A representation of the configurations and evolution of metamorphic mechanisms

Metamorphic mechanisms are members of the class of mechanisms that are able to change their configurations sequentially to meet different requirements. The paper introduces a comprehensive symbolic matrix representation for characterizing the topology of one of these mechanisms in a single configuration using general information concerning links and joints. Furthermore, a matrix representation of an original metamorphic mechanism that has the ability to evolve is proposed by uniting the matrices representing all of the mechanism's possible configurations. The representation of metamorphic kinematic joints is developed in accordance with the variation laws of these mechanisms. By introducing the joint variation matrices derived from generalized operations on the related symbolic adjacency matrices, evolutionary relationships between mechanisms in adjacent configurations and the original metmaorphic mechanism are made distinctly. Examples are provided to demonstrate the validation of the method.

Composition of infinitary structures

Настоящая статья является продолжением рассмотрения полиморфных свойств троичных символьных матриц (TSM - Ternary Symbolic Matrix) над алфавитом $A=\{0,1,2\}$ как биекций кратчайших $k$-мерных путей между антиподальными вершинами ($skap$-путей) в $n$-кубе. Отображение TSM на структуру $k$-арного глобального дерева ($GTk$) определено как генетическое пространство $T(k)$ $skap$-путей. Автоморфизм TSM индуцирует нумерацию вершин $T(k)$ множеством натуральных чисел $\mathbb{N}$. С позиций такой структуры рассматриваются арифметическая геометрия $skap$-путей и свойства симметричности простых чисел относительно натуральных. В основу исследования симметричности простых предложены разностный таблоид DT (Difference Tabloid) и конструктивный метод оценки его наполнения как индикатора метрических отношений между натуральными и простыми числами. The infinitary structure of an $n$-cube, global $k$-ary trees, and natural numbers are considered as a single genetic structure. A number of geometric characteristics of the shortest paths in an $n$-cube are specified and the properties of prime number symmetry among the natural numbers are studied on the basis of this structure.

“The Five Fallen” as a Meta-image of Polish Culture in the Mid-19th Century

The article is devoted to the national manifestations of 1861 and its consequences. Photography played an important role in warming up the atmosphere of Warsaw street, especially the photos of "The Fallen Five" taken by Karol Beyer (1818–1877), a famous Warsaw photographer. The author analyses the ways of using photos in the context of contemporary visual and performative practices. The tableau representing the fallen becomes a cultural meta-image – a representation linking current emotions and a durable symbolic matrix, documentary features of photography, and persistent religious-national patterns.

On the fixation probability of superstars

The Moran process models the spread of genetic mutations through populations. A mutant with relative fitness r is introduced and the system evolves, either reaching fixation (an all-mutant population) or extinction (no mutants). In a widely cited paper, Lieberman et al. (2005 Evolutionary dynamics on graphs. Nature 433 , 312–316) generalize the model to populations on the vertices of graphs. They describe a class of graphs (‘superstars’), with a parameter k and state that the fixation probability tends to 1− r − k as the graphs get larger: we show that this is untrue as stated. Specifically, for k =5, we show that the fixation probability (in the limit, as graphs get larger) cannot exceed 1−1/ j ( r ), where j ( r )= Θ ( r 4 ), contrary to the claimed result. Our proof is fully rigorous, though we use a computer algebra package to invert a 31×31 symbolic matrix. We do believe the qualitative claim of Lieberman et al. —that superstar fixation probability tends to 1 as k increases—and that it can probably be proved similarly to their sketch. We were able to run larger simulations than the ones they presented. Simulations on graphs of around 40 000 vertices do not support their claim but these graphs might be too small to exhibit the limiting behaviour.

Zhuangzi’s concept of harmony and its cultural implications

Tamkang UniversityThe ancient Daoist philosopher Zhuangzi (beginning of the 4th century BC), whose views comprise the core of the book ascribed to him, offers a profound concept of harmony, the basic condition of which is the differential relation within the continuity of universal change. Harmony for Zhuangzi is the predetermined or rather in-determined power of self-affection which constitutes the nature of life. As such it stands for the symbolic matrix of experience anticipating the world of things. This idea of harmony lies at the origin of creativity and style in culture. Zhuangzi’s philosophy is neither nihilistic nor apologetic in relation to actual cultures but provides, as it were, a comment on the conditions of the formation of culture.