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This paper meticulously revisit and study the flux geometry of any compact oriented manifold $(M; W)$. We generalize several well-known factorization results, exhibit some orbital conditions for the study of flux geometry, give a proof of…

The relation between the infinite-dimensional 3-algebras and the dispersionless KdV hierarchy is investigated. Based on the infinite-dimensional 3-algebras, we derive two compatible Nambu Hamiltonian structures. Then the dispersionless KdV…

Let $(W,H,\mu)$ be the classical Wiener space. Assume that $U=I_W+u$ is an adapted perturbation of identity, i.e., $u:W\to H$ is adapted to the canonical filtration of $W$. We give some sufficient analytic conditions on $u$ which imply the…

A classical result of Hardy and Littlewood says that if $f=u+iv$ is analytic in the unit disk $\mathbb{D}$ and $u$ is in the harmonic Bergman space $a^p$ ($0<p<\infty$), then $v$ is also in $a^p$. This complements a celebrated result of M.…

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, and the proper choice of integration constants should be the one dictated by the associated Inverse…

Continuing our investigation into the Hierarchical Reference Theory of fluids for thermodynamic states of infinite isothermal compressibility kappa[T] we now turn to the available numerical evidence to elucidate the character of the partial…

Given a positive integer $k$, we investigate the $k$-redcibility of self-maps in the monoid $\AA^k(X\vee Y)$, consisting of self-maps that induce isomorphisms on homology groups up to degree $k$. In general, verifying $k$-reducibility is a…

The Bieberbach estimate, a pivotal result in the classical theory of univalent functions, states that any injective holomorphic function $f$ on the open unit disc $D$ satisfies $|f"(0)|\leq 4 |f'(0)|$. We generalize the Bieberbach estimate…

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of…

Let $I=[0,1)$, $b\in \{2,3,\ldots\}$ and $f:I\to I$ be an injective piecewise $\frac{1}{b}$-affine map, that is, assume that there exists a partition of $I$ into intervals $I_1,\ldots,I_n$ such that $\vert f(x)-f(y)\vert\le\frac1b \vert…

The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators…

In this note, we briefly discuss how singular KAM Theory - which was worked out in a previous work by L.B. and L.C. for the mechanical case $\frac12 |y|^2+\varepsilon f(x)$ - can be extended to convex real analytic nearly integrable…

This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars.…

The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its $L^2$ boundedness. In particular, a famous proof of Melnikov-Verdera [18] relies upon an iconic symmetrization identity of…

A novel Hamiltonian system in n dimensions which admits the maximal number 2n-1 of functionally independent, quadratic first integrals is presented. This system turns out to be the first example of a maximally superintegrable Hamiltonian on…

The first author's geometric Hopf invariant of a stable map $F:\Sigma^{\infty}X \to \Sigma^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):\Sigma^{\infty}X \to \Sigma^{\infty}(Y \wedge Y)$ constructed by an explicit difference…

For a finite dimensional Lie algebra $\mathfrak{g}$, the Duflo map $S\mathfrak{g}\rightarrow U\mathfrak{g}$ defines an isomorphism of $\mathfrak{g}$-modules. On $\mathfrak{g}$-invariant elements it gives an isomorphism of algebras.…

We study the interior problem of tomography. The starting point is the Gelfand-Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function $f$ along a collection of lines. Pick one such…

We discuss the dynamics of smooth diffeomorphisms of the disc with vanishing topological entropy which satisfy the mild dissipation property introduced in [CP]. In particular it contains the H\'enon maps with Jacobian up to 1/4. We prove…

We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer's metric. Namely: (1) we resolve the Kapovich-Polterovich question by showing…