Abstract. Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification.
We prove that if
f
f
is a reduced homogeneous polynomial of degree
d
d
, then its
F
F
-pure threshold at the unique homogeneous maximal ideal is at least
1
d
−
1
\frac {1}{d-1}
. We show, furthermore, that its
F
F
-pure threshold equals
1
d
−
1
\frac {1}{d-1}
if and only if
f
∈
m
[
q
]
f\in \mathfrak m^{[q]}
and
d
=
q
+
1
d=q+1
, where
q
q
is a power of
p
p
. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such “extremal singularities”, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are “extremal”, for example, in terms of the configurations of lines they can contain.
We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.
Let R be a standard graded polynomial ring that is finitely generated over a field, and let I be a homogenous prime ideal of R. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of R/I t , as t grows arbitrarily large. Such rings are known as thickenings of R/I. We consider R = F[X] where F is a field of characteristic 0, X is a 2 × m matrix, and I is the ideal generated by size two minors. We give concrete constructions for the local cohomology modules of thickenings of R/I. Bizarrely, these local cohomology modules can be described using the Taylor series of natural log.
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