Involve: A Journal of Mathematics Articles (Project Euclid)
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The latest articles from Involve: A Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 13:11 EDTThu, 19 Oct 2017 13:11 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Algorithms for finding knight's tours on Aztec diamonds
https://projecteuclid.org/euclid.involve/1508433088
<strong>Samantha Davies</strong>, <strong>Chenxiao Xue</strong>, <strong>Carl Yerger</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 5, 721--734.</p><p><strong>Abstract:</strong><br/> A knight’s tour is a sequence of knight’s moves such that each square on the board is visited exactly once. An Aztec diamond is a square board of size [math] where triangular regions of side length [math] have been removed from all four corners. We show that the existence of knight’s tours on Aztec diamonds cannot be proved inductively via smaller Aztec diamonds, and explain why a divide-and-conquer approach is also not promising. We then describe two algorithms that aim to efficiently find knight’s tours on Aztec diamonds. The first is based on random walks, a straightforward but limited technique that yielded tours on Aztec diamonds for all [math] apart from [math] . The second is a path-conversion algorithm that finds a solution for all [math] . We then apply the path-conversion algorithm to random graphs to test the robustness of our algorithm. Online supplements provide source code, output and more details about these algorithms. </p>projecteuclid.org/euclid.involve/1508433088_20171019131139Thu, 19 Oct 2017 13:11 EDTUlrich partitions for two-step flag varietieshttps://projecteuclid.org/euclid.involve/1513087854<strong>Izzet Coskun</strong>, <strong>Luke Jaskowiak</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 3, 531--539.</p><p><strong>Abstract:</strong><br/>
Ulrich bundles play a central role in singularity theory, liaison theory and Boij–Söderberg theory. It was proved by the first author together with Costa, Huizenga, Miró-Roig and Woolf that Schur bundles on flag varieties of three or more steps are not Ulrich and conjectured a classification of Ulrich Schur bundles on two-step flag varieties. By the Borel–Weil–Bott theorem, the conjecture reduces to classifying integer sequences satisfying certain combinatorial properties. In this paper, we resolve the first instance of this conjecture and show that Schur bundles on [math] are not Ulrich if [math] or [math] .
</p>projecteuclid.org/euclid.involve/1513087854_20171212091053Tue, 12 Dec 2017 09:10 ESTNew algorithms for modular inversion and representation by the form $x^2 + 3xy + y^2$https://projecteuclid.org/euclid.involve/1513097137<strong>Christina Doran</strong>, <strong>Shen Lu</strong>, <strong>Barry R. Smith</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 541--554.</p><p><strong>Abstract:</strong><br/>
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representing prime numbers by the binary quadratic form [math] . The Euclidean algorithm is commenced with inputs from one of the families, and the first remainder less than a predetermined size produces the modular inverse or representation.
</p>projecteuclid.org/euclid.involve/1513097137_20171212114550Tue, 12 Dec 2017 11:45 ESTNew approximations for the area of the Mandelbrot sethttps://projecteuclid.org/euclid.involve/1513097138<strong>Daniel Bittner</strong>, <strong>Long Cheong</strong>, <strong>Dante Gates</strong>, <strong>Hieu Nguyen</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 555--572.</p><p><strong>Abstract:</strong><br/>
Due to its fractal nature, much about the area of the Mandelbrot set [math] remains to be understood. While a series formula has been derived by Ewing and Schober (1992) to calculate the area of [math] by considering its complement inside the Riemann sphere, to date the exact value of this area remains unknown. This paper presents new improved upper bounds for the area based on a parallel computing algorithm and for the 2-adic valuation of the series coefficients in terms of the sum-of-digits function.
</p>projecteuclid.org/euclid.involve/1513097138_20171212114550Tue, 12 Dec 2017 11:45 ESTBases for the global Weyl modules of $\mathfrak{sl}_n$ of highest weight $m\omega_1$https://projecteuclid.org/euclid.involve/1513097139<strong>Samuel Chamberlin</strong>, <strong>Amanda Croan</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 573--581.</p><p><strong>Abstract:</strong><br/>
We utilize a theorem of B. Feigin and S. Loktev to give explicit bases for the global Weyl modules for the map algebras of the form [math] of highest weight [math] . These bases are given in terms of specific elements of the universal enveloping algebra, [math] , acting on the highest weight vector.
</p>projecteuclid.org/euclid.involve/1513097139_20171212114550Tue, 12 Dec 2017 11:45 ESTLeverage centrality of knight's graphs and Cartesian products of regular graphs and path powershttps://projecteuclid.org/euclid.involve/1513097140<strong>Roger Vargas</strong>, <strong>Abigail Waldron</strong>, <strong>Anika Sharma</strong>, <strong>Rigoberto Flórez</strong>, <strong>Darren A. Narayan</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 583--592.</p><p><strong>Abstract:</strong><br/>
In 2010, Joyce et al. defined the leverage centrality of a graph as a means to analyze connections within the brain. In this paper we investigate this property from a mathematical perspective and determine the leverage centrality for knight’s graphs, path powers, and Cartesian products.
</p>projecteuclid.org/euclid.involve/1513097140_20171212114550Tue, 12 Dec 2017 11:45 ESTEquivalence classes of $\mathrm{GL}(p, \mathbb{C})\times \mathrm{GL}(q, \mathbb{C})$ orbits in the flag variety of $\mathfrak{gl}(p+q, \mathbb{C})$https://projecteuclid.org/euclid.involve/1513097141<strong>Leticia Barchini</strong>, <strong>Nina Williams</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 593--623.</p><p><strong>Abstract:</strong><br/> We consider the pair of complex Lie groups [math] and the finite set [math] . The moment map [math] of the [math] -action on the cotangent bundle [math] maps each conormal bundle closure [math] onto the closure of a single nilpotent [math] -orbit, [math] . We use combinatorial techniques to describe [math] . </p>projecteuclid.org/euclid.involve/1513097141_20171212114550Tue, 12 Dec 2017 11:45 ESTGlobal sensitivity analysis in a mathematical model of the renal insterstitiumhttps://projecteuclid.org/euclid.involve/1513097142<strong>Mariel Bedell</strong>, <strong>Yilin Lin</strong>, <strong>Emmie Román-Meléndez</strong>, <strong>Ioannis Sgouralis</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 625--649.</p><p><strong>Abstract:</strong><br/>
The pressure in the renal interstitium is an important factor for normal kidney function. Here we develop a computational model of the rat kidney and use it to investigate the relationship between arterial blood pressure and interstitial fluid pressure. In addition, we investigate how tissue flexibility influences this relationship. Due to the complexity of the model, the large number of parameters, and the inherent uncertainty of the experimental data, we utilize Monte Carlo sampling to study the model’s behavior under a wide range of parameter values and to compute first- and total-order sensitivity indices. Characteristically, at elevated arterial blood pressure, the model predicts cases with increased or reduced interstitial pressure. The transition between the two cases is controlled mostly by the compliance of the blood vessels located before the afferent arterioles.
