Edge-enhanced imaging is critical for visualizing weakly absorbing and transparent objects. Extending this functionality into the mid-infrared (MIR) region enables chemical sensitivity and improved imaging performance for biomedical, material, and remote-sensing applications. Here, we present a wide-field MIR edge-enhanced upconversion imaging system that integrates vortex-pump complex-amplitude engineering with aperiodic quasi-phase matching. In contrast to the bright-field modality, the wide-field edge-enhanced operation shows sensitive dependence on the crystal position relative to the Fourier plane. The system achieves single-shot operation with a 25-mm field of view and 79-$μ$m spatial resolution, yielding a record-high space-bandwidth product of $7.9 \times 10^4$. We show that this capability enables direct visualization of phase gradients in transparent optical elements and enhances structural contrast in biological specimens. The demonstrated architecture combines high sensitivity, spectral specificity, and robust edge detection, offering a promising route toward advanced MIR imaging in industrial inspection and biomedical diagnostics.

Transformer-based document cross-encoder rerankers are a central component of modern information retrieval systems. Despite their success, these models suffer from high computational costs due to processing long query-document sequences at inference time. A known approach to improve efficiency is token compression, which consists of aggregating groups of tokens together in the initial embedding layer, reducing the effective number of tokens, and making the computation faster. While token compression has proven to be successful for bi-encoder retrievers, we empirically observed that this approach may be ineffective for cross-encoder rerankers. In this paper, we propose Layer-wise Token Compression (LTC), which applies adaptive token pooling at intermediate transformer layers. Through extensive ablation studies on MS MARCO passage and document ranking tasks, we demonstrate that compression at middle layers preserves ranking quality while increasing inference QPS by up to 25% for passage ranking and up to 116% for document ranking. We also extend LTC to listwise LLM rerankers and show that the same approach can be easily applied to long-context listwise reranking, where the QPS improvements are even greater. More surprisingly, when applying rerankers trained on short passages to long-document ranking tasks, models trained with compression outperform their uncompressed counterparts, suggesting that compression may act as a beneficial regularizer that encourages length-invariant representations.

We return to and refine Zwiebach's formulation of closed string field theory (CSFT) built around non-critical backgrounds [1,2], restricting our attention to genus zero. The structure involves a special string state $F$ that encodes the failure of worldsheet BRST invariance, and a metric-dependent descent operator $\mathcal{B}$ adapted to the Weyl frame. We construct the mixed moduli spaces needed for the classical BV action, prove their existence, and extend the Sen-Zwiebach background independence argument to first order off of the conformal locus. We apply the formalism to the mildest deviation away from criticality - worldsheet CFTs with nonzero central charge: we consider both D=26-$ε$ dimensional flat space and linear dilaton profiles in bosonic string theory, focusing for simplicity on building solutions that depend on only one of the D dimensions.

While the standard $Λ$CDM model succeeds on large cosmological scales, it faces persistent small-scale challenges, including the core-cusp problem, the diversity of galaxy rotation curves, and the tight correlation between dark matter and baryons observed in the Tully-Fisher relation. To address these issues, we recently proposed an empirical law where the effective dark matter energy density is directly correlated with the baryonic gravitational potential, $ρ_{\rm DM} \propto Φ_b^2$, which reproduces observed rotation curves and resolves the core-cusp and diversity problems. To provide a theoretical foundation for this empirical law, we construct an effective field theory (EFT) introducing massive scalar, vector, and tensor mediators between baryons and a dark sector field $χ$. We demonstrate that aligning the mediator couplings to a specific ratio (4:6:3) with degenerate masses cancels the additional fifth forces acting on baryons up to $\mathcal{O}(v^2)$. We then show that this theoretical framework originates from a 5-dimensional (5D) spacetime. Treating the baryonic source as a 5D null fluid reveals that the three mediators emerge from a single 5D symmetric tensor field. By confining the field $χ$ to a 4D brane, we show that its interaction with these mediators generates a pressureless energy density ($ρ_{\rm int} \propto Φ_b^2$) that yields the empirically required baryon-correlated profile. Consequently, the field $χ$ exhibits a scale-dependent transition: on cosmological scales, its mass energy acts as standard Cold Dark Matter (CDM), whereas on galactic scales, its interaction energy governs local dynamics. Finally, by evaluating the dynamical boundary of this localized interaction region, we provide a physical interpretation that yields the relation $μ= K M_b^{-3/2}$, offering a theoretical basis for the Tully-Fisher relation.

