Nonradial sign changing solutions to Lane Emden equation

In this paper we prove the existence of continua of nonradial solutions for the Lane-Emden equation. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane Emden problem has a very rich and complex structure.Comment: 13 p

Stable fixed points of combinatorial threshold-linear networks

Combinatorial threshold-linear networks (CTLNs) are a special class of neural networks whose dynamics are tightly controlled by an underlying directed graph. In prior work, we showed that target-free cliques of the graph correspond to stable fixed points of the dynamics, and we conjectured that these are the only stable fixed points allowed. In this paper we prove that the conjecture holds in a variety of special cases, including for graphs with very strong inhibition and graphs of size $n \leq 4$. We also provide further evidence for the conjecture by showing that sparse graphs and graphs that are nearly cliques can never support stable fixed points. Finally, we translate some results from extremal combinatorics to upper bound the number of stable fixed points of CTLNs in cases where the conjecture holds.Comment: 22 pages, 5 figure