Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue ...measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and strong-faithfulness has been proposed and assumed to achieve uniform or high-dimensional consistency. In contrast to the plain faithfulness assumption, the set of distributions that is not strong-faithful has nonzero Lebesgue measure and in fact, can be surprisingly large as we show in this paper. We study the strong-faithfulness condition from a geometric and combinatorial point of view and give upper and lower bounds on the Lebesgue measure of strong-faithful distributions for various classes of directed acyclic graphs. Our results imply fundamental limitations for the PC-algorithm and potentially also for other algorithms based on partial correlation testing in the Gaussian case.
We provide a characterization of the non-singular Hermitian variety of PG(4,q2) as a hypersurface of degree q+1 over GF(q2) with q7+q5+q2+1 rational points, which does not contain linear subspaces of ...dimension greater than 1 and having exactly one line in common with at least a plane of PG(4,q2).
A compact set
E
⊂
C
N
satisfies the Markov inequality if the supremum norm on
E
of the gradient of a polynomial
p
can be estimated from above by the norm of
p
multiplied by a constant polynomially ...depending on the degree of
p
. This inequality is strictly related to the Bernstein approximation theorem, Schur-type estimates and the extension property of smooth functions. Additionally, the Markov inequality can be applied to the construction of polynomial grids (norming sets or admissible meshes) useful in numerical analysis. We expect such an inequality with similar consequences not only on polynomially determining compacts but also on some nowhere dense sets. The primary goal of the paper is to extend the above definition of Markov inequality to the case of compact subsets of algebraic varieties in
C
N
. Moreover, we characterize compact sets satisfying such a Markov inequality on algebraic hypersurfaces as well as on certain varieties defined by several algebraic equations. We also prove a division inequality (a Schur-type inequality) on these sets. This opens up the possibility of establishing polynomial grids on algebraic sets. We also provide examples that complete and ilustrate the results.
Darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces, called now the Darboux theory of ...integrability. In 1979 Jouanolou successfully improved the Darboux theory of integrability characterizing the existence of rational first integrals, for this he used sophisticated tools of algebraic geometry. The aim of this paper is to improve the classical result of Darboux and the new one of Jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces. Additionally our proof of the improved result of Jouanolou is much simpler and elementary than the original one. Some examples show that the improved Darboux theory of integrability with multiplicity is much useful than the classical one.
We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we ...prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.
This paper, based on Bihan and Sottile’s method which reduces a polynomial system to its Gale dual system and then bounds the number of solutions of this Gale system, proves that a real coefficient ...polynomial system with n equations and with n variables involving n+k+1 monomials has fewer than 27e53+8890∏i=0k−1(2i(n−1)+1) positive solutions and 27e103+8890∏i=0k−1(2i(n−1)+1) non-degenerate non-zero real solutions. This dramatically improves F. Bihan and F. Sottile’s bounds of e2+342k2nk and e4+342k2nk respectively. Using the new upper bound for positive solutions, we establish restrictions to the sum of the Betti numbers of real piecewise algebraic hypersurfaces and real piecewise algebraic curves. A new bound on the number of compact components of algebraic hypersurfaces in R>n is also given.
In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations in an arbitrary number of variables. The method uses rational parametrizations of ...algebraic (hyper)surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classes of equations having rational solutions. However, the method can also be used for finding non-rational solutions.
Darboux theory of integrability was established by Darboux in 1878, which provided a relation between the existence of first integrals and invariant algebraic hypersurfaces of vector fields in
R
n
or
...C
n
with
n
⩾
2
. Jouanolou 1979 improved this theory to obtain rational first integrals via invariant algebraic surfaces using sophisticated tools of algebraic geometry. Recently in J. Llibre, X. Zhang, Darboux theory of integrability in
C
n
taking into account the multiplicity, J. Differential Equations, in press this theory was improved taking into account not only the invariant algebraic hypersurfaces but also their multiplicity. In this paper we will show that if the hyperplane at infinity for a polynomial vector field in
R
n
has multiplicity larger than 1, we can improve again the Darboux theory of integrability. We also show some difficulties for obtaining an extension of this result to polynomial vector fields in
C
n
.
A real
n
-dimensional homogeneous polynomial
f
(
x
)
of degree
m
and a real constant
c
define an algebraic hypersurface
S
whose points satisfy
f
(
x
)
=
c
. The polynomial
f
can be represented by
A
x
...m
where
A
is a real
m
th order
n
-dimensional supersymmetric tensor. In this paper, we define rank, base index and eigenvalues for the polynomial
f
, the hypersurface
S
and the tensor
A
. The rank is a nonnegative integer
r
less than or equal to
n
. When
r
is less than
n
,
A
is singular,
f
can be converted into a homogeneous polynomial with
r
variables by an orthogonal transformation, and
S
is a cylinder hypersurface whose base is
r
-dimensional. The eigenvalues of
f
,
A
and
S
always exist. The eigenvectors associated with the zero eigenvalue are either recession vectors or degeneracy vectors of positive degree, or their sums. When
c
⁄
=
0
, the eigenvalues with the same sign as
c
and their eigenvectors correspond to the characterization points of
S
, while a degeneracy vector generates an asymptotic ray for the base of
S
or its conjugate hypersurface. The base index is a nonnegative integer
d
less than
m
. If
d
=
k
, then there are nonzero degeneracy vectors of degree
k
−
1
, but no nonzero degeneracy vectors of degree
k
. A linear combination of a degeneracy vector of degree
k
and a degeneracy vector of degree
j
is a degeneracy vector of degree
k
+
j
−
m
if
k
+
j
≥
m
. Based upon these properties, we classify such algebraic hypersurfaces in the nonsingular case into ten classes.
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