Let $F(G)$ be the number of spanning forests in a graph $G$ and $\mathcal{C}(n,d)$ be the set of all connected $d$-regular simple graphs of order $n$. Define $\widehat{f}_{d}=\liminf_{n\rightarrow \infty}\{F(G)^{1/n}:G\in \mathcal{C}(n,d)\}$. Let $n_i$ be the number of vertices of degree $i$ in $G$. In this paper we give two lower bounds for $F(G)$ in terms of $n_i$ in connected graphs whose vertex degrees belong to $\{2,3\}$ and $\{2,3,4\}$, respectively. Furthermore, we determine the exact values of $\widehat{f}_3$ and $\widehat{f}_4$.

In this paper, we propose certain assumptions on the principal curvatures for a closed minimal hypersurface $M^5$ in $\mathbf{S}^6$ to be isoparametric, provided that the functions $S, f_3,f_4$ are constants. Our result removes the nonnegative scalar curvature assumption as in Tang and Yan \cite{TY}. Finally, as a rigidity result, if $M^5\subset \mathbf{S}^6$ has a point with exactly two distinct principal curvatures, then it must be a Clifford torus.

The gradient expansion and the separate universe approach provide an effective description of inflationary soft modes after coarse-graining shorter-wavelength degrees of freedom. We formulate a locality condition on the quantum state, requiring that the hard-mode state in each local universe depend on the soft modes only through the local soft-mode values in the same patch. When this condition is satisfied, the coarse-grained soft-mode dynamics remains local, and loop corrections from hard modes to superhorizon correlators of the adiabatic curvature perturbation are perturbatively suppressed. This provides a model-independent diagnosis of when enhanced corrections due to hard modes can invalidate the gradient expansion. We further show that the same locality condition implies a generalized soft theorem, from which the standard consistency relations follow under additional assumptions. This formulation clarifies the origin of possible deviations from the standard consistency relations in multi-field systems or in a non-attractor background. We also show that the locality condition guarantees the absence of infrared divergences for the correlators of operators invariant under a large gauge transformation. Thus, locality of the hard-mode state provides a unified criterion for the effective description of inflationary soft modes, generalized soft theorems, the suppression of hard-mode loop corrections, and the infrared regularity of observable correlators.

We propose a reformulation of the conformal perturbation theory of the correlation functions in $J\bar{J}$-deformed CFTs as a dressing on the deformed operators, that matches both bare and renormalized perturbation theory. The key is to use the Riemann bilinear identity to convert the deformation into a dressing and a large-cycle integral for higher genus. Based on the proposal, we calculate the deformation of partition functions on the torus and higher genus Riemann surfaces, which can be written as kernel integrals that preserve modular invariance or covariance. We also calculate the flow of the conformal weights and conserved charges along the deformation. Based on this flow and the modular $S$-transformation, we propose a criterion for constructing dressed operators. We test our formalism and results by studying the $O(2, 2)$ theories and strings on the TsT background.

Bayesian inference provides a natural framework for updating knowledge as new information becomes available, often in a sequential manner by incorporating datasets in stages or reusing previous posteriors as priors. In practice, this is commonly implemented using a factorized update in which datasets are treated as conditionally independent. When datasets are statistically correlated, however, this approximation becomes inconsistent with the joint likelihood and can lead to biased posterior estimates. In this work, we investigate this issue in a controlled setting using pseudo-data with a tunable covariance structure. We compare joint inference, factorized sequential updating, and a formulation based on the exact conditional likelihood. We show that factorized updates reproduce the joint posterior only in the limit of conditional independence, and otherwise lead to systematic deviations that grow with the correlation strength, while conditional updates remain consistent with the joint result. To interpret these deviations, we introduce an information decomposition that separates contributions into components that are new and components that are redundant across datasets. We show that correlations induce a structured, parameter-dependent redistribution of information, governed by the overlap of dataset sensitivities. The resulting mismatch between marginal and conditional information quantitatively explains the observed deviations. These results provide a practical diagnostic for assessing the consistency of sequential Bayesian inference with correlated datasets and highlight the need for a consistent treatment of correlations within a common probabilistic framework.

