The unconditional, i.e. without assuming validity of RH, sharp limit relationship (as p tends to infinity) is found between the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals and maximal values of Gronwall numbers G(N) among all integers whose greatest prime factor is p and which are divided by any prime q<p. The proofs are based on the properties of G(N) studied in previous author's preprints.

We investigate the impact of an induced mass term $Δ$ on the current density in graphene subjected to a space- and time-dependent periodic potential $U(x,t)$. By solving the Dirac equation and deriving both the quasi-energy spectrum and the corresponding eigenspinors, we obtain explicit analytical expressions for the current density, which exhibits a clear dependence on $Δ$. We show that $Δ$ acts as a tunable control parameter that governs the amplitude, sign, and resonance structure of Josephson-like current oscillations. For normal incidence and a purely time-periodic potential, our results reveal that the oscillations within the energy gap gradually diminish as the mass term $Δ$ increases. This suppression leads to a weakening of the Josephson-like effect typically observed in such systems. When the potential $U(x,t)$ is periodic in both space and time, the behavior becomes more complex. The current density can take either positive or negative values depending on the magnitude of the induced gap, and it generally decreases over time. As a result, the resonance phenomena--prominent at lower gap values--become progressively less significant as $Δ$ increases. These findings underscore the tunable nature of light-matter interactions and quantum transport in gapped graphene, suggesting potential applications in terahertz (THz) nanoelectronic devices and optically controlled quantum switches.

In this paper, we analyze $m$-dimensional ($m$D) convolutional codes with finite support, viewed as a natural generalization of one-dimensional (1D) convolutional codes to higher dimensions. An $m$D convolutional code with finite support consists of codewords with compact support indexed in $\mathbb{N}^m$ and taking values in $\mathbb{F}_{q}[z_1,\ldots,z_m]^n$, where $\mathbb{F}_{q}$ is a finite field with $q$ elements. We recall a natural upper bound on the free distance of an $m$D convolutional code with rate $k/n$ and degree~$δ$, called $m$D generalized Singleton bound. Codes that attain this bound are called maximum distance separable (MDS) $m$D convolutional codes. As our main result, we develop new constructions of MDS $m$D convolutional codes based on superregularity of certain matrices. Our results include the construction of new families of MDS $mD$ convolutional codes of rate $1/n$, relying on generator matrices with specific row degree conditions. These constructions significantly expand the set of known constructions of MDS $m$D convolutional codes.

We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all remaining edges follow the Erdős--Rényi model $\mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $\mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies that $\mathcal{G}(n, p, d, k)$ and $\mathcal{G}(n, p)$ are indistinguishable at the level of the marginals, and the signal lies entirely in the edge dependencies induced by the local geometry. We investigate both the information-theoretic and computational limits of detection. On the information-theoretic side, our upper bounds follow from three tests based on signed triangle counts: a global test, a scan test, and a constrained scan test; our lower bounds follow from two complementary methods: truncated second moment via Wishart--GOE comparison, and tensorization of KL divergence. These results together settle the detection threshold at $d = \widetildeΘ(k^2 \vee k^6/n^3)$ for fixed $p$, and extend the state-of-the-art bounds from the full model (i.e., $k = n$) for vanishing $p$. On the computational side, we identify a computational--statistical gap and provide evidence via the low-degree polynomial framework, as well as the suboptimality of signed cycle counts of length $\ell \geq 4$.

Density functional theory at finite temperatures often relies on the zero-temperature approximation, which uses a ground-state exchange-correlation functional with thermalized densities. This approach, however, neglects the explicit temperature dependence of the exchange-correlation free energy -- a key factor in regimes such as warm dense matter, where both electronic and thermal effects are significant. In this work, we introduce the entropy-corrected zero-temperature approach, in which the exchange-correlation entropy is extracted using the generalized thermal adiabatic connection formula to construct a thermal correction to the standard zero-temperature approximation. Using a uniform electron gas parametrization, we compare this approach to the finite-temperature adiabatic connection and demonstrate that it performs best at lower densities. This provides a useful complement to zero-temperature density functional approximations, which generally perform better at moderate-to-large densities. We further identify a density-dependent intersection between the adiabatic connection curves, revealing a dependence on the ground state correlation energy and correlation potential. Additionally, extension of the entropy corrected approach applied as a local density approximation--like temperature correction to the zero temperature approximation is discussed.

