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Articles 1 - 30 of 120
Full-Text Articles in Physical Sciences and Mathematics
Hypersurfaces With Nonnegative Ricci Curvature In Hyperbolic Space, Vincent Bonini, Shiguang Ma, Jie Qing
Hypersurfaces With Nonnegative Ricci Curvature In Hyperbolic Space, Vincent Bonini, Shiguang Ma, Jie Qing
Mathematics
Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier (Proc Symp Pure Math 54(3):37–44, 1993).
On Nonnegatively Curved Hypersurfaces In Hyperbolic Space, Vincent Bonini, Shiguang Ma, Jie Qing
On Nonnegatively Curved Hypersurfaces In Hyperbolic Space, Vincent Bonini, Shiguang Ma, Jie Qing
Mathematics
In this paper we prove a conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.
Invariant Subspaces Of Compact Operators And Related Topics, Weston Mckay Grewe
Invariant Subspaces Of Compact Operators And Related Topics, Weston Mckay Grewe
Mathematics
The invariant subspace problem asks if every bounded linear operator on a Banach space has a nontrivial closed invariant subspace. Per Enflo has shown this is false in general, however it is known that every compact operator has an invariant subspace. The purpose of this project is to explore introductory results in functional analysis. Specifically we are interested in understanding compact operators and the proof that all compact operators on a Hilbert space have an invariant subspace. In the process of doing this we build up many examples and theorems relating to operators on a Hilbert or Banach space. Continuing …
Weakly Horospherically Convex Hypersurfaces In Hyperbolic Space, Vincent Bonini, Jie Qing, Jingyong Zhu
Weakly Horospherically Convex Hypersurfaces In Hyperbolic Space, Vincent Bonini, Jie Qing, Jingyong Zhu
Mathematics
In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces ϕ:Mn→Hn+1 and a class of conformal metrics on domains of the round sphere Sn . Some of the key aspects of the correspondence and its consequences have dimensional restrictions n≥3 due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of Sn . In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the …
Hypersurfaces In Hyperbolic Space With Support Function, Vincent Bonini, José M. Espinar, Jie Qint
Hypersurfaces In Hyperbolic Space With Support Function, Vincent Bonini, José M. Espinar, Jie Qint
Mathematics
Based on [previous publication*], we develop a global correspondence between immersed hypersurfaces ϕ:Mn→Hn+1ϕ:Mn→Hn+1 satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e2ρgSne2ρgSn on domains Ω in the boundary SnSn at infinity of Hn+1Hn+1, where ρ is the horospherical support function, ∂∞ϕ(Mn)=∂Ω∂∞ϕ(Mn)=∂Ω, and Ω is the image of the Gauss map G:Mn→SnG:Mn→Sn. To do so we first establish results on when the Gauss map G:Mn→SnG:Mn→Sn is injective. We also …
An Analysis Of Strategic Treatment Interruptions During Imatinib Treatment Of Chronic Myelogenous Leukemia With Imatinib-Resistant Mutations, Dana Paquin, David Sacco, John Shamshoian
An Analysis Of Strategic Treatment Interruptions During Imatinib Treatment Of Chronic Myelogenous Leukemia With Imatinib-Resistant Mutations, Dana Paquin, David Sacco, John Shamshoian
Mathematics
Chronic myelogenous leukemia (CML) is a cancer of the white blood cells that results from increased and uncontrolled growth of myeloid cells in the bone marrow and the accumulation of these cells in the blood. The most common form of treatment for CML is imatinib, a tyrosine kinase inhibitor. Although imatinib is an effective treatment for CML and most patients treated with imatinib do attain some form of remission, imatinib does not completely eradicate all leukemia cells, and if treatment is stopped, all patients eventually relapse (Cortes, 2005). In Kim (2008), the authors developed a mathematical model for the dynamics …
The Stratified Structure Of Spaces Of Smooth Orbifold Mappings, Joseph E. Borzellino, Victor Brunsden
The Stratified Structure Of Spaces Of Smooth Orbifold Mappings, Joseph E. Borzellino, Victor Brunsden
Mathematics
We consider four notions of maps between smooth C∞ orbifolds , with compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of Cr maps between and with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of Cr maps between and with the Cr topology carries the …
Hilbert Space Theory And Applications In Basic Quantum Mechanics, Matthew Gagne
Hilbert Space Theory And Applications In Basic Quantum Mechanics, Matthew Gagne
Mathematics
We explore the basic mathematical physics of quantum mechanics. Our primary focus will be on Hilbert space theory and applications as well as the theory of linear operators on Hilbert space. We show how Hermitian operators are used to represent quantum observables and investigate the spectrum of various linear operators. We discuss deviation and uncertainty and briefly suggest how symmetry and representations are involved in quantum theory.
