For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in
K}$ which is invariant, in law, under unitary conjugation, we
give general sufficient conditions for central limit theorems
for random variables of the type $\operatorname{Tr}(\mathbf{A}_k
\mathbf{M})$, where the matrix $\mathbf{M}$ is deterministic
(such random variables include for example the normalized matrix
entries of the $\mathbf{A}_k$'s). A consequence is the asymptotic
independence of the projection of the matrices $\mathbf{A}_k$
onto the subspace of null trace matrices from their projections
onto the orthogonal of this subspace. These results are used
to study the asymptotic behavior of the outliers of a spiked
elliptic random matrix. More precisely, we show that the fluctuations
of these outliers around their limits can have various rates
of convergence, depending on the Jordan Canonical Form of the
additive perturbation. Also, some correlations can arise between
outliers at a macroscopic distance from each other. These phenomena
have already been observed
with random matrices
from the Single Ring Theorem.
We study comparison properties in the category $\mathrm{Cu}$ aiming to
lift results to the C*-algebraic setting. We introduce a new
comparison property and relate it to both the CFP and $\omega$-comparison.
We show differences of all properties by providing examples,
which suggest that the corona factorization for C*-algebras might
allow for both finite and infinite projections. In addition,
we show that R{\o}rdam's simple, nuclear C*-algebra with a finite
and an infinite projection does not have the CFP.
We study the Bishop-Phelps-Bollobás property (BPBp for short)
for compact operators. We present some abstract techniques which
allows to carry the BPBp for compact operators from sequence
spaces to function spaces. As main applications, we prove the
following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$
has the BPBp for compact operators, then so do $(C_0(L),Y)$ for
every locally compact Hausdorff topological space $L$ and $(X,Y)$
whenever $X^*$ is isometrically isomorphic to $\ell_1$.
If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$
has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$
for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$
has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional
or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any
positive measure $\nu$ and $1\lt p\lt \infty$.
For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact
operators, then so does $(X,L_p(\mu,Y))$ for every positive measure
$\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X,Y)$
has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$
for every $\sigma$-finite positive measure $\mu$ and $(X,C(K,Y))$
for every compact Hausdorff topological space $K$.
This note provides a correct proof of the result claimed by the
second author that locally compact normal spaces are collectionwise
Hausdorff in certain models obtained by forcing with a coherent
Souslin tree. A novel feature of the proof is the use of saturation
of the non-stationary ideal on $\omega_1$, as well as of a strong
form of Chang's Conjecture. Together with other improvements,
this enables the consistent characterization of locally compact
hereditarily paracompact spaces as those locally compact, hereditarily
normal spaces that do not include a copy of $\omega_1$.
This paper introduces a class of abstract linear representations
on
Banach convolution function algebras over
homogeneous spaces of compact groups. Let $G$ be a compact group
and $H$ be a closed subgroup of $G$.
Let $\mu$ be the normalized $G$-invariant measure over the compact
homogeneous space $G/H$ associated to the
Weil's formula and $1\le p\lt \infty$.
We then present a structured class of abstract linear representations
of the
Banach convolution function algebras $L^p(G/H,\mu)$.
Let $\mathbf{F}_{q}[T]$ be the ring of polynomials over the finite
field of $q$ elements, and $Y$ be a large integer. We say a polynomial
in $\mathbf{F}_{q}[T]$ is $Y$-smooth if all of its irreducible
factors
are of degree at most $Y$. We show that a ternary additive equation
$a+b=c$ over $Y$-smooth polynomials has many solutions. As an
application,
if $S$ is the set of first $s$ primes in $\mathbf{F}_{q}[T]$ and
$s$ is large, we prove that the $S$-unit equation $u+v=1$ has at
least $\exp(s^{1/6-\epsilon}\log q)$ solutions.
