Papers

We present a quaternionic proof that the quadratic form $t^2+2x^2+5y^2+10z^2$ represents all positive integers, and that the form $t^2+x^2+7y^2+7z^2$ represents all natural numbers which are neither equal to $3 \cdot 7^\ell$ nor equal to $6 \cdot 7^\ell$, for any $\ell \in \mathbb{N}_0$. To do this we introduce a technique that may be useful for other purposes, and that we call "metaconjugation".

We prove that the connected sum of a possibly noncompact area enlargeable manifold $M_1$ with an arbitrary spin manifold $M_2$ of the same dimension admits no complete Riemannian metric of uniformly positive scalar curvature. This extends a theorem of Wang–Zhang, where $M_1$ is assumed closed, to noncompact enlargeable summands; in this generality the uniform positivity hypothesis enters the argument in an essential way. We also prove a spectral analogue of the generalized Geroch conjecture in terms of the $\gamma$-spectral constant: for $\gamma>(\dim M_1-1)/(4\dim M_1)$, such a connected sum carries no complete metric with positive $\gamma$-spectral constant. The proofs are based on a covering connected sum construction together with the scalar-cowaist and spectral-cowaist inequalities.

We prove that the bracket width of the Lie algebra of vector fields on any smooth affine algebraic variety of dimension $n$ is at most $(n+1)^2$. We give improved bounds for some families of $\mathbb{C}^*$-varieties, in particular for $\mathrm{SL}_n(\mathbb{C})$ and for the Koras–Russell cubic threefold.

We study the fundamental problem of estimating the dependence structure of a centered stationary Gaussian process under local differential privacy (LDP). In this setting, the spectral density characterizes the dependence structure of the data and is the quantity to be estimated. Our main contribution is to close the open $\alpha^2$-versus-$\alpha^4$ gap between the previously known lower and upper bounds on the minimax rate. Specifically, we establish a minimax lower bound showing that, over Sobolev-type classes of spectral densities, the effective sample size in the high-privacy regime is $N\alpha^4$, rather than the usual $N\alpha^2$ arising for independent observations. This additional privacy cost is caused by the temporal dependence between the observations rather than by their marginal distributions. The proof relies on a contraction bound for privatized dependent Gaussian observations. Our second contribution is a matching upper bound, free of the polylogarithmic losses present in previous work. Rather than applying a generic privatization scheme to classical estimators, we construct a problem-specific procedure attaining the rate identified by our lower bound. Beyond closing the gaps in spectral density estimation, we apply the tools developed for this problem to several related questions. We (i) close the logarithmic gap for fixed-lag autocovariance estimation, (ii) show that the $\alpha^4$ cost arises locally around every spectral density bounded away from zero, and (iii) establish that classical asymptotic equivalence with an independent Gaussian experiment generally fails under LDP.

We show that the higher Legendrian homology of any Legendrian link given by the rainbow closure of a positive braid must necessarily vanish. The result is proven in the most general setting: for any number of strands and crossings, integrally and with one basepoint per component of the Legendrian link.

We characterize all maximally charged black hole spacetimes in $3+1$ dimensions. More precisely, given any initial data set $(M,g,k)$ saturating the mass-charge inequality that satisfies the charged dominant energy condition, we show that $(M,g,k)$ must arise from an isometric embedding into a Majumdar-Papapetrou spacetime.

The transition toward Open Radio Access Networks (O-RANs) is reshaping how cellular infrastructure is deployed, managed, and optimized. This paper investigates the energy-aware joint placement and migration of cloud-native functions (CNFs) in an O-RAN edge cloud. We consider both a Single-CU-UP association model and a slice-aware Multi-CU-UP relaxation, in which distinct slice-flow groups of the same distributed unit (DU) may be assigned to different Centralized Unit User Plane (CU-UP) processing targets under one Centralized Unit Control Plane (CU-CP). For brevity, these scenarios are referred to as Single-CU and Multi-CU, respectively; Multi-CU never denotes multiple CU-CP associations. We formulate the problem as a Mixed-Integer Linear Program (MILP) that minimizes server, transmission, wake-up, and migration energy while satisfying server-resource capacities and one-way delay requirements over the F1 user-plane interface (F1-U) between each DU and its selected CU-UP in a fat-tree edge data center. To improve computational scalability, we also develop a deterministic k-means-based heuristic that approximates the MILP decisions without requiring repeated exact optimization. Over the evaluated 24-hour workload, the theoretical Multi-CU relaxation reduces modeled energy consumption by 5.7% relative to the Single-CU baseline. For the Multi-CU case, the proposed heuristic remains within approximately 9.7% of the proposed MILP, demonstrating a favorable trade-off between energy efficiency and computational tractability.

