Papers

We present a localized rooted-operator argument for Weil positivity in the real odd logarithmic channel, retaining the polar rank-one term throughout. The three gates are structural levels rather than a serial dependency: Gate I is an independent strict cellular/shell positivity theorem and diagnostic; Gate II carries the load-bearing operator argument; Gate III uses compression, closure, and the restricted odd Weil criterion. In Gate II, the inherited parent response is placed in the same post-old-core/common-cut primal form in which the arithmetic mismatch is charged. The audited MASTER-P2 same-block splice combines the aligned reference contribution with the actual Schur response before the target-ground coefficient is used, while the literal-slot/true-ground mismatch is paid explicitly by a leakage term. The source elimination is written as one exact common-source Schur completion on the same harmonic vector, and the endpoint fold is used only as unitary coordinate transport, so it creates no second source debit. An outward-rounded finite computation and analytic tail give a strictly positive MASTER margin for k >= 7. Starting from the rigorously certified physical coefficient-2 endpoint Y=7, Schur induction yields positivity at the integer endpoints; zero-extension gives non-negativity at every finite support radius. Closure and the restricted odd Weil criterion then give the conclusion stated in the main theorem.

We consider infinite walks in $\mathbb{N}^k$ with standard unit basis vector steps that avoid $t$ collinear points, and show that these walks exist for $(k,t) \in \{(6,3), (4,4), (3,7)\}$. In particular, our construction for $k = 3$ improves the previous bound $189$, obtained by Lidbetter, to $7$. Our results also imply the existence of infinite words over small finite alphabets that are weakly abelian squarefree (resp., weakly abelian cubefree, weakly abelian 6th-power-free).

Let $\mathbb{D}$ be a division ring with an involution $x \mapsto x^\star$, and $n \geq 2$ be an integer. Denote by $\mathcal{H}_n(\mathbb{D})$ the set of all $n$-by-$n$ Hermitian matrices with entries in $\mathbb{D}$, and by $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ the set of all matrices $A-A^\star$ with $A \in \mathcal{M}_n(\mathbb{D})$. Here, we give a complete solution to the following problem: Determine all group homomorphisms from $\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ (respectively, from $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ unless $(-)^\star$ is the identity) that take every matrix to a right linear combination of its columns. The solution to this problem was already known when $(-)^\star$ is the identity, and the novelty here lies in the generalization to arbitrary involutions, and in particular in the noncommutative case. These results are to be used in a subsequent article on subspaces of Hermitian matrices of bounded rank, and on large spaces of diagonalisable matrices.

We prove a generalization of the version, due to A. Gabrielov, of the Łojasiewicz inequality for not necessarily restricted semi-Pfaffian sets. Unlike the most variants of the Łojasiewicz inequality, it describes the rate of growth of a Pfaffian function on a semi-Pfaffian set $X$ not only in a neighbourhood of its zero set in the domain of $X$ but also in a neighbourhood of zeroes on the boundary of the domain. Gabrielov's version is essentially one-dimensional, we generalize it to a multidimensional case.

In this paper, we prove the long-time existence of the heterotic $G_2$ flow with bounded Ricci curvature, torsion tensor and dilaton. Besides, we also get the forward and backward uniqueness results for the solution.

This paper develops and analyzes an optimal-order fractional backward collocation method for adjoint Volterra integro-differential equations with weakly singular kernels. The backward Volterra structure, together with the weakly singular kernel, induces fractional-power singularities at the terminal endpoint, thereby reducing the classical regularity of the exact solution and causing order deterioration in standard polynomial collocation methods. We first establish a regularity result that characterizes the terminal singular behavior of the solution, showing that the solution is continuously differentiable, whereas its second derivative may exhibit a weak singularity at the terminal endpoint. Motivated by this regularity structure, we introduce a terminally graded mesh and construct a fractional backward collocation scheme whose local approximation space is adapted to the endpoint singularity. Rigorous convergence and superconvergence estimates are derived for both the solution and its derivative. With suitable choices of the fractional parameter and the mesh-grading exponent, the proposed method attains the optimal convergence orders dictated by the local approximation degree. Numerical experiments confirm the theoretical predictions and demonstrate the accuracy and effectiveness of the method for adjoint weakly singular Volterra integro-differential equations.