Nonlinear Schwarz methods are a type of nonlinear domain decomposition method used as an alternative to Newton's method for solving discretized nonlinear partial differential equations. In this article, the first parallel implementation of a two-level nonlinear Schwarz method leveraging the GDSW-type coarse spaces from the Fast and Robust Overlapping Schwarz (FROSch) framework in Trilinos is presented. This framework supports both additive and hybrid two-level nonlinear Schwarz methods and makes use of modifications to the coarse spaces constructed by FROSch to further enhance the robustness and convergence speed of the methods. Efficiency and excellent parallel performance of the software framework are demonstrated by applying it to two challenging nonlinear problems: the two-dimensional lid-driven cavity problem at high Reynolds numbers, and a Neo-Hookean beam deformation problem. The results show that two-level nonlinear Schwarz methods scale exceptionally well up to 9\,000 subdomains and are more robust than standard Newton-Krylov-Schwarz solvers for the considered Navier-Stokes problems with high Reynolds numbers or, respectively, for the nonlinear elasticity problems and large deformations. The new parallel implementation provides a foundation for future research in scalable nonlinear domain decomposition methods and demonstrates the practical viability of nonlinear Schwarz techniques for large-scale simulations.

This work presents a modular, Python-based simulator that simplifies the evaluation of novel vehicle control and coordination algorithms in complex traffic scenarios while keeping the implementation overhead low. It allows researchers to focus primarily on developing the control and coordination strategies themselves, while the simulator manages the setup of complex road networks, vehicle configuration, execution of the simulation and the generation of video visualizations of the results. It is thereby also well-suited to support control education by allowing instructors to create interactive exercises providing students with direct visual feedback. Thanks to its modular architecture, the simulator remains easily customizable and extensible, lowering the barrier for conducting advanced simulation studies in vehicle and traffic control research.

Gaussian process (GP) emulators have become essential tools for approximating complex simulators, significantly reducing computational demands in optimization, sensitivity analysis, and model calibration. While traditional GP emulators effectively model continuous and Gaussian-distributed simulator outputs with homogeneous variability, they typically struggle with discrete, heteroskedastic Gaussian, or non-Gaussian data, limiting their applicability to increasingly common stochastic simulators. In this work, we introduce a scalable Generalized Deep Gaussian Process (GDGP) emulation framework designed to accommodate simulators with heteroskedastic Gaussian outputs and a wide range of non-Gaussian response distributions, including Poisson, negative binomial, and categorical distributions. The GDGP framework leverages the expressiveness of DGPs and extends them to latent GP structures, enabling it to capture the complex, non-stationary behavior inherent in many simulators while also modeling non-Gaussian simulator outputs. We make GDGP scalable by incorporating the Vecchia approximation for settings with a large number of input locations, while also developing efficient inference procedures for handling large numbers of replicates. In particular, we present methodological developments that further enhance the computation of the approach for heteroskedastic Gaussian responses. We demonstrate through a series of synthetic and empirical examples that these extensions deliver the practical application of GDGP emulators and a unified methodology capable of addressing diverse modeling challenges. The proposed GDGP framework is implemented in the open-source R package dgpsi.

In this note we show how the consistent truncations on half-supersymmetric branes of Leung and Stelle and Lin, Skrzypek and Stelle fit into the general exceptional generalised geometry analysis of Cassani \emph{et al.}. Each solution defines a torsion-free $Spin(n)$ structure in the $Spin(n,n)\times \mathbb{R}^+$ generalised geometry introduced by Strickland--Constable, where $n$ is the dimension of the space transverse to the brane. Embedding this into the appropriate exceptional generalised geometry then defines the truncation. As a by-product we derive a new consistent truncation on the IIA NS5-brane to six-dimensional $\mathcal{N}=(2,0)$ supergravity coupled to a tensor mutliplet, and new consistent truncations on the D6- and D7-branes to seven- and eight-dimensional pure half-maximal supergravity respectively.

