Papers

Semantic feedback can repair a noisy first decode without retransmitting the whole message, but current end-to-end evaluations hide where a feedback design fails. An oracle that sees both the source-side state and receiver observation may describe a useful correction with few bits, while the receiver, which sees only its own observation, may not know which correction to request. We formulate these as distinct direct and receiver-constrained coding problems, connect repair distortion to retained local utility and prove a random-access separation: one direct bit is exact although receiver utility vanishes with the number of source-conditioned alternatives. A synthetic ladder verifies the exact law. In paired populations of eight Reuters text-JSCC codecs, the trained recurrent population retains over three times more direct 16-bit repair value than the trained DeepSC population despite comparable full-repair opportunity. Nevertheless, an entropy-valid receiver audit across all eight recurrent checkpoints does not resolve against repetition, source-only or shuffled-index controls. The results show that compressible repair is an upper bound, not evidence of deployable feedback, and motivate reporting both rate curves under matched resources.

Let $\mathcal{H}_{\mathcal{Q}}$ be the field of unary germs at $0^+$ definable in the quasianalytic o-minimal structure $\mathbb{R}_{\mathcal{Q}}$ of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of $\mathcal{H}_{\mathcal{Q}}$ into the field of natural-support generalized Laurent series. Every unary germ has a unique such asymptotic expansion, and bounded germs have expansions with only nonnegative exponents. A germ admits a convergent generalized Laurent-series representation exactly when its canonical expansion converges; in that case, every convergent representation with well-ordered support is canonical. We also prove that $\mathbb{R}_{\mathcal{Q}}$ is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic $\mathbb{R}_{\mathcal{Q}}$-definable germ whose canonical generalized power series diverges. This gives a negative answer to a question of Dembner asking whether every unary germ definable in a branching structure admits a convergent generalized power-series representation. Nevertheless, every bounded unary $\mathbb{R}_{\mathcal{Q}}$-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of $\mathcal{H}_{\mathcal{Q}}$.

Each quasi-static load step of phase-field fracture is a bound-constrained minimization of a nonconvex, coupled displacement-damage energy under an irreversibility bound on the damage. Monolithic Newton stalls once the nonlinearity localizes at the advancing crack front, and staggered (alternate-minimization) schemes converge slowly there. We present an on-demand nonlinear-elimination preconditioned trust-region Newton method: an energy Steihaug-Toint trust region, a primal-dual active set for irreversibility, and a bound-constrained field-split sweep that eliminates an algebraically-identified "hard set" spanning both fields before each step. The elimination is applied on demand – triggered by the coupled Newton's own stalling and otherwise skipped – so the method reduces to monolithic Newton at no surcharge where the step is already healthy. We find the robustness to come from the energy trust region: with that globalization fixed, monolithic Newton already completes every loading history without cutbacks, where residual-merit Newton death-spirals, alternate minimization stalls, and the full-field sweep loses robustness. Against that well-globalized baseline, the on-demand elimination cuts outer nonlinear iterations by 19-25% (brittle) and 17% (ductile), with always-on elimination reaching 26-28% and about $39\%$ at the ductile nucleation step. Measured machine-independently, as a full-mesh-equivalent assembly-work proxy rather than wall-clock, it is competitive with – not faster than – monolithic Newton (within about 10%), whereas an always-on sweep adds up to 30%. Nonlinear elimination is thus an iteration-reduction mechanism whose overhead the on-demand gate bounds, with no demonstrated total-work advantage over well-globalized monolithic Newton.

An annular liquid jet may enclose a volume of gas and, when surface tension dominates over inertia, close on the symmetry axis at a finite distance. In the one-dimensional membrane model, closure provides an algebraic relation among the state variables at the free end. Earlier domain-adaptive schemes differentiate this relation to obtain an evolution equation for the domain length. We instead absorb the endpoint closure identically in the dependent variable and discretise the reduced system by Chebyshev collocation. Convergence is geometric, published steady data are recovered, and the unsteady calculations are checked by componentwise off-grid residuals, temporal refinement, and an independent upwind finite-difference discretisation. For jets of finite thickness, the convergence length decreases with nozzle thickness when the Froude number, Weber number, nozzle angle, and pressure coefficient are held fixed and the terminal zero remains transversal; this is a parameter-specific model result rather than a universal experimental thickness law. The forced response has a pressure-coefficient maximum near a Strouhal number of 0.093, whereas the convergence-length gain increases monotonically over the globally admissible frequency interval studied. Most importantly, a global positivity check changes the interpretation of the first finite-time event. Under body-force forcing, the mean radius first reaches zero at an interior point while the tracked endpoint remains transversal. The interior contact is quadratic, and the independent finite-difference scheme converges to the same first-contact time. After this event, the tracked endpoint is no longer the first physical closure point, so the single-interval annular-jet representation is no longer globally admissible.

