Papers

Blumer and Quadrelli introduced a family $\mathcal{F}_2$ of two-relator pro-$p$ groups obtained from a Demuškin group by imposing the commutativity of two generators which are not paired in the Demuškin relation. They proved the triple and quadruple Massey vanishing properties and asked whether the same holds in every length. We introduce the $z$-profile \(v_h=(\alpha_h(z_1),\alpha_h(z_2))\in\Fp^2\) of a defined $n$-fold Massey product. Definability forces $\det(v_h,v_{h+1})=0$, so the nonzero profile entries decompose into support blocks carrying projective directions in $\PP^1(\mathbb{F}_p)$. After the known endpoint reduction, profiles with exactly $r$ blocks are counted by $\binom{n-1}{2r}$, and $r$ blocks first occur in length $2r+1$. We also prove an all-length conditional reduction for Dwyer's lifting problem: if the added commuting relator can be made exact in a lift, then the remaining central defect of the Demuškin relator can be removed by an endpoint correction. The correction preserves the power term $x_1^q$ for every parameter allowed in $\mathcal{F}_2$. In length five, this yields vanishing whenever all three interior profile vectors $v_2,v_3,v_4$ are nonzero, for every prime $p$. The resulting classification identifies the remaining five-fold support types and the compatibility mechanisms governing them.

We establish higher-order Gaussian upper bounds for the heat semigroups associated with a broad class of sectorial maximally subelliptic quadratic forms on manifolds with boundary. Near non-characteristic boundary points and in the interior, we obtain pointwise bounds for all mixed derivatives of the heat kernel in time and along the Hörmander vector fields, expressed in the associated Carnot–Carathéodory geometry. The results apply to operators of arbitrary even order, systems, nonsymmetric forms, and boundary conditions beyond the Dirichlet case.

Let $F$ be a global field and let $p$ be an odd prime with $p\neq\operatorname{char}F$. Assuming first that $\mu_p\subset F$, we record in a uniform global-field form the length-four common-slot chain for equal degree-$p$ symbol classes and emphasize the signed normalization adapted to explicit norm constructions. Thus, from \[(a,b)_p=(c,d)_p\in\textrm{Br}(F)[p]\] one obtains $x,y\in F^\times$ such that \[(a,b)_p=(x^{-1},b)_p=(x,y)_p=(c^{-1},y)_p=(c,d)_p.\] For number fields, the underlying chain is the length-four chain lemma of Gille–Szamuely, based on Tate's simultaneous local–global theorem. The point developed here is that its signed form yields four compatible norm equations that can be used constructively. For the extraspecial central product $H_{p^3}*H_{p^3}$ over a $C_p^4$-Kummer extension, the central-embedding obstruction is $(a,b)_p-(c,d)_p$, while the four norm equations supplied by the chain assemble, under a natural independence hypothesis on the auxiliary Kummer classes, into an explicit factorized radical realization of the central product. Finally, when the ground field does not contain $\mu_p$, we show by a restriction–corestriction argument that the central-embedding obstruction is detected after passage to the cyclotomic extension $F(\mu_p)$. Over that field the problem is Kummer, and the obstruction is again the difference of the two symbol classes.

In this work we propose an approach to construct a fully asynchronous unsourced random access communications (URA) system. Besides the common URA features such as the absence of the user identification information in the packet and the decoding oriented to receiving the content of the messages, there is no limitation on the user transmission timing. The active users can transmit their packet any time on demand. The proposed system belongs to the class of preamble-payload URA formats. Contrary to the typical role of the preamble to serve as a temporary user identification, the preamble serves for the purpose of timing acquisition. The proposed system also employs a very small pool of preambles to resolve collisions of packets transmitted simultaneously.We provide a performance analysis and simulation results for the proposed system and demonstrate that for a wide range of signal-to-noise ratios (SNR)s the supported numbers of active users are close to the numbers of users supported by a slotted URA system counterpart.