We present an automated, GPU-accelerated framework for many-body perturbation theory (MBPT) calculations of the zero-temperature nuclear equation of state (EOS) based on chiral nucleon-nucleon (NN) and three-nucleon (3N) interactions. Automated diagram generation and evaluation enable the computation of all diagrams up to fifth order in the MBPT expansion at the normal-ordered two-body level in infinite matter, with residual three-body contributions explicitly included up to third order. Multi-GPU acceleration of 3N normal ordering, a novel Monte Carlo integrator (called PVegas), and further advances in high-performance computing enable us to evaluate all 840 fifth-order diagrams with controlled numerical uncertainties. We investigate the MBPT convergence up to fifth order in pure neutron matter (PNM) and symmetric nuclear matter (SNM) for two sets of chiral interactions, study neutron star matter, and present fourth-order results for asymmetric matter including normal-ordered 3N forces. The framework enables systematic MBPT studies with harder interactions and benchmarks against nonperturbative methods. It can be further extended to finite-temperature EOS calculations and to improved uncertainty quantification using emulation and resummation techniques.

Unconventional physical computing is producing many novel and exotic devices that can potentially be used in a computational mode. Currently, these tend to be used to implement traditional models of computation, such as boolean logic circuits, or neuromorphic approaches. This runs the risk of failing to exploit the devices to their full potential. Here we describe a methodology for deriving a model of computation and domain specific language more closely matched to a given physical device's capabilities, and illustrate it with a case study of bosonic computing as implemented by a physical multi-component interferometer.

Fault-tolerant spanners are fundamental objects that preserve distances in graphs even under edge failures. A long line of work culminating in Bodwin, Dinitz, Robelle (SODA 2022) gives $(2k-1)$-stretch, $f$-fault-tolerant spanners with $O(k^2 f^{\frac{1}{2}-\frac{1}{2k}} n^{1+\frac{1}{k}} + k f n)$ edges for any odd $k$. For any $k = \tilde{O}(1)$, this bound is essentially optimal for deterministic spanners in part due to a known folklore lower bound that \emph{any} $f$-fault-tolerant spanner requires $Ω(nf)$ edges in the worst case. For $f \geq n$, this $Ω(nf)$ barrier means that any $f$-fault tolerant spanners are trivial in size. Crucially however, this folklore lower bound exploits that the spanner \emph{is itself a subgraph}. It does not rule out distance-reporting data structures that may not be subgraphs. This leads to our central question: can one beat the $n \cdot f$ barrier with fault-tolerant distance oracles? We give a strong affirmative answer to this question. As our first contribution, we construct $f$-fault-tolerant distance oracles with stretch $O(\log(n)\log\log(n))$ that require only $\widetilde{O}(n\sqrt{f})$ bits of space; substantially below the spanner barrier of $n \cdot f$. Beyond this, in the regime $n \leq f \leq n^{3/2}$ we show that by using our new \emph{high-degree, low-diameter} decomposition in combination with tools from sparse recovery, we can even obtain stretch $7$ distance oracles in space $\widetilde{O}(n^{3/2}f^{1/3})$ bits. We also show that our techniques are sufficiently general to yield randomized sketches for fault-tolerant ``oblivious'' spanners and fault-tolerant deterministic distance oracles in bounded-deletion streams, with space below the $nf$ barrier in both settings.

The world's power systems are undergoing a rapid transformation, shifting away from carbon-intensive power generation to renewable sources. As a result, electricity is being transported over ever longer distances, while the intrinsic system inertia provided by thermal power plants decreases. Together, these developments raise the probability of cascading line failures and reduce the stability of the system after a system split. In this article, we assess the risk of cascading failures and system splits in the European power grid for different carbon reduction scenarios. We analyze the most likely and most dangerous splits, and identify critical transmission infrastructures and we discuss potential countermeasures that can address the problem of cascades. Our results show that while the risks of splits causing power failures rises with decarbonization, it can be mitigated cost efficiently.

