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PF:2026.000023 · immutable v1 · math.GR — Group Theory · Uploaded by Bernardo Di Siena
Abstract. We give an elementary affirmative answer to Problem~20.73 of the
Kourovka Notebook. More precisely, we show that every countably
infinite group admits a factorization into two infinite subsets in
which each element has a unique representation. The main ingredient
is an inductive extension lemma for groups in which every nonidentity
element has only finitely many square roots. The general case follows
by combining this lemma with elementary reductions involving abelian
subgroups, infinite-index subgroups, and the FC-centre.
PF:2026.000017 · immutable v2 · math.GR — Group Theory · Uploaded by Ernesto Ingrosso
Abstract. We prove that every torsion-free locally graded group in which each
subgroup is either pronormal or contained in the pro-norm is abelian.
Consequently, no infinite simple locally graded group has this
property. We also extend the classification of insoluble locally
$c$-graded groups with the same property to all locally graded
groups. The proof combines results on pronormal subgroups with the
theory of finitely generated HNN-free groups.
PF:2026.000002 · immutable v3 · math.GR — Group Theory · Uploaded by Marco Trombetti
Abstract. For every prime $p$ we construct a family of balanced finite two-generator $p$-groups whose nilpotency class is arbitrarily large. This gives an affirmative answer to Problem 17.113 in the Kourovka Notebook.
PF:2026.000019 · immutable v1 · math.GR — Group Theory · Uploaded by Ernesto Ingrosso
Abstract. We answer a question of Brescia, Ersoy and Kuzucuoglu concerning the
subgroup lattice of a $\kappa$-existentially closed group. We prove,
for every infinite cardinal $\kappa$, that such a group contains an
elementary abelian $2$-subgroup of rank $\kappa$. We then construct an
explicit ordered set of cardinality $\kappa$ without endpoints, every
nonempty open interval of which has cardinality $\kappa$. Indexing
a basis of the elementary abelian subgroup by this set gives a strictly
increasing chain of subgroups with exactly $\kappa$ members between
any two distinct members. No regularity assumption on $\kappa$ is
required.
PF:2026.000018 · immutable v1 · math.CO — Combinatorics · Uploaded by Zhanhe Zhang
Abstract. For a graph $F$, the $F$-degree of a vertex counts the copies of $F$ containing it, without requiring the copies to be induced. We determine the eventual regularity spectrum of connected patterns of order at least three: stars admit no regular $F$-irregular hosts, whereas every non-star $F$ admits a connected $r$-regular $F$-irregular host for every sufficiently large integer $r$. These hosts can be chosen to have diameter two and $r+o(r)$ vertices, asymptotically optimal to first order. The construction uses complements of sparse layered regular graphs. Rooted-homomorphism localization and quotient expansion isolate the leading contributions to $F$-degree differences; spectral amplification separates competing terms, and a scale-preserving choice of mixed-radix parameters realizes each prescribed regularity.
PF:2026.000016 · immutable v1 · math.GR — Group Theory · Uploaded by Marco Trombetti
Abstract. For every oracle $X\subseteq\mathbb N$ we construct a three-generated group $G_X$ with an $X$-computably enumerable presentation such that a finitely generated group has an $X$-computably enumerable presentation if and only if it embeds as a finitely generated subgroup of a quotient of a finite free product of copies of $G_X$ by finitely many relators. This gives an affirmative answer to Kourovka Problem 20.33.
PF:2026.000014 · immutable v1 · math.GM — General Mathematics · Uploaded by Ronald de Haan
Abstract. An allocation of indivisible goods is MMS-fair if every agent receives at least
her maximin share, and such allocations need not exist.
We classify the complexity of exact maximin shares for additive valuations,
with the values encoded in binary or in unary.
With values in binary, computing a maximin share is $\mathrm{OptP}$-complete
and computing an MMS-fair allocation is $\mathrm{FP}^{\mathrm{NP}}$-complete, already for two agents with identical valuations.
With values in binary, deciding whether an MMS-fair allocation exists is $\mathrm{P}^{\mathrm{NP}}$-complete,
already for four agents holding three distinct valuations.
With values in unary and the number of agents part of the input,
computing a maximin share is $\mathrm{OptP}[O(\log n)]$-complete
and computing an MMS-fair allocation is complete for Chen and Toda's class $\mathrm{FNP}/\!\!/\mathrm{OptP}[O(\log n)]$, again with identical valuations.
In the same setting, deciding existence is $\mathrm{P}^{\mathrm{NP}}[O(\log n)]$-complete.
An MMS-fair allocation always exists if all agents but one share a valuation,
so the two existence results need agents with differing valuations, and the two computation results do not.
