to prove a conjecture of Frenkel and Reshetikhin in arXiv:q-alg/9708006 in types $B_\ell$, $C_\ell$, and for some nodes in other types. This framework opens up new possibilities for studying explicitly fields in the deformed $W$-algebra $\mathbf{W}_{q,t}(\mathfrak{g})$.

We prove that the occupation measures of Brownian motions conditioned to have large intersections converge weakly, up to spatial shifts, to the measure whose density is the square of an optimizer of the Gagliardo-Nirenberg inequality. We do so by proving a large deviation principle (LDP) for Brownian occupation measures conditioned either on large self-intersections or large mutual intersections. To this end, we derive a compact LDP for unconditioned Brownian occupation measures, generalizing the work of Mukherjee and Varadhan. We also prove the LDP for Brownian occupation measures tilted by their intersections in the same topology. A key tool of independent interest is an exponentially good approximation of the intersection measure tested against all bounded measurable functions, from which we further get the LDP for the intersection measure of independent Brownian motions.

DAG-Rider popularized a new paradigm of DAG-BFT protocols, separating dissemination from consensus: all nodes disseminate transactions as blocks that reference previously known blocks, while consensus is reached by electing certain blocks as leaders. This design yields high throughput but confers optimal latency only to leader blocks; non-leader blocks cannot be committed independently. We present Lemonshark, an asynchronous DAG-BFT protocol that reinterprets the DAG at a transactional level and identifies conditions where commitment is sufficient -- but not necessary -- for safe results, enabling nodes to finalize transactions before official commitment, without compromising correctness. Compared to the state-of-the-art asynchronous BFT protocol, Lemonshark reduces latency by up to 65\%.

Gauss--Bonnet inflation typically affects the dynamics over an extended portion of the trajectory, making it difficult to isolate a controlled imprint at CMB scales. We consider a trajectory-selective coupling \(μ(ϕ,X)\) that gates the Gauss--Bonnet sector in phase space, enabling the higher-curvature contribution to be localized within a finite e-fold window while remaining negligible elsewhere. We identify stable inflationary solutions consistent with this localization and enforce standard ghost and gradient stability conditions for both scalar and tensor perturbations. For these viable backgrounds we compute pivot-scale observables and examine their dependence on the overall Gauss--Bonnet strength and on the kinetic gating. The framework offers a controlled route for realizing localized higher-curvature effects with predictable consequences for CMB-scale measurements.

The propagation of light through a turbulent flow field around an aircraft results in optical distortions commonly known as aero-optic effects. The development of methods to mitigate these effects requires large amounts of realistic aero-optic data. However, methods for obtaining this data, including experiment, computational fluid dynamics, and simple phase screen algorithms (e.g., boiling flow), each have significant drawbacks such as high cost, high computation, limited quantity, and/or inaccurate statistics. More recently, data-driven algorithms have been proposed that are computationally efficient and can synthesize aero-optic data to match the statistics of measured data, but these approaches still have drawbacks including limited quality, inaccurate statistics, and the use of complicated algorithms. In this paper, we introduce ReVAR (Re-whitened Vector AutoRegression), a data-driven algorithm for generating synthetic aero-optic data that matches the statistics of measured data. A key contribution in this algorithm is Long-Range AutoRegression, a linear predictive model that combines a standard autoregression with a set of low-pass filters of the data to fit both short-range and long-range temporal statistics. ReVAR uses Long-Range AR together with a spatial re-whitening step to convert measured aero-optic data to temporally and spatially un-correlated white noise. ReVAR can then generate synthetic aero-optic data by reversing this process using white noise input. Using two measured turbulent boundary layer data sets, we demonstrate that ReVAR better matches the measured data's temporal power spectrum and other key metrics than do two conventional phase screen generation methods and an existing single time-lag autoregressive model.