We classify, for every prime power $q$ and every $e\geqslant2$, the permutation binomials $X^r(X^{q-1}+a)$ over $\mathbb F_{q^e}$. Writing $\ell_j(q)=(q^j-1)/(q-1)$, such a binomial is a permutation if and only if $\gcd(r,q-1)=1$, $(-a)^{\ell_e(q)}\ne1$, and $r\ell_h(q)\equiv1\pmod{\ell_e(q)}$ for some $1\leqslant h<e$ coprime to $e$. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from $(X^{q^h}+aX)\circ X^r$ for a suitable $h$. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family $X^r(X^{d(q-1)}+a)$ in the coprime-index case $\gcd(d,\ell_e(q))=1$. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas' theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-$q$ digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.

We study random linear projections of a finite-field subset for which every fiber has cardinality close to its mean. We bound the mean fiber size needed to ensure that all fibers satisfy a prescribed relative discrepancy, with a prescribed failure probability. For $S\subseteq\mathbb F_q^n$ projected to $\mathbb F_q^b$, one theorem gives three regimes: at fixed discrepancy and failure probability, sufficient mean fiber sizes are $O(q2^b)$ for arbitrary $q$, $O(q^2)$ when $q$ is at least a suitable constant multiple of $b$, and $O_q(b)$ for fixed $q$. The resulting entropy loss over fixed fields is $h-b=\log_q h+O(1)$, where $h=\log_q|S|$ is the input entropy. This matches the order of the binary obstruction of Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos (1999); we give a quantitative random-source refinement over every fixed field. Our proof combines a quotient-and-average counting lemma with the local balanced/unbalanced argument of Dhar and Dvir (arXiv:2204.01665) and Furstenberg estimates of Dhar and Dvir and Kumar and Mon (arXiv:2609.17020). We apply these bounds in the reduction of Ordentlich, Regev, and Weiss (arXiv:2311.04644) to improve their $O(n^3)$ bound for smooth lattice coverings to $O(n^2)$. For each fixed convex body $K\subseteq\mathbb R^n$, a Haar-Siegel random lattice of covolume one has the number of lattice points in every translate of $K$ within a prescribed relative error of $\operatorname{vol}(K)$, with prescribed high probability, once $\operatorname{vol}(K)\ge Cn^2$ and $n$ is sufficiently large. The constant and dimension cutoff depend only on the error and failure probability. Complements of higher-rank Kakeya sets of Kopparty, Lev, Saraf, and Sudan (arXiv:1003.3736) show that no hashing guarantee for arbitrary subsets can yield a smaller order in the same reduction.

For the Dirichlet-type space $\mathcal D(\rho^{(1)},\rho^{(2)})$ on the unit bidisc $\mathbb D^2,$ where $\rho^{(1)}$ and $\rho^{(2)}$ are finite positive Borel measures on the closed unit disc $\overline{\mathbb D},$ we show that the Lévy measure $\nu_{(\mathscr M_z,1)}$ associated with the completely alternating multisequence $\left\{\|z^\alpha\|_{\mathcal D(\rho^{(1)},\rho^{(2)})}^2 \right\}_{\alpha\in\mathbb Z_+^2}$ admits the explicit representation \begin{equation*} d\nu_{(\mathscr M_z,1)}(x)=\frac{1}{1-x_1}d((S_{*}\rho^{(1)})\times\delta_1)(x)+\frac{1}{1-x_2}d(\delta_1\times (S_{*}\rho^{(2)}))(x) \end{equation*} on $[0,1]^2\backslash\{(1,1)\},$ where $S_{*}\rho^{(i)},$ $i=1,2$, denotes the pushforward measure of $\rho^{(i)}$ by the map $S:\overline{\mathbb D}\rightarrow [0,1]$ defined by $S(z)=|z|^2,$ $z\in \overline{\mathbb D},$ and $\delta_1$ denotes the Dirac measure at 1.