We survey computations and tools concerning the mapping class group of a compact, oriented, connected 3-manifold $M$. We provide a guide to the literature and sketch proofs for various families of irreducible and geometric 3-manifolds. We also consider JSJ and prime decompositions of 3-manifolds, and consequences for their mapping class groups.

In this article, we establish a new dimensional reduction result in (Galilean or Lorentzian) Robertson Walker spacetimes with Euclidean fiber. In this context, we prove that every unchanged direction observer can be embedded into a three dimensional totally geodesic timelike submanifold.

Dual quasi-mixing and dual mixing of a measure preserving transformation are uniform individual ratio limit properties of the transfer operator which entail the quasi-mixing and mixing properties of Krickeberg (respectively). The real restrictions of normalised, parabolic, non-Möbius inner functions of the upper half plane preserve Lebesgue measure and are dual quasi mixing; such a restriction being dual mixing iff it is exact and singular dual quasi-mixing otherwise.

We present a meshfree numerical solver for the incompressible Navier-Stokes equations on oriented curved surfaces that are represented by surface point clouds. On curved surfaces, numerical challenges pertaining to stiffness and pressure-velocity coupling are exacerbated. Moreover, vector calculus on curved surfaces differs from its Euclidean counterpart. The presented method operates on surface point clouds in an Eulerian frame of reference without requiring a computational grid or mesh. It achieves consistent approximation in space and time with high order of accuracy; we demonstrate up to order six. The incompressibility constraint is locally imposed as a weak artificial compressibility approximation, avoiding global matrix inversion. We show that the method provides consistent and convergent approximations of surface vector fields and differential operators. We study the relationship between error, spatial resolution, and artificial Mach number and characterize the frequency spectrum of the artificial oscillations. We provide numerical solutions of the incompressible Navier-Stokes equations on symmetric surfaces, such as the sphere and torus, and on parametric and non-parametric asymmetric surfaces. Since the proposed method works directly on unstructured surface point clouds, it provides a promising approach for simulations on image-derived geometries, such as in biological morphogenesis from microscopy videos.

We study the algebraic structure of $(6,3)$ biangular Parseval frames. The two-distance property yields adjacency matrices $A_1,A_2$ whose span forms a three-dimensional commutative algebra, and adjoining the commutator $[A_1,A_2]$ produces a three-dimensional Lie algebra $\g$. The Gram matrix $G = I + c_1 A_1 + c_2 A_2$ induces a derivation $\alpha(X) = [G,X]$ on $\g$, equipping $(\g, [\cdot,\cdot], \alpha)$ with a Hom–Lie algebra structure. We compute all structure constants explicitly in terms of the strongly regular graph parameters $(k_1,\lambda_1,\mu_1,k_2,\lambda_2,\mu_2)$ and the frame angles $(c_1,c_2)$, and use this framework to derive explicit bounds for the partial frame operators arising in Marcus–Spielman–Srivastava (MSS) partitions. These bounds depend directly on the structure constants and refine the universal MSS estimate in the regime of highly unbalanced partitions.

We confirm a conjecture of Brause, Randerath, Rautenbach and Schiermeyer (2016) by proving a localized lower bound on the independence number of a graph that strengthens the classical bounds of Fajtlowicz (1978) and of Caro (1979) and Wei (1981), which in turn settles a conjecture by Bertram and Horák (1996). Our proof is based on a new Motzkin–Straus-type inequality involving local clique numbers and the independence number. We then apply the developed methods to study spectral and algebraic measures of graph bipartiteness. In particular, we extend a theorem of Brandt (1998) on spectral bipartiteness from regular $K_{r+1}$-free graphs to all $K_{r+1}$-free graphs, we improve a general upper bound for the least signless Laplacian eigenvalue of $K_{r+1}$-free graphs, and we disprove a conjecture of de Lima, Nikiforov and Oliveira (2016) in the case of $K_4$-free graphs.