In the Gale-Shapley model of two-sided matching, it is well known that for generic preferences, the outcomes for each side can vary dramatically in the male-optimal vs. female-optimal stable matchings. In this paper, we show that under a widely used characterization of similarity in rankings, even a weak correlation in preferences guarantees assortative matching with high probability as the market size tends to infinity. It follows that the men's average ranking of women and the women's average ranking of men are asymptotically equivalent in all stable matchings with high probability, as long as the market imbalance is not too extreme.

Barren plateaus present a major challenge in the training of variational quantum algorithms (VQAs), particularly for large-scale discretizations of nonlinear partial differential equations. In this work, we introduce a domain decomposition framework to mitigate barren plateaus by localizing the cost functional. Our strategy is based on partitioning the spatial domain into overlapping subdomains, each associated with a localized parameterized quantum circuit and measurement operator. Numerical results for the time-independent Gross-Pitaevskii equation show that the domain-decomposed formulation, allowing subdomain iterations to be interleaved with optimization iterations, exhibits improved solution accuracy and stable optimization compared to the global VQA formulation.

Nonlinear spectroscopy provides a unique perspective to understand time-resolved molecular dynamics under vibrational strong coupling (VSC). Herein, equilibrium-nonequilibrium cavity molecular dynamics simulations are performed to compute the two-dimensional (2D) infrared-infrared-Raman (IIR) spectroscopy of liquid water under VSC. In conventional computational chemistry practices, accurate molecular spectra are often constructed by using an advanced molecular dipole or polarizability model to post-process molecular dynamics trajectories evolved under a computationally efficient potential. By contrast, this work highlights the necessity of employing a consistent dipole surface model in both CavMD simulations and spectroscopic post-processing. While utilizing inconsistent dipole models only mildly influences the linear polariton spectrum, it severely distorts 2D spectra in wide frequency regions. With a consistent dipole-induced-dipole model, compared to the outside-cavity molecular 2D-IIR spectrum, the cavity 2D-IIR spectrum splits the OH stretch band to a pair of polariton branches along only the IR (not Raman) axis, while fading molecular signals at other frequency regions. This work provides the foundation for employing direct CavMD simulations to construct 2D spectra of realistic molecules under VSC.

The controlled propagation of spin textures at bifurcations is a critical challenge for racetrack-based logic devices. Here, we investigate the effect of longitudinal and transverse magnetic fields on the propagation of magnetic antivortices at bifurcations within the stripe domain pattern of a reconfigurable NdCo/NiFe racetrack in order to control the preferred antivortex trajectory. Magnetic Transmission X-ray Microscopy experiments were employed to correlate the observed propagation path with the local magnetic configuration. We demonstrate that Zeeman coupling to the magnetization components at the bifurcation core enables switching of the preferred propagation branch using low-amplitude transverse magnetic fields, without modifying the global stripe domain configuration that defines the guiding racetrack landscape. In-plane magnetic anisotropy provides an additional mechanism to break the symmetry between the upper and lower bifurcation branches by tuning the relative orientation between the stripe domain pattern and the longitudinal magnetic fields.

Given a $\vartheta$-Anosov representation into a real reductive group $G$, we construct a natural resonance spectrum associated with the representation. This spectrum is a complex analytic variety of codimension $1$ in $(\mathfrak{a}_\vartheta^*)_{\mathbb{C}}$, the complexified dual of the split component of the associated Levi group $L_\vartheta < G$. We reinterpret several objects from the theory of Anosov representations within this spectral framework and investigate, in higher rank, questions that are classically related to Ruelle-Pollicott theory in the rank-one setting. In particular, the ``leading resonance'' -- which is now a hypersurface -- is identified with the critical hypersurface of the representation. As a corollary of our work, we prove that the zeta functions and Poincaré series associated with Anosov representations admit a meromorphic extension to $(\mathfrak{a}_\vartheta^*)_{\mathbb{C}}$. We also establish sharp mixing estimates for the refraction flow under a Diophantine condition on the representation. Most of our results concerning Anosov representations are obtained as a byproduct of a general theory of free Abelian cocycles over hyperbolic flows. This article is intended as a foundational work toward more advanced results such as higher-rank quantum/classical correspondence, the detection of topological invariants of representations via the value at zero of Poincaré series or the order of vanishing of zeta functions, sharp counting results for the Lyapunov spectrum, etc.

