A Structured Systems Approach for Optimal Actuator-Sensor Placement in Linear Time-Invariant Systems

In this paper we address the actuator/sensor allocation problem for linear time invariant (LTI) systems. Given the structure of an autonomous linear dynamical system, the goal is to design the structure of the input matrix (commonly denoted by $B$) such that the system is structurally controllable with the restriction that each input be dedicated, i.e., it can only control directly a single state variable. We provide a methodology that addresses this design question: specifically, we determine the minimum number of dedicated inputs required to ensure such structural controllability, and characterize, and characterizes all (when not unique) possible configurations of the \emph{minimal} input matrix $B$. Furthermore, we show that the proposed solution methodology incurs \emph{polynomial complexity} in the number of state variables. By duality, the solution methodology may be readily extended to the structural design of the corresponding minimal output matrix (commonly denoted by $C$) that ensures structural observability.Comment: 8 pages, submitted for publicatio

On linear series with negative Brill-Noether number

Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basic object of study is $\mathcal{W}^r_{d,g}$, the moduli space of smooth genus $g$ curves with a choice of degree $d$ line bundle having at least $(r+1)$ independent global sections. The Brill-Noether theorem asserts that the map $\mathcal{W}^r_{d,g} \rightarrow \mathcal{M}_g$ is surjective with general fiber dimension given by the number $\rho = g - (r+1)(g-d+r)$, under the hypothesis that $0 \leq \rho \leq g$. One may naturally conjecture that for $\rho < 0$, this map is generically finite onto a subvariety of codimension $-\rho$ in $\mathcal{M}_g$. This conjecture fails in general, but seemingly only when $-\rho$ is large compared to $g$. This paper proves that this conjecture does hold for at least one irreducible component of $\mathcal{W}^r_{d,g}$, under the hypothesis that $0 < -\rho \leq \frac{r}{r+2} g - 3r+3$. We conjecture that this result should hold for all $0 < -\rho \leq g + C$ for some constant $C$, and we give a purely combinatorial conjecture that would imply this stronger result.Comment: 16 page

On defect groups for generalized blocks of the symmetric group

In a paper of 2003, B. K\"ulshammer, J. B. Olsson and G. R. Robinson defined $\ell$-blocks for the symmetric groups, where $\ell >1$ is an arbitrary integer. In this paper, we give a definition for the defect group of the principal $\ell$-block. We then check that, in the Abelian case, we have an analogue of one of M. Brou\'e's conjectures.Comment: 18 page

Theorems, Problems and Conjectures

These notes are designed to offer some (perhaps new) codicils to related work, a list of problems and conjectures seeking (preferably) combinatorial proofs. The main items are Eulerian polynomials and hook/contents of Young diagram, mostly on the latter. The new additions include items on Frobenius theorem and multi-core partitions; most recently, some problems on (what we call) colored overpartitions. Formulas analogues to or in the spirit of works by Han, Nekrasov-Okounkov and Stanley are distributed throughout. Concluding remarks are provided at the end in hopes of directing the interested researcher, properly.Comment: 14 page