We combine the spherical-clothoid reconstruction theorem of Lucas and Ortega-Yagues with an explicit spinorial parametrization of the spherical clothoid and a Kampe de Feriet evaluation of Kummer-product integrals. This gives explicit Cartesian coordinates, in arclength, for every nonhelical three-dimensional clothoid. Kummer asymptotics then yield explicit asymptotic lines, including their gamma-digamma offsets and the angle between their axes.

Bondy and Szwarcfiter defined $\mathrm{ex}^*(n,F)$ as the largest number of edges in an $n$-vertex graph whose edge set partitions into induced copies of $F$; for $F=2K_2$ the deficiency $\binom{n}{2}-\mathrm{ex}^*(n,2K_2)$ is $\Theta(n^{3/2})$. We study the ordered relaxation at fixed matching size, in which each part need only be induced in the union of itself with the parts that follow it; write $\mathrm{ORS}_n(r)$ for the largest number of parts, so that $r\,\mathrm{ORS}_n(r)$ is the ordered analogue of $\mathrm{ex}^*(n,rK_2)$. Our main tool is a characterisation valid for every $r$: an ordered decomposition into induced $r$-matchings is a sequence of steps that start from $K_n$ and repeatedly delete a perfect matching from $2r$ vertices currently spanning a clique. Reading a decomposition backwards turns a condition about the ordering into a reachability question that an exhaustive search can settle. For $r=2$ we determine $\mathrm{ORS}_n(2)$ exactly at orders five through nineteen, where it takes the values $1,3,5,8,11,14,19,23,28,34,40,47,54,62,70$, and we confine $\mathrm{ORS}_{20}(2)$ to $\{78,79\}$. The counting bound $\lfloor n(n-4)/4\rfloor$ is attained at orders five through nine and at eleven, and missed by exactly one part at every other order below twenty, so order eleven is an isolated exception, not a parity effect. Across this range the ordered deficiency equals $\frac32n+O(1)$, and along powers of two a dyadic construction keeps it below $O(n\log n)$; whether it is linear for all $n$ is our main open question. The structural results are formalised in Lean 4, and the searches are certified by fail-closed sweeps and an independent checker.

The binary-tree height recursion defines an algebra $\mathcal{H}$ on $\mathbb{N}\cup\{-\infty\}$, with join given by $\max$ and product \[ a\star b=\max\{a,b\}+1. \] We show that weighted evaluation of a labelled tree depends only on the greatest depth of each label. Single-tree profiles are exactly the vectors satisfying the binary Kraft inequality, while finite joins realize every vector in $\bigl(\mathbb{N}\cup\{-\infty\}\bigr)^n$; hence the $n$-variable term operations form the free algebra $\mathcal{H}^n$. We also classify $\mathcal{H}$'s compatible semilattice operation, subalgebras, endomorphisms, congruences, and finite quotients, and recover the dyadic-composition spectrum at the full-linear boundary.

Let $\Pp=\{2,3,5,7,\ldots\}$ denote the set of prime numbers. We prove that if every sufficiently large positive even integer can be represented as a difference of two primes, then there is no Borel probability measure $\mu$ on $\R$ for which \(\left\{e^{2\pi i p x}:p\in\Pp\right\}\) is an orthonormal basis of $L^2(\mu)$. In particular, under the Polignac conjecture, the prime numbers $\Pp$ cannot be a spectrum (i.e., the set of frequencies of an exponential orthonormal basis) of any probability measure on $\R$.