</p>projecteuclid.org/euclid.involve/1513097142_20171212114550Tue, 12 Dec 2017 11:45 ESTSums of squares in quaternion ringshttps://projecteuclid.org/euclid.involve/1513097143<strong>Anna Cooke</strong>, <strong>Spencer Hamblen</strong>, <strong>Sam Whitfield</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 651--664.</p><p><strong>Abstract:</strong><br/>
Lagrange’s four squares theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question for quaternion rings, focusing on squares of elements of quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infinitely many quaternion rings, and give global upper and lower bounds.
</p>projecteuclid.org/euclid.involve/1513097143_20171212114550Tue, 12 Dec 2017 11:45 ESTOn the structure of symmetric spaces of semidihedral groupshttps://projecteuclid.org/euclid.involve/1513097144<strong>Jennifer Schaefer</strong>, <strong>Kathryn Schlechtweg</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 665--676.</p><p><strong>Abstract:</strong><br/>
We investigate the symmetric spaces associated to the family of semidihedral groups of order [math] . We begin this study by analyzing the structure of the automorphism group and by determining which automorphims are involutions. We then determine the symmetric spaces corresponding to each involution and the orbits of the fixed-point groups on these spaces.
</p>projecteuclid.org/euclid.involve/1513097144_20171212114550Tue, 12 Dec 2017 11:45 ESTSpectrum of the Laplacian on graphs of radial functionshttps://projecteuclid.org/euclid.involve/1513097145<strong>Rodrigo Matos</strong>, <strong>Fabio Montenegro</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 677--690.</p><p><strong>Abstract:</strong><br/>
We prove that if [math] is a complete, noncompact hypersurface in [math] , which is the graph of a real radial function, then the spectrum of the Laplace operator on [math] is the interval [math] .
</p>projecteuclid.org/euclid.involve/1513097145_20171212114550Tue, 12 Dec 2017 11:45 ESTA generalization of Eulerian numbers via rook placementshttps://projecteuclid.org/euclid.involve/1513097146<strong>Esther Banaian</strong>, <strong>Steve Butler</strong>, <strong>Christopher Cox</strong>, <strong>Jeffrey Davis</strong>, <strong>Jacob Landgraf</strong>, <strong>Scarlitte Ponce</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 691--705.</p><p><strong>Abstract:</strong><br/>
We consider a generalization of Eulerian numbers which count the number of placements of [math] rooks on an [math] chessboard where there are exactly [math] rooks in each row and each column, and exactly [math] rooks below the main diagonal. The standard Eulerian numbers correspond to the case [math] . We show that for any [math] the resulting numbers are symmetric and give generating functions of these numbers for small values of [math] .
</p>projecteuclid.org/euclid.involve/1513097146_20171212114550Tue, 12 Dec 2017 11:45 ESTThe $H$-linked degree-sum parameter for special graph familieshttps://projecteuclid.org/euclid.involve/1513097147<strong>Lydia East Kenney</strong>, <strong>Jeffrey Powell</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 4, 707--720.</p><p><strong>Abstract:</strong><br/>
For a fixed graph [math] , a graph [math] is [math] -linked if any injection [math] can be extended to an [math] -subdivision in [math] . The concept of [math] -linked generalizes several well-known graph theory concepts such as [math] -connected, [math] -linked, and [math] -ordered. In 2012, Ferrara et al. proved a sharp [math] (or degree-sum) bound for a graph to be [math] -linked. In particular, they proved that any graph [math] with [math] vertices and [math] is [math] -linked, where [math] is a parameter maximized over certain partitions of [math] . However, they do not discuss the calculation of [math] in their work. In this paper, we prove the exact value of [math] in the cases when [math] is a path, a cycle, a union of stars, a complete graph, and a complete bipartite graph. Several of these results lead to new degree-sum conditions for particular graph classes while others provide alternate proofs of previously known degree-sum conditions.
</p>projecteuclid.org/euclid.involve/1513097147_20171212114550Tue, 12 Dec 2017 11:45 ESTStability analysis for numerical methods applied to an inner ear modelhttps://projecteuclid.org/euclid.involve/1513135626<strong>Kimberley Lindenberg</strong>, <strong>Kees Vuik</strong>, <strong>Pieter W. J. van Hengel</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 181--196.</p><p><strong>Abstract:</strong><br/>
Diependaal, Duifhuis, Hoogstraten and Viergever investigated three time-integration methods to solve a simplified one-dimensional model of the human cochlea. Two of these time-integration methods are dealt with in this paper, namely fourth-order Runge–Kutta and modified Sielecki. The stability of these two methods is examined, both theoretically and experimentally. This leads to the conclusion that in the case of the fourth-order Runge–Kutta method, a bigger time step can be used in comparison to the modified Sielecki method. This corresponds with the conclusion drawn in the article by Diependaal, Duifhuis, Hoogstraten and Viergever.
</p>projecteuclid.org/euclid.involve/1513135626_20171212222716Tue, 12 Dec 2017 22:27 ESTThree approaches to a bracket polynomial for singular linkshttps://projecteuclid.org/euclid.involve/1513135627<strong>Carmen Caprau</strong>, <strong>Alex Chichester</strong>, <strong>Patrick Chu</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 197--218.</p><p><strong>Abstract:</strong><br/>
In this paper we extend the Kauffman bracket to singular links. Specifically, we define a polynomial invariant for singular links, and in doing this, we consider three approaches to our extended Kauffman bracket polynomial: (1) using skein relations involving singular link diagrams, (2) using representations of the singular braid monoid, (3) via a Yang–Baxter state model. We also study some properties of the extended Kauffman bracket.
