Community Health Workers (CHWs) play a critical role in delivering primary healthcare services in low-resource settings, yet sustaining their training and performance remains a persistent challenge. Prior research has explored digital and game-based approaches for CHW training. However, limited work has synthesized longitudinal design insights into generalizable guidelines for interactive health interventions. Building on a four-year design-based research program involving multiple game-based refresher training systems, including quiz-based mobile apps, physical and augmented reality games, card-based games, and location-based games, we examine which design guidelines support sustained engagement, learning transfer, and contextual appropriateness in CHW training. We conducted a mixed-methods analysis across deployments with Accredited Social Health Activists and Anganwadi Workers in India, including interviews, field observations, and usage logs. Through thematic synthesis, we derive eight design guidelines addressing contextual realism, adaptive learning, hybrid interaction, social motivation, explainability, professional identity, and ethical considerations. Our findings contribute actionable design knowledge for researchers and practitioners developing interactive health interventions in low-resource healthcare contexts.

Digital health technologies are increasingly used to improve healthcare access and delivery worldwide. However, many healthcare applications are designed for environments with stable infrastructure, high digital literacy, and strong institutional support. These assumptions often do not hold in low-resource contexts where healthcare delivery often depends on community health workers, caregivers, and informal care networks. Designing effective healthcare applications for such environments requires attention to infrastructural constraints, cultural contexts, language diversity, and usability challenges. This Birds of a Feather session aims to bring together researchers, designers, and practitioners interested in healthcare application design in low-resource contexts. The session will provide an informal forum for discussing challenges encountered in the design and deployment of digital health technologies in underserved settings, sharing field experiences, and identifying opportunities for collaboration within the Interactive Health (IH) community.

We study the iteration that replaces a planar hexagon by the hexagon formed by joining the midpoints of consecutive edges. While this iteration quickly drives any polygon toward a point and their shapes asymptotically regularize, we show a stronger and unexpected rigidity holds for hexagons: from the second iterate onward, the centroids of the filled hexagons all lie exactly on a fixed line. This exact colinearity reflects a special algebraic feature of the hexagonal case and does not hold generally for any other polygons.

Let $\mathfrak g$ be a finite simple Lie algebra, and let $r$ denote the ratio of the square length of long roots to that of short roots. Let $\wp>2r$ be an integer and $ζ$ a primitive $\wp$-th root of unity. Denote by $\mathcal U_ζ(\widehat{\mathfrak g})$ the Lusztig big quantum affine algebra at root of unity defined by divided powers. In this paper, we establish a current algebra presentation of $\mathcal U_ζ(\widehat{\mathfrak g})$. Based on this presentation, we construct a $\mathbb Z_\wp$-module quantum vertex algebras $V_{\wp,τ}^\ell(\mathfrak g)$ for each integer $\ell$. Moreover, we establish a fully faithful functor from the category of smooth weighted $\mathcal U_ζ(\widehat{\mathfrak g})$-modules of level $\ell$ to the category of $(\mathbb Z_\wp,χ_ϕ)$-equivariant $ϕ$-coordinated quasi-modules of $V_{\wp,τ}^\ell(\mathfrak g)$, where $χ_ϕ:\mathbb Z_\wp\to\mathbb C^\times$ is the group homomorphism defined by $s\mapsto ζ^s$. We also determine the image of this functor. The structure $V_{\wp,τ}^\ell(\mathfrak g)$ is substantially different from that of affine vertex algebras. We realize $V_{\wp,τ}^\ell(\mathfrak g)$ as a deformation of a simpler quantum vertex algebra $V_{\wp,\varepsilon}^\ell(\mathfrak g)$ by using vertex bialgebras, and decompose $V_{\wp,\varepsilon}^\ell(\mathfrak g)$ into a Heisenberg vertex algebra and a more interesting quantum vertex algebra determined by a quiver.