For a finite abelian $p$-group $A$ of rank $d=\dim A/pA$, let
$\mathbb{M}_A := \log_p |A|^{1/d}$ be its
(logarithmic) mean exponent. We study the behavior of
the mean exponent of $p$-class groups in pro-$p$ towers $\mathrm{L}/K$
of number fields. Via a combination of results from analytic
and algebraic number theory, we construct infinite tamely
ramified pro-$p$ towers in which the mean exponent of $p$-class
groups remains bounded. Several explicit
examples are given with $p=2$. Turning to group theory, we
introduce an invariant $\underline{\mathbb{M}}(G)$ attached to a finitely generated
pro-$p$ group $G$; when $G=\operatorname{Gal}(\mathrm{L}/\mathrm{K})$, where $\mathrm{L}$ is the Hilbert
$p$-class field tower of a number field $K$, $\underline{\mathbb{M}}(G)$ measures
the asymptotic behavior of the mean exponent of $p$-class groups
inside $\mathrm{L}/\mathrm{K}$. We compare and contrast the behavior of this
invariant in analytic versus non-analytic groups. We exploit
the interplay of group-theoretical and number-theoretical perspectives
on this invariant and explore some open questions that arise
as a result, which may be of independent interest in group theory.
We study a second-order ordinary differential
equation coming from the Kepler problem on $\mathbb{S}^2$. The
forcing term under consideration is a piecewise constant with
singular nonlinearity which changes sign. We establish necessary
and
sufficient conditions to the existence and multiplicity of
$T$-periodic solutions.
The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}\mathbb{P}^n$ are computed
along with
the Atiyah-Hirzebruch spectral sequence for
$ER(2)^*(\mathbb{C}\mathbb{P}^\infty)$.
This, along with other papers in this series, gives
us the $ER(2)$-cohomology of all Eilenberg-MacLane spaces.
I construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty$
of real functions with logarithmic generalized power series as
asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation
and $\log$-composition; in particular, $\mathcal{F}$ is a Hardy field.
Moreover, the field $\mathcal{F} \circ (-\log)$ of germs at $0^+$ contains
all transition maps of hyperbolic saddles of planar real analytic
vector fields.
Let $\Theta ^{[j]}$ be an analogue of the Ramanujan theta operator
for Siegel modular forms.
For a given prime $p$, we give the weights of elements of mod
$p$ kernel of $\Theta ^{[j]}$,
where the mod $p$ kernel of $\Theta ^{[j]}$ is the set of all
Siegel modular forms $F$ such that $\Theta ^{[j]}(F)$ is congruent
to zero modulo $p$.
In order to construct examples of the mod $p$ kernel of $\Theta
^{[j]}$ from any Siegel modular form,
we introduce new operators $A^{(j)}(M)$ and show the modularity
of $F|A^{(j)}(M)$ when $F$ is a Siegel modular form.
Finally, we give some examples of the mod $p$ kernel of $\Theta
^{[j]}$ and the filtrations of some of them.
Let $W$ be a compact simply connected triangulated manifold with
boundary and $K\subset W$ be a subpolyhedron. We construct an
algebraic model of the rational homotopy type of $W\backslash K$ out of
a model of the map of pairs $(K,K \cap \partial W)\hookrightarrow
(W,\partial W)$ under some high codimension hypothesis.
We deduce the rational homotopy invariance of the configuration
space
of two points in a compact manifold with boundary under 2-connectedness
hypotheses. Also, we exhibit
nice explicit models of these configuration spaces for a large
class
of compact manifolds.
We address the classification problem for graph $C^*$-algebras of
finite graphs (finitely many edges and vertices), containing
the class of Cuntz-Krieger algebras as a
prominent special case. Contrasting earlier work, we do not assume
that the graphs satisfy the standard condition (K), so that the
graph
$C^*$-algebras may come with uncountably many ideals.
