Papers

We study the singular limit of a nonlocal conservation law incorporating both non-locality in density and velocity. We derive classical solutions of the underlying Burgers equation as a singular limit. The derivation is valid up until the maximal time before formation of shocks. The main result answers a question raised in [10].

Let $\mathcal F$ be a graph class that is closed under topological minors and 1-sums, has decidable membership, contains a triangle, and is not the class of all graphs. Recently, Lee, Liu, and Tsai [ICALP 2026] showed that the edge-decomposition problem into $k \geq 3$ elements of $\mathcal F$ is NP-hard. In particular, their general hardness reduction covers a long-standing problem on outerthickness (when $\mathcal F$ is the class of outerplanar graphs). On the other hand, it is well known that decomposing a graph into forests is polynomial-time solvable, as implied by work of Edmonds [J. Res. Natl. Bur. Stand. B. 1965]. In this paper, we take a first step toward determining the complexity of edge-decomposition problems into just two graphs (the case $k=2$). We consider the simplest possible graph class $\mathcal F$ satisfying the criteria above: the triangular forests, that is, graphs in which every 2-connected component is a triangle. We prove that determining whether a graph can be edge-decomposed into two triangular forests is NP-complete.

From the perspective of continuum thermodynamics, we revisit the pressure oscillation problem in finite-volume methods for multiphase real fluids and clarify the physical counterpart of the Real Fluid Quasi-Conservative (RFQC) method. The pressure oscillation in conservative finite-volume methods originates from their implicit thermodynamic equilibrium assumption, whereas recovering an oscillation-free pressure requires additional physical information. The RFQC method achieves this by evolving the affine parameters xi and E0 of the isentropic internal-energy-pressure relation along pathlines, while the thermodynamic re-projection converts the deviation from the isentropic trajectory into an internal-energy error, thereby ensuring the thermodynamic consistency and numerical stability of the method. We then investigate the applicability limit of the RFQC method and identify a Liquid-upwind Anomaly (LUA) in extreme phase-change cases. For a Riemann problem involving liquid-vapor phase change, a numerical anomaly may occur if a liquid-upwind translational velocity is initially superimposed. Theoretical analysis reveals that this anomaly is initiated by the jump in the affine slope xi during phase change, which delays pressure rise in the downstream vapor cell. Concurrently, the re-projection removes the positive pressure increment, repeatedly generating large internal-energy errors and trapping the vapor cell in a cycle of delayed pressure recovery. The analysis indicates that the LUA is a start-up anomaly, which can be resolved by introducing a regularization strategy at the initial discontinuity. With the proposed regularization strategy, the RFQC method is equipped with enhanced accuracy and robustness for extreme thermodynamic flows, such as sonic phase-change jets.

Classes of graphs excluding a path and a biclique as induced subgraphs are extensively studied in the literature. One of the key structural results for such graphs is a Ramsey-type result due to Galvin, Rival, and Sands (1982), establishing the existence of a function $f$ bounding the maximum length of a path in terms of clique number $\omega$. We improve the best known bound on $f$ to a function that is a singly exponential in $\omega^c$, for some constant $c$, which we show is best possible, up to optimizing $c$. Our approach also has consequences for treedepth. In particular, we show that, for graphs excluding a path and a biclique as induced subgraphs, treedepth is bounded by a polynomial function of clique number. In turn, this result implies that every hereditary graph class that admits a function bounding treedepth of graphs in the class in terms of clique number, admits a polynomial such function. This gives a treedepth analogue of a recent result on pathwidth due to Hajebi (2025).

