Papers

This paper proposes novel optimal-control-based guidance laws for ground-to-air missiles with time-varying acceleration bounds. In such engagements, as the missile climbs in altitude, its acceleration bound decreases, which may lead to acceleration saturation and significant miss distances if not explicitly accounted for. The proposed guidance laws incorporate hard acceleration command constraints directly into a linear-quadratic optimal-control framework, in contrast to conventional unbounded or softly constrained approaches. Analytically based guidance laws are developed for linear zero-order and first-order strictly proper missile dynamics with arbitrary-order linear target dynamics. Unlike the constant hard-bound case with minimum-phase missile dynamics, time-varying acceleration command bounds permit an initial unsaturated interval in which the proposed guidance laws can anticipate future saturation and reshape the acceleration profile accordingly. This enables earlier maneuvers when the missile possesses greater low-altitude maneuverability, fundamentally altering the structure of the optimal solution. The proposed approach is evaluated in nonlinear simulations and compared with equivalent unbounded and softly constrained optimal guidance laws. The results demonstrate substantially improved interception performance under saturation, reduced tuning requirements compared to softly constrained guidance laws, and enhanced capability in challenging engagement scenarios.

The Arnold conjecture is a classic and important conjecture in the field of symplectic geometry, which describes the estimation of the number of periodic solutions of the Hamiltonian quantity on any symplectic manifold, which is equivalent to the symplectic version of Morse's theory. In the past development process, Andreas Floer was the first to propose Floer's theory, and solved the situation of non-degenerate Hamiltonian quantities under monotonic symplectic manifolds, and then Kenji Fukaya and Kaoru Ono solved the estimation of non-degenerate Hamiltonian quantities under rational coefficients in 1999. In 2023, Bai-Xu solved the estimation under the integer coefficient, and the non-degenerate version of Arnold's conjecture was completely resolved, but the degenerate version of Arnold conjecture did not progress so smoothly. In this paper, we have compiled some work on the degenerate version of Arnold conjecture, and based on these works, a new proof idea is proposed. By combining the action of quantum cohomology on Floer cohomology, combined with a universal energy estimation of the trajectories of $bar{partial}_H$-holomorphic in the paper, we can prove the degenerate Arnold conjecture in some special cases under the framework of Floer's theory.

In this paper, we demonstrate that twisted Hermitian-Einstein metrics on holomorphic vector bundles exist without obstruction. More precisely, for an arbitrary holomorphic vector bundle $E$ over a compact Kähler manifold $(M,\omega_g)$, we prove that the twisted Hermitian-Einstein equation $$\Lambda_{\omega_g}\left(\sqrt{-1}R^h\right) = \lambda h + P$$ admits a unique smooth solution $h$, provided that $P\in\Gamma(M,E^*\otimes\bar{E}^*)$ is positive-definite and $\lambda<\lambda_E^-$. The constant $\lambda_E^-$ is intrinsically associated with the stability constant of $E$. This result extends the classical Donaldson-Uhlenbeck-Yau (DUY) theorem for stable bundles and, in the limit $P\rightarrow0$, gives a new proof of the DUY theorem. As an application, we obtain an intrinsic Chern number inequality for unstable vector bundles: $$\int_M \left((r-1)c_1(E)^2 - 2rc_2(E)\right) \wedge \omega_g^{n-2} \leq \Bigl\lfloor \frac{r^2}{4} \Bigr\rfloor \frac{(\lambda_E^+-\lambda_E^-)^2}{4\pi^2 n^2} \int_M \omega_g^n.$$

We study the semiclassical wave front set of vector-valued microlocal solutions to the pseudo-differential system with a non-Hermitian matrix-valued symbol. The main result establishes the propagation of the semiclassical wave front sets under a generalized microhyperbolicity condition for the real part of the symbol and the non-negativity for the imaginary part. The proof is based on Hörmander's positive commutator method.