For a knot $K\subset S^3$, let $S(K)$ denote the set of knot types represented by simple closed curves on a minimal genus Seifert surface of $K$. We study the directed relation $K\to J$ defined by $J\in S(K)$, which we call the \emph{hosting relation}, and call its symmetric part friendship. This gives a new framework for describing how knots appear on minimal genus Seifert surfaces of other knots. A classical result of Lyon implies that the family of torus knots is a universal host family: every non-trivial knot is hosted by some torus knot. In contrast, a central result of this paper is that no knot is a universal host: for every knot $K$, there exists a knot $J$ such that \[ J\notin S(K). \] Thus universal hosting occurs at the level of families, but never at the level of a single knot. We also study explicit examples of hosting and friendship. In particular, we describe the hosting set of the trefoil in terms of primitive slope classes on its once-punctured torus fiber, and use this description to obtain concrete friendship and non-friendship phenomena. For example, we show that $3_1$ and $8_{19}$ are friends, whereas $3_1$ and $4_1$ are not. These results provide a framework for studying universal host phenomena, hosting, and friendship among knots on minimal genus Seifert surfaces, and suggest further connections with graph-theoretic, rigidity, and categorical aspects of knot theory.

Low Earth orbit (LEO) satellites play an essential role in intelligent Earth observation by leveraging artificial intelligence models. However, limited onboard memory and excessive inference delay prevent the practical deployment of large language models (LLMs) on a single satellite. In this paper, we propose a communication-efficient collaborative LLM inference scheme for LEO satellite networks. Specifically, the entire LLM is split into multiple sub-models, with each deployed on a satellite, thereby enabling collaborative LLM inference via exchanging intermediate activations between satellites. The proposed scheme also leverages the pipeline parallelism mechanism that overlaps sub-model inference with intermediate activation transmission, thereby reducing LLM inference delay. An adaptive activation compression scheme is designed to mitigate cumulative errors from multi-stage model splitting while preserving inference accuracy. Furthermore, we formulate the LLM inference delay minimization problem by jointly optimizing model splitting and compression ratios under onboard memory and inference accuracy constraints. The problem is transformed into a shortest-path search problem over a directed acyclic graph that edge weights explicitly quantify the inference delay induced by model splitting and compression strategies, which is solved via a modified A Star-based search algorithm. Extensive simulation results indicate that the proposed solution can reduce inference delay by up to 42% and communication overhead by up to 71% compared to state-of-the-art benchmarks, while maintaining the inference accuracy loss of less than 1%.

Quasi one dimensional materials provide a compelling platform where reduced dimensionality stabilizes intertwined topological and superconducting phases. Here we report superconductivity in a new Bi based quasi 1D compound, PtPb3Bi, which hosts a nontrivial electronic structure. It exhibits type II superconductivity below 3.01(1) K. Heat capacity and transverse field muon spin rotation relaxation (muSR) measurements demonstrate a fully gapped isotropic s wave state with moderate electron phonon coupling, while zero field muSR confirms the preservation of time reversal symmetry (TRS). Transport measurements reveal low carrier mobility with diffusive normal state transport. Electronic structure calculations show strong dispersion along the quasi 1D direction and relatively flatter bands in the transverse plane, giving rise to pronounced Fermi surface nesting in the kx-ky plane. Consistent with this, the compound undergoes a charge density wave transition at 280(1) K. The flow of Wannier charge centers, together with surface state dispersion, establishes nontrivial band topology. These results identify PtPb3Bi as a new quasi 1D superconductor with nontrivial electronic structure and a promising candidate for topological superconductivity.

We compute the logarithmic asymptotics of the non-existence probability (and more generally the lower-tail probability) for a wide variety of combinatorial problems for a range of parameters in the `critical regime' between the regime amenable to hypergraph container methods and that amenable to Janson's inequality. Examples include lower tails and non-existence probabilities for subgraphs of random graphs and for $k$-term arithmetic progressions in random sets of integers. Our methods apply in the general framework of estimating the probability that a $p$-random subset of vertices in a $k$-uniform hypergraph induces significantly fewer hyperedges than expected. We show that under some simple structural conditions on the hypergraph and an upper bound on $p$ determined by a phase transition in the hard-core model on the infinite $k$-uniform, $Δ$-regular, linear hypertree, this probability can be accurately approximated by the Bethe free energy evaluated at the unique fixed point of a Belief Propagation operator on the hypergraph.