</p>projecteuclid.org/euclid.involve/1513135627_20171212222716Tue, 12 Dec 2017 22:27 ESTSymplectic embeddings of four-dimensional ellipsoids into polydiscshttps://projecteuclid.org/euclid.involve/1513135628<strong>Madeleine Burkhart</strong>, <strong>Priera Panescu</strong>, <strong>Max Timmons</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 219--242.</p><p><strong>Abstract:</strong><br/>
McDuff and Schlenk recently determined exactly when a four-dimensional symplectic ellipsoid symplectically embeds into a symplectic ball. Similarly, Frenkel and Müller recently determined exactly when a symplectic ellipsoid symplectically embeds into a symplectic cube. Symplectic embeddings of more complicated sets, however, remain mostly unexplored. We study when a symplectic ellipsoid [math] symplectically embeds into a polydisc [math] . We prove that there exists a constant [math] depending only on [math] (here, [math] is assumed greater than [math] ) such that if [math] is greater than [math] , then the only obstruction to symplectically embedding [math] into [math] is the volume obstruction. We also conjecture exactly when an ellipsoid embeds into a scaling of [math] for [math] , and conjecture about the set of [math] such that the only obstruction to embedding [math] into a scaling of [math] is the volume. Finally, we verify our conjecture for [math] .
</p>projecteuclid.org/euclid.involve/1513135628_20171212222716Tue, 12 Dec 2017 22:27 ESTCharacterizations of the round two-dimensional sphere in terms of closed geodesicshttps://projecteuclid.org/euclid.involve/1513135629<strong>Lee Kennard</strong>, <strong>Jordan Rainone</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 243--255.</p><p><strong>Abstract:</strong><br/>
The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere.
</p>projecteuclid.org/euclid.involve/1513135629_20171212222716Tue, 12 Dec 2017 22:27 ESTA necessary and sufficient condition for coincidence with the weak topologyhttps://projecteuclid.org/euclid.involve/1513135630<strong>Joseph Clanin</strong>, <strong>Kristopher Lee</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 257--261.</p><p><strong>Abstract:</strong><br/>
For a topological space [math] , it is a natural undertaking to compare its topology with the weak topology generated by a family of real-valued continuous functions on [math] . We present a necessary and sufficient condition for the coincidence of these topologies for an arbitrary family [math] . As a corollary, we give a new proof of the fact that families of functions which separate points on a compact space induce topologies that coincide with the original topology.
</p>projecteuclid.org/euclid.involve/1513135630_20171212222716Tue, 12 Dec 2017 22:27 ESTPeak sets of classical Coxeter groupshttps://projecteuclid.org/euclid.involve/1513135631<strong>Alexander Diaz-Lopez</strong>, <strong>Pamela E. Harris</strong>, <strong>Erik Insko</strong>, <strong>Darleen Perez-Lavin</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 263--290.</p><p><strong>Abstract:</strong><br/>
We say a permutation [math] in the symmetric group [math] has a peak at index [math] if [math] and we let [math] . Given a set [math] of positive integers, we let [math] denote the subset of [math] consisting of all permutations [math] where [math] . In 2013, Billey, Burdzy, and Sagan proved [math] , where [math] is a polynomial of degree [math] . In 2014, Castro-Velez et al. considered the Coxeter group of type [math] as the group of signed permutations on [math] letters and showed that [math] , where [math] is the same polynomial of degree [math] . In this paper we partition the sets [math] studied by Billey, Burdzy, and Sagan into subsets of permutations that end with an ascent to a fixed integer [math] (or a descent to a fixed integer [math] ) and provide polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie types [math] and [math] into [math] , we partition these groups into bundles of permutations [math] such that [math] has the same relative order as some permutation [math] . This allows us to count the number of permutations in types [math] and [math] with a given peak set [math] by reducing the enumeration to calculations in the symmetric group and sums across the rows of Pascal’s triangle.
</p>projecteuclid.org/euclid.involve/1513135631_20171212222716Tue, 12 Dec 2017 22:27 ESTFox coloring and the minimum number of colorshttps://projecteuclid.org/euclid.involve/1513135632<strong>Mohamed Elhamdadi</strong>, <strong>Jeremy Kerr</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 291--316.</p><p><strong>Abstract:</strong><br/>
We study Fox colorings of knots that are 13-colorable. We prove that any 13-colorable knot has a diagram that uses exactly five of the thirteen colors that are assigned to the arcs of the diagram. Due to an existing lower bound, this gives that the minimum number of colors of any 13-colorable knot is 5.
</p>projecteuclid.org/euclid.involve/1513135632_20171212222716Tue, 12 Dec 2017 22:27 ESTCombinatorial curve neighborhoods for the affine flag manifold of type $A_1^1$https://projecteuclid.org/euclid.involve/1513135633<strong>Leonardo C. Mihalcea</strong>, <strong>Trevor Norton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 317--325.</p><p><strong>Abstract:</strong><br/>
Let [math] be the affine flag manifold of Lie type [math] . Its moment graph encodes the torus fixed points (which are elements of the infinite dihedral group [math] ) and the torus stable curves in [math] . Given a fixed point [math] and a degree [math] , the combinatorial curve neighborhood is the set of maximal elements in the moment graph of [math] which can be reached from [math] using a chain of curves of total degree [math] . In this paper we give a formula for these elements, using combinatorics of the affine root system of type [math] .
</p>projecteuclid.org/euclid.involve/1513135633_20171212222716Tue, 12 Dec 2017 22:27 ESTTotal variation based denoising methods for speckle noise imageshttps://projecteuclid.org/euclid.involve/1513135634<strong>Arundhati Bagchi Misra</strong>, <strong>Ethan Lockhart</strong>, <strong>Hyeona Lim</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 327--344.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a new algorithm based on total variation for denoising speckle noise images. Total variation was introduced by Rudin, Osher, and Fatemi in 1992 for regularizing images. Chambolle proposed a faster algorithm based on the duality of convex functions for minimizing the total variation, but his algorithm was built for Gaussian noise removal. Unlike Gaussian noise, which is additive, speckle noise is multiplicative. We modify the original Chambolle algorithm for speckle noise images using the first noise equation for speckle denoising, proposed by Krissian, Kikinis, Westin and Vosburgh in 2005. We apply the Chambolle algorithm to the Krissian et al. speckle denoising model to develop a faster algorithm for speckle noise images.
