Over the last decade, advances in GPU hardware have been driven in large part by the demands of real-time graphics, culminating in dedicated hardware ray tracing cores (RT cores). These units accelerate ray scene intersection queries directly in hardware, making physically based ray tracing algorithms increasingly practical for interactive applications. This paper compares and analyzes the performance of two ray-based rendering algorithms: forward path tracing (PT) and wavefront path tracing (WPT). GPU-based PT computes the color of each pixel by having each thread trace a single path to completion, naturally leading to a megakernel approach - while WPT maintains state buffers between specialized kernel invocations to trace path stages simultaneously. We find that WPT affords a ~16% speedup over PT in our implementation. By analyzing traces from NVIDIA Nsight Graphics, we attributed this speedup to WPT's improved cache locality compared to PT. We also find that our implementation does not achieve maximum GPU throughput across any of its units, suggesting that communication and memory latency, as well as synchronization, are the limiting factors. Finally, we address potential algorithmic improvements and future work for real-time path tracing implementation for practical applications.

The condensation of photons into a macroscopically populated ground state, a superradiant phase transition (SRPT), is one of the most striking predictions of cavity quantum electrodynamics (cQED), yet has resisted experimental realization in equilibrium for over fifty years. Whether such a transition can survive at all in light-matter coupled systems is still debated, with a widely established \textit{No-Go} theorem ruling it out across several models. Graphene cyclotron transitions ultrastrongly coupled to terahertz (THz) cavities have been at the heart of this debate. At leading order, the linear Dirac dispersion of electrons does not generate the diamagnetic $\vec{A}^2$ term that enforces the theorem, making graphene the cleanest candidate for a No-Go-evading phase transition. Here, we present the first THz spectroscopic measurements of an encapsulated monolayer graphene flake ultrastrongly coupled to a single complementary split-ring resonator. By tuning the carrier density we sweep the system from weak coupling into the ultrastrong regime, reaching a normalized coupling $Ω_\text{R}/ω_\text{cav} \approx 0.4$. This is well into the range where a second order SRPT would manifest by a softening of the lower polariton branch, which we do not observe. The full polariton dispersion is instead quantitatively reproduced by a Hopfield Hamiltonian derived from first principles using a near-field model that accounts for the sub-wavelength character of the cavity. Our results establish an experimental baseline for predictions of cavity-driven phase transitions in two-dimensional Dirac systems, and rule out a No-Go-evading SRPT in graphene Landau polaritons up to the strongest couplings accessible today, with direct implications for proposals invoking vacuum-induced order in solid-state cQED.

The automorphism group of a code is the group of permutations that map a code to itself. Berman codes are a class of binary linear codes characterized by two integer parameters $n\geq 2$ and $m\geq 1$, and this class includes the Reed-Muller codes as well. The class of Berman codes and their duals were recently shown to achieve the capacity of the binary erasure channel. A number of abelian codes that arise from the intersection and subspace sums of Berman and Dual Berman codes were also identified recently, for odd $n\geq 3$. A subclass of these abelian codes was shown to have good short block-length performance for AWGN channels, with efficient decoding algorithms. In this work, we identify the exact automorphism group for Berman codes and their duals. Further, we find the exact automorphism group for the above mentioned abelian codes, when $n\geq 5$. In the case of such abelian codes with $n=3$, we present partial characterizations of the automorphism groups for a large collection of parameter choices, and complete characterizations for a few.

Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete geometric elements (usually triangles). Recently, physics-informed neural networks (PINNs) have emerged as a continuous, mesh-free alternative that does not suffer from FEM's sensitivity to mesh quality or geometric discretization errors. We present PINNSur, a simple framework for using PINNs on curved surfaces: we train a neural field to approximate the surface's normals, and then we express surface differential operators using their projection from $\mathbb{R}^3$ onto the surface. Since every orientable manifold has well-defined normals, our method is suitable for all such surfaces, regardless of curvature or topology, enabling many geometry processing applications. Moreover, despite their empirical success in solving PDEs in flat Euclidean domains, PINNs lack convergence guarantees to the true solution of the underlying PDE, and there is limited systematic experimental evidence demonstrating such convergence. This gap restricts their adoption as reliable solvers compared to established methods like FEM, where convergence to the true solution is well understood and theoretically grounded. These surface PDEs are particularly challenging to solve convergently, as one must not only deal with the convergence of the function approximation, but also with the convergence of the geometric approximation of the surface itself. In this work, we empirically investigate the convergence behavior of PINNs for solving surface PDEs by introducing a simple empirical convergence test.

