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Register a new userA semigroup is finite if and only if it is chain-finite and antichain-finite
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A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xyin{x,y}$ ($xy otin{x,y}$) for any (distinct) elements $x,yin S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $sqrt[infty]{e}={xin S:exists ninmathbb N;;(x^n=e)}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.
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A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $G$. A group $G$ is then said to be CCA if every connected Cayley graph on $G$ is CCA. We show that a finite simple group is CCA if and only if it has no element of order 4. We also show that many 2-groups are non-CCA.
We compute the cogrowth series for Baumslag-Solitar groups $mathrm{BS}(N,N) = < a,b | a^N b = b a^N > $, which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.
Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.
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Pascal Fries Ernst Strungmannn Institute for Neuroscience in Cooperation with Max Planck Society
2021
The field of in-vivo neurophysiology currently uses statistical standards that are based on tradition rather than formal analysis. Typically, data from two (or few) animals are pooled for one statistical test, or a significant test in a first animal is replicated in one (or few) further animals. The use of more than one animal is widely believed to allow an inference on the population. Here, we explain that a useful inference on the population would require larger numbers and a different statistical approach. The field should consider to perform studies at that standard, potentially through coordinated multi-center efforts, for selected questions of exceptional importance. Yet, for many questions, this is ethically and/or economically not justifiable. We explain why in those studies with two (or few) animals, any useful inference is limited to the sample of investigated animals, irrespective of whether it is based on few animals, two animals or a single animal.
Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results from the bottom level up by determining completely the non-cyclic finite p-groups whose number of subgroups among p-groups of a given order is minimal.
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