Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT

Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch, Ramgoolam, and Wen [Nuclear Phys.\,B {\bf 870} (2013), 530--581] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an `equivalent' form of the famous Harvey's theorem on the cyclic groups of automorphisms of compact Riemann surfaces. Our main tool is an explicit formula for the number of solutions of restricted linear congruence recently proved by Bibak et al. using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions

On an almost-universal hash function family with applications to authentication and secrecy codes

Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. MMH$^*$, which was shown to be $\Delta$-universal by Halevi and Krawczyk in 1997, is a well-known universal hash function family. We introduce a variant of MMH$^*$, that we call GRDH, where we use an arbitrary integer $n>1$ instead of prime $p$ and let the keys $\mathbf{x}=\langle x_1, \ldots, x_k \rangle \in \mathbb{Z}_n^k$ satisfy the conditions $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $t_1,\ldots,t_k$ are given positive divisors of $n$. Then via connecting the universal hashing problem to the number of solutions of restricted linear congruences, we prove that the family GRDH is an $\varepsilon$-almost-$\Delta$-universal family of hash functions for some $\varepsilon<1$ if and only if $n$ is odd and $\gcd(x_i,n)=t_i=1$ $(1\leq i\leq k)$. Furthermore, if these conditions are satisfied then GRDH is $\frac{1}{p-1}$-almost-$\Delta$-universal, where $p$ is the smallest prime divisor of $n$. Finally, as an application of our results, we propose an authentication code with secrecy scheme which strongly generalizes the scheme studied by Alomair et al. [{\it J. Math. Cryptol.} {\bf 4} (2010), 121--148], and [{\it J.UCS} {\bf 15} (2009), 2937--2956].Comment: International Journal of Foundations of Computer Science, to appea

Cacophonous Settler Grounded Normativity: Interrelationality With The More-than-human World As A Path For Decolonial Transformation

This paper proposes settler grounded normativity as a path for transforming understandings of relationships, so that settlers might become better able to engage in decolonization. The term ‘grounded normativity’ comes from Coulthard (2014). I define grounded normativity as understanding that: (1) humans are interrelated with the more-than-human world; (2) interrelationality demands that humans develop ethical relationships with others; and (3) these ethical relationships can be learned and enacted through engagement with the more-than-human world. Settler grounded normativity puts an additional focus on understanding interrelatedness with settler colonialism and how to transform settler/Indigenous relations in pursuit of ethics and justice. The three chapters of this paper reflect the three tenets of grounded normativity. Chapter One surveys Indigenous accounts of interrelationality, as well as examples of where interrelationality continues to undergird Western understandings. Chapter One also addresses where settlers and Indigenous peoples remain interrelated with settler colonialism. Chapter Two is a theoretical analysis of how interrelationality demands ethical relations, considering identity, difference, and indistinction frameworks for ethical consideration developed by Calarco (2015). Chapter Two ultimately argues that interrelationality ought to be understood as demanding ethical relations through cacophony―a concept developed by Byrd (2011). Chapter Three argues that settlers ought to pursue active roles in decolonization in order to enact ethical relations against settler colonialism. Chapter Three includes interviews with settlers engaging in decolonial work, in order to explore questions regarding how settlers ought to participate in decolonization ethically and effectively. The paper closes by considering future work that needs to be done to develop settler grounded normativity and preliminarily addresses how ethical relations might be learned and enacted through engagement with the more-than-human world

"A Canadian Climate of Mind: Passages from Fur to Energy and Beyond" by Timothy B. Leduc

Review of A Canadian Climate of Mind: Passages from Fur to Energy and Beyond by Timothy B. Leduc (2016, McGill-Queen's UP).Find full text in .pdf below

Doctor of Philosophy

dissertationFemoroacetabular impingement (FAI) describes subtle structural abnormalities, including femoral asphericity and acetabular overcoverage, which reduce clearance in the hip joint. FAI is a common cause of hip pain for young, athletic adults. The first theme of this dissertation investigated if FAI morphology is more prevalent in athletes and if physical exams could be used to identify individuals with underlying FAI morphology. In a cohort of collegiate football players, 95% were found to have radiographic abnormalities consistent with those seen in FAI patients. This finding not only suggests that athletes, such as football players, may have an increased risk for developing symptomatic FAI, but also highlights that FAI morphology may frequently occur in asymptomatic subjects. In the same cohort, radiographic measures of femoral asphericity and femoral head-neck offset were mildly correlated to maximum internal rotation. As such, athletes with diminished internal rotation in whom hip pain develops should be evaluated for FAI. Altered articulation in FAI hips is believed to cause chondrolabral damage and may lead to osteoarthritis, but FAI kinematics have not been accurately quantified. To this end, the second theme of this dissertation focused on developing, validating, and applying a dual fluoroscopy and model-based tracking protocol to accurately quantify three-dimensional in vivo hip kinematics. In a cadaver experiment, model-based tracking was compared to the reference standard, dynamic radiostereometric analysis. Model-based tracking was found to have a positional error less than 0.48 mm and rotational error was less than 0.58°. The methodology was then applied to evaluate a cohort of asymptomatic control subjects and three patients with differing FAI morphology. The results, which represent the most accurate data collected on hip kinematics to date, demonstrate that hip articulation is a highly complex process, including translation, pelvic motion, no bone contact, and labrum involvement in large ranges of motion. Collected data provide necessary baseline results for future comparison studies and could be used to validate computer simulations of impingement, guide pre-operative planning, and serve as boundary conditions in finite element models investigating chondrolabral mechanics

Restricted linear congruences

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, with $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $a_1,t_1,\ldots,a_k,t_k, b,n$ ($n\geq 1$) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of $a_i=t_i=1$ $(1\leq i \leq k)$. Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [{\it Duke Math. J.} {\bf 39} (1972), 521--527] gave a nice explicit formula for the number of such solutions when $(a_1,\ldots,a_k)=t_i=1$ $(1\leq i \leq k)$. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.Comment: Journal of Number Theory, to appea

A Note on Negative Tagging for Least Fixed-Point Formulae

We consider proof systems with sequents of the formU |- F for proving validity of a propositional modal mu-calculus formula F over a set U of states in a given model. Such proof systems usually handle fixed-point formulae through unfolding, thus allowing such formulae to reappear in a proof. Tagging is a technique originated by Winskel for annotating fixed-point formulae with information about the proof states at which these are unfolded. This information is used later in the proof to avoid unnecessary unfolding, without having to investigate the history of the proof. Depending on whether tags are used for acceptance or for rejection of a branch in the proof tree, we refer to ``positive'' or ``negative'' tagging, respectively. In their simplest form, tags consist of the sets U at which fixed-point formulae are unfolded. In this paper, we generalise results of earlier work by Andersen, Stirling and Winskel which, in the case of least fixed-point formulae, are applicable to singleton U sets only