This paper develops a unified analytical framework for determinant identities under finite-rank perturbations of square matrices that remains valid without invertibility assumptions. In contrast to classical inverse-based formulations, the approach is based on an adjugate-driven additive representation, which extends naturally to singular matrices and yields explicit, non-asymptotic formulas. Building on this representation, we derive recursive and multiplicative expressions describing the evolution of determinant and log-determinant quantities under successive rank-one updates. These results reveal a structural interpretation in which determinant-based quantities evolve as cumulative measures of independent directions, providing a precise decomposition of incremental contributions. To address the singular case, we develop a systematic extension based on the Drazin inverse and the pseudodeterminant, leading to closed-form identities that isolate the contribution of the nonzero spectrum. In particular, we obtain a generalized determinant formula that can be viewed as a singular counterpart of the matrix determinant lemma. The spectral impact of low-rank perturbations is analyzed, yielding explicit conditions governing eigenvalue shifts and stability preservation. The proposed framework establishes a direct analytical link between matrix perturbation theory and system-theoretic concepts. In particular, we show that the pseudodeterminant of controllability Gramians admits a multiplicative decomposition that explicitly quantifies the incremental expansion of the reachable subspace under successive inputs. This leads to a unified interpretation of information accumulation, uncertainty reduction, and reachability in both full-rank and rank-deficient linear systems.

This paper studies an $α$-robust utility maximization problem where an investor faces an intractable claim -- an exogenous contingent claim with known marginal distribution but unspecified dependence structure with financial market returns. The $α$-robust criterion interpolates between worst-case ($α=0$) and best-case ($α=1$) evaluations, generalizing both extremes through a continuous ambiguity attitude parameter. For weighted exponential utilities, we establish via rearrangement inequalities and comonotonicity theory that the $α$-robust risk measure is law-invariant, depending only on marginal distributions. This transforms the dynamic stochastic control problem into a concave static quantile optimization over a convex domain. We derive optimality conditions via calculus of variations and characterize the optimal quantile as the solution to a two-dimensional first-order ordinary differential equation system, which is a system of variational inequalities with mixed boundary conditions, enabling numerical solution. Our framework naturally accommodates additional risk constraints such as Value-at-Risk and Expected Shortfall. Numerical experiments reveal how ambiguity attitude, market conditions, and claim characteristics interact to shape optimal payoffs.

We extend earlier international efforts to optimise hexahedral-based spectral element methods on GPUs and vectorised CPUs to mixed element meshes additionally involving prismatic, pyramidic, and tetrahedral shapes using tensorial expansions. We demonstrate that common finite element operators (such as the mass and Helmholtz matrices) benefit from alternative implementation strategies depending on the element shape, choice of polynomial order, and system architecture in order to achieve optimal performance. In addition, we introduce a new approach/interpretation to efficiently evaluate more complex operations involving inner products with the derivative of the expansions as part of the integrand such as the stiffness matrix. This approach seeks to maximise operations using the collocation properties of the nodal tensorial expansion associated with classical quadrature rules. Our GPU performance tests demonstrate that the throughput of the Helmholtz operator on tetrahedral elements is at most 2.5 times slower than on hexahedral elements, despite tetrahedra having a factor of six greater floating-point operations.