Lie algebra $\mathfrak{sl}(2)$ can be realised by vector fields on $\mathbb{R}^1\ni x$ with polynomial coefficients $1$, $-2x$, $-x^2$; their Wronskian determinants yield the Lie bracket. Likewise, the monomials $1$, $\ldots$, $x^k/k!$, $\ldots$, $x^N/N!$ span finite-dimensional strong homotopy (SH) Lie algebras with the Wronskians $\mathbf{1} \wedge \partial_x \wedge \ldots \wedge \partial_x^{N-1}$ as the $N$-ary brackets. Over dimension $d=2$ with $\mathbb{R}^2\ni(x,y)$ and for the generalised complete Wronskian $W^{d=2}_{k=1}=\mathbf{1}\wedge \partial_x \wedge \partial_y$ of differential order $k=1$ as the ternary bracket, the finite-dimensional polynomial SH-Lie algebras are spanned by $\langle 1$, $x$, $y$, $p\rangle$ with $p\in\{x^2$, $xy$, $y^2\}$. We explicitly describe all finite-dimensional polynomial SH-Lie algebras $\Bbbk_k[{\boldsymbol{x}}]\subseteq \mathcal{A} \subseteq \Bbbk[x^1,\ldots,x^d]$ (over $\Bbbk=\mathbb{R}$ or $\mathbb{C}$) with the complete generalised Wronskians $W^{d\geqslant 1}_{k\geqslant 1}$ of order $k$ as $N$-ary bracket: $N=\binom{d+k}{d}$. We obtain a factorisation formula for the generalised Vandermonde determinants which show up in the structure constants of the polynomial algebras $\mathcal{A}$.

We report a comparative analysis of three dynamic neural network (NN) architectures -- NAR, NARX, and NIO -- to evaluate their efficiency in estimating the critical-measurement-strength parameter ($c_{crit}$) characterizing topological phase transitions in geometric phases induced by weak measurements. Our results demonstrate that the NARX architecture achieves superior predictive fidelity, reaching a Mean Squared Error (MSE) of $10^{-27}$ -- the limit of numerical precision -- at an optimal delay of $d=1$. This exceptional performance implies the identification of a perfect functional identity, suggesting that the relationship between winding numbers $W$ and $c_{crit}$ is mathematically deterministic. We observe a "complexity paradox" where the NARX model's accuracy collapses at higher delays ($d=4$), a phase-sensitivity that confirms the model captures a high-precision dynamic mapping rather than a trivial pattern. While the NAR model remains robust for local-trend capture, the NIO architecture fails to accurately resolve the phase transition despite increased neuronal capacity. These findings underscore that both autoregressive feedback and immediate exogenous context are essential for the exact characterization of topological phases, establishing NARX as a robust framework for deriving governing laws in complex quantum systems, where analytical solutions remain elusive.

Disclination lines in three-dimensional nematic liquid crystals generically form closed loops whose topology is classified by homotopy theory. While this classification successfully captures global topological features, it does not encode the geometry of the defect profile along the loop, which can strongly influence defect dynamics. Here, we propose a geometric description of nematic disclination loops using the Clifford algebra Cl(3,0). This approach naturally captures the geometry of the local defect profile, as well as changes along the loop, which is mathematically a SU(2) holonomy. Simulations of the dynamics of defect loops with specified geometries embedded in nematic liquid crystals demonstrate that loops nucleate the growth of "topological blobs" of defects, which later dissipate leaving uniform nematic textures. Self-twist of the defect profile leads to nucleation of additional linking disclination lines, with a simple arithmetic relation between total self-twist and linking number. In contrast, loops with an even number of discrete profile transitions generate patterns with threading between loops, but no linking. These results establish a direct connection between the geometric holonomy of a disclination loop and its subsequent evolution, and may be extendable to more complex order parameter manifolds, such as cholesterics or smectics.