We develop the Itô calculus of Brownian motion, machine-checked in Lean~4 over Mathlib and the \lean{BrownianMotion} package. On a bounded interval $[0,T]$ the Itô integral is built as a Hilbert-space isometry, from a predictable-rectangle $\pi$-system through the density of simple adapted processes. Realized as a process, it is a continuous $L^2$ martingale. One structural identity drives this: the integral at time $t$ is the conditional-expectation projection of its terminal value onto $\F_t$, and from it adaptedness, the martingale property, the contraction bound, and both the terminal and time-indexed Itô isometries follow as corollaries. On this integral we prove Itô's formula for $C^3$ functions with bounded derivatives, including the time-dependent form $df = f_x\,dB + (f_t + \tfrac12 f_{xx})\,dt$, by a discrete-to-continuous argument through weighted quadratic variation with explicit $L^2$ remainder bounds. We then pass from the $L^2$ theory to the pathwise. The integral process has an almost-surely continuous modification, and its everywhere-continuous representative is a local martingale for the null-augmented Brownian filtration; gluing the bounded-horizon representatives along the half-line yields the Itô integral as a continuous local martingale on all of $\R_{\ge 0}$, the form it takes in the classical theory. To our knowledge these are the first machine-checked constructions of the Itô integral and of Itô's formula in any proof assistant, and the first to reach a pathwise-continuous local martingale. The boundary is explicit. The $L^2$ integral and Itô's formula are developed on $[0,T]$ with bounded-derivative integrands; the unrestricted $C^2$ formula, integrators beyond brownian motion, and right-continuity of the filtration lie outside the development.

We classify finite simple Novikov conformal algebras over an algebraically closed field of characteristic zero. The classification consists of the current conformal algebra over the base field and a one-parameter list of Virasoro-like conformal algebras. A semisimple finite Novikov conformal algebra is proved to be a direct sum of simple ones. We also describe deformations and central extensions of finite simple Novikov conformal algebras.

We develop a finite-window audit and transfer framework for the local regularity problem of the three-dimensional incompressible Navier–Stokes equations. The paper has two logically separate components. First, a finite-scale critical ledger estimate shows that a persistent non-CKN branch cannot survive without cumulative untaxed critical supply or accumulated localization leakage. Second, a local-to-clean transfer theorem shows how a clean finite-window anti-phantom gap gives a localized coercive estimate for Navier–Stokes defect packages, provided three structural inputs hold: quotient-lifting stability, componentwise detection comparison, and a normalized residual budget. The technical contribution is an explicit bookkeeping of the residuals that can obstruct this transfer. Pressure tails, cutoff leakage, truncation loss, nonlinear cutoff mismatch, reproduction drift, gauge mismatch, and profit discrepancy are separated as finite-window ledger entries and then assembled into a concrete absorption criterion. The main transfer theorem is algebraic once these entries satisfy the stated normalized bounds; the PDE-facing work left open is precisely to prove the corresponding component compatibility estimates against the localized quotient distance. Thus the article is a conditional reduction. Its conclusion is structural: any surviving obstruction must either defeat the stated compatibility/growth controls or appear as an NS-realizable, cleaned, scale-critical, combined-invisible, profitable, reproducible defect cascade.

We solve the inverse problem for Ising models on reduced planar graphs in a disk, i.e., recovering the edge coupling constants from the boundary spin correlations. A recursive solution to this problem was provided by Galashin–Pylyavskyy. Our solution is nonrecursive. It is based on an Ising analog of the chamber ansatz which asserts that the inverse map factors through variables on the graph that transform under the Ising Y-$\Delta$ move according to the discrete CKP equation.

The Graded Classification Conjecture asserts that the graded Grothendieck group $K_0^{\mathrm{gr}}$ is a complete invariant for the classes of Leavitt path algebras and graph $C^*$-algebras. The conjecture remains open, as neither a proof nor a counterexample is currently known. In this article, we extend the study of this invariant to the class of vertex-weighted Leavitt path algebras. We show that $K_0^{\mathrm{gr}}$ distinguishes weighted Leavitt path algebras from ordinary (unweighted) Leavitt path algebras. We further prove that an isomorphism between the graded Grothendieck groups of weighted Leavitt path algebras induces an isomorphism between the corresponding semilattices of vertex-generated ideals. In addition, we show that $K_0^{\mathrm{gr}}$ classifies the classical Leavitt algebras $L_K(n,n+k)$. Next, we prove that $K_0^{\mathrm{gr}}$ is a full functor on the category of all weighted Leavitt path algebras. Consequently, in the special case where all weights are equal to $1$, we recover the lifting theorem established independently by Arnone and Vas for Leavitt path algebras. This confirms one direction of the Graded Classification Conjecture for Leavitt path algebras.

