Decomposable Model Spaces And A Topological Approach To Curvature, 2021 Wheaton College

#### 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 two examples, we provide ...

The Optimal Double Bubble For Density $R^P$, 2021 Yale University

#### The Optimal Double Bubble For Density $R^P$, 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 **R**^{n} *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 ...

(R1514) Nano Continuous Mappings Via Nano M Open Sets, 2021 Government Arts College (Autonomous); Annamalai University

#### (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, 2021 Beni-Suef University

#### (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, 2021 Ordu University

#### (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, 2021 Clemson University

#### 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, 2021 College of the Holy Cross

#### 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 ...

Image-Based Microbiome Profiling Differentiates Gut Microbial Metabolic States, 2021 Illinois State University

#### 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, 2021 Illinois State University

#### 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.

On 𝜃- -Closed Sets And 𝜃- -Continuous Functlons, 2021 * Faculty of Applied Sciences, Taiz University

#### 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, 2021 University of Massachusetts Amherst

#### 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, 2021 College of the Holy Cross

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

*Mathematics Department Faculty Scholarship*

A hypersurface *M* in **R**^{n} or S^{n} 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 **R**^{n }or *S ^{n}* 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 setting ...

Representing The Derivative Of Trace Of Holonomy, 2021 The Graduate Center, City University of New York

#### 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 ...

Clifford Harmonics, 2021 The Graduate Center, City University of New York

#### 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 ...

Differentiability Of The Liouville Map Via Geodesic Currents, 2021 The Graduate Center, City University of New York

#### 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 ...

A Geometric Model For Real And Complex Differential K-Theory, 2021 The Graduate Center, City University of New York

#### 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.

Centralizers Of Abelian Hamiltonian Actions On Rational Ruled Surfaces, 2021 The University of Western Ontario

#### 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 ...

From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, 2021 Wayne State University

#### From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar

*Mathematics Faculty Research Publications*

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting *structure from the* (non-summarized) *fMRI data itself* are heretofore nonexistent ...

Topology Optimization Of 2d Structures With Multiple Displacement Constraints, 2021 Syracuse University

#### Topology Optimization Of 2d Structures With Multiple Displacement Constraints, Patricio Uarac Pinto

*English Language Institute*

The use of topology optimization in the design process in Civil Engineering can lead to innovative building shapes that not only fulfill structural requirements but also open new opportunities for arquitectonics.

Results On Nonorientable Surfaces For Knots And 2-Knots, 2021 University of Nebraska-Lincoln

#### Results On Nonorientable Surfaces For Knots And 2-Knots, Vincent Longo

*Dissertations, Theses, and Student Research Papers in Mathematics*

A classical knot is a smooth embedding of the circle into the 3-sphere. We can also consider embeddings of arbitrary surfaces (possibly nonorientable) into a 4-manifold, called knotted surfaces. In this thesis, we give an introduction to some of the basics of the studies of classical knots and knotted surfaces, then present some results about nonorientable surfaces bounded by classical knots and embeddings of nonorientable knotted surfaces. First, we generalize a result of Satoh about connected sums of projective planes and twist spun knots. Specifically, we will show that for any odd natural n, the connected sum of the n-twist ...