3D Gaussian Splatting (3DGS) has become a leading technique for real-time neural rendering and 3D scene reconstruction, but its rendering cost remains too high for many latency-sensitive scenarios. In particular, the rasterization stage in 3DGS dominates end-to-end rendering time, during which the renderer repeatedly evaluates each Gaussian's contribution to each covered pixel, making this stage compute-bound. At the same time, modern GPUs provide high-throughput Tensor Cores for low-precision matrix operations, yet existing 3DGS systems execute rasterization entirely on CUDA cores and leave Tensor Cores idle. We find that 3DGS rendering can be executed in FP16 with negligible quality degradation, suggesting a promising opportunity for Tensor Core acceleration. However, exploiting Tensor Cores for 3DGS is non-trivial because rasterization does not naturally match their execution model. Existing 3DGS rasterization is expressed as irregular per-pixel scalar operations, whereas Tensor Cores require dense, regular, and reuse-rich matrix workloads. Moreover, conventional tile-by-tile execution fails to exploit Gaussian reuse across neighboring tiles, resulting in repeated data loading and thus high data movement overhead. To this end, we present TensorGS, a 3DGS acceleration framework using Tensor Cores. TensorGS tensorizes the dominant rasterization computation into Tensor-Core-compatible matrix operations and introduces cross-tile grouping to improve Gaussian reuse, amortize overhead, and increase Tensor Core utilization. Experimental results show that TensorGS improves end-to-end rendering performance by 1.65$\times$ while preserving image quality.

This article examines the case of two papers published in Naturwissenschaften by the physicist Max Planck that were retrospectively marked as retracted on Springer digital platform. Rather than originating in scientific fraud, these withdrawals appear to result from contemporary digitization and copyright-management procedures applied anachronistically to historical publications. Through an investigation of the circulation history of Planck 1940 and 1942 philosophical essays, the article shows that republication across multiple formats was a common and legitimate practice within the scientific publishing culture of the early 20th century. Such practices only became problematic with the later transformation of the scientific article into a countable and proprietary unit within systems of bibliometric evaluation and commercial academic publishing. This article argues that contemporary notions such as duplicate publication and self-plagiarism are historically situated categories that cannot be applied retrospectively without distorting the historical record. More broadly, the Planck case reveals how digital scholarly infrastructures controlled by large commercial publishers can limit the accessibility of the scientific past. Ironically, the original papers remain accessible today through the nonprofit digital platform Internet Archive rather than through the publisher that originally issued the journal.

Oscillator Ising machines (OIMs) are often viewed as physical systems that perform gradient descent on an energy landscape encoding Ising solutions. Here, we show that this interpretation is not generic and breaks down in a broad class of oscillator implementations. We establish that gradient-flow dynamics require a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response. Deviations from this condition, which we term harmonic misalignment, introduce even components in the pairwise interaction function, leading to non-conservative phase dynamics and precluding a gradient-flow description. We introduce a normalized metric for this non-gradient contribution and evaluate it across representative oscillator models relevant to OIMs. This metric reveals substantial non-gradient contributions in ring oscillators and across other hardware-realistic oscillator models. These findings identify harmonic misalignment as a fundamental mechanism for the breakdown of energy-based dynamics in OIMs and motivate nonequilibrium analysis and algorithms that explicitly account for and potentially exploit non-gradient behavior.

A train of periodic optical pulses gives an optical frequency "comb" that acts as a precise ruler for light measurement due to its equally spaced frequencies. Today, such pulses last millionths of a billionth of a second (Femtoseconds/fs) and associated comb spans billions of frequencies (Terahertz), similar to a discretized rainbow ranging from ultraviolet to infrared light. This is the core technology in many optic-based applications like atomic clocks and secure communication. Despite its obvious value, these remain mostly confined to research labs for being complex, expensive, and power-hungry. One promising solution is to use laser mode-locking: a technique that forces a laser to emit short coherent pulses. While chip-size systems have already been demonstrated, this approach still lacks flexibility and performance in repetition rate and bandwidth simultaneously. This research proposal leverages the industrialization of integrated photonic chips to develop a first-ever all-integrated two-coloured pulsed source with durations of a few hundred fs. It will engineer a novel turn-key device that will 1) pioneer the demonstration of modelocked pulses at two separate central frequencies originating from the same laser, 2) fit on a fingertip, and 3) be compatible with generic foundry processes, and thus, mass-manufacturable. Sustained generation of such pulses is intricate with little knowledge about light-material interaction at this scale. The chip will emit two broadband combs using one control parameter. These combs will have synergetic comb properties and coupled through one gain medium. Thus, we will create a new ultra-broadband comb resulting in unprecedented phase correlation between the two sub-combs. Simultaneously, the device will be significantly smaller, lighter, cheaper, and more power-efficient than its free-space rivals, reducing the gap between lab and market.

