In this paper, we present the numerical analysis of an elliptic eigenvalue problem in which the essential boundary condition is imposed weakly by means of the Nitsche method. The resulting discrete eigenvalue problem is studied within the framework of compact operator theory. We prove norm convergence of the discrete solution operator and derive error estimates for the eigenvalues and eigenfunctions, with rates depending on the chosen Nitsche variant. In addition, we develop an a posteriori error analysis and propose a residual-based estimator suitable for adaptive refinement. Several numerical experiments are presented to assess the convergence, stability and robustness of the method, including the influence of the Nitsche stabilization parameter and the performance of the adaptive strategy.

We give the first polynomial-time constant-factor approximation of the Gromov--Hausdorff distance $d_{GH}$ between finite point sets in the Euclidean plane; in fixed Euclidean dimension such an approximation was previously known only on the line (Majhi, Vitter, and Wenk, 2024). Its engine is the bijective (bottleneck) Gromov--Hausdorff distance $d_{GH}^{bij}$: for two equal-size sets the least additive distortion $\max_{i,j}|d_X(i,j) - d_Y(σi, σj)|$ of a bijection $σ$ equals $2\,d_{GH}^{bij}$, which we likewise approximate within an absolute constant. Approximating additive distortion goes back to Hall and Papadimitriou (2005), who gave a $2$-approximation on the line and observed approximation within $3$ to be NP-hard in dimension three; the planar case they left open is the one we settle. A fat-or-collinear dichotomy drives both bounds: a fat set is aligned by a single rigid motion, while a near-collinear set is split into clusters matched along their dendrogram in one flat, scale-free pass, with relative orientations and per-node reflection signs -- at every scale of the dendrogram -- recovered by global cuts. Relaxing bijections to correspondences yields $d_{GH}$ itself, which reduces to a lone within-cluster-multiplicity kernel -- the pairs an optimal correspondence collapses -- that the same theory closes. Matching lower bounds -- a dimension drop, a multiplicity gap, and a reflection barrier acting at every scale -- show each ingredient is necessary.

The paper proposes a new approach to address the bounded-thrust powered-descent problem while ensuring ground-collision avoidance. A time-dependent polynomial approximation of the mass is employed to formulate a bounded linear-quadratic optimal-control problem that minimizes the thrust-acceleration control effort, terminal miss, and terminal velocity error. The resulting approximation is used to impose a hard constraint on the horizontal thrust profile while keeping the vertical thrust profile unconstrained. The key idea is a hierarchical separation of the thrust allocation, which enables analytical ground-collision avoidance under bounded thrust. Unlike existing bounded-thrust powered-descent approaches based on numerical optimization and trajectory-shaping constraints, the proposed method provides explicit analytical collision-avoidance conditions. Building on this formulation, the guidance law predicts the switching times between saturated and unsaturated arcs and shapes the thrust-acceleration profile to achieve a soft landing, even when the controller remains saturated over extended portions of the trajectory. Owing to its analytical nature, the guidance law is computationally efficient, and its continuous thrust profile facilitates real-time implementation. The proposed method was evaluated over a grid of perturbed initial conditions in realistic simulations, demonstrating accurate collision-free soft-landing performance. The results highlight the importance of combining saturation-aware guidance with ground-collision avoidance under bounded thrust.

We deal with fields with certain operators - introduced by the author and Kowalski - which we call $\mathcal{B}$-fields. We are mostly interested in $\mathcal{B}$-fields which are existentially closed, possibly in some restricted category of $\mathcal{B}$-fields. This in particular includes seeking for a model companion, but also is related to so-called pseudo algebraically closed structures. We prove a very general result saying that in many cases being existentially closed (in a generalized sense) is an elementary property. This encompasses, generalizes and simplifies many results from the literature. We study the resulting first-order theories, most importantly we study dividing lines and quantifier elimination.