We establish the normalized covariance bound conjectured by Anari et al. (2026, Conjecture 3) for fixed-rank external-field measures. Let $d\ge2$ and $m\in[d-1]$. For $w\in(0,+\infty)^d$, let $\mathsf S$ be an $m$-subset of $[d]$ with rank-$m$ external-field law $\mathbb P(\mathsf S=S)=\frac{\prod_{i\in S}w_i}{e_m(w)},S\subseteq[d],|S|=m,$ where $e_m(w):=\sum_{T\subseteq[d], |T|=m}\prod_{\ell\in T}w_\ell$ is the $m$th elementary symmetric polynomial in $w_1,\dots,w_d$. Let $X:=(X_1,\dots,X_d)^\top$ be its indicator vector, i.e., $X_i=\mathbb I\{i\in\mathsf S\},i\in\{1,\dots,d\}.$ Let $\Sigma:=\mathrm{Cov}(X)$, put $v_i:=\Sigma_{ii}$ for each $i\in[d]$, and define $v:=(v_1,\dots,v_d)^\top,D:=\mathrm{diag}(v),V:=\sum_{i=1}^dv_i.$ We prove $\Sigma\succeq D-\frac{vv^\top}{V}.$ This improves the coefficient $1/2$ in the first version of our work (Cesari and Colomboni, 2026, Corollary 1.3) to the optimal universal value $1$. As a first corollary, we improve the coefficient in the pseudoinverse bound of Bacchiocchi et al. (2026, Lemma 2) from $2$ to the optimal value $1$. Specializing this bound to coordinate differences gives an alternative proof of our effective-resistance theorem from the first version of our work (Cesari and Colomboni, 2026, Theorem 1.1). The framework of Anari et al. (2026, Theorem 1 and Corollary 2) also yields unconditional correlated-sampling guarantees with stretch $6$ on the hypersimplex and $12$ on its at-most variant. Unconditional constant-stretch guarantees were first established in the first version of our work (Cesari and Colomboni, 2026, Corollaries 1.4 and 1.5), with constants $16$ and $32$, which we improve here to $6$ and $12$. These improved constants strengthen the positive resolution, established in the first version of our work, of the constant-stretch question posed by Naor et al. (2026, Theorem 2 and Section 5).

Let $X$ be a Fano $3$-fold with even canonical class which satisfies the generalized Bogomolov-Gieseker inequality, such as $\mathbb P^3$. We express Donaldson-Thomas invariants counting Gieseker semistable sheaves of rank $r$, where $r > 0$, on $X$ in terms of those counting sheaves of rank $0$ and pure dimension $2$. This implements an analogue of the programme initiated by S. Feyzbakhsh and R. Thomas in the case of Calabi-Yau varieties. The methods include $K$-theoretic Donaldson-Thomas theory and, quite unexpectedly, the use of certain combinatorial properties of vertex algebras.

In this paper, we show that the piecewise gradient associated with a representation of a continuous piecewise-$C^{p}$ function is a selection of a conservative field. Specifically, we prove that a set-valued map whose selections include the piecewise gradients, also called associated gradients, has the chain rule property along Lipschitz curves. As a consequence, continuous piecewise smooth functions are path differentiable and their Clarke subdifferentials satisfy the chain rule property. These results establish a connection between a representation-based calculus for nonsmooth automatic differentiation and the conservative-field framework. From an algorithmic perspective, we show that bounded iterates of the stochastic piecewise-gradient method converge to the set of conservative critical points. With an additional Lebesgue-null interface condition, a generic stepsize scaling, and a generic initialization, this result holds for the Clarke critical set. Finally, we demonstrate that, even with the interface condition, the graph of a Lipschitz continuous and piecewise-$C^{\infty}$ function need not admit a $C^1$ stratification satisfying Whitney's condition (a). Thus, the analysis developed here does not fall within the setting of Whitney stratifiable functions (e.g., semialgebraic functions).

We derive new formulas for quantities involved in the study of uniformly convex spaces, linear and nonlinear.

In this note we show that if the sphere $\mathbb{S}^n$ is covered by $k$ open sets with $n \geq 2k-2$, then one of these sets contains a path with antipodal endpoints. This is best possible in the sense that the statement fails for $n < 2k-2$. The result can be seen as a spherical analogue of a well-known conjecture of Norine on edge-colourings of the discrete hypercube.