Ruddlesden-Popper (RP) nickelates have recently emerged as a new family of high-temperature superconductors. In bilayer RP nickelates, magnetic excitations with large exchange couplings have been observed, supporting a spin-mediated pairing mechanism. Whether comparable spin correlations persist in trilayer nickelates, however, remains unknown. Here, we present a Ni $L$-edge resonant inelastic X-ray scattering (RIXS) study of La$_4$Ni$_3$O$_{10}$ single crystals. While the orbital excitations remain similar to those of La$_3$Ni$_2$O$_{7}$, the collective spin excitations in La$_4$Ni$_3$O$_{10}$ exhibit a comparable bandwidth of about $60$ meV but substantially suppressed spectral weight, implying a weaker electronic correlation in the trilayer compounds. Our results underscore the three-dimensional and multi-orbital electronic character in La$_4$Ni$_3$O$_{10}$, highlighting important differences from the bilayer nickelates. These findings provide crucial insights into the evolution of magnetism across the RP nickelate family and its connection to superconductivity.

We consider an optimal dividend payout problem for an insurance company whose surplus follows the classical Cramér-Lundberg model. The dividend rate is subject to a ratcheting constraint (i.e., it must be nondecreasing over time), and the company may inject capital at a proportional cost to avoid ruin. This problem gives rise to a stochastic control problem with a self-path-dependent control constraint, costly capital injections, and jump-diffusion dynamics. The associated Hamilton-Jacobi-Bellman (HJB) equation is a partial integro-differential variational inequality featuring both a nonlocal integral term and a gradient constraint. We develop a systematic probabilistic and PDE-based approach to solve this HJB equation. By discretizing the space of admissible dividend rates, we construct a sequence of approximating regime-switching systems of ordinary integro-differential equations. Through careful a priori estimates and a limiting argument, we prove the existence and uniqueness of a \emph{strong solution} in a suitable space. This regularity result is fundamental: it allows us to characterize the optimal dividend policy via a switching free boundary and to construct an explicit optimal feedback control strategy. To the best of our knowledge, this is the first complete solution -- comprising both the value function and an implementable optimal strategy -- for a dividend ratcheting problem with capital injection under the Cramér-Lundberg model. Our work advances the mathematical theory of optimal stochastic control beyond the standard viscosity solution framework, providing a rigorous foundation for dividend policy design in economics.

Plane waves and pp-waves are well-known universal metrics that solve all metric-based gravitational field equations. Similarly, the Kerr-Schild-Kundt class of metrics is almost universal: all metric-based gravitational field equations reduce to a linear scalar partial differential equation that always admits a solution. Here, we add a new member to this class of metrics and show that nonzero constant curvature pp-wave metrics are also almost universal. They reduce the generic gravity field equations to those of cosmological Einstein-Maxwell theory with null dust. The background of the pp-waves has the topology $\mathbb{R}^{1,1}\times S^{2}$ and provides the missing partner to the Nariai metric with ${\rm dS}^{2}\times S^{2}$ and the Bertotti-Robinson metric with ${\rm AdS}^{2}\times S^{2}$ topologies. These quantum-protected metrics are of clear interest. We exemplify our results by using the quadratic and cubic gravity theories.

In this paper, we study estimation of parameters in a two-parameter Potts model with $q$ colors and coupling matrix $A_N$. We characterize concrete sufficient conditions for existence of the pseudo-likelihood estimator of the Potts model, in terms of the local magnetic fields, and give sufficient conditions for the validity of the above characterization. We then provide sufficient criteria for estimation of both parameters at the optimal rate $\sqrt{N}$. In particular, if $A_N$ is the scaled adjacency matrix of a graph $G_N$, then we show that joint estimation is possible if either $G_N$ has bounded degree or is irregular. In contrast, we give an example of a graph sequence $G_N$ which is approximately regular and dense, where no consistent estimator exists. We also show that one-parameter estimation at the optimal rate $\sqrt{N}$ holds under much milder conditions when the other parameter is known. Along the way, we develop a concentration result for mean-field Potts models using the framework of nonlinear large deviations. Compared to the Ising case, our results for the Potts case require a novel analysis across multiple colors.