</p>projecteuclid.org/euclid.involve/1513135634_20171212222716Tue, 12 Dec 2017 22:27 ESTA new look at Apollonian circle packingshttps://projecteuclid.org/euclid.involve/1513135635<strong>Isabel Corona</strong>, <strong>Carolynn Johnson</strong>, <strong>Lon Mitchell</strong>, <strong>Dylan O’Connell</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 10, Number 2, 345--360.</p><p><strong>Abstract:</strong><br/>
We define an abstract Apollonian supergasket using the solution set of a certain Diophantine equation, showing that the solutions are in bijective correspondence with the circles of any concrete supergasket. Properties of the solution set translate directly to geometric and algebraic properties of Apollonian gaskets, facilitating their study. In particular, curvatures of individual circles are explored and geometric relationships among multiple circles are given simple algebraic expressions. All results can be applied to a concrete gasket using the curvature-center coordinates of its four defining circles. These techniques can also be applied to other types of circle packings and higher-dimensional analogs.
</p>projecteuclid.org/euclid.involve/1513135635_20171212222716Tue, 12 Dec 2017 22:27 ESTOn halving-edges graphshttps://projecteuclid.org/euclid.involve/1513775038<strong>Tanya Khovanova</strong>, <strong>Dai Yang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 1--11.</p><p><strong>Abstract:</strong><br/>
In this paper we study halving-edges graphs corresponding to a set of halving lines. Particularly, we study the vertex degrees, path, cycles and cliques of such graphs. In doing so, we study a vertex-partition of said graph called chains which are equipped with interesting properties.
</p>projecteuclid.org/euclid.involve/1513775038_20171220080402Wed, 20 Dec 2017 08:04 ESTKnot mosaic tabulationhttps://projecteuclid.org/euclid.involve/1513775039<strong>Hwa Jeong Lee</strong>, <strong>Lewis Ludwig</strong>, <strong>Joseph Paat</strong>, <strong>Amanda Peiffer</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 13--26.</p><p><strong>Abstract:</strong><br/>
In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs in liquid helium II for example. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tame knots. In this article, we consider the mosaic number of a knot, which is a natural and fundamental knot invariant defined in the knot mosaic system. We determine the mosaic number for all eight-crossing or fewer prime knots. This work is written at an introductory level to encourage other undergraduates to understand and explore this topic. No prior knowledge of knot theory is assumed or required.
</p>projecteuclid.org/euclid.involve/1513775039_20171220080402Wed, 20 Dec 2017 08:04 ESTExtending hypothesis testing with persistent homology to three or more groupshttps://projecteuclid.org/euclid.involve/1513775040<strong>Christopher Cericola</strong>, <strong>Inga Jo Johnson</strong>, <strong>Joshua Kiers</strong>, <strong>Mitchell Krock</strong>, <strong>Jordan Purdy</strong>, <strong>Johanna Torrence</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 27--51.</p><p><strong>Abstract:</strong><br/>
We extend the work of Robinson and Turner to use hypothesis testing with persistent homology to test for measurable differences in shape between the spaces of three or more groups. We conduct a large-scale simulation study to validate our proposed extension, considering various combinations of groups, sample sizes and measurement errors. For each such combination, the percentage of p-values below an [math] -level of 0.05 is provided. Additionally, we apply our method to a cardiotocography data set and find statistically significant evidence of measurable differences in shape between the spaces corresponding to normal, suspect and pathologic health status groups.
</p>projecteuclid.org/euclid.involve/1513775040_20171220080402Wed, 20 Dec 2017 08:04 ESTMerging peg solitaire on graphshttps://projecteuclid.org/euclid.involve/1513775041<strong>John Engbers</strong>, <strong>Ryan Weber</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 53--66.</p><p><strong>Abstract:</strong><br/>
Peg solitaire has recently been generalized to graphs. Here, pegs start on all but one of the vertices in a graph. A move takes pegs on adjacent vertices [math] and [math] , with [math] also adjacent to a hole on vertex [math] , and jumps the peg on [math] over the peg on [math] to [math] , removing the peg on [math] . The goal of the game is to reduce the number of pegs to one.
We introduce the game merging peg solitaire on graphs , where a move takes pegs on vertices [math] and [math] (with a hole on [math] ) and merges them to a single peg on [math] . When can a configuration on a graph, consisting of pegs on all vertices but one, be reduced to a configuration with only a single peg? We give results for a number of graph classes, including stars, paths, cycles, complete bipartite graphs, and some caterpillars.
</p>projecteuclid.org/euclid.involve/1513775041_20171220080402Wed, 20 Dec 2017 08:04 ESTLabeling crossed prisms with a condition at distance twohttps://projecteuclid.org/euclid.involve/1513775042<strong>Matthew Beaudouin-Lafon</strong>, <strong>Serena Chen</strong>, <strong>Nathaniel Karst</strong>, <strong>Jessica Oehrlein</strong>, <strong>Denise Troxell</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 67--80.</p><p><strong>Abstract:</strong><br/>
An L ( 2,1 ) -labeling of a graph is an assignment of nonnegative integers to its vertices such that adjacent vertices are assigned labels at least two apart, and vertices at distance two are assigned labels at least one apart. The [math] -number of a graph is the minimum span of labels over all its L(2,1)-labelings. A generalized Petersen graph (GPG) of order [math] consists of two disjoint cycles on [math] vertices, called the inner and outer cycles , respectively, together with a perfect matching in which each matching edge connects a vertex in the inner cycle to a vertex in the outer cycle. A prism of order [math] is a GPG that is isomorphic to the Cartesian product of a path on two vertices and a cycle on [math] vertices. A crossed prism is a GPG obtained from a prism by crossing two of its matching edges; that is, swapping the two inner cycle vertices on these edges. We show that the [math] -number of a crossed prism is 5, 6, or 7 and provide complete characterizations of crossed prisms attaining each one of these [math] -numbers.
</p>projecteuclid.org/euclid.involve/1513775042_20171220080402Wed, 20 Dec 2017 08:04 ESTNormal forms of endomorphism-valued power serieshttps://projecteuclid.org/euclid.involve/1513775043<strong>Christopher Keane</strong>, <strong>Szilárd Szabó</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 81--94.</p><p><strong>Abstract:</strong><br/>
We show for [math] , and an [math] -dimensional complex vector space [math] that if an element [math] has constant term similar to a Jordan block, then there exists a polynomial gauge transformation [math] such that the first [math] coefficients of [math] have a controlled normal form. Furthermore, we show that this normal form is unique by demonstrating explicit relationships between the first [math] coefficients of the Puiseux series expansion of the eigenvalues of [math] and the entries of the first [math] coefficients of [math] .