The population of short-period exo-Neptunes is thought to be shaped by an interplay between different dynamical mechanisms, such as orbital migration and tidal effects, and photoevaporation. We can gain insight into these processes by studying observables such as the stellar obliquity. Here we study the Rossiter-McLaughlin (RM) effect and measure the projected obliquity, $λ$, of the Neptunian ridge planet WASP-156 b. We analyse new ESPRESSO and MAROON-X spectroscopic transit observations, and new NGTS photometry simultaneous to the ESPRESSO data. Our analyses show an aligned orbit ($λ=-8\pm16^\circ$, based on the ESPRESSO observations), in contrast to a previous report of a highly misaligned orbit. We also find the star's projected rotational velocity to be $v \sin i_\mathrm{\star}\leq2$ km/s from spectral line modelling and $v \sin i_\mathrm{\star}=0.40\pm0.11$ km/s from the RM modelling. This is lower than the previously reported value of $\sim4$ km/s, which could partly explain the previously derived polar orbit. We also update the system's orbital parameters and rule out Jupiter-mass companions within 5 au using long-term radial velocity data. The planet's aligned and circular orbit ($e<0.16$ at $3σ$), and lack of nearby massive companions, are consistent with in situ formation or early disc-driven migration. Our findings move WASP-156 b from a tentative cluster of close-in Neptunes in polar orbits to the group of aligned Neptunes.

Sequence models, and particularly Linear Recurrent Neural Networks (LRNNs) of the form $\mathbf{h}_{k+1} = \mathbf{W} \mathbf{h}_{k} + \mathbf{y}_k + \mathbf{b}$, are widely applicable in time-series analysis for dynamical systems, yet, as black-box algorithms, much is unknown about why they perform well. In this work, we leverage Takens' embedding theorem, which provides conditions under which partially observed time series organized into delay-coordinate vectors can faithfully represent the original system's dynamics, as a theoretical framework for explaining how and why sequence models preserve and reconstruct dynamical systems. For LRNNs, concatenating output states into delay-coordinate vectors gives rise to a ``delay" matrix $\mathbb{M}_{n,m}\in \mathbb{C}^{(nm) \times (n+1)m}$: a block matrix consisting of identity matrices $\mathbf{I} \in \mathbb{R}^{m \times m}$ repeated $n$ times along the main diagonal and weight matrices $\mathbf{W} \in \mathbb{C}^{m \times m}$ featured $n$ times along the super-diagonal. $\mathbb{M}_{n,m}$ relates the delay-coordinates of the input time series to those of the LRNN output states, and, for $\mathbb{M}_{n,m}$ to be an embedding, it must be full row-rank. We provide explicit conditions for $\mathbb{M}_{n,m}$ to be full row-rank and prove the condition number of $\mathbb{M}_{n,m}$ and determinant of $\mathbb{M}_{n,m} \mathbb{M}_{n,m}^*$--measures of embedding stability--are bounded independent of $n$, at least for certain ranges of $\mathbf{W}$'s singular values: namely, when $σ_{\max}(\mathbf{W}) \le 1$. This result explains why the spectrum of $\mathbf{W}$ for trained LRNNs tends to converge to within the unit circle.

