In this paper we prove uniform a priori estimates for transmission problems with constant coefficients on two subdomains, with a special emphasis for the case when the ratio between these coefficients is large. In the most part of the work, the interface between the two subdomains is supposed to be Lipschitz. We first study a scalar transmission problem which is handled through a converging asymptotic series. Then we derive uniform a priori estimates for Maxwell transmission problem set on a domain made up of a dielectric and a highly conducting material. The technique is based on an appropriate decomposition of the electric field, whose gradient part is estimated thanks to the first part. As an application, we develop an argument for the convergence of an asymptotic expansion as the conductivity tends to infinity.

In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of any order on a d-dimensional polyhedral domain. One of the main tools for the analysis is a recently introduced smoothed Poincaré lifting operator [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]. For forms of order 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p-version and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, Nédélec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory.

Haar random states are fundamental objects in quantum information theory and quantum computing. We study the density matrix resulting from sampling $t$ copies of a $d$-dimensional quantum state according to the Haar measure on the orthogonal group. In particular, we analytically compute its spectral decomposition. This allows us to compute exactly the trace distance between $t$-copies of a real Haar random state and $t$-copies of a complex Haar random state. Using this we show a lower-bound on the approximation parameter of real-valued state $t$-designs and improve the lower-bound on the number of copies required for imaginarity testing.

We study a unified semi-analytical framework to study null geodesics and weak-field light deflection in generic static, spherically symmetric spacetimes of the form \(ds^2 = -α(r,δ)\,dt^2 + γ(r,δ)\,dr^2 + β(r,δ)\,dΩ_2^2,\) where $α$, $β$, and $γ$ encode model-dependent deviations from Schwarzschild gravity inspired from [Phys.Rev.D 112 (2025) 12, 124072]. Starting from the exact first-order orbit equation, we derive a compact master equation for the impact-parameter-dependent trajectory $u(φ)\equiv 1/r(φ)$ and obtain a model-independent expression for the bending angle $α(b)$ in terms of generic metric functions and their derivatives. This master equation is then solved semi-analytically by three complementary techniques: (i) the homotopy perturbation method (HPM), (ii) the variational iteration method (VIM), and (iii) a calibrated impulse (single-kick) approximation expressed directly in terms of the effective gravitational potential. As nontrivial test beds we consider a scalar-hairy Reissner-Nordström-like black hole where the scalar hair enters as $Q_s$ in an effective charge parameter. Then we derive closed-form expressions for the deflection angle, identify the leading scalar-hair, and compare the accuracy and convergence properties of HPM, VIM, and the impulse method against the standard Schwarzschild limits. Our results show that the generic formulation in $(α,β,γ)$ can efficiently accommodate a broad class of modified gravity black hole solutions. We further supplement the analytic treatment with a compact numerical results against the exact null-geodesic integral near the photon sphere in order to delineate the practical range of validity of the three approximation schemes.

Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct torus-invariant points. In this paper, we derive an explicit combinatorial formula for the Chow weight of $\widetilde{X}$ in terms of the underlying toric manifold $X$ and the symplectic cuts of its associated Delzant polytope. As an application, we study toric blow-ups of the projective plane and compare their Chow stability with that of blow-ups at general points.

We solve the one-dimensional Helmholtz equation in several scenarios using the quantum annealer provided by D-Wave Systems within a pseudospectral framework, where the solution is encoded in an appropriate set of basis functions. We evaluate the performance of different encoding strategies based on algebraic conditioning and adiabatic considerations, and benchmark their performance against the classical simulated annealing algorithm. In particular, we analyze the minimum energy gap, the so-called dynamic range, and the mean squared error to assess the numerical stability, consistency, and accuracy of the solutions returned by each strategy. Our work highlights the importance of developing custom embedding techniques ensuring well-conditioned algebraic systems. In particular, we show that encoding strategies producing full-rank matrices with reduced dynamic ranges enhance the performance of the quantum annealer even under polychromatic driving and nontrivial initial conditions. We further discuss the potential of developing hybrid quantum-classical schemes designed to simultaneously satisfy algebraic conditioning and adiabatic requirements.