We present a hybrid adiabatic algorithm for maximum independent set (MIS) using Rydberg atom arrays. We engineer local controls that preferentially excite atoms with few neighbors, which represent graph nodes with small degrees. Numerical simulations show that the designed pulses accelerate convergence to the MIS state and suppress population in trap states. We obtain higher success probabilities than traditional global controls and a $25\%$ reduction in fidelity decay rate as problem hardness increases.

Informative cluster size (ICS) and informative subgroup size (ISS) can distort marginal association estimates when the number of observed units, or their distribution across outcome-defined categories, is related to the outcomes under study. This issue is especially relevant for paired outcomes, where the observed association can depend on cluster size, paired-category composition, and the process by which units become available for analysis. We propose three weighted estimating approaches for marginal association between paired outcomes in clustered data. The weights are derived from within-cluster resampling arguments and extend inverse cluster-size and subgroup-size weighting to paired outcome categories. We also modify an existing ISS testing procedure by utilizing Stouffer's method to reduce computational burden. To evaluate the methods, we develop a simulator for clustered paired outcomes that separates unit-level association, latent cluster-level association, and outcome-dependent retention. Simulations show that pair-based weighting can reduce bias when association arises through unit-level dependence and subgroup composition is informative, but can attenuate association carried by latent cluster-level structure. Typical inverse-cluster weighting remains more stable when the association is primarily cluster-level. Application to NHANES oral-health data shows small positive periodontal and caries associations overall, with filled-surface outcomes showing stronger ISS evidence and greater sensitivity to pair-based weighting than decayed-surface outcomes. These results indicate that marginal association under ICS and ISS should be interpreted in relation to the source of association, observed-unit structure, and assumptions used to choose the weighting scheme.

These notes have been written for a series of lectures to be given at the 44th Finnish Summer School on Probability and Statistics in Lammi, Finland, from 25th to 29th May, 2026. They contain an introduction to Wiener chaos decomposition in finite dimension, a construction of Gaussian fields on the torus, including white noise and the Gaussian free field, and applications to the $Φ^4$ model. They do not cover other important aspects of the topic, such as stochastic integration, stochastic PDEs and Malliavin calculus.

Solving non-linear Diophantine systems lies at the mathematical core of integer optimization and cryptography. While the general unbounded problem is undecidable, even over bounded integer domains it remains classically intractable in the worst case. In this work, we introduce a fully reversible quantum algorithmic framework tailored to solve arbitrary polynomial Diophantine equations over bounded integer domains. The core of our approach is the explicit, gate-level synthesis of an evaluation oracle for amplitude amplification. By coherently evaluating polynomial constraints via in-place two's complement arithmetic and routing operations into a single recycled accumulator, this garbage-free strategy achieves a compact and scalable synthesis of the underlying non-linear arithmetic. Through analytical derivations and empirical circuit simulations, we prove that the overall spatial complexity is bounded by $q = \mathcal{O}((n + d^2)\log_2 N)$ logical qubits for $n$ variables, maximum degree $d$, and interval length $N$. The non-Clifford Toffoli depth is upper-bounded by $\mathcal{O}(q^2)$. This structural scaling exponent remains invariant to the variable count, modulated linearly only by the coefficients' Hamming weights. By moving beyond abstract black-box assumptions, this explicit architectural synthesis guarantees that the necessary quantum arithmetic acts as a bounded polynomial overhead. This ensures a quadratic speedup over classical exhaustive search, whether retrieving a unique assignment or dynamically enumerating an unknown number of solutions.

This note studies parity vectors and paradoxical sequences in the accelerated Collatz iteration $T(n) = (3n+1)/2$ for $n$ odd, $T(n) = n/2$ for $n$ even. Building on Rozier and Terracol (arXiv:2502.00948, 2025), Terras (1976), Lagarias (1985), and Tao (2019), we prove three theorems and add one numerical observation. The first is a sharp finitary form of Terras's parity-vector density; the second is a closed-form analytic count of paradoxical $Ω_k(n)$ for each fixed length $k$. The third is a density-zero theorem for bounded-length paradoxical sequences with explicit constant. As for the numerical piece, among the seven $(j, q)$ pairs that show up in the Rozier-Terracol enumeration with first term $n \le 10^9$, every paradoxical reduced ratio $q/j$ turns out to be a left convergent, a left semiconvergent, or a Stern-Brocot mediant of adjacent convergents/semiconvergents of $\log_3 2$. The three theorems are unconditional. The fourth observation is verified for $n \le 10^7$ and conjectured for all $n$. We make no claim toward the Collatz conjecture or Terras's coefficient-stopping-time conjecture.