Many computational problems possess the following three features: (1) the problem could be solved efficiently if a linear operator could be diagonalized, (2) direct diagonalization is infeasible due to scale, but, (3) the operator is known to commute with a permutation since the problem is (a) symmetric with respect to a change of coordinates, or (b) the operator is block circulant. For example, high-resolution template matching demands repeated convolution of large, high resolution images with an exhaustive list of related templates. The associated linear operator could be compressed, via a low rank approximation, if diagonalized. The required decomposition expensive, but, the entire problem is symmetric to in-plane rotations. In these cases, the eigenfunctions are constrained by the symmetry. These constraints allow efficient decomposition. We illustrate a parallelized algorithm that allows fast diagonalization of any block-circulant matrix. When the index space can be partitioned into $l$ classes of $m$ interchangeable elements, the algorithm reduces storage costs by a factor of $m$ and, if $w$ workers are available, the algorithm demands $\mathcal{O}(l^2m\log(m)/w) + \mathcal{O}(ml^3/w)$ floating point operations per worker yielding a $m^2$ speedup. We use this procedure to decompose a high precision template matching matrix. We compare runtime on a reduced problem, where the fast procedure ran 205 times faster per recovered feature, for 22.5 times as many features. On a typical template matching matrix, the decomposition required 25 times less time than computing the matrix entries. We also demonstrate decomposition of a $\sim$30 times larger matrix that covers the complete set of possible projections represented in a cryo-EM image at 2~Å resolution in 14 minutes. This procedure is more stable and allows recovery of in-plane rotations to arbitrary precision.

In 1988, Jaeger conjectured that every bridgeless cubic graph $G$ admits a Petersen coloring; that is, a map $E(G) \to E(P)$ mapping any two adjacent edges of $G$ to two adjacent edges of the Petersen graph $P$. A positive resolution of Jaeger's conjecture would have immediately resolved several other famous and long-standing problems in graph theory. In July 2026, a 68-vertex counterexample was announced on X. Shortly afterwards, Putman independently presented two non-isomorphic 112-vertex counterexamples, relying solely on computer-assisted verification. In this paper, we present two counterexamples of order $52$, currently the smallest known, and provide a purely theoretical proof. In the second part, we construct an infinite family of cyclically $4$-edge-connected cubic graphs without a Petersen coloring for every even order at least $60$. Additionally, through computational verification, we show that any counterexample must have order at least $40$. Moreover, we show that our counterexamples provide a negative answer to other related problems. Finally, we conclude the paper by discussing key open problems and highlighting avenues for future work.

We characterize the subsets of a finite distributive lattice that arise as the minimizer sets of modular functions. It is immediate that the minimizer set of any modular function is a decomposition-closed sublattice. Our main theorem establishes the converse: every decomposition-closed sublattice is the minimizer set of a modular function. Our proof is constructive. Using Birkhoff's representation, we decompose the problem along the intervals determined by an arbitrary maximal chain of the given sublattice. The key ingredient is a weight assignment on connected difference posets that yields local modular functions vanishing precisely at the endpoints of each interval. We also provide an example showing that this characterization fails for nondistributive lattices.

We provide an explicit mathematical characterization of regional characteristics and the resulting epidemiology to evaluate whether elimination or suppression is feasible using targeted measures, such as case isolation and contact tracing. We derive an epidemic model where community members can be infected by either travellers or other community members, and with constraints on the public health resources available to support isolation of arriving travellers and infected community members. We prove that the optimal controls are to immediately implement public health measures at their maximum levels. We find that elimination or suppression strategies are feasible with targeted measures if public health capacity is high, the transmission rate is low, or if the arrival rate of infected travellers is low. Our results illustrate that a particular country that implemented a mitigation strategy during the pandemic may not necessarily have achieved better outcomes if it had instead implemented elimination because elimination may not have been feasible in countries with low public health capacity to support control measures, high transmission rates, or high traveller arrival rates. Combinations of regional characteristics determine whether an elimination, suppression or mitigation strategy without resurgence is feasible. We find that regions with the low traveller arrival rates and the capacity to implement and enforce isolation or quarantine of arriving travellers are less likely to exceed their capacity to contact trace and isolate community members. Travel measures prevent only a small number of community infections via their direct effect, but indirectly, by protecting contact tracing capacity, travel measures can prevent epidemic resurgence and substantially reduce the number of people infected during a pandemic before a vaccine or therapy is developed.

Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain $\Omega$ with real-analytic Jordan boundary, which is not a disc and for which there exist $k\in(27.4381178838,27.4381198839)$ and a nonzero real-valued function $u\in C^\omega(\overline\Omega)$ satisfying $(\Delta+k^2)u=0$ in $\Omega$, with $u=0, \partial_\nu u=\text{constant}\ne0$ on $\partial\Omega$. Thus the overdetermined Dirichlet–Neumann data do not characterize the disc without an additional sign assumption on $u$. The domain has dihedral symmetry of order $26$, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies $\widehat{\sigma_{\partial\Omega}}(k\omega)=0$ for $\omega\in\mathbb S^1$. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton–Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu–Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At the Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior–boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.