We establish effective intrinsic ergodicity for renewal-type potentials on one-sided \(S\)-gap shifts. Inducing on the one-symbol cylinder \([1]\) reduces the system to a full shift over the alphabet \(S\), where the induced potential becomes a one-symbol potential and the equilibrium measure is Bernoulli. The associated renewal equation has a unique solution \(P\), and under the condition \(P>ϕ(0^\infty)\) (automatic when \(S\) is infinite), we show that \(P\) is the topological pressure and that the potential admits a unique equilibrium state \(μ_ϕ\). Our main result is an effective intrinsic ergodicity estimate: invariant measures whose free energy is within \(Δ\) of the pressure are \(O(\sqrtΔ)\)-close to \(μ_ϕ\) when tested against Hölder observables. As an application, every finite-word cylinder of positive \(μ_ϕ\)-measure yields a uniform pressure gap for the set of orbits avoiding that cylinder, leading in the entropy case to strict entropy and Hausdorff-dimension gaps.

In the present manuscript, we study an inverse problem related to a semilinear dynamical Schr{ö}dinger equation with lower order terms, in a bounded domain of $\Rb^{1+n},n\geq 2$. Our focus is on determination of the time-dependent coefficients appearing in the aforementioned equation, from the boundary measurements of the solutions. More precisely, we establish the {pointwise reconstruction} formulae for determining the time-dependent coefficients of linear and nonlinear terms from the knowledge of Dirichlet-to-Neumann map. Since the concerned non-linear Schrödinger equation possesses a trivial solution, we linearize the equation around the trivial solution and use the asymptotic solutions (\textit{with concentrated amplitudes}) of the linearized problem for reconstructing the aforementioned coefficients. To be more specific, we use first-order linearization to reconstruct vector and scalar potentials associated with the coefficients of linear terms and the higher-order linearization technique is used to reconstruct coefficients of nonlinearity. The nonlinear equation considered in this manuscript can be seen as a generalization of the Gross-Pitaevskii equation (GPE), which is employed to describe the dynamics of dilute Bose-Einstein condensates (BEC).

A central objective of machine learning is to identify structure and patterns in data. Advances in data acquisition have increasingly produced datasets whose observations possess rich geometric form, giving rise to shape spaces that encode variability in object geometry. Such datasets arise across a wide range of disciplines, including biology, medicine, anthropology, and computer vision, where subtle geometric differences often carry important scientific information. Traditional machine learning methods, however, are frequently ill-equipped to account for the nonlinear geometric structure underlying these data. This survey synthesizes a rapidly growing body of work on shape space analysis, which provides a mathematical and computational framework for the study of geometric data. Drawing on ideas from differential geometry, statistics, and machine learning, we organize the literature around a common analytical pipeline: shape representation and parameterization, the rigorous construction of robust geodesic metrics, statistical analysis on shape spaces, and geometry-aware learning methods. We discuss how these tools enable the characterization of shape variability, the comparison of geometric objects, and the analysis of structural trajectories across populations and time. To illustrate the breadth of the field, we highlight applications spanning multiple scales of biological organization, including studies of subcellular morphology and primate tooth evolution. Across these and many other domains, researchers face common challenges arising from complex, nonlinear, and often unaligned geometric variation. The review concludes by identifying key theoretical and computational challenges, as well as emerging opportunities driven by increasingly large and diverse geometric datasets.

We compute the exact value of the Ramsey number $R(K_4-e,K_7)$. It is equal to 28.

In this note, we prove a tight lower bound on symmetric extended formulations for the spanning tree polytope of the complete graph. More precisely, let $P_{ST}(K_n)$ be the spanning tree polytope of $K_n$. We show that, for all $n\ge13$, every symmetric extended formulation for $P_{ST}(K_n)$ has at least $\binom n3$ inequalities. Since the classical Martin formulation has a symmetric formulation of size $O(n^3)$, this gives \[ \operatorname{xcs}(P_{ST}(K_n))=Θ(n^3). \]

