Multiview geometry is the study of
two-dimensional images of three-dimensional scenes, a foundational subject in computer vision.
We determine a universal Gröbner basis for the multiview ideal of $n$ generic cameras.
As the cameras move, the multiview varieties vary in a family of dimension $11n-15$.
This family is the distinguished component of a multigraded Hilbert scheme
with a unique Borel-fixed point.
We present a combinatorial study
of ideals lying on that Hilbert scheme.
We introduce the concept of a rare element in a non-associative normed
algebra and show that the existence of such element is the only obstruction
to continuity of a surjective homomorphism from a non-associative Banach
algebra to a unital normed algebra with simple completion. Unital
associative algebras do not admit any rare element and hence automatic
continuity holds.
Let $G$ be a locally compact group. Let $A_{M}(G)$ ($A_{0}(G)$)denote
the closure of $A(G)$, the Fourier algebra of $G$ in the space of
bounded (completely bounded) multipliers of $A(G)$.
We call a locally compact group M-weakly amenable if
$A_M(G)$
has a
bounded approximate identity. We will show that when $G$ is M-weakly
amenable, the algebras $A_{M}(G)$ and $A_{0}(G)$ have
properties that are characteristic of the Fourier algebra of an
amenable group. Along the way we show that the sets of tolopolically
invariant means associated with these algebras have the same
cardinality as those of the Fourier algebra.
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.
We study locally compact quantum groups $\mathbb{G}$ through the
convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})),
\triangleright)$. We prove that the reduced quantum group
$C^*$-algebra $C_0(\mathbb{G})$ can be recovered from the convolution
$\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$-module
$\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is
equal to $C_0(\mathbb{G})$. On the other hand, we show that the left
$T(L_2(\mathbb{G}))$-module $\langle T(L_2(\mathbb{G}))\triangleright
K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product
$C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is
a much larger $C^*$-subalgebra of $B(L_2(\mathbb{G}))$.
We establish a natural isomorphism between the completely bounded
right multiplier algebras of $L_1(\mathbb{G})$ and
$(T(L_2(\mathbb{G})), \triangleright)$, and settle two invariance
problems associated with the representation theorem of
Junge-Neufang-Ruan (2009). We characterize regularity and discreteness
of the quantum group $\mathbb{G}$ in terms of continuity properties of
the convolution $\triangleright$ on $T(L_2(\mathbb{G}))$. We prove
that if $\mathbb{G}$ is semi-regular, then the space
$\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right
$\mathbb{G}$-continuous operators on $L_2(\mathbb{G})$, which was
introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$-subalgebra
of $B(L_2(\mathbb{G}))$. In the representation framework formulated by
Neufang-Ruan-Spronk (2008) and Junge-Neufang-Ruan, we show that the
dual properties of compactness and discreteness can be characterized
simultaneously via automatic normality of quantum group bimodule maps
on $B(L_2(\mathbb{G}))$. We also characterize some commutation
relations of completely bounded multipliers of $(T(L_2(\mathbb{G})),
\triangleright)$ over $B(L_2(\mathbb{G}))$.
In this paper we use the recent developments in the
representation theory of locally compact quantum groups,
to assign, to each locally compact
quantum group $\mathbb{G}$, a locally compact group $\tilde {\mathbb{G}}$ which
is the quantum version of point-masses, and is an
invariant for the latter. We show that ``quantum point-masses"
can be identified with several other locally compact groups that can be
naturally assigned to the quantum group $\mathbb{G}$.
This assignment preserves compactness as well as
discreteness (hence also finiteness), and for large classes of quantum
groups, amenability. We calculate this invariant for some of the most
well-known examples of
non-classical quantum groups.
Also, we show that several structural properties of $\mathbb{G}$ are encoded
by $\tilde {\mathbb{G}}$: the latter, despite being a simpler object, can carry very
important information about $\mathbb{G}$.
Nous considérons les perturbations $H := H_{0} + V$ et $D := D_{0} +
V$ des Hamiltoniens libres $H_{0}$ de Pauli et $D_{0}$ de Dirac en
dimension 3 avec champ magnétique non constant, $V$ étant un
potentiel électrique qui décroît super-exponentiellement dans la
direction du champ magnétique. Nous montrons que dans des espaces de
Banach appropriés, les résolvantes de $H$ et $D$ définies sur le
demi-plan supérieur admettent des prolongements méromorphes. Nous
définissons les résonances de $H$ et $D$ comme étant les pôles de
ces extensions méromorphes. D'une part, nous étudions la
répartition des résonances de $H$ près de l'origine $0$ et d'autre
part, celle des résonances de $D$ près de $\pm m$ où $m$ est la
masse d'une particule. Dans les deux cas, nous obtenons d'abord des
majorations du nombre de résonances dans de petits domaines au
voisinage de $0$ et $\pm m$. Sous des hypothèses supplémentaires,
nous obtenons des développements asymptotiques du nombre de
résonances qui entraînent leur accumulation près des seuils $0$ et
$\pm m$. En particulier, pour une perturbation $V$ de signe défini,
nous obtenons des informations sur la répartition des valeurs propres
de $H$ et $D$ près de $0$ et $\pm m$ respectivement.
In this paper, we study the reduced loci of special cycles on local
models of the Shimura variety for $\operatorname{GU}(1,n-1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat-Tits
theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.
We define partial differential (PD in the following), i.e., field
theoretic analogues of Hamiltonian systems on abstract symplectic
manifolds and study their main properties, namely, PD Hamilton
equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in
standard multisymplectic approach to Hamiltonian field theory, in our
formalism, the geometric structure (kinematics) and the dynamical
information on the ``phase space''
appear as just different components of one single geometric object.