Elementary Orbifold Differential Topology, Joseph E. Borzellino, Victor Brunsden
Elementary Orbifold Differential Topology, Joseph E. Borzellino, Victor Brunsden
Mathematics
Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f:O→P between smooth orbifolds O and P. We show that Sardʼs theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence of a smooth orbifold map imposes on local isotropy groups. As an application, we prove a Borsuk no retraction theorem for compact orbifolds with boundary and some obstructions to the existence of real-valued orbifold maps from local model orbifold …
When Is A Trigonometric Polynomial Not A Trigonometric Polynomial?, Joseph E. Borzellino, Morgan Sherman
When Is A Trigonometric Polynomial Not A Trigonometric Polynomial?, Joseph E. Borzellino, Morgan Sherman
Mathematics
No abstract provided.
Temporal Topos And U-Singularities, Goro C. Kato
Temporal Topos And U-Singularities, Goro C. Kato
Mathematics
Several papers and books by C. Isham, C.Isham-A. Doering, F. Van Oystaeyen, A.Mallios-I. Raptis, C. Mulvey, and Guts and Grinkevich, have been published on the methods of categories and sheaves to study quantum gravity. Needless to say, there are well-written treatises on quantum gravity whose methods are non-categorical and non-sheaf theoretic. This paper may be one of the first papers explaining the methods of sheaves with minimally required background that retains experimental applications. Temporal topos (t-topos) is related to the topos approach to quantum gravity being developed by Prof. Chris Isham of the Oxford-Imperial research group (with its foundations inthe …
An Axiomatic Approach To The Non-Linear Theory Of Generalized Functions And Consistency Of Laplace Transforms, Todor D. Todorov
An Axiomatic Approach To The Non-Linear Theory Of Generalized Functions And Consistency Of Laplace Transforms, Todor D. Todorov
Mathematics
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of …
Completeness Of The Leibniz Field And Rigorousness Of Infinitesimal Calculus, James F. Hall, Todor D. Todorov
Completeness Of The Leibniz Field And Rigorousness Of Infinitesimal Calculus, James F. Hall, Todor D. Todorov
Mathematics
We present a characterization of the completeness of the field of real numbers in the form of a collection of ten equivalent statements borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. As an application we exploit one of our results to argue that the Leibniz infinitesimal calculus in the 18th century was already a rigorous branch of mathematics – at least much more rigorous than most contemporary mathematicians prefer to believe. By advocating our particular historical point of view, we hope to …
Strategic Treatment Interruptions During Imatinib Treatment Of Chronic Myelogenous Leukemia, Dana C. Paquin, Peter S. Kim, Peter P. Lee, Doron Levy
Strategic Treatment Interruptions During Imatinib Treatment Of Chronic Myelogenous Leukemia, Dana C. Paquin, Peter S. Kim, Peter P. Lee, Doron Levy
Mathematics
Although imatinib is an effective treatment for chronic myelogenous leukemia (CML), and nearly all patients treated with imatinib attain some form of remission, imatinib does not completely eliminate leukemia. Moreover, if the imatinib treatment is stopped, most patients eventually relapse (Cortes et al. in Clin. Cancer Res. 11:3425–3432, 2005). In Kim et al. (PLoS Comput. Biol. 4(6):e1000095, 2008), the authors presented a mathematical model for the dynamics of CML under imatinib treatment that incorporates the anti-leukemia immune response. We use the mathematical model in Kim et al. (PLoS Comput. Biol. 4(6):e1000095, 2008) to study and numerically simulate strategic treatment interruptions …
Another Proof Of The Existence A Dedekind Complete Totally Ordered Field, James F. Hall, Todor D. Todorov
Another Proof Of The Existence A Dedekind Complete Totally Ordered Field, James F. Hall, Todor D. Todorov
Mathematics
We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis.