The Wigner-Poisson system is a deterministic phase-space model for quantum kinetic electron dynamics, but high-dimensional simulations are limited by the full 3D3V phase space and the nonlocal Wigner potential. We develop a structure-preserving, sampling-based adaptive-rank solver in hierarchical Tucker format for finite-$H$ regimes in which Wigner-Poisson solutions exhibit exploitable low-rank structure. The central difficulty is that adaptive compression can destroy the Fourier-Hermitian tensor symmetry required for a real inverse velocity transform and can break discrete global conservation laws. We address these issues with a Fourier-Hermitian-symmetry-aware sampling and mapping procedure and a global moment correction enforcing mass, momentum, and self-consistent total energy. Numerical tests for two-stream instability and strong Landau damping in 2D2V and 3D3V show roundoff-level conservation, preservation of the real-valued inverse transform, and approximately linear scaling with respect to the number of grid points per coordinate over the tested rank range. The results demonstrate that long-time 3D3V Wigner-Poisson simulations can be performed without assembling the full phase-space tensor.

This paper studies the identification of an unknown number of stationary point sources in a two-dimensional heat equation from boundary flux measurements. Unlike many reconstruction approaches that assume the number of sources to be known in advance, we develop a determinant-based counting method that extracts this number directly from the measured data. In the unit disk, the Laplace-transformed and normalized boundary flux admits a Fourier moment representation whose low-frequency limit has a finite exponential-sum structure. This structure leads to a family of Hankel matrices and associated determinant characteristics. We prove that, under a generic determinant lifting condition, the vanishing order of the Hankel determinant at the zero Laplace frequency changes exactly when the Hankel order exceeds the true number of sources. Consequently, the source number is characterized by the first nonzero contour count of the determinant characteristic through the argument principle. We further establish a Rouché-type stability result showing that the determinant zero count is preserved under sufficiently small perturbations induced by measurement noise, boundary discretization, and time truncation. After the source number is identified, the source locations and strengths are recovered from the low-frequency moment sequence by an annihilating-polynomial and Vandermonde reconstruction procedure. Numerical experiments confirm the predicted count pattern, demonstrate robustness with respect to contour selection, illustrate the role of the Rouché margin under noise and near-degenerate configurations, and validate the subsequent recovery of source locations and strengths.

The Constructor–Blocker game is played on the edge set of $K_n$. Two players alternately claim previously unclaimed edges. Constructor aims to maximize the number of copies of a target graph $H$ in her graph while keeping it $F$-free throughout the game, whereas Blocker aims to minimize this number. When both players play optimally and Constructor moves first, the final number of copies of $H$ in Constructor's graph is called the score of the game and is denoted by $g(n,H,F)$. Recently, Balogh, Chen, and English systematically studied this game for non-bipartite forbidden graphs $F$. However, bipartite forbidden graphs present additional difficulties. In this paper, we focus on the case in which the forbidden graph is an even cycle. First, using a finite-field-geometric construction, we confirm a conjecture of Balogh, Chen and English by proving \[ g(n,K_3,C_4)=\Theta(n^{3/2}). \] We also establish two-sided bounds for $g(n,K_3,C_{2k})$ that relate the game score to classical extremal numbers. Our proofs reveal a structural distinction between the cases $k=2$ and $k\ge3$: a vertex-duplication method works for longer even cycles, but it necessarily creates $4$-cycles and therefore cannot be used in the $C_4$-free game. Finally, we determine the exact order of $g(n,C_t,C_{2k})$ for all $k\ge2$ and all $t\ge4$.

Quantum Phase Estimation (QPE) is the quantum algorithmic workhorse for computing ground state energies of quantum Hamiltonians with quantum computers. Ground state energy calculation of physical systems is perhaps the most promising use case for quantum computing in terms of scientific and commercial value with a plausible path to outperformance of classical alternatives. This path, however, hinges on the availability of initial states for QPE with significant overlap with the true ground state. Using extensive (classical) numerical computations, we study whether the NISQ-era algorithm VQE (Variational Quantum Eigensolver) could be used to efficiently prepare high-overlap states of disordered fully-connected anisotropic Heisenberg spin glass quantum Hamiltonians with up to $15$ qubits. We find that (i) – consistent with widely held, but rarely numerically illustrated beliefs – VQE is generally unable to efficiently converge to the ground state for our Hamiltonians, which is a well-known issue with VQE due to a variety of factors including vanishing gradients and local minima; (ii) low energy states do not necessarily have large ground-state overlap, but there is typically a correlation between the two measures; (iii) adding more than three layers to the VQE ansatz neither improves overlap nor the energies found; and (iv) the best-found overlap scaling as a function of the Hamiltonian system size is not strongly exponentially decreasing, suggesting potential for VQE to be a heuristic state preparation algorithm for QPE.