Convolutional codes are a class of error-correcting codes that performs very well over erasure channels with low delay requirements. In particular, Maximum Distance Profile (MDP) convolutional codes, which are defined to have optimal column distances, are able to correct a maximal number of erasures in decoding windows of fixed sizes. However, the required field size in the known constructions for MDP convolutional codes increases rapidly with the code parameters. On the other hand, if the code parameters are small, larger bursts of erasures cannot be corrected. In this paper, we present a new class of convolutional codes, which we call Pseudo-MDP convolutional codes. By definition these codes can correct large bursts of erasures within a prescribed time-delay and still keep part of the advantageous properties of MDP convolutional codes, in the sense that we require some but not all column distances to be optimal. This release in the condition on the column distances allows us to construct Pseudo-MDP convolutional codes over fields of smaller size than those required for MDP convolutional codes with the same code parameters.

A conjecture of Verstraëte states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for $\ell = 4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger's construction for extremal $C_{10}$-free graphs.

The rational design of new materials emerges as an important direction to explore new topological materials, which is based on the understanding of the correlation between crystal and electronic structures. In this paper, we perform a comprehensive study on the crystal and electronic structures in LaAgAs2 through a combination of single-crystal x-ray diffraction (XRD), quantum oscillation, and angle-resolved photoemission spectroscopy (ARPES) experimental measurements, and density functional theory (DFT) calculations. Single-crystal XRD measurements reveal that LaAgAs2 crystallizes into a HfCuSi2-derived structure with the square net distorted into cis-trans chains. Quantum oscillation measurements reveal two frequencies with small effective masses and quasi-two-dimensional (2D) characters. ARPES measurements reveal an electronic structure strikingly different from the square-net-based semimetals, such as LaAgAs2. The Fermi surface is quasi-two-dimensional (2D), with Dirac-like hole pockets at the zone center and a quasi-1D elliptical electron pocket at the zone boundary. Based on the DFT calculations, the measured electronic structure can be well understood regarding the cis-trans distortion, which transforms the two-dimensional square net-derived Dirac bands into quasi-1D trivial bands. Intriguingly, multiple topological states can be identified around the zone center, including a nontrivial Z2 topological surface state and a bulk Dirac state. Our study clarifies the impact of cis-trans distortion and identifies LaAgAs2 as a topological material with multiple topological states near the Fermi level, providing a guideline for intentionally designing new topological materials.

Experimental inferences of cross-field particle flux at the separatrix, $Γ_{\perp}^\mathrm{sep}$, show rapid growth near H-mode and L-mode density limits at high magnetic field on Alcator C-Mod. Increases in $Γ_{\perp}^\mathrm{sep}$ correlate well with proximity to high density operational boundaries as proposed by the separatrix operational space model. $Γ_{\perp}^\mathrm{sep}$ grows as the L-mode density limit and the H-L-mode back transition boundaries are approached, consistent with expectations of plasma instability-driven turbulence suggested by theory, confirming the power dependence of density limits. $Γ_{\perp}^\mathrm{sep}$ is well-organized by the characteristic wavenumber for resistive ballooning mode turbulence, $k_\mathrm{RBM}$, from interchange-drift-Alfvén fluid turbulence theory, with additional dependence on the cylindrical safety factor, $\hat{q}_\mathrm{cyl}$, yielding an empirical limit to plasma operation of $k_\mathrm{RBM}^{2}\hat{q}_\mathrm{cyl} = 1$. This limit corresponds to the point where the perpendicular heat flux, $Q_{\perp}$, reaches the level of the parallel heat flux, $Q_{\parallel}$, i.e. $Q_{\perp} \approx Q_{\parallel}$, beyond which point thermal equilibrium is not satisfied, resulting in a fold catastrophe.