For a simplicial poset $P$, Stanley assigned the face ring $A_P$, which is the quotient of the polynomial ring $S:=K[t_x \mid x \in P \setminus \{\widehat{0} \}]$ by the ideal $I_P$. This is a generalization of Stanley-Reisner rings, but $S$ and $A_P$ are not standard graded in this case, and $I_P$ is not a monomial ideal. To establish the foundation of the theory on local cohomology $H_{I_p}^i(S)$ and its injective resolution, we give an explicit description of the graded injective envelope ${}^*\! E_S(S/\mathfrak{p}_x)$, where $\mathfrak{p}_x$is the prime ideal associated with $x \in P$, and analyze their behavior in the graded dualizing complex.

We investigate baryonic vortices as topological excitations in rotating nuclear matter within the framework of chiral perturbation theory. We identify two distinct configurations: local and global vortices, both carrying the baryon number as the topological charge associated with the third homotopy group $π_3(S^3)$. For the local vortex, similar to the vortex Skyrmion in a finite isospin chemical potential, charged pions form the condensate on the boundary and have a phase winding, while the neutral pion varies along the rotation axis inside the vortex core. On the other hand, a global vortex is formed by the condensate and phase winding of the neutral pion, while the charged pions vary on the inside along the rotation axis. Crucially, although global vortices are usually discarded in infinite systems due to logarithmic divergence in energy, we demonstrate that the finite-size constraint dictated by causality in a rotating frame regularizes the divergence physically, rendering the global vortex a viable excitation. We reveal an energetic competition between global and local vortex states, under the tunable parameters of rotation, system size, and baryon chemical potential. Our results suggest that the previously overlooked global vortex can play a significant role in the topological structure of rotating dense QCD matter.

A fundamental obstacle to scalable solid-state quantum networks is the lack of a universal interface providing strong light-matter coupling, deterministic nonreciprocal photon routing, and efficient extraction. Here we propose a plasmonic platform overcoming these challenges using a Tomonaga-Luttinger liquid (TLL) in a single-walled carbon nanotube (SWCNT) microtoroid. The TLL's collective bosonic excitations are kinematically protected against backscattering by a large valley-momentum mismatch, guaranteeing robust chiral spin-momentum locking unattainable in dielectric cavities. This 1D protection enables deterministic routing of circularly polarized photons from a quantum emitter (e.g., a nitrogen-vacancy center) into distinct propagation channels. By aligning the emitter's symmetry axis, parasitic π transitions are geometrically forbidden. Furthermore, residual atomic-scale backscattering is suppressed to ~100 Hz via electrostatic gating and annealing. To overcome the severe mode mismatch between the CNT plasmon and optical fiber, we introduce a graded plasmonic-photonic mode converter, providing a path to near-unity extraction efficiency. Using a tripod-STIRAP scheme, we demonstrate high-fidelity, magnetically tunable spin-photon entanglement. Our analysis confirms operation deep in the strong-coupling regime, with cooperativities C > 100 and chiral contrast exceeding 20 dB. This wavelength-agnostic architecture is compatible with any solid-state emitter, establishing a scalable blueprint for robust, nonreciprocal quantum nodes in a global quantum internet.

We consider an eigenvalue problem of the form \begin{equation*} \left\{\begin{array}{rclll} -Δ_{p} u -Δ_{q} u&=& λK(x)|u|^{p-2}u & \mbox{ in } Ω^e u&=&0\qquad \quad &\mbox{ on } \partial Ω u(x) &\to& 0 &\mbox{ as } |x| \to \infty\,, \end{array}\right. \end{equation*} where $Ω^e$ is the exterior of a simply connected, bounded domain $Ω$ in $\mathbb{R}^N$, $p, q \in (1, N)$ with $p \neq q$, $0 < K \in L^{\infty}(Ω^e) \cap L^{\frac{N}{p}}(Ω^e)$, and $λ\in \mathbb{R}$. We establish the existence of an unbounded set of the principal eigenvalues and corresponding eigenfunctions. Moreover, we establish the regularity, positivity and the asymptotic profiles of these eigenfunctions with respect to the eigenvalue parameter $λ$. We use the {\em fibering method} of S.~I. Pohozaev to prove our results.

