arXiv subjects

Per Åhag

Publications and source records attributed to Per Åhag.

12 recordsLinked to original sources

arXiv2026

Square Functions, Complete Crouzeix Conjecture in Dimension Three, and the Clouâtre-Ostermann-Ransford conjecture

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre-Ostermann-Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. We also obtain sharp column and row square-function inequalities, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

math.CV↗

arXiv2026

On $(p,q)$-binomial coefficient ratios for complex parameters

We prove local asymptotics for near-central complex $(p,q)$-binomial coefficient moduli ratios allowing an imaginary parameter perturbation of order $n^{-3/4}$ at a $\sqrt{n}$ length scale from the centre. Moreover, we obtain ratio asymptotics for a smaller imaginary perturbation of order $n^{-5/4}$ at the length scale $n^{3/4}$. These results were obtained by reducing the two-parameter coefficients to just one parameter, giving a branch-free logarithmic representation of the second-order ratio and, hence, uniform complex curvature asymptotes for near-central ratios.

math.CA↗

arXiv2026

Quartic Fourier-Laplace limits for $(p,q)$-Rogers-Szegő polynomials

We prove asymptotic formulae for positive real $(p,q)$-Rogers-Szegő polynomials when the quadratic term in the expansion about the middle coefficient vanishes. For coefficient indices whose distance from $n/2$ is of order $n^{3/4}$, the coefficient ratios have a quadratic-quartic exponential limit. A uniform bound valid for all indices allows the ratios to be summed and gives locally uniform convergence of the centred generating polynomials to a Fourier--Laplace integral. We also obtain an asymptotic formula for the sum of the coefficients, convergence of all rescaled moments, weak convergence of the normalised coefficient measures, and convergence with multiplicity of the zeros tending to $w=1$.

math.CA↗

arXiv2026

Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces

Let $H$ be a complex Hilbert space and let $Ω\subset H$ be a domain. In infinite dimensions, there is no canonical complex Monge--Ampère operator and no basis-free determinant of the Levi form. Hence, a determinant-type characterization of maximal plurisubharmonic functions is not immediate. We propose to use the normalized determinants of Fujii and Seo: for a bounded strictly positive operator $A$ and a unit vector $x\in H$, we set $Δ_x(A):=\exp\bigl(\langle (\log A)x,x\rangle\bigr)$, and we extend this naturally to non-invertible positive operators. We show that, for strictly positive operators, inequalities for $Δ_x$ precisely describe the chaotic order $\log A\ge \log B$, and we combine this observation with Kantorovich--Specht type bounds for positive operators. For $u\in \mathcal{PSH}(Ω)\cap C^2(Ω)$ we define the \emph{Fujii--Seo determinant density} \[ \operatorname{FSD}(u)(a):=\inf_{\|x\|=1}Δ_x\!\bigl(D'D''u(a)\bigr),\qquad a\inΩ, \] and identify it with the lower spectral endpoint $\infσ(D'D''u(a))$. Thus, $\operatorname{FSD}(u)$ is precisely the infimum of the spectrum of the Levi form, and its vanishing gives a basis-independent criterion for pointwise degeneracy of the Levi form. We prove that maximality implies $\operatorname{FSD}(u)\equiv 0$, give sufficient global degeneracy criteria for maximality, and establish several comparison principles for $C^2$ plurisubharmonic functions, including results under uniform ellipticity bounds on the Levi form.

math.CV↗

arXiv2026

PolyNODE: Variable-dimension Neural ODEs on M-polyfolds

Neural ordinary differential equations (NODEs) are geometric deep learning models based on dynamical systems and flows generated by vector fields on manifolds. Despite numerous successful applications, particularly within the flow matching paradigm, all existing NODE models are fundamentally constrained to fixed-dimensional dynamics by the intrinsic nature of the manifold's dimension. In this paper, we extend NODEs to M-polyfolds (spaces that can simultaneously accommodate varying dimensions and a notion of differentiability) and introduce PolyNODEs, the first variable-dimensional flow-based model in geometric deep learning. As an example application, we construct explicit M-polyfolds featuring dimensional bottlenecks and PolyNODE autoencoders based on parametrised vector fields that traverse these bottlenecks. We demonstrate experimentally that our PolyNODE models can be trained to solve reconstruction tasks in these spaces, and that latent representations of the input can be extracted and used to solve downstream classification tasks. The code used in our experiments is publicly available at https://github.com/turbotage/PolyNODE .

