arXiv subjects

Diana Schneiderová

Publications and source records attributed to Diana Schneiderová.

6 recordsLinked to original sources

arXiv2026

Stable $q$-Hermitian coordinate calculus: Capelli--PBW transport, mixed polarization algebra, and localization obstructions

We extend a finite-support $q$-Hermitian radial calculus to Clifford-valued coordinate polynomials. For $N$ bosonic Hermitian vector labels and finite fermionic support, the stable range $m\ge 2N$ admits a canonical Gram--harmonic normal form by Howe separation. A label--Capelli inverse normalizes the Clifford contractions, while a finite Wick--Chevalley conjugation lifts the divided-power scalar gauge to the full Clifford PBW module. Coupling the normalized contractions to contractible fermionic seed complexes gives two conjugate polynomial-preserving $q$-Hermitian coordinate families. They are square-zero, anticommute within each polarization, restrict exactly to the radial PBW calculus, have scalar boundary trace $[2m-2n]_q$, and recover the classical Hermitian super Dirac pair as $q\to1$. For the mixed polarizations we introduce a relative triangular transport. The Grassmann relations for the normalized Capelli contractions hold on the full coordinate module and yield a labelwise factorization. Hence all higher-filtration mixed terms are finite subset products of explicit local defects, and the mixed polarization algebra closes for arbitrary finite label sets and finite fermionic support. The classical complex structure is common to both polarizations, whereas the two full transported Clifford--Weyl products are distinct for $0<q<1$. We also show that the fermionic Cartan denominator is forced in the natural polynomial first-order Berezin--Weyl class and determine the minimal factorwise Ore localization needed for the conjugate all-charge homotopies. The $q$-dependent Cartan orbit is nonresonant for $m\ge\max\{2N,n\}$. Finally, fixed finite-order differential and finite nonzero-shift realizations on the undeformed coordinate algebra are ruled out.

math.CV↗

arXiv2026

A Chain- and Diagram-Level Semantics for Morphological Calculus Refinement, monodromy, and bivector orbit decompositions

Morphological calculus represents decompositions of geometric objects by polynomial-like expressions in a symbol for the real line, but scalar operations suppress the incidence and attachment maps needed to reconstruct the space. We formulate a finite semantics using script chain complexes. The cell-count polynomial satisfies $$ \mathcal M_S(t)=\mathcal P_{S,\mathbb K}(t)+(1+t)\mathcal B_{S,\mathbb K}(t), $$ where $\mathcal P$ is the Poincar'e polynomial and $\mathcal B$ records boundary ranks. Over $\mathbb Z$, Smith labels separate unit pairs, representing refinement overhead, from non-unit pairs carrying torsion. We distinguish Cartesian products from bundles, derive the monodromy defect for mapping tori, and replace scalar gluing by a finite bar construction. Mapping cones, joins, and double mapping cylinders arise as reduced models. We apply the construction to bivector discrepancies in Sommen's calculations. In dimension four, Hodge decomposition identifies the unit bivector sphere with $S^2*S^2$; the angular excess is a contractible relative complex from inserting the rank-two midpoint. In dimension five, the $SO(5)$-action on $S(Λ^2\mathbb R^5)=S^9$ has principal orbit $SO(5)/T^2$ and singular orbits $\widetilde G_2(\mathbb R^5)$ and $\mathbb{CP}^3$. For standard Bruhat cell structures, the reduced double-mapping-cylinder inventory differs from sphere homology by seven unit-labelled pairs. We also treat Borel--Moore realizations of selected noncompact symbols, define division as an action-certified partial operation, and describe the orbit-type face diagram for unit bivectors in dimension six. There the correction polynomial has nonnegative coefficients, while compatibility of the face attachments remains open. These examples separate subdivision and attachment data from quotient actions and support conditions suppressed by scalar notation.

math.AG↗

arXiv2026

Optimal location of small favourable regions for Robin eigenvalues with indefinite weights