In adaptive-sampling control, the control frequency can be adjusted during task execution. Ensuring that these changes do not jeopardize the safety of the system being controlled requires attention. We introduce robust M-step hold model predictive control (MPC) to address this. Our formulation provides robust constraint satisfaction for an uncertain discrete-time system model with a fixed sampling time subject to an adaptable multi-step input hold (referred to as M-step hold). We show how to ensure recursive feasibility of the MPC utilizing M-step hold extensions of robust invariant sets, and demonstrate how to enable safe adaptive-sampling control via the online selection of M. We evaluate the utility of the robust M-step hold MPC formulation in a cruise control example.

By using Wald formalism, we show that the identification between the bulk and boundary conserved quantities induced by the perturbation of generic non-electromagnetic matter field holds not only on top of the asymptotically flat stationary spacetimes but also on top of the asymptotically AdS stationary ones. We further show that such an identification reduces to the familiar form for the test point particle by viewing it as the limiting case of general matter.

We give a near-linear time 4-coloring algorithm for planar graphs, improving on the previous quadratic time algorithm by Robertson et al. from 1996. Such an algorithm cannot be achieved by the known proofs of the Four Color Theorem (4CT). Technically speaking, we show the following significant generalization of the 4CT: every planar triangulation contains linearly many pairwise non-touching reducible configurations or pairwise non-crossing obstructing cycles of length at most 5 (which all allow for making effective 4-coloring reductions). The known proofs of the 4CT only show the existence of a single reducible configuration or obstructing cycle in the above statement. The existence is proved using the discharging method based on combinatorial curvature. It identifies reducible configurations in parts where the local neighborhood has positive combinatorial curvature. Our result significantly strengthens the known proofs of 4CT, showing that we can also find reductions in large ``flat" parts where the curvature is zero, and moreover, we can make reductions almost anywhere in a given planar graph. This also opens possibilities for extensions to higher surfaces since we can find such flat parts in any large-width triangulation of any fixed surface. From a computational perspective, the old proofs allowed us to apply induction on a problem that is smaller by some additive constant. The inductive step took linear time, resulting in a quadratic total time. With our linear number of reducible configurations or obstructing cycles, we can reduce the problem size by a constant factor. Our inductive step takes $O(n\log n)$ time, yielding a 4-coloring in $O(n\log n)$ total time. To efficiently handle a linear number of reducible configurations, we need them to be sufficiently robust to be useful in other applications. All our reducible configurations are what is known as D-reducible.

In recent years it has been shown how approximate bosonization can be used to justify the random phase approximation for the correlation energy of interacting fermions in a mean-field scaling limit. At the core is the interpretation of particle-hole excitations close to the Fermi surface at bosons. The main two approaches however differ in emphasizing collective degrees of freedom (particle-hole pairs delocalized over patches on the Fermi surface) or particle-hole pairs exactly localized in momentum space. Both methods lead to equal precision for the correlation energy with regular interaction potentials. This poses the question how big the influence of delocalizing particle-hole pairs really is. In the present note we show that a description with few, completely collective bosonic degrees of freedom only yields an upper bound of about 92% of the optimal value. Nevertheless it is remarkable that such a simple approach comes that close to the optimal bound.

The influence of pressure on finite-nuclear-size corrections to atomic energy levels and electron-capture decay rate is investigated in confined hydrogenlike ions. The ions are modeled inside an impenetrable spherical cavity, with a Gaussian distribution used to represent the nuclear charge distribution. For each confinement radius used to simulate external pressure, the energies and wave functions of the lowest-lying bound states are determined by numerically solving the Dirac equation via the kinetically balanced generalized pseudospectral method. In contrast to unconfined ions, both the FNS corrections and electron-capture decay rates increase markedly under pressure and exhibit parallel trends with increasing confinement. Pressure also removes level degeneracies and alters the relative magnitudes of FNS corrections across different bound states. Moreover, the nuclear charge radius is found to significantly affect the pressure-enhanced electron-capture decay rate.

