‎In this paper‎, ‎we study translation invariant surfaces in the‎ ‎3-dimensional Heisenberg group $rm Nil_3$‎. ‎In particular‎, ‎we‎ ‎completely classify translation invariant surfaces in $rm Nil_3$‎ ‎whose position vector $x$ satisfies the equation $Delta x = Ax$‎, ‎where $Delta$ is the Laplacian operator of the surface and $A$‎ ‎is a $3 times 3$-real matrix‎.

In this paper we de ne a metric on the collection of all translation invarinat spaces on a locally compact abelian group and we study some properties of the metric space.

‎In this paper‎, ‎we investigate the invariant elements of the dual mod $p$ Steenrod subalgebra ${mathcal{A}_p}^*$ under the conjugation map $chi$ and give bounds on the dimensions of $(chi-1)({mathcal{A}_p}^*)_d$‎, ‎where $({mathcal{A}_p}^*)_d$ is the dimension of ${mathcal{A}_p}^*$ in degree $d$‎.

Let S be a locally compact foundation semigroup with identity and                          be its semigroup algebra. Let X be a weak*-closed left translation invariant subspace of    In this paper, we prove that  X  is invariantly  complemented in   if and  only if  the left ideal  of    has a bounded approximate identity. We also prove that a foundation semigroup with identity S is left amenab...

‎in this paper‎, ‎we study translation invariant surfaces in the‎ ‎3-dimensional heisenberg group $rm nil_3$‎. ‎in particular‎, ‎we‎ ‎completely classify translation invariant surfaces in $rm nil_3$‎ ‎whose position vector $x$ satisfies the equation $delta x = ax$‎, ‎where $delta$ is the laplacian operator of the surface and $a$‎ ‎is a $3 times 3$-real matrix‎.

We experimentally demonstrate {0.6 dB of photocurrent noise reduction below the shot-noise limit ({1.4 dB without losses) in a phase-sensitive ampliier (nonlinear Sagnac interferometer) using two orthogonally polarized pulses to cancel the GAWBS noise.

A compact, orthogonally polarized, gain-switched a-cut Ho:LuLiF4 laser with intra-cavity pumping by a self-Q-switched Tm:YAP is demonstrated here for the first time. The π-polarization at 2052 nm and σ-polarization 2066 were experimentally observed maximum output power values of 299 mW 126 mW, respectively, two polarization directions always kept mutually orthogonal as pump increased. ratio bet...