We develop a rigorous theory for a structure-preserving discretisation of the incompressible Euler and Navier--Stokes equations, based on discrete exterior calculus on prismatic Delaunay--Voronoi meshes over closed Riemannian manifolds. The central result is a selection principle: exact algebraic conservation at the discrete level is not merely a fidelity property but rules out entire classes of weak solutions that other discretisations reach unconditionally. We establish this in four regimes. \emph{Smooth solutions}: convergence at rate $\mathcal{O}(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|^{β_d})$, uniformly in viscosity $ν\ge 0$, with $β_3 = 0$ and $β_2 = 1$; first order on general meshes and second order on meshes with centroid proximity and reconstruction symmetry. \emph{Leray--Hopf weak regime}: subsequential $L^2$ limits are weak solutions of the viscous system. \emph{Inviscid measure-valued regime}: limits are conservative measure-valued Euler solutions; their concentration defect vanishes above the Onsager threshold $α> 1/3$ \emph{provided the discrete solutions admit a uniform $C^{0,α}$ bound there}. \emph{Dissipative regime}: no subsequence converges to an energy-dissipating Euler solution at any regularity, a structural exclusion that follows from exact discrete energy conservation and distinguishes the scheme. The gap $1/3 < α< 1$, where energy conservation and defect-free convergence hold but uniqueness remains open, isolates the central open problem of inviscid fluid dynamics.

This paper studies a stochastic mean-field linear-quadratic Stackelberg differential game with random coefficients. The interaction between mean-field terms and random coefficients precludes the direct use of conventional decoupling techniques. We apply an extended Lagrange multiplier method to derive an affine operator representation of the follower's optimal response. The induced leader problem is then formulated as a generalized stochastic LQ control problem with operator-valued coefficients, and the Stackelberg optimal control is characterized through a Riccati-free coupled FBSDE system. We further develop a Deep FBSDE Picard Solver that preserves the Stackelberg order through follower-response learning, response-sensitivity extraction, leader optimization, and neural augmented Lagrangian enforcement of mean-field consistency constraints. Numerical studies covering convergence diagnostics, discretization sensitivity, Riccati calibration, ablation tests, stability under control perturbations, Stackelberg--Nash comparisons, and a financial application support the effectiveness of the proposed framework.

Large curvature perturbations generated during slow first-order phase transitions are a promising source of primordial black holes. However, recent analyses suggested that the mechanism is ruled out once the density contrast and the formation threshold are evaluated in the same gauge. In this work, we show that this mechanism remains viable: after a supercooled transition, reheating can be sufficiently slow that the Universe enters an early matter-dominated era, during which even small overdensities grow and collapse into primordial black holes.

This paper presents a PAC-Bayes framework for learning controllers for unknown stochastic linear discrete-time systems, where the system parameters are drawn from a fixed but unknown distribution. We derive a data-dependent high probability bound on the performance of any learned (stochastic) controller, and propose novel efficient learning algorithms with theoretical guarantees, which can be implemented for both finite and infinite controller spaces. Compared to prior work, our bound holds for unbounded quadratic cost. In the special case where LQG is optimal, our numerical results suggest that the learned controllers achieve comparable performance to LQG.

Augmented Reality (AR) is increasingly utilized to guide users through complex spatial tasks in domains such as manufacturing, non-destructive testing, and surgery. These applications often require strict compliance with 5D+ trajectories using rotation-symmetric tools (3D position, 2D orientation, and movement speed). However, the sensori-motor baselines of untrained users during these multidimensional tracing tasks, along with the cognitive-motor trade-offs induced by varying visual feedback paradigms, remain underexplored. We present a controlled within-subjects user study (N=30) evaluating three distinct AR UI concepts for trajectory guidance, both with and without explicit orientation constraints. We analyzed spatial, orientational, and speed compliance based on the internal AR tracking, which was validated against a high-precision external optical tracking system to rule out hardware drift. By segmenting the execution into transient and steady-state phases and applying Aligned Rank Transform (ART) ANOVA, we isolated the interaction effects between visual design and task complexity. Alongside subjective metrics (NASA-TLX, SUS), our results establish conservative performance baselines for novice users performing freehand 5D trajectory following. We reveal orientation-induced cognitive-motor trade-offs and identify mitigating UI synergies. Ultimately, we provide empirical baselines and actionable design guidelines for developing effective AR guidance systems.