Completeness Of Ordered Fields, James Forsythe Hall
Completeness Of Ordered Fields, James Forsythe Hall
Mathematics
The main goal of this project is to prove the equivalency of several characterizations of completeness of Archimedean ordered fields; some of which appear in most modern literature as theorems following from the Dedekind completeness of the real numbers, while a couple are not as well known and have to do with other areas of mathematics, such as nonstandard analysis. Continuing, we study the completeness of non-Archimedean fields, and provide several examples of such fields with varying degrees of properties, using nonstandard analysis to produce some relatively "nice" (in particular, they are Cantor complete) final examples. As a small detour, …
Correspondences Of Hypersurfaces In Hyperbolic Poincaré Manifolds And Conformally Invariant Pdes, Vincent Bonini, José M. Espinar, Jie Qing
Correspondences Of Hypersurfaces In Hyperbolic Poincaré Manifolds And Conformally Invariant Pdes, Vincent Bonini, José M. Espinar, Jie Qing
Mathematics
On a hyperbolic Poincaré manifold, we derive an explicit relationship between the eigenvalues of Weyl-Schouten tensor of a conformal representative of the conformal infinity and the principal curvatures of the level sets of the associated geodesic defining function. This considerably simplifies the arguments and generalizes the results of Gálvez, Mira and the second author. In particular, we obtain the equivalence between Christoffel-type problems for hypersurfaces in a hyperbolic Poincar´e manifold and scalar curvature problems on the conformal infinity.
Consecutive Patterns: From Permutations To Column-Convex Polyominoes And Back, Don Rawlings, Mark Tiefenbruck
Consecutive Patterns: From Permutations To Column-Convex Polyominoes And Back, Don Rawlings, Mark Tiefenbruck
Mathematics
We expose the ties between the consecutive pattern enumeration problems associated with permutations, compositions, column-convex polyominoes, and words. Our perspective allows powerful methods from the contexts of compositions, column-convex polyominoes, and of words to be applied directly to the enumeration of permutations by consecutive patterns. We deduce a host of new consecutive pattern results,including a solution to the (2m+1)-alternating pattern problem on permutations posed by Kitaev.
U-Singularity And T-Topos Theoretic Entropy, Goro C. Kato
U-Singularity And T-Topos Theoretic Entropy, Goro C. Kato
Mathematics
We will give descriptions of u-singularities as we introduce the notion of t-topos theoretic entropies. The unifying methodology for a u-singularity is the universal mapping property of an inverse or direct limit. The qualitative, conceptual, and structural analyses of u-singularities are given in terms of inverse and direct limits of micro decompositions of a presheaf and coverings of an object in t-site in the theory of temporal topos.
Urcohomologies And Cohomologies Of N -Complexes, Naoya Hiramatsu, Goro Kato
Urcohomologies And Cohomologies Of N -Complexes, Naoya Hiramatsu, Goro Kato
Mathematics
For a general sequence of objects and morphisms, we construct two N-complexes. Then we can define cohomologies (i, k)-type of the N-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of N-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of N-complexes are given.
Multiscale Registration Of Planning Ct And Daily Cone Beam Ct Images For Adaptive Radiation Therapy, Dana C. Paquin, Doron Levy, Lei Xing
Multiscale Registration Of Planning Ct And Daily Cone Beam Ct Images For Adaptive Radiation Therapy, Dana C. Paquin, Doron Levy, Lei Xing
Mathematics
Adaptive radiation therapy (ART) is the incorporation of daily images in the radiotherapy treatment process so that the treatment plan can be evaluated and modified to maximize the amount of radiation dose to the tumor while minimizing the amount of radiation delivered to healthy tissue. Registration of planning images with daily images is thus an important component of ART. In this article, the authors report their research on multiscale registration of planning computed tomography (CT) images with daily cone beam CT (CBCT) images. The multiscale algorithm is based on the hierarchical multiscale image decomposition of E. Tadmor, S. Nezzar, and …
Full Algebra Of Generalized Functions And Non-Standard Asymptotic Analysis, Todor D. Todorov, Hans Vernaeve
Full Algebra Of Generalized Functions And Non-Standard Asymptotic Analysis, Todor D. Todorov, Hans Vernaeve
Mathematics
We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution. We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article …
Multiscale Deformable Registration Of Noisy Medical Images, Dana C. Paquin, Doron Levy, Lei Xing
Multiscale Deformable Registration Of Noisy Medical Images, Dana C. Paquin, Doron Levy, Lei Xing
Mathematics
Multiscale image registration techniques are presented for the registration of medical images using deformable registration models. The techniques are particularly effective for registration problems in which one or both of the images to be registered contains significant levels of noise. A brief overview of existing deformable registration techniques is presented, and experiments using B-spline free-form deformation registration models demonstrate that ordinary deformable registration techniques fail to produce accurate results in the presence of significant levels of noise. The hierarchical multiscale image decomposition described in E. Tadmor, S. Nezzar, and L. Vese’s, ”A multiscale image representation using hierarchical (BV, L2 …
Local Geometry Of Zero Sets Of Holomorphic Functions Nears The Torus, Jim Agler, John E. Mccarthy, Mark Stankus
Local Geometry Of Zero Sets Of Holomorphic Functions Nears The Torus, Jim Agler, John E. Mccarthy, Mark Stankus
Mathematics
We call a holomorphic function f on a domain in Cn locally toral at the point P in the n-torus if the intersection of the zero set of f with the n-torus has dimension n−1 at P. We study the interplay between the structure of locally toral functions and the geometry of their zero sets.