The stochastic series expansion (SSE) algorithm is one of the most powerful quantum Monte Carlo methods and has been extensively applied to the study of quantum many body systems. Its efficiency is particularly enhanced with a deterministic loop update scheme in the study of the S=1/2 quantum spin systems that preserve SU(2) spin rotational symmetry. Once the symmetry is broken, such as by an external field, a directed loop method is typically required, resulting in a significant reduction in efficiency. Inspired by the SSE approach developed for the quantum Ising model, we introduce a deterministic loop SSE method that is particularly suited for antiferromagnetic systems under a staggered magnetic field. This method enables separate investigations of longitudinal and transverse modes in magnetically ordered phases arising from spontaneous symmetry breaking. We benchmark the performance of our algorithm against the standard directed loop approach applied to the antiferromagnetic Heisenberg chain and demonstrate that our method substantially reduces CPU time per Monte Carlo step, thereby can outperform the directed loop algorithm in efficiency.

Carrier multiplication (CM) describes a strong charge-carrier interaction process in semiconductor colloidal nanocrystals (NCs), wherein two band-edge excitons are simultaneously created by an absorbed photon with at least twice the bandgap energy (2 Eg). While being fundamentally intriguing, it has been exclusively utilized to enhance the light-to-electricity conversion efficiencies in the photodetector and solar-cell devices. In this report, we have synthesized the core/shell perovskite FAPbI3/NdF3 NCs with a biexciton recombination lifetime of ~3.9 ns, and demonstrated that a CM efficiency of ~25.7% can be achieved under the ~355 nm laser excitation (~2.21 Eg). This CM occurrence leads to a two-fold reduction in the optical gain threshold, as compared to that obtained under the ~640 nm laser excitation (~1.23 Eg). When combined with the single-exciton and zero-threshold optical gain schemes previously developed for semiconductor colloidal NCs, the CM effect introduced here would further mitigate the optical-pumping requirement for the routine operation of continuous-wave lasing.

We propose an effective Bethe ansatz for solving quantum many-body systems near an integrable point. Our approach retains the functional form of the Bethe wave function while renormalizing the Bethe roots to account for integrability-breaking interactions. These effective roots are determined by minimizing physically motivated cost functions. The resulting off-shell Bethe states serve as approximate eigenstates of the non-integrable models. We assess the quality of the approximation using various physical observables, including the energy eigenvalue, state fidelity, and bipartite entanglement entropy. Our tests show that for models with weak integrability-breaking, the effective Bethe ansatz provides a high-quality approximation to the exact eigenstates over a wide range of deformation parameters. In contrast, for models with strong integrability-breaking interactions, the efficacy of the effective Bethe ansatz degrades relatively quickly as the deformation parameter increases. The efficacy of the method thus offers a useful probe for characterizing the strength of integrability breaking. Within its regime of accuracy, it also provides a new representation of the eigenstates of nearly integrable models, enabling one to exploit the algebraic structure inherited from integrability.

We introduce the space $\mathcal{W}^{s,p}(\mathbb{D})$ of analytic functions $u$ on the unit disc such that the radial restrictions $u_{r}(ξ):=u(rξ)$ satisfy the Gagliardo seminorm-only bound \[ \sup_{0<r<1}[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}<\infty, \] with no $\emph{a priori}$ control of $\sup_{r}\|u_{r}\|_{L^{p}(\mathbb{S}^{1})}$. Our main result shows that this assumption already forces $u\in H^{p}(\mathbb{D})$ and that the radial boundary trace $u^{*}$ belongs to $W^{s,p}(\mathbb{S}^{1})$, with $u_{r}\to u^{*}$ in $W^{s,p}(\mathbb{S}^{1})$ as $r\to1^{-}$. The key mechanism combines the mean-value property (which pins the constant mode at $u(0)$) with a fractional Poincar$é$ inequality on $\mathbb{S}^{1}$, recovering $L^{p}$ control from oscillation alone. As a consequence, the trace map $u\mapsto u^{*}$ is a surjective isomorphism $\mathcal{W}^{s,p}(\mathbb{D})\xrightarrow{\sim}B^{s}_{p,p,+}(\mathbb{S}^{1})$ with explicit norm equivalence.