Efficient quantum arithmetic operations are essential building blocks for complex quantum algorithms, yet few theoretical designs have been implemented in practical quantum programming frameworks. This paper presents the first complete implementation of the T-count optimized non-restoring quantum square root algorithm using the Qrisp quantum programming framework. The algorithm, originally proposed by Thapliyal et al., offers better resource efficiency compared to alternative methods, achieving reduced T-count and qubit requirements while avoiding garbage output. Our implementation validates the theoretical resource estimates, confirming a T-count of 14n-14 and T-depth of 5n+3 for n-bit inputs. The modular design approach enabled by Qrisp allows construction from reusable components including reversible adders, subtractors, and conditional logic blocks built from fundamental quantum gates. The three-stage algorithm - comprising initial subtraction, iterative conditional addition/subtraction, and remainder restoration is successfully translated from algorithmic description to executable quantum code. Experimental validation across multiple test cases confirms correctness, with the circuit producing accurate integer square roots and remainders. This work demonstrates the practical realizability of resource-optimized quantum arithmetic algorithms and establishes a foundation for implementing different arithmetic operations in modern quantum programming frameworks.

Under the standard model of hierarchical structure formation, the overall geometry of galaxy clusters is better described by a triaxial ellipse than a sphere. As a result, applying spherically-symmetric models can result in significant biases. These biases can be mitigated by fitting a triaxial model, requiring deep multiprobe data and a set of physically motivated models to describe them. Here we present a multiprobe triaxial analysis methodology based on the data available for galaxy clusters in the Cluster Heritage project with XMM-Newton - Mass Assembly and Thermodynamics at Endpoint of structure formation (CHEX-MATE), which includes X-ray data from XMM-Newton, SZ data from Planck and ACT, and WL data from Subaru. This work builds on our previous development of a gas-only X-ray and SZ triaxial fitting formalism in Paper I. We apply our approach to the CHEX-MATE cluster PSZ2 G313.33+61.13 (Abell 1689) and find that it is elongated along the line of sight relative to the plane of sky by a factor of $\mathcal{R}_{LP} = 1.27 \pm 0.02$. As a result, the WL mass obtained from our triaxial fit, $\text{M}_{200c}=(13.69_{-1.41}^{+1.56})\times10^{14} \text{M}_{\odot}$, is significantly lower than the value of $(17.77_{-1.75}^{+2.00})\times10^{14} \text{M}_{\odot}$ obtained from a spherically-symmetric fit that otherwise employs the same methodology. Our triaxial fit finds a concentration of $c_{200c}=8.55_{-1.61}^{+2.20}$, consistent with the spherically-symmetric value of $9.99_{-1.78}^{+2.26}$, which suggests that the unexpectedly high concentration in Abell 1689 is not due to triaxiality and orientation. We also measure the non-thermal pressure fraction at radii between 0.18-1.37 Mpc, finding a minimum of approximately 20 per cent at intermediate radii increasing to near 30 per cent at both the smallest and largest radii, and with a typical measurement precision of $\pm$5 per cent.

We present assembly-theory, a Rust package for computing assembly indices of covalently bonded molecular structures. This is a key complexity measure of assembly theory, a recent theoretical framework quantifying selection across diverse systems, most importantly chemistry. assembly-theory is designed for researchers and practitioners alike, providing (i) extensible, high-performance implementations of assembly index calculation algorithms, (ii) comprehensive benchmarks against which current and future algorithmic improvements can be tested, and (iii) Python bindings and RDKit-compatible data loaders to support integration with existing computational pipelines.

