arXiv subjects

Junkai Qiu

Publications and source records attributed to Junkai Qiu.

3 recordsLinked to original sources

arXiv2026

Exact Decision and a Surjective Stabilizer Atlas for Affine Coprime-Factor Segments

We consider simultaneous internal stabilization of square proper real-rational plants of arbitrary but fixed finite input-output dimension $n$, arranged as an affine right-coprime factor segment, under explicit nondegeneracy hypotheses. After a classical normalization, the remaining global spectral-cut constraint and the reconstruction constraints at finite poles and at infinity are encoded, without loss, as a finite-dimensional semialgebraic seed, using a function-level double-Cayley factorization. For input coefficients in an effectively presented real closed field, that seed yields an exact existence decision that does not enumerate controller McMillan degree; the decision engine is classical quantifier elimination. The same encoding, now ranging over rational radii, admissible seeds, and residual unimodular and rational Schur parameters, yields a surjective atlas of all proper real-rational common stabilizers of the segment. The results concern this structured class and are compatible with the rational undecidability of simultaneous stabilization of three general plants.

math.OC↗

arXiv2026

A counterexample to vertex interpolation by planar $C^1$ cubic splines

We construct a nondegenerate conforming straight-line triangulation of a polygonal disk on which some vertex data admit no continuously differentiable piecewise polynomial interpolant of total degree at most three. The triangulation has 41 vertices and 56 triangles, and the graph induced by its interior vertices is a tree with eight arms of length two. Every cubic $C^1$ spline on this triangulation satisfies an explicit linear relation with integer coefficients among its vertex values. We derive the relation by a weighted sum of Bernstein--B\'ezier smoothness equations and verify it using rational coordinates and weights. This disproves Alfeld's conjecture on vertex interpolation. The nonexistence proof does not require a matrix rank computation. A separate computation in exact arithmetic gives rank 40 for the vertex evaluation map and dimension 107 for the spline space, attaining the classical dimension lower bound for this triangulation.

math.NA↗

arXiv2026

Infinite rational distance sets in affine general position: constructions in every dimension

For every integer $d\geq 1$, we construct a countably infinite set $X_d\subset\mathbb{R}^d$ in affine general position, with all pairwise distances rational. When $d$ is odd, $X_d$ may also be chosen so that no $d+2$ points lie on a common sphere. The construction is uniform in $d$: positive Chebyshev square decompositions produce harmonic curves on spheres whose points corresponding to rational parameter values have pairwise rational distances. A divided-difference factorization of the affine determinant shows that sufficiently short arcs are locally convex, and stereographic projection produces the odd-dimensional examples. We also construct infinite rational distance sets in $\mathbb{Q}^d$ in affine general position for every even $d$, and in general position for every $d\equiv 1\pmod 4$. For every $d\geq 1$ and $n\geq d+1$, taking and rescaling suitable finite subsets gives $n$-point integral point sets in affine general position. A suitable ordered choice yields integral-distance realizations of all cyclic polytopes. In dimension three, we give an explicit rational parametrization and obtain infinitely many pairwise non-similar primitive $n_3$-clusters for every $n\geq 4$.

math.CO↗