We study the long-time dynamics near Couette flow for the 2D Euler equations in unbounded geometries. We identify a mechanism of nonlinear inviscid damping at Yudovich regularity, distinct from classical phase mixing: velocity decay driven by spatial evacuation of vorticity. The mechanism leads to a geometry-sign classification of the dynamics. In the infinite channel, small nonnegative bounded-vorticity perturbations undergo global damping without derivative assumptions. In the whole plane, small nonnegative perturbations exhibit enhanced dispersion and velocity damping, while non-positive perturbations remain confined and do not damp, even when arbitrarily small in Gevrey classes. Moreover, damping can coexist with infinite-time growth of vorticity derivatives, even under arbitrary shear modulation. The classification is obtained through an interplay between new Lyapunov functionals and Hamiltonian conservation.

Two competitors choose starting vertices for independent, constant-speed random walks, and each site is acquired by its first visitor. We study the spatial geometry of the resulting first-passage location game. On every finite path the optimal strategies are exactly the distributions supported on the central vertices. The proof combines reflecting-boundary harmonic barriers with a parameter-uniform aggregate estimate for product-chain exit probabilities. After diffusive rescaling, the complete two-start payoff landscape converges uniformly to the game between independent reflected Brownian motions. Its unique equilibrium concentrates at the midpoint, with explicit cubic stability. Beyond paths, attaching two leaves to every vertex of a clique of order k produces a 3k-vertex graph on which every exact optimal strategy randomizes over all k clique vertices; for k=2, this is a six-vertex tree with no pure equilibrium. The continuum best response to an endpoint is uniquely determined. A first-passage random-ranking representation relates the finite game to maximal lotteries without identifying it with nonstrategic painting, deterministic Voronoi allocation, or absorbing-trap placement. Parameter-uniform statements follow from analytic arguments or symbolic polynomial identities; identified finite exceptions and numerical enclosures have reproducible certificates.

Let $G$ be a quasi-split reductive group over a finite field and let $\^G$ be the Langlands dual group. Assuming the derived subgroup of $G$ is almost simple, we use techniques from the geometric Langlands program to relate special values of Bessel functions for generic principal series representations of $G$ to the trace of Frobenius acting on Kloosterman sheaves of Heinloth-Ngô-Yun for $\^G$. We also give explicit examples in essentially all possible cases, including exceptional groups.

A result of Cembranos states that a non-trivial injective tensor product of a $C$-space contains a complemented copy of $c_{0}$, and in particular fails the Grothendieck property. We establish a positive analogue of the Cembranos theorem for tensor products of Banach lattices. If $E$ and $F$ are infinite dimensional Banach lattices, with $E$ containing $c_{0}$ and $E^{\ast}$ or $F^{\ast}$ having the bounded positive approximation property, then the Wittstock tensor product $E\widetilde{\otimes}_{\vert{\varepsilon}\vert}F$ contains a complemented copy of $c_{0}$. If $F$ is in addition reflexive, then $E\widetilde{\otimes}_{\vert{\varepsilon}\vert}F$ fails the positive Grothendieck property.