We study stability forms of two classical integral inequalities: Hardy's integral inequality and the integral Carleman inequality, also known as the Pólya–Knopp inequality. The sharp constants in these two inequalities are $(p')^p$ and $e$, respectively, but neither inequality has a non-trivial extremizer in its natural function space. Their formal extremal functions are $c x^{-1/p}$ and $c/x$, respectively. We first derive a deficit identity for Hardy's integral inequality. In particular, when $p=2$, the deficit is exactly a squared norm. We then prove a distance stability estimate for the integral Carleman inequality, whose remainder term directly measures a local weighted Hellinger-type distance from the family $\{c/x:c\ge0\}$. Together these results illustrate the same stability phenomenon: although the classical integral inequalities have no genuine extremizers, their deficits still measure the deviation from the corresponding families of virtual extremals.

Surrogate models are central to scientific machine learning, where they enable fast prediction, simulation, inference, and control for complex physical systems. For time-dependent problems, however, accurate interpolation of training trajectories is not sufficient: reliable surrogates should also respect the conservation laws, invariants, admissibility conditions, and dissipative structures that give those trajectories physical meaning. We introduce Physics-conforming Latent Twins, a framework for learning latent surrogate solution operators whose dynamics satisfy selected physical principles by design. The method builds on the Latent Twin formulation by jointly learning an encoder, a decoder, and a latent flow map between arbitrary time-indexed states, while constraining the latent dynamics to preserve or dissipate prescribed structural quantities. We develop a constraint-transfer viewpoint that connects physical structure in the original state space with compatible constraints in latent space, and prove structure-preservation bounds showing how latent enforcement improves control of physical defects after decoding. We also derive algebraic conditions for latent flow maps that preserve linear and quadratic invariants or enforce dissipative inequalities. Numerical experiments on representative ODE and PDE benchmarks demonstrate improved constraint satisfaction, structural fidelity, and qualitative long-time behavior while maintaining accurate surrogate prediction.

Uncrewed aerial vehicles (UAVs) are increasingly being employed for data collection tasks, thanks to their high mobility and easy deployment, acting as aerial platforms to collect data from ground devices (GDs). This study considers a secure underlay data collection system assisted by dual UAVs and focuses on the joint design of the UAVs' three-dimensional (3D) flight paths, the power of the jamming UAV, the power of GDs, and the scheduling of the underlay GDs in the context of an aerial eavesdropper. The highly coupled objective function and non-convex constraints make the formulated problem more complicated to solve. We first utilize an approximate lower bound on the expected spectral efficiency to streamline the solution process. The average secrecy spectral efficiency (ASSE) is maximized by jointly designing the 3D trajectory of the UAVs, the transmit power of GDs, and the user scheduling. The optimization problem is decomposed into four subproblems using block coordinate descent, with each of them into manageable convex optimization tasks by incorporating slack variables and employing successive convex approximation methods. The numerical results validate the effectiveness of our proposed approach, demonstrating that the design of UAV 3D trajectories remarkably improves the ASSE of the considered system.

Let $S\subset \{1,2,\ldots,n\}$ be a Sidon set with $|S|=n^{1/2}+O(n^{1/2-\delta})$ for some fixed $\delta>0$. This article provides the following expected asymptotic formula $$ \sum_{\substack{a\in S\\ a\equiv r\pmod{m}}} a^\ell =\frac{1}{m(\ell+1)}n^{\ell+1/2} +o\left(n^{\ell+1/2}\right), $$ where $m\geq 1$, $0\leq r<m$, and $\ell\geq 0$ are three integers. This removes the additional hypothesis in a previous residue class asymptotic formula by the author. The proof uses the Fourier uniformity of extremal Sidon sets due to Ortega and Prendiville.

In differential geometry, the underlying mathematical structure behind the electroweak theory is a trivial principal $(SU(2)\times U(1))$–bundle over $\R^4$ with the Minkowski metric. The aim of this paper is to show that, using non–commutative geometry, the electroweak theory can be described using a trivial quantum principal $SU(2)$–bundle over $C^\infty_\C(\R^4)$ with the Minkowski metric.

