Článek interpretuje Humovu teorii asociace idejí především se zřetelem k podobnosti jako jednomu z principů asociace a k obecným idejím (či pojmům) jako zásadnímu produktu asociace. Na základě této interpretace autor tvrdí, že Humovo pojetí podobnosti a obecných termínů není podmíněno přijetím tzv. „mýtu daného“. V důsledku přijetí tohoto předpokladu však vyvstávají nové otázky, zejména proč asociací vznikají zrovna ty obecné pojmy, které reálně vznikají, a jak dochází k jejich intersubjektivnímu sdílení. Tyto otázky vedou k nutnosti doplnit obrázek mysli jako „zrcadla přírody“ ze začátku první knihy Humova Pojednání o lidské přirozenosti obrázkem mysli jako „zrcadla druhých“ z jeho druhé knihy., The article interprets Hume’s theory of association of ideas, primarily with respect to resemblance as one of the principles of association and to general ideas (or concepts) as a principal consequence of association. On the basis of this interpretation, the author argues that Hume’s conception of resemblance and general terms is not conditioned by the acceptance of the so-called “myth of the given”. As a result of accepting this assumption, however, new questions arise; in particular, why is it that just those general concepts arise that in fact arise and how are they intersubjectively shared. These questions lead to the need to supplement the image of the mind as a “mirror of nature” from the beginning of Hume’s A Treatise of Human Nature, book I, with the image of the mind as a “mirror of others” from book II., and Der vorliegende Artikel bietet eine Interpretation von Humes Theorie der Assoziation von Ideen, insbesondere im Hinblick auf die Ähnlichkeit als eines der Assoziationsprinzipien und auf die allgemeinen Ideen (bzw. Begriffe) als Grundprodukte der Assoziation. Auf Grundlage dieser Interpretation stellt der Autor die Behauptung auf, dass Humes Auffassung der Ähnlichkeit und der allgemeinen Begriffe nicht durch die Annahme des sog. „Mythos des Gegebenen“ bedingt ist. Infolge der Annahme dieser Voraussetzung treten jedoch neue Fragen auf, insbesondere die Frage, warum durch Assoziationen ausgerechnet jene allgemeinen Begriffe entstehen, die real entstehen, und wie es zu deren intersubjektiven Mitteilung kommt. Diese Fragen führen zu der Notwendigkeit, das Bild vom Denken als „Spiegel der Natur“ am Anfang des ersten Buches von Humes Abhandlung über die menschliche Natur durch das Bild des Denkens als „Spiegel der Anderen“ aus dem zweiten Buch zu ergänzen.
For integers $m > r \geq0$, Brietzke (2008) defined the $(m,r)$-central coefficients of an infinite lower triangular matrix $G=(d, h)=(d_{n,k})_{n,k \in\mathbb{N}}$ as $ d_{mn+r,(m-1)n+r}$, with $n=0,1,2,\cdots$, and the $(m,r)$-central coefficient triangle of $G$ as $G^{(m,r)} = (d_{mn+r,(m-1)n+k+r})_{n,k \in\mathbb{N}}. $ It is known that the $(m,r)$-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G=(d,h)$ with $h(0)=0$ and $d(0), h'(0)\not= 0$, we obtain the generating function of its $(m,r)$-central coefficients and give an explicit representation for the $(m,r)$-central Riordan array $G^{(m,r)}$ in terms of the Riordan array $G$. Meanwhile, the algebraic structures of the $(m,r)$-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of $m$ and $r$. As applications, we determine the $(m,r)$-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach., Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He., and Obsahuje bibliografii
It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $(1)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $(2)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings.
In the present paper we are concerned with convergence in $\mu $-density and $\mu $-statistical convergence of sequences of functions defined on a subset $D$ of real numbers, where $\mu $ is a finitely additive measure. Particularly, we introduce the concepts of $\mu $-statistical uniform convergence and $\mu $-statistical pointwise convergence, and observe that $\mu $-statistical uniform convergence inherits the basic properties of uniform convergence.
Let $R$ be a ring and $M$ a right $R$-module. $M$ is called $ \oplus $-cofinitely supplemented if every submodule $N$ of $M$ with $\frac{M}{N}$ finitely generated has a supplement that is a direct summand of $M$. In this paper various properties of the $\oplus $-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus $-cofinitely supplemented modules is $\oplus $-cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$-module is $\oplus $-cofinitely supplemented. In addition, if $M$ has the summand sum property, then $M$ is $\oplus $-cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of $M$.
A sign pattern $A$ is a $\pm $ sign pattern if $A$ has no zero entries. $A$ allows orthogonality if there exists a real orthogonal matrix $B$ whose sign pattern equals $A$. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for $\pm $ sign patterns with $n-1 \le N_-(A) \le n+1$ to allow orthogonality.
We prove a separable reduction theorem for $\sigma $-porosity of Suslin sets. In particular, if $A$ is a Suslin subset in a Banach space $X$, then each separable subspace of $X$ can be enlarged to a separable subspace $V$ such that $A$ is $\sigma $-porous in $X$ if and only if $A\cap V$ is $\sigma $-porous in $V$. Such a result is proved for several types of $\sigma $-porosity. The proof is done using the method of elementary submodels, hence the results can be combined with other separable reduction theorems. As an application we extend a theorem of L. Zajíček on differentiability of Lipschitz functions on separable Asplund spaces to the nonseparable setting.