We find that in this generality, stable isomorphism of graph
$C^*$-algebras does not coincide with the geometric notion of Cuntz
move equivalence. However, adding a modest condition on the
graphs, the two notions are proved to be mutually equivalent and
equivalent to the $C^*$-algebras having isomorphic $K$-theories. This
proves in turn that under this condition, the graph
$C^*$-algebras are in fact classifiable by $K$-theory, providing in
particular complete classification when the $C^*$-algebras in question
are either of real rank zero or type I/postliminal. The key ingredient
in obtaining these results is a characterization of Cuntz move
equivalence using the adjacency matrices of the graphs.
Our results are applied to discuss the classification problem
for the quantum lens spaces defined by Hong and Szymański,
and to complete the classification of graph $C^*$-algebras associated to
all simple graphs with four vertices or less.
Culler and Vogtmann defined a simplicial space $O(g)$ called
outer space to study the outer automorphism group
of the free group $F_g$. Using representation theoretic methods,
we give an embedding of $O(g)$ into the analytification of $\mathcal{X}(F_g,
SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$ character variety
of $F_g,$ reproving a result of Morgan and Shalen. Then we show
that every point $v$ contained in a maximal cell of $O(g)$ defines
a flat degeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to
a toric variety $X(P_{\Gamma})$. We relate $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ and $X(v)$ topologically by showing that there
is a surjective, continuous, proper map $\Xi_v: \mathcal{X}(F_g,
SL_2(\mathbb{C})) \to X(v)$. We then show that this map is a
symplectomorphism on a dense, open subset of $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ with respect to natural symplectic structures
on $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this
way, we construct an integrable Hamiltonian system in $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ for each point in a maximal cell of $O(g)$,
and we show that each $v$ defines a topological decomposition
of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived from the decomposition
of $X(P_{\Gamma})$ by its torus orbits. Finally, we show that
the valuations coming from the closure of a maximal cell in $O(g)$
all arise as divisorial valuations built from an associated projective
compactification of $\mathcal{X}(F_g, SL_2(\mathbb{C})).$
We introduce the tracial Rokhlin property for a conditional expectation
for an inclusion
of unital C*-algebras $P \subset A$ with index finite, and show
that an action $\alpha$
from a finite group $G$ on a simple unital C*-algebra $A$ has
the tracial Rokhlin property
in the sense of N. C. Phillips
if and only if the canonical conditional expectation $E\colon
A \rightarrow A^G$ has the tracial
Rokhlin property.
Let $\mathcal{C}$ be a class of infinite dimensional stably
finite separable unital C*-algebras
which is closed under the following conditions:
(1)
If $A \in {\mathcal C}$ and $B \cong A$, then $B \in \mathcal{C}$.
(2)
If $A \in \mathcal{C}$ and $n \in \mathbb{N}$, then $M_n(A) \in \mathcal{C}$.
(3)
If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection,
then $pAp \in \mathcal{C}$.
Suppose that any C*-algebra in $\mathcal{C}$ is weakly semiprojective.
We prove that if $A$ is a local tracial $\mathcal{C}$-algebra
in the sense of Fan and Fang and a conditional expectation
$E\colon A \rightarrow P$ is of index-finite type with the tracial
Rokhlin property, then $P$ is a unital
local tracial $\mathcal{C}$-algebra.
The main result is that if $A$ is simple, separable, unital
nuclear, Jiang-Su absorbing
and $E\colon A \rightarrow P$ has the tracial Rokhlin property,
then $P$ is Jiang-Su absorbing.
As an application, when an action $\alpha$
from a finite group $G$ on a simple unital C*-algebra $A$ has
the tracial Rokhlin property,
then for any subgroup $H$ of $G$ the fixed point algebra $A^H$
and the crossed product algebra
$A \rtimes_{\alpha_{|H}} H$ is Jiang-Su absorbing.
We also show that the strict comparison property for a Cuntz
semigroup $W(A)$
is hereditary to $W(P)$ if $A$ is simple, separable, exact,
unital,
and $E\colon A \rightarrow P$ has the tracial Rokhlin property.
We consider three point positive masses moving on $S^2$ and $H^2$.
