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The units of a number field k act naturally on the real vector space k ⊗ Q R , and so on open subsets of (k ⊗ Q R) ⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k , whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for totally complex number fields has been published. Here we give a quick way to compute Shintani domains for totally complex quartic number fields. To construct these domains we exploit the existence of a forward attractor and backward repeller set for this action. We prove that the number of polyhedral cones needed for the Shintani domain is bounded by an absolute constant, a fact previously known only for cubic or quadratic fields. We also show that our algorithm for finding a Shintani domain runs in polynomial time and we give a table describing the results of running the algorithm on more than 168,000 fields. [ABSTRACT FROM AUTHOR]

Any complex-analytic vector bundle E admits naturally defined homotheties ϕ α , α ∈ C ⁎ , i.e. ϕ α is the multiplication of a local section by a complex number α. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to E. Further, we compute the automorphism supergroup of a Π-symmetric super-Grassmannian Π Gr n , k , and, using Galois cohomology, we classify the real structures on Π Gr n , k and compute the corresponding supermanifolds of real points. [ABSTRACT FROM AUTHOR]

In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

This paper is to study what we call twisted regular representations for vertex operator algebras. Let V be a vertex operator algebra, let σ 1 , σ 2 be commuting finite-order automorphisms of V and let σ = (σ 1 σ 2) − 1. Among the main results, for any σ -twisted V -module W and any nonzero complex number z , we construct a weak σ 1 ⊗ σ 2 -twisted V ⊗ V -module D σ 1 , σ 2 (z) (W) inside W ⁎. Let W 1 , W 2 be σ 1 -twisted, σ 2 -twisted V -modules, respectively. We show that P (z) -intertwining maps from W 1 ⊗ W 2 to W ⁎ are the same as homomorphisms of weak σ 1 ⊗ σ 2 -twisted V ⊗ V -modules from W 1 ⊗ W 2 into D σ 1 , σ 2 (z) (W). We also show that a P (z) -intertwining map from W 1 ⊗ W 2 to W ⁎ is equivalent to an intertwining operator of type ( W ′ W 1 W 2 ) , which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each τ -twisted V -module M with τ any finite-order automorphism of V , the coefficients of the q -graded trace function lie in D τ , τ − 1 (− 1) (V) and generate a τ ⊗ τ − 1 -twisted V ⊗ V -submodule isomorphic to M ⊗ M ′. [ABSTRACT FROM AUTHOR]

We introduce a family of rational functions R N (a , d 0 , d 1 , ... , d N) with the following property. Let d 0 , d 1 , ... , d N be equal respectively to the value of some function f (s) and the values of its first N derivatives calculated at a certain complex number a lying not too far from a zero ρ of this function. It is expected that the value of R N (a , d 0 , d 1 , ... , d N) is very close to ρ. We demonstrate this phenomenon on several numerical instances with the Riemann zeta function in the role of f (s). For example, for N = 10 and a = 0.6 + 14 i we have | R 10 (a , d 0 , d 1 , ... , d 10) − ρ 1 | < 10 − 18 where ρ 1 = 0.5 + 14.13... i is the first non-trivial zeta zero. Also we define rational functions R N , n (a , d 0 , d 1 , ... , d N) which (under the same assumptions) have values which are very close to n − ρ , that is, to the terms from the Dirichlet series for the zeta function calculated at its zero. In the case when a is between two consecutive zeros, say ρ l and ρ l + 1 , functions R N , n (a , d 0 , d 1 , ... , d N) approximate neither of n − ρ l no n − ρ l + 1 ; nevertheless, they allow us to approximate the sum n − ρ l + n − ρ l + 1 and the product n − ρ l n − ρ l + 1 and hence to calculate both n − ρ l and n − ρ l + 1 by solving corresponding quadratic equation. [ABSTRACT FROM AUTHOR]

• Using the structural dynamic modification theory, the general damped system is considered a perturbation of the un-damped system, where the modification is defined by the damping matrix. C • The complexity of the mode shapes depends on the level of orthogonality of the normal mode shapes with respect to damping matrix. C. • The modal parameters of general damped systems (poles, complex modal masses, length of mode shapes) can be expressed as linear combination of the corresponding un-damped modal parameters. • If the mass density of the system is constant, the modal mass is always equal to the product between the total mass of the structure and the magnitude of the length squared, but with a phase to introduce a complex number. In this paper the modal mass is considered in the case of general damping where it is well-known that the modal mass becomes a complex valued quantity. The aim of this paper is to present a definition of the complex modal mass, which leads to an easier physical understanding of this modal parameter. It is demonstrated that based on the structural modification theory, the complex modal mass can be expressed as a linear combination of the un-damped modal masses, where a complex coefficient matrix can be defined that depends on the general damping matrix and on the mode shapes of the damped and the un-damped system. An important finding is that an apparent mass can be defined so that the modal mass is always equal to the product between the apparent mass and the length squared. The paper includes a 2-DOF example that illustrates the influence of the amount of general damping on modal quantities that are normally considered non-sensitive to damping. [ABSTRACT FROM AUTHOR]

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to R I I -recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices. [ABSTRACT FROM AUTHOR]