This paper presents a 1-bit reconfigurable intelligent surface (RIS) fabricated using a three-layer structure. It employs a manual layer stackup incorporating an optimal air gap to reduce the effective dielectric losses while using a low-cost FR4 substrate. The new design of the unit cells of the proposed RIS is outlined, with each unit cell featuring a PIN-diode-based, compact, simplified biasing network that simplifies the control circuit while maintaining distinct $\boldsymbol{0^\circ/180^\circ \pm 20^\circ}$ phase states between ON/OFF conditions. The designed RIS is in the form of a $\boldsymbol{10\times10}$ array with a compact size of $\boldsymbol{2.9λ_g \times 2.9λ_g}$. Additionally, a phase-gradient coding scheme is presented and utilized that achieves measured beam steering up to $\boldsymbol{\pm30^\circ}$ in both anechoic and noisy environments. Controlled and driven by an Arduino-cum-digital interface, the proposed RIS exhibits measured reflected wave gain enhancement of about 9\,dB over an incident wave angular range of $\boldsymbol{\pm 30^\circ}$. Furthermore, the design is also experimentally validated by transmitting quadrature phase-shift keying-modulated symbols via the RIS-assisted wireless channel. The proposed RIS works for the range 3.38--3.67\,GHz (8.3\%), and is suitable for deployment for the 5G n78 \mbox{band (3.5\,GHz).}

In this work, we have analyzed the nuclear structure and several prospective decay characteristics of the $^{240-259}$Es$_{99}$ isotopes. For this we use Relativistic Mean Field model (RMF) with NL-SH and NL3* force parameter in an axially deformed oscillator basis. In structural properties, we have analyzed binding energy (B.E.), skin thickness ($r_{np}$) , charge radius ($r_c$), one neutron separation energy ($S_{1n}$), two neutron separation energy ($S_{2n}$), differential variation of two neutron separation energy ($dS_{2n}$), the single particle energy and its variation with quadrupole deformation parameter of Es isotopes. We have also estimated the $α$-decay, $β$-decay and cluster decay half lives of Es isotopes to analyze the shell structure and also to predict the suitable decay mode among them. The $α$-decay half-life periods are calculated using the MUDL and AKRE formulae using both our calculated Q-values and empirically assessable Q-values. In a similar manner, we have computed the half-lives of cluster decay using Universal Decay Law and HOROI formula. A longer decay half-life indicates a shell stabilized parent nucleus, while a small parent half-life suggests the shell stability of the daughter. This study provides us the insights regarding the structural changes with the change in neutron number enabling us to predict shell closures and nuclear stability. We found a shell/sub-shell closure at N = 154 for the NL-SH parameter set. This research aids in our comprehension of Es isotopes' shell structure and decay mechanism.

Thumb gestures provide an effective and unobtrusive input modality for wearable and always-available human-machine interaction. Wrist-worn surface electromyography (sEMG) has emerged as a promising approach for compact and wearable human-machine interfaces. However, compared to forearm sEMG, the impact of electrode configuration on wrist-based decoding performance remains understudied. We systematically investigated electrode configuration strategies for wrist-based thumb-movement recognition using high-density (HD) and low-density (LD) sEMG measurement systems. We considered factors such as muscle region, reference scheme, channel count, and spatial density of the electrode. Experimental results show that 1) extensor-side electrodes outperform flexor-side electrodes (HD: 0.871 vs. 0.821; LD: 0.769 vs. 0.705); 2) monopolar recordings consistently outperform bipolar configurations (15 channel with HD monopolar vs. LD bipolar: 0.885 vs. 0.823); and 3) increasing channel count enhances performance, but exhibits diminishing returns. We further show that electrode spatial distribution introduces a trade-off between spatial coverage and compactness. The findings suggest that the effectiveness of wrist-worn sEMG systems depends less on the deployment of a large number of electrodes in a broad sensing area and more on the optimization of electrode placement and the referencing scheme. This work provides practical guidelines for developing efficient wrist-worn sEMG-based gesture recognition systems.

