Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking

We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera. The number of robots that can be uniquely identified is limited by the number of colours available, $q$, the number of lights on each robot, $k$, and the number of consecutive lights the camera can see, $\ell$. For a given set of parameters, we would like to maximise the number of robots that we can use. We model this as a combinatorial problem and show that it is equivalent to finding the maximum number of disjoint $k$-cycles in the de Bruijn graph $\text{dB}(q,\ell)$. We provide several existence results that give the maximum number of cycles in $\text{dB}(q,\ell)$ in various cases. For example, we give an optimal solution when $k=q^{\ell-1}$. Another construction yields many cycles in larger de Bruijn graphs using cycles from smaller de Bruijn graphs: if $\text{dB}(q,\ell)$ can be partitioned into $k$-cycles, then $\text{dB}(q,\ell)$ can be partitioned into $tk$-cycles for any divisor $t$ of $k$. The methods used are based on finite field algebra and the combinatorics of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied Mathematic

Two Reflected Gray Code based orders on some restricted growth sequences

We consider two order relations: that induced by the m-ary reflected Gray code and a suffix partitioned variation of it. We show that both of them when applied to some sets of restricted growth sequences still yield Gray codes. These sets of sequences are: subexcedant or ascent sequences, restricted growth functions, and staircase words. In each case we give efficient exhaustive generating algorithms and compare the obtained results

Loopless Algorithms to Generate Maximum Length Gray Cycles wrt. k-Character Substitution

Given a binary word relation $\tau$ onto $A^*$ and a finite language $X\subseteq A^*$, a $\tau$-Gray cycle over $X$ consists in a permutation $\left(w_{[i]}\right)_{0\le i\le |X|-1}$ of $X$ such that each word $w_{[i]}$ is an image under $\tau$ of the previous word $w_{{[i-1]}}$. We define the complexity measure $\lambda_{A,\tau}(n)$, equal to the largest cardinality of a language $X$ having words of length at most $n$, and s.t. some $\tau$-Gray cycle over $X$ exists. The present paper is concerned with $\tau=\sigma_k$, the so-called $k$-character substitution, s.t. $(u,v)\in\sigma_k$ holds if, and only if, the Hamming distance of $u$ and $v$ is $k$. We present loopless (resp., constant amortized time) algorithms for computing specific maximum length $$\sigma_k$-Gray cycles.Comment: arXiv admin note: text overlap with arXiv:2108.1365

Combinatorial generation via permutation languages

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an $n$-element set by adjacent transpositions; the binary reflected Gray code to generate all $n$-bit strings by flipping a single bit in each step; the Gray code for generating all $n$-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an $n$-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into $n$ rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group~$S_n$. Recently, Pilaud and Santos realized all those lattice congruences as $(n-1)$-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope

The word problem and combinatorial methods for groups and semigroups

The subject matter of this thesis is combinatorial semigroup theory. It includes material, in no particular order, from combinatorial and geometric group theory, formal language theory, theoretical computer science, the history of mathematics, formal logic, model theory, graph theory, and decidability theory. In Chapter 1, we will give an overview of the mathematical background required to state the results of the remaining chapters. The only originality therein lies in the exposition of special monoids presented in §1.3, which uni.es the approaches by several authors. In Chapter 2, we introduce some general algebraic and language-theoretic constructions which will be useful in subsequent chapters. As a corollary of these general methods, we recover and generalise a recent result by Brough, Cain & Pfei.er that the class of monoids with context-free word problem is closed under taking free products. In Chapter 3, we study language-theoretic and algebraic properties of special monoids, and completely classify this theory in terms of the group of units. As a result, we generalise the Muller-Schupp theorem to special monoids, and answer a question posed by Zhang in 1992. In Chapter 4, we give a similar treatment to weakly compressible monoids, and characterise their language-theoretic properties. As a corollary, we deduce many new results for one-relation monoids, including solving the rational subset membership problem for many such monoids. We also prove, among many other results, that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. In Chapter 5, we study context-free graphs, and connect the algebraic theory of special monoids with the geometric behaviour of their Cayley graphs. This generalises the geometric aspects of the Muller-Schupp theorem for groups to special monoids. We study the growth rate of special monoids, and prove that a special monoid of intermediate growth is a group

Building the American Republic, Volume 2

Building the American Republic combines centuries of perspectives and voices into a fluid narrative of the United States. Throughout their respective volumes, Harry L. Watson and Jane Dailey take care to integrate varied scholarly perspectives and work to engage a diverse readership by addressing what we all share: membership in a democratic republic, with joint claims on its self-governing tradition. It will be one of the first peer-reviewed American history textbooks to be offered completely free in digital form. Visit buildingtheamericanrepublic.org for more information. The American nation came apart in a violent civil war less than a century after ratification of the Constitution. When it was reborn five years later, both the republic and its Constitution were transformed. Volume 2 opens as America struggles to regain its footing, reeling from a presidential assassination and facing massive economic growth, rapid demographic change, and combustive politics. The next century and a half saw the United States enter and then dominate the world stage, even as the country struggled to live up to its own principles of liberty, justice, and equality. Volume 2 of Building the American Republic takes the reader from the Gilded Age to the present, as the nation becomes an imperial power, rethinks the Constitution, witnesses the rise of powerful new technologies, and navigates an always-shifting cultural landscape shaped by an increasingly diverse population. Ending with the 2016 election, this volume provides a needed reminder that the future of the American republic depends on a citizenry that understands—and can learn from—its history

