Some early modern scholars believed that Scripture provided more certain knowledge than all secular authorities – even Aristotle – or investigating nature as such. In this paper, I analyse one such attempt to establish the most reliable knowledge of nature: the so-called Mosaic physics proposed by the Reformed encyclopaedist Johann Heinrich Alsted. Although in his early works on Physica Mosaica Alsted declares that his primary aim is proving the harmony that exists between various traditions of natural philosophy, namely between the Mosaic and the Peripatetic approaches, and despite the fact that his biblical encyclopaedia of 1625 was intended to be based on a literal reading of the Bible, he never truly abandoned the Aristotelian framework of physics. What is more, in his mature encyclopaedia of 1630, he eventually openly preferred Aristotle to other naturalphilosophical traditions. I argue, therefore, that Alsted’s bold vision of Mosaic physics remained unfulfi lled and should be assessed as an unsuccessful project of early modern natural philosophy. and Někteří raně novověcí učenci byli přesvědčeni, že Písmo poskytuje jistější poznání nežli všechny světské autority – s Aristotelem včele – či zkoumání samotné přírody. Ve své studii se zabývám jednou z takových snah o získání toho nejspolehlivějšího vědění o přírodě: takzvanou mosaickou fyzikou, jak ji koncipoval reformovaný encyklopedista Johann Heinrich Alsted. Ačkoliv ve svých raných dílech, jež se Physica Mosaica týkají, Alsted za svůj nejvyšší cíl prohlašuje dokázání souladu mezi různými podobami přírodní fi losofi e, především mezi mosaickým a peripatetickým přístupem, a navzdory tomu, že jeho biblická encyklopedie z roku 1625 měla být založena na doslovném čtení Bible, Alsted ve skutečnosti nikdy neopustil aristotelská východiska přírodní fi losofi e. Co je ještě pozoruhodnější, Alstedova vrcholná encyklopedie z roku 1630 již přímo upřednostňuje aristotelismus před jinými přírodně-fi losofi ckými přístupy. Tvrdím proto, že Alstedova smělá vize mosaické fyziky zůstala nenaplněna a měla by být hodnocena jako neúspěšný projekt raně novověké přírodní filosofie.
On a bounded $q$-pseudoconvex domain $\Omega $ in $\mathbb {C}^{n}$ with a Lipschitz boundary, we prove that the $\bar {\partial }$-Neumann operator $N$ satisfies a subelliptic $(1/2)$-estimate on $\Omega $ and $N$ can be extended as a bounded operator from Sobolev $(-1/2)$-spaces to Sobolev $(1/2)$-spaces.
We prove that compactness of the canonical solution operator to $\bar \partial $ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[\mathcal P,\bar M]$ defined on the whole $L^2_{(0,1)}(\Omega ),$ where $\bar M$ is the multiplication by $\bar z$ and $\mathcal P $ is the orthogonal projection of $L^2_{(0,1)}(\Omega )$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar \partial $-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by $z $ and $\bar z$.
Let $k$ be a positive integer, and let $G$ be a simple graph with vertex set $V(G)$. A {\it $k$-dominating set} of the graph $G$ is a subset $D$ of $V(G)$ such that every vertex of $V(G)-D$ is adjacent to at least $k$ vertices in $D$. A {\it $k$-domatic partition} of $G$ is a partition of $V(G)$ into $k$-dominating sets. The maximum number of dominating sets in a $k$-domatic partition of $G$ is called the {\it $k$-domatic number} $d_k(G)$. \endgraf In this paper, we present upper and lower bounds for the $k$-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number $d(G)=d_1(G)$.
In this article we prove for $1<p<\infty $ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega $. More precisely, $\Omega $ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^1$-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $\Omega $ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection as a Fourier multiplier operator.
In this paper, we define the $M_\alpha $-integral of real-valued functions defined on an interval $[a,b]$ and investigate important properties of the $M_{\alpha }$-integral. In particular, we show that a function $f\colon [a,b]\rightarrow R$ is $M_{\alpha }$-integrable on $[a,b]$ if and only if there exists an $ACG_{\alpha }$ function $F$ such that $F'=f$ almost everywhere on $[a,b]$. It can be seen easily that every McShane integrable function on $[a,b]$ is $M_{\alpha }$-integrable and every $M_{\alpha }$-integrable function on $[a,b]$ is Henstock integrable. In addition, we show that the $M_{\alpha }$-integral is equivalent to the $C$-integral.