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Register a new userSome consequences of Schanuels Conjecture
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During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define $E_0 = overline{mathbb{Q}}$ Inductively, for $n geq 1$, define $E_n$ as the algebraic closure of the field generated over $E_{n-1}$ by the numbers $exp(x)=e^x$, where $x$ ranges over $E_{n-1}$. Let $E$ be the union of $E_n$, $n geq 0$. Show that Schanuels Conjecture implies that the numbers $pi, log pi, log log pi, log log log pi, ldots $ are algebraically independent over $E$. b) Try to get a (conjectural) generalization involving the field $L$ defined as follows. Define $L_0 = overline{mathbb{Q}}$. Inductively, for $n geq 1$, define $L_n$ as the algebraic closure of the field generated over $L_{n-1}$ by the numbers $y$, where $y$ ranges over the set of complex numbers such that $e^yin L_{n-1}$. Let $L$ be the union of $L_n$, $n geq 0$. We were able to prove that Schanuels Conjecture implies $E$ and $L$ are linearly disjoint over $overline{mathbb{Q}}$.
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We prove the Mumford--Tate conjecture for those abelian varieties over number fields whose extensions to C have attached adjoint Shimura varieties that are products of simple, adjoint Shimura varieties of certain Shimura types. In particular, we prove the conjecture for the orthogonal case (i.e., for the $B_n$ and $D_n^R$ Shimura types). As a main tool, we construct embeddings of Shimura varieties (whose adjoints are) of prescribed abelian type into unitary Shimura varieties of PEL type. These constructions implicitly classify the adjoints of Shimura varieties of PEL type.
Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s in mathbb N$. We investigate the Tate conjecture on the special fiber of $G(U(r,s) times U(s,r))$-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of $G(U(1,s) times U(s,1))$. We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.
This paper summarizes recent progresses in our theoretical understanding of particle acceleration at relativistic shock waves and it discusses two salient consequences: (1) the maximal energy of accelerated particles; (2) the impact of the shock-generated micro-turbulence on the multi-wavelength light curves of gamma-ray burst afterglows.
Strongly Turing determinacy, or $mathrm{sTD}$, says that for any set $A$ of reals, if $forall xexists ygeq_T x (yin A)$, then there is a pointed set $Psubseteq A$. We prove the following consequences of Turing determinacy ($mathrm{TD}$) and $mathrm{sTD}$: (1). $mathrm{ZF+TD}$ implies weakly dependent choice ($mathrm{wDC}$). (2). $mathrm{ZF+sTD}$ implies that every set of reals is measurable and has Baire property. (3). $mathrm{ZF+sTD}$ implies that every uncountable set of reals has a perfect subset. (4). $mathrm{ZF+sTD}$ implies that for any set of reals $A$ and any $epsilon>0$, (a) there is a closed set $Fsubseteq A$ so that $mathrm{Dim_H}(F)geq mathrm{Dim_H}(A)-epsilon$. (b) there is a closed set $Fsubseteq A$ so that $mathrm{Dim_P}(F)geq mathrm{Dim_P}(A)-epsilon$.
The paper compares the asymptotic of the expressions $frac {1} {x} sumlimits_{n leq x} {f(n)}$ and $sumlimits_{n leq x} {frac {f(n)} {n}}$, $frac {1} {x} sumlimits_{p leq x} {f(p)}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$. The asymptotic of sums $sumlimits_{n leq x} {frac {f(n)} {n}}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$ ($n,p$ - respectively, positive and prime numbers) are determined if the asymptotic of sums are known, respectively: $sumlimits_{n leq x} {f(n)}$,$sumlimits_{p leq x} {f(p)}$.
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