We study the non-equilibrium steady state of a weakly anharmonic chain of $N$ oscillators driven at its boundary by Langevin thermostats at unequal temperatures. Under a perturbative weak-anharmonicity condition, we prove a full-gradient logarithmic Sobolev inequality whose constant is independent of the chain length $N$. For homogeneous pinned chains, an additional quantitative regularity assumption yields a boundary space-time logarithmic Sobolev inequality and relative-entropy decay on the same $O(N^3)$ relaxation time scale as the harmonic chain. The proof extracts a finite-dimensional Gaussian component from the boundary noise and compares conditional terminal-state laws by a change of variables. The estimates are uniform over bounded positive temperatures and require no near-equilibrium assumption on their difference.

We prove that the Post Correspondence Problem for finitely generated free groups is undecidable, even when one of the two homomorphisms is injective and has finite-index image. This resolves a longstanding open problem in algorithmic group theory. The proof proceeds through a connection with finite-state transducers. Given a cyclic tag system $\mathcal C$, we effectively construct a finite partial deterministic inverse transducer $\mathcal T_{\mathcal C}$ whose fixed-point set is nontrivial if and only if $\mathcal C$ halts. We then associate to any such transducer two homomorphisms $g,h\colon F_Y\longrightarrow F_A,$ with $h$ injective, such that their equalizer is nontrivial precisely when the transducer has a nontrivial fixed loop. As an immediate consequence, the rank of these equalizers cannot be computed in general, answering a question posed by Stallings in 1984. We further prove that there is no algorithm which decides whether the fixed subgroup of a virtual endomorphism of a finitely generated free group is trivial. Finally, we apply the main result to show that the stabilizer problem is undecidable for free subgroups of $\operatorname{SL}_4(\mathbb Z)$, and that the upper-right-corner problem is undecidable for free subgroups of $\operatorname{SL}_5(\mathbb Z)$, even when the given generators are promised to form a free basis, improving on recent results of Breuillard and Kocharyan.

We study continuous paths in the Chabauty topology on the set $\mathcal{D}_n$ of torsion free discrete subgroups of the isometry group of $n$-dimensional hyperbolic space. We prove a combination theorem for paths in $\mathcal{D}_n$, which allows us to construct an exotic path of discrete subgroups along which no two subgroups are isomorphic. We also introduce a technique we refer to as "chromatography" to prove a decomposition theorem that characterizes paths of convex cocompact groups in $\mathcal{D}_3$.

We investigate traveling wave fronts for an acid-mediated tumor invasion model with density-dependent degenerate diffusion. The model is a partially diffusive PDE–ODE system of Gatenby–Gawlinski type, in which the tumor diffusion coefficient $D(U)$ is allowed to be a general decreasing function satisfying $D(1)=0$. This degeneracy causes the traveling wave equation for the tumor component to lose uniform ellipticity near the healthy state, and hence standard arguments for nondegenerate reaction diffusion systems are not directly applicable. To overcome this difficulty, we introduce a nonlinear change of variables which removes the degeneracy from the highest-order term of the tumor equation. For each fixed admissible tumor profile, the acid profile is represented by a Green kernel, while the healthy-tissue profile is obtained from an explicit integral formula. The transformed tumor profile is then constructed as the stationary limit of a uniformly parabolic auxiliary problem. By combining comparison principles, local Schauder estimates, carefully chosen super- and sub-solutions, and the Schauder fixed-point theorem, we prove the existence of traveling wave fronts for every wave speed $\theta\ge 2\sqrt{rD(0)}$. The resulting wave connects the tumor-dominant state $(0,1,1)$ at $z=-\infty$ to the healthy state $(1,0,0)$ at $z=+\infty$. We further establish strict pointwise bounds, monotonicity of all wave components, and one-sided exponential asymptotic estimates in both the transformed variable and the original traveling-wave variable.

