arXiv preprints from January 1, 2026 through September 5, 2026 — 14:06:36 EST
We construct an explicit finite chain of non-aspherical group presentation complexes $K(\mathcal X)\subset K(\mathcal Y)\subset K(\mathcal Z)$ in which $K(\mathcal Y)$ is Cockcroft, both inclusion-induced maps on $π_2$ are zero, and the pair $(K(\mathcal Y),K(\mathcal X))$ has the identity property. This answers a question of...
Let $G$ be a stratified Lie group and $\mathcal L$ its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step,...
We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in Weis' characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal...
We solve John Milnor's problem: among all convex regions of a given area on the sphere, the round disk requires the greatest distortion in cartographic projections onto the plane.
A unit-area translation surface is called \emph{maximal} if it maximizes the length of the shortest saddle connection among all surfaces in the same stratum. We investigate whether a non-maximal translation surface can be continuously deformed into a maximal one while the systole increases strictly monotonically. Although such a...
A scalar field alone does not determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural. Realification illustrates this point: an orthogonal complex structure J^2 = -I selects the physical real operators and the balanced composite. Quaternionic quantum mechanics has an...
In these lecture notes, we discuss various `hydrodynamic LLN' and `CLT' scaling limits, among others, in types of stochastic interacting particle systems, connecting `microscopic' behaviors to continuum laws. Via `short stories', the aim is to present some of the `basics' for students and those entering the field, as a complement to...
A notable feature of confined active matter systems is the tendency for motile particles to accumulate near solid boundaries. In various linear models with no-flux boundary conditions, this accumulation is realized through the development of sharp boundary layers at small particle diffusivity $κ$. In this paper, we present the first...
Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $σ^{-1}τ$ has a fixed point for all $σ,τ\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-sized intersecting families of...
Accounting for information flow on the path space of trajectories of a nonnegative martingale yields exact variational identities for it, even at arbitrary random times. This recovers the widely used classical concentration inequalities, from Ville to PAC-Bayes, and measures what each one discards. The tail a bound controls is itself...
We study large-sample properties of higher-order Markov chains on a finite alphabet $Σ$ when the order $m_n$ is allowed to grow with the sequence length $n$. By embedding the process into a first-order chain on $Σ^{m_n}$ and exploiting return-time decompositions, we establish a central limit theorem for additive functionals...
This paper investigates the estimation of the regression operator in function-on-function regression models. While traditional research has predominantly focused on linear models or their immediate nonlinear extensions, we propose a neural operator approach to accommodate general regression operators under mild smoothness assumptions....
Composite likelihood (CL) methods provide a computationally efficient alternative to full likelihood inference for complex multivariate models by replacing the joint likelihood with a product of lower-dimensional marginal or conditional components. Like the MLE, however, the maximum CL estimator (MCLE) is highly sensitive to data...
Bayesian posteriors can converge under model refinement even when the exact sensitivities used by gradient-based samplers do not. We study this mismatch for discretely scale-invariant rough potentials and its consequences for Metropolized Hamiltonian proposals. For Weierstrass truncations, adjacent classical-force increments grow...
In this work we analyze the eigendecomposition of the Hessian matrix of the Robin function $\mathcal{R}(x)$ for the spectral fractional Laplacian in orthogonally invariant domains. We prove that, if $Ω$ a smooth bounded convex domain invariant under the action of an orthogonal transformation $\mathcal{O}$ then, for...
Let $Φ$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $Ξ$-function, set $s(t)=Φ(\sqrt t)$, and define the first Laguerre expression $f(t)=s'(t)^2-s(t)s''(t)$. Csordas and Dimitrov (2000) conjectured that $\log f$ is strictly concave on $(0,\infty)$; Csordas (2015) later restated the assertion as...
We propose a local Legendre frame method for the accurate computation of Caputo fractional derivatives of order \(0<α<1\). On each local subinterval, the function is represented by a restricted Legendre frame obtained from scaled Legendre polynomials on an extended interval. The local coefficients are computed from equispaced samples...
Let $A$ be a commutative simple algebra in a braided finite tensor category $\mathcal{B}$. We identify the largest transparent subalgebra of $A$ as the algebra induced by a central lift of the free-module functor. This identification gives formulas for the Frobenius-Perron dimension and the Müger center of the category of local...
We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of...
We determine the critical probability for Bernoulli bond percolation on essentially any finite graph. Namely, letting $λ(G)$ denote the spectral radius (maximum eigenvalue) of $G$, we prove that the critical probability is at $1/λ(G)$: above this probability there is typically a component of order $Ω(λ(G))$, whereas below it all...
Let $π: M \to B$ be an elliptic fibration over $B = D^2$ or $B = S^2$ with $n$ nodal fibers over $Δ\subseteq B$. We study the universal liftable braids for $π$: those braids that admit a fiber-preserving lift to $M$ for all choices of coordinates on $(B,Δ)$. When $B = S^2$, we show that nontrivial universal braids do not exist by...
The goal of this paper is to prove an upper bound for the $L^4$ norm of a trigonometric polynomial whose spectrum is a nontrivial strictly monotone polynomial with integer coefficients of degree three and higher, via its $L^2$ norm. Our condition on the coefficients of the trigonometric polynomial in question is that they form a...
We consider the spherical Borel Eisenstein series induced from the trivial representation for a split semisimple linear algebraic group over a number field. We prove that its regularization at the special point corresponding to half the weighted marking of a distinguished coadjoint nilpotent orbit in the Langlands dual Lie algebra is...
The Erdős Matching Conjecture is governed by two competing ways of excluding $s+1$ disjoint edges: one may concentrate all edges on fewer than $k(s+1)$ vertices, or force every edge to meet a fixed $s$-set. We determine a near-optimal range in which the second construction is extremal. For every fixed $k\ge2$, there is $s_0(k)$ such...
We study the 4D Hessian conjecture in Lorentzian signature. For a polynomial potential $φ$ in 4 real variables whose Hessian matrix has inertia index 1 and determinant $-1$, we define a pivot of $φ$ as a direction vector $v$ such that the double derivative $D^2_vφ$ is a constant function. We then prove that the gradient mapping of...