Classical simulation of many-body quantum systems remains economical only when wavefunction amplitudes stay localized in the working basis. Fixed-basis sparse-state simulators scale memory as $\mathcal{O}(k)$ by keeping the largest computational-basis amplitudes; however, fidelity drops once entanglement or basis rotations spread weight across the Hilbert space. In this work, we introduce an algorithm called Basis-Adaptive Sparse-State Simulation (BASS), which updates each qubit's local representation basis during execution rather than locking the computational basis for the entire circuit. Before truncation, each qubit is rotated into the eigenbasis of its single-qubit reduced density matrix, following the natural-orbital idea from quantum chemistry, so the retained amplitudes stay clustered. We prove that top-$k$ selection is uniquely optimal for one-step truncation in any fixed basis and that the one-body reduced-density-matrix eigenbasis is a stationary product basis for the inverse participation ratio (PR), with a residual bounded by local entanglement coherence. We perform a systematic benchmarking over a variety of quantum circuits and demonstrate that the ratio \(k/\text{PR}_Z\) (sparse budget over computational participation ratio) serves as an indicator for regimes in which adaptive measurement bases provide a performance advantage. On structured brickwork circuits, BASS achieves substantially higher fidelity than the fixed-basis approach, while incurring only a moderate increase in wall-clock time in the memory-limited regime. Moreover, for disordered Ising circuits, BASS systematically provides an improvement of approximately one order of magnitude in state overlap at a fixed computational budget.

We prove that every perfectoid tower can be decomposed into a fiber product of perfectoid towers that are either $p$-torsion free or perfect of characteristic $p$. As an application, we show that separated perfectoid towers are reduced. We also establish the tilting invariance of étale cohomology and Koszul homology for perfectoid towers.

The recent development of differentiable simulation codes for particle accelerators has enabled gradient-based workflows that promise finer control and more realistic modeling of accelerator facilities. However, when using reverse-mode automatic differentiation, the memory usage continuously increases during the simulation, and can potentially exceed the available hardware memory -- especially when costly space charge computation is included. To study the memory requirements for differentiable simulations, we have implemented space charge in Cheetah, a PyTorch-based beam tracking code that supports reverse-mode differentiation. We find that the memory usage for reverse-mode differentiation grows linearly with the number of macroparticles and cells, and that it is proportional to the number of space charge kicks involved in the simulation. This general scaling can be used to evaluate whether a given differentiable simulation is feasible given hardware memory constraints.

The first part of this article is devoted to characterizing the cocycles $α$ of a finite group $G$ that give rise to faithful projective representations of $G$. We prove that a $p$-group $G$ admits a faithful irreducible projective representation if and only if the cohomology class $[α]$ does not lie in the image of the inflation map $\operatorname{inf}: \mathrm{H}^2\!\left(G / N, \mathbb{C}^{\times}\right) \longrightarrow \mathrm{H}^2\!\left(G, \mathbb{C}^{\times}\right)$ for any non-trivial central subgroup $N$ of $G$. In the case where $[α] \in \operatorname{Im}(\operatorname{inf})$, we determine a criterion such that a direct sum of irreducible $α$-representations is faithful. We conclude this part by describing the behaviour of cocycles $α$ that yield faithful irreducible representations for direct products of groups. In the second part, we introduce the notion of the projective embedding degree of a finite group $G$, defined as the smallest integer $n$ such that $G$ embeds into $\mathrm{PGL}_n(\mathbb{C})$; equivalently, it is the smallest $n$ such that $G$ has a faithful complex projective representation of degree $n$. We also define the analogous notion of the irreducible projective embedding degree of $G$. These invariants have been investigated for several classes of groups, including direct products of groups, finite abelian groups, extra-special $p$-groups, Heisenberg groups, and groups of order $p^3$, $p^4$ (for primes $p$), and $p^5$ (for $p \geq 5$).

We characterize perfectoid towers in terms of conormal cones rather than torsion parts. This result is deduced from a refined study of the relationship between torsion with respect to a principal ideal and the associated conormal cone, building on the work of O. Gabber and L. Ramero.

In noncentrosymmetric metals such as $C_{3v}$ topological-insulator surfaces, moiré heterobilayers, and zincblende crystals, point-group symmetry can forbid the linear and quadratic electric-field-induced orbital magnetization, leaving the cubic response as the leading signal. Using a Ward-complete finite-momentum cubic Kubo kernel with an antisymmetric linear-in-$q$ projection, we show that the dc response separates into three quantum-geometric channels. These are a mixed electric-magnetic positional-shift quadrupole, a quantum-metric drift term, and an orbital-moment octupole. The three contributions share the same point-group symmetry but differ in their lifetime, frequency, and gate fingerprints. For a warped $C_{3v}$ surface the metric channel obeys the cutoff-independent law $\barχ_G \propto μ^{-2}$. We propose third-harmonic magneto-optical Kerr spectroscopy as an experimental route.

