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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- inhibitors and a peptide vaccine. Additionally, we show that combination treatment has limited effectiveness on low antigenicity aggressive tumors and that using TGF- 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.
We define a collection of functions 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 -local moves, and proves that -local moves correspond to the maps acting on standard Young tableaux. We also prove that the graph of -local moves is a connected, graded poset with unique minimal and maximal elements. We then extend this discussion to functions that mimic reflections in the Weyl group of type . 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.
A graph is apex if it can be made planar by deleting a vertex, that is, there exists such that is planar. We also define several related notions; a graph is edge apex if there exists such that is planar, and contractionapex if there exists such that is planar. Additionally we define the analogues with a universal quantifier: for all , is planar; for all , is planar; and for all , 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 such that is nonplanar, and for all , is nonplanar) and determine the corresponding minor minimal graphs.
We prove a nesting phenomenon for twisted Frobenius extensions. Namely, suppose are graded superrings such that and are both twisted Frobenius extensions of , is contained in the center of , and is projective over . Our main result is that, under these assumptions, is a twisted Frobenius extension of . This generalizes a result of Pike and the second author, which considered the case where is a field.
The notion of -graph was introduced by Bretto et al. and has interesting properties. This graph is related to a group and a set of generators of and is denoted by . In this paper, we consider several types of groups and study the existence of Hamiltonian and Eulerian paths and circuits in .
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.
Let be the family of complex-valued polynomials of the form with . The Gauss–Lucas theorem guarantees that the critical points of 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 and the interior of , in which critical points of cannot occur. Furthermore, each inside the unit disk and outside of the two desert regions is the critical point of at most two polynomials in .
Given real numbers , we form the sequence of polynomials satisfying the four-term recurrence
with the initial conditions and . We find necessary and sufficient conditions on and under which the zeros of are real for all , and provide an explicit real interval on which is dense, where is the set of zeros of .
This paper extends three results from classical finite frame theory over real or complex numbers to binary frames for the vector space . 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.
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