Inspired by interior-point methods (IPM) for structured convex optimization, Kannan and Narayanan introduced the Dikin walk for sampling uniformly from polytopes in 2009. As in IPMs, the Dikin walk is affine-invariant, and its convergence is governed by the barrier geometry used to define its local proposal. They showed that the Dikin walk with the logarithmic barrier for a polytope in $\mathbb{R}^{d}$ with $m$ linear inequalities mixes in $md$ iterations. In 2017, Chen, Dwivedi, Wainwright, and Yu improved this to $d^{2.5}$ using a Lewis-weight barrier, and conjectured that the correct mixing time should be $d^{2}$. We make progress toward this conjecture by improving the previous $d^{2.5}$-mixing bound. For exponential sampling over a polytope, we prove that the Dikin walk with a scaled Lee–Sidford metric mixes from a warm start in $d^{2.25}$ iterations. This also yields an improved cold-start complexity via a known annealing framework. The main technical ingredient is improved average self-concordance of the Lee–Sidford metric, which gives high acceptance probability for the Metropolis filter along a random Dikin proposal. While previous analyses were effectively limited to second-order control due to technical difficulties, we develop a principled higher-order analysis. The proof combines a selective higher-order expansion of recursive bottleneck terms, a moving orthonormal-frame calculus for higher derivatives of the Lewis weights, and Wiener-chaos decompositions via multiple stochastic integrals to control the resulting Gaussian polynomials.

We analyse reverse-mode automatic differentiation (AD) for discrete probabilistic programs. Our construction is formulated in the framework of Combinatory Homomorphic Automatic Differentiation (CHAD), treating AD as a structure-preserving transformation of programs, guided by a denotational semantics. The main case study is the finite atomic distribution monad, whose computations have finite support and differentiable weights. The key point is that differentiating probabilistic programs requires cotangents to flow backwards not only through deterministic computations, but also through the probabilistic structure itself. We define the corresponding reverse-mode code transformation and prove its correctness, for handled real-output programs, by a categorical logical-relations argument. Although the paper focuses on finite discrete probability, the construction gives a reusable pattern for differentiating discrete-output algebraic effects, including finite multiset non-determinism (e.g., from fork-join parallelism), exceptions, and writer-style accumulation (e.g., for in-place accumulation of high-dimensional vectors). More broadly, we view this work as a foundational step towards extending CHAD to richer probabilistic languages and to other algebraic effects with handlers.

Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if $k$ verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in $k$, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable $\alpha \sim G$ (de Finetti), the exact cascade posterior is $\ell_k = \ell_0 - \ln m_k$, with $m_k$ the $k$-th moment of $G$. Then: (i) $\ell_k$ is concave in $k$ for every non-degenerate $G$ – the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta$(a,b)$ latents, failure decays polynomially, $1-r_k \asymp k^{-b}$, with correlation parameter $\rho_v = 1/(a+b+1)$; (iii) a blind-spot atom of mass $1-\pi$ at $\alpha=1$ caps the evidence extractable from any number of gates at $-\ln(1-\pi)$ nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary ($\beta \sim H$) yields a trichotomy – gates eventually always help, plateau, or actively harm – decided by the upper-tail exponents of $G$ and $H$, with closed-form crossover $k^\dagger$. The mechanism is survivorship: errors surviving gates are the high-$\alpha$ ones. The theory is measurable: $R$ repeated verdicts per instance identify the first $R$ moments of $G$, so two verdicts identify $\rho_v$; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at $k=5$ and ~3000x at $k=10$; the correlated fit at $R=8$ tracks held-out depths. The practical lever is decorrelation – changing model family, modality, or evidence source – not adding gates.

In this note, we study $L^{2}$-harmonic forms on complete simply-connected Riemannian manifolds with non-positive sectional curvature. We first establish an a priori $L^{\infty}$-estimate for such forms via Moser iteration, under the curvature bounds $-K\leq\mathrm{sec}_{g}\leq0$. We then prove that any $L^{2}$-harmonic form which is also $L^{1}$-integrable must vanish identically. Consequently, on the universal cover of a closed non-positively curved manifold, the $k$-th $L^{2}$-Betti number vanishes if and only if every $L^{2}$-harmonic $k$-form is $L^{1}$-integrable. This criterion reformulates a topological vanishing statement as an analytic integrability condition.