An ensemble model of turbulence based on states with constant flux in wavenumber space was proposed in [K. Yoshida, Phys. Rev. E, 106, 045106 (2022)]. The justification of this ensemble model relies on the conjecture that almost all states with constant flux correspond to turbulence states. To verify this conjecture, a numerical search for states with constant enstrophy flux in wavenumber space over finite time intervals in two-dimension turbulence is conducted using a Monte Carlo method. Properties of the obtained states, such as energy spectrum and spectra related to fourth-order moments, are examined and compared with those of turbulence states obtained from direct numerical simulations. The dependence of the obtained states on the time interval and the initial conditions used in the numerical search is also discussed.

The Brenner-Navier-Stokes-Fourier (BNSF) system, introduced by Howard Brenner, was developed to address some deficiencies in the classical Navier-Stokes-Fourier system, based on the concept of volume velocity. We consider the one-dimensional BNSF system in Lagrangian mass coordinates, incorporating temperature-dependent transport coefficients, which yields a more physically realistic framework. We establish the existence and uniqueness of monotone traveling wave solutions (or viscous shocks) to the BNSF system with any positive $C^2$ dissipation coefficients, provided that the shock amplitude is sufficiently small. We utilize geometric singular perturbation theory as in the constant coefficient case [13]; however, due to the arbitrary nonlinearities of the coefficients, we employ the implicit function theorem, which grants robustness to our approach. This work is motivated by [12], which proves a contraction property of any large solutions to the BNSF system around the traveling wave solutions. Thus, we also derive some quantitative estimates on the traveling wave solutions that play a fundamental role in [12].

We present DeepCHART (Deep learning for Cosmological Heterogeneity and Astrophysical Reconstruction via Tomography), a deep learning framework designed to reconstruct the three-dimensional dark matter density field at redshift $z=2.5$ from Ly$α$ forest spectra. Leveraging a 3D variational autoencoder with a U-Net architecture, DeepCHART performs fast, likelihood-free inference, accurately capturing the non-linear gravitational dynamics and baryonic processes embedded in cosmological hydrodynamical simulations. When applied to joint datasets combining Ly$α$ forest absorption and coeval galaxy positions, the reconstruction quality improves further. For current surveys, such as Subaru/PFS, CLAMATO, and LATIS, with an average transverse sightline spacing of $d_\perp=2.4h^{-1}$cMpc, DeepCHART achieves high-fidelity reconstructions over the density range $0.4<Δ_{\rm DM}<15$, with a voxel-wise Pearson correlation coefficient of $ρ\simeq 0.77$. These reconstructions are obtained using Ly$α$ forest spectra with signal-to-noise ratios as low as 2 and instrumental resolution $R=2500$, matching Subaru/PFS specifications. For future high-density surveys enabled by instruments such as ELT/MOSAIC and WST/IFS with $d_\perp\simeq 1h^{-1}\mathrm{cMpc}$, the correlation improves to $ρ\simeq 0.90$ across a wider dynamic range ($0.25<Δ_{\rm DM}<40$). The framework reliably recovers the dark matter density PDF as well as the power spectrum, with only mild suppression at intermediate scales. In terms of cosmic web classification, DeepCHART successfully identifies 81% of voids, 75% of sheets, 63% of filaments, and 43% of nodes. We propose DeepCHART as a powerful and scalable framework for field-level cosmological inference, readily generalisable to other observables, and offering a robust, efficient means of maximising the scientific return of upcoming spectroscopic surveys.

We study the problem of determining the minimal genus of a simple finite connected graph. We present an algorithm which, for an arbitrary graph $G$ with $n$ vertices and $m$ edges, determines the orientable genus of $G$ in $O(n(4^m/n)^{n/t})$ steps where $t$ is the girth of $G$. This algorithm avoids difficulties that many other genus algorithms have with handling bridge placements which is a well-known issue. The algorithm has a number of useful properties for practical use: it is simple to implement, it outputs the faces of an optimal embedding, and it iteratively narrows both upper and lower bounds. We illustrate the algorithm by determining the genus of the $(3,12)$ cage (which is 17); other graphs are also considered.