Chen and Xie recently proved, using the Ashikhmin–Barg lemma on minimal vectors, that every binary linear $[n,k,d]$ code with $k=n-2d+2+v$ ($v\ge 0$) has no codewords of weight in the interval $[2d-v,\,2d-1]$. Their argument uses two of the five basic properties of minimal vectors established by Ashikhmin and Barg (1998). In this note we utilize the third property, the disjoint-support decomposition of non-minimal codewords in binary codes, to generate a mirror vanishing band on the other side of $2d$: if $A_{d+1}=\cdots=A_{d+t}=0$ for some $t\ge 1$ and $k\ge n-2d+1$, then $A_w=0$ for all $w\in[2d+1,\,2d+t]$. Combining the two bands, the number of nonzero weights of such a code is at most $n-d-v-2t+1$, improving the Chen–Xie bound $n-d+1-v$ by $2t$.

This paper establishes the existence of three distinct weak solutions for a class of nonhomogeneous quasilinear inclusion systems, which are governed by locally Lipschitz functionals within Orlicz-Sobolev spaces over unbounded domains $\mathbb{R}^{N}$.By employing a new three critical points theorem established by Wu-Zhou in [29], we extend the nonsmooth critical point theory to handle scenarios with nonlinear growth conditions.Our analysis imposes a precise structural condition on the nonlinear term $H$, requiring it to exhibit superlinear growth while maintaining subcritical asymptotic behavior. This condition establishes a delicate balance between growth restrictions and functional analytic requirements. The novelties of this work include the construction of suitable energy functionals that are consistent with both the structure of Orlicz-Sobolev spaces and the characteristics of nonlinear growth. Through rigorous variational analysis and careful estimation techniques, we establish the existence of three weak solutions under these generalized conditions, thereby significantly expanding the applicability of critical point methods in nonstandard function spaces.

We analyse the essential spectrum of ${\mathcal M} = {\operatorname{curl}} {\operatorname{curl}}$ acting on divergence-free vector fields in unbounded domains of $\mathbb{R}^3$. We show that $\sigma_e({\mathcal M}) = [0,+\infty)$ in quasi-conical domains and $\sigma_e({\mathcal M}) \neq \emptyset$ in quasi-cylindrical domains. For horn-shaped domains with circular cross-section and eventually mean-convex boundary, we establish that $\sigma_e({\mathcal M}) = \emptyset$, independently of their volume. For horns with annular cross-section, the transverse normal harmonic field determines an effective one-dimensional Schrödinger operator $H_V$ with $\sigma_e({\mathcal M}) \supseteq \sigma_e(H_V)$. Finally, for a concrete family of perforated exponential horns with a double-exponential hole, we show that depending on the rate of shrinking of the hole at infinity, either $\sigma_e({\mathcal M}) = \emptyset$, or $\sigma_e({\mathcal M}) = [\gamma^2, +\infty)$, or $\sigma_e({\mathcal M}) = [0,+\infty)$.

We study reconstruction in autoencoders that apply the same forward map before and after setting the observed coordinates to zero. For equal odd input and hidden dimensions $d\geq 3$, among orientation-preserving diffeomorphisms whose Jacobian singular values lie in $[m,M]$, we show that the least uniform reconstruction-derivative error is $\max\{1-M(M-m)/2,0\}$, with affine maps attaining this sharp bound at every prescribed depth. A translated radial rotation can nevertheless reconstruct any prescribed ball exactly with singular values arbitrarily close to one, motivating additional conditions for a finite-data bound. We test this prediction on a 798,452-point terrestrial LiDAR forest scan. At input scale $0.05$, the mean theoretical bound is $0.155$, about $84\%$ of the mean normalized training error $0.185$ across four spatial regions, two depths, and three seeds. At this scale, adding one hidden coordinate reduces the mean reconstruction error below $6\times10^{-6}$.