We identify different dust features in our compilation of infrared spectra for 267 planetary nebulae (PNe) from the Spitzer, ISO, and IRAS telescopes. We classify 209 objects according to their dust type: mixed dust (MD), oxygen-rich dust (ORD), carbon-rich dust (CRD), PNe with only polycyclic aromatic hydrocarbons (PAHs) in their spectra (oPAH), and featureless (F) PNe. We study statistically the distributions of surface brightness and diameter of PNe with different types of dust as well as their distributions in the Galaxy.We find that both MD and ORD PNe are closer to the Galactic centre than CRD and oPAH PNe, and that the Galactic distributions of each pair of groups are statistically compatible, suggesting that they have similar progenitors. Since oPAH PNe have, on average, larger diameters and lower surface brightness than CRD PNe, we suggest that oPAH PNe are evolved CRD PNe. On the other hand, F PNe have the lowest surface brightness and the largest diameters, suggesting they could contain evolved PNe from any initial type of dust. Among the PNe with silicates, we find that only a few ORD PNe have just amorphous silicates in their spectra, and their distributions of Galactocentric distances and Galactic heights suggest that they had low-mass progenitors. MD PNe with both amorphous and crystalline silicates have the largest surface brightness and the smallest diameters and might be the earliest stages of PNe with the most massive and metal-rich progenitors.

Known symmetry groups are insufficient to describe the various couplings among spin, charge, and spatial degrees of freedom in fermionic systems. To address this problem, we introduce spin-charge groups (SCGs), which provide a unified framework for fermionic symmetries. SCGs incorporate spin and charge operations as `internal' symmetries, spatial and temporal operations as `external' symmetries, together with their couplings and projective twists. After deriving the general group structure of SCGs, we explore their applications in concrete physical systems, including $^3$He superfluids, charge-4e superconductors, collinear magnets with spin-fluxes, and superconductors with coexisting magnetic orders. We show that SCGs can enforce additional band degeneracies, Chern numbers and cross spin-charge responses. Hence SCGs provide a symmetry-based route toward the classification and exploration of new phases of matter even when strong interactions are included.

We study canonical one-step neighboring shuffle experiments for finite-output epsilon_0-LDP d-ary channels along growing alphabets, with frequency estimation and mechanism design under a pairwise chi-squared budget. The pairwise likelihood-ratio law nu_{ab,d} (pushforward of the row ratio under the null row) is the governing invariant: the canonical shuffled histogram experiment is exactly equivalent to the quotient multinomial experiment generated by nu_{ab,d}. Alphabet growth improves canonical shuffled privacy iff the worst pairwise law collapses to delta_1. We prove a sharp pure-LDP endpoint principle for the pairwise chi-squared, construct full-support obstruction families saturating it, and establish a diluting/persistent dichotomy with explicit finite-n hockey-stick bounds. The worst-case pairwise budget chi_*(W) governs a two-regime Assouad lower bound for arbitrary estimators in the i.i.d. multinomial model. Symmetrization reduces the uniform-point Fisher criterion to permutation-equivariant channels. Calibrated GRR is not optimal; in the low-budget regime, augmented GRR is optimal among permutation-equivariant channels.

The d(e,e'p) cross section was measured at momentum transfers $Q^2 = $ 0.8, 2.1 and 3.5 $(GeV/c)^2$ covering a wide range of proton kinematics at each $Q^2$ setting that made it possible to study this reaction as a function of missing momentum as well as a function of the neutron laboratory recoil angle $θ_{nq}$. Missing momentum distributions were determined for fixed values of $θ_{nq}$ up to missing momenta of 0.65 $GeV/c$. For the two larger momentum transfer settings, the characteristics of the experimental momentum distributions confirm the theoretical prediction that final state interactions (FSI) contribute maximally around a $θ_{nq} \sim 70^\circ$, while for $θ_{nq} < 45^\circ$ FSI are significantly reduced. The data at reduced FSI settings were best reproduced by calculations using the CD-Bonn potential wave functions.

