Quantum key distribution (QKD) could help to share secure key between two distant peers. In recent years, twin-field (TF) QKD has been widely investigated because of its long transmission distance. One of the popular variants of TF QKD is sending-or-not-sending (SNS) QKD, which has been experimentally verified to realize 1000-km level fibre key distribution. In this article, the authors introduce phase postselection into the SNS protocol. With this modification, the probability of selecting "sending" can be substantially improved. The numerical simulation shows that the transmission distance can be improved both with and without the actively odd-parity pairing method. With discrete phase randomization, the variant can have both a larger key rate and a longer distance.

Using Hubbard U corrected density functional theory calculations, lattice Monte-Carlo, and spin-Monte-Carlo simulations, we investigate the impact of dopant clustering on the magnetic properties of WSe2~doped with period four transition metals. We use manganese (Mn) and iron (Fe) as candidate n-type dopants and vanadium (V) as the candidate p-type dopants, substituting the tungsten (W) atom in WSe2. Specifically, we determine the strength of the exchange interaction in the Fe-, Mn-, and V-doped WSe2~ in the presence of clustering. We show that the clusters of dopants are energetically more stable than discretely doped systems. Further, we show that in the presence of dopant clustering, the magnetic exchange interaction significantly reduces because the magnetic order in clustered WSe2~becomes more itinerant. Finally, we show that the clustering of the dopant atoms has a detrimental effect on the magnetic interaction, and to obtain an optimal Curie temperature, it is important to control the distribution of the dopant atoms.

Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$ for any $u,v$, where $G_{uv}$ is the graph obtained by applying Kelmans operation from $u$ to $v$; (II) $\mathcal{P}(G) <\mathcal{P}(G+e)$ for any edge $e\notin E(G)$. Let $P_k$ be a path of order $k$, $\mathcal{C}_{\geq k}$ the set of all cycles of length at least $k$ and $M_{k+1}$ a matching containing $k+1$ independent edges. In this paper, we mainly prove the following three results: (i) Let $n\geq k\geq 5$ and let $t=\left\lfloor\frac{k-1}{2}\right\rfloor$. Let $G$ be a $2$-connected $n$-vertex $\mathcal{C}_{\geq k}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\in \mathcal{G}^1_{n,k}=\{W_{n,k,s}=K_{s}\vee ((n-k+s)K_1\cup K_{k-2s}): 2\leq s\leq t\}$. (ii) Let $n\geq k\geq 4$ and let $t=\left\lfloor\frac{k}{2}\right\rfloor-1$. Let $G$ be a connected $n$-vertex $P_{k}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\in \mathcal{G}^2_{n,k}=\{W_{n,k-1,s}=K_{s}\vee ((n-k+s+1)K_1\cup K_{k-2s-1}): 1\leq s\leq t\}.$ (iii) Let $G$ be a connected $n$-vertex $M_{k+1}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\cong K_n$ when $n=2k+1$ and $G\in \mathcal{G}^3_{n,k}=\{K_s\vee ((n-2k+s-1)K_1\cup K_{2k-2s+1}):1\leq s\leq k\}$ when $n\geq 2k+2$. Directly derived from these three main results, we obtain a series of applications in Turán-type problems, generalized Turán-type problems, powers of graph degrees in extremal graph theory, and problems related to spectral radius, and signless Laplacian spectral radius in spectral graph theory.

