Given a Hamiltonian system $ (M,\omega, G,\mu) where $(M,\omega)$ is symplectic manifold, $G$ compact connected Lie group acting on with moment map \mu:M \rightarrow\mathfrak{g}^{*}$, then one may construct the quotient $(M//G, \omega_{red})$ $M//G := \mu^{-1}(0)/G$. Kirwan used norm-square of map, $|\mu|^2$, as G-equivariant Morse function $M$ to derive formulas for rational Betti numbers $M//...

In this paper, we introduce the notions of expansion of ideals in $MV$-algebras, $ (tau,sigma)- $primary, $ (tau,sigma)$-obstinate  and $ (tau,sigma)$-Boolean  in $ MV- $algebras. We investigate the relations of them. For example, we show that every $ (tau,sigma)$-obstinate ideal of an $ MV-$ algebra is $ (tau,sigma)$-primary  and $ (tau,sigma)$-Boolean. In particular, we define an expansion $ ...

We have developed N = 1 supersymmetric nonlinear realization methods, which realize global symmetry breaking in N = 1 supersymmetric theories. The target space of nonlinear sigma models with a linear model origin is a Gorbit, which is a non-compact non-homogeneous Kähler manifold. We show that, if and only if the orbit is open, it includes a compact homogeneous Kähler manifold as a submanifold,...

Haver’s near-selection theorem deals with approximate selections of Hausdorff continuous CE-valued mappings defined on $\sigma $-compact metrizable $C$-spaces. In the present paper, we show that it is valid for all paracompact $C$-spaces, which answers in

Let $\Omega \subset \mathbb R^d$ be a $C^1$ domain or, more generally, Lipschitz with small constant and $A(x)$ $d \times d$ uniformly elliptic, symmetric matrix coefficients. Assume $u$ is harmonic in $\Omega$, or greater generality solves $\operatorname{div}(A(x)\nabla u)=0$ vanishes on $\Sigma = \partial\Omega \cap B$ for some ball $B$. We study the dimension of singular set $\Sigma$, partic...

Abstract In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The is symmetric with respect to a measure induced by Liouville conformal field theory. Using theory of Dirichlet forms, construct weak solution associated equation area on flat torus, full “$L^1$ regime” $\sigma < \sigma _{L^1}=2 \sqrt \pi $ where noise strength. We also describe main necessar...

In this paper, we show that any compact quasi k-Yamabe gradient soliton must have constant \(\sigma _{k}\)-curvature. Moreover, provide a certain condition for to be gradient, and noncompact solitons, present geometric rigidity under decay on the norm of field.

We study supersymmetric $AdS_5\times \Sigma$ solutions, for $\Sigma$ being a topological disk with non-trivial $U(1)$ holonomy at the boundary or "half-spindle", in seven-dimensional $N=2$ gauged supergravity coupled to three vector multiplets. consider compact and non-compact gauge groups, $SO(4-p,p)$, $p=0,1,2$, find number of solutions $SO(2)\times SO(2)$ $SO(2)_{\text{diag}}$ residual symme...