Research
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Research
Research
31,036+ open-access research outputs.
In 1967 Hajnal and Juh{\'a}sz showed that the cardinality of a first-countable Hausdorff space with the countable chain condition has cardinality at most $\mathfrak{c}$, the cardinality of the real li…
Building on the genus-3 reduction $C_A : w^2 = \lambda^8 + A \lambda^4 + 1$ established in our companion paper (arXiv:2604.09328), we give an unconditional proof of the perfect-cuboid conjecture ("Con…
The impossibility of eliminating hallucination, understood here as incorrect definite answers, in sufficiently expressive yes-or-no formal domains is an immediate consequence of classical undecidabili…
We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term sati…
A skew Bollob\'{a}s system $\mathcal{P}=\{(A_i,B_i):1\leq i\leq m\}$ is a collection of pairs of disjoint subsets of $[n]$ such that $A_i\cap B_j\ne\emptyset$ for any $1\leq i<j\leq m$. Denote by $S_1…
Let $G = \mathrm{SL}(n,\mathbb{C})$, let $B$ be a fixed Borel subgroup, and let $P \supset B$ be a parabolic subgroup determined by a composition $(c_1,\dots,c_k)$ of $n$. Write $P'$ for the derived g…
We prove a Positivstellensatz for operator-valued noncommutative polynomials that are positive on matrix convex sets. Specifically, let $p$ be an operator-valued polynomial in $B(H)\otimes C<x>$ of de…
We study groups endowed with Alexandroff topologies and show that no non-discrete Alexandroff topology can turn a group into a topological group. This settles negatively the basic existence problem fo…
Coisotropic A-branes were introduced by Kapustin--Orlov to enlarge the Fukaya category of a symplectic manifold in a way that aligns with predictions from homological mirror symmetry. From a mathemati…
We define the notion of couple density $(D, \mathbf b)$ where $D$ is a non-empty subset of $\mathbb Z^{m}$ and $ \mathbf b$ a fixed element in $\{0, \cdots, q-2\}^{m};$ We determine a minimum in terms…
This paper explores the restriction behavior of silting-induced $t$-structures and co-$t$-structures on triangulated categories endowed with metrics. For compactly generated triangulated categories ad…
Based on discrete observations, we develop a test to infer if the volatility function $\sigma(\cdot)$ within the nonparametric Gaussian white noise model $dY_t = \sigma(t)dW_t$ is constant. The testin…
Let $q=p^n$, $r\in \mathbb{Z}_{\ge 2}$, $e=q-1$, and $k=\frac{q^r-1}{e}$. In this paper, we study the cyclotomic numbers $(a,b)_{q-1}$ over $\mathbb{F}_{q^r}$. We prove that $(a,b)_{q-1}\le \left\lcei…
We prove the congruence $\sum_{1 \leq k < \sqrt{N}} \sigma_0 (N - k^2) \equiv 0 \pmod 4$, where $\sigma_0(m)$ denotes the number of positive divisors of $m$, for $N = An + B$ with $(A,B) \in \{ (16,14…
In the previous paper, the author showed that for a smooth family $X \to \mathbb{X} \to B$ of a homotopy $K3$ surface, the obstruction for the tangent bundle along the fibers $T_B \mathbb{X}$ to have …
We consider the two families of even polynomials $\Xi_n$ and $\Lambda_n$ studied in~\cite{TallaWaffo2026arxiv2602.16761}, together with the rescaled polynomials $\widetilde{\Xi}_n(x):=\Xi_n(\sqrt{x})$…
We give a metaplectic proof of Hilbert reciprocity, and hence of quadratic reciprocity, in which the local phase is the Kashiwara--Maslov phase of a triple of Lagrangians. In rank two the phase of the…
Motivated by branching problems for principal series representations of the Lie group $G = GL(n,\mathbb R)$, we consider all pairs $(G', P)$ with $G'$ being the Levy factor of a parabolic subgroup of …
A finite set $S \subset \mathbb{Z}$ is a \emph{Sidon set} if its pairwise differences are distinct. A \emph{perfect difference set} (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n +…
We develop a Koszul-theoretic framework for comparing classical Alexander-type invariants with infinitesimal invariants arising from finite-type commutative differential graded algebra models. The cen…
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