We train a two-layer transformer encoder to classify rational elliptic curves $E/\mathbb{Q}$ of conductor $\leq 10000$ as either rank 0 or rank 1 from the first 128 normalized Frobenius traces. We achieve >99% accuracy on both classes, and accuracy is essentially unchanged on test curves with no isogeny or quadratic-twist relative in the training set. We then apply techniques from mechanistic interpretability such as attention analysis, linear probing, activation patching, logit attribution, and neuron-level circuit analysis to reverse-engineer the algorithm the (centroid in function space) model learned. We find that a sparse circuit of 20 out of 512 layer-1 MLP neurons is sufficient for rank prediction under a linear probe with an AUROC of 0.992 at plateau, implementing a push-pull detector architecture of rank-0 and rank-1 detectors with a one-sided readout. However, we notice that the model has sub-optimal readout problems indicating a mismatch in rank-order between the readout pathway and the discriminative circuit. Critically, the learned input weights of the top discriminating neuron match the Mestre-Nagao sum heuristic weights $\log(p)/(p\cdot \log{B})$ with a Spearman coefficient $r = 0.997$ and Pearson coefficient $r = 0.952$: the model has learnt a result from analytic number theory from the Frobenius trace data alone. We additionally find that all 50 independently trained models concentrate CLS attention on prime positions at 2-50$\times$ the rate of composite positions. The CLS embedding encodes $\log{L(E,1)}$ with $R^2 = 0.962\pm 0.011$ across the 50 models (after controlling for the conductor). Activation patching analysis reveals that attention weights are dissociated from causal information flow. Additionally, the 50 solutions from training are near-identical in function space (with pairwise agreement $>$98.8%) despite large weight space barriers.

We develop an unstaggered, potential-based particle-in-cell method for the nonrelativistic Vlasov-Maxwell system in the Lorenz gauge. The field update is written as a Crank-Nicolson discretization of first-order wave systems for the scalar potential, the vector potential, and their time derivatives. The charge density is not deposited directly; instead, it is advanced from the discrete continuity equation using the current deposited from the particles. This opens up algorithmic flexibility with a range of innovation, including unstaggered mesh layouts that preserve the Lorenz gauge and Gauss's law at the discrete level. In the potential formulation, this source ordering also permits preservation of the Lorenz gauge and Gauss's law at the discrete level. To extend the paradigm to an energy-conserving formulation, we introduce a consistent orbit-averaged scatter, gather, and particle push. For energy consistency, the update of the canonical momentum is modified by replacing the pointwise midpoint derivative of the vector potential with an orbit-averaged discrete gradient of the mesh-interpolated vector potential consistent with the orbit-average maps. This construction satisfies an exact finite-difference chain rule along each particle orbit. As a result, the particle work equals the mesh work appearing in the Crank-Nicolson field-energy balance, yielding exact total-energy conservation up to nonlinear solver tolerance and roundoff. We demonstrate exact energy conservation of the method in 3D on the cold two-stream instability.

Stigmergic coordination has many medium-specific formalizations. This paper formulates a medium-agnostic comparison layer whose comparison object is the abstract enabled-response signature. For metadata refinements, we prove that a coarse quotient is response-adequate exactly when each quotient fiber is response-aligned. When that fails, there is a canonical coarsest response-adequate repair, and every response-adequate repair must retain at least the corresponding fiberwise number of response classes. On the dynamic side, quotient-compatible matched writes preserve one-step behavior, freshness-conditional actions yield a one-step non-liftability obstruction, and the matched one-step result lifts to finite serial traces. These results are instantiated for a tuple-space medium and a timestamped virtual-stigmergic dataspace. The result is a medium-agnostic comparison framework with one explicit positive/negative theorem cycle; richer pathwise and concurrent semantics remain outside scope.

