Given a vertex operator algebra V, one can construct two associative algebras, the Zhu A(V) and C2-algebra R(V). This gives rise to abelian categories A(V)-Mod R(V)-Mod, in addition category of admissible modules V. In case V is rational C2-cofinite, V-modules all A(V)-modules are equivalent. However, when not rational, connection between these unclear. The goal this paper study triplet W(p), a...

We show that the category of finite-length generalized modules for singlet vertex algebra M(p), p∈Z>1, is equal to OM(p) C1-cofinite M(p)-modules, and this admits algebraic braided tensor structure Huang-Lepowsky-Zhang. Since includes uncountably many typical which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend previous work categories atypical M(...

In the paper as a new application of the Jacquet-Langlands correspondence we connect the transfer operators for different cofinite Fuchsian groups by comparing the corresponding Selberg zeta functions. 1 Transfer operator for cofinite Fuchsian groups and Selberg’s zeta function In this section we give a short summary of the generalization of Mayer’s theory [1] by following Morita [2]. In Mayer’...

In the early 1970’s S.Tennenbaum proved that all countable models of PA+∀1−Th(N) are embeddable into the reduced product N /F , where F is the cofinite filter. In this paper we show that if M is a model of PA− + ∀1 − Th(N), and |M | = א1, then M is embeddable into N/D, where D is any regular filter on ω.

We prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. This extends earlier results by Hejhal [10] and Strömbergsson [34] in dimension 2. Our proofs use spectral methods, and lead to precise estimates on the rate of convergence to equidistribution.

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ, including an explicit finite presentation for Γ. Let Γ ⊂ PSL2(R) be a Fuchsian group, a discrete group of orientationpreserving isometries of the upper half-plane H with hyperbolic metric d. A fundamental domain for Γ is a closed domain D ⊂ H such th...

We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that is exactly as random as the halting probability of a universal machine equipped with an oracle for the second jump of the halti...

If S is a cofinite set of positive integers, an “S-restricted composition of n” is a sequence of elements of S, denoted ~λ = (λ1, λ2, . . . ), whose sum is n. For uniform random S-restricted compositions, the random variable B(~λ) = ∏ i λi is asymptotically lognormal. (A precise statement of the theorem includes an error term to bound the rate of convergence.) The proof is based upon a combinat...

We investigate the problem of deciding whether a system linear equations, together with divisibility conditions on variables, has solution over holomorphy subrings global fields. obtain decidability results when we allow poles at cofinite set places, and undecidability finite places.