arXiv subjects

Zongjian Han

Publications and source records attributed to Zongjian Han.

7 recordsLinked to original sources

arXiv2026

Rough paths below the Young threshold: an exact scale calculus and the locality phase transition at one quarter

Since Young's 1936 theorem, irregular integration has been organized around the threshold 1/2: above it the path determines the integral, while below it higher-order data are needed. For fractional Brownian motion, H = 1/4 is the threshold for the canonical Gaussian enhancement, although geometric rough lifts exist for every H > 0. We prove that H = 1/4 is instead the exact threshold for measurable locality. For d-dimensional fractional Brownian motion with independent components, d at least 2, if 0 < H <= 1/4, no positive-measure Borel set supports even one finite off-diagonal second-level coordinate satisfying Chen's relation and measurable interval by interval from path increments. No moment, Holder, geometricity, stationarity, or scaling assumption is imposed. If 1/4 < H <= 1/2, every full-law local rough-path lift in the standard Holder range is automatically square-integrable and hence classified without an assumed L2 condition; for H > 1/2, every finite-step enhancement with natural graded Holder bounds is uniquely the Young signature. We also introduce an exact scale calculus below classical differentiability. Matched dyadic differences recover normalized derivatives with sharp O(epsilon^2) error, exact localized inversion, and lossless reconstruction. Multiplication and smooth functional calculus transport exactly to scale coordinates, while the corona quotient yields an exact universal derivation. Reinserting the Holder amplitude and using Fourier-normal ordering produces strong geometric lifts for every positive input regularity and every lower rough exponent, with explicit ultraviolet rates and stability. Thus rough lifts exist below one quarter, but no measurable lift on a positive-measure domain can be interval-local there. The results separate existence from local recoverability and show that nondifferentiability does not destroy exact differential information.

math.FA↗

arXiv2026

The completion of a continuous inverse algebra need not be a continuous inverse algebra

In 2006 Neeb asked whether the Hausdorff completion of a continuous inverse algebra must again be a continuous inverse algebra. The noncommutative case remained open, while the commutative case was known to be true. We give a negative answer after twenty years. We construct a Hausdorff metrizable locally m-convex complex continuous inverse algebra whose completion is a Fr\'echet locally m-convex algebra, where inversion stays continuous but the set of invertible elements is not open. The construction uses finite-support sequences in a dense nil subalgebra of a Jacobson-semisimple Banach algebra. Finite support makes every element nilpotent, giving continuous inversion without a locally uniform bound on nilpotence indices. In the completion, shifting a fixed noninvertible element to later and later coordinates produces noninvertible elements converging to the identity. Thus completion destroys exactly the local spectral stability at the identity. The counterexample is necessarily noncommutative and marks the precise boundary of the commutative completion theorem.

math.FA↗

arXiv2026

A Counterexample To Universal Character Density For Compact Quantum Groups

In 1987, Woronowicz asked whether the linear span of irreducible characters is norm dense in the cocommutative part of the ambient C-star algebra of a compact quantum group. After thirty-nine calendar years, we give a negative answer to the universal form of this question, even for compact matrix quantum groups of Kac type. Starting from an infinite finitely generated simple group with property T and a finite bicharacter twist, we construct a compact quantum group whose universal C-star algebra is a full group C-star algebra. A Kazhdan projection is shown to remain cocommutative under the twisted coproduct, while a vector state arising from an induced representation separates this projection from every algebraic cocommutative element. Quantitatively, the distance from the Kazhdan projection to the closed linear span of irreducible characters is at least one half. Moreover, the projection lies in the kernel of the reducing morphism. The obstruction is therefore genuinely universal and disappears after passage to the reduced compact quantum group, where the known character-density theorem remains valid. The construction identifies a sharp boundary between universal and reduced character theory and shows that Kac symmetry alone does not control cocommutative elements in the universal completion.

math.QA↗

arXiv2026

Completeness of the Model Space Does Not Force Regularity for Infinite-Dimensional Lie Groups

We give negative solutions to the two basic completeness problems for locally convex Lie groups: whether a Lie group modelled on a Mackey-complete locally convex space must be regular, and whether it must at least possess a smooth exponential function. Both counterexamples have complete model spaces. We first construct a contractible complex analytic BCH--Lie group $H$, modelled on the complete Silva space $\mathbb C^{(\mathbb N)}$, for which $\exp_H$ is a global homeomorphism although $H$ is not $C^0$-semiregular. The failure is witnessed by smooth controls converging to zero in a fixed finite-dimensional subspace; their evolutions exist uniquely on $[0,1)$ but have no endpoint at time $1$. The obstruction is a multiplicative graded escape in the principal unit group of a complete continuous inverse algebra. We then suspend one such control: the translation action on $C^\infty(S^1,H)$ converts the time-dependent obstruction into a single element of a semidirect-product Lie algebra. The resulting Lie group is modelled on a complete Hausdorff locally convex space and contains an element which generates no one-parameter subgroup. Hence it admits no exponential function. Thus completeness forces neither non-autonomous evolution nor autonomous exponentiation.

