Ensuring the safe and efficient operation of Advanced Air Mobility (AAM) in low-altitude airspace requires a reliable, robust, and resilient surveillance system capable of continuously detecting, identifying, and tracking aircraft under both normal and off-nominal conditions. To address this need, this study develops a comprehensive 3R modeling framework, reliability, robustness, and resilience, for the Surveillance for Advanced Air Mobility (SAM) system, with a focus on the optimal design and operation of a multi-type sensor network. Under normal operating conditions, the reliability model determines the baseline sensor types, quantities, and locations required to satisfy surveillance coverage and detection requirements. To address external perturbations, such as adverse weather conditions or sudden increases in AAM traffic demand, the robustness model identifies additional sensor requirements needed to maintain system performance. Furthermore, for surveillance outages caused by primary sensor failures, the resiliency model develops backup sensor deployment and dispatch strategies to provide temporary surveillance coverage, minimize operational disruptions, and support the safe continuation of AAM operations.

We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.

We determine the limiting density of the largest sum-free subset of the lattice cube $\{1,2,\ldots,n\}^d$ for all $d$, thus resolving the natural conjecture that it is constructed by two appropriate hyperplane slices.

It is a well-known theorem of homotopy type theory, originally due to Voevodsky, that function extensionality holds inside any univalent universe. We consider a weaker variant of the univalence axiom, asserting that the wild category formed by the universe is univalent, which we call categorical univalence. We show that categorical univalence does not imply function extensionality by an analysis of Von Glehn's polynomial model construction, which produces models of Martin-Löf type theory that always refute function extensionality. We find in particular that when the base model has a univalent universe, its polynomial model has a universe that is categorically univalent but lacks function extensionality.

We propose a conjectural $q$-analogue of the classical duality for iterated integrals on $\mathbb{P}^{1}$ minus four points, arising from the involutive Möbius transformation which exchanges the four marked points in pairs. To this end, we introduce iterated $q$-integrals with position-dependent $q$-shifts of the parameters and define a functional on admissible words in the six pairwise letters. The conjecture states that this functional is invariant under a natural anti-automorphism of the word algebra. We relate the conjecture to Yamamoto's duality for one-variable multiple $q$-polylogarithms. Finally, we prove the conjecture in several special cases.

Let $p$ be an odd prime. We describe a method to compute the Schur multiplier of finite $p$-groups $G$ of nilpotency class $2$ such that $G/[G,G]$ is isomorphic to direct product of copies of $\mathbb{Z}_{p^s}$ for $s \in \mathbb{N}$, generalizing a method of Blackburn and Evens, who treated the case $s=1$. As an application, we investigate which abelian $p$-groups can occur as the Schur multiplier of a non-abelian $p$-group. We further introduce the notions of $s$-special $p$-groups of rank $k$ generalizing the notion of special $p$-groups of rank $k$. We study the structural properties, compute the Schur multipliers of $s$-special $p$-groups of rank $1$.

We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form) similarity exponents. We prove that the exact Euler solution corresponding to the ground state implosion profile is stable to radially symmetric perturbations, as a solution to the full nonlinear compressible Euler equations, modulo a one-dimensional compatibility condition on the initial data. For perturbations of the Euler solution corresponding to the ground state implosion profile of a monatomic or diatomic gas, that do not obey any symmetry assumptions, we provide a complete characterization of the set of initial data that yield nonlinear stability.

In this paper, we study the long time behavior of solutions of Klein-Gordon-Zakharov system. We show that there exists a solution with special characteristics, which we shall refer to as a dipole solution, that is, there exists a solution $\vec{u}$ such that $$\left\|\vec{u}(t)-\sum_{k=1}^{2}\vec{R}_{k}\right\|_{X} \to 0 \, \, \text{as}\, \, t\to \infty,$$ where $\vec{R}_{k}$ represents a solitary wave for each $k$, with a translation $z_k$ with respect to its position, satisfying that $$|z_1(t)-z_2(t)| \sim 2\log(t)\, \, \text{as} \, \, t\to \infty.$$ Our approach will initially focus on the spectral analysis of the Hamiltonian operator associated with our system. Subsequently, we aim to establish a coercivity estimate that will allow us to derive conditions ensuring the existence of our solution. It is important to note that, in this problem, our objective is to obtain approximate solutions by solving a final data problem. These approximate solutions will then be used, through uniform estimates and compactness results, to derive the desired conclusions via density arguments.