PLAnetary Transits and Oscillations of stars (PLATO) is the ESA M3 space mission dedicated to detect and characterise transiting exoplanets including information from the asteroseismic properties of their stellar hosts. The uninterrupted and high-precision photometry provided by space-borne instruments such as PLATO require long preparatory phases. An exhaustive list of tests are paramount to design a mission that meets the performance requirements, and as such, simulations are an indispensable tool in the mission preparation. To accommodate PLATO's need of versatile simulations prior to mission launch - that at the same time describe accurately the innovative but complex multi-telescope design - we here present the end-to-end PLATO simulator specifically developed for the purpose, namely PlatoSim. We show step-by-step the algorithms embedded into the software architecture of PlatoSim that allow the user to simulate photometric time series of CCD images and light curves in accordance to the expected observations of PLATO. In the context of the PLATO payload, a general formalism of modelling, end-to-end, incoming photons from the sky to the final measurement in digital units is discussed. We show the strong predictive power of PlatoSim through its diverse applicability and contribution to numerous working groups within the PLATO Mission Consortium. This involves the on-going mechanical integration and alignment, performance studies of the payload, the pipeline development and assessments of the scientific goals. PlatoSim is a state-of-the-art simulator that is able to produce the expected photometric observations of PLATO to a high level of accuracy. We demonstrate that PlatoSim is a key software tool for the PLATO mission in the preparatory phases until mission launch and prospectively beyond.

Quantum chemistry is envisioned as an early and disruptive application for quantum computers. Yet, closer scrutiny of the proposed algorithms shows that there are considerable difficulties along the way. Here, we propose two criteria for evaluating two leading quantum approaches for finding the ground state of molecules. The first criterion applies to the variational quantum eigensolver (VQE) algorithm. It sets an upper bound to the level of imprecision/decoherence that can be tolerated in quantum hardware as a function of the targeted precision, the number of gates and the typical energy contribution from states populated by decoherence processes. We find that decoherence is highly detrimental to the accuracy of VQE and performing relevant chemistry calculations would require performances that are expected for fault-tolerant quantum computers, not mere noisy hardware, even with advanced error mitigation techniques. Physically, the sensitivity of VQE to decoherence originates from the fact that, in VQE, the spectrum of the studied molecule has no correlation with the spectrum of the quantum hardware used to perform the computation. The second criterion applies to the quantum phase estimation (QPE) algorithm, which is often presented as the go-to replacement of VQE upon availability of (noiseless) fault-tolerant quantum computers. QPE requires an input state with a large enough overlap with the sought-after ground state. We provide a criterion to estimate quantitatively this overlap based on the energy and the energy variance of said input state. Using input states from a variety of state-of-the-art classical methods, we show that the scaling of this overlap with system size does display the standard orthogonality catastrophe, namely an exponential suppression with system size. This in turns leads to an exponentially reduced QPE success probability.

In recent years, large language models have been responsible for great advances in the field of artificial intelligence (AI). ChatGPT in particular, an AI chatbot developed and recently released by OpenAI, has taken the field to the next level. The conversational model is able not only to process human-like text, but also to translate natural language into code. However, the safety of programs generated by ChatGPT should not be overlooked. In this paper, we perform an experiment to address this issue. Specifically, we ask ChatGPT to generate a number of program and evaluate the security of the resulting source code. We further investigate whether ChatGPT can be prodded to improve the security by appropriate prompts, and discuss the ethical aspects of using AI to generate code. Results suggest that ChatGPT is aware of potential vulnerabilities, but nonetheless often generates source code that are not robust to certain attacks.

The randomized unbiased estimators of Rhee and Glynn (Operations Research:63(5), 1026-1043, 2015) can be highly efficient at approximating expectations of path functionals associated with stochastic differential equations (SDEs). However, there is a lack of algorithms for calculating the optimal distributions with an infinite horizon. In this article, based on the method of Cui et.al. (Operations Research Letters: 477-484, 2021), we prove that, under mild assumptions, there is a simple representation of the optimal distributions. Then, we develop an adaptive algorithm to compute the optimal distributions with an infinite horizon, which requires only a small amount of computational time in prior estimation. Finally, we provide numerical results to illustrate the efficiency of our adaptive algorithm.

