202 research outputs found
Undecidability in a free *-algebra
No full textPutinar, Mihai. (2007). Undecidability in a free *-algebra. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/4368
Extremal positive pluriharmonic functions on Euclidean balls
No full text1 online resource (PDF, 11 pages)Jafari, Farhad; Putinar, Mihai. (2007). Extremal positive pluriharmonic functions on Euclidean balls. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/178890
Positive polynomials on projective limits of real algebraic varieties
No full textKuhlmann, Salma; Putinar, Mihai. (2007). Positive polynomials on projective limits of real algebraic varieties. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/4365
Polynomial optimization on odd-dimensional spheres
No full textD'Angelo, John P.; Putinar, Mihai. (2007). Polynomial optimization on odd-dimensional spheres. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/4369
A panorama of positivity. I: dimension free
Get PDFThis survey contains a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included
Recommended from our members
Hermitian Sums of Squares Modulo Hermitian Ideals
Get PDFIn this work we study the problem of writing a Hermitian polynomial as a Hermitian sum of squares modulo a Hermitian ideal. We investigate a novel idea of Putinar-Scheiderer to obtain necessary matrix positivity conditions for Hermitian polynomials to be Hermitian sums of squares modulo Hermitian ideals. We show that the conditions are sufficient for a class of examples making a connection to the operator-valued Riesz-Fejer theorem and block Toeplitz forms. The work fits into the larger themes of Hermitian versions of Hilbert's 17-th problem and characterizations of positivity
Construction principles of tight wavelet frames in connection with linear system theory
No full textThis dissertation explores the connection between two mathematical fields: wavelet analysis, a subfield of time-frequency analysis, and linear system theory, which has its origins in the modeling of real-world systems using dynamical systems. In the former, we focus on the construction of multivariate tight wavelet frames via the Unitary and Oblique Extension Principles. In the latter, we are particularly interested in the state-space approach and its ties to the Nevanlinna-Pick interpolation and operator theory.
In 2015, Scheiderer, Charina, Putinar, and Stöckler linked wavelet frames constructed via the Unitary Extension Principle to so-called realizations from linear system theory. This dissertation aims to generalize this connection to frames based on the Oblique Extension Principle, potentially aiding the construction and study of such frames.
After brief introductions to the two fields mentioned above and to results on the factorization of trigonometric polynomials, we first consider the problem in the univariate case. Here, we can solve the problem both by bringing it back to the setting of the Unitary Extension Principle and by using a generalized version of the Nevanlinna-Pick interpolation.
These univariate results lead to a multivariate result: Using the Kronecker product, we construct so-called separable multivariate frames and connect them to linear system theory for any dimension.
Finally, in the bivariate case, we have to restrict ourselves to a specific subset of frames. For this subset, we use a parameterized version of the univariate results to solve the bivariate problem and again describe connections to the Nevanlinna-Pick interpolation. We illustrate all constructions with examples
Recommended from our members
Roots and Critical Points of Complex Polynomials: Applications of Algorithms in Real Algebra, Moment Theory, Convex Analysis, Optimization, and Positive Polynomials to a Conjecture in Pure Mathematics
Get PDFConjecture 0.1 (Conjecture of Blagovest Sendov (1958)): For a complex polynomial of degree two or more with all its roots contained within the closed unit disk, each root has a critical point within unit distance. We introduce a countable collection of conjectures – one for each degree – by transferring into languages of Real Algebra. For fixed degree, each conjecture is decidable. Thus, we consider decidable statements in Real Algebra. For each of these decidable statements, we seek for certificates. We provide variations of this theme for different contexts: Positivstellensatz, Nichtnegativstellensatz and real radical ideal membership. In our context, our positivity certificates (or membership certificates) provide proof when achieved. We also find plenty of novel numerical evidence substantiating our conjectures
Recommended from our members
Mini-Workshop: Applied Koopmanism
Get PDFKoopman and Perron–Frobenius operators are linear operators that encapsulate dynamics of nonlinear dynamical systems without loss of information. This is accomplished by embedding the dynamics into a larger infinite-dimensional space where the focus of study is shifted from trajectory curves to measurement functions evaluated along trajectories and densities of trajectories evolving in time. Operator-theoretic approach to dynamics shares many features with an optimization technique: the Lasserre moment–sums-of-squares (SOS) hierarchies, which was developed for numerically solving non-convex optimization problems with semialgebraic data. This technique embeds the optimization problem into a larger primal semidefinite programming (SDP) problem consisting of measure optimization over the set of globally optimal solutions, where measures are manipulated through their truncated moment sequences. The dual SDP problem uses SOS representations to certify bounds on the global optimum. This workshop highlighted the common threads between the operator-theoretic dynamical systems and moment–SOS hierarchies in optimization and explored the future directions where the synergy of the two techniques could yield results in fluid dynamics, control theory, optimization, and spectral theory
An invariant Kähler metric on the tangent disk bundle of a space-form
Get PDFWe find a family of Kähler metrics invariantly defined on the radius r0 > 0 tangent disk bundle TM_r0 of any given real space-form M or any of its quotients by discrete groups of isometries. Such metrics are complete in the non-negative
curvature case and non-complete in the negative curvature case. If dim M = 2 and M has constant sectional curvature K nonvanishing, then the Kähler manifolds TM_r0 have holonomy SU(2); hence they are Ricci-flat. For M = S^2, just this
dimension, the metric coincides with the Stenzel metric on the tangent manifold TS^2 , giving us a new most natural description of this well-known metric
- …