</p>projecteuclid.org/euclid.involve/1513775043_20171220080402Wed, 20 Dec 2017 08:04 ESTContinuous dependence and differentiating solutions of a second order boundary value problem with average value conditionhttps://projecteuclid.org/euclid.involve/1513775044<strong>Jeffrey Lyons</strong>, <strong>Samantha Major</strong>, <strong>Kaitlyn Seabrook</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 95--102.</p><p><strong>Abstract:</strong><br/>
Using a few conditions, continuous dependence, and a result regarding smoothness of initial conditions, we show that derivatives of solutions to the second order boundary value problem [math] , [math] , satisfying [math] , [math] , where [math] and [math] with respect to each of the boundary data [math] , [math] , [math] , [math] , [math] solve the associated variational equation with interesting boundary conditions. Of note is the second boundary condition, which is an average value condition.
</p>projecteuclid.org/euclid.involve/1513775044_20171220080402Wed, 20 Dec 2017 08:04 ESTOn uniform large-scale volume growth for the Carnot–Carathéodory metric on unbounded model hypersurfaces in $\mathbb{C}^2$https://projecteuclid.org/euclid.involve/1513775045<strong>Ethan Dlugie</strong>, <strong>Aaron Peterson</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 103--118.</p><p><strong>Abstract:</strong><br/>
We consider the rate of volume growth of large Carnot–Carathéodory metric balls on a class of unbounded model hypersurfaces in [math] . When the hypersurface has a uniform global structure, we show that a metric ball of radius [math] either has volume on the order of [math] or [math] . We also give necessary and sufficient conditions on the hypersurface to display either behavior.
</p>projecteuclid.org/euclid.involve/1513775045_20171220080402Wed, 20 Dec 2017 08:04 ESTVariations of the Greenberg unrelated question binary modelhttps://projecteuclid.org/euclid.involve/1513775046<strong>David P. Suarez</strong>, <strong>Sat Gupta</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 119--126.</p><p><strong>Abstract:</strong><br/>
We explore different variations of the Greenberg unrelated question RRT model for a binary response. In one of the variations, we allow multiple independent responses from each respondent. In another variation, we use inverse sampling. It turns out that both of these variations produce more efficient models, a fact validated by both theoretical comparisons as well as extensive computer simulations.
</p>projecteuclid.org/euclid.involve/1513775046_20171220080402Wed, 20 Dec 2017 08:04 ESTGeneralized exponential sums and the power of computershttps://projecteuclid.org/euclid.involve/1513775047<strong>Francis N. Castro</strong>, <strong>Oscar E. González</strong>, <strong>Luis A. Medina</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 127--142.</p><p><strong>Abstract:</strong><br/>
Today’s era can be characterized by the rise of computer technology. Computers have been, to some extent, responsible for the explosion of the scientific knowledge that we have today. In mathematics, for instance, we have the four color theorem, which is regarded as the first celebrated result to be proved with the assistance of computers. In this article we generalize some fascinating binomial sums that arise in the study of Boolean functions. We study these generalizations from the point of view of integer sequences and bring them to the current computer age of mathematics. The asymptotic behavior of these generalizations is calculated. In particular, we show that a previously known constant that appears in the study of exponential sums of symmetric Boolean functions is universal in the sense that it also emerges in the asymptotic behavior of all of the sequences considered in this work. Finally, in the last section, we use the power of computers and some remarkable algorithms to show that these generalizations are holonomic; i.e., they satisfy homogeneous linear recurrences with polynomial coefficients.
</p>projecteuclid.org/euclid.involve/1513775047_20171220080402Wed, 20 Dec 2017 08:04 ESTCoincidences among skew stable and dual stable Grothendieck polynomialshttps://projecteuclid.org/euclid.involve/1513775048<strong>Ethan Alwaise</strong>, <strong>Shuli Chen</strong>, <strong>Alexander Clifton</strong>, <strong>Rebecca Patrias</strong>, <strong>Rohil Prasad</strong>, <strong>Madeline Shinners</strong>, <strong>Albert Zheng</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 143--167.</p><p><strong>Abstract:</strong><br/>
The question of when two skew Young diagrams produce the same skew Schur function has been well studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the [math] -theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons.
</p>projecteuclid.org/euclid.involve/1513775048_20171220080402Wed, 20 Dec 2017 08:04 ESTA probabilistic heuristic for counting components of functional graphs of polynomials over finite fieldshttps://projecteuclid.org/euclid.involve/1513775049<strong>Elisa Bellah</strong>, <strong>Derek Garton</strong>, <strong>Erin Tannenbaum</strong>, <strong>Noah Walton</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 1, 169--179.</p><p><strong>Abstract:</strong><br/>
Flynn and Garton (2014) bounded the average number of components of the functional graphs of polynomials of fixed degree over a finite field. When the fixed degree was large (relative to the size of the finite field), their lower bound matched Kruskal’s asymptotic for random functional graphs. However, when the fixed degree was small, they were unable to match Kruskal’s bound, since they could not (Lagrange) interpolate cycles in functional graphs of length greater than the fixed degree. In our work, we introduce a heuristic for approximating the average number of such cycles of any length. This heuristic is, roughly, that for sets of edges in a functional graph, the quality of being a cycle and the quality of being interpolable are “uncorrelated enough”. We prove that this heuristic implies that the average number of components of the functional graphs of polynomials of fixed degree over a finite field is within a bounded constant of Kruskal’s bound. We also analyze some numerical data comparing implications of this heuristic to some component counts of functional graphs of polynomials over finite fields.
</p>projecteuclid.org/euclid.involve/1513775049_20171220080402Wed, 20 Dec 2017 08:04 ESTFinding cycles in the $k$-th power digraphs over the integers modulo a primehttps://projecteuclid.org/euclid.involve/1513775055<strong>Greg Dresden</strong>, <strong>Wenda Tu</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 181--194.</p><p><strong>Abstract:</strong><br/>
For [math] prime and [math] , let us define [math] to be the digraph whose set of vertices is [math] such that there is a directed edge from a vertex [math] to a vertex [math] if [math] . We find a new way to decide if there is a cycle of a given length in a given graph [math] .
</p>projecteuclid.org/euclid.involve/1513775055_20171220080420Wed, 20 Dec 2017 08:04 ESTEnumerating spherical $n$-linkshttps://projecteuclid.org/euclid.involve/1513775056<strong>Madeleine Burkhart</strong>, <strong>Joel Foisy</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 195--206.</p><p><strong>Abstract:</strong><br/>
We investigate spherical links: that is, disjoint embeddings of 1-spheres and 0-spheres in the 2-sphere, where the notion of a split link is analogous to the usual concept. In the quest to enumerate distinct nonsplit [math] -links for arbitrary [math] , we must consider when it is possible for an embedding of circles and an even number of points to form a nonsplit link. The main result is a set of necessary and sufficient conditions for such an embedding. The final section includes tables of the distinct embeddings that yield nonsplit [math] -links for [math] .