We study properties of vertex (operator) algebras associated with 3d H-twisted $\mathcal{N}=4$ supersymmetric gauge theories with a boundary. The vertex operator algebras (VOAs) are defined by BRST cohomologies of currents with symplectic bosons, complex fermions, and bc-ghosts. We point out that VOAs for 3d $\mathcal{N}=4$ abelian gauge theories are fermionic extensions of VOAs associated with toric hyper-Kähler varieties. From this relation, it follows that the VOA associated with the 3d mirror of $N$-flavor $U(1)$ SQED is a fermionic extension of a $W$-algebra $W^{-N+1}(\mathfrak{sl}_N, f_{\text{sub}})$. For $N=3$, we explicitly compute the OPE of elements in the BRST cohomology and find a new algebra that is a fermionic extension of a Bershadsky-Polyakov algebra $W^{-2}(\mathfrak{sl}_3, f_{\text{sub}})$. We also suggest an expression for the vacuum character of the fermionic extension of $W^{-N+1}(\mathfrak{sl}_N, f_{\text{sub}})$ predicted by 3d $\mathcal{N}=4$ mirror symmetry.

Kempe equivalence is a classical and fundamental notion in graph coloring theory. In the present paper we establish a connection between Kempe equivalence and quadratic stable set ring, which are toric rings associated to graphs. In fact, we characterize when the stable set ring of a graph is quadratic by using Kempe equivalence. As an application, we relate our theorem to the theory of perfectly contractile graphs, a hereditary subclass of perfect graphs introduced by Bertschi. In particular, our characterization implies that the conjecture of Everett and Reed on perfectly contractile graphs entails the conjecture of the authors and Shibata on quadratic stable set rings. Furthermore, we show that the stable set rings of several important subclasses of perfectly contractile graphs including weakly chordal graphs are quadratic. Finally, we propose a new combinatorial conjecture characterizing perfectly contractile graphs purely in terms of Kempe equivalence on replication graphs.

This work proposes a statistical model for crossover trials with multiple skewed responses measured in each period. A 3 $\times$ 3 crossover trial data where different drug doses were administered to subjects with a history of seasonal asthma rhinitis to grass pollen is used for motivation. In each period, gene expression values for ten genes were measured from each subject. It considers a linear mixed effect model with skew normally distributed random effect or random error term to model the asymmetric responses in the crossover trials. The paper examines cases (i) when a random effect follows a skew-normal distribution, as well as (ii) when a random error follows a skew-normal distribution. The EM algorithm is used in both cases to compute maximum likelihood estimates of parameters. Simulations and crossover data from the gene expression study illustrate the proposed approach. Keywords: Crossover design, Mixed effect models, Skew-normal distribution, EM algorithm.

In 1933 von Neumann proved a beautiful result that one can approximate a point in the intersection of two convex sets by alternating projections, i.e., successively projecting on one set and then the other. This algorithm assumes that one has access to projection operators for both sets. In this note, we consider the much weaker setup where we have only access to linear minimization oracles over the convex sets and present an algorithm to find a point in the intersection of two convex sets.

Drafts are sequential round-robin allocation procedures for distributing heterogeneous and indivisible objects among agents subject to some priority order (e.g., allocating players' contract rights to teams in professional sports leagues). Agents report ordinal preferences over objects and bundles are partially ordered by pairwise comparison. We provide a simple characterization of draft rules: they are the only allocation rules that are respectful of a priority (RP), envy-free up to one object (EF1), non-wasteful (NW), and resource monotonic (RM). RP and EF1 are crucial for competitive balance in sports leagues. We also prove two related impossibility theorems showing that the competitive-balance axioms RP and EF1 are generally incompatible with strategy-proofness. Nevertheless, draft rules satisfy maxmin strategy-proofness. If agents may declare some objects unacceptable, then draft rules are characterized by RP, EF1, NW, and RM, in conjunction with individual rationality and truncation invariance.