We establish uniqueness of blow-ups, with sharp quantitative convergence, for several classes of singular minimizing cones in the Alt-Phillips problem, in the range $γ\in(0,2)$. As a consequence, we obtain uniqueness at every free boundary point for $γ\in(1,2)$ in dimensions $d=2,3,4$, and for $γ\in\left(1,\frac32\right)$ in dimensions $d\geq 5$. The proof is based on three new logarithmic epiperimetric inequalities. The sharp distinction between polynomial and logarithmic convergence is governed by a finite-dimensional integrability condition (sub-integrability) for the spherical linearized problem. We prove this sharpness for radial and cylindrical cones through an explicit integrability and bifurcation analysis, and show that logarithmic convergence may be sharp even in dimension two. In contrast, we show that the one-dimensional cone is exceptional: although the integrability condition fails, the convergence is polynomial. Finally, we characterize the minimality of the radial cone in terms of $γ$ and $d$ by means of a one-dimensional calibration argument, exhibiting in dimension $d\geq6$ a nontrivial regime in which the radial cone is stable but not minimizing.

Joint prediction sets for multivariate time series should control a single event while adapting to cross-coordinate dependence. We study filtered conformal ellipsoids: a frozen state-space filter emits a one-step predictive mean and covariance, and split-conformal calibration is applied to the resulting Mahalanobis scores. The filter is used to choose the ellipsoid shape; conformal calibration chooses the scalar radius, so the construction benefits from a learned predictive covariance without relying on Gaussian tail probabilities for coverage. The main difficulty is that filtered scores are dependent and learned recurrent filters need not contract in their raw hidden state; we therefore analyse contraction in an observable predictive-law quotient that identifies hidden states producing the same future sequence of emitted Gaussian laws. Under a stable Bayes Gaussian-projection filter, covariance bounds, and a finite-horizon observability Fisher condition, small excess Gaussian negative log-likelihood implies contraction of the learned emitted laws. Combined with a threshold-autocovariance envelope this yields a Chebyshev-type approximate coverage bound for filtered split-conformal prediction under dependence; a sharper Bernstein-type bound requires an additional geometric-mixing concentration assumption. Under Gaussian oracle realisability we also obtain a near-oracle log-volume comparison within the class of conditionally valid Gaussian ellipsoid rules. We instantiate the framework with a GCN-GRU filter with diagonal-plus-low-rank covariance. On moderate-size graph-native traffic benchmarks (METRLA-$20$ and PEMSBAY-$50$), the learned filter gives sharper at-target ellipsoids than static-covariance and non-filter baselines; at full-graph scale and on non-graph-native datasets, factor and copula baselines can be stronger.

The well known phenomenon of exploding and vanishing gradients in deep neural networks is analyzed using multiplicative ergodic theory. The effect of adding a residual connection is explained in this context. Specifically, a characterization of Liapunov exponents due to Furstenberg and Kifer is exploited in order to make a precise statement about the Liapunov spectrum and the effect of residual connections on it.

Subwavelength elastic resonators can concentrate wave energy at length scales far below the incident wavelength, but their behavior becomes especially delicate when two resonators almost touch. In this paper, we give a rigorous analysis of a two-dimensional dimer made of two high-contrast hard inclusions embedded in a soft elastic matrix. The analysis confronts two features that are absent from the corresponding three-dimensional theory: the logarithmic low-frequency singularity of the two-dimensional elastic fundamental solution and the possible non-invertibility of the static single-layer potential. We overcome these difficulties by proving the invertibility of the correct frequency-dependent leading-order operator and then using it to reduce the resonance problem to a finite-dimensional system. For generally convex resonators satisfying natural symmetry assumptions, we derive six subwavelength resonant frequencies and identify their dependence on the material contrast $δ$ and the inter-inclusion distance $\varepsilon$. We further quantify the resonant field concentration in the narrow gap. In the regime $\varepsilon=\Ocal(δ^β)$, $0<β<2$, the gradients of the eigenmodes display sharply classified blow-up behavior: some modes attain the stronger rate $\Ocal(1/\varepsilon)$ at the closest point of the gap, while others blow up at the rate $\Ocal(1/\sqrt{\varepsilon})$ away from the centerline; the remaining mode is governed by a boundary mismatch mechanism. These results uncover resonance-induced singularities that are markedly stronger and more structured than those in static or non-resonant elasticity, and they provide a framework for analyzing larger clusters of closely spaced elastic subwavelength resonators.

