We study Muirhead-type generalizations of families of inequalities due to Newton, Maclaurin and others. Each family is defined in terms of a commonly used basis of the ring of symmetric functions in n variables. Inequalities corresponding to elementary symmetric functions and power sum symmetric functions are characterized by the same simple poset which generalizes the majorization order. Some ...

Experimental demonstrations of two edge-pumped zigzag slab lasers are presented. The Nd:YAG slab laser generated 127 W of multimode output power with 300W of pump power. Preliminary results with a Yb:YAG slab produced 46 W of output power with 315W of pump power. The edge-pumped slab design permits symmetric conduction cooling and efficient pump absorption and accepts large-numerical-aperture p...

The symmetric group S2n and the hyperoctaheadral group Hn is a Gelfand triple for an arbitrary linear representation φ of Hn. Their φ-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet are always to be a Gelfand triple. Furthermore we study the rela...

In the study of the algebra NCSym of symmetric functions in noncommutative variables, Bergeron and Zabrocki found a free generating set consisting of power sum symmetric functions indexed by atomic partitions. On the other hand, Bergeron, Reutenauer, Rosas, and Zabrocki studied another free generating set of NCSym consisting of monomial symmetric functions indexed by unsplitable partitions. Can...

We interpret divided power structures on the homotopy groups of simplicial commutative rings as having a counterpart in divided power structures on chain complexes coming from a non-standard symmetric monoidal structure.