</p>projecteuclid.org/euclid.involve/1513775056_20171220080420Wed, 20 Dec 2017 08:04 ESTDouble bubbles in hyperbolic surfaceshttps://projecteuclid.org/euclid.involve/1513775057<strong>Wyatt Boyer</strong>, <strong>Bryan Brown</strong>, <strong>Alyssa Loving</strong>, <strong>Sarah Tammen</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 207--217.</p><p><strong>Abstract:</strong><br/>
We seek the least-perimeter way to enclose and separate two prescribed areas in certain hyperbolic surfaces.
</p>projecteuclid.org/euclid.involve/1513775057_20171220080420Wed, 20 Dec 2017 08:04 ESTWhat is odd about binary Parseval frames?https://projecteuclid.org/euclid.involve/1513775058<strong>Zachery J. Baker</strong>, <strong>Bernhard G. Bodmann</strong>, <strong>Micah G. Bullock</strong>, <strong>Samantha N. Branum</strong>, <strong>Jacob E. McLaney</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 219--233.</p><p><strong>Abstract:</strong><br/>
This paper examines the construction and properties of binary Parseval frames. We address two questions: When does a binary Parseval frame have a complementary Parseval frame? Which binary symmetric idempotent matrices are Gram matrices of binary Parseval frames? In contrast to the case of real or complex Parseval frames, the answer to these questions is not always affirmative. The key to our understanding comes from an algorithm that constructs binary orthonormal sequences that span a given subspace, whenever possible. Special regard is given to binary frames whose Gram matrices are circulants.
</p>projecteuclid.org/euclid.involve/1513775058_20171220080420Wed, 20 Dec 2017 08:04 ESTNumbers and the heights of their happinesshttps://projecteuclid.org/euclid.involve/1513775059<strong>May Mei</strong>, <strong>Andrew Read-McFarland</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 235--241.</p><p><strong>Abstract:</strong><br/>
A generalized happy function, [math] maps a positive integer to the sum of its base [math] digits raised to the [math] -th power. We say that [math] is a base- [math] , [math] -power, height- [math] , [math] -attracted number if [math] is the smallest positive integer such that [math] . Happy numbers are then base-10, 2-power, 1-attracted numbers of any height. Let [math] denote the smallest height- [math] , [math] -attracted number for a fixed base [math] and exponent [math] and let [math] denote the smallest number such that every integer can be written as [math] for some nonnegative integers [math] . We prove that if [math] is the smallest nonnegative integer such that [math] ,
d
=
⌈
g
(
e
)
+
1
1
−
(
b
−
2
b
−
1
)
e
+
e
+
p
e
,
b
⌉
,
and [math] , then [math] .
</p>projecteuclid.org/euclid.involve/1513775059_20171220080420Wed, 20 Dec 2017 08:04 ESTThe truncated and supplemented Pascal matrix and applicationshttps://projecteuclid.org/euclid.involve/1513775060<strong>Michael Hua</strong>, <strong>Steven B. Damelin</strong>, <strong>Jeffrey Sun</strong>, <strong>Mingchao Yu</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 243--251.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce the [math] (with [math] ) truncated, supplemented Pascal matrix, which has the property that any [math] columns form a linearly independent set. This property is also present in Reed–Solomon codes; however, Reed–Solomon codes are completely dense, whereas the truncated, supplemented Pascal matrix has multiple zeros. If the maximum distance separable code conjecture is correct, then our matrix has the maximal number of columns (with the aforementioned property) that the conjecture allows. This matrix has applications in coding, network coding, and matroid theory.
</p>projecteuclid.org/euclid.involve/1513775060_20171220080420Wed, 20 Dec 2017 08:04 ESTHexatonic systems and dual groups in mathematical music theoryhttps://projecteuclid.org/euclid.involve/1513775061<strong>Cameron Berry</strong>, <strong>Thomas M. Fiore</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 253--270.</p><p><strong>Abstract:</strong><br/>
Motivated by the music-theoretical work of Richard Cohn and David Clampitt on late-nineteenth century harmony, we mathematically prove that the [math] -group of a hexatonic cycle is dual (in the sense of Lewin) to its [math] / [math] -stabilizer. Our points of departure are Cohn’s notions of maximal smoothness and hexatonic cycle, and the symmetry group of the 12-gon; we do not make use of the duality between the [math] / [math] -group and [math] -group. We also discuss how some ideas in the present paper could be used in the proof of [math] / [math] - [math] duality by Crans, Fiore, and Satyendra ( Amer. Math. Monthly 116 :6 (2009), 479–495).
</p>projecteuclid.org/euclid.involve/1513775061_20171220080420Wed, 20 Dec 2017 08:04 ESTOn computable classes of equidistant sets: finite focal setshttps://projecteuclid.org/euclid.involve/1513775062<strong>Csaba Vincze</strong>, <strong>Adrienn Varga</strong>, <strong>Márk Oláh</strong>, <strong>László Fórián</strong>, <strong>Sándor Lőrinc</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 271--282.</p><p><strong>Abstract:</strong><br/>
The equidistant set of two nonempty subsets [math] and [math] in the Euclidean plane is the set of all points that have the same distance from [math] and [math] . Since the classical conics can be also given in this way, equidistant sets can be considered as one of their generalizations: [math] and [math] are called the focal sets. The points of an equidistant set are difficult to determine in general because there are no simple formulas to compute the distance between a point and a set. As a simplification of the general problem, we are going to investigate equidistant sets with finite focal sets. The main result is the characterization of the equidistant points in terms of computable constants and parametrization. The process is presented by a Maple algorithm. Its motivation is a kind of continuity property of equidistant sets. Therefore we can approximate the equidistant points of [math] and [math] with the equidistant points of finite subsets [math] and [math] . Such an approximation can be applied to the computer simulation, as some examples show in the last section.