We prove that computing approximate stationary points of min-max optimization over the hypercube is PPAD-hard for quadratic polynomials. This holds even when the polynomials are multilinear, each variable appears in at most three monomials, and the approximation factor is inverse polynomial. As a direct consequence, we obtain the first PPAD-hardness results for two-team zero-sum polymatrix games.

We give a bound for the norm of each primary genus zero Gromov-Witten invariant in any smooth projective variety. This bound depends only on the maximum degree of its defining polynomials, the norms of the differential forms representing each insertion, the number of marked points and the degree of the curves. These bounds grow factorially in the number of marked points and a multiple of the degree. This implies that the Borel transform of the genus zero primary generating function converges absolutely in a particular translation of the ample cone. To prove these bounds, we start with Siebert's formula, expressing genus zero Gromov-Witten invariants in terms of the Segre class of the normal cone of the underlying moduli space inside the space of curves mapping to projective space and the Chern classes of the bundle whose fiber over a curve is the space of sections of the pulled back normal bundle. We then use this description to bound each primary Gromov-Witten invariant in terms of an intermediate count of curves called the D-volume. A deformation to the normal cone argument is then used to compare such an intermediate count with one in projective space.

While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales. We propose a compact spectral representation that distills the persistent Laplacian into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion. Across benchmark datasets including MNIST, QM-3D, and SKEMPI WT, we demonstrate that this reduced feature space captures the essential predictive signal of the full spectrum, and in some cases outperforms it, while significantly reducing computational overhead and preventing the noise introduced by higher-frequency eigenvalues. Our results suggest that these invariants provide a principled, fixed-length interface between spectral geometry and topological learning.

The theory of total positivity was shown by Lusztig to be intrinsically linked to the canonical basis with its positivity properties. When we restrict ourselves to studying total positivity just for the set of lower-triangular unipotent Toeplitz matrices, say in type $A$, then there is a similar link with the quantum cohomology rings of flag varieties and the Schubert bases and their positivity properties. Namely, this builds on a theory of Dale Peterson that gives a uniform Lie-theoretic description of all of the quantum cohomology rings $qH^*(G/P)$. In a precursor to this paper, the Schubert basis and quantum parameters in $qH^*(SL_n/B)$, which restrict to positive-valued functions on totally positive Toeplitz matrices, were analysed with respect to their limiting behaviour as $n\to\infty$, uncovering a novel connection with the classical Edrei theorem on parametrising the infinite totally positive Toeplitz matrices. In this paper we study the Grassmannian case, using the conventions from the $SL_{n}/B$ setting as a guide, and we determine the quantum parameter and Schubert class asymptotics in different scenarios. Along the way, we obtain a new interpretation of the strange duality involution on the localised quantum cohomolgy ring of the Grassmannian. Finally, we prove an asymptotic formula for quantum parameters in a partial flag setting, and we furthermore formulate some conjectures concerning partial flag varieties and related quantum cohomology asymptotics.

A covariant reconstruction framework for electromagnetic Kantowski--Sachs (KS) geometries in teleparallel $F(T)$ gravity is developed using the coframe/spin-connection (CSC) formalism and the Coley--Landry invariant approach. In a restricted Maxwell-compatible branch, the electromagnetic conservation laws strongly constrain the anisotropic KS scale factors and lead to the scaling $ρ_{\mathrm{em}}\propto A_3^{-4}$. The corresponding symmetric and antisymmetric field equations are derived and used to reconstruct the functional form of $F(T)$ directly from the KS dynamics. Power-law and exponential ansätze generate distinct invariant reconstruction branches associated with electric, magnetic, and transverse electromagnetic sectors. The exponential branch naturally admits reduced teleparallel de Sitter limits and shifted models of the form $F(T)=f(T_0-T)$. The reconstructed branches describe anisotropic cosmological sectors together with local BH-interior-like sectors that may reproduce reduced BH-interior-like or RN--dS-type behaviors at the level of the KS dynamics. These branches are organized through the invariant coframe/spin-connection classification and screened using the necessary leading-order viability conditions $F_T>0$ and $F_{TT}>0$. The local and branch-dependent nature of the construction is emphasized throughout.

