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In this paper we show the existence of the minimal solution to the multidimensional Lambert-Euler inversion, a multidimensional generalization of $[-e^{-1} ,0)$ branch of Lambert W function $W_0(x)$. Specifically, for a given nonnegative irreducible symmetric matrix $V in mathbb{R}^{k times k}$, we show that for ${bf u}in(0,infty)^k$, if equation $$y_j exp{-{bf e}_j^T V {bf y} } = u_j ~~~~~~forall j=1,...,k,$$ has at least one solution, it must have a minimal solution ${bf y}^*$, where the minimum is achieved in all coordinates $y_j$ simultaneously. Moreover, such ${bf y}^*$ is the unique solution satisfying $rholeft(V D[y^*_j] right) leq 1$, where $D[y^*_j]={sf diag}(y_j^*)$ is the diagonal matrix with entries $y^*_j$ and $rho$ denotes the spectral radius. Our main application is in the vector-multiplicative coalescent process. It is a coalescent process with $k$ types of particles and vector-valued weights that begins with $alpha_1n+...+alpha_k n$ particles partitioned into types of respective sizes, and in which two clusters of weights ${bf x}$ and ${bf y}$ would merge with rate $({bf x}^{sf T} V {bf y})/n$. We use combinatorics to solve the corresponding modified Smoluchowski equations, obtained as a hydrodynamic limit of vector-multiplicative coalescent as $n to infty$, and use multidimensional Lambert-Euler inversion to establish gelation and find a closed form expression for the gelation time. We also find the asymptotic length of the minimal spanning tree for a broad range of graphs equipped with random edge lengths.
We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some combinatorial coalescent processes: a time reversal of a fragmentation of Cayley trees or a parki
In this paper we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a correspon
The star transform is a generalized Radon transform mapping a function of two variables to its integrals along star-shaped trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical mo
We study admissible and equivalence point transformations between generalized multidimensional nonlinear Schrodinger equations and classify Lie symmetries of such equations. We begin with a wide superclass of Schrodinger-type equations, which include
In this paper we reconsider the problem of the Euler parametrization for the unitary groups. After constructing the generic group element in terms of generalized angles, we compute the invariant measure on SU(N) and then we determine the full range o