Vizing and Gupta showed that the chromatic index $χ'(G)$ of a graph $G$ is bounded above by $Δ(G) + μ(G)$, where $Δ(G)$ and $μ(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined this bound, proving that $χ'(G) \leq Δ(G) + \left\lceil μ(G)/\left\lfloor g(G)/2 \right\rfloor \right\rceil$, where $g(G)$ is the girth of the graph $G$. A {\it ring graph} is a graph obtained from a cycle by duplicating some edges. The equality in Steffen's bound is achieved by ring graphs of the form $μC_g$, obtained from an odd cycle $C_g$ by duplicating each edge $μ$ times. We answer two questions posed by Stiebitz et al. regarding the characterization of graphs which achieve Steffen's bound. In particular, we show that if $G$ is a critical graph which achieves Steffen's bound with $g(G)\geq 5$ and $χ'(G)\geq Δ+2$, then $G$ must be a ring graph of odd girth.

This work presents a data-driven framework for fast forward and inverse analysis in topology optimization (TO) by combining Rank Reduction Autoencoders (RRAEs) with neural latent-space mappings. The methodology targets the efficient approximation of the relationship between optimized geometries and their corresponding mechanical responses or Quantity of Interest (QoI), with a particular focus on compliance-minimized linear elastic structures. High-dimensional TO results are first compressed using RRAEs, which encode the data into a low-rank approximation via Singular Value Decomposition (SVD), obtained in this sense the most important features that approximate the data. Separate RRAE models are trained for geometry and for different types of QoIs, including scalar metrics, one-dimensional stress fields, and full two-dimensional von Mises stress distributions. The resulting low-dimensional latent coefficients of the latent space are then related through multilayer perceptrons to address both direct problems -- predicting structural responses from geometry -- and inverse problems -- recovering geometries from prescribed performance targets. The proposed approach is demonstrated on a benchmark TO problem based on a half MBB beam, using datasets generated via density-based Solid Isotropic Material with Penalization (SIMP) optimization. Numerical results show that the framework enables accurate and computationally efficient surrogate models, with increasing robustness and fidelity as richer QoIs are considered. The methodology also provides a foundation for generative mechanical design by enabling the synthesis of new geometries and responses through latent-space exploration.

In the previous parts of this work, we established the Prequantum Groupoid $\mathbf{T}_ω$ as the universal geometric container for quantum mechanics. This approach, which we call the "Geometric Quantization by Paths" (GQbP) framework, replaces the traditional construction of principal bundles with the distillation of the space of histories. In this third part, we cross the "Threshold of Analysis" by constructing the intrinsic observable algebra of the system. The harmonic oscillator is treated here as a validation case, demonstrating that the standard resolution via complex polarization and half-forms is naturally integrated into the GQbP framework. Starting from the complexified groupoid, we define the algebra using symplectic half-densities to ensure a canonical convolution product. We then show that the transition to a polarized representation forces a factorization of these densities. The action of the symmetry group on the polarized half-forms generates a divergence term, which we identify as the source of the zero-point energy of the harmonic oscillator, $E_0 = n\hbar/2$. This derivation resolves the "Metaplectic Anomaly" as a necessary geometric consequence of the intrinsic quantization process.

Machine learning is increasingly used to guide branch-and-cut (B&C) for mixed-integer linear programming by learning score-based policies for selecting branching variables and cutting planes. Many approaches train on local signals from lookahead heuristics such as strong branching, and linear programming (LP) bound improvement for cut selection. Training and evaluation of the learned models often focus on local score accuracy. We show that such local score-based methods can lead to search trees exponentially larger than optimal tree sizes, by identifying two sources of this gap. The first is that these widely used expert signals can be misaligned with overall tree size. LP bound improvement can select a root cut set that yields an exponentially larger strong branching tree than selecting cuts by a simple proxy score, and strong branching itself can be exponentially suboptimal (Dey et al., 2024). The second is that small discrepancies can be amplified by the branch-and-bound recursion. An arbitrarily small perturbation of the right-hand sides in a root cut set can change the minimum tree size from a single node to exponentially many. For branching, arbitrarily small score discrepancies, and differences only in tie-breaking, can produce trees of exponentially different sizes, and even a small number of decision differences along a trajectory can incur exponential growth. These results show that branch-and-cut policies trained and learned using local expert scores do not guarantee small trees, thus motivating the study of data-driven methods that produce policies better aligned with tree size rather than only accuracy on expert scores.

