Geometry and Topology Commons^™

Decomposable Model Spaces And A Topological Approach To Curvature, Kevin M. Tully

Rose-Hulman Undergraduate Mathematics Journal

This research investigates a model space invariant known as k-plane constant vector curvature, traditionally studied when k=2, and introduces a new invariant, (m,k)-plane constant vector curvature. We prove that the sets of k-plane and (m,k)-plane constant vector curvature values are connected, compact subsets of the real numbers and establish several relationships between the curvature values of a decomposable model space and its component spaces. We also prove that every decomposable model space with a positive-definite inner product has k-plane constant vector curvature for some integer k>1. In …

The Optimal Double Bubble For Density 𝑟ᵖ, Jack Hirsch, Kevin Li, Jackson Petty, Christopher Xue

Rose-Hulman Undergraduate Mathematics Journal

In 2008 Reichardt proved that the optimal Euclidean double bubble---the least-perimeter way to enclose and separate two given volumes---is three spherical caps meeting along a sphere at 120 degrees. We consider Rⁿ with density r^p, joining the surge of research on manifolds with density after their appearance in Perelman's 2006 proof of the Poincaré Conjecture. Boyer et al. proved that the best single bubble is a sphere through the origin. We conjecture that the best double bubble is the Euclidean solution with the singular sphere passing through the origin, for which we have verified equilibrium (first variation …

Interpretation Of De Sitter Space Of Second Kind, Abdulaziz Artikbaev, Botirjon Mamadaliyev

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

In a five-dimensional pseudo-Euclidean space of index two, the geometry on its sphere is studied. The equivalence of the geometry on a sphere of imaginary radius on de Sitter space is shown. The interpretation of the geometry on a sphere of imaginary radius, inside the sphere of imaginary radius of the Minkowski four-dimensional space, is implemented. We study a curve in a five-dimensional pseudo-Euclidean space of index two and determine the membership condition of the curve to a sphere of imaginary radius.

(R1514) Nano Continuous Mappings Via Nano M Open Sets, A. Vadivel, A. Padma, M. Saraswathi, G. Saravanakumar

Applications and Applied Mathematics: An International Journal (AAM)

Nano M open sets are a union of nano θ semi open sets and nano δ pre open sets. The properties of nano M open sets with their interior and closure operators are discussed in a previous paper. In this paper, we discuss about nano M-continuous and nano M-irresolute functions are introduced in a nano topological spaces along with their continuous and irresolute mappings. Also, nano M-open and nano M-closed functions are introduced and compare with their near open and closed mappings in a nano topological spaces. Further, nano M homeomorphism is also discussed in nano …

(R1519) On Some Geometric Properties Of Non-Null Curves Via Its Position Vectors In \Mathbb{R}_1^3, Emad Solouma, Ibrahim Al-Dayel

Applications and Applied Mathematics: An International Journal (AAM)

In this work, the geometric properties of non-null curves lying completely on spacelike surface via its position vectors in the dimensional Minkowski 3-space \mathbb{R}_1^3 are studied. Also, we give a few portrayals for the spacelike curves which lie on certain subspaces of \mathbb{R}_1^3. Finally, we present an application to demonstrate our insights.

(R1499) Family Of Surfaces With A Common Bertrand D-Curve As Isogeodesic, Isoasymptotic And Line Of Curvature, Süleyman Şenyurt, Kebire Hilal Ayvacı, Davut Canlı

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we establish the necessary and sufficient conditions to parameterize a surface family on which the Bertrand D-partner of any given curve lies as isogeodesic, isoasymptotic or curvature line in \mathbb{E}^3. Then, we calculate the fundamental forms of these surfaces and determine the developability and minimality conditions with the Gaussian and mean curvatures. We also extend this idea on ruled surfaces and provide the required conditions for those to be developable. Finally, we present some examples and graph the corresponding surfaces.

Acceleration Skinning: Kinematics-Driven Cartoon Effects For Articulated Characters, Niranjan Kalyanasundaram

All Theses

Secondary effects are key to adding fluidity and style to animation. This thesis introduces the idea of “Acceleration Skinning” following a recent well-received technique, Velocity Skinning, to automatically create secondary motion in character animation by modifying the standard pipeline for skeletal rig skinning. These effects, which animators may refer to as squash and stretch or drag, attempt to create an illusion of inertia. In this thesis, I extend the Velocity Skinning technique to include acceleration for creating a wider gamut of cartoon effects. I explore three new deformers that make use of this Acceleration Skinning framework: followthrough, centripetal stretch, and …

Practical Geometry, Christopher Clavius S.J., John B. Little

Holy Cross Bookshelf

John B. Little is the translator.

This is a Latin to English translation of Geometria Practica by Chrisopher Clavius, S.J. (1538-1612), the preeminent Jesuit mathematician and mathematical astronomer of his time. The first edition of Geometria Practica appeared in 1604. This translation is of the second edition from 1606, produced by the printshop of Johann Albin in Mainz.