We obtain a complete characterization of the class of conjugation operators $C$ on $H^2$ that map shift-invariant subspaces to shift-invariant subspaces. Alongside, we also obtain the existence and non-existence of conjugations that send shift-invariant subspaces to coinvariant subspaces.

Symmetric monoidal categories have been understood since the 1960's and are central to many branches of mathematics. In particular, the construction of spectra from symmetric monoidal categories is at the heart of algebraic $K$-theory. This construction starts from either categories with an action by a suitable operad $\sP$ or with suitable functors from the category $\sF$ of finite sets to the category $\mathbf{Cat}$ of categories. Infinite loop space theory, which codifies these constructions, led to the invention of $\infty$-categories. So why the title? We shall prove that the $2$-category of symmetric monoidal categories is equivalent (in fact very nearly isomorphic) both to a $2$-category of $\sP$-pseudoalgebras and to an isomorphic $2$-category of strictly special $\sF$-pseudoalgebras. This equivalence underlies a streamlined equivariant and multiplicative enhancement of infinite loop space theory, but it should be of independent interest.

Networked integrated sensing and communication (ISAC) exploits cooperation among multiple ground base stations (GBSs) to support safe uncrewed aerial vehicle (UAV) flight in low-altitude wireless networks (LAWNs). Existing studies mainly focus on communication enhancement or target parameter estimation, while the detection reliability of non-cooperative targets in the UAV forward region remains insufficiently investigated. To address this issue, this paper proposes a safety-aware forward detection design in networked ISAC, where multiple GBSs jointly support UAV downlink communication, state estimation, and non-cooperative target detection within the forward region of interest (ROI). First, the forward ROI is determined by the UAV position, velocity, and safe braking distance, and is voxelized to characterize target-existence states. Then, the Cramér-Rao lower bound (CRLB) for UAV state estimation and the forward-ROI miss-detection probability are derived, and their scaling laws are characterized: In detail, the UAV state-estimation CRLB approximately decreases as $\ln^{-2}J$ with the number of cooperative GBSs $J$, while the forward-ROI miss-detection probability follows an exponential-form scaling law as $\lambda_{t}D_{f}\ln^{-2}J$. Furthermore, a safety-aware resource optimization problem is formulated to jointly configure the sensing pilot ratio, transmit power, and beam direction, balancing UAV state-estimation performance and forward detection reliability under the communication-rate constraint. Simulation results show that, compared with the baseline scheme without forward detection, the proposed design reduces the average miss-detection probability and the corresponding sensing-induced collision risk by $17.05\%$, while introducing only limited state-estimation performance degradation, reflected by a $14.82\%$ increase in the average CRLB.

Path homology plays a central role in digraph topology and GLMY theory more generally. Unfortunately, the computation of the path homology of a digraph $G$ is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In this paper we introduce an inductive method of constructing elements of the path homology chain modules $\Omega_n(G;R)$ from elements in the preceding two dimensions. This proceeds via the formation of what we call upper and lower extensions, that are parametrised by certain labelled multigraphs which we introduce and call face multigraphs. The inductive elements we construct generate $\Omega_*(G;R)$ when $R$ has characteristic $2$. With characteristic $0$ coefficients, the inductive elements at least generate $\Omega_i(G;R)$ for $i=0,1,2,3$. In low dimensions, the inductive elements coincide with the natural generators, and when the digraph contains no multisquares, the inductive elements coincide with the basis elements produced by Fu and Ivanov. Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph $G$. We employ inductive elements to construct explicit generators of $\Omega_3(G;R)$ for a ring $R$ of characteristic $0$ or $2$, answering an open question posed by Grigor'yan. Several universal coefficient statements for path homology are obtained as a byproduct.