For any closed minimal hypersurface $M$ in the $N+1$-dimensional Euclidean sphere $S^{N+1}$, $-N$ is an eigenvalue of the stability operator. In this paper we show that the multiplicity of this eigenvalue for the Carlotto and Schulz minimal embedding $X_{CS}^{n}:S^{n-1}\times S^{n-1}\times S^{1}\to S^{2n}$ is at least $2n+1+n^2$. We conjecture that if $n>2$, then the stability index of $X_{CS}^{n}$ is $\frac{1}{3} \left(n^3+9 n^2+11 n+3\right)$ and for the hypertorus in $S^4$ (case $n=2$) the stability index is $27$. We numerically verify the conjecture for the first 100 values of $n$. We also numerically verify that Yau's conjecture on the first eigenvalue of the Laplacian holds when $2\le n\le 260$.

For a digraph $D$, a subset $Q\subseteq V(D)$ is called a $q$-kernel if $Q$ is an independent set and all vertices in $V(D)$ are reachable from $Q$ via a directed path of length at most $q$. Given integers $q\geq 2$ and $δ\geq 1$, Spiro arXiv:2404.07305 [math.CO] posed the question: what is the smallest constant $c_{δ,q}$ such that every digraph $D$ with minimum in-degree $δ$ has a $q$-kernel of size at most $c_{δ,q}|V(D)|$? We show the constants $c_{δ,q}$ are monotone in both $δ$ and $q$, and we improve upon the known upper bounds for $c_{δ,q}$. Our main results show $\frac{1}{δ+1} \leq c_{δ,q}\leq \frac{1}{\lfloor\sqrt{δ+1}\rfloor+1}$ for all $q \geq 3$ and $δ\geq 1$, and $ c_{δ,q}=\frac{1}{δ+1}$ whenever $δ\geq 1$ and $q \geq \left\lceil\frac{3δ}{2}\right\rceil + 1$.

Encoding phylogenetic networks by suitable substructures is a central problem in phylogenetic combinatorics. We study encodings based on least common ancestor (LCA) constraints. For a directed acyclic graph (DAG) $G$ with leaf set $X$, we consider the relation on pairs of leaves in which $(ab,xy)$ records that the LCAs of $a,b$ and $x,y$ are well-defined and that the former is a descendant of the latter. We first identify precisely which part of $G$ is determined by this relation. To this end, we compare the canonical DAG constructed from the LCA relation with the 2-regularization of $G$, obtained by removing all vertices that are not LCAs of one or two leaves and then deleting shortcut edges. We prove that these two DAGs are isomorphic. Hence the obstruction to encoding a graph by its LCA relation is exactly the information lost under 2-regularization. This yields a general reconstruction principle, which we apply to several natural classes of phylogenetic networks. In particular, we show that shortcut-free 2-LCA-relevant DAGs, phylogenetic trees, regular level-1 networks, regular networks with binary clustering systems, regular networks whose clustering systems are closed weak hierarchies, strong-phylogenetic normal networks, separated phylogenetic normal networks, and binary normal networks are encoded by their LCA relations. We also introduce a sparse triple-like restriction consisting only of comparisons of the form $(ab,ac)$, where $a,b,c\in X$ are pairwise distinct. For graphs with the 2-LCA property, we show that this sparse relation, together with the leaf set, determines the full LCA relation after a natural closure operation. Consequently, several of the above classes can be reconstructed, up to isomorphism, from the sparse relation in polynomial time.

We classify the dynamics of radial $H^1$ solutions at the (radial) ground state threshold for the $3d$ cubic NLS in the presence of a repulsive inverse-square potential.