Rapid data assimilation is required for real-time prediction and control in digital twins. For many physical systems, the data assimilation task requires the solution of a physics-constrained inverse problem, which is often computationally intractable in real time using traditional physics solvers. This work presents a reduced-basis neural operator approach to enable real-time inverse problem solutions in the digital twin setting. Our approach specifically targets the large class of problems with spatiotemporal dynamics governed by partial differential equations (PDEs) that are parameterized nonlinearly with respect to model parameters $m$, and linearly with respect to inversion parameters $q$. Based on this physical structure, our neural operator approximates the nonlinear map from the model parameters $m$ to the parameter-to-observable operator $\mathcal{F}(m)$ in a reduced subspace. Since the output of the neural operator is the parameter-to-observable operator itself (manifested as a matrix), we refer to this approach as NEural Matrix Operator (NEMO). With NEMO, for new given $m$, we enable a closed-form inverse problem solution for $q$ in a reduced subspace. We apply NEMO in two real-world applications: contaminant transport initial condition identification, and hypersonic vehicle load identification. We show that NEMO delivers high quality inverse problem solutions for data assimilation in real time, with over three orders of magnitude speedup compared to constructing the reduced operator with the PDE solver. Further, NEMO demonstrates comparable inverse performance to a state-of-the-art multiple-input neural operator, while reducing online computational complexity by over an order of magnitude and providing real-time uncertainty quantification.

We study pointwise convergence of polynomial ergodic averages over short intervals whose left endpoints tend to infinity. For a polynomial orbit of degree $d\geq2$ and doubly lacunary starting times, we prove $L^p$ variational estimates, and hence almost-everywhere convergence, for $1<p<\infty$ in the range $c>(d-1)/d$. This gives the first pointwise ergodic theorem for polynomial orbits along short intervals. We also show that the endpoint $L^1$ fails along every infinite subsequence. In a different direction, we prove that substantially denser sequences of starting times exhibit the strong sweeping-out property.

We consider the SIS contact process on a Poisson system of independent Brownian motions in $\mathbb{R}^d$ of intensity $\eta$. Two particles are in contact when their distance is at most $2r$. An infected particle transmits the infection to a susceptible particle in contact with it at rate $\lambda\in(0,\infty]$, and recovers at rate $\mu\in(0,\infty]$, after which it is susceptible again. When $\lambda=\infty$, recovery acts only while a particle is isolated. Let $\eta_c^{\mathrm{B}}$ be the critical intensity of the static Boolean model, below which the contact graph has only finite components at every fixed time. We show that for $d\ge2$, with motion, at every intensity strictly below $\eta_c^{\mathrm{B}}$, the infection survives and spreads at positive speed once the recovery rate $\mu$ is small enough, depending on the intensity; that is, started from a single infected particle, at time $t$ there is an infected particle at distance of order $t$ from the origin. In particular there exists $\mu_\dagger>0$ such that $\eta_c(\infty,\mu)<\eta_c^{\mathrm{B}}$ for every $\mu<\mu_\dagger$, and as $\mu\downarrow0$ the set of intensities at which the infection survives extends to all of $(0,\eta_c^{\mathrm{B}})$. We also show that the critical density is positive at every recovery rate, with a lower bound that does not depend on the infection rate, and that for $d\ge2$, at every intensity and every finite infection rate, the infection survives and spreads at positive speed once $\mu$ is small enough.

Let $\mathbb{H}=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$, and let $N(\mathbb{H})$ denote the Nevanlinna class in $\mathbb{H}$, consisting of meromorphic functions representable as quotients of two bounded analytic functions in $\mathbb{H}$. We construct a nonconstant meromorphic function $F$ on $\mathbb{C}$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb{R}$ and $F|_{\mathbb{H}}\notin N(\mathbb{H})$. Thus, omitting three distinct values of the Riemann sphere in a half-plane does not force a meromorphic function on $\mathbb{C}$ to be of bounded type there. This provides a counterexample to Nevanlinna's century-old half-plane problem.