We define a new morphism on the fundamental group of the space of Lagrangian submanifolds, which takes into account the Lagrangian flux and monodromy. We study the discreteness of its image and relate this discreteness to the ($C^\infty$) topology of the Hamiltonian orbit of the Lagrangian in question. We completely describe this new morphism for Lagrangian tori in $\mathbb{R}^4$ and $\mathbb{C}P^2$ and give explicit constructions. Similar constructions allow us to study the shape invariant of the Clifford torus in $\mathbb{C}P^n$. Finally, we upgrade our results on Lagrangian tori in $\mathbb{R}^4$ from the $C^\infty$ topology to the Hausdorff one.

For an oriented graph $G$, denote by $fvs(G)$ the minimum number of vertices whose deletion from $G$ makes it acyclic. We show that an oriented graph $G$ on $n$ vertices and $m$ arcs satisfies $fvs(G) \le \frac{2n+m+h}{9}$ where $h$ denotes the number of connected components of $G$ that belong to a special class of oriented graphs. This result has three consequences. First, when $G$ is planar, we obtain that $fvs(G) \le \frac{2n+m}{9}$. In particular, this implies that $fvs(G) \le \frac{5n-6}{9}$ for any planar oriented graph $G$, improving the best known upper bound of $\frac{3n}{5}$~[Borodin, Discrete Mathematics, 1979]. Then, applying this inequality to the planar digraphs without directed triangles, we get that $fvs(G) \le \frac{6n-8}{13}$, which improves the current best bound of $\frac{n}{2}$~[Li and Mohar, SIAM Journal on Discrete Mathematics, 2017]. Finally, when $G$ has maximum degree 6, we have $fvs(G) \le \frac{4n}{7}$ and this bound is tight, answering a conjecture of Ai, Gutin, Liu, Yeo and Zhou~[arXiv:2512.01676, 2025].

We show short-time existence and uniqueness up to reparametrization for a large class of higher-order geometric flows of curves and networks obtained as $L^2$-gradient flow of higher-order functionals involving derivatives of the curvature. Additionally, we prove a long-time existence result via energy methods.

We establish all order long-time asymptotic expansions for heat kernels on nilpotent coverings and for prime closed geodesics in fixed central classes of nilpotent quotients of compact hyperbolic surfaces. The exact lattice-side input is the finite-dimensional rational Floquet-Bloch theory of the companion paper: rational Kirillov restrictions give exact finite-dimensional fibers, and a generalized Pytlik functional gives exact Fourier-inversion and normalized-trace identities. At a rational parameter $p/q$ the decomposition is exact, and the fluctuation of the fiber integrand is controlled only by $q$. Hence the large-denominator comparison with the smooth Kirillov or Schrödinger normal form is uniform on the rational support of the Pytlik functional; irrational parameters do not enter the rigorous trace argument. For general nilpotent models, coefficient-weighted spectral sums are justified to every fixed order by positive Rockland estimates, the Plancherel-Mellin formula, and a trace-level order-balance argument. In contrast with approaches which usually give leading terms or integrated Edgeworth-type asymptotics, the method gives genuinely local, pointwise higher-order heat-kernel expansions. The same representation-theoretic quantity governs the leading term in the closed-geodesic asymptotics, producing a nilpotent Chebotarev-type phenomenon. The Heisenberg model is computed to the first correction term, and the Engel model is represented through the resolvent and heat-kernel calculus of the quartic oscillato

Voltage-source converter high-voltage direct current (VSC-HVDC) links offer controllable active and reactive power output, making them a promising asset for emergency voltage support. This paper presents an analytical method for adjusting the active power setpoint of a VSC-HVDC station to maximise loadability during voltage-stressed conditions. By exploiting the geometric structure of converter capability limits, a closed-form expression for the optimal setpoint is derived under combined current and voltage constraints. The method requires local voltage measurements and an estimate of a wide-area voltage angle difference, making it suitable for real-time emergency control without global optimisation. Validation on the Nordic Test System confirms that the analytically predicted optimum is consistent with the setpoint yielding the highest loadability, and that adjustments of active power setpoint can yield a more-than-proportional increase in loadability. The results further indicate robustness to angle estimate uncertainty.

