arXiv subjects

Bryce Christopherson

Publications and source records attributed to Bryce Christopherson.

3 recordsLinked to original sources

arXiv2026

A Refinement in \v{C}ech Cohomology of Coron's Necessary Condition

Coron established a homological obstruction to continuous feedback stabilization of nonlinear control systems $\dot{x}=f(x,u)$ with $f \in C(\Omega,\mathbb{R}^n)$ and $f(0,0)=0$, showing that local asymptotic stabilizability implies the induced homomorphism $f_*$ satisfies $f_*\big(H_{n-1}(\Sigma_\epsilon)\big)=H_{n-1}(S^{n-1})$, where $\Sigma_\epsilon:=\Big(\big(\mathbb{B}_\epsilon^{\mathbb{R}^n}(0)\times\mathbb{B}_\epsilon^{\mathbb{R}^m}(0)\big)\cap \Omega\Big)\setminus f^{-1}(0)$. In this paper, we refine Coron's necessary condition using \v{C}ech cohomology and the Vietoris-Begle mapping theorem. Specifically, we prove that the closed version of $\Sigma_\epsilon$ must be a \v{C}ech cohomology $(n-1)$-sphere and that the restriction of $f$ to this subset induces an isomorphism on its \v{C}ech cohomology groups in all degrees. This strengthens Coron's condition from a constraint on the top class to a full cohomological rigidity statement.

math.OC↗

arXiv2026

Brockett Openness Profiles and Gain-Limited Feedback Stabilization

Brockett's necessary condition asserts that a continuously stabilizable nonlinear control system must have a vector field that is open at the equilibrium. We show that the quantitative data behind this openness condition constrains the possible growth of stabilizing feedbacks. To a system vector field $f$, we associate its openness profile $\Omega_f(r)=\sup\{\rho:\mathbb{B}_\rho(0)\subset f(\mathbb{B}_r(0,0))\}$, so that Brockett's condition becomes $\Omega_f(r)>0$ for all sufficiently small $r>0$. If a feedback $u$ satisfies $\|u(x)\|\leq d(\|x\|)$, then the openness profile of the closed-loop field $F_u(x)=f(x,u(x))$ satisfies $\Omega_{F_u}(r)\leq \Omega_f\!\left(\sqrt{r^2+d(r)^2}\right)$. Consequently, any prescribed lower openness rate for the closed-loop dynamics yields a necessary lower bound on the feedback growth. For systems with $\Omega_f(r)\lesssim r^q$, linear-rate closed-loop openness forces $d(r)\gtrsim r^{1/q}$, and this exponent is sharp in elementary polynomial examples. Thus Brockett's condition is not merely a binary topological obstruction; its quantitative profile governs gain requirements for stabilizing feedback.

math.OC↗

arXiv2025

Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors

The Ramsey number $R(s,t)$ is the least integer $n$ such that any coloring of the edges of $K_n$ with two colors produces either a monochromatic $K_s$ in one color or a monochromatic $K_t$ in the other. If $s=t$, we say that the Ramsey number $R(s,s)$ is diagonal. The threshold Ramsey multiplicity of a diagonal Ramsey number $R(s,s)$, denoted $m(s,s)$ or $m_2(s)$, is the smallest number of copies of a monochromatic $K_s$ that can be found in any coloring of the edges of $K_{R(s,s)}$. For instance, $m_2(2)=1$, $m_2(3)=2$, and $m_2(4)=9$. We derive upper bounds for multicolor, off-diagonal threshold Ramsey multiplicities. In the diagonal two-color case, the resulting bounds improve the elementary random-coloring estimate for $5\leq s\leq8$. In particular, we recover the known value $m(3,3,3)=5$ and obtain the bound $m(3,3,4)\leq 10$. We conclude with a general framework for seeking further improvements.

math.CO↗