An Analysis Of The Application Of Simplified Silhouette To The Evaluation Of K-Means Clustering Validity, 2017 Dublin Institute of Technology

#### An Analysis Of The Application Of Simplified Silhouette To The Evaluation Of K-Means Clustering Validity, Hector-Hugo Franco-Penya, John D. Kelleher, John Pugh, Robert Ross

*Conference papers*

Silhouette is one of the most popular and effective internal measures for the evaluation of clustering validity. Simplified Silhouette is a computationally simplified version of Silhouette. However, to date Simplified Silhouette has not been systematically analysed in a specific clustering algorithm. This paper analyses the application of Simplified Silhouette to the evaluation of k-means clustering validity and compares it with the k-means Cost Function and the original Silhouette from both theoretical and empirical perspectives. The theoretical analysis shows that Simplified Silhouette has a mathematical relationship with both the k-means Cost Function and the original Silhouette, while empirically, we show that ...

Rigorous Debates Over Debatable Rigor: Monster Functions In Introductory Analysis, 2017 Colorado State University-Pueblo

#### Rigorous Debates Over Debatable Rigor: Monster Functions In Introductory Analysis, Janet Heine Barnett

*Analysis*

No abstract provided.

The Definite Integrals Of Cauchy And Riemann, 2017 Ursinus College

#### The Definite Integrals Of Cauchy And Riemann, Dave Ruch

*Analysis*

Rigorous attempts to define the definite integral began in earnest in the early 1800's. One of the pioneers in this development was A. L. Cauchy (1789-1857). In this project, students will read from his 1823 study of the definite integral for continuous functions . Then students will read from Bernard Riemann's 1854 paper, in which he developed a more general concept of the definite integral that could be applied to functions with infinite discontinuities.

Solutions Of The System Of Operator Equations $Bxa=B=Axb$ Via The *-Order, 2017 Ferdowsi University of Mashhad

#### Solutions Of The System Of Operator Equations $Bxa=B=Axb$ Via The *-Order, Mehdi Vosough, Mohammad Sal Moslehian

*Electronic Journal of Linear Algebra*

In this paper, some necessary and sufficient conditions are established for the existence of solutions to the system of operator equations $BXA=B=AXB$ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions, it is proved that an operator $X$ is a solution of $BXA=B=AXB$ if and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{ \leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover, the general solution of the equation above is obtained ...

Weak Factorizations Of The Hardy Space H1(RN) In Terms Of Multilinear Riesz Transforms, 2017 Washington University in St. Louis

#### Weak Factorizations Of The Hardy Space H1(RN) In Terms Of Multilinear Riesz Transforms, Ji Li, Brett D. Wick

*Mathematics Faculty Publications*

This paper provides a constructive proof of the weak factorization of the classical Hardy space in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of (the dual of ) via commutators of the multilinear Riesz transforms.

Spaces Of Dirichlet Series With The Complete Pick Property, 2017 Washington University in St Louis

#### Spaces Of Dirichlet Series With The Complete Pick Property, John E. Mccarthy, Orr Moshe Shalit

*Mathematics Faculty Publications*

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s,u)=∑a_{n}n^{−s−u¯}, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space H_{d}^{2} in *d* variables, where *d* can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are ...

Observations On Convexity, 2017 Stephen F Austin State University

#### Observations On Convexity, Chad A. Huckaby

*Electronic Theses and Dissertations*

This thesis will explore convexity as it pertains to sets of complex-valued functions. These include preliminary looks at established linear and polynomially convex hulls, along with the development of new types of convex hulls. These types will include, but are not limited to the hulls determined by inversions, shift inversions, and Mobius transformations. A convex hull must be preceded by the set of functions involved. These hulls are the smallest convex sets that contain the original set. Justifications and precise definitions are included within the body of the work.

The Loewner Equation And Weierstrass' Function, 2017 University of Tennessee, Knoxville

#### The Loewner Equation And Weierstrass' Function, Gavin Ainsley Glenn

*University of Tennessee Honors Thesis Projects*

No abstract provided.