We prove the compactness of weighted Sobolev trace operators in outward cuspidal domains by using composition operators on Sobolev spaces. This result allows us to formulate the non-linear Steklov problem in outward cuspidal domains in a correct functional setting and to establish the existence of its non-trivial solution.

We study recursive maximum likelihood estimation for stochastic interacting particle systems based on continuous observation of a single particle. In this regime, consistent estimation of the finite-particle log-likelihood is not possible, even in the limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. We thus seek to optimise the stationary log-likelihood of the limiting mean-field system. We achieve this via a form of stochastic gradient estimate in continuous time, with stochastic gradient estimates computed using the continuous trajectory of the single observed particle, alongside a virtual interacting particle system and a virtual tangent interacting particle system, which are integrated with the online parameter estimate. For fixed numbers of real and virtual particles, we show that the resulting algorithms drive the gradient of a finite-particle surrogate objective to zero as $t\to\infty$. We then prove that, in the iterated limit $t\to\infty$ followed by $N,M\to\infty$, these surrogate gradients converge uniformly to the gradient of the stationary log-likelihood of the limiting mean-field system, yielding convergence to its stationary points. We illustrate the method on several numerical examples, including a model with quadratic confinement and interaction potentials, a model of interacting FitzHugh--Nagumo neurons, and a stochastic Kuramoto model.

Let ${\mathcal H}_1$ be a finite dimensional complex Hilbert space. Let $ψ\mapsto Z(ψ)$ be a canonical anti-commutation relations (CAR) field over ${\mathcal H}_1$ acting irreducibly on a Hilbert space ${\mathord{\mathscr K}}$. The $*$-algebra ${\mathscr A}_{{\mathcal H}_1}$ generated by the $Z(ψ)$, $ψ\in {\mathcal H}_1$, is simply all operators on ${\mathscr K}$. However, the CAR field endows ${\mathscr A}_{{\mathcal H}_1}$ with additional structure, and we are concerned with quantum operations whose acting in harmony with this structure. In particular, there is a gauge automorphism group generated by ``second quantizing'' $ψ\mapsto e^{it}ψ$. The fixed point algebra of the gauge group, ${\mathscr G}_{{\mathcal H}_1}$, is a sub-algebra of ${\mathscr A}_{{\mathcal H}_1}$ studied by Araki and Wyss. It contains the density matrices of an important class of states, the gauge invariant Gaussian states, ${\mathfrak S}_{GIG}$. Our focus is on semigroups $\{e^{t{\mathscr L}}\}_{t\geq 0}$ of quantum operations on ${\mathscr A}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. Each $e^{t{\mathscr L}}$ is one-to-one, and our first main result is a structure theorem forsuch quantum operations on ${\mathscr G}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. We apply this to study semigroups of quantum operations on ${\mathscr G}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. Our second main result is a structure theorem showing that they are parameterized by pairs $(G,A)$ where $G$ is a contraction semigroup generator on ${\mathcal H}_1$, and $0 \leq A \leq -G -G^*$. We then show that each of these semigroups has a natural extension to the full CAR algebra ${\mathscr A}_{{\mathcal H}_1}$. Further results are obtained under further assumptions on the pair $(G,A)$.

We develop a theory of completeness for weight structures on stable categories, dual to the theory of complete t-structures. As in the bounded case, we show that complete weight structures are determined by their weight heart, giving rise to a universal construction $A \mapsto K(A)$ that assigns a complete weight category to an additive category and recovers classical examples such as homotopy categories of chain complexes. We also give a general construction of weight structures on presentable stable categories generated by a small set of objects, generalizing a result of Bondarko. This recovers the standard weight structure on spectra and an exotic one related to Anderson duality. We identify their completions with Bousfield--Kan completions arising in Adams-type spectral sequences. To treat naturally occurring examples - such as derived categories of abelian categories and module categories over ring spectra - which are often only partially weight complete, we introduce the notion of weak t-structures. Within this framework, we prove that any stable category equipped with compatible weight and weak t-structures, and satisfying left weight completeness and right t-completeness, can be reconstructed from its heart via a two-step completion process $A \mapsto \widehat{K}(A)$.