We study the negative spectrum of the Laplacian on a metric graph with general vertex matching conditions and with two length scales: a compact core whose edges have length of order a small parameter $\epsilon$, together with finitely many edges of infinite length. As $\epsilon\to0$, some negative eigenvalues may escape to $-\infty$, and we describe precisely how. There are exactly two rates of escape, $\epsilon^{-1}$ and the fractional rate $\epsilon^{-2/3}$. We determine the number of eigenvalues of each rate, together with their leading coefficients, explicitly from the vertex conditions. The analysis rests on the Dirichlet-to-Neumann map of the graph and on an implicit Rellich-type theorem, that identifies the power-law rates of the solution branches of a nonlinear 2-parameter matrix pencil with the leading orders of a one-parameter Hermitian family.

Countable compactness need not be preserved by finite products. Motivated by compactness notions for maps indexed by finite subsets of $\omega$, we introduce cascade countable compactness for arbitrary barriers. For $\mathcal B=[\omega]^2$, this is the previously studied notion of being doubly countably compact. We show that, in ZFC, if $G$ is a Hausdorff topological group and $1\leq k<\omega$, then $k$-cascade countable compactness of $G$ implies that $G^k$ is countably compact. Cascade countable compactness for the Schreier barrier implies that $G^\omega$ is countably compact. In contrast, we construct a Tychonoff space that is $n$-cascade countably compact for every $n<\omega$ but has a non-countably compact square, and a Hausdorff Boolean group $H$ that is $\mathcal B$-countably compact for every barrier $\mathcal B$ but whose square is not countably compact. We also construct a Hausdorff Boolean group without nontrivial convergent sequences that is $\mathcal B$-cascade countably compact for every barrier $\mathcal B$. Finally, we obtain a subspace $X\subseteq\beta\omega$ such that $X^\kappa$ is $n$-cascade countably compact for every $\kappa<\mathfrak h$ and every $n<\omega$, whereas $\exp X$ is not pseudocompact.

Let $\Omega_n$ denote the set of $n\times n$ doubly stochastic matrices. Kim and Roush conjectured in 1981 that, for $n=2k+1>1$, $ \max_{A\in\Omega_{2k+1}}\operatorname{per}(I-A)=3\cdot 2^{k-2}$. They proposed the block construction $A_\star=\frac12(J_3-I_3)\oplus P_2^{\oplus(k-1)}$, where $P_2=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Here $J_3$ is the $3\times3$ all-ones matrix. They did not claim uniqueness. We fully prove their conjecture and classify equality: the maximizers are exactly the simultaneous-permutation conjugates of $A_\star$.

We study the long-time asymptotics of heat kernels on Abelian covers of compact manifolds and on Abelian covers of principal bundles. For Abelian covers, we establish three complementary asymptotic expansions: a distributional expansion describing correlations of the heat semigroup, a local pointwise expansion of the heat kernel on compact subsets, and a global expansion valid on the natural diffusive scale, revealing the Gaussian profile governing heat propagation. We then extend these results to horizontal heat kernels on principal bundles under natural holonomy and curvature assumptions. In this setting, we show that the leading asymptotics are entirely determined by the geometry of the underlying Abelian cover, while the nontrivial fiber modes decay exponentially fast due to a uniform spectral gap. The proof combines Floquet theory on Abelian covers with the Borel–Weil decomposition on principal bundles.

Given a finite group $G$ and a factorization of its order, $|G|= a_1 a_2,$ we show that a sufficient condition for there to exist subsets $A_1,\,A_2\subseteq G$ such that $|A_1| = a_1,$ $|A_2| = a_2,$ and $G = A_1\,A_2,$ is that there exist a chain of subgroups $\{e\} = G_0 < \dots < G_n = G$ such that $a_1$ is the product of some subfamily of the indices $|G_i:G_{i-1}|$ (and hence $a_2$ is the product of the complementary subfamily).