While Macroscopic Fluctuation Theory (MFT) has been highly successful in analyzing non-equilibrium steady states, its application to non-steady-state processes remains limited. In this study, we apply MFT to the relaxation process of one-dimensional boundary-driven diffusive systems coupled to particle reservoirs at both ends. We exactly derive the current variance for systems with a constant diffusion coefficient and arbitrary mobility, as well as the cumulant generating function for the current in Reflective Brownian Motion (RBM). Our results demonstrate that non-steady current fluctuations during the approach to a steady state can be quantitatively described within the MFT framework.

This paper establishes a quantitative and structural framework for civilizational continuity under rapid, non-linear ecological transitions. Utilizing empirical data on Earth's Energy Imbalance (EEI), currently measured at 1.36 W/m^2, we demonstrate that the planetary system is on a deterministic trajectory toward a new ecological equilibrium, breaching the 1.5 degrees C thermodynamic threshold in less than 6.5 years. We show that mitigating direct anthropogenic waste heat emissions is statistically insufficient, as 35/36-ths of the annual energy imbalance stems from atmospheric radiative forcing rather than localized thermal output. Consequently, preserving legacy centralized socioeconomic frameworks is unfeasible. We present the structural case for the "Eco-Civilization Paradigm" -- a decentralized network of autonomous, modular eco-communities engineered to insulate core life-support functions, localized energy networks, and edge computation from cascading macro-system failures. Finally, we outline the systemic requirements for this transition, including the realignment of asset valuation frameworks from geography-dependent to engineering-dependent metrics, the mobilization of youth-driven innovation capital, and the strategic reallocation of global industrial manufacturing capacity to safeguard civilizational survival.

The onset of nuclear effects in light-ion collisions is studied by measuring the ratio of $D^0$ meson production between NeNe and OO collisions at a center-of-mass energy per nucleon pair of $5.36\,{\rm TeV}$ recorded by the LHCb detector. The $D^0$ meson yields are measured differentially in transverse momentum ($p_{\rm T}$) for $0.5<p_{\rm T}<20\,{\rm GeV}$ in the rapidity ($y$) region $2.0<y<4.5$, and are normalized to the total number of recorded inelastic nucleus-nucleus collisions in each data sample. The resulting production ratio shows evidence of variation as a function of $p_{\rm T}$, which is inconsistent with predictions based on nuclear modification of nucleon structure alone. This measurement is consistent with the onset of quark-gluon plasma production with increasing nucleus size in high-energy nuclear collisions.

Evidence syntheses and meta-analyses are used to inform clinical practice guidelines and health economic evaluations. However, heterogeneity of treatment effects poses a significant challenge. Conventional meta-analysis addresses heterogeneity through random-effect assumptions, which are not supported by design and lead to estimates that may not apply to any real-world population. Causally-interpretable meta-analysis (CIMA) offers a rigorous framework for specification, identification, and estimation of causal effects when combining information from multiple randomized trials. Initial development of CIMA focused on using individual data from randomized trials, but such data are often unavailable in practice. Here, we propose a new version of CIMA that only requires aggregate data from trials, addressing the limitations of traditional meta-analysis methods while relying only on aggregate data. The method leverages the trials' reported estimates of marginal and one-at-a-time subgroup treatment effects and descriptive statistics for baseline covariates to build moment equations for identifying and estimating a parametric conditional average treatment effect (CATE) function. The average treatment effect in a new target population is obtained by marginalizing the CATE function over the individual covariate data that defines the target population. The method can also be used to obtain causally-interpretable indirect treatment comparisons in the target population. We establish the asymptotic properties of the method, assess its finite-sample performance in simulation studies, and illustrate the application of the method by re-analyzing a published meta-analysis for SGLT2 inhibitors in patients with heart failure.