Spin-polarized beams are important for some nuclear and high-energy physics experiments, such as those planned for the future Electron-Ion Collider (EIC). However, maintaining polarization during the acceleration of a charged-particle beam is difficult because the periodic nature of circular accelerators leads to spin-orbit resonances where the spin-precession frequency is a sum of integer multiples of the orbital frequencies. Usually, the dominant depolarization mechanisms are first-order spin-orbit resonances and the depolarization associated with crossing such a resonance can be computed using the Froissart-Stora formula. However, accelerating polarized hadron beams to high energy requires special magnet structures called Siberian snakes. When these are implemented to maintain a spin-precession frequency of one-half the revolution frequency, there will be no first-order spin-orbit resonance crossings. The dominant depolarization mechanisms are then higher-order spin-orbit resonances. The Froissart-Stora formula can be applied to higher-order resonances when the slope of the amplitude-dependent spin tune is constant. However, the slope of the amplitude-dependent spin tune often changes at the moment of resonance crossing. This work introduces a generalization of the Froissart-Stora formula which is applicable when the slope changes in this manner. The applicability of this formula is demonstrated through tracking simulations of a higher-order resonance crossing in both a toy model and the Relativistic Heavy Ion Collider (RHIC). It is additionally shown that the Froissart-Stora formula is mathematically equivalent to the Landau-Zener formula for the diabatic transition probability in two-level systems with a linearly increasing energy gap and constant coupling. This work therefore also extends the Landau-Zener formula to the case of changing slope.

Communication-aware control is essential to reduce costs and complexity in large-scale networks. This work proposes a method to design dissipativity-augmented output feedback controllers with reduced online communication. The contributions of this work are three fold: a generalized well-posedness condition for the controller network, a convex relaxation for the constraints that infer stability of a network from dissapativity of its agents, and a synthesis algorithm integrating the Network Dissipativity Theorm, alternating direction method of multipliers, and iterative convex overbounding. The proposed approach yields a sparsely interconnected controller that is both robust and applicable to networks with heterogeneous nonlinear agents. The efficiency of these methods is demonstrated on heterogeneous networks with uncertain and unstable agents, and is compared to standard $\cH_\infty$ control.

We study the linear-quadratic optimal control problem for infinite-dimensional dissipative systems with possibly indefinite cost functional. Under the assumption that a storage function exists, we show that this indefinite optimal control problem is equivalent to a linear-quadratic optimal control problem with a nonnegative cost functional. We establish the relationship between the corresponding value functions and present the associated operator Lur'e equation. Finally, we illustrate our results with several examples.

The deployment of extremely large-scale antenna array (ELAA) in sixth-generation (6G) communication systems introduces unique challenges for efficient near-field channel estimation. To tackle these issues, this paper presents a theory-guided approach that incorporates angular information into an attention-based estimation framework. A piecewise Fourier representation is proposed to implicitly encode the near-field channel's inherent nonlinearity, enabling the entire channel to be segmented into multiple subchannels, each mapped to the angular domain via the discrete Fourier transform (DFT). Then, we develop a joint subchannel-spatial-attention network (JSSAnet) to extract the spatial features of both intra- and inter-subchannels. To guide theoretically the design of the joint attention mechanism, we derive upper and lower bounds based on approximation criteria and DFT quantization loss mitigation, respectively. Following by both bounds, a JSSA layer of an attention block is constructed to assign independent and adaptive spatial attention weights to each subchannel in parallel. Subsequently, a feed-forward network (FFN) of an attention block further captures and refines the residual nonlinear dependencies across subchannels. Moreover, the proposed JSSA map is linearly computed via element-wise product combining large-kernel convolutions (DLKC), maintaining strong contextual learning capability. Numerical results verify the effectiveness of embedding sparsity information into the attention network and demonstrate JSSAnet achieves superior estimation performance compared with existing methods.