An Eulerian-relative equilibrium, is a relative equilibrium where
the three masses are on the same geodesic, in this paper we analyze
the spectral stability of these kind of orbits where the mass
at the middle is arbitrary and the masses at the ends are equal
and located at the same distance from the central mass. For the
case of $S^2$, we found a positive measure set in the set of
parameters where the relative equilibria are spectrally stable,
and we give a complete classification of the spectral stability
of these solutions, in the sense that, except on an algebraic
curve in the space of parameters, we can determine if the corresponding
relative equilibria is spectrally stable or unstable.
On $H^2$, in the elliptic case, we prove that generically all
Eulerian-relative equilibria are unstable; in the particular
degenerate case when the two equal masses are negligible we get
that the corresponding solutions are spectrally stable. For the
hyperbolic case we consider the system where the mass in the
middle is negligible, in this case the Eulerian-relative equilibria
are unstable.
For a Shimura variety of Hodge type with hyperspecial level
structure at a prime~$p$, Vasiu and Kisin constructed a smooth
integral model (namely the integral canonical model) uniquely
determined by a certain extension property. We define and study
the Ekedahl-Oort stratifications on the special fibers of those
integral canonical models when $p\gt 2$. This generalizes
Ekedahl-Oort stratifications defined and studied by Oort on moduli
spaces of principally polarized abelian varieties and those
defined and studied by Moonen, Wedhorn and Viehmann on good
reductions of Shimura varieties of PEL type. We show that the
Ekedahl-Oort strata are parameterized by certain elements $w$ in
the Weyl group of the reductive group in the Shimura datum. We
prove that the stratum corresponding to $w$ is smooth of dimension
$l(w)$ (i.e. the length of $w$) if it is non-empty. We also
determine the closure of each stratum.
We prove the Gross-Deligne conjecture on CM periods for motives
associated with $H^2$ of certain surfaces fibered over the projective
line. Then we prove for the same motives a formula which expresses
the $K_1$-regulators in terms of hypergeometric functions ${}_3F_2$,
and obtain a new example of non-trivial regulators.
The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager
invariant subspace theorem for operators on certain Banach spaces
of functions on a multiply connected domain in $\mathbb{C}$. The norms
for these spaces are either the usual Lebesgue and Hardy space
norms or certain continuous gauge norms.
In the Hardy space case the expected corollaries include the
characterization of the cyclic vectors as the outer functions
in this context, a demonstration that the set of analytic multiplication
operators is maximal abelian and reflexive, and a determination
of the closed operators that commute with all analytic multiplication
operators.
The paper constitutes a comprehensive study of ten classes of
self-maps on metric spaces $\langle X,d\rangle$ with the local
and pointwise (a.k.a. local radial) contraction properties.
Each of those classes appeared previously in the literature in
the context of fixed point theorems.
We begin with presenting an overview
of these fixed point results, including concise self contained
sketches of their proofs.
Then, we proceed with a discussion of the relations among the
ten classes of self-maps with domains $\langle X,d\rangle$
having various topological properties which often appear in the
theory of fixed point theorems: completeness, compactness, (path)
connectedness, rectifiable path connectedness, and $d$-convexity.
The bulk of the results presented in this part consists of examples
of maps that show non-reversibility of the previously established
inclusions between theses classes.
Among these examples, the most striking is a differentiable auto-homeomorphism
$f$ of a compact perfect subset $X$ of $\mathbb R$
with $f'\equiv 0$, which constitutes also a minimal dynamical
system.
We finish with discussing a few remaining open problems on weather
the maps with specific pointwise contraction
properties must have the fixed points.
