## Term papers on algebra

Our method is particularly useful for PDEs for which anisotropic mesh elements are beneficial and we demonstrate it with a Navier--Stokes simulation. Code for our method can be found at this URL. In , Alexander Shapovalov posed an old twist on the classical coin weighing problem by asking for strategies that manage to conceal the identities of specific coins while providing general information on the number of fake coins.

In , Diaco and Khovanova studied various cases of these "discreet strategies" and introduced the revealing factor, a measure of the information that is revealed. In this paper we discuss a natural coin weighing strategy which we call the sorting strategy: divide the coins into equal piles and sort them by weight. We study the instances when the strategy is discreet, and given an outcome of the sorting strategy, the possible number of fake coins.

We prove that in many cases, the number of fake coins can be any value in an arithmetic progression whose length depends linearly on the number of coins in each pile. We also show the strategy can be discreet when the number of fake coins is any value within an arithmetic subsequence whose length also depends linearly on the number of coins in each pile. We arrive at these results by connecting our work to the classic Frobenius coin problem.

In addition, we calculate the revealing factor for the sorting strategy. Schaposnik University of Illinois at Chicago , On the geometry of regular icosahedral capsids containing disymmetrons arXiv. Icosahedral virus capsids are composed of symmetrons, organized arrangements of capsomers. There are three types of symmetrons: disymmetrons, trisymmetrons, and pentasymmetrons, which have different shapes and are centered on the icosahedral 2-fold, 3-fold and 5-fold axes of symmetry, respectively. In the present paper we incorporate disymmetrons to obtain a geometric classification of icosahedral viruses formed by regular penta-, tri-, and disymmetrons.

For every class of solutions, we further provide formulas for symmetron sizes and parity restrictions on h, k, and T numbers. We also present several methods in which invariants may be used to classify a given configuration. In this paper, we study a particular class of combinatorial game motivated by previous research conducted by Professor James Propp, called Games of No Strategy , or games whose winners are predetermined. Finding the number of ways to play such games often leads to new combinatorial sequences and involves methods from analysis, number theory, and other fields.

They characterized these graphs in the smallest two cases.

We use this to extend Oh and Speyer's characterization of these graphs to the smallest four cases, and we present a conjecture on a bound on the maximal order of these graphs. In addition, we fully characterize certain classes of these graphs in the special cases of cycles and trees.

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In , a new variety of the well-known problem of identifying a counterfeit coin using a balance scale was introduced in the sixth International Kolmogorov Math Tournament. This paper offers a comprehensive overview of this new problem by presenting it in the context of the traditional coin weighing puzzle and then explaining what makes the new problem mathematically unique. Two weighing strategies described previously are used to derive lower bounds for the optimal number of admissible situations for given parameters.

Additionally, a new weighing procedure is described that can be adapted to provide a solution for a broad spectrum of initial parameters by representing the number of counterfeit coins as a linear combination of positive integers. In closing, we offer a new form of the traditional counterfeit coin problem and provide a lower bound for the number of weighings necessary to solve it. Circular planar graphs are used to model electrical networks, which arise in classical physics. Associated with such a network is a network response matrix, which carries information about how the network behaves in response to certain potential differences.

Circular planar graphs can be organized into equivalence classes based upon these response matrices. In each equivalence class, certain fundamental elements are called critical. Additionally, it is known that equivalent graphs are related by certain local transformations. Next, we consider positivity phenomena, studying how testing the signs of certain circular minors can be used to determine if a given network response matrix is associated with a particular equivalence class.

In particular, we prove a conjecture by Kenyon and Wilson for some cases.

As the mesh size of the lattice approaches zero while the polygon remains fixed, a random tiling approaches a deterministic limit shape. An important phenomenon that occurs with the convergence towards a limit shape is the formation of frozen facets ; that is, areas where there are asymptotically tiles of only one particular type. This inscribed curve is defined as the frozen boundary. The goal of this project was to understand the purely algebraic approach, elaborated on in a paper by Kenyon and Okounkov, to the problem of explicitly computing the frozen boundary.

We will present our results for a number of special cases we considered. Efficient primality testing is fundamental to modern cryptography for the purpose of key generation. Different primality tests may be compared using their runtimes and rates of non-witnesses.

Binomial Theorem Expansion, Pascal's Triangle, Finding Terms & Coefficients, Combinations, Algebra 2

An important and ongoing topic of research is the study of infectious diseases and the speed at which these diseases spread. Modeling the spread and growth of such diseases leads to a more precise understanding of the phenomenon and accurate predictions of spread in real life.

We consider a long-range infection model on an infinite regular binary tree. Tiling-harmonic functions are a class of functions on square tilings that minimize a specific energy. These functions may provide a useful tool in studying square Sierpinski carpets. In this paper we show two new Maximum Modulus Principles for these functions, prove Harnack's Inequality, and give a proof that the set of tiling-harmonic functions is closed.

