Cocyclic Hadamard Matrices: An Efficient Search Based Algorithm

This dissertation serves as the culmination of three papers. “Counting the decimation classes of binary vectors with relatively prime fixed-density presents the first non-exhaustive decimation class counting algorithm. “A Novel Approach to Relatively Prime Fixed Density Bracelet Generation in Constant Amortized Time presents a novel lexicon for binary vectors based upon the Discrete Fourier Transform, and develops a bracelet generation method based upon the same. “A Novel Legendre Pair Generation Algorithm expands upon the bracelet generation algorithm and includes additional constraints imposed by Legendre Pairs. It further presents an efficient sorting and comparison algorithm based upon symmetric functions, as well as multiple unique Legendre Pairs

Gray code order for Lyndon words

International audienceAt the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order

Exercise as Labour: Quantified Self and the Transformation of Exercise into Labour

The recent increase in the use of digital self-tracking devices has given rise to a range of relations to the self often discussed as quantified self (QS). In popular and academic discourse, this development has been discussed variously as a form of narcissistic self-involvement, an advanced expression of panoptical self-surveillance and a potential new dawn for e-health. This article proposes a previously un-theorised consequence of this large-scale observation and analysis of human behaviour; that exercise activity is in the process of being reconfigured as labour. QS will be briefly introduced, and reflected on, subsequently considering some of its key aspects in relation to how these have so far been interpreted and analysed in academic literature. Secondly, the analysis of scholars of “digital labour” and “immaterial labour” will be considered, which will be discussed in relation to what its analysis of the transformations of work in contemporary advanced capitalism can offer to an interpretation of the promotion and management of the self-tracking of exercise activities. Building on this analysis, it will be proposed that a thermodynamic model of the exploitation of potential energy underlies the interest that corporations have shown in self-tracking and that “gamification” and the promotion of an entrepreneurial selfhood is the ideological frame that informs the strategy through which labour value is extracted without payment. Finally, the potential theoretical and political consequences of these insights will be considered

Creation of Cornerstone: financial plan

Treball Final de Grau en Finances i Comptabilitat. Codi: FC1049. Curs acadèmic: 2019/2020The purpose of the study is to verify the viability of the start-up of the Cornerstone company, importing and importing and selling products in the fashion sector, specifically accessories: necklaces and bracelets. Throughout the analysis, the components that intervene in the creation of a company can be observed, from the figure of the entrepreneur and its impact on the country's economy, going through the creation of a business plan, its financial analysis and ending with the creation of said project

궤도 조화 이론과 순환체 현상

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2021.8. 국웅.Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point locus in a complex space \CC^n. The cyclic sieving phenomenon is a notion in enumerative combinatorics which encapsulates the fixed-point structure of the action of a finite cyclic group $C$ on a finite set $X$ in terms of root-of-unity evaluations of an auxiliary polynomial $X(q)$. In this thesis, we apply orbit harmonics to prove a variety of cyclic sieving results. This includes cyclic sieving results involving enumerations of combinatorial objects such as words, graphs or matrices, and symmetric functions such as Hall--Littlewood polynomials or Macdonald polynomials.궤도 조화 이론은 유한집합 $X$에 대한 선형군 $G$가 작용할 때, $X$를 복소공간 \CC^n안의 $G$-안정한 점 자취로 이해함으로써 $X$에 대한 $G$의 작용을 등급환 구조를 갖는 다항식환으로 이해할 수 있게 해주는 조합적 표현론 분야의 도구이다. 순환체현상은 유한 순환 군 $C$가 유한집합 $X$에 작용할 때 고정점에 대한 정보를 특정 다항식의 단위근에서의 값을 통해 알아낼 수 있는 현상에 관한 세는 조합론 분야의 주제이다. 본 학위논문에서는 궤도 조화 이론을 적용하여 다양한 순환체 현상을 증명하는 방법을 다룬다. 특히, 조합적인 대상인 단어, 그래프, 행렬의 개수와 관련한 , 다른 한 편으로는 홀-리틀우드 다항식이나 맥도날드 다항식과 같은 대칭함수 이론의 중요한 예시들과 관련한 순환체현상을 제시한다.1 Introduction 1 2 Preliminaries 7 2.1 Combinatorics 7 2.2 Symmetric functions 9 2.3 Representation theory of the symmetric group 12 2.4 Complex reflection groups 13 2.5 Regular sequences 15 3 Classical results and sieving generating theorem 16 3.1 Springer’s theorem on regular elements 16 3.2 A theorem of Morita-Nakajima via Orbit Harmonics 19 3.3 Loci and Sieving 22 4 The Functional Loci 25 4.1 The Quotient Rings 26 4.2 Bicyclic Sieving 30 4.3 The subgroup $G=\mathfrak{S}_n$ 32 4.4 The subgroup $G=C_n$ 34 4.5 The subgroup $G=H_r$ 36 5 Other Combinatorial Loci 40 5.1 The Tanisaki Locus 40 5.2 The Springer Locus 42 5.3 Loci of colored words 44 6 Macdonald polynomials and cyclic sieving 49 6.1 Main theorems 49 6.2 Modules of Garsia and Haiman 52 6.3 Diagonal orbit harmonics and cyclic sieving 53 6.4 Orbit harmonics and Garsia–Haiman module 57 6.5 Proofs of main theorems 58 6.5.1A proof of Theorem 6.1.1 58 6.5.2A proof of Theorem 6.1.2 60 7 Concluding remarks 64 7.1 A conjecture for Macdonald polynomials for rectangles 64 7.2 Other combinatorial loci 65 7.3 CSP for bracelets 66 Abstract (in Korean) 72 Acknowledgement (in Korean) 73박

Musical Haptics

Haptic Musical Instruments; Haptic Psychophysics; Interface Design and Evaluation; User Experience; Musical Performanc