We prove that the set of complex dyadic orthonormal wavelets is path connected in $\textup{L}^2(\mathbb{R})$. This answers, in the affirmative, a long-standing and frequently addressed problem in wavelet theory.
6 unreviewed2 full-paper AI checks
PF:2026.000009ProofForum Algebra & Number Theory Group Theory v1
We prove an integrated computability matrix for exceptional Brunnian words in finite marked group presentations. The matrix separates common-representative literal deletion from quotient killing, arbitrary markings from genuine finite sets, and supplied-word recognition from positive-only and total witness search. We also determine the exact arithmetical upper boundary, identify a sharp one-relator obstruction, and use a sphere benchmark to show that the two Brunnian numerators are genuinely different.
Let $\psi$ be a pure state of a two-dimensional quantum spin system, and
assume that its GNS representation satisfies approximate Haag duality (AHD)
in the sense of Ogata. Using product-state stabilization to make the cone
von Neumann algebras properly infinite, the cone sectors of $\psi$ form a
braided Buchholz--Fredenhagen (BF) $C^*$-tensor category $\operatorname{BF}(\psi)$. We show how $\operatorname{BF}(\psi)$ changes under
three operations. (i) A quantum cellular automaton (QCA) of finite spread
preserves AHD and gives a linear braided equivalence. (ii) A spatial
reflection gives a linear equivalence that reverses the braiding. (iii) An
on-site time reversal gives an anti-linear braided equivalence; in other
words, the complex conjugate of $\operatorname{BF}(\psi)$ is braided equivalent to the BF
category of the time-reversed state. For QCA we give a direct proof that AHD
is preserved, with explicit constants. We also compare the QCA result with
related results of Ogata, Naaijkens--Ogata, Jones, and
Bhardwaj--Brisky--Chuah--Kawagoe--Keslin--Penneys--Wallick.
6 unreviewed2 full-paper AI checks
PF:2026.000007ProofForum Algebra & Number Theory Group Theory v1
For every order $n\geq3$ we construct an irreducible elementary net of modules over a principal ideal domain in an algebraic extension of its fraction field that is not closed. The example is defined over $\mathbb Z$ inside the cubic field generated by a root of $x^3-x-1$. This gives a negative answer to Kourovka Problem 19.48.
3 unreviewed1 full-paper AI check
PF:2026.000006ProofForum Algebra & Number Theory Group Theory v1
We answer both parts of Kourovka Problem 19.9. First, for $r\geq4$ the finitely generated linear group $\operatorname{Sp}_{2r}(\mathbb Z)$ has a non-linear non-abelian tensor square, by combining the Brown--Loday description for perfect groups with Deligne's non-residually finite universal central extension. Second, for every $n>3$ the tensor square $B_n\otimes B_n$ of the braid group is linear. The latter follows from the homology map $H_2(B_n;\mathbb Z)\to H_2(S_n;\mathbb Z)$ and the standard splitting of the tensor square when the abelianization has no $2$-torsion.
4 unreviewed1 full-paper AI check
PF:2026.000005ProofForum Algebra & Number Theory Group Theory v1
We prove that the near Frattini subgroup of every knot group is trivial. The argument first identifies the lower and upper near Frattini subgroups for finitely generated groups, and then shows that every nontrivial element of a finitely generated torsion-free virtually RFRS group is a near generator. Virtual specialness of knot groups then gives the result. This answers both parts of Problem 19.1 of the Kourovka Notebook.