In this paper, we study the twisted Fourier-Mukai partners of abelian surfaces. Following the work of Huybrechts [doi:10.4171/CMH/465], we introduce the twisted derived equivalence between abelian surfaces. We show that there is a twisted derived Torelli theorem for abelian surfaces over algebraically closed fields with characteristic $\neq 2,3$. Over complex numbers, the derived isogenies correspond to rational Hodge isometries between the second cohomology groups, which is in analogy to the work of Huybrechts and Fu-Vial on K3 surfaces. Their proof relies on the global Torelli theorem over $\mathbb{C}$, which is missing in positive characteristics. To overcome this issue, we firstly extend a trick given by Shioda on integral Hodge structures, to rational Hodge structures, $\ell$-adic Tate modules and $F$-crystals. Then we make use of Tate's isogeny theorem to give a characterization of the twisted derived equivalences between abelian surfaces via isogenies. As a consequence, we show the two abelian surfaces are principally isogenous if and only if they are derived isogenous.

We theoretically investigate the properties of ultra-cold dipolar atoms in radially coupled, concentric annular traps created by a potential barrier. The non-rotating ground-state phases are investigated across the superfluid-supersolid phase transition, revealing a particle imbalance between the two rings and a preferential density modulation in the outer ring in the absence of rotation. Near the phase transition on the superfluid side, applying rotation can induce density modulations in either ring, depending on the angular momentum and barrier strength. For low angular momentum, such rotation-induced density modulation forms in the outer ring, while for high angular momentum and weak barriers, it emerges in the inner ring. Rotation can lead to persistent currents and the nucleation of a vortex residing either at the center (central vortex) or at the ring junction (Josephson vortex). Josephson vortices can also form at the junctions of the localized density sites induced by rotation in the inner ring, a behavior that is unique to our system. By switching off the trap and allowing the system to expand, distinct interference patterns emerge, which can be analyzed to identify and distinguish between various vortex configurations, and thus can be observed in current state-of-the-art experiments.

We report a higher-order neuromorphic Ising machine that exhibits superior scalability compared to architectures based on quadratization, while also achieving state-of-the-art quality and reliability in solutions with competitive time-to-solution metrics. At the core of the proposed machine is an asynchronous autoencoder architecture that captures higher-order interactions by directly manipulating Ising clauses instead of Ising spins, thereby maintaining resource complexity independent of interaction order. Asymptotic convergence to the Ising ground state is ensured by sampling the autoencoder latent space defined by the spins, based on the annealing dynamics of the Fowler-Nordheim quantum mechanical tunneling. To demonstrate the advantages of the proposed higher-order neuromorphic Ising machine, we systematically solved benchmark combinatorial optimization problems such as MAX-CUT and MAX-SAT, comparing the results to those obtained using a second-order Ising machine employing the same annealing process. Our findings indicate that the proposed architecture consistently provides higher quality solutions in shorter time frames compared to the second-order model across multiple runs. Additionally, we show that the techniques based on the sparsity of the interconnection matrix, such as graph coloring, can be effectively applied to higher-order neuromorphic Ising machines, enhancing the solution quality and the time-to-solution. The time-to-solution can be further improved through hardware co-design, as demonstrated in this paper using a field-programmable gate array (FPGA). The results presented in this paper provide further evidence that autoencoders and Fowler-Nordheim annealers are sufficient to achieve reliability and scaling of any-order neuromorphic Ising machines.

We analyse the typical structure of games in terms of the connectivity properties of their best-response graphs. Our central result shows that, among games that are `generic' (without indifferences) and that have a pure Nash equilibrium, all but a small fraction are \emph{connected}, meaning that every action profile that is not a pure Nash equilibrium can reach every pure Nash equilibrium via best-response paths. This has important implications for dynamics in games. In particular, we show that there are simple, uncoupled, adaptive dynamics for which period-by-period play converges almost surely to a pure Nash equilibrium in all but a small fraction of generic games that have one (which contrasts with the known fact that there is no such dynamic that leads almost surely to a pure Nash equilibrium in \emph{every} generic game that has one). We build on recent results in probabilistic combinatorics for our characterisation of game connectivity.