Critical current density is the key parameter for the practical application of superconductivity. In this study, 3 MeV proton irradiation experiments were conducted on HgBa$_2$Ca$_2$Cu$_3$O$_8$$_+$$_δ$ single crystals to introduce pinning centers. The critical current density is found to be strongly enhanced after the irradiation with its maximum at a dose of 1$\times$10$^{16}$/cm$^2$, where the self-field critical current density at 2 K is enhanced from 5.5 MA/cm$^2$ to 26 MA/cm$^2$. At 77 K, the self-field critical current density for all irradiated crystals is over 0.1 MA/cm$^2$. The power-law dependence of the critical current density on the magnetic field is observed after irradiation, with a large power-law exponent $α$ close to 1. A monotonic magnetic field dependence of the normalized magnetic relaxation rate is observed, which could be attributed to the low irreversibility field caused by the large anisotropy in Hg1223 single crystals. Through the analysis of the pinning force density of the crystal before and after irradiation, a clear mechanism change has been observed.

TRAPPIST-1 is an ultra-cool dwarf that flares frequently. These flares shape the surrounding planets' high-energy irradiation environments, with consequences for atmospheric chemistry and escape, and they can contaminate transmission spectroscopy of those planets. A quantitative flare-frequency distribution (FFD) spanning the full energy range is therefore essential for both interpreting JWST spectra and modeling the planets' irradiation histories. Here we present a unified FFD over four orders of magnitude in energy by jointly analyzing $\approx$87\,hr of JWST/NIRISS and JWST/NIRSpec time-series spectroscopy together with $\approx$74\,days of \textit{Kepler}/K2 photometry. To enable a consistent comparison across these heterogeneous datasets, we convert all events to energies in the TESS bandpass. For the Kepler-to-TESS conversion we adopt a cooler flare continuum appropriate for ultra-cool dwarfs ($T_{\rm flare}=3500$\,K). After correcting for flare-detection sensitivities, the combined JWST+K2 cumulative FFD is consistent with a single power law, $N(\ge E_\mathrm{TESS})\propto E_\mathrm{TESS}^{-β}$, with $β=0.753$ over $E_{\rm TESS}\simeq10^{29}$-$10^{33}$\,erg. The slope of the distribution indicates that the time-averaged flare energy budget is dominated by rare, high-energy events rather than by the more numerous low-energy flares. Moreover, we found that strong flares with energies $E_\mathrm{TESS} > 10^{32}$~erg occur once every 25 days, about an order of magnitude more frequently than inferred from previous TRAPPIST-1/analog FFD estimates. This elevated rate of energetic flares has important implications for atmospheric escape, photochemistry, and habitability assessments of the TRAPPIST-1 planets.

Let $d\ge 3$ and $p>0$. Let $\|x\|_p$ denote the $\ell_p$ (quasi-)norm of a $d$-dimensional vector $x$. Holevo and Utkin \cite{HU26} conjectured that for $0<p\le 1$, \[ \min \left\{\frac{\|x\|_p}{\|x\|_2}:\vec{0}\neq x\in\mathbb R^d,\ \sum_{i=1}^d x_i=0\right\} =2^{1/p-1/2}; \] for $1<p<2$, \[ \min \left\{\frac{\|x\|_p}{\|x\|_2}:\vec{0}\neq x\in\mathbb R^d,\ \sum_{i=1}^d x_i=0\right\} = \min\left\{2^{1/p-1/2},\left(\frac{(d-1)^{p/2}+(d-1)^{1-p/2}}{d^{p/2}}\right)^{1/p}\right\}; \] and for $2<q<\infty$ \[ \max\left\{\frac{\|x\|_q}{\|x\|_2}:\vec{0}\neq x\in\mathbb R^d,\ \sum_{i=1}^d x_i=0\right\} = \max\left\{2^{1/q-1/2},\left(\frac{(d-1)^{q/2}+(d-1)^{1-q/2}}{d^{q/2}}\right)^{1/q}\right\}. \] They proved the $d=3$ case in \cite{HU26}. In this paper, we confirm the conjecture of the remaining cases $d\ge 4$.