Several coefficient-adapted methods provide optimal-order approximation for elliptic equations with rough coefficients. Prominent examples include localized orthogonal decomposition, multiscale spectral GFEM, and constraint energy-minimizing GMsFEM. Their proven accuracy, however, is obtained by allowing the localization radius or the local spectral dimension to grow as the coarse scale \(H\) tends to zero. Classical MsFEM has an FEM-like local construction, but its available analysis does not give a coefficient-uniform \(\bigO(H)\) energy estimate over the full bounded-contrast measurable coefficient class. Motivated by this gap, we formalize an FEM-like notion of structural scalability. A chosen spatially local basis has uniformly bounded overlap, hence \(\bigO(1)\) stiffness entries per row, and every anchored local span uses coefficient information from only \(\bigO(1)\) coarse-element layers. We prove that no deterministic construction satisfying fixed bounds on the support radius, coefficient-information radius, and local multiplicity can converge uniformly over the coefficient class. In fact, its worst-case \(L^2\)-to-energy Galerkin error remains bounded below by a positive constant independent of \(H\). The lower bound is established using a fixed finite family of smooth periodic coefficients and smooth right-hand sides. The proof combines coefficients that coincide on local patches, a finite-dimensional approximation lower bound for corrector fields, a positive-density mesh argument, and strong periodic corrector convergence. Thus uniform optimal accuracy requires at least one local construction parameter to grow or requires coefficient information beyond the fixed-visibility model.

We study signature controlled differential equations (Sig-CDEs), that is, path-dependent controlled differential equations (CDEs) whose vector fields factor through the signature map. Working on spaces of stopped Hölder paths, we develop an existence, uniqueness, and stability theory for general path-dependent CDEs, and translate these pathwise well-posedness criteria into conditions on the corresponding signature functionals. We then prove dynamic universality: simply parametrized Sig-CDEs approximate the solution path of any well-posed path-dependent CDE arbitrarily well, uniformly over bounded sets of controls and initial histories, with global variants obtained using weighted spaces. Within this framework, entire maps of group-like elements provide a specific class of Sig-CDEs. Using a new class of limiting tensor spaces, we recast Sig-CDEs as infinite-dimensional classical CDEs and prove their well-posedness via a gauge-type scaling argument, thereby establishing a principled way to lift generic path-dependent dynamics. Lastly, we study truncated Sig-CDEs as finite-dimensional differential equations on step$-N$ Lie groups under intrinsic conditions, that is, with well-posedness formulated in terms of the underlying group metric.

In this paper, we consider two explicit Gaussian solutions to the constant (or parameter-independent) quantum Yang–Baxter equation and produce the corresponding bialgebras using the Faddeev–Reshetikhin–Takhtajan construction (FRT). Additionally, we twist these two Gaussian solutions, via Zhang twists and corresponding 2-cocycle twists, to obtain solutions to the Yang–Baxter equation which are not necessarily Gaussian.

The causal action principle for causal fermion systems is analyzed for a minimizing measure whose support is assumed to have the structure of a smooth manifold $\tilde{M}$. The concept of osculating vacua is introduced. It is shown that the Lagrangian induces on $\tilde{M}$ a Lorentzian metric. Moreover, the Euler-Lagrange equations of the causal action imply that the Ricci tensor must satisfy the Einstein equations of General Relativity for an energy-momentum tensor given in terms of a power expansion in the regularization length. The gravitational coupling constant is found to be the square of the regularization length. Our methods provide a systematic procedure for deriving corrections to the Einstein equations. The paper includes a self-contained introduction to causal variational principles and the causal action principle. Most geometric structures (connection, Riemannian metric and curvature) are introduced and analyzed in the general setting of causal variational principles for an arbitrary dimension of $\tilde{M}$. The Lorentzian setting works only for causal fermion systems and is worked out only in four spacetime dimensions.