Eventually constant set-valued representations of a quiver are set-theoretic analogues of nilpotent representations. In recent work by Green-Holmes-Im, the authors enumerated eventually constant set-valued representations for equioriented cyclic quivers using the directed matrix-tree theorem. In this paper, we extend this enumeration to finite quivers without sinks for which every vertex is the target of sufficiently long paths. We encode the representations as directed acyclic graphs and introduce a recursive source-removal method for certain classes of directed acyclic graphs. This yields a strictly upper triangular matrix enumerator in the incidence algebra of the subset lattice. To compute the cardinality of the eventually constant representations, we compress this enumerator to a matrix indexed by cardinality vectors, the set-theoretic analogues of dimension vectors. We conclude by specializing the formulas to the Jordan quiver and recovering the multisymmetric generating polynomial for the cyclic quiver without using the matrix-tree theorem.

Extending earlier results of Nešetřil and Rödl [Selective graphs and hypergraphs, Ann. Discrete Math. 3 (1978), 181--189], we show that for every ordered graph $F$ there exist an ordered graph $H$ and a system $\mathscr{H}_F$ of induced copies of $F$ such that every colouring of the edges of $H$ yields a canonically coloured copy of $F$ from $\mathscr{H}_F$ and any two copies from $\mathscr{H}_F$ intersect either in a vertex or an edge or not at all. As a consequence, this allows us to construct, for any given ordered graph $F$, canonical Ramsey graphs $H$ enjoying additional structural properties. In particular, $H$ can have the same clique number as $F$ and, provided $F$ is not bipartite, the same odd girth. Moreover, if $F$ is connected, then the copies of $F$ from $\mathscr{H}_F$ are not only induced, but their pairs of vertices also have the same distances in $H$ as in $F$.

Dense Gaussian networks provide degree-4 interconnection topologies with small diameter and regular structure, making them suitable for efficient one-to-all broadcasting. However, node failures can disrupt the broadcast process when faulty nodes occupy internal forwarding positions. This paper proposes a lightweight fault-tolerant broadcasting method based on dynamic source relocation, or re-rooting. Instead of constructing redundant spanning trees or backup routing structures, the proposed method selects a new source node so that the faulty nodes are located at graph distance \(k\), the network diameter, from the new source. Consequently, faulty nodes become leaf-level nodes in the broadcast process and are not required to forward the message. For the single-fault case, the new source is selected directly from the graph-distance-\(k\) boundary of the faulty node. For the two-fault case, we prove that for any pair of faulty nodes in \(G(k+(k+1)i)\), there exists a node whose graph distance from both faulty nodes is exactly \(k\). The source-selection procedure requires \(O(k)\) time. Since the original one-to-all broadcast completes in \(k\) parallel steps and the relocation distance is at most \(k\), the proposed method completes in at most \(2k\) steps in the worst case. We also show that the two-fault guarantee does not generally extend to arbitrary three-fault configurations by giving a counterexample in \(G(3+4i)\). Simulation results confirm complete delivery to all non-faulty nodes under the tested one- and two-node failure scenarios, while the baseline broadcast may fail when faulty nodes occur at internal forwarding positions.

Classical Gelfand duality provides an equivalence between commutative C-star algebras and topological spaces, but fails to furnish a geometric object for noncommutative operator systems. To address this, we introduce a categorified notion of spectrum that captures the full operator-semantic structure. For an operator system A, we construct a spectral stack, denoted Spec(A), over the site of its commutative contexts. The construction proceeds in three stages. First, we encode the syntax of A via a colored operad called the synergy operad. Second, we aggregate local semantic data using a left Kan extension, providing an explicit coend formula. Third, we enforce descent via sheafification, yielding a stack that satisfies higher gluing conditions. We prove that Spec(A) satisfies a Yoneda-style universal property, making it the initial descent-complete semantic realization of A. This yields a contravariant functor from operator systems to spectral objects that admits a right adjoint, the global sections functor. Under semantic generation and descent completeness, the counit of this adjunction is an equivalence, establishing a reconstruction theorem: A is isomorphic to the endomorphisms of the structure sheaf on its spectrum. We further provide a recognition theorem characterizing the essential image of this functor and prove Morita invariance of the associated quasi-coherent sheaf categories. The construction naturally recovers classical Gelfand spectra and Bohrification as truncations of a Postnikov tower. Explicit computations for matrix algebras, Pauli systems, and the Mermin-Peres square yield a quantitative invariant of contextuality extracted from the inertia stack of Spec(A). Comparison theorems show that our framework subsumes Gelfand duality, Bohrification, and Tannaka reconstruction as special cases.