We introduce and study the Combined Van-and-Mopeds Routing Problem with Time Windows (CVMRPTW), a novel variant of the heterogeneous fleet vehicle routing problem motivated by last-mile parcel delivery in dense European cities. In the CVMRPTW, a single van serves as a mobile base from which on-demand mopeds are deployed to reach customers located in areas inaccessible or inefficient for vans. The primary objective is to minimise the number of mopeds used while guaranteeing that all delivery time windows are met. We propose a Mixed-Integer Linear Programming (MILP) formulation that captures vehicle synchronisation, capacity tracking, and time-window compliance through 22 structured constraint families, and extend it to three operationally motivated variants: the standard model, the active-waiting-van variant, and the common-depot variant. A lexicographic two-stage optimisation framework allows secondary objectives-route duration, combined travel time, and combined distance-to be optimised without sacrificing the primary objective. Three graph-sparsification preprocessing techniques reduce the number of variables and constraints by up to 25%. For instances beyond the practical limits of exact solvers, we develop the Cluster-Based Combined Routing (CBCR) heuristic, which decomposes the problem into a clustering phase, a cluster-level exact solve, and a route-reconstruction phase. Computational experiments on real-world instances derived from OpenStreetMap data for Lisbon and Stuttgart show that the MILP solves instances with up to 48 customers to feasibility within one hour, and that the CBCR heuristic extends tractability to instances with up to 80 customers under favourable conditions.

Condensed mathematics is a framework developed by Clausen and Scholze for handling algebraic structure in the presence of topological information, using methods from homological algebra and sheaf theory. This text is the author's Master's thesis and closely follows Scholze's lecture notes on Condensed Mathematics; no original mathematical results are claimed by the author. This text is primarily expository and split into two parts. The first part offers an introduction to the subject, covering its basic definitions, constructions, and motivating examples. It emphasizes accessibility and includes many details intended to assist readers approaching the theory for the first time. The second part aims to show a particular application of the framework. Using the machinery of analytic rings and the corresponding notion of complete condensed modules, we work towards a proof of coherent duality in the case of affine schemes.

We first prove that two matrix semirings $\mathbf{M}_n(S_1)$ and $\mathbf{M}_n(S_2)$ are equationally equivalent whenever additively idempotent semirings $S_1$ and $S_2$ are equationally equivalent. We then prove an embedding theorem for matrix semirings $\mathbf{M}_n(S)$ over an additively idempotent semiring $S$: for all $n \geq 2$, $\mathbf{M}_n(S)$ embeds into $\mathbf{M}_{n+1}(S)$. This yields an ascending chain of varieties $\mathsf{V}(\mathbf{M}_2(S)) \leq \mathsf{V}(\mathbf{M}_3(S)) \leq \cdots$, which is strictly ascending when $S$ is the two-element distributive lattice. Finally, we show that every variety in the interval $[\mathsf{V}(S_c(abc)), \mathsf{V}(\mathbf{M}_n(S_7))]$ is nonfinitely based (i.e., has no finite basis for its identities), where $S_c(abc)$ is an eight-element flat semiring and $S_7$ is the unique nonfinitely based three-element additively idempotent semiring. Consequently, $\mathbf{M}_n(S_7)$ is nonfinitely based, yielding an ascending chain $\mathsf{V}(\mathbf{M}_2(S_7)) \leq \mathsf{V}(\mathbf{M}_3(S_7)) \leq \cdots$; moreover, every variety in $[\mathsf{V}(S_7), \mathsf{V}(\mathbf{M}_n(S_7))]$ is also nonfinitely based, and this interval contains at least countably infinitely many distinct varieties.

Wireless foundation models offer a path toward reusable channel state information (CSI) intelligence for sixth-generation (6G) systems. However, existing generic-backbone adaptation and CSI pretraining methods often treat CSI as task tensors rather than propagation-conditioned channel responses, thereby failing to capture the intrinsic time-frequency-spatial geometry of wireless environments. This paper presents a channel-adaptive roadmap toward CSI-native foundation models, proposing a unified framework that aligns pretraining, positional modeling, and attention control with three channel requirements: scale-aware heterogeneous exposure, physical time-frequency-antenna coordinates, and correlation-bounded token interaction. Extensive experiments demonstrate the superiority of the proposed framework across three dimensions: zero-shot generalization, reducing NMSE by more than 4 dB across spatial-temporal-frequency tasks; scale extrapolation, yielding up to a 5.4 dB gain under 8 times unseen antenna scaling; and inference efficiency, accelerating mobility-aware processing by up to 18.8%. A system-level evaluation with Sionna SYS further shows that the proposed framework uses only 7.01% of dense-pilot overhead, reaches -18.64 dB average NMSE, and improves average net spectral efficiency by 36.6% over dense LMMSE and 15.5% over WiFo, indicating that CSI-native representation learning can support pilot-efficient radio access.