In 2023, Wyman and Xi asked “Can you hear your location on a manifold?” We pose a related question, asking if probabilistic data can be used to recover location within a manifold. More precisely, we ask if location can be recovered from the distribution of exit times of Brownian sample paths. We show that exit time moments determine location up to symmetry for a variety of two-dimensional domains. Beginning with (convex) domains whose boundaries are conic curves, we show that in elliptic, parabolic, and hyperbolic domains, the first two exit time moments are sufficient to detect location up to symmetry. We establish similar results in unbounded wedge domains and equilateral triangular domains. We show that the full sequence of exit time moments determines location in rectangular domains. We end with two location detection results on general planar domains. The key tool driving our work is a “Poisson hierarchy,” which establishes a connection between exit time moments of Brownian motion and solutions to a family of PDE problems.

A quiver representation consists of a collection of vector spaces along with a set of arrows, which are linear maps between these spaces. In this work, we study quiver representations in equioriented type $A$ which are also chain complexes; that is, in which consecutive arrows compose to zero. We show that orbits of these representations under a change of basis action are in bijection with mixed dimer covers of a $2 \times n$ grid graph. The latter object can be endowed with a partial order which is a distributive lattice, and we show that the degeneration order on the orbits of chain complexes is a coarsening of this partial order. In addition, we use recent matrix formulae of Claussen and Ovenhouse to enumerate these orbits. This also computes the Kostant partition function applied to height-restricted, type $A$ roots. When the dimension vector is uniform, we discuss a correspondence with paths of a beam of light bouncing between glass plates and give an explicit generating function.

We prove the generalized Geroch conjecture, and the corresponding hyperbolic conjecture, in dimension two.

We prove that a standard conic bundle $X \to \mathbb P^2_{\mathbb C}$ whose discriminant is very general of degree at least 18 admits no rational multisections of degree two. This is the first step towards proving a conjecture of Iskovskikh that there are conic bundle threefolds that are not unirational, since to prove that $X$ is not unirational, it suffices to show that there are no rational multisections of any degree. Proving Iskovskikh's conjecture would provide the first example of a rationally connected variety that is not unirational.

We prove a rectifiability theorem for harmonic maps into DM-complexes. More specifically, if $u$ is a harmonic map from a $n$-dimensional Riemannian domain to an $N$-dimensional NPC DM-complex $Y$, the $k$-th singular stratum of the singular set is countably $k$-rectifiable for all $k \in \{0,...,n-2\}.$ Our proof follows the framework in \cite{dm}. This extends the rectifiability result for the singular set of harmonic maps into a $F$-connected complex in \cite{dees} and generalizes the rectifiability theorem with respect the singular strata of harmonic maps into a $F$-connected complex in \cite{bd}.

We explicitly compute and tabulate the suspension map for cohomology operations, applying it to the dimensional reduction of anomalies and higher group structures of generalised symmetries in quantum field theories and lattice systems. In a quantum system with a discrete $p$-form symmetry, we can restrict the symmetry defects to a codimension-$q$ subspace of spacetime, or equivalently confine the gauge field to a slab whose thickness approaches zero. This results in a $(p-q)$-form symmetry on the subspace. We determine the fate of two important properties of symmetries under this process. First, we compute the anomaly of the reduced symmetry, or the higher anomaly of the original symmetry, which acts as an obstruction to higher gauging and onsiteability. Second, for two symmetries forming a higher-group structure, encoded as a Postnikov class, we compute the reduced higher-group structure, which may or may not be trivial. Both cases are captured by the $q$-fold iteration of the cohomology suspension $\Omega$, also known as the loop functor or transgression, in the cohomology of Eilenberg–MacLane spaces. Classical results state that $\Omega$ is an isomorphism in a stable range and annihilates mixed anomalies. To calculate $\Omega$ over a base ring which is not a field, we make use of small models for the chain DGAs of Eilenberg–MacLane spaces, pioneered by Cartan and Moore, whose results we review and extend. The critical added step is the improvement of a mod $p$ chain contraction to the $p$-local integers, obtained via a $p$-adic series. We automate the computations in a software package emcm, and provide extensive tables of the results. Finally, to make contact with explicit cochain formulations of cohomology operations, we demonstrate how to calculate cup products and certain cup-$i$ products in this framework.