In this work, we develop a scalable approach for a flexible latent factor model for high-dimensional dynamical systems. Each latent factor process has its own correlation and variance parameters, and the orthogonal factor loading matrix can be either fixed or estimated. We utilize an orthogonal factor loading matrix that avoids computing the inversion of the posterior covariance matrix at each time of the Kalman filter, and derive closed-form expressions in an expectation-maximization algorithm for parameter estimation, which substantially reduces the computational complexity without approximation. Our approach has several applications, including noise filtering for high-dimensional time series, estimating nonseparable covariance structure between different time series, and estimating latent physical processes from real-world measurements. Extensive simulated studies illustrate higher accuracy and scalability of our approach compared to alternatives. Furthermore, by applying our method to geodetic measurements to estimate slow slip events from geodetic data in the Cascadia region, our estimated slip better agrees with independently measured seismic data of tremor events. The substantial acceleration from our method enables the use of massive noisy data for geological hazard quantification and other applications.

We investigate the existence and uniqueness of non-Markovian second-order backward stochastic differential equations with an uncertain terminal horizon and establish comparison principles under the assumption that the driver is Lipschitz continuous. The terminal time is both random and exogenous, and it may not be adapted to the Brownian filtration, leading to a singular jump in the 2BSDE decomposition. We also provide a connection between this new class of 2BSDE and a fully nonlinear PDE in a Markovian setting. Our theoretical results are applied to non-Markovian stochastic control problems in two settings: (1) when an agent seeks to maximize utility from a payoff received at an uncertain terminal time by controlling both the drift and volatility of a diffusion process; and (2) when the agent contends with volatility uncertainty stemming from an external source, referred to as Nature, and optimizes the drift in a worst-case scenario for the ambiguous volatility. We term this class of problems erratic stochastic control, reflecting the dual uncertainty in both model parameters and the timing of the terminal horizon.

Let $X$ be a singular, projective variety. For every $p>0$, $H^{2p}(X,\mathbb{Q})$ is equipped with a mixed Hodge structure. The elements of $\mathrm{Gr}^W_{2p}H^{2p}(X,\mathbb{Q}) \cap H^{p,p} \mathrm{Gr}^W_{2p}H^{2p}(X,\mathbb{C})$ will be called Hodge (p,p)-classes. The purpose of this article, is to study the Bloch-Gillet-Soulé (BGS) cycle class map from the $p$-th operational Chow group $A^p(X)$ to the space of $(p,p)$-Hodge classes. We show that if $p=1$ and $X$ is a normal surface with at worst rational singularities, then the BGS cycle class map is surjective. This extends the Lefschetz $(1,1)$-theorem to the setup of rational surface singularities. However, the BGS map is not always surjective. For this reason we introduce extended operational Chow group $A^p_{\mathrm{ext}}(X)$ which contains the operational Chow group. We show that the BGS cycle class map extends to $A^p_{\mathrm{ext}}(X)$. Moreover, if $p=1$ and $X$ has at worst isolated singularity (not necessarily a surface), then the extended BGS map is surjective. This further extends the Lefschetz $(1,1)$-theorem to the case of isolated singularities.