The problem of quantum tomography is to estimate an unknown quantum state $ρ$ from a measurement of $n$ copies of $ρ$. One can ask which tomography protocol, i.e.\ which choice of multi-copy measurement, gives the best possible estimate of $ρ$. To do so, we characterize tomography protocols by their \emph{rate function}, which governs the exponential rate at which a protocol assigns probability to a particular estimate $σ$ of the true state $ρ$. This rate function is a quantum mechanical generalization of the classical relative entropy between the true state and its estimate, and depends on the choice of protocol. It is bounded by the quantum relative entropy, and we show that this bound is sharp: for any $ρ$ and $σ$ we construct a family of protocols whose rate functions converge to the quantum relative entropy $D(σ\|ρ)$. We consider the family of covariant tomography protocols; these are the basis independent state estimation schemes that assume no prior information about $ρ$ and $σ$. Keyl described a specific tomography protocol based on Schur sampling, and conjectured that among all covariant tomography protocols it has the largest possible rate function for all $σ$ and $ρ$. We prove this conjecture. The resulting rate function is an annealed version of quantum relative entropy, due to the cost of learning the eigenbasis in covariant quantum state tomography.

In this paper, we investigate the existence of positive supersolutions for the following mixed local-nonlocal Lane-Emden type equation: $$ -Δu+(-Δ)^s u=u^q\quad\text{in }\mathbb R^n, $$ where $n\geq 3$, $s\in(0,1)$, and $q>1$. More precisely, we prove that the equation admits positive distributional supersolutions if and only if $q>\frac{n}{n-2s}$. In the process, we establish several novel properties of mixed local-nonlocal operators, including sharp asymptotic estimates for the fundamental solution, a maximum principle in the distributional sense, and an equivalent integral inequality for supersolutions.

We provide examples of spaces satisfying generalized combinatorial covering properties such as the Hurewicz, Menger, and $γ$-properties in an uncountable setting. Our approach is motivated by canonical constructions from the classical countable case, including the examples of Bartoszyński and Shelah separating the Hurewicz property from $σ$-compactness, the examples of Tsaban and Zdomskyy separating the Hurewicz and Menger properties, and Tsaban's construction of a nontrivial set of reals with the $γ$-property. We focus on the genuinely nontrivial aspects of these higher-cardinal generalizations, uncovering several open problems whose nature appears substantially different from that of their countable counterparts.

Functional optimization problems are typically solved by optimizing the parameters of a fixed representation, such as a neural network, resulting in highly nonconvex losses that complicate both training and theoretical analysis. An interesting alternative is functional gradient descent (FGD), that is, gradient descent directly in function space, which benefits from strong convergence results and admits a clean theory. However, FGD is difficult to implement in practice because functional gradients are infinite-dimensional, and thus cannot be fully computed nor stored in memory. Existing implementations therefore rely on fixed approximations, which introduce approximation error. We propose a new, theoretically-grounded FGD algorithm that adapts the representation of the functional gradients over the course of optimization. By explicitly incorporating this approximation into the analysis, we establish convergence to a stationary point (for smooth losses) and to a global minimizer (under smoothness + a Polyak-Lojasiewicz-type condition) regardless of our approximations. To the best of our knowledge, this is the first implementable FGD method with such guarantees in a general setting. We demonstrate the effectiveness of our method on regression, numerical solution of PDEs, and modern computer vision. Across settings, our method consistently outperforms both FGD with fixed approximations and neural network baselines in efficiency and accuracy.