Sheaf Theoretic Formulation Of Entanglement, Goro C. Kato
Sheaf Theoretic Formulation Of Entanglement, Goro C. Kato
Mathematics
A formulation in terms of sheaf theoretic (or categorical) notions for quantum entanglement is given with direct experimental consequences. The notions from sheaf theory and category theory give structural theory, i.e., qualitative theory, as a candidate for quantum gravity. Its advantage is the following: it provides not only space-time background independent, but also scale independent.This theory is called the theory of temporal topos (or simply t-topos theory).
Microcosm To Macrocosm Via The Notion Of A Sheaf (Observers In Terms Of T-Topos), Goro Kato
Microcosm To Macrocosm Via The Notion Of A Sheaf (Observers In Terms Of T-Topos), Goro Kato
Mathematics
The fundamental approach toward matter, space and time is that particles (either objects of macrocosm or microcosm), space and time are all presheafified. Namely, the concept of a presheaf is most fundamental for matter, space and time. An observation of a particle is represented by a morphism from the observed particle (its associated presheaf) to the observer (its associated presheaf) over a specified object (called a generalized time period) of a t-site (i.e. a category with a Grothendieck topology). This formulation provides a scale independent and background space-time free theory (since, for the t-topos theoretic formulation, space and time are …
A Positive Mass Theorem On Asymptotically Hyperbolic Manifolds With Corners Along A Hypersurface, Vincent Bonini, Jie Qing
A Positive Mass Theorem On Asymptotically Hyperbolic Manifolds With Corners Along A Hypersurface, Vincent Bonini, Jie Qing
Mathematics
In this paper we take an approach similar to that in [13] to establish a positive mass theorem for spin asymptotically hyperbolic manifolds admitting corners along a hypersurface. The main analysis uses an integral representation of a solution to a perturbed eigenfunction equation to obtain an asymptotic expansion of the solution in the right order. This allows us to understand the change of the mass aspect of a conformal change of asymptotically hyperbolic metrics.
A Lost Theorem: Definite Integrals In An Asymptotic Setting, Ray Cavalcante, Todor D. Todorov
A Lost Theorem: Definite Integrals In An Asymptotic Setting, Ray Cavalcante, Todor D. Todorov
Mathematics
No abstract provided.
A Manifold Structure For The Group Of Orbifold Diffeomorphisms Of A Smooth Orbifold, Joseph E. Borzellino, Victor Brunsden
A Manifold Structure For The Group Of Orbifold Diffeomorphisms Of A Smooth Orbifold, Joseph E. Borzellino, Victor Brunsden
Mathematics
For a compact, smooth Cr orbifold (without boundary), we show that the topological structure of the orbifold diffeomorphism group is a Banach manifold for 1 ≤ r < ∞ and a Fréchet manifold if r = ∞. In each case, the local model is the separable Banach (Fréchet) space of Cr (C ∞, resp.) orbisections of the tangent orbibundle.
Deformable Image Registration With Inclusion Of Auto-Detected Homologous Tissue Features, Y. Xie, Lei Xing, Dana C. Paquin, Doron Levy, T. Yang
Deformable Image Registration With Inclusion Of Auto-Detected Homologous Tissue Features, Y. Xie, Lei Xing, Dana C. Paquin, Doron Levy, T. Yang
Mathematics
No abstract provided.