In this paper, we are devoted to obtain some results on p-biharmonic map from gradient Ricci soliton, especially on two dimensional cigar soliton.

Inspired by the Bohlin transformation relating the planar harmonic oscillator to the Kepler problem, a variant of the Eisenhart lift is studied, in which a Lagrangian conservative dynamical system with d degrees of freedom is embedded into timelike geodesics of a conformally flat metric on a (d+2)-dimensional space-time of the Lorentzian signature. The uplift is used to construct novel examples of conformally flat metrics admitting higher rank Killing tensors.

Multiple-precision floating-point branch-free algorithms can significantly accelerate multi-component arithmetic implemented by combining hardware-based binary64 and binary32, particularly for triple- and quadruple-precision computations. In this study, we achieved benchmark results on x86 and ARM CPU platforms to quantify the accelerations achieved in linear computations and polynomial evaluation by integrating these algorithms.

Quantum channels generally reduce the distinguishability of quantum states, limiting information transmission and processing. Previous work introduced a protocol capable of increasing the distinguishability of states after the action of a specific quantum channel. Here we show how to systematically determine the maximal distinguishability gain achievable by this method. We develop an algorithm that identifies the optimal implementation of the protocol and applies to arbitrary quantum channels in a straightforward manner. Using this approach, we demonstrate that a weakly non-Markovian channel can effectively be converted into an essentially non-Markovian one through a distillation-like process. We further analyze the quantitative features of the optimized protocol, characterizing the conditions under which the enhancement is most pronounced. Our results provide a general framework to assess and optimize distinguishability recovery in open quantum systems.

Threshold-free cluster enhancement (TFCE) integrates cluster extent across thresholds to improve voxel-wise neuroimaging inference, but permutation testing makes it prohibitively slow for large datasets. Probabilistic TFCE (pTFCE) uses analytical Gaussian random field (GRF) p-values but discretises the threshold grid. Exact TFCE (eTFCE) eliminates discretisation via a union-find data structure but still requires permutations. We combine eTFCE's union-find for exact cluster-size retrieval with pTFCE's analytical GRF inference. The union-find builds the cluster hierarchy in one pass over sorted voxels and enables exact size queries at any threshold; GRF theory then converts these sizes to analytical p-values without permutations. Validation on synthetic phantoms (64^3, 80 subjects): FWER controlled at nominal level (0/200 null rejections, 95% CI [0.0%, 1.9%]); power matches baseline pTFCE (Dice >= 0.999); smoothness error below 1%; concordance r > 0.99. On UK Biobank (N=500) and IXI (N=563), significance maps form strict subsets of reference R pTFCE, which supports conservative error control. Implemented in pytfce (pip install pytfce): baseline completes whole-brain VBM in ~5s (75x faster than R pTFCE), hybrid in ~85s (4.6x faster) with exact cluster sizes; both >1000x faster than permutation TFCE.