The $\upmu$RWELL-PICOSEC detector, which combines a $\upmu$RWELL gaseous amplification structure with a Cherenkov radiator and photocathode, is a novel approach to acheive fast and precise timing in gaseous detectors. With timing precision at the level of tens of picoseconds, this technology is particularly suited for time-of-flight (TOF) applications in particle physics and potentially medical imaging. Beam tests with a 150~GeV/$c$ muon beam have been carried out on a large-area (10~$\times$~10~cm$^{2}$) prototype equipped with a cesium iodide (CsI) photocathode. Using an oscilloscope-based single-channel readout, timing measurements on two individual pads of the detector have yielded resolutions of $\approx$ 48 ps and $\approx$ 52 ps under different biasing conditions respectively.

Reconfigurable antenna technology, such as movable antennas (MAs) and rotatable antennas (RAs), has emerged as a promising solution to enhance the communication performance of wireless systems by exploiting the new degree of freedom (DoF) in antenna reconfiguration. However, existing RA designs mostly consider the array-wise or antenna-wise rotation only, which constrain their great potential in the wide-range radiation pattern control. To overcome this limitation, we propose a new hierarchical rotational six-dimensional MA (HR-6DMA) architecture to improve downlink coverage, which exploits array-wise rotation for global orientation adjustment and individual antenna rotation for fine-grained radiation refinement. Based on this array architecture, we then formulate an optimization problem to maximize the minimum beamforming gain over a target region by jointly optimizing the two-level rotations and transmit beamforming. To solve this non-convex problem, an efficient algorithm is proposed, where the transmit beamforming and per-antenna rotation are optimized via alternating optimization under any feasible array rotation, followed by a low-complexity linear search to determine the optimal array rotation. Last, numerical results show that the proposed HR-6DMA significantly improves the minimum beamforming gain over fixed and single-level rotatable arrays.

Microscopic heat engines operate in regimes where thermodynamic quantities fluctuate strongly, making stochastic effects an essential aspect of their performance. However, existing geometric formulations of finite-time thermodynamics primarily characterize average dissipation and do not systematically capture its fluctuations. Here, we develop a unified geometric framework that consistently describes both the mean dissipated availability and its fluctuations. In the linear-response regime, we show that these quantities are governed by metric tensors constructed from equilibrium correlation functions, providing a common geometric structure for dissipation and its fluctuations. This framework yields geometric bounds on both the mean and variance of the dissipated availability, and thereby on the efficiency and its fluctuations. The formalism applies broadly to stochastic systems, including Markov jump processes and overdamped and underdamped Brownian dynamics, establishing a unified geometric description across microscopic heat engines.

We study deterministic exploration by a single agent in $T$-interval-connected graphs, a standard model of dynamic networks in which, for every time window of length $T$, the intersection of the graphs within the window is connected. The agent does not know the window size $T$, nor the number of nodes $n$ or edges $m$, and must visit all nodes of the graph. We consider two visibility models, $KT_0$ and $KT_1$, depending on whether the agent can observe the identifiers of neighboring nodes. We investigate two fundamental questions: the minimum window size that guarantees exploration, and the optimal exploration time under sufficiently large window size. For both models, we show that a window size $T = Ω(m)$ is necessary. We also present deterministic algorithms whose required window size is $O(ε(n,m)\cdot m + n \log^2 n)$, where $ε(n,m) = \frac{\ln n}{1 + \ln m - \ln n}$. These bounds are tight for a wide range of $m$, in particular when $m = n^{1+Θ(1)}$. The same algorithms also yield optimal or near-optimal exploration time: we prove lower bounds of $Ω((m - n + 1)n)$ in the $KT_0$ model and $Ω(m)$ in the $KT_1$ model, and show that our algorithms match these bounds up to a polylogarithmic factor, while being fully time-optimal when $m = n^{1+Θ(1)}$. This yields tight bounds when parameterized solely by $n$: $Θ(n^3)$ for $KT_0$ and $Θ(n^2)$ for $KT_1$.