math.DG↗

arXiv2025

Preventing Model Collapse via Contraction-Conditioned Neural Filters

This paper presents a neural network filter method based on contraction operators to address model collapse in recursive training of generative models. Unlike \cite{xu2024probabilistic}, which requires superlinear sample growth ($O(t^{1+s})$), our approach completely eliminates the dependence on increasing sample sizes within an unbiased estimation framework by designing a neural filter that learns to satisfy contraction conditions. We develop specialized neural network architectures and loss functions that enable the filter to actively learn contraction conditions satisfying Assumption 2.3 in exponential family distributions, thereby ensuring practical application of our theoretical results. Theoretical analysis demonstrates that when the learned contraction conditions are satisfied, estimation errors converge probabilistically even with constant sample sizes, i.e., $\limsup_{t\to\infty}\mathbb{P}(\|\mathbf{e}_t\|>\delta)=0$ for any $\delta>0$. Experimental results show that our neural network filter effectively learns contraction conditions and prevents model collapse under fixed sample size settings, providing an end-to-end solution for practical applications.

cs.LG↗

arXiv2024

Rota-Baxter operators, differential operators, pre- and Novikov structures on groups and Lie algebras

Rota-Baxter operators on various structures have found important applications in diverse areas, from renormalization of quantum field theory to Yang-Baxter equations. Relative Rota-Baxter operators on Lie algebras are closely related to pre-Lie algebras and post-Lie algebras. Some of their group counterparts have been introduced to study post-groups, skew left braces and set-theoretic solutions of Yang-Baxter equations, but searching suitable notions of relative Rota-Baxter operators on groups with weight zero and pre-groups has been challenging and has been the focus of recent studies, by provisionally imposing an abelian condition. Arising from the works of Balinsky-Novikov and Gelfand-Dorfman, Novikov algebras and their constructions from differential commutative algebras have led to broad applications. Finding their suitable counterparts for groups and Lie algebras has also attracted quite much recent attention. This paper uses one-sided-inverse pairs of maps to give a perturbative approach to a general notion of relative Rota-Baxter operators and differential operators on a group and a Lie algebra with limit-weight. With the extra condition of limit-abelianess on the group or Lie algebra, we give an interpretation of relative Rota-Baxter and differential operators with weight zero. These operators motivate us to define pre-groups and Novikov groups respectively as the induced structures. The tangent maps of these operators are shown to give Rota-Baxter and differential operators with weight zero on Lie algebras. The tangent spaces of the pre-Lie and Novikov Lie groups are pre-Lie algebras and Novikov Lie algebras, fulfilling the expected property. Furthermore, limit-weight relative Rota-Baxter operators on groups give rise to skew left braces and then set-theoretic solutions of the Yang-Baxter equation.

math.QA↗

arXiv2024

Rota-Baxter groups with weight zero and integration on topological groups

Rota-Baxter groups with weights $\pm 1$ have attracted quite much attention since their recent introduction, thanks to their connections with Rota-Baxter Lie algebras, factorizations of Lie groups, post- and pre-Lie algebras, braces and set-theoretic solutions of the Yang-Baxter equation. Despite their expected importance from integrals on groups to pre-groups and Yang-Baxter equations, Rota-Baxter groups with weight zero and other weights has been a challenge to define and their search has been the focus of several attempts. By composing an operator with a section map as a perturbation device, we first generalize the notion of a Rota-Baxter operator on a group from the existing case of weight $\pm 1$ to the case where the weight is given by a pair of maps and then a sequence limit of such pairs. From there, two candidates of Rota-Baxter operators with weight zero are given. One of them is the Rota-Baxter operator with limit-weight zero detailed here, with the other candidate introduced in a companion work. This operator is shown to have its tangent map the Rota-Baxter operator with weight zero on Lie algebras. It also gives concrete applications in integrals of maps with values in a class of topological groups called $\RR$-groups, satisfying a multiplicative version of the integration-by-parts formula. In parallel, differential groups in this framework is also developed and a group formulation of the First Fundamental Theorem of Calculus is obtained.

math.QA↗