Spacecraft rendezvous on elliptical orbits must hold a safety corridor under navigation noise, unmodelled perturbations, and thrust errors driven by propellant mass uncertainty. This paper develops a tube-based model predictive controller for the Yamanaka-Ankersen linear time-varying dynamics, carrying constraint-tightening tube methods from circular to elliptical orbits. A horizon-dependent, element-wise error bound propagates the actual closed-loop matrices along the prediction window, so early prediction steps keep nearly the full corridor that a constant-width tube would surrender. A multiplicative-to-additive conversion folds mass and thrust uncertainty into the same tightening recursion, and a Perron-Frobenius spectral-radius condition supplies a computable certificate that the tightening converges, with an explicit input-to-state stability gain. In a paired Monte Carlo campaign on a Mars sample-return orbit, the tube controller cuts mean corridor violations by more than an order of magnitude against a nominal predictive baseline at comparable fuel cost. A nonlinear truth-model test with oblateness perturbations well beyond the assumed disturbance budget leaves the corridor unviolated, and when the design envelope is exceeded the tightened problem becomes infeasible and the controller reverts to a saturated linear fallback, making the loss of guarantee explicit rather than silent. The construction requires only element-wise arithmetic and an open-source quadratic-programming solver.

Toric real loci are distinguished lagrangian submanifolds of toric symplectic manifolds. We develop a polyhedral model for toric real loci, called a kaleidoscope, based on the restriction of the moment map. This construction provides a direct geometric description of the topology of toric real loci and leads to simple criteria for orientability, together with transparent formulas for the Euler characteristic. We illustrate the theory with numerous examples and count, in every complex dimension up to 9, the smooth toric Fano varieties whose real loci are orientable. The kaleidoscope construction offers a simple, visual, and effective framework for understanding the topology of toric real loci through the combinatorics of their moment polytopes.

For a planar graph $G = (V, E)$ embedded in $\mathbb{R}^2$, let $\mathcal{F}(G)$ denote the set of faces of $G$. Then $G$ is called a \textit{$C_n$-face-magic} graph if there exists a bijection $f: V(G) \to \{1, 2, \dots, |V(G)|\}$ such that for any $F \in \mathcal{F}(G)$ with $F \cong C_n$, the sum of all the vertex labels along $C_n$ is a constant $c$. In this paper, we prove that all polyominoes are $C_4$-face-magic.

We consider the problem of PDE discovery from possibly noisy observations under the hypothesis that the underlying dynamic is symmetric in all rigid motions. Rather than using a generic library of derivative monomials, we leverage this assumption to construct libraries whose candidate terms are themselves rigid-motion-equivariant, and combine them with sparse regression to benchmark such libraries over five different equations. The advantage is most pronounced when the noise itself breaks rigid-motion symmetry (e.g., radially or axially varying noise), and when the ambient spatial dimension increases, in which case they are also less resource intensive.

We prove that the surface of second-type lines on a general cubic fourfold has fundamental group of order two. Its universal cover, constructed by Huybrechts, is obtained by considering the two ramification points of the Gauss map along each second-type line.

Building on prior results [1], we introduce vector-valued modular resurgent series, whose components exhibit a single infinite tower of singularities in the Borel plane, trivial secondary resurgent series, and Stokes constants given by linear combinations of the coefficients of a vector of $L$-functions. We extend the paradigm of modular resurgence to this setting, emphasizing the role of the Stokes constants and the interplay between the associated vectors of $q$-series and Dirichlet series, and describing the resulting symmetry relating canonical pairs of vector-valued modular resurgent series. Moreover, we conjecture that certain vectors of $q$-series with modular resurgent asymptotics are vector-valued quantum modular forms and can be reconstructed via median resummation. Finally, we show that vectors of $q$-Pochhammer symbols, previously considered in [2], and Eichler integrals of vector-valued modular forms of weight $1/2$ and $3/2$ can be studied within the framework of vector-valued modular resurgence. While the first case amounts to a convenient repackaging of scalar modular resurgence, the second involves a non-trivial representation of the modular group and therefore illustrates the necessity of the vector-valued framework; we establish its modular resurgent structure in general, and work it out in full detail for the unary theta series, whose Eichler integrals are the false theta functions of quantum topology.

We study graphical mean curvature flow in arbitrary codimension over the half-space and over smooth bounded domains, subject to homogeneous Dirichlet boundary conditions. At the scaling-critical Lipschitz regularity, we prove local well-posedness for initial graphs that can be approximated in $W^{1,\infty}$ by smooth profiles compatible with the boundary condition. Within this class, if the initial Lipschitz seminorm is sufficiently small, the corresponding solution is global; on a bounded domain, it also converges exponentially to the flat graph. Positive-time regularization is quantified by time-weighted Hölder estimates whose weighted quantities remain bounded as $t\downarrow0$. The main analytic ingredient is a boundary Schauder theory for variable-coefficient parabolic systems. On the half-space, it combines coefficient freezing with parity extensions and boundary identities intrinsic to the graphical system. On curved domains, localization and boundary flattening lead to anisotropic estimates, from which normal derivatives are recovered recursively.