</p>projecteuclid.org/euclid.involve/1513775062_20171220080420Wed, 20 Dec 2017 08:04 ESTZero divisor graphs of commutative graded ringshttps://projecteuclid.org/euclid.involve/1513775063<strong>Katherine Cooper</strong>, <strong>Brian Johnson</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 283--295.</p><p><strong>Abstract:</strong><br/>
We study a natural generalization of the zero divisor graph introduced by Anderson and Livingston to commutative rings graded by abelian groups, considering only homogeneous zero divisors. We develop a basic theory for graded zero divisor graphs and present many examples. Finally, we examine classes of graphs that are realizable as graded zero divisor graphs and close with some open questions.
</p>projecteuclid.org/euclid.involve/1513775063_20171220080420Wed, 20 Dec 2017 08:04 ESTThe behavior of a population interaction-diffusion equation in its subcritical regimehttps://projecteuclid.org/euclid.involve/1513775064<strong>Mitchell G. Davis</strong>, <strong>David J. Wollkind</strong>, <strong>Richard A. Cangelosi</strong>, <strong>Bonni J. Kealy-Dichone</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 297--309.</p><p><strong>Abstract:</strong><br/>
A model interaction-diffusion equation for population density originally analyzed through terms of third-order in its supercritical parameter range is extended through terms of fifth-order to examine the behavior in its subcritical regime. It is shown that under the proper conditions the two subcritical cases behave in exactly the same manner as the two supercritical ones unlike the outcome for the truncated system. Further, there also exists a region of metastability allowing for the possibility of population outbreaks. These results are then used to offer an explanation for the occurrence of isolated vegetative patches and sparse homogeneous distributions in the relevant ecological parameter range where there is subcriticality for a plant-groundwater model system, as opposed to periodic patterns and dense homogeneous distributions occurring in its supercritical regime.
</p>projecteuclid.org/euclid.involve/1513775064_20171220080420Wed, 20 Dec 2017 08:04 ESTForbidden subgraphs of coloring graphshttps://projecteuclid.org/euclid.involve/1513775065<strong>Francisco Alvarado</strong>, <strong>Ashley Butts</strong>, <strong>Lauren Farquhar</strong>, <strong>Heather M. Russell</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 311--324.</p><p><strong>Abstract:</strong><br/>
Given a graph [math] , its [math] -coloring graph has vertex set given by the proper [math] -colorings of the vertices of [math] with two [math] -colorings adjacent if and only if they differ at exactly one vertex. Beier et al. ( Discrete Math. 339 :8 (2016), 2100–2112) give various characterizations of coloring graphs, including finding graphs which never arise as induced subgraphs of coloring graphs. These are called forbidden subgraphs, and if no proper subgraph of a forbidden subgraph is forbidden, it is called minimal forbidden. In this paper, we construct a finite collection of minimal forbidden subgraphs that come from modifying theta graphs. We also construct an infinite family of minimal forbidden subgraphs similar to the infinite family found by Beier et al.
</p>projecteuclid.org/euclid.involve/1513775065_20171220080420Wed, 20 Dec 2017 08:04 ESTComputing indicators of Radford algebrashttps://projecteuclid.org/euclid.involve/1513775066<strong>Hao Hu</strong>, <strong>Xinyi Hu</strong>, <strong>Linhong Wang</strong>, <strong>Xingting Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 325--334.</p><p><strong>Abstract:</strong><br/>
We compute higher Frobenius–Schur indicators of Radford algebras in positive characteristic and find minimal polynomials of these linearly recursive sequences. As a result of the work of Kashina, Montgomery and Ng, we obtain gauge invariants for the monoidal categories of representations of Radford algebras.
</p>projecteuclid.org/euclid.involve/1513775066_20171220080420Wed, 20 Dec 2017 08:04 ESTUnlinking numbers of links with crossing number 10https://projecteuclid.org/euclid.involve/1513775067<strong>Lavinia Bulai</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 335--353.</p><p><strong>Abstract:</strong><br/>
We investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinking numbers of all but two of the 287 prime, nonsplit links with crossing number 10.
</p>projecteuclid.org/euclid.involve/1513775067_20171220080420Wed, 20 Dec 2017 08:04 ESTOn a connection between local rings and their associated graded algebrashttps://projecteuclid.org/euclid.involve/1513775068<strong>Justin Hoffmeier</strong>, <strong>Jiyoon Lee</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 2, 355--359.</p><p><strong>Abstract:</strong><br/>
We study a class of local rings and a local adaptation of the homogeneous property for graded rings. While the rings of interest satisfy the property in the local case, we show that their associated graded [math] -algebras do not satisfy the property in the graded case.
</p>projecteuclid.org/euclid.involve/1513775068_20171220080420Wed, 20 Dec 2017 08:04 ESTA mathematical model of treatment of cancer stem cells with immunotherapyhttps://projecteuclid.org/euclid.involve/1513775073<strong>Zachary J. Abernathy</strong>, <strong>Gabrielle Epelle</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 361--382.</p><p><strong>Abstract:</strong><br/>
Using the work of Shelby Wilson and Doron Levy (2012), we develop a mathematical model to study the growth and responsiveness of cancerous tumors to various immunotherapy treatments. We use numerical simulations and stability analysis to predict long-term behavior of passive and aggressive tumors with a range of antigenicities. For high antigenicity aggressive tumors, we show that remission is only achieved after combination treatment with TGF- [math] inhibitors and a peptide vaccine. Additionally, we show that combination treatment has limited effectiveness on low antigenicity aggressive tumors and that using TGF- [math] inhibition or vaccine treatment alone proves generally ineffective for all tumor types considered. A key feature of our model is the identification of separate cancer stem cell and tumor cell populations. Our model predicts that even with combination treatment, failure to completely eliminate the cancer stem cell population leads to cancer recurrence.
</p>projecteuclid.org/euclid.involve/1513775073_20171220080435Wed, 20 Dec 2017 08:04 ESTRNA, local moves on plane trees, and transpositions on tableauxhttps://projecteuclid.org/euclid.involve/1513775074<strong>Laura Del Duca</strong>, <strong>Jennifer Tripp</strong>, <strong>Julianna Tymoczko</strong>, <strong>Judy Wang</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 383--411.</p><p><strong>Abstract:</strong><br/>
We define a collection of functions [math] on the set of plane trees (or standard Young tableaux). The functions are adapted from transpositions in the representation theory of the symmetric group and almost form a group action. They were motivated by local moves in combinatorial biology, which are maps that represent a certain unfolding and refolding of RNA strands. One main result of this study identifies a subset of local moves that we call [math] -local moves, and proves that [math] -local moves correspond to the maps [math] acting on standard Young tableaux. We also prove that the graph of [math] -local moves is a connected, graded poset with unique minimal and maximal elements. We then extend this discussion to functions [math] that mimic reflections in the Weyl group of type [math] . The corresponding graph is no longer connected, but we prove it has two connected components, one of symmetric plane trees and the other of asymmetric plane trees. We give open questions and possible biological interpretations.