For a prime $p$, let $c(p)$ denote the limiting fraction of $n\times n$ matrices over $\mathbb{F}_p$ whose determinant is a primitive root modulo $p$. The quantity $c(p)$ is a natural multiplicative deformation of the totient ratio $φ(p-1)/(p-1)$ and inherits its distributional behaviour over the primes. Existence and continuity of the limiting law follow from the shifted-prime Erdős--Wintner--Hildebrand framework. We prove the following new results: unconditionally, $\inf_p c(p)=0$ and the sharp order is $\min_{p\le x}c(p)\asymp 1/\log\log x$; the reciprocal satisfies $\limsup_{p\to\infty, \, p\text{ prime}} 1/(c(p)\log\log p)=e^γ$, and no smaller constant suffices. We give a complete proof, combining an adaptation of Erdős's argument with the Jessen--Wintner pure-type dichotomy, that the limiting distribution is purely singular, and strengthen this to $\dim_H(μ_G)=0$, i.e. the limiting measure is carried by a Borel set of Hausdorff dimension zero. The distribution has full topological support $[0,\tfrac12]$ and admits a Bernoulli-product representation indexed by the odd primes. The moments are given by convergent Euler products, and the Mellin transform $\mathbb{E}[X^s]$ extends to an entire function of $s$, non-vanishing on $\operatorname{Re}(s)>0$. Near the right endpoint, $1-G(\tfrac12-\varepsilon)\simκ/\log(1/\varepsilon)$ with an explicit constant $κ$. The quantity $1/c(p)$ equals the dimension-uniform expected rejection-sampling overhead in the reduction from learning with errors (LWE) to PRIM-LWE in lattice-based cryptography. The explicit bounds yield concrete overhead estimates for all primes appearing in current NIST post-quantum standards and representative NTT-friendly moduli.

Atomic force nanolithography provides a precise method for sculpting magnetic thin films, enabling controlled engineering of magnetic anisotropy in soft ferromagnets at the microscale. We demonstrate that nanoscale groove arrays patterned into permalloy (Ni80Fe20) films induce a robust in-plane uniaxial anisotropy, with the easy axis aligned along the groove direction. The anisotropy field is shown to increase with decreasing groove period and increasing engraving depth, offering continuous tunability of magnetic hardness within a single fabrication step. Artificially engraved microstructures further allow domain configurations and domain-wall trajectories to be directed along predefined pathways, exemplified by the creation of a chessboard-like magnetic landscape. Owing to its adaptability to diverse ferromagnetic materials and arbitrary corrugation geometries, this approach provides a versatile platform for tailoring in-plane magnetic anisotropy. Concrete applications are demonstrated in the design of magnonic elements and anisotropic magnetoresistance sensors.

This paper is the second in a series by the author and collaborators devoted to the study of geometric and analytic properties of nonlinear Lebesgue spaces, that is, L^p spaces of mappings taking values in arbitrary metric spaces. The present article formalizes the pointwise description of their geometric properties -- their length structure, bounds on their Alexandrov curvature as well as the definition of a speed for absolutely continuous curves despite the lack of differential structure. To obtain this pointwise description, we first prove a nonlinear analogue of the Fubini--Lebesgue theorem, which yields an identification of L^p curves in nonlinear Lebesgue spaces to mappings taking values in the space of L^p curves. This identification of L^p curves then enables a similar identification for absolutely continuous curves, from which the pointwise description of the geometric properties of nonlinear Lebesgue spaces follows.

We prove that embedded infinite planar maps in ergodic scale-free environments are close to their circle packing and Riemann uniformization embedding on a large scale, as long as suitable moment and connectivity conditions are satisfied. Ergodic scale-free environments were earlier considered by Gwynne, Miller and Sheffield (2018) in the context of the invariance principles for random walk, and they arise naturally in the study of random planar maps and Liouville quantum gravity.

Electromagnetic (EM) shielding is widely used to suppress radiated emissions and limit passive EM side-channel leakage. However, shielding does not address active probing, where an adversary injects external radio-frequency (RF) signals and observes the device's reflective response. This work studies whether such impedance-modulated backscattering persists when radiated emissions are suppressed by shielding. By injecting controlled RF signals and analyzing the reflections, we demonstrate that state-dependent impedance variations remain observable at frequencies outside the shields' primary attenuation band. Using processors implemented on FPGA and microcontroller prototypes, and evaluating workload profiles under three industry-standard shields, we find that passive EM measurements lose discriminative power under shielding, while backscattering responses remain separable. These results indicate that active RF probing can expose execution-dependent behavior even in shielded systems, motivating the need to consider active impedance-based probing within hardware security evaluation flows.