Conservation laws are time-invariant properties that constrain many physical systems. For systems of chemical reactions, the law of mass conservation constrains how atoms flow between chemical species. Chemical reaction networks can display emergent conservation not explained by mass conservation: these hidden symmetries arise instead from coupled kinetics. Kinetic invariants emerge when branching reactions with proportional rates cause species concentrations to evolve in lockstep. We detect emergent conservation in a simplified atmospheric chemical mechanism of ozone formation through a data-driven analysis of simulated concentrations, a result matching the theoretical kinetic explanation. Surveying 35 widely used atmospheric chemical mechanisms spanning five orders of magnitude in complexity, we discover emergent conservation in 15 mechanisms. Kinetic invariants constrain the intrinsic dimensionality of chemical systems: mechanisms with emergent conservation evolve in lower-dimensional spaces than their size suggests. Identifying emergent conservation can provide theoretical bounds for exact mechanism reduction and uncover kinetic symmetries in atmospheric chemistry.

Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize the Gorenstein flat-cotorsion modules over $T_R(M)$, showing that a $T_R(M)$-module $(X, u)$ is Gorenstein flat-cotorsion if and only if $u$ is monomorphic and $\Coker u$ is a Gorenstein flat-cotorsion $R$-module. As applications, we describe the Gorenstein flat-cotorsion modules over some trivial extension rings and Morita context rings.

In this article we provide a classification and description of compact Riemann surfaces admitting a triangular action of a group of order $2p^2,$ where $p$ is an odd prime number. We obtain that all such Riemann surfaces are isomorphic to curves defined over the rational numbers. As a by-product, we derive a classification of orientably-regular hypermaps whose orientation-preserving automorphism group has order $2p^2.$

We propose a temperature-resolved Monte Carlo (MC) approach to identify the temperature regimes in which low-energy helium-burning reaction rates most strongly affect nucleosynthesis in very massive stars that undergo pair-instability supernovae (PISNe). By performing MC simulations of PISNe, we quantify how temperature-dependent variations in key helium-burning reaction rates, i.e., the triple-$α$ and $^{12}{\rm C}(α,γ)^{16}{\rm O}$ rates, influence $^{56}{\rm Ni}$ synthesis. Thousands of stellar evolution calculations using $\texttt{MESA}$ reveal that both the $^{12}{\rm C}(α,γ)^{16}{\rm O}$ and triple-$α$ reactions exhibit their strongest sensitivity at $T \simeq 2.5 \times 10^{8}\,{\rm K}$, but with opposite correlation signs. We show that this temperature corresponds to the regime in which the ratio of the sampled rate multipliers is most clearly imprinted on the pre-carbon-burning C/O composition. This demonstrates that PISN nucleosynthesis can probe helium-burning reaction rates in specific low-temperature regimes.

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting object, the $(d+2)$-angulated cluster category, and the $d$-almost positive subcategory of the derived category (the higher analogue of the category of two-term complexes of projectives). We show that the third one, introduced by the second-named author, is the $d$-exangulated quotient of the other two, introduced by Oppermann and Thomas, by the ideals generated by morphisms factoring through morphisms from injective to projective objects, thus providing an algebraic connection between the two models of Oppermann-Thomas. This is a $d$-exangulated version in type $A$ of a result of Brüstle and Yang and its interpretation by the first-named author together with Fang, Palu, Plamondon and Pressland. It also explains a well-known coincidence between the number of 2-term silting complexes in type $A_{n}$ and of tilting modules in type $A_{n+1}$ from the $0$-Auslander perspective. We expect this to serve as a prototypical example of the $0$-Auslander correspondence in higher homological algebra.

In quantum purity amplification, one is given $n$ copies of a noisy quantum state $ρ\in \mathbb{C}^{d \times d}$ and asked to prepare $k$ copies of its principal eigenstate $|v_d\rangle$. Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples $n$ tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if $ρ$'s eigenvalues are sorted $p_1 \leq \cdots \leq p_d$ and $p_{d-1} < p_d$, then \begin{equation*} n = O\Big(k + \frac{k}δ \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big) \end{equation*} copies suffice to output a state with fidelity at least $1-δ$ with $|v_d^{\otimes k}\rangle$. Our bound holds for arbitrary spectra, and is independent of the dimension $d$. In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams.

