This study deals with a controversy between Leibniz and Clarke concerning the relativity of space. Although substantivalism, i.e. an approach treating space as a substance, is to be indicated as the main target of Leibniz’s attack, it has usually been replaced by Newtonian absolutism instead, as a proper opposition to Leibniz’s relationalism. However, such absolutism has not been defined ontologically, but dynamically, as if the difference between their conceptions consisted of a different approach to the inertiallity of motion. However, this would mean that while Leibniz intended to reduce all motion to an inertial one, Newton reduced it to a noninertial one instead, or that only one of them acknowledged the existence of noninertial motion at all. Nevertheless, none of them actually denied the existence of noninertial motion, and although all motion indeed seemed noninertial to Newton, Leibniz never responded to such a challenge in the course of their correspondence. and Předmětem této studie je polemika mezi Leibnizem a Clarkem ohledně relativity prostoru. Přestože za cíl Leibnizova útoku by se v kontextu této polemiky patřilo označit v prvé řadě substantivalismus, tj. přístup nakládající s prostorem jako se substancí, obvykle bývá do opozice vůči Leibnizovu relacionalismu kladen spíše newtonovský absolutismus. Vzhledem k tomu, že takto pojatý absolutismus nebývá vymezován ontologicky, nýbrž dynamicky, musel by v takovém případě rozdíl mezi jejich pojetími spočívat v odlišném přístupu k inercialitě pohybu. To by tudíž jinými slovy znamenalo, že zatímco Leibniz veškerý pohyb redukoval na inerciální, Newton jej redukoval naopak na neinerciální, případně, že pouze jeden z nich uznával existenci neinerciálních pohybů. Existenci neinerciálních pohybů však ve skutečnosti nepopíral žádný z nich, a přestože Newtonovi se neinerciálním jevil být vskutku veškerý pohyb vůbec, Leibniz na takovou výzvu v rámci clarkovské korespondence již nereagoval.
A matrix whose entries consist of elements from the set $\lbrace +,-,0\rbrace $ is a sign pattern matrix. Using a linear algebra theoretical approach we generalize of some recent results due to Hall, Li and others involving the inertia of symmetric tridiagonal sign matrices.
The inertia of an $n$ by $n$ symmetric sign pattern is called maximal when it is not a proper subset of the inertia of another symmetric sign pattern of order $n$. In this note we classify all the maximal inertias for symmetric sign patterns of order $n$, and identify symmetric sign patterns with maximal inertias by using a rank-one perturbation.
The inertia set of a symmetric sign pattern $A$ is the set $i(A)=\lbrace i(B) \mid B=B^T \in Q(A)\rbrace $, where $i(B)$ denotes the inertia of real symmetric matrix $B$, and $Q(A)$ denotes the sign pattern class of $A$. In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern $A$ in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns $A$ with zero diagonal that require unique inertia.