58 research outputs found
Every Ternary Quintic is a Sum of Ten Fifth Powers
To our knowledge at the time of writing, the maximum Waring rank for the set
of all ternary forms of degree (with coefficients in an algebraically
closed field of characteristic zero) is known only for . The best upper
bound that is known for is twelve, and in this work we lower it to ten.Comment: Relevant information added in the footnote (1) at p.
The asymptotic leading term for maximum rank of ternary forms of a given degree
Let be the maximum Waring rank for the set of
all homogeneous polynomials of degree in indeterminates with
coefficients in an algebraically closed field of characteristic zero. To our
knowledge, when , the value of is known
only for . We prove that
as a consequence of the upper bound
.Comment: v1: 10 pages. v2: extended introduction and some mistakes correcte
Surfaces with triple points
In this paper we compute upper bounds for the number of ordinary triple
points on a hypersurface in and give a complete classification for degree
six (degree four or less is trivial, and five is elementary). But the real
purpose is to point out the intricate geometry of examples with many triple
points, and how it fits with the general classification of surfaces
Real rank geometry of ternary forms
We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics, we determine the real rank boundary: It is a hypersurface of degree 168. For quartics, sextics and septics, we identify some of the components of the real rank boundary. The real varieties of sums of powers are stratified by discriminants that are derived from hyperdeterminants
Generic Power Sum Decompositions and Bounds for the Waring Rank
A notion of open rank, related with generic power sum decompositions of
forms, has recently been introduced in the literature. The main result here is
that the maximum open rank for plane quartics is eight. In particular, this
gives the first example of , such that the maximum open rank for degree
forms that essentially depend on variables is strictly greater than the
maximum rank. On one hand, the result allows to improve the previously known
bounds on open rank, but on the other hand indicates that such bounds are
likely quite relaxed. Nevertheless, some of the preparatory results are of
independent interest, and still may provide useful information in connection
with the problem of finding the maximum rank for the set of all forms of given
degree and number of variables. For instance, we get that every ternary forms
of degree can be annihilated by the product of pairwise
independent linear forms.Comment: Accepted version. The final publication is available at
link.springer.com: http://link.springer.com/article/10.1007/s00454-017-9886-
Seeking for the maximum symmetric rank
We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds
Study of the best linear approximation of nonlinear systems with arbitrary inputs
System identification is the art of modelling of a process (physical, biological,
etc.) or to predict its behaviour or output when the environment condition
or parameter changes. One is modelling the input-output relationship of a system,
for example, linking temperature of a greenhouse (output) to the sunlight intensity
(input), power of a car engine (output) with fuel injection rate (input). In linear
systems, changing an input parameter will result in a proportional increase in the
system output. This is not the case in a nonlinear system. Linear system identification
has been extensively studied, more so than nonlinear system identification.
Since most systems are nonlinear to some extent, there is significant interest in this
topic as industrial processes become more and more complex.
In a linear dynamical system, knowing the impulse response function of a
system will allow one to predict the output given any input. For nonlinear systems
this is not the case. If advanced theory is not available, it is possible to approximate
a nonlinear system by a linear one. One tool is the Best Linear Approximation
(Bla), which is an impulse response function of a linear system that minimises the
output differences between its nonlinear counterparts for a given class of input. The
Bla is often the starting point for modelling a nonlinear system. There is extensive
literature on the Bla obtained from input signals with a Gaussian probability
density function (p.d.f.), but there has been very little for other kinds of inputs.
A Bla estimated from Gaussian inputs is useful in decoupling the linear dynamics
from the nonlinearity, and in initialisation of parameterised models. As Gaussian
inputs are not always practical to be introduced as excitations, it is important to
investigate the dependence of the Bla on the amplitude distribution in more detail.
This thesis studies the behaviour of the Bla with regards to other types of signals,
and in particular, binary sequences where a signal takes only two levels. Such an
input is valuable in many practical situations, for example where the input actuator
is a switch or a valve and hence can only be turned either on or off.
While it is known in the literature that the Bla depends on the amplitude
distribution of the input, as far as the author is aware, there is a lack of comprehensive
theoretical study on this topic. In this thesis, the Blas of discrete-time
time-invariant nonlinear systems are studied theoretically for white inputs with an arbitrary amplitude distribution, including Gaussian and binary sequences. In doing
so, the thesis offers answers to fundamental questions of interest to system engineers,
for example: 1) How the amplitude distribution of the input and the system
dynamics affect the Bla? 2) How does one quantify the difference between the
Bla obtained from a Gaussian input and that obtained from an arbitrary input?
3) Is the difference (if any) negligible? 4) What can be done in terms of experiment
design to minimise such difference?
To answer these questions, the theoretical expressions for the Bla have been
developed for both Wiener-Hammerstein (Wh) systems and the more general Volterra
systems. The theory for the Wh case has been verified by simulation and physical
experiments in Chapter 3 and Chapter 6 respectively. It is shown in Chapter 3
that the difference between the Gaussian and non-Gaussian Bla’s depends on the
system memory as well as the higher order moments of the non-Gaussian input.
To quantify this difference, a measure called the Discrepancy Factor—a measure of
relative error, was developed. It has been shown that when the system memory is
short, the discrepancy can be as high as 44.4%, which is not negligible. This justifies
the need for a method to decrease such discrepancy. One method is to design a random
multilevel sequence for Gaussianity with respect to its higher order moments,
and this is discussed in Chapter 5.
When estimating the Bla even in the absence of environment and measurement
noise, the nonlinearity inevitably introduces nonlinear distortions—deviations
from the Bla specific to the realisation of input used. This also explains why more
than one realisation of input and averaging is required to obtain a good estimate of
the Bla. It is observed that with a specific class of pseudorandom binary sequence
(Prbs), called the maximum length binary sequence (Mlbs or the m-sequence), the
nonlinear distortions appear structured in the time domain. Chapter 4 illustrates
a simple and computationally inexpensive method to take advantage this structure
to obtain better estimates of the Bla—by replacing mean averaging by median
averaging.
Lastly, Chapters 7 and 8 document two independent benchmark studies separate
from the main theoretical work of the thesis. The benchmark in Chapter 7 is
concerned with the modelling of an electrical Wh system proposed in a special session
of the 15th International Federation of Automatic Control (Ifac) Symposium on
System Identification (Sysid) 2009 (Schoukens, Suykens & Ljung, 2009). Chapter 8
is concerned with the modelling of a ‘hyperfast’ Peltier cooling system first proposed
in the U.K. Automatic Control Council (Ukacc) International Conference
on Control, 2010 (Control 2010)
The monic rank
We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone . We show that the monic rank is finite and greater than or equal to the usual -rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree is the sum of th powers of forms of degree . Furthermore, in the case where is the cone of highest weight vectors in an irreducible representation--this includes the well-known cases of tensor rank and symmetric rank--we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances
- …