The aim of this paper is to develop a deformation theory of monoid schemes, generalising the approach developed by Grillet. The core idea of this approach is to introduce the notion of a system of abelian groups, as the naive approach to exactness does not work for monoids. We first study the case of monoid sheaves (functors over a poset into the category of monoids) and prove a classification theorem in this setting, showing that the coextensions of a monoid functor with a system of abelian groups is a symmetric categorical group and equivalent to the one obtained by the abelian group homomorphism $[\mathcal{C}^0 \to \mathsf{ker}\partial^1]$, thereby linking with cohomology of certain types of complexes, as expected. We then move towards monoid schemes, which are a type of a monoid sheaf, but where localisations now allow us to develop our most noteworthy result: We show that coextensions can be seen in a natural way as a stack of symmetric categorical groups. We will mention a few mild implications of this, but leave the deeper uses of stack theory in this setting for later papers.

We revisit the variational principle for relativistic perfect fluids in a manifestly covariant formulation based on differential forms, with particular attention to the boundary data required for a well-posed action principle. For timelike flows, the formulation reproduces the standard formulation of Schutz and Brown. We then examine the extension obtained by replacing proper time normalization of the velocity with a nullity constraint, with the he Bi{\v c}ák-Kucha{\v r} null dust action emerging as a special case. Requiring the complete action to be invariant under local rescalings of the null flow vector imposes the condition that the energy density depend only on the entropy per particle. For this class of null flows, the conventional equilibrium condition, i.e., that the energy density is convex, fails at non-zero temperatures, therefore limiting the interpretation of the fluid as an ordinary thermodynamic system.

Let $G$ and $H$ be connected graphs that are 2-isomorphic. It is known that their Laplacian matrices are congruent by a unimodular matrix $U$. In this paper we show (Thm. \ref{thm:main2}) that $U$ is a bijection between certain spaces of harmonic vectors on the vertices of $G$ and $H$. In particular (Cor. \ref{cor:main1}) if $u$ is a harmonic vector with respect to vertices $c, d$ in $H$ and the 2-isomorphism maps edge $(a,b)$ in $G$ to edge $(c,d)$ in $H$, then $uU$ is a harmonic vector with respect to vertices $a, b$ in $G$.

False data injection attacks (FDIAs) threaten smart-grid operation by manipulating measurement data and misleading power system state estimation. Although recent data-driven methods have shown promising performance, many existing approaches rely on a single graph-filtering mechanism and therefore struggle to adapt to diverse cyberattack patterns. This paper proposes a topology-aware encoder-only transformer mixture-of-experts framework for joint FDIA detection and localization. The proposed model integrates Laplacian positional encoding and diffusion-biased self-attention to capture global spatial dependencies, while a densely routed mixture-of-experts module composed of Chebyshev and autoregressive moving-average graph convolutional experts adaptively applies complementary graph filters based on the current grid state. The framework is evaluated on the IEEE 118- and 300-bus systems using the New York Independent System Operator load profile. Experimental results demonstrate that the proposed approach achieves up to 93.91% detection F1 score and 84.97% localization F1 score, outperforming existing benchmark methods while maintaining low false-alarm rates.

We study $q$-analogs of finite free convolutions and their interaction with families of $q$-hypergeometric polynomials. First, we revisit the $q$-multiplicative finite free convolution, previously introduced in the literature, and show that it acts naturally on $q$-hypergeometric polynomials: the convolution of two such polynomials remains within the same class, with parameters obtained by concatenation. This observation provides a simple mechanism for constructing large families of $q$-hypergeometric polynomials whose zeros are real and whose logarithmic mesh is controlled. We illustrate it with an example of multiple little $q$-Jacobi polynomials of the first kind. A result of independent interest is also an alternative definition of the $q$-multiplicative convolution in terms of $q$-differential operators. Motivated by the additive finite free convolution, we introduce a $q$-additive finite free convolution and study its algebraic and analytic properties. Although this convolution does not preserve real-rootedness in general, we show that a natural modification involving a $q$-multiplicative convolution restores the preservation of real roots and interlacing for polynomials with bounded logarithmic mesh. Finally, we develop a systematic method to translate product identities of $q$-hypergeometric functions into convolution identities for $q$-hypergeometric polynomials. This approach yields several explicit formulas for $q$-additive convolutions and produces new families of real-rooted $q$-hypergeometric polynomials.