We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang--bang indefinite weight $κ\mathbf 1_D-\mathbf 1_{Ω\setminus D}$, and ask where a favourable region $D$ of prescribed small volume $|D|=δ$ should be located. Put $\varepsilon=δ^{1/N}$ and $τ_δ=α_δ\varepsilon$. We prove that there is a finite threshold $τ_*=τ_*(N,κ)$, independent of the ambient domain, such that $δ^{2/N}Λ_δ(α_δ)\toΛ_{\mathbb H}(τ)$ whenever $τ_δ\toτ<\infty$. If $τ<τ_*$, optimal small regions concentrate at the boundary. If $τ>τ_*$, their concentration centres move to distances much larger than $\varepsilon$ from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At $τ=τ_*$, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime $τ_δ=τ_*+σ\varepsilon+o(\varepsilon)$, we determine the first-order competition between the boundary and interior configurations. Tangential symmetry of compact threshold optimisers reduces the geometric correction to mean curvature. In particular, every fixed finite Robin coefficient is asymptotically in the boundary regime. Numerical experiments illustrate the boundary--interior transition, the three cases in the first-order selection law, and the curvature-dependent boundary location; for $N=2$ and $κ=1$ they suggest a transition near $τ_*\approx3.2$.

math.AP↗

arXiv2026

Discrete q-Hermitian Clifford analysis

We develop a discrete $q$-Hermitian Clifford calculus based on coordinatewise Jackson differences on multiplicative $q$-lattices. The resulting Hermitian Jackson--Dirac operators are nilpotent, factor a $q$-Laplacian, and have occupancy-dependent Euler anticommutators. These give explicit Fischer projectors and homotopies, but the local calculus is not controlled by total degree. We prove a two-face Cauchy--Kovalevskaya theorem for polynomials and extend it to a $q$-analytic Jackson--Fischer class. A divided-power conjugation recovers scalar Euler relations and transfers the classical joint Fischer decomposition and dimension formulas. The Jackson--Fischer completion is a vector-valued $q$-Fock space whose polarized nullspaces have projected reproducing kernels; the joint kernel is obtained by alternating projections. We also determine the linear symmetry group of the multiplicative lattice.

math.CV↗

arXiv2026

One-dimensional optimisation of indefinite-weight principal eigenvalues with asymmetric Robin parameters and a Schrödinger-type perturbation

We study the minimisation of the positive principal eigenvalue for an indefinite-weight problem with asymmetric Robin parameters. The model is motivated by diffusive logistic equations in spatially heterogeneous environments, where the weight describes allocatable favourable resources and the Robin parameters measure boundary loss. After recalling the variational setting and the bang--bang reduction, we analyse the one-dimensional optimisation problem: the optimal favourable set is an interval, and the placement problem is reduced to a branchwise criterion. The key analytical tool is a shape-derivative formula for $a\mapstoλ(a)$, which shows that interior candidates are characterised by equality of the endpoint values of the positive eigenfunction, equivalently by the coupled transfer-matrix equations $f=0$ and $g=0$. We also introduce a Schrödinger-type extension with a fixed nonnegative background potential. In the coercive case we establish the corresponding principal-eigenvalue and bang--bang results, and in one dimension with constant potential we prove a compactness-type stability result showing that minimisers for small background potential converge, along subsequences, to minimisers of the unperturbed problem. No placement classification is claimed for general positive background potential. The computations are presented as numerical illustrations generated with an adaptive root-search protocol.

math.SP↗

arXiv2026

Asymptotic minimization of the first Dirichlet eigenvalue in a square with two hard obstacles

We study an asymptotic minimization problem for the first Dirichlet eigenvalue in the square $Q=(-1,1)^2$, where two equal hard circular obstacles of radius $r$ move inside $Q$ with disjoint interiors; tangency between the obstacles and with the outer boundary is allowed. We prove that, as $r\to0$, every minimizing configuration approaches, up to the symmetries of the square and interchange of the holes, a pair of corner-tangent obstacles at adjacent corners, with displacement $o(r)$ from the corresponding corner-tangent centers. A uniform $u$-capacity localization argument shows that any obstacle whose individual eigenvalue excess is $O(r^4)$ must lie in a fixed $O(r)$ neighborhood of a corner, thereby excluding interior, open-side, and intermediate approaches to the boundary. At the corner scale, the leading coefficient is the full $u$-capacity. The proof is then organized by geometric branches. The side-tangent one-hole branch is asymptotically driven to the adjacent corner; an exact polarization argument shows that the adjacent corner-tangent pair has smaller first eigenvalue than the opposite pair; and configurations concentrated near a single corner are excluded by a scalar one-hole comparison. Additivity for well-separated corner defects then completes the distinct-corner analysis. We also include reproducible finite element computations that support the branch ordering predicted by the proof.

math.AP↗