We characterize the Local Langlands Correspondence (LLC) for
inner forms of $\operatorname{GL}_n$ via the Jacquet-Langlands Correspondence
(JLC) and compatibility with the Langlands Classification. We
show that LLC satisfies a natural compatibility with parabolic
induction and characterize LLC for inner forms as a unique family
of bijections $\Pi(\operatorname{GL}_r(D)) \to \Phi(\operatorname{GL}_r(D))$ for each $r$,
(for a fixed $D$) satisfying certain properties. We construct
a surjective map of Bernstein centers $\mathfrak{Z}(\operatorname{GL}_n(F))\to
\mathfrak{Z}(\operatorname{GL}_r(D))$
and show this produces pairs of matching distributions in the
sense of Haines. Finally, we construct explicit Iwahori-biinvariant
matching functions for unit elements in the parahoric Hecke
algebras
of $\operatorname{GL}_r(D)$, and thereby produce many explicit pairs of matching
functions.
We investigate the smoothing problem of limit linear series of
rank one on an enrichment of the notions of nodal curves and
metrized complexes called saturated metrized complexes. We give
a finitely verifiable full criterion for smoothability of a limit
linear series of rank one on saturated metrized complexes, characterize
the space of all such smoothings, and extend the criterion to
metrized complexes. As applications, we prove that all limit
linear series of rank one are smoothable on saturated metrized
complexes corresponding to curves of compact-type, and prove
an analogue for saturated metrized complexes of a theorem of
Harris and Mumford on the characterization of nodal curves contained
in a given gonality stratum. In addition, we give a full combinatorial
criterion for smoothable limit linear series of rank one on saturated
metrized complexes corresponding to nodal curves whose dual graphs
are made of separate loops.
We study a relation between distinction and special values of
local invariants for representations of the general linear group
over a quadratic extension of $p$-adic fields.
We show that the local Rankin-Selberg root number of any pair
of distinguished representation is trivial and as a corollary
we obtain an analogue for the global root number of any pair
of distinguished cuspidal representations. We further study the
extent to which the gamma factor at $1/2$ is trivial for distinguished
representations as well as the converse problem.
The $\operatorname{SL}(2,\mathbb C)$-representation varieties of punctured surfaces
form natural families parameterized by monodromies at the punctures.
In this paper, we compute the loci where these varieties are
singular for the cases of one-holed and two-holed tori and the
four-holed sphere. We then compute the de Rham cohomologies
of these varieties of the one-holed torus and the four-holed
sphere when the varieties are smooth via the Grothendieck theorem.
Furthermore, we produce the explicit Gauss-Manin connection
on the natural family of the smooth $\operatorname{SL}(2,\mathbb C)$-representation
varieties of the one-holed torus.
In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with
superharmonic weights induced by positive Borel measures $\mu$
on
the open unit disk. We establish the Alexander-Taylor-Ullman
inequality for $\mathcal{D}_\mu$ spaces and we characterize the cases where
equality occurs.
We define a class of weighted Hardy spaces $H_{\mu}^{2}$ via
the balayage of the measure $\mu$.
We show that $\mathcal{D}_\mu$
is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a
Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the
reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite
sum of point mass measures. We consider the boundary
behavior and inner-outer factorization of functions in $\mathcal{D}_\mu$.
We also characterize the boundedness and
compactness of composition operators on $\mathcal{D}_\mu$.
If the Hasse invariant of a $p$-divisible group is small enough,
then one can construct a canonical subgroup inside its $p$-torsion.
We prove that, assuming the existence of a subgroup of adequate
height in the $p$-torsion with high degree, the expected properties
of the canonical subgroup can be easily proved, especially the
relation between its degree and the Hasse invariant. When one
considers a $p$-divisible group with an action of the ring of
integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$,
then much more can be said. We define partial Hasse invariants
(they are natural in the unramified case, and generalize a construction
of Reduzzi and Xiao in the general case), as well as partial
degrees. After studying these functions, we compute the partial
degrees of the canonical subgroup.
We will give a representation-theoretic proof for the multiplication
formula
in the Ringel-Hall algebra
$\mathfrak{H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$. As a first
application, we see immediately the existence of Hall polynomials
for cyclic quivers, a fact established
by J. Y. Guo and C. M. Ringel,
and derive a recursive formula
to compute them.