We determine the branching of irreducible unitary representations of the basic classical Lie superalgebras $\mathfrak{sl}(m|n)$, $\mathfrak{osp}(m|2n)$, and $F(4)$ under restriction to their even subalgebras. The proof combines Dirac inequalities for the Huang-Pandžić-Dirac operator $\operatorname{D}_{\mathfrak{g},\mathfrak{g}_{\bar{0}}}$ with a super analogue of Freudenthal's recursion derived from the Dirac formalism. We extend the Dirac method to positive systems that are not adapted to the relevant real form by transporting the Dirac inequality through odd reflections and determining the resulting correction terms. This yields finite recursive procedures that determine the occurring even constituents and their branching multiplicities, together with closed formulas for the latter, uniformly in the degree of atypicality. For $\mathfrak{sl}(m|n)$, the corrected Dirac inequalities also give a characterization of unitarity for all positive systems relevant to unitary representations.

We study smooth strongly convex–strongly concave minimax optimization in the deterministic unconstrained setting, without assuming a bilinear or separable structure. Although existing multi-loop methods attain near-optimal condition-number dependence, standard single-loop methods generally exhibit a substantial complexity gap. To close this gap, we propose the Single-Loop Predictor–Corrector Extragradient Method with Damped Momentum (PCE-DM), which combines an extragradient prediction–correction scheme with a novel auxiliary feedback recursion for the weaker-curvature variable. PCE-DM uses fixed parameters and two new full-gradient evaluations per iteration after one initialization query, while requiring no inner solves, accuracy schedules, or staged restarts. We develop a Lyapunov analysis that controls the predictor–corrector mismatch through corrected-gradient increments and establish last-iterate linear convergence. Specifically, PCE-DM computes an $\varepsilon$-accurate relative solution, measured by the squared Euclidean distance to the saddle point, within $\mathcal{O}\!\left(\sqrt{\kappa_x\kappa_y} \log(2\kappa_x\kappa_y/\varepsilon)\right)$ full-gradient queries. This result closes the condition-number complexity gap between standard single-loop methods and near-optimal multi-loop methods, matching the known lower-bound order up to logarithmic factors while retaining fixed, explicit single-loop updates. Numerical experiments on regularized linear regression and AUC maximization demonstrate the computational efficiency of PCE-DM.

Time series of random objects, such as covariance matrices, probability distributions, and functional data, call for forecasting methods that do not rely on standard arithmetic operations. We introduce geodesic exponential smoothing, a generalization of exponential smoothing to time series in Hadamard spaces: the forecast level moves a fixed fraction of the way along the geodesic toward each new observation. The smoothing parameter is estimated by minimizing the average squared distance between observations and their forecasts. We further introduce an innovations mechanism under which each observation has conditional Fréchet mean equal to the current level, providing the metric-space analog of the innovations state-space model. In contrast to autoregressive models for object-valued time series, the framework involves a single scalar parameter, assumes no stationarity, and updates online in constant time per observation. Under this mechanism, we establish sample-path properties of the generative process via the quasilinearization available in Hadamard spaces, and prove almost-sure consistency of the smoothing-parameter estimator. Three real-data applications, spanning covariance-matrix, distributional, and functional time series, assess the forecasting performance of the method against structurally heavier alternatives.

We prove that every traceless real or complex matrix $A$ can be written as $A=BC-CB$, with factors of the same size and over the same field satisfying $\|B\|\,\|C\|\le K\|A\|$, where $K$ is an absolute constant. The proof combines an approximate-rank dichotomy with stable commutator representations obtained from quantum expansion.

We introduce a matrix generalization of the cone of Harvey and Lawson and prove that it is an exact special Lagrangian manifold. We further show that it belongs to a family of exact special Lagrangian manifolds that foliate the balanced manifold arising in deep learning.

We compute the groups of minimal nondegenerate extensions of the real symmetric fusion categories $\mathrm{sVec}_{\mathbb R}$ and $\mathrm{sVec}_{\mathbb H}$. We find that they are both isomorphic to the Klein-four group $(\mathbb{Z}/2\mathbb{Z})^2$. Along the way, we classify all nondegenerately braided fusion categories over $\mathbb R$ that have Frobenius-Perron dimension 4, and we determine exactly which of these categories is a Drinfeld center. We end the paper by discussing a homotopy-theoretic conjecture that organizes all these structures.