Quantum computing is evolving so rapidly that it forces us to revisit, rewrite, and update the foundations of the theory. \emph{Basic Quantum Algorithms} revisits the earliest quantum algorithms. The journey began in 1985 with Deutsch attempting to evaluate a function at two domain points simultaneously. Then, in 1992, Deutsch and Jozsa created a quantum algorithm that determines whether a Boolean function is constant or balanced. The following year, Bernstein and Vazirani realized that essentially the same algorithm could be used to identify a specific Boolean function within a set of linear Boolean functions. In 1994, Simon introduced a novel quantum algorithm that determines whether a function is one-to-one or two-to-one exponentially faster than any classical algorithm for the same problem. That same year, Shor developed two groundbreaking quantum algorithms for integer factoring and calculating discrete logarithms, posing a threat to widely used cryptographic methods. In 1995, Kitaev proposed an alternative formulation based on phase estimation that proved valuable in numerous applications. The following year, Grover devised a quantum search algorithm that is quadratically faster than its classical counterpart. More than a decade later, Harrow, Hassidim, and Lloyd proposed a quantum algorithm for solving systems of linear equations, now known as the HHL algorithm. With an emphasis on the circuit model, this work provides a detailed description of all these remarkable algorithms.

We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds proved previously by Fodor and V\'ıgh (2018).

A comprehensive gyrokinetic simulation model has been implemented in the global toroidal gyrokinetic code (GTC) and verified for studying low-frequency waves and turbulence in magnetic fusion plasmas by treating all kinetic-MHD processes on an equal footing. A theoretical framework has been formulated to unify various methods for efficiently solving the electron drift kinetic equation in multiscale simulations by separating electron responses into analytic and non-analytic parts based on the smallness parameter of electron-to-ion mass ratio. The model can be reduced to the ideal MHD model with both the linear dispersion relation and the nonlinear ponderomotive force in theory and simulation. The model is used for the verification and validation of simulating internal kink modes in the DIII-D tokamak with accurate calculations of equilibrium parallel current and compressible magnetic perturbation. A large simulation database has been generated to train a surrogate model to predict the kink instability. Statistical analysis shows that the radial location of safety factor q=1 flux-surface and the plasma beta inside the q=1 surface are the most important parameters for predicting the kink instability.

The study of graphs associated with of various algebraic structures is an emerging topic in algebraic graph theory. Recently, the concept of nonzero component graph of a finite dimensional vector space $Γ(\mathbb{V})$ was put forward by Das \cite{5}. In this paper, we study some degree based topological indices over $Γ(\mathbb{V})$ the derived graphs of $Γ(\mathbb{V})$.

Let $i: \mathrm{L} \hookrightarrow \mathrm{X}$ be a compact Kähler Lagrangian in a holomorphic symplectic variety $\mathrm{X}/\mathbf{C}$. We use deformation quantisation to show that the endomorphism differential graded algebra $\mathrm{RHom}\big(i_*\mathrm{K}_\mathrm{L}^{1/2},i_*\mathrm{K}_\mathrm{L}^{1/2}\big)$ is formal. We prove a generalisation to pairs of Lagrangians, along with auxiliary results on the behaviour of formality in families of $\mathrm{A}_\infty$-modules.

In this paper, we introduce the notion of a four-angle Hopf module for a Hom-Hopf algebra $(H,β)$ and show that the category $\!^{H}_{H}\mathfrak{M}^{H}_{H}$ of four-angle Hopf modules is a monoidal category with either a Hom-tensor product $\otimes_{H}$ or a Hom-cotensor product $\Box_{H}$ as a monoidal product. We study the category $\mathcal{YD}^{H}_{H}$ of Yetter-Drinfel'd modules with bijective structure map can be organized as a braided monoidal category, in which we use a new monoidal structure. Finally, We prove an equivalence between the monoidal category $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\otimes_{H})$ or $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$ of four-angle Hopf modules, and the monoidal category $\mathcal{YD}^{H}_{H}$ of Yetter-Drinfel'd modules, and furthermore, we give a braiding structure of the monoidal categorys $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\otimes_{H})$ (and $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$).