We will further use the formula and the construction of a certain
monomial base for $\mathfrak{H}_\Delta(n)$ given
by Deng, Du, and Xiao
together with the double Ringel--Hall algebra realisation of
the quantum loop algebra $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_n)$
given by
Deng, Du, and Fu
to develop some algorithms and to compute the canonical basis
for $\mathbf{U}_v^+(\widehat{\mathfrak{g}\mathfrak{l}}_n)$. As examples,
we will show explicitly the part of the canonical basis
associated with modules of Lowey length at most $2$ for the quantum
group $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_2)$.
We provide general inequalities that compare the surface area
$S(K)$ of a convex body $K$ in ${\mathbb R}^n$
to the minimal, average or maximal surface area of its hyperplane
or lower dimensional projections. We discuss the
same questions for all the quermassintegrals of $K$. We examine
separately the dependence of the constants
on the dimension in the case where $K$ is in some of the classical
positions or $K$ is a projection body.
Our results are in the spirit of the hyperplane problem, with
sections replaced by projections and volume by
surface area.
Consider a finite sequence of linear contractions $S_{j}(x)=\varrho
x+d_{j}$ and
probabilities $p_{j}\gt 0$ with $\sum p_{j}=1$. We are interested
in the
self-similar measure $\mu =\sum p_{j}\mu \circ S_{j}^{-1}$, of
finite type.
In this paper we study the multi-fractal analysis of such measures,
extending the theory to measures arising from non-regular probabilities
and
whose support is not necessarily an interval.
Under some mild technical assumptions, we prove that there exists
a subset
of supp$\mu $ of full $\mu $ and Hausdorff measure, called the
truly
essential class, for which the set of (upper or lower) local
dimensions is a
closed interval. Within the truly essential class we show that
there exists
a point with local dimension exactly equal to the dimension of
the support.
We give an example where the set of local dimensions is a two
element set,
with all the elements of the truly essential class giving the
same local
dimension. We give general criteria for these measures to be
absolutely
continuous with respect to the associated Hausdorff measure of
their support
and we show that the dimension of the support can be computed
using only
information about the essential class.
To conclude, we present a detailed study of three examples. First,
we show
that the set of local dimensions of the biased Bernoulli convolution
with
contraction ratio the inverse of a simple Pisot number always
admits an
isolated point. We give a precise description of the essential
class of a
generalized Cantor set of finite type, and show that the $kth$
convolution
of the associated Cantor measure has local dimension at $x\in
(0,1)$ tending
to 1 as $k$ tends to infinity. Lastly, we show that within a
maximal loop
class that is not truly essential, the set of upper local dimensions
need
not be an interval. This is in contrast to the case for finite
type measures
with regular probabilities and full interval support.
Let $k$ be a number field. We describe the category of Laumon
$1$-isomotives over $k$ as the universal category in the sense
of Nori associated with a quiver representation built out of
smooth proper $k$-curves with two disjoint effective divisors
and a notion of $H^1_\mathrm{dR}$ for such "curves with modulus".
This result extends and relies on the theorem of J. Ayoub
and L. Barbieri-Viale that describes Deligne's category
of $1$-isomotives in terms of Nori's Abelian category of motives.
In this paper, we prove the spherical fundamental lemma for
metaplectic group $Mp_{2n}$ based on the formalism of endoscopy
theory by J.Adams, D.Renard and Wen-Wei Li.
We prove that for every surface $\Sigma$ of Euler genus $g$,
every edge-maximal embedding of a graph in $\Sigma$ is at most
$O(g)$ edges short of a triangulation of $\Sigma$. This provides
the first answer to an open problem of Kainen (1974).
In this paper we study a class of second order fully nonlinear
elliptic equations
containing gradient terms on compact Hermitian manifolds and
obtain a priori estimates under
proper assumptions close to optimal.
The analysis developed here should
be useful to deal with other Hessian equations containing gradient
terms in other contexts.