Hexagon

Submitted works

math.CT — Category Theory

The third unit in the adjoint tower of group involutions

Contributed by Zhenyu Sun

The functor $G\mapsto\operatorname{Hom}(C_2,G)$ determines an adjoint tower through successive Eilenberg--Moore comparison functors. We prove that the unit of the third adjunction need not be an isomorphism, answering a question of Bergman. For the generalized dihedral group $H=(C_4\times C_4)\rtimes C_2$ , with inversion action, we compute $F_3V_3H\cong H\times C_2$ , with counit the first projection. The unit at $V_3H$ is injective, with underlying domain and codomain of cardinalities $20$ and $40$ . For an arbitrary group $G$ , let $E(G)=F_2V_2G$ and let $K_G$ be the kernel of the second counit. We prove that $K_G$ is central and that $F_3V_3G\cong E(G)/K_G[2]$ , where $K_G[2]$ consists of the elements of $K_G$ whose square is one. The third unit at $V_3G$ is an isomorphism precisely when $K_G$ has no element of order four. A central extension of $H$ of order $128$ supplies such an element and yields the stated description of $F_3V_3H$ .

A mix of human-written and AI-generated textHuman understanding: all partsadjoint towercentral extensiongeneralized dihedral groupinvolutory quandlemonadic decomposition

Oct 20, 2026PDF ↗View work →

hexagon:2610.00410v1

math.LO — Logic

Relative $P$ -Ideals of Hypergraphs and Ultrafilter Spectra

Contributed by Zhenyu Sun

We study ideals of hypergraphs of finite uniformity under countable upper bounds modulo finite vertex covers. The canonical ideal generated by a free filter and finite weak colorings has this relative $P$ -property exactly when the filter is the intersection of finitely many $P$ -points. The classification is preserved by adjoining hypergraphs that are locally finite after a finite vertex deletion. The corresponding extension of a trace ideal induces capture ideals on increasing sequences. These are $\sigma$ -ideals exactly for $P$ -filters, and their uniformities are $\max\{\pi\chi(\mathcal{F}),\mathfrak{d}\}$ . For an ultrafilter $\mathcal{U}$ , the arity-two capture ideal equals the Louveau ideal exactly when $\mathcal{U}$ is a $P$ -point. Successive capture ideals decrease with the arity, and equality characterizes $Q$ -points. Every strict inclusion is witnessed by a closed set. Thus a non- $Q$ $P$ -point yields a decreasing sequence of $\sigma$ -ideals, distinguished by closed sets, with equal uniformities. We also establish invariance under locally finite witness enlargements and analyze Stone kernels and countable block products. Consistently, nested ray-positive $P$ -sets have the same singleton endpoint spectrum and different ray transversal numbers.

A mix of human-written and AI-generated textHuman understanding: all partsHypergraph idealLouveau idealP-pointQ-pointStone dualityreaping numberultrafilter

Oct 20, 2026PDF ↗View work →

hexagon:2610.00406v1

math.OC — Optimization and Control

Classical regularity and attainment in portfolio choice with nonhomothetic utility

Contributed by Yuanhong Wei, Zhenyu Sun

We study classical regularity and attainment in terminal-utility portfolio choice with unspanned factors, nonnegative wealth, and exposures that are square-integrable in expectation. A joint estimate for logarithmic marginal utility controls relative risk aversion and factor sensitivity. In several dimensions, a risk-price component uniformly separated from the factor-hedging span gives classical solvability and a unique strong optimal portfolio on every finite horizon. A scalar comparison equation treats correlated single-factor markets. For state-independent $C^4$ utility, the preference assumptions reduce to positive lower and upper bounds on relative risk aversion. In an orthogonal single-factor market, a uniform finite-order condition allows the risk price to vanish and change sign. Subelliptic gradient estimates and a parabolic equation in the factor variable yield a classical value; verification and strict concavity give strong existence for its continuous feedback. In both settings, the optimal dual coefficient has a deterministic bound uniform over initial states and multipliers. Attainment has a separate integrability boundary. For deterministic opportunities with a finite static optimum, the prescribed strategy class attains the value exactly when the optimal payoff lies in $L^2$ under the physical probability. An explicit nonhomothetic utility gives a horizon and wealth classification with smooth finite values and nonattainment. Further results identify wealth-dependent purchasing-power hedges and a strict increase in the dual value under factor-information restrictions.

Primarily human-written textHuman understanding: all partsNonhomothetic utilityclassical regularityconvex dualitydegenerate HJB equationssquare-integrable strategiesstochastic factors

Oct 20, 2026PDF ↗View work →

hexagon:2610.00395v1

math.OC — Optimization and Control

Additive Rounding for LP-Tight Three-Item Bin Packing

Contributed by Ruixue Yuan, Da Teng, Zhenyu Sun

We consider bin-packing instances with $3m$ items of sizes in $(1/4,1/2)$ , total size $m$ , and configuration linear programming value $m$ . Let $d$ be the relative dimension of the polytope of zero-sum multiplicity patterns obtained by centering sizes at $1/3$ . We determine the maximum additive integrality gap for every $d\le10$ . It is zero for $d\le4$ and one for $5\le d\le10$ . More generally, a representation of the multiplicity vector by at most five zero-sum patterns admits an integral packing, and a representation by at most eleven patterns requires at most one additional bin. The integral-packing result has a structural proof; the additive-one result combines structural restrictions with exact finite enumeration. For arbitrary $d$ , we construct an LP-tight subinstance with at most $6d+3$ items and no smaller additive gap. An integer representation preserving the signs of triple sums then yields a finite rational family attaining the worst gap, which is computable in time $2^{O((d+1)^2)}$ . Finally, we construct a sharp family with simplex pattern polytopes and a linearly growing zero-sum matching deficit. Its eleven-dimensional member violates the zero-sum remainder condition used in the ten-dimensional proof. Thus the bound of eleven fractional patterns in that condition is sharp.

Primarily human-written textHuman understanding: all partsbin packingconfiguration linear programminghypergraph matchinglattice polytopeszero-sum patterns

Oct 20, 2026PDF ↗View work →

Submitted works

The third unit in the adjoint tower of group involutions

Relative PPP-Ideals of Hypergraphs and Ultrafilter Spectra

Classical regularity and attainment in portfolio choice with nonhomothetic utility

Additive Rounding for LP-Tight Three-Item Bin Packing

Relative $P$ -Ideals of Hypergraphs and Ultrafilter Spectra