We construct an intrinsic q-deformation of the vector derivative on radial algebras. The construction is not obtained from a coordinate realization by replacing ordinary partial derivatives with one-variable Jackson derivatives; that coordinatewise procedure does not preserve radial subalgebras. Instead, for each distinguished vector variable $x$ and each finite set of auxiliary variables $Y\subset S\setminus\{x\}$, we define a q-Cartan derivative $\partial^Y_{x,q}$ on $R(\{x\}\cup Y)$ using the $x$-relative scalar variables $x^2$ and $\{x,y\}$, $y\in Y$. We prove two Fischer-type theorems. First, an exterior Fischer operator has a triangular anticommutator with explicit resonance factors; after inverting them one obtains a global Green operator and an exterior direct-sum decomposition. Second, using full left multiplication by $x$, we prove the monogenic Fischer decomposition after localization by finite-block determinants. We also describe the first denominator factors: the one-vector and two-vector factors are explicit, while the general determinant factors split by $x$-support. A degree-zero support-rank obstruction shows that a universal unlocalized theorem for all real $0<q<1$ cannot hold without excluding q-resonances.

This is a review of progress on the question whether noetherian Hopf algebras always have finite injective dimension and related good homological properties. As well as discussing in detail the main results giving positive answers for particular classes of Hopf algebras, some consequences of such positive answers are also described. Full definitions and references are included, also sketches of some proofs. A considerable number of open questions are listed, additional to the original question, which itself remains open after 30 years.

In domains, categories, and toposes, the Sierpiński cone construction glues onto a space a universal closed point lying below all the other points. Although this is a lax colimit, it also enjoys a well-known right-handed universal property: the Sierpiński cone classifies partial maps defined on an open subspace. The situation proves more subtle in synthetic models of space based on extending homotopy type theory with an interval, as in several recent approaches to synthetic higher categories and domains: although globally it may well be the case that the Sierpiński cone classifies partial maps, this property cannot hold of all parameterised types without degenerating the theory. On the other hand, there are reflective subuniverses within which the classifying property nonetheless holds. We show that the largest subuniverse in which the Sierpiński cone classifies partial maps is the accessible localisation at a family of embeddings parameterised in the interval, and this subuniverse is contained within the Segal types; this containment is moreover strict in the sense that when the interval is non-trivial, it is not possible for all Segal types to lie in the subuniverse. We finally extend these results from Sierpiński cones to mapping cylinders, providing a new right-handed universal property for the latter.

Estimating network formation models with degree heterogeneity raises two problems in empirical networks. First, agents that send no links, receive no links, or link to all remaining agents can make the fixed-effects MLE fail to exist. Trimming these agents changes the estimation sample and induces selection bias. Second, the incidental-parameter problem biases common parameters and average partial effects. We resolve both issues through a penalized likelihood approach. Our leading specification is a directed network model with reciprocity, nesting the standard undirected and non-reciprocal directed models. The penalty guarantees finite-sample existence and yields bias corrections for coefficients and partial effects. We establish asymptotic results without imposing compactness on the fixed-effects. Allowing the fixed effects to diverge at a logarithmic rate, our asymptotic framework accommodates the degree sparsity ubiquitous in large empirical networks. A global trade application demonstrates that our estimator avoids selection bias and recovers robust parameters where conventional methods fail.

Building on \cite{daworrpap,dawpap}, we prove two Zilber-Pink-type statements in $Y(1)^n$, assuming a weak form of the Lang-Trotter conjecture for pairs of elliptic curves.

Communications in highly dynamic channels relying on training-based channel estimation experience a trade-off between increasing channel measurement accuracy by sending more frequent training sequences and increasing data rate by sending fewer training sequences. Simultaneously, most communication systems use forward error correction to enable error detection and correction at the receiver. This paper presents decoder-provided pilots for time-varying channels by using decoded codewords as training sequences to update the channel estimate at the receiver. In contrast to approaches such as data-aided channel estimation, decision-feedback equalization, joint channel estimation and error correction, and turbo equalization, the decoder-provided pilots approach is non-iterative, which is ideal for low-latency requirements in highly dynamic scenarios. Furthermore, it is modulation-, code-, and decoder-agnostic, meaning it can be implemented on top of virtually any communication system that uses forward error correction. From an information-theoretic perspective, we derive the fundamental limits of decoder-provided pilots' ability to simultaneously sense the channel and transmit data. Simulation results demonstrate that decoder-provided pilots significantly improve performance, that when coding across frequency, soft-output can further enhance performance, and that when coding across time, short codes can outperform long codes of the same rate in fast-fading channels.