In this paper, we present a new qualitative extension of the Hopf theorem (and a generalization of Borsuk-Ulam theorem), concerning continuous maps $f$ from a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We remove the assumption of a Riemannian structure and instead consider closed triangulable manifolds $M$ equipped with a topological notion of 'distant' points. We show that for any continuous map $f \colon M \to \mathbb{R}^n$, there exists a connected component in the space of $f$-neighbors (where a pair of points $a, b$ are $f$-neighbors if $f(a) = f(b)$) that contains both a pair of 'distant' points and a pair of identical points. This result yields further consequences for Lusternik-Schnirelmann and Tucker-type theorems, as well as a multidimensional extension of the mountain-climbing lemma, which in the special case of the standard Euclidean $2$-sphere, may be stated informally as follows. For any continuous distribution of temperature and pressure on Earth (assumed time-independent), there exists a pair of antipodal points with identical values such that travelers starting from these points can move and meet while, at each moment of their journey, experiencing matching 'climatic conditions' up to an arbitrarily small constant.

We propose and analyze a hybridized discontinuous Galerkin (HDG) method for the spherically symmetric Einstein--scalar system in Bondi gauge. After rewriting the model as a local first-order PDE--ODE system by introducing suitable scaled variables, we construct a semidiscrete scheme in which the element unknowns are computed locally and the coupling is carried by traces on the mesh skeleton. In the present radial setting, these traces can be eliminated recursively, so that only the main evolution variable is advanced in time, while the metric variables are recovered from discrete constraint relations. We prove local semidiscrete well-posedness, derive a global \(L^2\)--stability estimate, establish an optimal order \(L^2\) error bound for the main evolution variable for polynomial degree \(k\ge 1\), and obtain reconstruction error estimates for the metric variables and the associated mass functional. Numerical experiments verify the predicted spatial convergence rate and illustrate qualitative features of the Einstein--scalar dynamics, including large-data collapse profiles and smooth-pulse evolution.

We study the Cauchy problem for the nonlinear Schrödinger equation characterized by contrasting effects between the concentration at the origin of a critical Hardy potential and the intrinsic nonlocality of a Choquard nonlinearity. We prove the existence of a ground state solution through optimizers of an interpolation Hardy-Gagliardo-Nirenberg inequality and derive a non-existence result via Poho\v zaev identities. Using these results, we provide various criteria for the global existence and finite-time blow-up for the problem in the energy-subcritical regime. Finally, we establish a key compactness result, which enables us to obtain a characterization of finite-time blow-up solutions with minimal mass.

For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus E(H_i)$ as edges that complete a new copy of $F$. The running time of this $F$-process, denoted by $M_F(H_0)$, is the smallest $i$ with $H_i=H_{i+1}$. Bollobás proposed the problem of determining the maximum running time for $n\in\mathbb{N}$, i.e., $$M_F(n)=\max_{\vert V(H_0)\vert=n}M_F(H_0)\,.$$ Recently, Noel and Ranganathan initiated the study of this quantity for $k$-graphs. In this work, we determine the asymptotics of $M_F(n)$ for a large class of $k$-graphs. Given a graph $G=(V,E)$, the $k$-extension of $G$ is a $k$-graph $F^{(k)}(G)$ obtained from $G$ by enlarging each edge with a $(k-2)$-set of new vertices. We show that for every graph $G$ on $t$ vertices and every $k\geq 3$, $M_{F^{(k)}(G)}(n)\leq C_{k,t}$ for some constant $C_{k,t}$ depending only on $t$ and $k$.