Weil's criterion states that the Riemann Hypothesis (RH) is equivalent to the non-negativity of an explicit quadratic form Q(f) on test functions. For f supported in [-L,L] we study the normalized infimum lambda_min(L) = inf Q(f)/||f||^2 from both sides. Lower bounds (unconditional): positivity on bounded support is classical for supp f in [-(log 2)/2,(log 2)/2] (Yoshida; Connes-Consani). We prove by certified computation that Q(f) >= 8.9e-18 ||f||^2 for all supp f in [-0.8,0.8], i.e. autocorrelation support 1.6, 2.3 times the classical range. The proof rests on a one-stroke reduction: a pointwise envelope for the Weil symbol, with a comb constant shown optimal by Weyl equidistribution, converts positivity on the whole window into positive semidefiniteness of a single finite matrix. The odd parity sector shows that the certified positivity holds for arbitrary complex test functions of support 1.6, and that the window ground state is simple and even, the spectral hypothesis required by the operator-theoretic program of Connes, Consani, Moscovici and van Suijlekom. Upper bounds (unconditional): variational bounds are evaluated on the geometric side in interval arithmetic, with no appeal to zeros or RH, down to 3.2e-283 at L=2. They follow the empirical law -ln lambda_min(L) ~ 2 pi^2 N(T*)/ln N(T*), T* = 2 pi e^{2L}, with N the zero-counting function; 2 pi^2 matches the Landau-Widom eigenvalue-plunge rate. Under RH, lambda_min(L) <= exp(-L e^L) for all large L. Synthesis: at L=0.8 the two halves enclose the profile, 8.9e-18 <= lambda_min(0.8) <= 2.27e-17, both certified. We show why the positivity route cannot reach RH unassisted: any one-stroke certificate must resolve frequencies up to 2 pi e^{A_L}, A_L ~ 4e^L, a doubly exponential threshold that no pointwise bound on the prime comb lowers, while the spectral margin collapses at the Landau-Widom rate.

We show that the operadic structure on the cohomology of the poset of noncrossing 2-partitions is isomorphic to the Brace operad.

Repeated observations of one physical object need not constitute independent channel uses. We model partial nanopore protein rereads as a finite-alphabet channel with canonical content, persistent readout, and pass-local coverage and synchronization. For exact compound-pass data, matched inference approaches the equivalence-class canonical posterior, and sitewise excess Bayes risk admits an action-aware achievable exponent. In an aligned specialization, observation-local redraw can cause linear-in-$K$ growth in true-label negative log-likelihood (NLL). We derive order-$b$ projection-stability bounds and an exact passwise-fusion diagnostic. On a PASTOR-informed semi-synthetic hard-symbol channel, label-blind deterministic-mixture importance sampling (LB-IS) agrees with exact enumeration at $L=7$. At $L=24, K=10$, LB-IS meets every prespecified aggregate absolute marginal-posterior and score-agreement criterion against a fixed high-allocation reference in three selected conditions. Joint agreement holds for the representative and high-NLL conditions, while the near-zero condition remains inconclusive. Exact $L \leq 6$ benchmarks identify order 4 as the smallest tested common cap. At target scale, the reference supports selected unprojected functionals, while neither order 4 nor 5 attains joint agreement, defining a tested finite-memory boundary. Across 16 cells, the order-4 shared branch lowers NLL by 0.033-0.224 nats per residue relative to pass-local.

Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form $\sum_{i=1}^{m} a_iX_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n \geq 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1, a_2, \ldots a_m \in \mathbb{Z}$, $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z})$. He gave bounds on $f(n)$ too. Koo and Lee further improved the bounds in 2025. This paper is organized as follows. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. In the first section we give necessary and sufficient condition for a quadratic form $\sum_{i=1}^{m}a_iX_i^2$ to be universal over $M_2(\mathbb{Z}_p)$ for $p=2$ and for an odd prime $p$. Consequently we express matrices of over $\mathbb{Z}_p$ as sum of squares. For a positive integer $n \geq 3$, we define $f(n)$ to be the smallest positive integer $m$ such that for every $p$-adic integers $a_1,a_2, \ldots a_m$ with at least three of them units, the diagonal quadratic form $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z}_p)$. In the next section we find bounds on $f(n)$ for $n \geq 3$.