We introduce combinatorial and diagrammatic methods for representing properly embedded symplectic surfaces in 4-dimensional Weinstein domains. We show that positive ascending surfaces, which include complex curves in Stein domains and multisections of Lefschetz fibrations, can be placed in bridge position with respect to Islambouli–Starkston's bisection-with-divides structure on the Weinstein domain. We algorithmically relate various decompositions of such surfaces, including transverse banded unlink diagrams, quasipositive factorizations, bridge bisections with divides, shadow diagrams (curves on surfaces), and pointed monodromy factorizations. We also develop a new way to present branched covers of Weinstein domains along positive ascending surfaces, which, combined with work of Loi–Piergallini, recovers Islambouli–Starkston's result that every compact Weinstein domain admits a bisection with divides.

We show that the subleading collinear sector of Yang-Mills (YM) amplitudes is controlled by ordinary Einstein-Yang-Mills (EYM) amplitudes and their higher-derivative corrections. The complete set of strict subleading collinear limits associated with an equal-helicity collinear pair can be extracted from open-closed string disk amplitudes. The latter generate $\frac12 (n-3)!$ BCJ-like relations with non-linear kinematic coefficients, reducing the $(n-3)!$ collinear data to a basis of dimension $\frac12 (n-3)!$. This reduced basis can be represented by gravitational amplitudes in EYM theories and by their higher-derivative open-string corrections. At multiplicity $n$ this gravitational basis consists of the $(n-4)!$ independent EYM subamplitudes together with $\frac12 (n-5)(n-4)!$ higher-order corrections involving one graviton and $n-2$ gluons. The resulting decomposition is governed by unsigned Stirling numbers of the first kind: the ordinary EYM amplitudes correspond to the sector $\left[{n-3\atop 1}\right]$, the higher-order BCJ-like relations to even sectors $\left[{n-3\atop 2j}\right]$, and the higher-derivative EYM corrections to odd sectors $\left[{n-3\atop 2j+1}\right]$ with $j\geq 1$.

We define the integral profiles of all modules and introduce integral coefficient rings ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)$ for all finite-dimensional complex algebras $A$. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules $M\cong N$ are isomorphic if and only if their integral profiles are similar, i.e., $M\cong N$ if and only if $\displaystyle \int M \sim \int N$. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of $A$ (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of $A$ is finite if and only if the embedded integral profile of $\mathrm{top}(A)$ lies in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$; (6) the finitistic dimension of $A$ is finite if and only if each embedded integral profile of $M$ lying in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$ implies that its degree is less than or equal to a fixing integer $d\in\mathbb{N}^+$; (7) we provide two sufficient conditions, such that if a finite-dimensional complex algebra $A$ satisfies one of them, then its finitistic dimension is finite.

For mathematicians: In this first in a series of two papers, we give a mathematically rigorous classification of (sufficiently nice) $c=1$ vertex operator algebras (VOAs) and vertex operator superalgebras (VOSAs). We confirm the lore that any such VO(S)A is either a lattice VO(S)A $V_L$ associated to a rank-1 integral lattice $L$, or can be obtained as an orbifold thereof, i.e. a $G$-invariant subalgebra $V_L^G$ for some finite group $G$ of automorphisms. All such $G$ are known, allowing for an explicit enumeration of nice $c=1$ VO(S)As. A key ingredient in our approach is to establish a general criterion for nice VOAs, requiring knowledge only of the vacuum character, for testing when the simple modules with integer conformal dimension span a symmetric fusion subcategory which is braided tensor equivalent to $Rep(G)$. In our companion paper, we calculate the ribbon auto-equivalences of the representation categories of the nice $c=1$ VOAs, and leverage this to obtain the classification of nice bosonic and fermionic $c=1$ full conformal field theories. For physicists: We rigorously classify the chiral algebras that can arise in the holomorphic sector of a bosonic or fermionic rational $c=1$ conformal field theory (CFT) whose non-identity primaries all have positive conformal dimension. Thinking of chiral algebras as gapless boundary conditions of 3D topological quantum field theories (TQFTs), our result says that any such chiral algebra is either a holomorphic boundary of $U(1)_k$ Chern-Simons theory, or can be obtained by passing to the $G$-invariant states thereof for some finite group $G$ of symmetries. We explicitly enumerate these chiral algebras and also discuss their non-invertible symmetries. In a companion paper, we build on these results using techniques from the study of 3D TQFTs to classify full bosonic and fermionic rational $c=1$ CFTs.

