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Register a new userSome theorems on passing from local to global presence of properties of functions
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When given a class of functions and a finite collection of sets, one might be interested whether the class in question contains any function whose domain is a subset of the union of the sets of the given collection and whose restrictions to all of them belong to this class. The collections with the formulated property are said to be strongly join permitting for the given class (the notion of join permitting collection is defined in the same way, but without the words a subset of). Three theorems concerning certain instances of the problem are proved. A necessary and sufficient condition for being strongly join permitting is given for the case when, for some $n$, the class consists of the potentially partial recursive functions of $n$ variables, and the collection consists of sets of $n$-tuples of natural numbers. The second theorem gives a sufficient condition for the case when the class consists of the continuous partial functions between two given topological spaces, and the collection consists of subsets of the first of them (the condition is also necessary under a weak assumption on the second one). The third theorem is of a similar character but, instead of continuity, it concerns computability in the spirit of the one in effective topological spaces.
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We prove field quantifier elimination for valued fields endowed with both an analytic structure and an automorphism that are $sigma$-Henselian. From this result we can deduce various Ax-Kochen-Ersov type results with respect to completeness and the NIP property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first order theory and that this theory is NIP.
In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $mathcal F$ in a domain $Dsubset mathbb C,$ and for a positive constant $epsilon$, if for each $fin mathcal F$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$min{rho(a_f(z),b_f(z)), rho(b_f(z),c_f(z)), rho(c_f(z),a_f(z))}geq epsilon,$$ for all $zin D$, then $mathcal F$ is normal in $D$. Here, $rho$ is the spherical metric in $widehat{mathbb C}$. In this paper, we establish the high-dimension
We study the period doubling renormalization operator for dynamics which present two coupled laminar regimes with two weakly expanding fixed points. We focus our analysis on the potential point of view, meaning we want to solve $$V=mathcal{R} (V):=Vcirc fcirc h+V circ h,$$ where $f$ and $h$ are naturally defined. Under certain hypothesis we show the existence of a explicit ``attracting fixed point $V^*$ for $mathcal{R} $. We call $mathcal{R}$ the renormalization operator which acts on potentials $V$. The log of the derivative of the main branch of the Manneville-Pomeau map appears as a special ``attracting fixed point for the local doubling period renormalization operator. We also consider an analogous definition for the one-sided 2-full shift $S$ (and also for the two-sided shift) and we obtain a similar result. Then, we consider global properties and we prove two rigidity results. Up to some weak assumptions, we get the uniqueness for the renormalization operator in the shift. In the last section we show (via a certain continuous fraction expansion) a natural relation of the two settings: shift acting on the Bernoulli space ${0,1}^mathbb{N}$ and Manneville-Pomeau-like map acting on an interval.
We work with symmetric inner models of forcing extensions based on strongly compact Prikry forcing to extend some known results.
A low-dimensional version of our main result is the following `converse of the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph $K_6$ in 3-space: For any integer $z$ there are 6 points $1,2,3,4,5,6$ in 3-space, of which every two $i,j$ are joint by a polygonal line $ij$, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair disjoint 3-cycles except for ${123,456}$ is zero, and for the exceptional pair ${123,456}$ is $2z+1$. We prove a higher-dimensional analogue, which is a `converse of a lemma by Segal-Spie.z.
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