In preparing this translation we have made use of the electronic version of the 1606 edition of the Geometria Practica maintained by the Bayerische StaatsBibliothek. In particular, all of the figures have been copied from the scanned images here. The typesetting was …

Image-Based Microbiome Profiling Differentiates Gut Microbial Metabolic States, Sarwesh Rauniyar

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Topology And Ecology: Deducing States Of The Upper Mississippi River System, Killian Davis

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

ℂ-Motivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka

Mathematics Faculty Research Publications

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations.We compute the Steenrod algebra in this context, and we construct a “motivic modular forms” spectrum over ℂ.

On 𝜃- -Closed Sets And 𝜃- -Continuous Functlons, Amin Hamoud Saif, Nahid Mohammed Al-Showhati

Hadhramout University Journal of Natural & Applied Sciences

In topological spaces, the class of 𝜃-closed sets and 𝜃-continuous function have been introduced by Velicko and Fomin respectively. The purpose of this paper is to introduce and study these notions in grill topological spaces by giving the new classes of 𝜃- -closed sets and 𝜃- -continuous functions in grill topological space.

Equivariant Surgeries And Irreducible Embeddings Of Surfaces In Dimension Four, Andrew J. Havens

Doctoral Dissertations

We construct families of smoothly irreducible embeddings of surfaces in the 4-sphere, corresponding to a range of normal Euler numbers. We also describe a procedure to produce equivariant symplectic sums of real symplectic 4-manifolds. For explicit real symplectic involutions on pairs of symplectic 4-manifolds the conditions for the existence of equivariant symplectic sums can be detected combinatorially. Such sums are sought for potential new constructions of families of irreducible knotted surfaces in fixed four manifolds.

Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, Thomas E. Cecil

Mathematics Department Faculty Scholarship

A hypersurface M in Rⁿ or Sⁿ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in Rⁿ or Sⁿ can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective …

On Extensions And Restrictions Of Τ-Smooth And Τ-Maxitive Idempotent Measures, Muzaffar Eshimbetov

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

In the paper we investigate maps between idempotent measures spaces, τ-maxitive idempotent measures and their extensions and restrictions. For an idempotent measure we prove that its extension is τ-maxitive if and only if its restriction is τ-maxitive.

A Geometric Model For Real And Complex Differential K-Theory, Matthew T. Cushman

Dissertations, Theses, and Capstone Projects

We construct a differential-geometric model for real and complex differential K-theory based on a smooth manifold model for the K-theory spectra defined by Behrens using spaces of Clifford module extensions. After writing representative differential forms for the universal Pontryagin and Chern characters we transgress these forms to all the spaces of the spectra and use them to define an abelian group structure on maps up to an equivalence relation that refines homotopy. Finally we define the differential K-theory functors and verify the axioms of Bunke-Schick for a differential cohomology theory.

Differentiability Of The Liouville Map Via Geodesic Currents, Xinlong Dong

Dissertations, Theses, and Capstone Projects

For a conformally hyperbolic Riemann surface, the Teichmüller space is the space of quasiconformal maps factored by an equivalence relation, and it is a complex Banach manifold. The space of geodesic currents endowed with the uniform weak* topology is a subset of a Fréchet space of Hölder distributions. We introduce an appropriate topology on the space of Hölder distributions and this new topology coincides with the uniform weak* topology on the space of geodesic currents. The Liouville map of the Teichmüller space becomes differentiable in the Fréchet sense. In particular, the derivative of Liouville currents exists and belongs to the …

Clifford Harmonics, Samuel L. Hosmer

Dissertations, Theses, and Capstone Projects

In 1980 Michelsohn defined a differential operator on sections of the complex Clifford bundle over a compact Kähler manifold M. This operator is a differential and its Laplacian agrees with the Laplacian of the Dolbeault operator on forms through a natural identification of differential forms with sections of the Clifford bundle. Relaxing the condition that M be Kähler, we introduce two differential operators on sections of the complex Clifford bundle over a compact almost Hermitian manifold which naturally generalize the one introduced by Michelsohn. We show surprising Kähler- like symmetries of the kernel of the Laplacians of these operators in …

Representing The Derivative Of Trace Of Holonomy, Jeffrey Peter Kroll

Dissertations, Theses, and Capstone Projects

Trace of holonomy around a fixed loop defines a function on the space of unitary connections on a hermitian vector bundle over a Riemannian manifold. Using the derivative of trace of holonomy, the loop, and a flat unitary connection, a functional is defined on the vector space of twisted degree 1 cohomology classes with coefficients in skew-hermitian bundle endomorphisms. It is shown that this functional is obtained by pairing elements of cohomology with a degree 1 homology class built directly from the loop and equipped with a flat section obtained from the variation of holonomy around the loop. When the …

Centralizers Of Abelian Hamiltonian Actions On Rational Ruled Surfaces, Pranav Vijay Chakravarthy

Electronic Thesis and Dissertation Repository

In this thesis, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $CP^2$ blown up once under the presence of Hamiltonian group actions of either $S^1$ or finite cyclic groups. For Hamiltonian circle actions, we prove that the centralizers are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. We can show that the same holds for the centralizers of most finite cyclic groups in the Hamiltonian group $\Ham(M)$. …