We study first-passage percolation on the $\ell$-spread-out one-dimensional cycle of size $n$, where vertices are connected if their graph distance is at most $\ell$. We assign i.i.d.~non-negative random weights from a Weibull distribution $\omega_e \sim \mathrm{Exp}(1)^{1/\theta}$ to the edges for $\theta>0$ fixed. This paper investigates the transition in the asymptotic behavior of the passage time $T_n$ between two typical vertices and the hop-count of the optimal path as the connectivity parameter $\ell$ diverges with $n$. We identify two fundamentally distinct geometric regimes. In the mesoscopic regime ($1 \ll \ell \ll n$), the optimal path locally mimics a spatial branching random walk but remains globally constrained to a one-dimensional geometry. We establish a law of large numbers characterized by the front speed of a Crump–Mode–Jagers branching random walk, prove a central limit theorem with Gaussian fluctuations when $\ell\ll n^{1/4}$, and show that the expected hop-count grows proportionally with the spatial distance. In the macroscopic regime ($\ell \approx \lambda n$ for $\lambda \in (0,1/2)$), the graph becomes a highly connected mean-field network. We prove that the passage time collapses to a $\log n$ scale with constant order non-Gaussian fluctuations, explicitly determining the extreme-value limit driven by the collision of two independent non-spatial CMJ processes. We establish a law of large numbers for the hop-count. Finally, we rigorously trace the transition in the order of the mean of $T_n$ between these two regimes, demonstrating an order transition for the passage time across the critical connectivity threshold $\ell \asymp n/\log n$. Our results provide a comprehensive deterministic-range interpolation from spatial Gaussian fluctuations to mean-field extreme-value fluctuations.

We consider the problem of finding critical domains $\Omega\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (\Omega) = {\rm Per}\,(\Omega) + \frac 12 \iint_{\Omega\times \Omega } \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|\Omega|=V$. We look for smooth, embedded, compact surfaces $\partial\Omega$ that solve this problem. We construct an axially symmetric, non-minimizing solution that, for a sufficiently small $V>0$, resembles the union of two balls with volume $V/2$ connected by a tiny, approximately catenoidal neck with a width of the order $V^{\frac 43}$.

In this paper, we shed light on basic questions such as: What is the "right" underlying set on which a function space lives, and is this set uniquely determined? What are the appropriate morphisms to classify function spaces? When does a weighted composition operator induce an isomorphism, and when does it induce an isometric isomorphism? We first formulate suitable notions for domains of reflexive functional Banach spaces and study the corresponding notions of isomorphism between Hilbert function spaces. We then apply this framework to classify certain unitarily invariant spaces of holomorphic functions in several variables. Our main results extend Hartz's results from the complete Pick setting to the general setting of weighted Hardy spaces and establish a simple relation between the kernels of isomorphic spaces, improving also on earlier work of Ofek and Sofer in several directions. Additionally, we apply our results to investigate and estimate the Banach-Mazur distance of Hilbert function spaces.

For a positive integer $a$, let $b(a)$ be the smallest integer $b > a$ such that the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b}$ is smaller than the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b-1}$. Recently it was shown that the limit inferior $$\liminf_{a \to \infty} \left(\frac{b(a) - a}{\log a}\right)$$ exists and is positive. Here we find its exact value.