We show that the Lang-Trotter conjecture for pairs of elliptic curves implies new cases of the Zilber-Pink conjecture for curves in $\mathcal{A}_3$. Unlike previous results for curves in $\mathcal{A}_g$, our result does not rely on any assumption on intersections with the boundary, and in particular applies to potentially compact curves. The argument is based on the $G$-functions method of Yves André.

Networked systems operating under intermittent adverse conditions and long memory can remain stable on average while exhibiting rare but extreme trajectory-level excursions. We study linear regime-switching network dynamics with Volterra-type memory, formulated through a finite-dimensional lifted ordinary differential equation embedding. Despite finite-horizon annealed boundedness, we show that quenched amplification emerges generically from the interaction of regime persistence, memory accumulation, and non-normal lifted operator geometry. A lower bound on burst-size distributions reveals power-law tails whose exponent is determined by the ratio between unfavorable dwell-time rates and an operator-defined instantaneous growth parameter. This parameter is computable online via the Euclidean logarithmic norm of the lifted operator, yielding a practical early-warning indicator. Building on this structure, we introduce a dynamic data-driven intervention strategy that enforces contraction on demand along rare amplification channels, thereby shaping or truncating tail risk without altering exogenous regime statistics or typical system behavior. The results provide a geometrically grounded and operationally actionable framework for understanding and mitigating extreme events in memory-driven regime-switching systems.

Randomized-subspace methods reduce the cost of first-order optimization by using only low-dimensional projected-gradient information, a feature that is attractive in forward-mode automatic differentiation and communication-limited settings. While Nesterov acceleration is well understood for full-gradient and coordinate-based methods, obtaining accelerated methods for general subspace sketches that use only projected-gradient information and can improve over full-dimensional Nesterov acceleration in oracle complexity is technically nontrivial. We develop randomized-subspace Nesterov accelerated gradient methods for smooth convex and smooth strongly convex optimization under matrix smoothness and generic sketch moment assumptions. The key technical ingredient is a three-sequence formulation tailored to matrix smoothness, which recovers the corresponding classical Nesterov methods in the full-dimensional case. The resulting theory establishes accelerated oracle-complexity guarantees and makes explicit how matrix smoothness and the sketch distribution enter the complexity. It also provides a unified basis for comparing sketch families and identifying when randomized-subspace acceleration improves over full-dimensional Nesterov acceleration in oracle complexity.

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. The DP-chromatic number of a graph $G$, $χ_{_{DP}}(G)$, is the analogue of the chromatic number of $G$ in the DP context and is bounded above by the degeneracy of $G$ plus one. Over the last two years a plethora of authors have introduced variations on the notion of degeneracy and used these new ideas to give improved bounds on the DP-chromatic number of certain families of graphs. Fractional DP-coloring is a generalization of fractional list coloring introduced by Bernshteyn, Kostochka, and Zhu in 2019. In this paper we introduce two analogues of the degeneracy of a graph to the fractional context, each of which bound its fractional DP-chromatic number from above. We use these analogues to bound the fractional DP-chromatic number of a variety of graphs including unicyclic graphs, some complete bipartite graphs, and sparse graphs.

We introduce a new notion, that of a contextuality profile of a system of random variables. Rather than characterizing a system's contextuality by a single number, its overall degree of contextuality, we show how it can be characterized by a curve relating degree of contextuality to level at which the system is considered. A system is represented at level n if one only considers the joint distributions with no more than n variables, ignoring higher-order joint distributions. We show that the level-wise contextuality analysis can be used in conjunction with any well-constructed measure of contextuality. We present a method of concatenated systems to explore contextuality profiles systematically, and we apply it to the contextuality profiles for three major measures of contextuality proposed in the literature.

Large-scale integration of distributed energy resources has led to a rapid increase in the number of controllable devices and a significant change in system dynamics. This has necessitating the shift towards more distributed and scalable control strategies to manage the increasing system complexity. In this work, we address the problem of partitioning a low-inertia power network into dynamically coherent subsystems to facilitate the utilization of distributed control schemes. We show that an embedding of the power network using the spectrum of the linearized synchronization dynamics matrix results in a natural decomposition of the network. We establish the connection between our approach and the broader framework of spectral clustering using the Laplacian matrix of the admittance network. The proposed method is demonstrated on the IEEE 30-bus test system. We consider the robustness of the clusters by analyzing the sensitivity of the small eigenvalues and their corresponding eigenspaces to perturbations caused by variation in the steady-state operating points of the network.