</p>projecteuclid.org/euclid.involve/1513775074_20171220080435Wed, 20 Dec 2017 08:04 ESTSix variations on a theme: almost planar graphshttps://projecteuclid.org/euclid.involve/1513775075<strong>Max Lipton</strong>, <strong>Eoin Mackall</strong>, <strong>Thomas W. Mattman</strong>, <strong>Mike Pierce</strong>, <strong>Samantha Robinson</strong>, <strong>Jeremy Thomas</strong>, <strong>Ilan Weinschelbaum</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 413--448.</p><p><strong>Abstract:</strong><br/>
A graph is apex if it can be made planar by deleting a vertex, that is, there exists [math] such that [math] is planar. We also define several related notions; a graph is edge apex if there exists [math] such that [math] is planar, and contraction apex if there exists [math] such that [math] is planar. Additionally we define the analogues with a universal quantifier: for all [math] , [math] is planar; for all [math] , [math] is planar; and for all [math] , [math] is planar. The graph minor theorem of Robertson and Seymour ensures that each of these six notions gives rise to a finite set of obstruction graphs. For the three definitions with universal quantifiers we determine this set. For the remaining properties, apex, edge apex, and contraction apex, we show there are at least 36, 55, and 82 obstruction graphs respectively. We give two similar approaches to almost nonplanar (there exists [math] such that [math] is nonplanar, and for all [math] , [math] is nonplanar) and determine the corresponding minor minimal graphs.
</p>projecteuclid.org/euclid.involve/1513775075_20171220080435Wed, 20 Dec 2017 08:04 ESTNested Frobenius extensions of graded superringshttps://projecteuclid.org/euclid.involve/1513775076<strong>Edward Poon</strong>, <strong>Alistair Savage</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 449--461.</p><p><strong>Abstract:</strong><br/>
We prove a nesting phenomenon for twisted Frobenius extensions. Namely, suppose [math] are graded superrings such that [math] and [math] are both twisted Frobenius extensions of [math] , [math] is contained in the center of [math] , and [math] is projective over [math] . Our main result is that, under these assumptions, [math] is a twisted Frobenius extension of [math] . This generalizes a result of Pike and the second author, which considered the case where [math] is a field.
</p>projecteuclid.org/euclid.involve/1513775076_20171220080435Wed, 20 Dec 2017 08:04 ESTOn $G$-graphs of certain finite groupshttps://projecteuclid.org/euclid.involve/1513775077<strong>Mohammad Reza Darafsheh</strong>, <strong>Safoora Madady Moghadam</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 463--476.</p><p><strong>Abstract:</strong><br/>
The notion of [math] -graph was introduced by Bretto et al. and has interesting properties. This graph is related to a group [math] and a set of generators [math] of [math] and is denoted by [math] . In this paper, we consider several types of groups [math] and study the existence of Hamiltonian and Eulerian paths and circuits in [math] .
</p>projecteuclid.org/euclid.involve/1513775077_20171220080435Wed, 20 Dec 2017 08:04 ESTThe tropical semiring in higher dimensionshttps://projecteuclid.org/euclid.involve/1513775078<strong>John Norton</strong>, <strong>Sandra Spiroff</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 477--488.</p><p><strong>Abstract:</strong><br/>
We discuss the generalization, in higher dimensions, of the tropical semiring, whose two binary operations on the set of real numbers together with infinity are defined to be the minimum and the sum of a pair, respectively. In particular, our objects are closed convex sets, and for any pair, we take the convex hull of their union and their Minkowski sum, respectively, as the binary operations. We consider the semiring in several different cases, determined by a recession cone.
</p>projecteuclid.org/euclid.involve/1513775078_20171220080435Wed, 20 Dec 2017 08:04 ESTA tale of two circles: geometry of a class of quartic polynomialshttps://projecteuclid.org/euclid.involve/1513775079<strong>Christopher Frayer</strong>, <strong>Landon Gauthier</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 489--500.</p><p><strong>Abstract:</strong><br/>
Let [math] be the family of complex-valued polynomials of the form [math] with [math] . The Gauss–Lucas theorem guarantees that the critical points of [math] will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, the unit disk contains two “desert” regions, the open disk [math] and the interior of [math] , in which critical points of [math] cannot occur. Furthermore, each [math] inside the unit disk and outside of the two desert regions is the critical point of at most two polynomials in [math] .
</p>projecteuclid.org/euclid.involve/1513775079_20171220080435Wed, 20 Dec 2017 08:04 ESTZeros of polynomials with four-term recurrencehttps://projecteuclid.org/euclid.involve/1513775080<strong>Khang Tran</strong>, <strong>Andres Zumba</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 501--518.</p><p><strong>Abstract:</strong><br/>
Given real numbers [math] , we form the sequence of polynomials [math] satisfying the four-term recurrence
H
m
(
z
)
+
c
H
m
−
1
(
z
)
+
b
H
m
−
2
(
z
)
+
z
H
m
−
3
(
z
)
=
0
,
m
≥
1
,
with the initial conditions [math] and [math] . We find necessary and sufficient conditions on [math] and [math] under which the zeros of [math] are real for all [math] , and provide an explicit real interval on which [math] is dense, where [math] is the set of zeros of [math] .
</p>projecteuclid.org/euclid.involve/1513775080_20171220080435Wed, 20 Dec 2017 08:04 ESTBinary frames with prescribed dot products and frame operatorhttps://projecteuclid.org/euclid.involve/1513775081<strong>Veronika Furst</strong>, <strong>Eric P. Smith</strong>. <p><strong>Source: </strong>Involve: A Journal of Mathematics, Volume 11, Number 3, 519--540.</p><p><strong>Abstract:</strong><br/>
This paper extends three results from classical finite frame theory over real or complex numbers to binary frames for the vector space [math] . Without the notion of inner products or order, we provide an analog of the “fundamental inequality” of tight frames. In addition, we prove the binary analog of the characterization of dual frames with given inner products and of general frames with prescribed norms and frame operator.
</p>projecteuclid.org/euclid.involve/1513775081_20171220080435Wed, 20 Dec 2017 08:04 EST