On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, 2017 The University of Western Ontario

#### On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, Nadia Alluhaibi

*Electronic Thesis and Dissertation Repository*

The objective of the study is to investigate the behaviour of the inner products of vector-valued Poincare series, for large weight, associated to submanifolds of a quotient of the complex unit ball and how vector-valued automorphic forms could be constructed via Poincare series. In addition, it provides a proof of that vector-valued Poincare series on an irreducible bounded symmetric domain span the space of vector-valued automorphic forms.

Self-Interlacing Polynomials Ii: Matrices With Self-Interlacing Spectrum, 2017 Shanghai Jiaotong University

#### Self-Interlacing Polynomials Ii: Matrices With Self-Interlacing Spectrum, Mikhail Tyaglov

*Electronic Journal of Linear Algebra*

An $n\times n$ matrix is said to have a self-interlacing spectrum if its eigenvalues $\lambda_k$, $k=1,\ldots,n$, are distributed as follows: $$ \lambda_1>-\lambda_2>\lambda_3>\cdots>(-1)^{n-1}\lambda_n>0. $$ A method for constructing sign definite matrices with self-interlacing spectrum from totally nonnegative ones is presented. This method is applied to bidiagonal and tridiagonal matrices. In particular, a result by O. Holtz on the spectrum of real symmetric anti-bidiagonal matrices with positive nonzero entries is generalized.

Optimal Approximation Of Skorohod Integrals – Examples With Substandard Rates, 2017 University of Mannheim

#### Optimal Approximation Of Skorohod Integrals – Examples With Substandard Rates, Peter Parczewski

*Communications on Stochastic Analysis*

No abstract provided.

Existence And Stability For Stochastic Impulsive Neutral Partial Differential Equations Driven By Rosenblatt Process With Delay And Poisson Jumps, 2017 Université Mohamed I

#### Existence And Stability For Stochastic Impulsive Neutral Partial Differential Equations Driven By Rosenblatt Process With Delay And Poisson Jumps, Mouhamed Ait Ouahra, Brahim Boufoussi, El Hassan Lakhel

*Communications on Stochastic Analysis*

No abstract provided.

On Martingale Representation And Logarithmic-Sobolev Inequality For Fractional Brownian Bridge Measures, 2017 Dongbei University of Finance and Economics

#### On Martingale Representation And Logarithmic-Sobolev Inequality For Fractional Brownian Bridge Measures, Xiaoxia Sun, Feng Guo

*Communications on Stochastic Analysis*

No abstract provided.

Nash Twist And Gaussian Noise Measure For Isometric C1 Maps, 2017 Indian Statistical Institute

#### Nash Twist And Gaussian Noise Measure For Isometric C1 Maps, Amites Dasgupta, Mahuya Datta

*Communications on Stochastic Analysis*

No abstract provided.

On Constructing Some Membranes For A Symmetric Α-Stable Process, 2017 Vasyl Stefanyk Precarpathian National University

#### On Constructing Some Membranes For A Symmetric Α-Stable Process, M. M. Osypchuk, M.I. Portenko

*Communications on Stochastic Analysis*

No abstract provided.

A Clark-Ocone Type Formula Under Change Of Measure For Multidimensional L´Evy Processes, 2017 Keio University

#### A Clark-Ocone Type Formula Under Change Of Measure For Multidimensional L´Evy Processes, Ryoichi Suzuki

*Communications on Stochastic Analysis*

No abstract provided.

Anticipative Integrals With Respect To A Filtered Lévy Process And Lévy–Itô Decomposition, 2017 University of Toulouse III, Toulouse Institute of Mathematics UMR C5583, Toulouse, France

#### Anticipative Integrals With Respect To A Filtered Lévy Process And Lévy–Itô Decomposition, Nicolas Savy, Josep Vives

*Communications on Stochastic Analysis*

No abstract provided.

From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, 2017 Bar-Ilan University

#### From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, Mikhail Katz, Luie Polev

*Journal of Humanistic Mathematics*

In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting “the definition” to the students as a monolithic absolute. We hope that our experiences could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicates that students tend to favor infinitesimal definitions over epsilon-delta ones.

Euler's Rediscovery Of E With Instructor Notes, 2017 Ursinus College

#### Euler's Rediscovery Of E With Instructor Notes, Dave Ruch

*Analysis*

No abstract provided.

The Mean Value Theorem, 2017 Ursinus College