The longest common substring problem consists in finding a longest string that appears as a (contiguous) substring of two input strings. We consider the dynamic variant of this problem, in which we are to maintain two dynamic strings $S$ and $T$, each of length at most $n$, that undergo edit operations, i.e., substitutions, insertions, and deletions of letters, in order to be able to return a longest common substring after each edit. Amir, Charalampopoulos, Pissis, and Radoszewski [Algorithmica 2020] presented a solution for this problem that requires $\tilde{\mathcal{O}}(n^{2/3})$ time per update. This brought the challenge of determining whether there exists a solution with polylogarithmic update time or we should expect a polynomial (conditional) lower bound. We answer this question by designing an exponentially faster algorithm that processes each edit operation in amortized $O(\log^7 n)$ time with high probability. Our solution relies on exploiting the local consistency of the parsing of a collection of dynamic strings due to Gawrychowski, Karczmarz, Kociumaka, Lącki, and Sankowski [SODA 2018], and on maintaining two dynamic trees with labeled bicolored leaves. We complement our solution with a lower bound of $Ω(\log n/ \log\log n)$ for the update time of any polynomial-size data structure that maintains an LCS of two dynamic strings, and the same lower bound for the update time of any $\tilde{\mathcal{O}}(n)$-size data structure that maintains the LCS of a static and a dynamic string. Both lower bounds hold even allowing amortization and randomization. This requires extending Pătraşcu's reduction from the lopsided set disjointness problem to the butterfly reachability problem [SICOMP 2011].

The packing density of the regular cross-polytope in Euclidean $n$-space is unknown except in dimensions $2$ and $4$ where it is 1. The only non-trivial upper bound is due to Gravel, Elser, and Kallus (2011) who proved that for $n=3$ the packing density of the regular octahedron is at most $1-1.4\ldots\times 10^{-12}$. In this paper, we prove upper bounds for the packing density of the $n$-dimensional regular cross-polytope in the case that $n\geq 7$. We use a modification of Blichfeldt's method due to G. Fejes Tóth and W. Kuperberg (1993).

Mutualistic networks provide a powerful way to describe and analyse plant-pollinator communities and their structure over time. While these networks capture the complex interdependencies that link population fates across the season, they can be hard to untangle, preventing us from understanding the emergence of community-scale properties and responses to perturbation. Here, we address this problem by developing a structural model of a plant-pollinator community that explicitly incorporates seasonal turnover and the temporal nature of species interactions. We analyse our model using percolation methods from network science to derive simple analytical solutions linking network structure to emergent community diversity. Our findings reveal that temporal structure organises community diversity into distinct ecological phases, creating the potential for alternative high- and low-diversity states and bistable regimes. We demonstrate how this temporal structure mediates the nature of transitions between these states, determining whether systems undergo gradual shifts or abrupt, catastrophic collapses. Crucially, we show how this temporal structure reduces the robustness of plant-pollinator systems, creating bottlenecks that inhibit species persistence and increase susceptibility to secondary extinctions. Our results demonstrate that the temporal dynamics of plant-pollinator networks are central to mediating their fragility, highlighting the importance of accounting for time when considering community resilience.

Reservoir engineering has emerged as a powerful paradigm to realize non-reciprocal dynamics in open quantum many-body systems. Here, we show that density-density interactions can transfer bath-induced non-reciprocity between different degrees of freedom. Specifically, we investigate a one-dimensional lattice of spin-$\frac{1}{2}$ fermions with all-to-all Hatsugai-Kohmoto interactions in the presence of an engineered reservoir. We establish the exact solvability of the Lindbladian dynamics and show that the interplay between non-reciprocity and interactions qualitatively reshapes the dynamics of excitations. Remarkably, interactions induce directional drift even in spin sectors that are not directly coupled to the reservoir. By analyzing a driven-dissipative Fermi-Hubbard chain, we show that the same mechanism persists for local interactions. The Hatsugai-Kohmoto model thus emerges as a minimal, exactly solvable platform for interaction-mediated non-reciprocal many-body dynamics.