Nested integration problems arise in various scientific and engineering applications, including Bayesian experimental design, financial risk assessment, and uncertainty quantification. These nested integrals take the form $\int f\left(\int g(\boldsymbol{y},\boldsymbol{x})\mathrm{d}\boldsymbol{x}\right)\mathrm{d}\boldsymbol{y}$, for nonlinear $f$, making them computationally challenging, particularly in high-dimensional settings. Although widely used for single integrals, traditional Monte Carlo (MC) methods can be inefficient when encountering complexities of nested integration. This work introduces a novel multilevel estimator, combining deterministic and randomized quasi-MC (rQMC) methods to handle nested integration problems efficiently. In this context, the inner number of samples and the discretization accuracy of the inner integrand evaluation constitute the level. We provide a comprehensive theoretical analysis of the estimator, deriving error bounds demonstrating significant reductions in bias and variance compared with standard methods. The proposed estimator is particularly effective in scenarios where the integrand is evaluated approximately, as it adapts to different levels of resolution without compromising precision. We verify the performance of our method via numerical experiments, focusing on estimating the expected information gain of experiments. When applied to Gaussian noise in the experiment, a truncation scheme ensures finite error bounds. The results reveal that the proposed multilevel rQMC estimator outperforms existing MC and rQMC approaches, offering a substantial reduction in computational costs and offering a powerful tool for practitioners dealing with complex, nested integration problems across various domains.

We propose transformed Diffsuion-Wave fractional Physics-Informed Neural Networks (tDWfPINNs) for efficiently solving time-fractional diffusion-wave equations with fractional order $α\in(1,2)$. Conventional numerical methods for these equations often compromise the mesh-free advantage of Physics-Informed Neural Networks (PINNs) or impose high computational costs when computing fractional derivatives. The proposed method avoids first-order derivative calculations at quadrature points by introducing an integrand transformation technique, significantly reducing computational costs associated with fractional derivative evaluation while preserving accuracy. We conduct a comprehensive comparative analysis applying this integrand transformation in conjunction with both Monte Carlo integration and Gauss-Jacobi quadrature schemes across various time-fractional PDEs. Our results demonstrate that tDWfPINNs achieve superior computational efficiency without sacrificing accuracy. Furthermore, we incorporate the proposed approach into adaptive sampling approaches such as the residual-based adaptive distribution (RAD) for the time-fractional Burgers equation with order $α\in(1,2)$, which exhibits complex solution dynamics. The experiments show that the Gauss-Jacobi method typically outperforms the Monte Carlo approach; however, careful consideration is required when selecting the number of quadrature points. Overall, the proposed tDWfPINNs offer a significant advancement in the numerical solution of time-fractional diffusion-wave equations, providing an accurate and scalable mesh-free alternative for challenging fractional models.

We consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators $ L = \tfrac{1}{ ρ} ( -Δ+ V ) $ and divergence form operators $ L = -\operatorname{div} A \nabla $, on bounded domains. For these operators, we prove ordering inequalities between the Dirichlet and the Neumann eigenvalues, generalizing results of Levine--Weinberger and Friedlander--Filonov for the Laplacian. We take inspiration from their proofs and derive sufficient conditions on the coefficients of the operator that ensure that the inequalities remain valid.

This study explores the effects of Russia's full-scale invasion of Ukraine on the international collaboration of Ukrainian scholars. First and foremost, Ukrainian scholars deserve respect for continuing to publish despite life-threatening conditions, mental strain, shelling and blackouts. In 2022-2023, universities gained more from international collaboration than the NASU. The percentage of internationally co-authored articles remained unchanged for the NASU, while it increased for universities. In 2023, 40.8% of articles published by the NASU and 32,2% of articles published by universities were internationally co-authored. However, these figures are still much lower than in developed countries (60-70%). The citation impact of internationally co-authored articles remained statistically unchanged for the NASU but increased for universities. The highest share of internationally co-authored articles published by the NASU in both periods was in the physical sciences and engineering. However, the citation impact of these articles declined in 2022-2023, nearly erasing their previous citation advantage over university publications. Universities consistently outperformed the NASU in the citation impact of internationally co-authored articles in biomedical and health sciences across both periods. International collaboration can help Ukrainian scholars to go through this difficult time. In turn, they can contribute to the strengthening of Europe.