We prove Conjecture~2 of Bondarenko, Ortega-Cerdà, Radchenko, and Seip for the three-term recurrence attached to the Hörmander--Bernhardsson extremal function $φ$. More precisely, define \[ \widetilde u_{-1}=0,\qquad \widetilde u_0=1, \] and \[ \widetilde u_{n+1} = \frac{4n+2}{n+1}\bigl(n(n+1)-λ\bigr)\widetilde u_n + \frac{4n}{n+1}x\,\widetilde u_{n-1}. \] Then \[ \widetilde u_n(x,λ)\in\mathbb Z[x,λ] \qquad(n\ge0). \] The proof is a determinant comparison in the scaled Legendre basis. After sign reversal and central-binomial normalization, the recurrence becomes exactly the continuant recurrence of a finite tridiagonal compression. In particular, if $T_n(a,λ)$ denotes the $n$th BOCRS tridiagonal truncation, then \[ \widetilde u_{n+1}(a^2,λ)=\binom{2n+2}{n+1}\det T_n(a,λ). \] As consequences, we derive that \[ \left(\fracπ{4C}\right)^2 \quad\text{and}\quad -\frac{L_τ(1)}{2C} \] are not simultaneously rational, where \(C\) is the sharp point-evaluation constant for $PW^1$, $\pmτ_n$ are the nonzero zeros of $φ$, and $ L_τ(1)=\sum_{n\ge1}\frac{(-1)^n}{τ_n}.$ Finally, if we write $φ(z)=\sum_{n\ge0}c_n z^{2n},$ then \[ c_n\in C^n\,\mathbb Z[π^2,C,L_τ(1)] \qquad(n\ge0). \]

There is a well understood way of generating random coverings of a fixed manifold by sampling homomorphisms from the fundamental group of this manifold into the symmetric group. We prove a central limit theorem for the number of connected components of these random coverings when the fundamental group is nilpotent. This provides a nonabelian generalization of an earlier result by the author and Shannon Starr in the case of the torus where the fundamental group is a free abelian group of rank at least two. Our result relies on the work of du Sautoy and Grunewald on the subgroup growth zeta functions of nilpotent groups, and on Delange's generalization of the Wiener-Ikehara Tauberian theorem.

This study investigates the impact of educational comics as an active learning strategy in physics workshops for undergraduate students in Chemistry and Pharmacy and Biochemistry during the second semester of 2025. Conceptual understanding was assessed using the Force Concept Inventory (FCI), and student motivation and attitudes toward physics were evaluated through a Likert-type survey administered in pre- and post-test formats. The results show an average normalized gain of g = 0.21 on the FCI, corresponding to a low-to-medium range according to physics education research. A higher gain is observed in items directly related to the intervened content (g = 0.23) compared to non-intervened items (g = 0.19), suggesting that instructional design influences domain-specific conceptual development. At the motivational level, improvements are observed in student interest, self-efficacy, and perceived usefulness of physics, along with a reduction in negative emotional responses toward the subject. These findings indicate that educational comics can serve as an effective pedagogical scaffold, promoting positive learning dispositions and supporting targeted conceptual development in non-physics undergraduate contexts.

We study an inverse problem for the viscoacoustic wave equation, an integro-differential model describing wave propagation in viscoacoustic media with memory in the leading order term. The medium is characterized by a spatially varying sound speed and a space-time dependent memory kernel. Assuming that waves are generated by sources supported outside the region of interest, we consider exterior measurements encoded by the source-to-solution map. To study this inverse problem, we construct solutions concentrating near fixed geodesics and establish a corresponding propagation of singularities result for the semiclassical wave front set. These results are valid without any restriction on the underlying sound speed. Then, under certain geometric conditions, we prove that the exterior data uniquely determine not just the sound speed inside the domain but also all time derivatives at zero of the memory kernel. This involves a reduction to the lens rigidity and geodesic ray transform inverse problems. As an application, we establish uniqueness for the recovery of variable parameters in the extended Maxwell model.