Associated to each Tambara functor $T$ is its Nakaoka spectrum $\mathrm{Spec}(T)$, analogous to the Zariski spectrum of a commutative ring. We establish that this topological space is spectral. This result follows from an analysis of the notion of nilpotence in Tamabra functors. We prove that the nilradical of a Tambara functor $T$ (the intersection of all of its prime ideals) is computed levelwise, i.e. consists precisely of the nilpotent elements in $T$. In contrast to ordinary commutative algebra, the nilpotents of $T$ are not the same as the elements $x$ such that $T[1/x] = 0$; we therefore also give a classification of these elements. As a corollary, we observe that the set of these elements in $π_\star^s$ (the equivariant stable stems, viewed as an $\mathrm{RO}(G)$-graded Tambara functor) forms an ideal.

Graph attention networks (GATs) are widely used and often appear robust to noise in node covariates and edges, yet rigorous statistical guarantees demonstrating a provable advantage of GATs over non-attention graph neural networks~(GNNs) are scarce. We partially address this gap for node regression with graph-based errors-in-variables models under simultaneous covariate and edge corruption: responses are generated from latent node-level covariates, but only noise-perturbed versions of the latent covariates are observed; and the sample graph is a random geometric graph created from the node covariates but contaminated by independent Erdős--Rényi edges. We propose and analyze a carefully designed, task-specific GAT that constructs denoised proxy features for regression. We prove that regressing the response variables on the proxies achieves lower error asymptotically in (a) estimating the regression coefficient compared to the ordinary least squares (OLS) estimator on the noisy node covariates, and (b) predicting the response for an unlabelled node compared to a vanilla graph convolutional network~(GCN) -- under mild growth conditions. Our analysis leverages high-dimensional geometric tail bounds and concentration for neighbourhood counts and sample covariances. We verify our theoretical findings through experiments on synthetically generated data. We also perform experiments on real-world graphs and demonstrate the effectiveness of the attention mechanism in several node regression tasks.

Several applications of the QR-AAA algorithm, a greedy scheme for vector-valued rational approximation, are presented. The focus is on demonstrating the flexibility and practical effectiveness of QR-AAA in a variety of computational settings, including Stokes flow computation, multivariate rational approximation, function extension, the development of novel quadrature methods and near-field approximation in the boundary element method.

Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of entropy is captured by the family of Rényi entropies, parameterized by a real number $α$. Conditional entropy extends the notion of entropy by quantifying uncertainty from the viewpoint of an observer with access to potentially correlated side information. However, despite their significance and the emergence of various useful definitions, a complete characterization of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterization of conditional entropy, defined through a set of axioms that are essential for any operationally meaningful definition: additivity for independent random variables, invariance under relabeling, and monotonicity under conditional mixing channels. We prove that the most general form of conditional entropy is captured by a family of measures that are exponential averages of Rényi entropies of the conditioned distribution and parameterized by a real parameter and a probability measure on the positive reals. Finally, we show that these quantities determine the rate of transformation under conditional mixing and provide a set of second laws of quantum thermodynamics with side information for states diagonal in the energy eigenbasis.

The Gelfand-Kazhdan criterion is a fundamental tool for studying multiplicity-one properties of local periods of representations. However, it does not apply to many cases arising in the relative Langlands program. Generalizing the usual Gelfand-Kazhdan criterion, we formulate and prove a vector-valued Gelfand-Kazhdan criterion that fits into the general framework of the relative Langlands program. As an illustration of its effectiveness, we establish the multiplicity-one property for the local Asai Rankin-Selberg periods.

We establish $\mathrm{W}^{1,1}$-regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on $\mathrm{BV}$. Unlike classical examples such as the minimal surface integrand, we only require linear growth from below but not necessarily from above. This typically comes with a non-uniformly degenerate elliptic behaviour, for which our results extend the presently available bounds from the superlinear growth case in a sharp way.