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One of these Maximum Modulus Principles is used to show that bounded infinite tiling-harmonic functions must have arbitrarily long constant lines. Additionally, we give three sufficient conditions for tiling-harmonic functions to be constant. Finally, we explore comparisons between tiling and graph-harmonic functions, especially in regards to oscillating boundary values.

Describing the behavior of traffic via mathematical modeling and computer simulation has been a challenge confronted by mathematicians in various ways throughout the last century. In this project, we introduce various existing traffic flow models and present a new, probability-based model that is a hybrid of the microscopic and macroscopic views, drawing upon current ideas in traffic flow theory.

We examine the correlations found in the data of our computer simulation. We hope that our results could help civil engineers implement efficient road systems that fit their needs, as well as contribute toward the design of safely operating unmanned vehicles.

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Our work answers a question of Fried in all but finitely many cases, and makes significant progress towards answering a question of Ritt and a question of Lyubich and Minsky. Recently, several papers proving lower bounds for the performance of the Sum Of Squares Hierarchy on the planted clique problem have come out. In these matrices, the entries are not completely independent of each other, but rather depend upon a few edges of the input graph. In this paper, we study the norms of these locally random matrices. For both cases, we prove almost-tight probabilistic bounds on the asymptotic behavior of the norms of these matrices.

Each orientable surface with nonempty boundary can be associated with a planar model, whose edges can then be labeled with letters that read out a surface word. Then, the curve word of a free homotopy class of closed curves on a surface is the minimal sequence of edges of the planar model through which a curve in the class passes. The length of a class of curves is defined to be the number of letters in its curve word.

We fix a surface and its corresponding planar model. The most popular such app is Tor, which relies on a series of relays to protect anonymity. The Hecke algebra and rational Cherednik algebra of the group G r ,1, n are non-commutative algebras that are deformations of certain classical algebras associated to the group. These algebras have numerous applications in representation theory, number theory, algebraic geometry and integrable systems in quantum physics.

Consequently, understanding their irreducible representations is important. If the deformation parameters are generic, then these irreducible representations, called Specht modules in the case of the Hecke algebra and Verma modules in the case of the Cherednik algebra, are in bijection with the irreducible representations of G r ,1, n. However, while every irreducible representation of G r ,1, n is unitary, the Hermitian contravariant form on the Specht modules and Verma modules may only be non-degenerate. Thus, the signature of this form provides a great deal of information about the representations of the algebras that cannot be seen by looking at the group representations.

In this paper, we compute the signature of arbitrary Specht modules of the Hecke algebra and use them to give explicit formulas of the parameter values for which these modules are unitary. We also compute asymptotic limits of existing formulas for the signature character of the polynomial representations of the Cherednik algebra which are vastly simpler than the full signature characters and show that these limits are rational functions in t.

In addition, we show that for half of the parameter values, for each k , the degree k portion of the polynomial representation is unitary for large enough n.

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We consider visibility graphs involving bars and arcs in which lines of sight can pass through up to k objects. We prove a new edge bound for arc k-visibility graphs, provide maximal constructions for arc and semi-arc k-visibility graphs, and give a complete characterization of semi-arc visibility graphs. We also give the first thickness bounds for arc and semi-arc k-visibility graphs. Finally, we introduce a model for random semi-bar and semi-arc k-visibility graphs and analyze its properties.

Modern operating system kernels are written in lower-level languages such as C.

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Although the low-level functionalities of C are often useful within kernels, they also give rise to several classes of bugs. Kernels written in higher level languages avoid many of these potential problems, at the possible cost of decreased performance. This research evaluates the advantages and disadvantages of a kernel written in a higher level language. Go is a high-level programming language that supports the CSP style, which recommends splitting large tasks into several smaller ones running in independent "threads". In this study, the implemented Go network stack, called GoNet, was compared to a representative network stack written in C.

The GoNet code is more readable and generally performs better than that of its C stack counterparts. From this, it can be concluded that Go with CSP style is a viable alternative to C for the language of kernel implementations. The scalability of cache coherence protocols is a significant challenge in multicore and other distributed shared memory systems. Traditional snoopy and directory-based coherence protocols are difficult to scale up to many-core systems because of the overhead of broadcasting and storing sharers for each cacheline.

Tardis, a recently proposed coherence protocol, shows potential in solving the scalability problem, since it only requires O logN storage per cacheline for an N-core system and needs no broadcasting support. The original Tardis protocol, however, only supports the sequential consistency memory model. This limits its applicability in real systems since most processors today implement relaxed consistency models like Total Store Order TSO. Tardis also incurs large network traffic overhead on some benchmarks due to an excessive number of renew messages.

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Furthermore, the original Tardis protocol has suboptimal performance when the program uses spinning to communicate between threads. In this paper, we address these downsides of Tardis protocol and make it significantly more practical. Specifically, we discuss the architectural, memory system and protocol changes required in order to implement TSO consistency model on Tardis, and prove that the modified protocol satisfies TSO.

We also propose optimizations for better leasing policies and to handle program spinning.