Let $d\geq 4$ and let $R>0$. When $d=4$, assume that $R^2\in\mathbb{N}\setminus 4\mathbb{N}$; when $d\geq 5$, let $R^2\in\mathbb{N}$ be arbitrary. We prove the fixed-radius estimate $$\|A_R f\|_{\ell^{p'}(\mathbb{Z}^d)}\leq C_{d,p,\varepsilon}R^{-d(2/p-1)+\varepsilon}\|f\|_{\ell^p(\mathbb{Z}^d)}$$ for $(d+2)/d\leq p\leq 2$, where $p'$ is the Hölder conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler–Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$.

This work revisits the key sufficient conditions on a ring $A$ that guarantee the existence of the exact sequence \[ H_3(\mathrm{SM}_2(A), \mathbb{Z})\to H_3(\mathrm{SL}_2(A), \mathbb{Z}) \to \mathcal{RB}(A) \to 0 \] and of the isomorphism \[ H_3(\mathrm{SL}_2(A), \mathrm{SM}_2(A); \mathbb{Z}) \simeq \mathcal{RP}_1(A). \] We show that there are rings which satisfy the relations above but do not satisfy the condition that -1 is an square, for example some rings $\mathbb{Z}[\frac{1}{m}]$ and finite local rings. In addition, we study the special case of non-dyadic local fields with characteristic not 2, obtaining a refined Bloch-Wigner exact sequence when -1 is not an square.

Let $X$ be a smooth, projective, connected variety over a field $k$ containing a root of unity of order $p$. If $X$ has a Chern number prime to $p$, we show that every finite $p$-group acting on $X$ acts through a subgroup of $\operatorname{GL}_n(k)$, where $n=\dim X$. This allows one to transfer properties of representations of finite $p$-groups to their actions on $X$. We deduce a fixed-point theorem which, unlike earlier results of this kind, is sensitive to the arithmetic of the base field, and whose dimensional bound is sharp for each individual group (apart from one low-dimensional case when $p=2$). We also obtain restrictions on finite $p$-groups of birational automorphisms of rationally connected varieties; for instance the Cremona group $\operatorname{Cr}_n(\mathbb{Q})$ contains no element of order $p^2$ when $p \ge n+2$. More generally, we prove variants of these results in the presence of a Chern number not divisible by a given power of $p$. The method is based on the following observation about the essential $p$-dimension $\operatorname{ed}_p(G)$ of an affine algebraic group $G$: over a field of characteristic zero, the number $\operatorname{ed}_p(G) + \dim G$ is the least dimension of a smooth projective variety $Y$ with a generically free $G$-action such that the degree map $\operatorname{CH}_G(Y) \to \mathbb{F}_p$ from the equivariant Chow group is nonzero. The Chern numbers modulo $p$ of $Y$ lie in the image of this map; combined with Karpenko and Merkurjev's computation of the essential $p$-dimension of finite $p$-groups, this yields our results.

In this paper we develop quantitative approximation results for shallow neural networks constructed using a dictionary based on metaplectic operators. First, we extend the concept of Barron spaces by considering a symplectically motivated extension of the Fourier transform, known as the metaplectic transform. Then, after establishing embedding between metaplectic Barron spaces and Sobolev spaces we consider a neural metaplectic dictionary and we prove Monte-Carlo approximation bounds for metaplectic Barron functions using finite linear combinations of atoms of the dictionary. Finally, we validate the introduction of the neural metaplectic dictionary by devising a deep neural network architecture that uses as building blocks the atoms of the dictionary. We test it to approximate solutions of time-dependent Schrödinger equations, demonstrating better performance compared to classical phyisics informed neural networks architectures.

We study quantitative unique continuation for bi-Laplace equations \[\Delta^{2}u+V(x)u=0 \] by introducing some new weighted frequency functions. We establish quantitative three-ball inequalities and vanishing-order bounds for bounded and Hölder continuous potentials. Three-ball inequalities are built on rescaling invariant weighted frequency functions. The vanishing order results are shown by related, but different weighted frequency functions.