cs.LG↗

arXiv2025

Explicit Solutions in Isotropic Planar Elastostatics

Addressing the intricate challenges in plane elasticity, especially with non-vanishing traction and complex geometries, requires innovative methods. This paper offers a novel approach, drawing inspiration from the Neumann problem for the inhomogeneous Cauchy-Riemann equations. Our method applies to domains conformally equivalent to a unit disk or an annulus, focusing on deriving explicit solutions for the displacement field rather than the stress tensor, which distinguishes it from most traditional approaches. We explore solutions for specific classical cases to demonstrate its efficacy, such as a cardioid domain, a ring domain with a shifted hole, and a gear-like structure. This work enhances the toolkit for researchers and practitioners tackling isotropic planar elastostatic challenges with a unified and flexible approach.

cond-mat.mtrl-sci↗

arXiv2025

On manifold-like polyfolds as differential geometrical objects with applications in complex geometry

We argue for more widespread use of manifold-like polyfolds (M-polyfolds) as differential geometric objects. M-polyfolds possess a distinct advantage over differentiable manifolds, enabling a smooth and local change of dimension. To establish their utility, we introduce tensors and prove the existence of Riemannian metrics, symplectic structures, and almost complex structures within the M-polyfold framework. Drawing inspiration from a series of highly acclaimed articles by László Lempert, we lay the foundation for advancing geometry and function theory in complex M-polyfolds.

math.DG↗

arXiv2024

Continuity of Solutions to Complex Hessian Equations via the Dinew-Kołodziej Estimate

This study extends the celebrated volume-capacity estimates of Dinew and Kolodziej, providing a foundation for examining the regularity of solutions to boundary value problems for complex Hessian equations. By integrating the techniques established by Dinew and Kolodziej and incorporating recent advances by Charabati and Zeriahi, we demonstrate the continuity of the solutions.

math.CV↗

arXiv2024

Kiselman Minimum Principle and Rooftop Envelopes in Complex Hessian Equations

We initiate the study of $m$-subharmonic functions with respect to a semipositive $(1,1)$-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of $m$-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among $m$-subharmonic functions, expressed in terms of their asymptotic envelopes.

math.CV↗

arXiv2024

Geodesic connectivity and rooftop envelopes in the Cegrell classes

This study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in $\mathbb{C}^n$. We establish that solutions possessing comparable singularities to the complex Monge-Ampère equation are identical, affirmatively addressing a longstanding open question raised by Cegrell. This achievement furnishes the most general form of the Bedford-Taylor comparison principle within the Cegrell classes. Building on this foundational result, we explore plurisubharmonic geodesics, broadening the criteria for geodesic connectivity among plurisubharmonic functions with connectable boundary values. Our investigation also delves into the notion of rooftop envelopes, revealing that the rooftop equality condition and the idempotency conjecture are valid under substantially weaker conditions than previously established, a finding made possible by our proven uniqueness result. The paper concludes by discussing the core open problems within the Cegrell classes related to the complex Monge-Ampère equation.

math.CV↗

arXiv2023

Complex branches of a generalised Lambert $W$ function arising from $p,q$--binomial coefficients

The $ψ(x)$-function, which solves the equation $x = \sinh(aw)e^w$ for $0<a<1$, has a natural connection to the renowned Lambert $W$ function and also physical relevance through its connection to the Lenz-Ising model of ferromagnetism. We give a detailed analysis of its complex branches and construct Riemann surfaces from these under various conditions of $a$, unveiling intriguing new links to the Lambert $W$ function.

math.CV↗

arXiv2023

On a generalised Lambert $W$ branch transition function arising from $p,q$-binomial coefficients

With only a complete solution in dimension one and partially solved in dimension two, the Lenz-Ising model of magnetism is one of the most studied models in theoretical physics. An approach to solving this model in the high-dimensional case ($d>4$) is by modelling the magnetisation distribution with $p,q$-binomial coefficients. The connection between the parameters $p,q$ and the distribution peaks is obtained with a transition function $ω$ which generalises the mapping of Lambert $W$ function branches $W_0$ and $W_{-1}$ to each other. We give explicit formulas for the branches for special cases. Furthermore, we find derivatives, integrals, parametrizations, series expansions, and asymptotic behaviors.

math-ph↗