We consider the minimization problem \[ \min\left\{\int_{B_1}|Du|^2:\ u\in W^{1,2}(B_1;\mathbb R^n),\quad u=x\ \text{on }\partial B_1,\quad |u|\ge a\right\}, \quad 0<a<1. \] Figalli, Guerra, Kim, and Shahgholian proved that the canonical radial vortex is the unique global minimizer for $n\ge7$, and asked whether the same holds in dimensions \(3\le n \le6\). We answer this question affirmatively, thereby completing the global minimality and rigidity of the constraint map vortex in every dimension \(n\ge3\).

We prove that sparse domination of a bilinear form implies convex body domination. More precisely, if a bilinear form admits an $(r,s)$-sparse bound, then its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound. The proof relies on a randomization argument. We establish the result both for sparse families in a fixed dyadic lattice and for sparse families of arbitrary cubes. As an application, we deduce sparse domination for iterated commutators, with the local oscillations of the symbol appearing in the sparse form.

Explicit sequences of forms with the Hilbert series predicted by Fröberg's conjecture are difficult to construct beyond the almost complete intersection case. Over a field of characteristic zero, we prove that six forms of degree $d\ge4$ in four variables cannot have the predicted series if each is divisible by a power of a linear form with exponent at least $(15d+91)/16$. A power of the product of six quadric cones gives a nonzero element of the inverse system in a degree where the predicted series vanishes. This obstruction implies that every fixed-order Taylor truncation of the homogeneous Rogers–Szegő polynomial at $q=1$ fails to complete five powers of linear forms to the predicted series in all sufficiently large degrees. As a fixed-degree counterpart, we use exact certificates to prove the conjecture for any number of general quintics or septics; together with known results, this establishes the equal-degree conjecture in four variables for $1\le d\le9$.

We show that every smooth manifold $M$ of positive dimension admits fully supported distributions $u,v\in \mathcal D'(M)$ with $\mathrm{WF}(u)\cap({-}\mathrm{WF}(v))=\emptyset$ and $uv=0$. Neither factor can be continuous on any non-empty open set, but $u$ may be chosen of optimal Sobolev order $\dim M/2$, with wavefront set contained in any prescribed closed conical set $\Gamma\subset T^\ast M\setminus 0$ with non-empty and non-maximal fibers. Simultaneously, the wavefront set of $v$ may be confined to any compatible open conical set or, in geometric situations, to closed conical sets such as conormal bundles of foliations or ray bundles generated by locally conformally closed one-forms. We give positivity criteria and show that full-support zero divisor pairs accumulate at $(1,0)$ in Sobolev-microlocal topologies. The one-dimensional case uses a Fourier null series of Kozma-Olevskiĭ.

In this work, we investigate rapid prediction of the dominant $n{=}0$ vertical instability growth rate in C-Mod and SPARC equilibria, where nonrigid free boundary response models are too slow for control cycle use. Using a Physics Attention Transformer trained on MEQ-FGE-L labels, we predict both the scalar growth rate and the associated two dimensional perturbed toroidal current density. We find mean absolute errors of 5.4~s$^{-1}$ on held out C-Mod equilibria and 12.7~s$^{-1}$ on synthetic SPARC cases, with spatial eigenfunction errors near 5%. We also compared PAT with operator based ML models : FNO2D and DeepONet, where we found PAT predicts a much lower normalised growth rate error and improved spatial reconstruction. These results indicate that PAT can reproduce MEQ-FGE-L outputs at control relevant latency and could support future studies of growth rate headroom monitoring and proximity aware shape control.

A free spectrahedron is the matricial solution set of a free linear matrix inequality $L_A(X) = I-A_1 \otimes X_1 - \dots - A_g \otimes X_g \succeq 0$. In this dimension-free setting, free extreme points play the role of classical extreme points. In particular, every bounded real free spectrahedron is the matrix convex hull of its free extreme points. In this qualitative sense, free extreme points of free spectrahedra are abundant. However, quantifications of this abundance have remained elusive. In particular, outside simplices, it is not known whether there exist bounded real free spectrahedra that have finitely many free extreme points. A necessary condition for having finitely many free extreme points is that the classical spectrahedron defined by $L_A(x) \succeq 0$ is a polytope. We strengthen this necessary condition through two constructions. First, we provide a geometric construction of an infinite family of free extreme points at the second level of the maximal matrix convex set over a polygon with at least four sides. Second, we develop a graded face lifting technique for general bounded real free spectrahedra, which allows us to construct free extreme points of the full spectrahedron from free extreme points of the graded face lift. As corollaries, we obtain obstructions to minimal matrix convex sets over polytopes being free spectrahedra and show that much of the scalar extreme-point geometry of maximal matrix convex sets over polytopes can be captured by higher-level free extreme points.

