We investigate the zero sets of complex exponential polynomials in one variable with only one iteration. We characterize when such polynomials have the same zero set in terms of the radical ideals. Moreover we give a bound on the multiplicity of zeros.
Let G be a finite group. A group G is called a T group if its every subnormal subgroup is normal. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of (H^G), where (H^G) is the normal closure of H in G. Using the notion of Hall normally embedded subgroups, we characterize supersolvable groups and solvable T-group. First, we prove that if every cyclic subgroup of G of order prime or 4 is Hall normally embedded in G, then G is supersolvable with a well defined structure. Second, we prove that an A-group G is supersolvable if and only if its Sylow subgroups are products of cyclic Hall normally embedded subgroups of G. Final, we show that G is a solvable T-group if and only if every p-subgroup of G is Hall normally embedded in G, for all primes (pin pi (G)).
Let ({mathfrak {Nil}}) be the class of nilpotent groups and G be a group. We call G a meta-({mathfrak {Nil}})-Hamiltonian group if any of its non-({mathfrak {Nil}}) subgroups is normal. Also, we call G a para-({mathfrak {Nil}})-Hamiltonian group if G is a non-({mathfrak {Nil}}) group and every non-normal subgroup of G is either a ({mathfrak {Nil}})-group or a minimal non-({mathfrak {Nil}}) group. In this paper we investigate the class of finitely generated meta-({mathfrak {Nil}})-Hamiltonian and para-({mathfrak {Nil}})-Hamiltonian groups.
Martingale optimal transportation has gained significant attention in mathematical finance due to its applications in pricing and hedging. A key distinguishing factor between martingale optimal transportation and traditional optimal transportation is the concept of a peacock, which refers to a sequence of measures satisfying the convex order property. In the realm of traditional optimal transportation, the Wasserstein geometry, induced by a transportation problem with the p-th power of distance as the cost, provides valuable geometric insights. This motivates us to investigate the differences between Wasserstein geometries with and without the martingale constraint. As a first step, this paper focuses on studying the topological properties of convex order, with the aim of establishing a foundational understanding for further exploration of the geometric properties of martingale Wasserstein geometry.
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