With the rapid adoption of AI writing assistants in education, educators and researchers need empirical evidence to understand the impact on student writing and inform effective pedagogical design. Despite widespread use, we lack systematic understanding of how students engage with these tools during authentic writing tasks: when they seek assistance, what they ask, and how they incorporate AI-generated content into their essays. This gap limits evidence-based policy development and rigorous evaluation of generative AI's learning effects. To address this gap, we introduce NIRVANA, a dataset capturing how university students use generative AI while writing an analytical essay. The dataset includes 77 students who completed an essay task with access to ChatGPT, recording keystroke-level writing behavior, full ChatGPT conversation histories, and all text copied from ChatGPT, enabling a complete reconstruction of the writing process and revealing how AI assistance shapes student work. Our analysis identifies key behavioral patterns, including variation in ChatGPT query frequency and its relationship to essay characteristics such as length and readability. We identify four writing profiles based on students' contribution and revision patterns: Lead Authors, Collaborators, Drafters, and Vibe Writers. To support deeper investigation, we developed a replay interface that reconstructs the writing process; qualitative analysis of sampled replays demonstrates how this tool enables systematic examination of student-AI interactions.

Distributed drive electric vehicles offer superior yaw moment control for autonomous drifting in extreme maneuvers. Conventional drift analysis constructs stability boundaries from open loop equilibria points and assumes a fixed envelope structure. However, coupling among control inputs reshapes the phase plane and shifts saddle point location, which can invalidate open loop envelopes when used for closed loop drifting. To address this issue, a saddle point coordinate model is established in this paper by combining a nonlinear tire model with the handling diagram and explicitly accounting for road adhesion coefficient, longitudinal velocity, front wheel steering angle, and additional yaw moment. Based on saddle point properties, an extended dual envelope framework is constructed in the phase plane of slip angle and yaw rate. Using the convergence tendency of state points toward saddle points under bounded control inputs, the outer envelope defines a recoverable set under constraints on front wheel steering angle and additional yaw moment. The inner envelope characterizes the non-drifting stability region associated with unsaturated tire forces. Finally, a nonlinear model predictive control (NMPC) controller is developed using the extended dual envelope constraint. Hardware-in-the-loop experiments show that, compared with NMPC without envelope constraints, the proposed method enables smoother convergence toward the drift saddle point, reduces the steady-state tracking errors of vehicle speed, sideslip angle, and yaw rate by 33.07%, 71.18%, and 31.27%, respectively, and decreases the peak tracking error by 63.66% under road-friction mismatch.

We incorporate an arbitrarily high-order method for the Laplacian operator into the Spectral Difference method (SD). The resulting method is capable of capturing shocks thanks to its a-posteriori limiting methodology, and therefore it is able to survive scenarios in which the dissipative scales (viscous and diffusive) are not properly described. Moreover, it is capable of capturing these scales at lower resolution compared to lower-order methods and therefore attains convergence at lower resolution. We show that the method at hand has exponential convergence when describing smooth solutions and is able to recover a high-order solution when solving the dissipative scales.

In 2025, Bernardes, Erler and Firat proposed a novel, elegant expression for the symplectic form on phase space applicable to non-local theories. We show that this BEF symplectic structure can be derived directly from an $L_\infty$-Lagrangian by following the covariant phase space approach. Moreover, we establish a precise relation between the BEF symplectic structure and the Barnich--Brandt symplectic form for general finite-derivative theories. In particular, we prove that for theories with second-order equations of motion, the BEF symplectic structure coincides with the Barnich--Brandt construction, thereby explaining the emergence of the canonical corner term in general relativity within the BEF approach. We further argue that the BEF symplectic structure naturally encodes information about generic corner terms and some information about boundary conditions. In addition, we develop a general expression for the Hamiltonian in theories in $L_\infty$-form and present several explicit examples illustrating the construction.