PSR J1905+0154A is a binary millisecond pulsar located in the globular cluster (GC) NGC 6749. It was discovered in 2004 in a search for pulsars in GCs carried out with the Arecibo 305-m radio telescope. The pulsar has a spin period of 3.2 ms, an orbital period of 0.81 days, and is in a low-eccentricity orbit with a low-mass WD companion. Combining early Arecibo and latter Five Hundred meter Aperture Spherical Telescope (FAST) data, we were able to derive a phase-coherent timing solution for this pulsar, which now spans 20 years. This includes a precise measurement of the astrometric, spin and orbital parameters of the system. The small range of predicted accelerations expected from the gravitational field of this GC allows an estimate of the intrinsic spin-down: the inferred magnetic field at the surface (2.2 - 2.4 * 10^8 G) and characteristic age (2.8 - 3.5 Gyr) are typical of what one finds among MSPs in the Galactic field. The position of this pulsar coincides with the position of one of the very few candidate white dwarfs (WDs) in the whole HST dataset on this GC. The position of the companion in the colour-magnitude diagram is consistent with a Helium WD with a mass of 0.17 - 0.19 M_sun, a cooling age of 0.4 - 0.7 Gyr, and a surface temperature of 11,600 - 14,800 K. A comparison with the characteristic age of the pulsar indicates that at the time of Roche lobe detachment, the spin period was between 1.98 and 2.62 ms.. The velocity of the system relative to the GC, which is 4.5-sigma significant and an order of magnitude larger than the escape velocity, raises the possibility that, despite its location close to the centre of the GC, the pulsar might not be associated with it. Finally, our effort to confirm a second pulsar candidate in this GC did not yield a positive confirmation, nor the discovery of any additional pulsar in this GC.

The $n$-th Fibonacci cube $Γ_n$ is the subgraph of the hypercube $Q_n$ induced by binary strings with no two consecutive ones. We determine $π(Γ_n) = 2^n$ for $n \le 6$, so the pebbling number of $Γ_n$ equals that of the ambient hypercube $Q_n$ despite $Γ_n$ having far fewer vertices. The lower bound is a standard potential argument. For the upper bound, the Weight Function Lemma yields $2^n+1$ -- one too many -- so we close the gap by exhaustive MILP verification. We conjecture $π(Γ_n) = 2^n$ for all $n$.

We study the problem of locating violating principal minors in matrix families lying near the boundary of P-matrices. Rather than viewing this search problem purely through computational complexity, we analyze it from an information-accessibility perspective. We show that, despite strong underlying algebraic structure, the location of a violating subset may remain difficult to infer through local queries. In the sparse-violation regime, local observations typically provide only weak eliminative power, and polynomially many queries accumulate only vanishing mutual information about the hidden witness under the induced oracle model. Using mutual information and Fano's inequality, we characterize the resulting limitation on information acquisition. The analysis highlights a conceptual distinction between structure and accessibility: a problem may possess rich underlying structure while the information required to identify a hidden witness remains weakly inferable from observable responses.

Accurate models of merger remnants are increasingly important for gravitational-wave science, including precision tests of gravity with ringdown, inference of black-hole populations, and modeling hierarchical mergers. For eccentric binaries, remnant mass, spin, and recoil carry nontrivial imprints of eccentricity that are both physically informative and more challenging to model, yet remain less developed than in the quasi-circular case. We present two new models trained on numerical-relativity (NR) simulations of unequal-mass, non-spinning eccentric binary black holes: NRSurE_q4NoSpin_Remnant, which predicts remnant properties, and NRSurE_q4NoSpin_Dynamics, a time-domain surrogate for the evolution of eccentricity and mean anomaly. Both models are trained on NR simulations over a three-dimensional parameter space with mass ratios $q \leq 4$, eccentricity $e < 0.23$, and mean anomaly $\ell \in [0,2π)$ radians, where both $e$ and $\ell$ defined at $t=-1000M$ relative to peak amplitude and $M$ is the total mass. We highlight some applications, including the phenomenological impact of eccentricity on remnant properties and the enhancement or suppression of recoil. We also provide error estimates for all modeled quantities, supporting reliable use in current and future gravitational-wave parameter-estimation analyses. Both models will be made available through open-source codes.