The generating graph $\Gamma(G)$ of a group $G$ is the graph whose vertex set is $G$, where two distinct vertices are adjacent if and only if they generate $G$. In this paper, we systematically study the structure of generating graphs of finite abelian groups (non-cyclic) and determine the set of all generating pairs. Moreover, we give some structural characterizations, in particular, we determine conditions under which $\Gamma(G)$ is regular, characterize when the isolated vertices form a subgroup, and establish necessary and sufficient conditions for two non-isomorphic finite abelian groups $G$ and $H$ to satisfy $\Gamma(G)\cong \Gamma(H)$. Furthermore, we compute the spectra of the adjacency and Laplacian matrices of these graphs.

We completely solve Colombo's 1928 determinant problem. For distinct real $x_1,\ldots,x_N$, $N\geq 2$, and an integer $D\geq 1$, we prove that $\det[(x_j-x_i)^D]\neq 0$ if and only if $D\geq N-1$ and either $N$ is even or $D$ is even. The even-exponent case follows from Dyn–Goodman–Micchelli (1986); the remaining odd case is proved by a strict Pfaffian sign theorem. The new odd-exponent theorem and its complete proof chain have also been formalized in Lean 4.

For a finite multiset of integers, repeatedly choose two entries $a,b$ and replace them with $|a^{2}-b^{2}|$. Hickerson and Kleber asked whether, for every integer $n$, the set ${n,n+1,\ldots,n+35}$ can be reduced to zero. We give an explicit reduction. Together with their results for lengths 12 and 24, this gives, for every positive integer $L$, [ {n,n+1,\ldots,n+L-1}\text{ reduces to zero for every }n\in\mathbb Z \Longleftrightarrow 12\mid L\text{ and }L\ge 24. ]

We consider a fractional diffusion problem with temporal nonlocality acting on the diffusive flux. A coercive space–time variational formulation in Bochner-valued fractional Sobolev spaces is derived and the existence, uniqueness, and regularity of solutions are established. We further develop a conforming tensor-product Galerkin discretization and prove quasi-optimal error estimates in the anisotropic energy norm and improved convergence rates in weaker norms using duality arguments. In contrast to some space-time formulations for classical diffusion, the method preserves the causal structure of the evolution problem and leads to a time-stepping procedure with memory terms. On uniform time grids, the discrete history operator has a lower-triangular Toeplitz structure which enables an efficient implementation using fast recursive convolution techniques.

We construct explicit translational tiles without spectra in three finite abelian $p$-groups. The first is a $64$-point subset of $\Z_4^4\times\Z_2^2$. The other two are a $512$-point subset of $\F_2^{13}$ and a $2187$-point subset of $\F_3^9$. Consequently, the tile-to-spectral implication fails for finite abelian $p$-groups, and it already fails within the class of elementary abelian groups for both $p=2$ and $p=3$. Two elementary mechanisms organize the examples. A two-layer obstruction turns a spectral non-tile with two suitable tiling complements into a tile without a spectrum. A fiber–clique obstruction converts a family of tiling complements with controlled common Fourier zeros into an elementary abelian counterexample. All coordinate data are included. The finite claims are certified by three short, self-contained programs using exact integer arithmetic and exhaustive searches; the accompanying source files recompute every assertion used in the proofs.

This paper studies the problem of computing the stochastic probability (shortest code length) of the encoded vectors containing cluster structure using Normalized Maximum Likelihood (NML) model. This is of great theoretical and practical importance in data clustering based on Minimum Description Length (MDL) principle, such as for estimating the best number of clusters and best cluster structure for the data. Straightforward computation of the shortest code length of the vector containing cluster structure based on the NML model requires polynomial time with respect to the size of the vector and number of clusters. We show that this is a tractable problem by introducing a recursion formula for the efficient computation of normalizing constant from the NML model. The time complexity of the new formula is linear opposed to previous polynomial time with respect to the size of the vector and number of clusters.