Plavnik, Schopieray, Yu, and Zhang have drawn attention to those (automatically premodular) fusion subcategories of modular fusion categories which are submodules for the Galois action on the ambient category. In particular, they showed that a subcategory is a Galois submodule if and only if its centralizer is integral. In the other direction, Schopieray has conjectured that every premodular fusion category can be embedded as a Galois-closed subcategory of a modular category; Schopieray calls such an embedding a "Galois-modular extension." We prove Schopieray's conjecture for pseudounitary categories. Along the way we record some general comments about the minimal nondegenerate extension problem for braided fusion categories.

In this paper, we propose a new framework for solving a general dynamic optimal stopping problem without time consistency. A sophisticated solution is proposed and is well-defined for any time setting with general flows of objectives. A backward iteration is proposed to find the solution. The iteration works with an additional condition, which holds in interesting cases including the time inconsistency arising from non-exponential discounting. Even if the iteration does not work, the equilibrium solution can still be studied by a forward definition.

In the context of training neural network-based approximations of solutions of parameter-dependent PDEs, we investigate the effect of preconditioning via well-conditioned frame representations of operators and demonstrate a significant improvement on the performance of standard training methods. We also observe that standard representations of preconditioned matrices are insufficient for obtaining numerical stability and propose a generally applicable form of stable representations that enables computations with single- and half-precision floating point numbers without loss of precision.

We develop a continuum limit and mean-field theory for interacting particle systems (IPS) on self-similar networks, a new class of discrete models whose large-scale behavior gives rise to nonlocal evolution equations on fractal domains. This work extends the graphon-based framework for IPS, used to derive continuum and mean-field limits in the non-exchangeable setting, to situations where the spatial domain is fractal rather than Euclidean. The motivation arises from both physical models naturally formulated on fractals and real-world networks exhibiting hierarchical or quasi-self-similar structure. Our analysis relies on tools from fractal geometry, including Iterated Function Systems and self-similar measures. A central result is an explicit isomorphism between self-similar IPS and graphon IPS, which allows us to justify the continuum and mean-field limits in the self-similar setting. This connection reveals that macroscopic dynamics on fractal domains emerge naturally as limits of dynamics on appropriate discretizations of fractal sets. Another contribution of the paper is the derivation of optimal convergence rates for the discrete self-similar models. We introduce a scale of generalized Lipschitz spaces on fractals, extending the Nikolskii-Besov spaces used in the Euclidean setting, and obtain convergence estimates for discontinuous Galerkin approximations of nonlocal equations posed on fractal domains. These results apply to kernels with minimal regularity addressing models relevant in applications.

Kelley-Morse set theory KM is weaker than generally supposed and fails to prove several principles that may be desirable in a foundational second-order set theory. Even though KM includes the global choice principle, for example, (i) KM does not prove the class choice scheme, asserting that whenever every set $x$ admits a class $X$ with $φ(x,X)$, then there is a class $Z\subseteq V\times V$ for which $φ(x,Z_x)$ on every section. This scheme can fail with KM even in low-complexity first-order instances $φ$ and even when only a set of indices $x$ are relevant. For closely related reasons, (ii) the theory KM does not prove the Łoś theorem scheme for internal second-order ultrapowers, even for large cardinal ultrapowers, such as the ultrapower by a normal measure on a measurable cardinal. Indeed, the theory KM itself is not generally preserved by internal ultrapowers. Finally, (iii) KM does not prove that the $Σ^1_n$ logical complexity is invariant under first-order quantifiers, even bounded first-order quantifiers. For example, $\forall α{<}δ ψ(α,X)$ is not always provably equivalent to a $Σ^1_1$ assertion when $ψ$ is. Nevertheless, these various weaknesses in KM are addressed by augmenting it with the class choice scheme, thereby forming the theory KM+, which we propose as a robust KM alternative for the foundations of second-order set theory.