This is a survey on Pham-Brieskorn varieties and their various properties, targeted for graduate students.

For cumulative-count first-passage problems, the Laplace transform of the time $\mathcal T_N$ of the $N$th count is naturally expressed through powers of a one-count kernel. A Jordan defect of that kernel does not, however, automatically survive the complete sum over terminal phases. We give an exact three-state Markov-renewal example in which it does. At $s=1$, the one-count kernel has spectrum $\{1/2,1/8,1/8\}$, with a one-dimensional eigenspace at $1/8$, and for initial phase $1$, \[ \mathbb E_1[e^{-\mathcal T_N}] =\frac56\,2^{-N}+\frac{N+1}{6}\,8^{-N}. \] Thus the ordinary, unconditioned cumulative-count transform contains the explicit Jordan contribution $N/(6\,8^N)$ at every threshold $N\ge1$. For the same fixed stochastic process, the two eigenvalues meeting at $1/8$ unfold as \[ \lambda_\pm(s)=\frac18\pm\frac1{96}\sqrt{s-1} -\frac{131}{2304}(s-1)+O((s-1)^{3/2}), \] and the characteristic discriminant has a simple zero at $s=1$. We place the example beside two structures that suppress such a marginal signature: fixed post-count reset, which gives scalar renewal, and a common total holding rate, for which terminal summation scalarizes the matrix power. The model also admits a monitored-Lindblad realization, but the mechanism is entirely classical. The result provides a small exact benchmark for how generalized spectral modes can remain visible in finite-threshold first-passage statistics.

Viscosity solutions of eikonal equations need not enjoy regularity beyond Lipschitz continuity. In contrast, differentiable solutions exhibit stronger structural properties. In the Euclidean setting, building on observations of Caffarelli and Crandall (Comm. Partial Differential Equations $\textbf{35}$ (2010), 391–414), one finds that differentiable solutions of the eikonal equation $\| Df \| \equiv 1 \,\, \text{in } \mathcal{U} \subset \mathbb{R}^d$ are of class $C^{1, 1}_{\mathrm{loc}}(\mathcal U)$. Moreover, if $\mathcal{U} = \mathbb{R}^d$, then $f$ must be affine. This manuscript revisits these phenomena in the setting of (finite– and infinite–dimensional) Banach spaces. Our results show how the geometry of the underlying Banach space determines the regularity and affineness of Gateaux–differentiable solutions of eikonal equations. As a geometric application in finite dimensions, we characterize differentiable functions whose gradient ranges are constrained to boundaries of convex bodies.

The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we propose an EDMD-variant with a Fourier dictionary on the $d$-dimensional torus, where the data are sampled on an equispaced tensor grid. We show that the EDMD regression problem admits a unique closed-form solution and coincides with trigonometric interpolation of the Koopman image. This identification has two consequences. First, using Koopman invariance of Sobolev spaces, we transfer approximation-theoretic results for trigonometric interpolation to derive $L^p$-error bounds, $2 \le p \le \infty$, on approximations of the Koopman operator. In particular, the established bounds are of optimal order with explicit constants. Second, the EDMD matrix never has to be assembled, since its action reduces to (nonequispaced) fast Fourier transforms such that the EDMD-approximation can be evaluated matrix-free with quasi-linear cost in the dictionary size. We illustrate the results for the Kuramoto model of coupled oscillators and for the chaotic ABC flow with dictionaries of up to $10^7$ modes.

We prove that, in Zermelo–Fraenkel set theory with the axiom of Foundation removed, the statement that every vector space has a basis implies the Axiom of Choice ($\mathsf{AC}$), concluding that the classical equivalence between $\mathsf{AC}$ and the existence of bases does not require regularity. This result extends to set theory with atoms.