All four reductions share one framework, in which a common weight fixes the admissible partitions
and small charges encode a formula, and each uses one of two devices for satisfiability, one for each encoding.
The proofs were found, and this text was written, in interaction with Anthropic's Claude models.
The human author has checked the proofs in the main text, but not yet those in the appendix.
PF:2026.000013 · immutable v1 · math.OA — Operator Algebras; math.MP — Mathematical Physics · Uploaded by Ruizhi Liu
Abstract. Woronowicz's purification of a factor state on a $C^*$-algebra is characterized by a doubled extension that is pure, $j$-positive, and exact. Exactness is the system--ancilla commutant equality: in the GNS representation of the doubled state, the von Neumann algebra generated by the ancilla copy equals the commutant of the original-system algebra. In his 1972 paper Woronowicz explicitly observed that ``the existence of pure non exact states is not clear.'' We give a concrete construction showing that exactness is independent of purity and $j$-positivity, even after the marginal state is fixed.
The first model uses the finitary symmetric group $G=\operatorname{Sym}_{\mathrm{fin}}(\mathbb N_0)$ and the point stabilizer $K=\{g:g(0)=0\}$. On $\ell^2(G)\otimes\ell^2(\mathbb N_0)$ we construct commuting left and twisted-right representations whose vector state satisfies
\[
\varphi(\bar u_g\otimes u_h)=\mathbf 1_{\{g=h\in K\}}.
\]
The resulting state is pure and $j$-positive, and both marginals are the canonical trace. The projection $\mathbf 1\otimes\ket 0\!\bra 0$ lies in the commutant of the system algebra but not in the ancilla algebra. Thus exactness fails strictly. The same state admits the conditional-expectation formula
\[
\varphi(\bar a\otimes b)=\tau\!\left(E(a)^*E(b)\right),
\qquad E:R\to Q\subsetneq R,
\]
where $R$ is the hyperfinite $II_1$ factor and $Q$ is a proper subfactor. By the uniqueness of the separable hyperfinite $II_1$ factor, the construction restricts to two copies of the qubit UHF algebra. Placing the system copy and the ancilla copy on opposite half-chains realizes the doubled state as a pure state on the two-sided qubit chain. Both copy marginals are the product trace, while the system--ancilla commutant equality fails.
This establishes that exactness is an independent part of Woronowicz's characterization of canonical purification. The qubit-chain realization is spatially inhomogeneous, and its $j$-positive form is degenerate; translation-invariant and strictly $j$-positive variants are separate questions.
PF:2026.000012 · immutable v1 · math.QA — Quantum Algebra; math.CO — Combinatorics; math.RA — Rings and Algebras · Uploaded by Qiyue Tang
Abstract. We prove Conjecture 11 of Lee, Li, Rupel, and Zelevinsky on the support
of triangular basis elements in rank-two quantum cluster algebras.
PF:2026.000011 · immutable v1 · math.QA — Quantum Algebra; math.RT — Representation Theory; math.RA — Rings and Algebras · Uploaded by Qiyue Tang
Abstract. We prove quantum Laurent positivity for mutation-acyclic
skew-symmetrizable cluster algebras of arbitrary finite rank.
For every integral compatible quantization with invertible frozen
variables, each cluster variable has a Laurent expansion with
coefficients in $\mathbb N[v^{\pm1}]$ in every seed.
For an indecomposable rigid representation of an acyclic valued quiver,
we identify its Hall shuffle character with the character of a
self-dual simple quiver Hecke module. These modules are real and admit
affinizations; modules corresponding to the summands of a rigid direct
sum strongly commute. A Feigin localization and a comparison of graded
composition multiplicities establish positivity at an arbitrary
target seed.
PF:2026.000010 · immutable v1 · math.CA — Classical Analysis and ODEs; math.FA — Functional Analysis · Uploaded by Vjekoslav Kovac
Abstract. 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.
PF:2026.000009 · immutable v1 · math.GR — Group Theory · Uploaded by Lukas Böke
Abstract. 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.
PF:2026.000008 · immutable v1 · math.MP — Mathematical Physics; math.OA — Operator Algebras · Uploaded by Ruizhi Liu
Abstract. 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.
PF:2026.000007 · immutable v1 · math.GR — Group Theory · Uploaded by Mattia Brescia
Abstract. 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.
PF:2026.000006 · immutable v1 · math.GR — Group Theory · Uploaded by Marco Trombetti
Abstract. 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.
PF:2026.000005 · immutable v1 · math.GR — Group Theory · Uploaded by Marco Trombetti
Abstract. 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.