1,166 research outputs found
Clustering via kernel decomposition
Spectral clustering methods were proposed recently which rely on the eigenvalue decomposition of an affinity matrix. In this letter, the affinity matrix is created from the elements of a nonparametric density estimator and then decomposed to obtain posterior probabilities of class membership. Hyperparameters are selected using standard cross-validation methods
Asymptotic behavior of age-structured and delayed Lotka-Volterra models
In this work we investigate some asymptotic properties of an age-structured
Lotka-Volterra model, where a specific choice of the functional parameters
allows us to formulate it as a delayed problem, for which we prove the
existence of a unique coexistence equilibrium and characterize the existence of
a periodic solution. We also exhibit a Lyapunov functional that enables us to
reduce the attractive set to either the nontrivial equilibrium or to a periodic
solution. We then prove the asymptotic stability of the nontrivial equilibrium
where, depending on the existence of the periodic trajectory, we make explicit
the basin of attraction of the equilibrium. Finally, we prove that these
results can be extended to the initial PDE problem.Comment: 29 page
Subexponential solutions of scalar linear integro-differential equations with delay
This paper considers the asymptotic behaviour of solutions of the scalar
linear convolution integro-differential equation with delay
x0(t) = â
n Xi=1
aix(t â i) + Z t
0
k(t â s)x(s) ds, t > 0,
x(t) = (t), â t 0,
where = max1in i. In this problem, k is a non-negative function in L1(0,1)\C[0,1),
i 0, ai > 0 and is a continuous function on [â, 0]. The kernel k is subexponential
in the sense that limt!1 k(t)(t)â1 > 0 where is a positive subexponential function. A
consequence of this is that k(t)et ! 1 as t ! 1 for every > 0
PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators
In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and
handling of Partial Integral (PI) operators. The toolbox introduces a new class
of MATLAB object, opvar, for which standard MATLAB matrix operation syntax
(e.g. +, *, ' e tc.) is defined. PI operators are a generalization of bounded
linear operators on infinite-dimensional spaces that form a *-subalgebra with
two binary operations (addition and composition) on the space RxL2. These
operators frequently appear in analysis and control of infinite-dimensional
systems such as Partial Differential equations (PDE) and Time-delay systems
(TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add
operator positivity constraints, declare an objective function, and solve the
resulting optimization problem using a syntax similar to the sdpvar class in
YALMIP. Use of the resulting Linear Operator Inequalities (LOIs) are
demonstrated on several examples, including stability analysis of a PDE,
bounding operator norms, and verifying integral inequalities. The result is
that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable,
user-friendly and computationally efficient toolbox for parsing, performing
algebraic operations, setting up and solving convex optimization problems on PI
operators
On Delay-independent Stability of a class of Nonlinear Positive Time-delay Systems
We present a condition for delay-independent stability of a class of
nonlinear positive systems. This result applies to systems that are not
necessarily monotone and extends recent work on cooperative nonlinear systems.Comment: 9 page
Tensor Products, Positive Linear Operators, and Delay-Differential Equations
We develop the theory of compound functional differential equations, which
are tensor and exterior products of linear functional differential equations.
Of particular interest is the equation with a single delay, where the delay
coefficient is of one sign, say with .
Positivity properties are studied, with the result that if then
the -fold exterior product of the above system generates a linear process
which is positive with respect to a certain cone in the phase space.
Additionally, if the coefficients and are periodic of
the same period, and satisfies a uniform sign condition, then there
is an infinite set of Floquet multipliers which are complete with respect to an
associated lap number. Finally, the concept of -positivity of the exterior
product is investigated when satisfies a uniform sign condition.Comment: 84 page
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