We introduce a new causal spinfoam vertex for $4$d Lorentzian quantum gravity. The causal data are encoded in Toller $T$-matrices, which add to Wigner $D$-matrices $T^{(+)}+T^{(-)}=D$, and for which we provide a Feynman $\mathrm{i}\varepsilon$ representation. We discuss how the Toller poles cancel in the EPRL vertex, how the Livine-Oriti model is obtained in the Barrett-Crane limit, and how spinfoam causal data are distinct from Regge causal data. In the large-spin limit, we show that only Lorentzian Regge geometries with causal data compatible with the spinfoam data are selected, resulting in a single exponential $\exp(+\mathrm{i}\, S_{\mathrm{Regge}}/\hbar)$ and a new form of causal rigidity.

This article develops a control method for linear time-invariant systems subject to time-varying and a priori unknown cost functions, that satisfies state and input constraints, and is robust to exogenous disturbances. To this end, we combine the online convex optimization framework with a reference governor and a constraint tightening approach. The proposed framework guarantees recursive feasibility and robust constraint satisfaction. Its closed-loop performance is studied in terms of its dynamic regret, which is bounded linearly by the variation of the cost functions and the magnitude of the disturbances. The proposed method is illustrated by a numerical case study of a tracking control problem.

A 2-covering for a finite group $G$ is a set of proper subgroups of $G$ such that every pair of elements of $G$ is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group $G$ is called the 2-covering number and denoted by $σ_2(G).$ In \cite{gk} it is conjectured that if $G$ is solvable and not 2-generated, then $σ_2(G)=1+q+q^2,$ where $q$ is a prime power. We disprove this conjecture.

The aim of this paper is to establish well-posedness properties for hyperbolic PDEs on Fourier Lebesgue spaces. We consider hyperbolic operators with complex characteristics. Since our approach comes from harmonic analysis, we establish boundedness properties of Fourier integral operators with complex-valued phase functions on Fourier Lebesgue spaces, Besov spaces and Triebel-Lizorkin spaces. Indeed, these classes of operators serve as propagators of the considered PDE problems. In terms of the boundedness properties, we prove new results in the case where the canonical relation of the operator is assumed to satisfy the {\it spatial smooth factorization condition}

Given a polynomial $f\in\mathbb{C}[x_1,\ldots,x_n]$ and an integer $\ell\in\mathbb{Z}$, we study some properties of the b-function with respect to weights of the annihilating ideal Ann$(f^\ell)$. In some particular cases the expression of the b-function is given explicitly.

Conditional independence testing (CIT) is essential for reliable scientific discovery. It prevents spurious findings and enables controlled feature selection. Recent CIT methods have used machine learning (ML) models as surrogates of the underlying distribution. However, model-agnostic approaches require a train-test split, which reduces statistical power. We introduce Semi-knockoffs, a CIT method that can accommodate any pre-trained model, avoids this split, and provides valid p-values and false discovery rate (FDR) control for high-dimensional settings. Unlike methods that rely on the model-$X$ assumption (known input distribution), Semi-knockoffs only require conditional expectations for continuous variables. This makes the procedure less restrictive and more practical for machine learning integration. To ensure validity when estimating these expectations, we present two new theoretical results of independent interest: (i) stability for regularized models trained with a null feature and (ii) the double-robustness property.

This paper frames calculus as a global, centuries-long development rather than a subject that began only with Newton and Leibniz. Drawing on ideas from Greek, Indian, Islamic, and later European mathematics, it highlights how concepts like infinity, area, motion, and continuous change slowly evolved through solving problems and cultural exchange. I argue that bringing this history into the classroom helps students see calculus as more than a set of procedures: it becomes a story of human creativity and persistence. By revisiting the questions early mathematicians struggled with, students can better appreciate and better understand the core ideas behind the formulas they use today.

In this paper, we propose a general Tikhonov regularized second-order dynamical system with viscous damping, time scaling and extrapolation coefficients for the convex-concave bilinear saddle point problem. By the Lyapunov function approach, we show that the convergence properties of the proposed dynamical system depend on the choice of the Tikhonov regularization parameter. Specifically, when the Tikhonov regularization parameter tends to zero rapidly, the convergence rate of the primal-dual gap along the generated trajectory is O(1 over t squared times beta(t)); when the Tikhonov regularization parameter tends to zero slowly, the convergence rate of the primal-dual gap is o(1 over beta(t)). We also prove the strong convergence property of the trajectory generated by the Tikhonov regularized dynamical system to the minimum-norm solution of the convex-concave bilinear saddle point problem, and derive several integral estimates. In addition, the effectiveness of the proposed dynamical system is verified through a series of numerical experiments.