In this paper, we consider a class of first-order abstract retarded functional differential equations in Banach spaces, incorporating a time-dependent infinite delay governed by a regulated function. We establish the existence of mild solutions for the nonlinear equation and show that the family of solution maps for the linear equation forms a well-defined evolution family of bounded linear operators on an appropriate phase space. Furthermore, we leverage this evolution family to prove a variation of constants formula for the inhomogeneous linear problem.

This article proposes a Caputo fractional accelerated gradient descent (CFAGD) method for unconstrained optimization problems that is applicable to both smooth and a class of non-smooth objective functions. The proposed approach incorporates an adaptive (e̱ta}k)-parameter, which is heuristically updated throughout the iterative process to improve the search direction. Furthermore, the method employs the Caputo fractional derivative together with the adaptive (e̱ta}^k)-parameter, thereby preserving the memory characteristics associated with non-integer-order derivatives. A suitable step-size is selected via an inexact line-search technique based on the Armijo condition. The central idea is to scale the step-size by a positive parameter to improve the behavior of the iterates as they approach an optimal point, thereby generating a descent sequence. Under strong convexity and bounded Hessian assumptions, linear convergence of the proposed method is established. Numerical validations, including neural-network-based examples, further indicate that the CFAGD method can achieve faster and more stable performance than competing approaches.

We investigate the distribution of the zeros of the Euler's function and its integral and fractional derivatives.

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry–Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $\lambda_1=K$. We prove that, after removing a canonical collection of edges on which every $K$-eigenfunction is constant, the resulting graph has a canonical bundle structure. Its fibers are regular, have similar structure with hypercubes, and are Laplacian-cospectral with hypercubes, although they need not themselves be hypercubes. If the base graph is nontrivial, then it satisfies $\mathrm{CD}(K,\infty)$ and has first eigenvalue strictly greater than $K$. As a consequence, if the vertex degree in $G$ is constant along each canonical fiber, then every fiber is a hypercube and $G$ is a hypercube bundle. Conversely, for every $d\geq 4$, we construct Lichnerowicz-sharp graphs with non-hypercube canonical fibers of degree $d$.

We develop the structure theory of the deterministic $\sin^2$-type algorithm for complex cubic fields introduced in the companion paper, addressing the complex-signature case of Karpenkov's Problem 4. The selection rule is shown to be, exactly, the minimization of a conformal module: the hyperbolic cosine of the distance between the transverse complex structure of the state and the round point. All governing quantities are exact elements of the real embedding of the field and satisfy closed dual-type identities; in particular no isotropic candidate ever arises, and the transverse deviation lattice has exactly pinned covolume. We prove an unconditional soft-rebound lemma (the module can grow by at most the factor $\varphi^2 = 2.618\ldots$ in one step), a finiteness theorem for states of bounded module and height at fixed coordinate discriminant, with explicit static constants, and a per-field periodicity theorem under two named hypotheses: $(C_\kappa)$, contraction of the module in the high phase, partially reduced here to a fixed finite minimax over a five-parameter compact with rational objective; and (B), recurrence of bounded height, which we then prove under $(C_\kappa)$ alone: a height-descent theorem shows the height can never exceed $\max(H(s_0), C_H)$ with an explicit constant. The remaining program for per-field periodicity is reduced to (R) on the compact and to the proved stretched subcases. All proved statements and certificates are finite and exact. A machine-checked core of the paper is sealed in Lean 4, kernel-only, under the standard axioms: the analytic core of the height-descent theorem, the finiteness pigeonhole, the dual and conformal identity layer, and an abstract assembly theorem composing them through named interface hypotheses.