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We extend the Cauchy residue theorem to a large class of domains including differential chains that represent, via canonical embedding into a space of currents, divergence free vector fields and non-Lipschitz curves. That is, while the…
We applied the method of finite-part integration [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to evaluate in closed-form the exact one-loop integral representations of the Heisenberg-Euler Lagrangian from QED for a constant magnetic field…
Let $f_1$, $f_2$, and $f_3$ be holomorphic functions on a complex manifold and assume that the common zero set of the $f_j$ has maximal codimension, i.e., that it is a complete intersection. We prove that the iterated Mellin transform of…
Following Abbatiello et al. [ DCCDS-A (41), 2020], we introduce dissipative turbulent solutions to a simple model of a mixture of two non interacting compressible fluids {\tc filling a bounded domain with general non zero inflow/outflow…
This paper develops the theory of affine Euler-Poincar\'e and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids,…
We study complex Monge-Ampere equations on Hermitian manifolds, extending classical existence results of Yau and Aubin in the Kahler case, and those of Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As an…
The Riesz $z$-energy of a manifold $X$ is the integration of the distance between two points to the power $z$ over the product space $X\times X$. Considered as a function of a complex variable $z$, it can be generalized to a meromorphic…
In [3], Connes found a conformal invariant using Wodzicki's 1-density and computed it in the case of 4-dimensional manifold without boundary. In [14], Ugalde generalized the Connes' result to $n$-dimensional manifold without boundary. In…
This thesis is divided into three parts. In the first part, we give an introduction to J. Harrison's theory of differential chains. In the second part, we apply these tools to generalize the Cauchy theorems in complex analysis. Instead of…
We prove the asymptotic large volume expression of diagonal form factors in integrable models by evaluating carefully the diagonal limit of a non-diagonal form factor in which we send the rapidity of the extra particle to infinity.
We associate bicomplexes with several integrable models in such a way that conserved currents are obtained by a simple iterative construction. Gauge transformations and dressings are discussed in this framework and several examples are…
We establish a formula for the sum of the Lyapounov exponents of an holomorphic endomorphism of ${\bf P}^k$. For an holomorphic family of such endomorphisms we define the {\em bifurcation current} as $dd^cL$ and show that it vanishes when…
There are three types of Dolbeault complexes arising from representations of holonomy group on a Riemannian manifold, two of which are dual to each other. Such a complex is elliptic if and only if its generator satisfies an algebraic…
The Leibniz rule for fractional Riemann-Liouville derivative is studied in algebra of functions defined by Laplace convolution. This algebra and the derived Leibniz rule are used in construction of explicit form of stationary-conserved…
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that this construction can produce…
For a class of non compact Riemannian manifolds with ends, we give pseudo-differential expansions of bounded functions of the semi-classical Laplacian and study related Lp boundedness properties.
Given an ideal $\mathcal{J}$ on a complex manifold, Andersson and Wulcan constructed a current $R^\mathcal{J}$ such that the annihilator of $R^\mathcal{J}$ is $\mathcal{J}$, generalizing the duality theorem for Coleff-Herrera products. We…
In this paper we present a new approach to Morse theory based on the de Rham-Federer theory of currents. The full classical theory is derived in a transparent way. The methods carry over uniformly to the equivariant and the holomorphic…
Complex (affine) lines are a major object of study in complex geometry, but their symplectic aspects are not well understood. We perform a systematic study based on their associated Ahlfors currents. In particular, we generalize (by a…
Extending a recent theory developed on compact K\"ahler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds.…