We study the simultaneous identification of the Lamé parameters $\lambda$ and $\mu$ in three-dimensional static isotropic elasticity. For smooth coefficients, we establish uniqueness from the full Dirichlet-to-Neumann map when $\mu$ is axisymmetric, admits a multiplicative or additive separation on a product domain, or is quasianalytic in one fixed direction. These results require neither smallness of $\nabla\mu$ nor real analyticity of the coefficients, and impose no corresponding structural condition on $\lambda$. In the axisymmetric and separated cases, partial boundary data suffice to determine $\mu$.

We develop and analyze a Monte Carlo method for sampling killed anomalous diffusions obtained by time-changing Brownian motion with drift by the inverse of a subordinator. The method targets probabilistic representations of time-nonlocal, including time-fractional, Cauchy–Dirichlet problems on bounded domains. Since inverse subordinators can be sampled exactly in broad classes, while Brownian exit times are generally unavailable in arbitrary domains, we approximate the killed Brownian component by an Euler scheme with discrete boundary detection. We prove a square-root weak error bound with explicit dependence on the Laplace exponent of the subordinator, and derive mean-square and central limit results for the resulting Monte Carlo estimator. A numerical example in the disk and one in a high-dimensional anisotropic shell illustrate the theoretical rates, computation time, and the mesh-free character of the method.

We introduce a synthetic framework for doing local Riemannian geometry that is influenced by a reverse-engineering method for formalizing in Rocq. These new perspectives and techniques lead to a synthetic Taylor formula for a Riemannian metric in normal coordinates, which is the main contribution of this work.

Global existence and the large-time behavior of strong solutions to the one-dimensional planar compressible magnetohydrodynamic system in unbounded domains are investigated.The longitudinal viscosity is assumed to be a positive constant plus any nonnegative power of the density, or to depend on temperature through a power law. The heat conductivity is taken to be proportional to any nonnegative power of temperature, while the transverse viscosity and magnetic diffusivity are taken to be positive constants. Global existence, uniform-in-time estimates, and convergence to equilibrium are established without imposing any smallness conditions on the initial data. In the density-dependent case, any fixed nonnegative viscosity exponent is allowed; in the temperature-dependent case, the nonnegative viscosity exponent is required to be sufficiently small in terms of the initial data. An auxiliary elliptic equation is introduced to control magnetic-pressure oscillations through a square-root-in-time estimate. Combined with the estimates appropriate to each viscosity law, this yields uniform upper and positive lower bounds for the specific volume and temperature and asymptotic stability of the global strong solution.

We study logarithmic Poisson and de Rham cohomologies associated with a class of inhomogeneous free divisors in the affine plane. These divisors arise as inhomogeneous deformations of reduced normal crossing divisors and admit explicit bases for their modules of logarithmic vector fields in the sense of Saito. Using these bases, we construct the corresponding logarithmic Poisson structures and describe explicitly the induced Koszul bracket on logarithmic differential 1-forms within the framework of Lie-Rinehart algebras. We then determine the logarithmic Poisson cochain complex and compute its cohomology for the class under consideration. Furthermore, by means of the logarithmic Spencer complex, we identify the corresponding logarithmic de Rham complex and compute its cohomology. These computations provide explicit cohomological invariants for the class of inhomogeneous divisors considered and show how logarithmic Poisson and de Rham theories extend the corresponding constructions for reduced normal crossing divisors. In addition, we establish an explicit comparison between the logarithmic cohomological theories: they are naturally isomorphic in any degree $k \neq 1$, whereas in degree 1 the logarithmic de Rham cohomology group is a split one-dimensional extension of the logarithmic Poisson cohomology group.

We introduce the Hopf image of a right coaction of a Hopf algebra on an algebra as the smallest Hopf subalgebra through which the coaction factors. We establish its basic properties and show that the induced coaction on the Hopf image is inner-faithful. For coactions on Nichols algebras arising from Yetter–Drinfeld modules, we prove that the Hopf image is the smallest Hopf subalgebra detected by the underlying coaction on the Yetter–Drinfeld module. We give explicit examples of proper non-trivial Hopf images arising from bosonizations.