We present a pilot study on the collection and synchronisation of multimodal data for player experience investigation. We collected game telemetry, self-reported surveys, biometrics, and cued-retrospective think-aloud (C-RTA) data from 19 participants playing three Atari 2600 games. The study then uses the data to investigate difficulty in PX, showcasing a protocol for future multimodal research. The dataset obtained from the experiment, which is publicly available, shows potential as a rich, transformative source that can be used to investigate dynamic difficulty adjustment algorithms, game balancing strategies or broader explorations of games user research. The study findings suggest that the experimental approach holds strong potential for generalisation in future player experience studies.

We present a fully extrinsic, parametrization-free variant of tensor calculus on embedded, possibly evolving, submanifolds with boundary in arbitrary dimension and codimension. The proposed approach is component-free and, for general rank tensors, covers fundamental concepts such as tangential projection, extrinsic and covariant derivatives, the extrinsic Stokes' formula, and the Laplace-Beltrami operator. The distinctive features of the developed framework are its algorithmic recursivity and the transparency provided by the special row representation of tensors reminiscent of the recursive data structure of complete trees. Consequently, the suggested tensor calculus is amenable to computations and theoretical analysis, and the latter is demonstrated for general dimension and codimension through three standalone applications. First, we derive a new extrinsic conservation law, namely the principle of vanishing extrinsic momentum, for incompressible Euler flows on Riemannian manifolds. Second, we revisit the concept of Cauchy stress on embedded submanifolds with positive codimension and argue that the conservation of angular momentum implies tangentiality and symmetry of the stress tensors only when they are restricted to act on tangential orientations. Third, for evolving submanifolds, we introduce the material derivative of tensor fields of general rank in an extrinsic manner and derive an expression for the rate of change of the associated tensorial Dirichlet energy. The paper provides a practical notation and tools that are immediately usable in mathematical modeling, analysis of geometry-aware PDEs and in numerical methods on embedded submanifolds.

We prove for every $n\ge4$ the existence of an $n$-player game in normal form with integer payoffs that has a unique Nash equilibrium, which is fully mixed. In the equilibrium, each probability weight is an algebraic number of degree $\mathbin{!n}$ (the derangement number), and its minimal polynomial has Galois group $S_{\mathbin{!n}}$ and $\mathbin{!n}+1$ nonzero coefficients.

We establish a correspondence between conformal partial waves on flat, thermal, and defect backgrounds using the shadow formalism. We demonstrate that scalar one-point thermal blocks can be systematically obtained from their four-point flat-space and two-point defect counterparts by considering specific operator configurations. This framework allows us to derive the thermal Casimir equation as a diagonal reduction of the flat-space Casimir system without introducing chemical potentials. We further show that defect two-point blocks with a spin-$l$ exchange operator correspond to thermal one-point blocks for an external spin-$l$ operator.

We consider predictive density estimation under logarithmic score for $d$-dimensional infinitely divisible location models. Taking the formal Bayes predictive density under the Lebesgue prior as a benchmark, we study the Kullback-Leibler regret of competing Bayes predictive densities. Our main contribution is an exact entropy-energy identity: the integrated regret of a Bayes predictive density $\hat{p}^π$ under prior $π$ relative to the benchmark admits an exact representation as the Dirichlet-form energy of the square-rooted marginal distribution $\sqrt{M^π}$ for the symmetric Markov semigroup induced by the benchmark kernel. This converts regret comparisons into a potential-theoretic problem and yields a sharp recurrence/transience characterization of when the benchmark predictive density can or cannot be uniformly improved. We introduce an $\mathcal{A}$-harmonic class of improper priors -- defined through the generator $\mathcal{A}$ of the induced process -- and give explicit tail conditions -- an integral test on the induced marginal, equivalent to power-law prior decay in heavy-tailed models -- that guarantee admissibility of the resulting Bayes predictive density. We illustrate the theory with new results for several distributions.