ON THE RUN, July 6, 1978

Issue No. 2 Jeff Wells Runs for God\u27s Glory by Paul PerryJeff Wells may be fast, but he never runs alone. The near winner at Boston has a constant companion who spurs him on to greater achievements. AIM Runs In Protest by Fritz Busch The American Indian Movement has included running in its latest protest against the US Government. But many on Capitol Hill think AIM may be running for nothing. BB Power by Lee StraightWant to increase your power as a runner while reducing your pain? Many Japanese runners claim they are doing just that with tiny silver balls. The Running Comic by Ronnie Crystal Did you hear the one about the weakling who turned his life around by running? Comedian Fred Ttavalena was that weakling. Now the funny man runs daily-even if it means getting a few miles in between shows in Las Vegas.https://openprairie.sdstate.edu/on_the_run/1001/thumbnail.jp

Love and Marriage: Domestic Relations and Matrimonial Strategies Among the Enslaved in the Atlantic World

Love and Marriage: Domestic Relations and Matrimonial Strategies Among the Enslaved in the Atlantic World argues that the cultural and sociopolitical dimensions of slave marriage were primary issues for diasporic Africans, abolitionists, and proslavery apologists whose lives were intertwined by the cultural and economic connections that framed the Atlantic World throughout the seventeenth, eighteenth, and nineteenth centuries. Through analyzing the interplay between legislation, cultural practice, and political discourse in the early periods of colonial slavery, I first show how matrimonial patterns from Atlantic Africa and Britain were re-imagined by diasporic Africans enslaved in Bermuda, the British West Indies, and colonial North America. Subsequent chapters then illuminate how matrimonial precedents established in these interconnected British territories influenced how both free and enslaved Americans approached the legislative restrictions that characterized slave marriage in the nineteenth-century American South. While past analyses have addressed the social, cultural, and legal dimensions of slave matrimony in specific regions, slave marriage was imbedded within transatlantic discourses that influenced the cultural and political maneuvers of blacks and whites throughout the British Atlantic. Five of my eight chapters specifically concentrate upon the internal dynamics of slave marital relations, and reveal how African-descended peoples reckoned with the circumstances of slavery by creatively re-imagining ancestral marital practices and appropriating foreign customs in Anglophone slave societies. Additionally, I use gender, class, and sexuality as analytical paradigms to explore how the concepts of masculinity, femininity, domesticity, homosociality, social status, and domestic authority were re-imagined by Atlantic Africans and their descendants in the Americas. The remaining three chapters examine how British abolitionism in the early nineteenth century impacted slave legislation and reform in North America up to 1865. As British abolitionists gained public support in the early 1800s their actions spurred the simultaneous developments of a more vociferous North American abolitionist movement, as well as a formidable unit of proslavery apologists. Abolitionists cited the rupture of slaves\u27 domestic relations as the most abhorrent feature of the slave regime, while slavery\u27s apologists cited examples of lavish slave weddings to demonstrate their supposed paternalistic approach toward enslaved laborers. As slave societies crumbled around them, white southerners followed previous examples established by British reformers in proposing that slaves\u27 marital unions be legalized, hoping that this maneuver would promote the system\u27s survival by making it more palatable to both American and international critics. While the reforms were largely ineffective in curtailing the master\u27s authority, the appeal to the marital contract as a remedy for slaves\u27 circumstances is highly suggestive of the subject\u27s importance throughout the nineteenth century. For multiple generations slaves who were ensnared within Anglophone slavery survived multiple events that violated their marital rights, such as the passage from Atlantic Africa to the Americas; Caribbean sugar production; internal slave trades; and the turbulence of warfare. These multifaceted examples plainly reveal the violence done to slaves\u27 domestic relations, and in turn prompt a single, critical question: What does marriage mean to people who are unable to access a legally-recognized domestic relationship? By grounding my analysis in ex-slave narratives, memoirs, plantation journals, political tracts, and court cases I answer this query by revealing the intricate details of slaves\u27 private lives and how their maneuvers to secure companionship were influential in challenging and overturning the brutal conditions of Atlantic slavery

Building Bridges: Church Women United and Social Reform Work Across the Mid-Twentieth Century

Church Women United incorporated in December 1941 as an interdenominational and interracial movement of liberal Protestant women committed to social reform. The one hundred organizers represented ten million Protestant women across the United States. They organized with the express purposes of helping to bring peace on Earth and to develop total equality within all humanity. Church Women United was the bridge between the First and Second Wave of Feminism and the bridge between the Social Gospel and Social Justice Movements. Additionally they connected laterally with numerous social and religious groups across American society. As such, they exemplify the continuity and matrix of reform in American history. Because they worked to promote international peace, develop positive race relations, and advance women’s rights, their campaigns give us a model for how to rectify the social problems of today. These women used communal prayer, politics, education, and hands-on labor to promote their ideas. They originated in collective prayer and continued this tool, but they added letter writing campaigns, public education forums, and lobbying politicians at all levels including the president to advance their goals. They held massive campaigns to collect needed items for war-torn countries and natural disaster areas as well as acting as counselors to the needy. They raised public awareness of issues facing migrant laborers, inner-city residents, Native Americans, Japanese internment detainees, and then worked hard to ameliorate the worst of these problems. They promoted literacy around the world, as well as new agricultural techniques to address human conditions that were known to lead to political and social unrest. This dissertation covers the mid-twentieth century while being predominately focused on the years 1941-1968. This study is built upon multiple archives across the United States and oral histories of movement leaders. It is one of the first interdenominational studies focused on the work of women in social reform work. This dissertation enlarges our knowledge of feminism and social reform work