Retrieval-Augmented Generation (RAG) systems improve the factual grounding of large language models (LLMs) but remain vulnerable to retrieval poisoning, where adversaries seed the corpus with manipulated content. Prior work largely evaluates this threat under a simplified single-attacker assumption. In practice, however, high-value or high-visibility queries attract multiple adversaries with conflicting objectives. Motivated by real cases, we introduce the setting of competing attacks, in which multiple attackers simultaneously attempt to steer the same or closely related query toward different targets. We formalize this threat model and propose competitive effectiveness, a metric that quantifies an attacker's advantage under competition. Extensive experiments show that many strategies that succeed in the single-attacker regime degrade markedly under competition, revealing performance inversions and highlighting the limits of conventional metrics such as attack success rate and F1. Furthermore, we present PoisonArena, a standardized framework and benchmark for evaluating poisoning attacks and defenses under realistic, multi-adversary conditions.

A natural family of potentially 2-torsion elements in the integer homology cobordism group consists of splices of knots with their mirrors. We show that such 3-manifolds have locally trivial involutive Floer homology. We show some related families of splices also have locally trivial involutive Floer homology. Our arguments show that many gauge theoretic invariants also vanish on these 3-manifolds.

Dualising the construction of a polyhedral product, we introduce the notion of a polyhedral coproduct as a certain homotopy limit over the face poset of a simplicial complex. We begin a study of the basic properties of polyhedral coproducts, surveying the Eckmann-Hilton duals of various familiar examples and properties of polyhedral products. In particular, we show that polyhedral coproducts give a functorial interpolation between the wedge and cartesian product of spaces which differs from the one given by polyhedral products, and we establish a general loop space decomposition for these spaces which is dual to the suspension splitting of a polyhedral product due to Bahri, Bendersky, Cohen and Gitler.

We define real topological Hochschild homology of separated log schemes with involutions. We show that real topological Hochschild homology is $(\mathbb{P}^n,\mathbb{P}^{n-1})$-invariant, which leads to the definition of the motivic real topological Hochschild spectrum living in a certain $\mathbb{Z}/2$-equivariant logarithmic motivic category. We explore properties of real topological Hochschild homology that can be deduced from the logarithmic motivic homotopy theory. We also define the motivic real topological cyclic spectrum.

Algorithms for mutual exclusion aim to isolate potentially concurrent accesses to the same shared resources. Motivated by distributed computing research on programmable matter and population protocols where interactions among entities are often assumed to be isolated, Daymude, Richa, and Scheideler (SAND`22) introduced a variant of the local mutual exclusion problem that applies to arbitrary dynamic networks: each node, on issuing a lock request, must acquire exclusive locks on itself and all its persistent neighbors, i.e., the neighbors that remain connected to it over the duration of the lock request. Assuming adversarial edge dynamics, semi-synchronous or asynchronous concurrency, and anonymous nodes communicating via message passing, their randomized algorithm achieves mutual exclusion (non-intersecting lock sets) and lockout freedom (eventual success with probability 1). However, they did not analyze their algorithm's runtime. In this paper, we prove that any node will successfully lock itself and its persistent neighbors within O$(nΔ^3)$ open rounds of its lock request in expectation, where $n$ is the number of nodes in the dynamic network, $Δ$ is the maximum degree of the dynamic network, rounds are normalized to the execution time of the ``slowest'' node, and ``closed'' rounds when some persistent neighbors are already locked by another node are ignored (i.e., only ``open" rounds are considered).

We study the symplectic semi-characteristic of a closed 4n-dimensional symplectic manifold. First, using the even-degree part of the primitive cohomology, we define the symplectic semi-characteristic. Second, using a vector field with nondegenerate zero points, we prove a counting formula for the symplectic semicharacteristic. As corollaries of the counting formula, we obtain a vanishing property and the fact that the definition of the symplectic semi-characteristic is independent of the choices of symplectic forms.