We propose a novel laboratory test of gravity combining seismic-wave measurements with cosmic-ray muon detections. Quantum-gravity corrections to the anharmonic Debye model are derived, yielding a modified bulk modulus that encodes deviations from standard gravity. The usual dependence on density, a dominant source of uncertainty, is removed via muon tomography and seismic velocities measurement. We show that this setup can constrain gravity parameters at a level comparable to current laboratory experiments. Prospects for further improvements are briefly discussed.

Wan proved the rationality of partial toric $L$-functions using $\ell$-adic techniques. In this paper, we present a $p$-adic proof in the spirit of Dwork. We demonstrate that partial $L$-functions can be expressed as an alternating product of twisted Fredholm determinants. These twisted determinants appear to be intrinsic to the analytic structure of partial $L$-functions, and unlike their classical counterparts, twisted Fredholm determinants of completely continuous operators are not automatically $p$-adic entire functions. However, for partial $L$-functions they will be $p$-adic meromorphic. After proving rationality, we construct a $p$-adic cohomology theory and give a $p$-adic cohomological formula for partial toric $L$-functions. Last, we show they have a unique $p$-adic unit root which may be explicitly written in terms of $A$-hypergeometric series.

The Lorentz force exerted by a primordial magnetic field (PMF) on the coupled baryon-dark matter system may enhance total matter power at small scales after recombination. In the post-reionization (post-EoR) era, a weakly scale-dependent PMF of sub-nG strength is thus expected to influence the Lyman-$α$ (Ly$α$) power spectrum, the 21 cm power spectrum, and the Ly$α$-21 cm cross-spectrum at scales $k\gtrsim 1\:h/\textrm{Mpc}$. We investigate the prospects of constraining the PMF sector via these three cosmological observables, by employing SNR estimation and Fisher forecast on the PMF amplitude $B_0$ and spectral index $n_{\rm B}$, for a next-generation DESI-like spectroscopic survey and two upcoming 21 cm facilities, namely SKA1-Mid and PUMA. Our results indicate the possibility of constraining both PMF parameters with $\lesssim10\%$ relative errors through the uncontaminated 21 cm auto-spectrum as well as the Ly$α$-21 cm cross-spectrum probed with the DESI-like+SKA1-Mid combination. Indicatively, the Ly$α$-21 cm cross-correlation via DESI-like+SKA1-Mid is predicted to constrain a fiducial scenario $B_0=0.8$ nG and $n_{\rm B}=-2.9$ with $1σ$ errors $ΔB_0\approx 0.07$ nG and $Δn_{\rm B}\approx0.02$. The DESI-like+PUMA setup is predicted to fare relatively worse due to its restriction to larger scales, resulting in comparatively one order of magnitude relaxed error bounds for similar fiducials. Since the Ly$α$-21 cm cross-signal is expected to be largely insensitive to foreground contamination (unlike the 21 cm auto-spectrum), it may serve as an optimal foreground-immune post-EoR probe to constrain a weakly scale-dependent sub-nG PMF via future DESI-like+SKA1-Mid observations.

We propose a new conformal prediction method for time-series data with a guaranteed asymptotic conditional coverage rate, Sequential Conformalized Density Regions (SCDR), which is flexible enough to produce both prediction intervals and disconnected prediction sets, signifying the emergence of bifurcations. Our approach uses existing estimated conditional highest density predictive regions to form initial predictive regions. We then use a quantile random forest conformal adjustment to provide guaranteed coverage while adaptively changing to take the non-exchangeable nature of time-series data into account. We show that the proposed method achieves the guaranteed coverage rate asymptotically under certain regularity conditions. In particular, the method is doubly robust -- it works if the predictive density model is correctly specified and/or if the scores follow a nonlinear autoregressive model with the correct order specified. Simulations reveal that the proposed method outperforms existing methods in terms of empirical coverage rates and set sizes. We illustrate the method using two real datasets, the Old Faithful geyser dataset and the Australian electricity usage dataset. Prediction sets formed using SCDR for the geyser eruption durations include both single intervals and unions of two intervals, whereas existing methods produce wider, less informative, single-interval prediction sets.