The antecedent of Einstein field equations with Euclidean signature is the complex elliptic Monge-Ampère equation, CMA2. For elliptic CMA2 solutions determine hyper-Kähler, self-dual and therefore Ricci-flat metrics with Euclidean signature. We shall consider a symmetry reduction of CMA2 to solutions that admit only a single Killing vector. Systematic studies of vacuum metrics with one Killing ...

The rotational invariance of energy is ensured by the implicit sum over the indices (i, j) in the above expression. In the Fourier representation, the energy depends on the quan­ tities q , |u|, and (q · u) which are clearly independent of rotations. For other lattices, there are more elastic coefficients, since the energy should only be invariant under lattice rotations. For example, the symme...

In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationally invariant regions and obtain the possible isoperimetric solutions by stud...

We study the Laplacian-∞ path as an extreme case of the Laplacian-α random walk. Although, in the finite α case, there is reason to believe that the process converges to SLEκ, with κ = 6/(2α+ 1), we show that this is not the case when α = ∞. In fact, the scaling limit depends heavily on the lattice structure, and is not conformal (or even rotational) invariant.

Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D4, the symmetry group of a square. We examine pattern-avoiding permutations with 180◦ rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our resu...

Precise measurements of the microwave background anisotropy have confirmed the inflationary picture of approximately scale invariant, Gaussian primordial adiabatic density perturbations. However, there are some anomalies that suggest a small violation of rotational and/or translational invariance in the mechanism that generates the primordial density fluctuations. Motivated by this we study the...

Dynamics has been generalized to a noncommutative phase space. The noncommuting phase space is taken to be invariant under the quantum group GL q,p (2). The q-deformed differential calculus on the phase space is formulated and using this, both the Hamiltonian and Lagrangian forms of dynamics have been constructed. In contrast to earlier forms of q-dynamics, our formalism has the advantage of pr...

The rotational invariance of energy is ensured by the implicit sum over the indices (i, j) in the above expression. In the Fourier representation, the energy depends on the quantities q, |u|, and (q · u) which are clearly independent of rotations. For other lattices, there are more elastic coefficients, since the energy should only be invariant under lattice rotations. For example, the symmetry...

The rotational dimension is a minor-monotone graph invariant related to the of Euclidean space containing spectral embedding corresponding first nonzero eigenvalue Laplacian, which introduced by Göring, Helmberg and Wappler. In this paper, we study dimensions graphs contain large complete graphs. characterized its dimension. It will be obtained that chordal may made while keeping constant.