애니메이션 <벼랑 위의 포뇨>에 나타난 상징과 상상

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

Group theory, an area in mathematics, has undergone extensive research, but not exhaustively. In group theory, especially symmetric groups have been studied in terms of properties and actions. In this research, two groups of the Symmetric Group (Alternating and Dihedral Groups) are studied in terms of their properties and also their direct product between them. Also, ordered sets are studied in this research where their Cartesian product is looked at. Finally, this research combines the direct product of two symmetric groups (alternating group and dihedral group) and the Cartesian product of two ordered sets (<I>X</I>×<I>Y</I>) by group action. This research therefore focuses on determining the combinatorial properties (transitivity and primitivity), invariants (ranks and subdegrees), and structures (suborbital graphs) of this group action. To accomplish this, orbit-stabilizer theorem is used to compute transitivity, block action concept is applied to determine primitivity, and Cauchy-Frobenius lemma is used to compute ranks and subdegrees. For <i>n</i> ≥ 3, it has been confirmed that the group action is transitive and also imprimitive. The rank of this group action is 6 and the subdegrees are obtained using the formula of Theorem 3.3. This research adds new concepts in group theory which will be useful in other areas like graph theory and also application in real life situations.

Weights for symmetric and general linear groups

A foundational area of mathematics, group theory has broad applications in many scientific domains. An integrated review of its use in cryptography and chemistry is presented in this work. Group theory simplifies structural predictions in chemistry by making it easier to analyze molecular symmetries, molecular orbitals, and spectroscopy. Group theory in cryptography provides the security underpinnings of contemporary encryption techniques, especially through discrete logarithm issues, homomorphisms, and permutation groups. We examine group-theoretical isomorphisms, homomorphism stability, and representations using both recent and classic literature, emphasizing their dual applicability in the information and biological sciences. Opportunities for interdisciplinary advances, such as improved chemical modeling and post-quantum cryptography protocols, are suggested by the synergy between these domains.

Set-theoretic solutions of the Yang–Baxter equation, braces and symmetric groups

The usage of Machine Learning in analyzing and creating art has accelerated substantially over the last couple of years, being used to mass produce novel pieces on a scale that artists would never have imagined feasible previously. This paper will focus on the study of an important Computer Vision tool- Generative Adversarial Networks (GANs) and whether or not it can learn and replicate the visually distinctive, defining characteristics of different forms of visual art, specifically rural Indian art forms. Indian rural art is a rich, underexplored area of visual media that consists of highly stylized, distinctive characteristics. We trained GANs to learn, abstract, and reproduce the visual motifs of five different forms of Indian rural art to understand whether GANs can learn these defining artistic attributes unsupervised and further our understanding of the differences between human intuition and machine perception of art. The textural and feature extraction ability will be analyzed through human perception as well as Fréchet Inception Distance to evaluate the quality of the generated images. This will benefit artists by allowing them to create artworks in unexplored genres, as well as benefit commercial industries, such as home décor and fashion, through the creation of different products in various art forms in an efficient manner.

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements that are moved (not fixed) by any non-identity automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of an infinite sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains. We note connections of the subject to computational group theory, asymptotic group theory, highly regular structures, and the Graph Isomorphism problem, and list a number of open problems.

If V is a (possibly infinite) set, G a permutation group on $${V, v\in V}$$ , and Ω is an orbit of the stabiliser G v , let $${G_v^{\Omega}}$$ denote the permutation group induced by the action of G v on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G v and $${G_v^\Omega}$$ . If G is primitive and G v is finite, then by a theorem of Betten et al. (J Group Theory 6:415–420, 2003) we can conclude that every composition factor of the group G v is also a composition factor of the group $${G_v^{\Omega(v)}}$$ . In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If $${\Omega=u^{G_v}}$$ is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N ≤ N Sym(V)(G) such that the N-orbital $${\{(v^g,u^g) \mid u\in \Omega, g\in N\}}$$ is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G v is also a section of $${G_v^\Omega}$$ . To demonstrate that the topological assumptions on G and the simple sections of G v cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G v is isomorphic to the modular group $${{\rm PSL}(2,\mathbb{Z}) \cong C_2*C_3}$$ , which is known to have infinitely many finite simple groups among its sections.

We consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

In this paper I make the case that at least some tattoos are artworks. I go on to propose a definition of tattoo art that distinguishes it from other uses of tattooing, and from other forms of visual art. I argue that tattoo art is an art form that creates artworks in living skin, and that the living body is an essential component of and contributor to the artwork. This gives rise to several other distinctive features of tattoo art, in particular that it reconfigures in interesting ways the relationships between artist, artworks, and viewers. Like street art, tattoo art by its nature resists inclusion in and valuation through art institutions such as galleries and museums.

Comics are widely loved by fans around the world as a form of visual art and cultural communication. With the development of digitalization, automated comic content detection and segmentation and comic coloring systems have become important research directions for digital archiving, automatic translation, and visual content analysis. This paper proposes a composite network composed of efficient content extraction and colorization, which includes a comic extraction module and a comic colorization module based on an improved Generative Adversarial Network. It solves the problem of single performance and poor effect that has been present in previous models. In various performance comparison experiments, our model shows an excellent and robust performance.

The paper aims to trace some features of Psychedelic art, as a form of visual art in Judith Thompson’s play Lion in the Street (1990). These characteristics are explored against the backdrop of C. G. Jung’s theory of Individuation.Psychedelic art corresponds to the analysis of the play as it explores the thoughts about personalized perceptions and tries to reflect them. In other words, it is a way to externalize the internal feelings stored in the unconscious.The core of Jungian theory is “to experience the unconscious with its vital imagery, affective power and strong spiritual quality” (Martin, p. 257). The purpose of Jung’s Individuation process is to increase the individual’s consciousness to heal the splits between the conscious and the unconscious, bringing them to wholeness to overcome the trauma of the individual’s psyche. The three stages of Individuation: Personal Unconscious, Shadow (Anima/Animus), and Self, are employed to analyze the characters process of moving from the unconscious to the conscious to reach a new realm of consciousness that boosts wholeness and freedom.Judith Thompson (1954- ) is a renow

Historical Perception of Visual Art Forms in Anlo Traditional Marriage Rites in Ghana Godwin Gbadagba, Agbeyewornu Kofi Kemevor (PhD), Prosper Kwaku Asiedu Abstract In this study, an attempt was made to evaluate the symbolic relevance of visual art forms in Anlo traditional marriage rites in Ghana. The study population consisted of Custodians: Chiefs, Elders and married couples in Anloga and Keta. Through a non-probability sampling technique, precisely purposive technique, a sample size of twenty (20) respondents were chosen; thus, a chief, three elders and sixteen married couples. A multi approach data collection technique was employed which included observation, interview and focus group discussion for data collection. The study showed that marriage is a very important rite among the Anlo people, and every stage of the marriage process has visual effects which are symbolic.The philosophies and concepts behind these visual art forms become the norms and ethics which bind the society. Similarly, the society is taught to make art forms not just for their aesthetics but also as a way of preserving the culture of the people. The study recommended that inhabitants of Anlo should be educated on the importance of these artifacts. Besides, the preservation of cultural heritage should be the responsibility of every member of the society. Full Text: PDF DOI: 10.15640/ijaah.v8n1a7

We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(\Omega , \mu )$ admits an invariant $\sigma $-finite measure equivalent to $\mu $. Second, we prove the following de Finetti-type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^{M}$, where $Z$ is a Polish space, is a product measure.

We prove the lattice-ordered group analogues of two easy results from group theory. Theorem A Let G,H be lattice-ordered groups with soluble word problem. Then the free product of G and H (in the category of lattice-ordered groups) has soluble word problem. Theorem B Let G be a lattice-ordered group and H a convex sublattice subgroup of G. Then G can be `-embedded in L, where L has presentation 〈G, t : tht = h (h ∈ H)〉 in the category of lattice-ordered groups. If g ∈ G, then in L, [t, g] = 1 iff g ∈ H, and if f , g are finite subsets of G (which may overlap), then w(f ,g) 6= 1 in G implies w(tf t,g) 6= 1 in L. The proofs use permutation groups, a technique of Holland and McCleary, and the ideas used to prove the lattice-ordered group analogue of the Boone-Higman Theorem. —————————————– AMS Classification: 06F15, 20F60, 20B27.