We provide a comparison test for meromorphic extensions, i.e., if two series are ``close enough" then the existence of a meromorphic extension of one to the entire complex plane ensures a similar extension for the other. We use this result to generate new examples of Dirichlet series admitting meromorphic extensions. Moreover, we demonstrate that our requirements are optimal by constructing a collection of counterexamples where the series are ``close but not enough": one series admits a meromorphic extension while the other possesses a natural boundary.

We prove that under the heat semigroup $(P_τ)$ on the Boolean hypercube, any nonnegative function exhibits a uniform tail bound that is better than Markov's inequality. Specifically, for any $τ> 0$, $n \geq 1$, $η> e^3$, and $f: \{-1,1\}^n \to \mathbb{R}_+$ with $\int f dμ> 0$, we have \begin{align*} \mathbb{P}_{X \sim μ}\left( P_τf(X) > η\int f dμ\right) \leq c_τ\frac{ (\log \log η)^{\frac32} }{η\sqrt{\log η}}, \end{align*} where $μ$ is the uniform measure on the Boolean hypercube $\{-1,1\}^n$ and $c_τ$ is a constant that depends only on $τ$. This result resolves Talagrand's convolution conjecture up to a dimension-free $(\log \log η)^{\frac32}$ factor. Our proof uses the reverse heat process on the Boolean hypercube, a coupling construction with carefully engineered perturbations of jump rates and a time-smoothed anti-concentration estimate.

Controlled commands -- computations whose execution depends on a separate input -- play a central role in reversible Boolean circuits and quantum circuits. However, existing formalisms typically treat control only implicitly, entangled with other aspects of computation. From a semantic perspective, control is most naturally expressed in semisimple rig categories, which -- unlike standard circuit models such as props -- support both parallel and conditional composition. We present a construction that freely adjoins an explicit syntactic notion of control to a circuit theory specified as a suitable prop, subject to eight universally quantified equations. Our main result is that these equations are sound and complete for the intended semantics of control: the resulting theory satisfies a universal property, identifying it exactly as the circuit subtheory of the free semisimple rig completion. The proof combines coherence for rig categories with a new method based on induction over Gray codes. We illustrate the usefulness of the framework by showing that it simplifies several existing sound and complete axiomatisations of quantum circuits, isolating a small and conceptually clean set of generators and equations. In addition, the same equations yield a sound and complete axiomatisation of the multiply controlled Toffoli gate set, that is universal for reversible Boolean circuits.

We study a spectral analog of the Turán problem for simplicial complexes. Specifically, we consider the extremal problem of maximizing the signless Laplacian spectral radius among simplicial complexes without holes. We determine the structure of the simplicial complex attaining the maximum spectral radius, extending classical extremal results for graphs without cycles to the setting of higher-dimensional simplicial complexes. More generally, we establish an upper bound on the signless Laplacian spectral radius of simplicial complexes with prescribed Betti numbers. As an application, using the connection between the signless Laplacian spectral radius and the face numbers of a simplicial complex, we derive bounds on Turán numbers for both hypergraphs and simplicial complexes. Our technique involves the canonical Alexander dual of perfect matchings and coloring of simplicial complexes.

Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this work, we combine deep learning techniques and classic numerical methods for differential equations to address two challenging missing physics problems in dynamical systems theory: dynamics discovery and parameter estimation. The presented methods encode available information relating to the system dynamics into deep learning architectures, incorporating different assumptions on the known inputs and desired outputs in each case. Results demonstrate the effectiveness of the proposed approaches in making data-driven model predictions given corrupt system observations on a suite of test problems exhibiting oscillatory and chaotic dynamics. When comparing the performance of various numerical schemes, such as the Runge-Kutta and linear multistep families of methods, we observe promising results in predicting the system dynamics and estimating physical parameters, given appropriate choices of spatial and temporal discretization schemes and numerical method orders.

We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-partitioned network data. Our methods further generalize well-known algebraic results from the 1990s: namely, that the corresponding moves can be used to construct a regular triangulation for a generalization of the second hypersimplex. On the other hand, in contrast to the monochromatic case, we show that for \emph{simple} graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.