For a Coxeter element $c$ in a Weyl group $W$, we define the $c$-Coxeter flag variety $\operatorname{CFl}_c\subset G/B$ as the union of left-translated Richardson varieties $w^{-1}X^{wc}_w$. This is a complex of toric varieties whose geometry is governed by the lattice $\operatorname{NC}(W,c)$ of $c$-noncrossing partitions. We show that $\operatorname{CFl}_c$ is the common vanishing locus of the generalized Plücker coordinates indexed by $W\setminus\operatorname{NC}(W,c)$. We also construct an explicit affine paving of $\operatorname{CFl}_c$ and identify the $T$-weights of each cell in terms of $c$-clusters. This paving gives a GKM description of $H^\bullet(\operatorname{CFl}_c)$ and $H^\bullet_{T_{ad}}(\operatorname{CFl}_c)$ in terms of the induced Cayley subgraph on $\operatorname{NC}(W,c)$, and we show these rings are naturally isomorphic for different choices of $c$. In type $\mathrm{A}$, this recovers the quasisymmetric flag variety for a special $c$, and for general $c$ we show the cohomology ring has a presentation as permuted quasisymmetric coinvariants.

Let $φ$ denote a $k$-algebra endomorphism of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of positive characteristic $p$. We prove that $φ$ can be lifted to an endomorphism of the Weyl algebra $A_n(W_2(k))$ over the Witt vectors $W_2(k)$ of length two over $k$ if and only if $φ$ induces a Poisson morphism of the center of $A_n(k)$. Furthermore, we improve a result of Tsuchimoto, which enables us to conclude that these equivalent statements hold at least when ${\rm deg}(φ) < p$. In particular, we conclude that $φ$ is injective if ${\rm deg}(φ) < p$.

The broadcasting problem concerns the efficient dissemination of information in graphs. In classical broadcasting, a single originator vertex initially has a message to be transmitted to all vertices. Every vertex which has received the message informs at most one uninformed neighbor at each discrete time unit. In this paper, we introduce infinite families of series-parallel graphs with efficient broadcast times: graphs on $n$ vertices with broadcast time at most $\lceil\log_2 n \rceil + 1$ for any $n$, graphs on $n$ vertices with broadcast time $\lfloor \frac{3 \lceil \log_2 n \rceil}{2} \rfloor$ and maximum degree $\lceil \log_2 n \rceil - 1$ for any $n$, and broadcast graphs on up to $2^{k-1} + 2^{\lfloor \frac{k}{2} \rfloor }$ vertices with broadcast time $k$ for any $k$. We also introduce an infinite family of planar broadcast graphs on up to $2^{k-1} + 2^{\lfloor \frac{3k}{4} \rfloor - 1}$ vertices with broadcast time $k$ for any $k$, which improves the known lower bound on the maximum number of vertices in a planar broadcast graph.

We study the structure of seminoetherian modules. Seminoetherian modules over non-primitive hereditary noetherian prime rings are completely described.

We prove the existence of solitary waves in a lattice where all particles interact with each other by pair-wise repulsive forces that decay with distance. The variational existence proof is based on constrained optimization and provides a one-parameter family of unimodal solutions. We also describe the asymptotic behavior of large, fast, high-energy waves.

We classify all finite groups that have lifting property of mod $p$ representations to mod $p^2$ representations for all prime $p$.

We prove Mermin-Wagner-type theorems for quantum lattice systems in the presence of multipole symmetries. These theorems show that the presence of higher-order symmetries protects against the breaking of lower-order ones. In particular, we prove that the critical dimension in which the charge symmetry can be broken increases if the system admits higher multipole symmetries, e.g. $ d = 4 $ on the regular lattice $ \mathbb{Z}^d $ in the presence of dipole symmetry.