# On the Generalizations of the Cauchy-Schwarz-Bunyakovsky Inequality with
Applications to Elasticity

Dimitra Labropoulou
Department of Chemical Engineering
University of Patras
Patra 26504, Greece
<EMAIL_ADDRESS>
&Thanasis Labropoulos
Department of Mathematics
School of Applied Mathematics and Physical Sciences
National Technical University of Athens
Athens 15773, Greece
<EMAIL_ADDRESS>
&Panayiotis Vafeas
Department of Chemical Engineering
University of Patras
Patra 26504, Greece
<EMAIL_ADDRESS>
& Dimitris M. Manias
Department of Mathematics
Khalifa University of Science and Technology
Abu Dhabi 127788, United Arab Emirates
<EMAIL_ADDRESS>
Corresponding Author<EMAIL_ADDRESS>
###### Abstract

In this article we present both the discrete and the integral form of Cauchy-
Bunyakovsky-Schwarz ($\rm CBS$) inequality, some important generalizations in
the $n$-dimensional Euclidean space and in linear subspaces of it, as well as
the strengthened $\rm CBS$. The last $\rm CBS$ inequality plays an important
role in elasticity problems. A geometrical interpretation and a collection of
the most important proofs of it are, also, presented.

##  1 Introduction

A.-L. Cauchy Cauchy (1821) in his monograph published in 1821, presented an
inequality (for finite discrete sums), which subsequently played and continues
to play an important role in mathematics. In 1859, V.I. Bunyakovsky
Buni͡akovskiĭ (1859) extended this inequality to its integral version (that
is, for the case where we have continuous summation). Some years later, in
1888, H.A. Schwarz Schwarz (1890) established the general form of this
inequality, valid in vector spaces endowed with dot product, the so-called
"Cauchy-Bunyakovsky-Schwarz inequality" (abbreviated CBS inequality).

The CBS inequality is one of the most famous inequalities in mathematics, on
the one hand, because of its close relations with a multitude of other
important inequalities of mathematics and, on the other hand, because of its
utility in a particularly wide and varied range of applications in almost all
branches of classical and modern mathematics, such as Real and Complex
Analysis, Numerical Analysis, Hilbert Space Theory, Differential Equations,
Probability Theory, Statistics, Number Theory, etc. More details on this topic
we can see in the works of DragomirDragomir (2003) and SteeleSteele (2004).

Additionally, new applications of various branches of mathematics are
constantly appearing, where the CBS inequality itself or CBS-type inequalities
play a significant role. B${\rm\hat{i}}$rsan and B${\rm\hat{i}}$rsan Birsan
and Birsan (2011) presented a CBS-type inequality in both the discrete and the
integral forms. The integral version of this inequality appears in the study
of mechanical properties of thin elastic rods. Also, Achchab and MaitreAchchab
and Maitre (1996) presented the Strengthened Cauchy - Bunyakowski - Schwarz
inequality scheme, which plays a fundamental role in the convergence rate of
multilevel iterative methods. Its optimal constant plays an important role in
elasticity problems (see MargenovMargenov (1994), Achchab et al.Achchab et al.
(2001), Achchab and MaitreAchchab and Maitre (1996), DragomirDragomir (2003),
Jung and MaitreJung and Maitre (1999)). Besides, this handy theory can be
readily applied to more recent developments in anisotropic elasticity, wherein
even nowadays novel methods have been introduced in the work of Labropoulou et
al.Labropoulou et al. (2021).

In this paper we attempt a presentation of the basic versions of the CBS
inequality, an inequality of CBS type, the integral version of which appears
in the study of mechanical properties of thin elastic rods Bîrsan and
Altenbach (2011); Birsan and Birsan (2011), demonstrating that the
Strengthened CBS inequality is very useful in elasticity problems. Finally, in
the Appendix we present an anthology of clever and fairly short proofs of the
CBS inequality (see also the work of Wu and Wu Wu and Wu (2009)).

##  2 Cauchy-Bunyakovsky-Schwarz Inequality

###  2.1 Cauchy Inequality

Cauchy’s inequality for non-negative real numbers is expressed as follows.

######  Proposition 1.

If $\alpha_{1},\,\alpha_{2},\,...,\alpha_{n}$ are non-negative real numbers,
then

$\frac{{\alpha_{1}+\alpha_{2}+...+\alpha_{\nu}}}{\nu}\geqslant\sqrt[\nu]{{\alpha_{1}\cdot\alpha_{2}...\cdot\alpha_{\nu}}}~{}.$
(1)

If in addition the numbers $\alpha_{1},\,\alpha_{2},\,...,\alpha_{n}$ are
different from zero, then

$\frac{{\alpha_{1}+\alpha_{2}+...+\alpha_{\nu}}}{\nu}\geqslant\sqrt[\nu]{{\alpha_{1}\cdot\alpha_{2}...\cdot\alpha_{\nu}}}\geqslant\frac{\nu}{{\dfrac{1}{{\alpha_{1}}}+\dfrac{1}{{\alpha_{2}}}+...+\dfrac{1}{{\alpha_{\nu}}}}}~{}.$
(2)

Equality is true only when $\alpha_{1}=\alpha_{2}=...=\alpha_{\nu}$.

For a nice proof, we propose the book of Rassias Rassias (2017).

Historical Notation 1***All Historical Notations are found in the Wikipedia
biographies of Cauchy, Schwarz, Bunyakovsky, Holder and Minkowski. . _This
inequality first appeared in Augustin-Louis Cauchy’s book (1789-1857) entitled
"Coursd ’Analysis of the Polycole Polytechnique" (1821) for students at the
Ecole Polytechnique in Paris. The method of proving inequality is a "kind of
mathematical induction". A very beautiful proof of this inequality is
presented in the book (see [Mathematics, Th. Rassias p. 271]) and is due to
the great Hungarian mathematician George Polya (1887-1985). It is worth noting
that this inequality has been given a particularly large amount of evidence._

Historical Notation 2. _Baron Augustin-Louis Cauchy, French mathematician and
pioneer of Analysis, was born on August 21, 1789 and died on May 23, 1857. He
graduated as a civil engineer in 1807 at the age of 18 from the Polytechnic
School of Paris, where he was a professor. He began to formulate and prove the
theorems of infinite calculus in a rigorous way, rejecting any heuristic
principle of the generality of algebra used by older writers. He defined the
sequence in infinite terms, gave several important theorems in complex
analysis, and began the study of transpositional groups in abstract algebra.
Cauchy, being a thoughtful mathematician, greatly influenced modern
mathematicians as well as later ones._)

Before formulating the Cauchy-Schwarz-Bunyakovsky inequality we will recall
the concept of the inner product in $\mathbb{R}^{n},\,\,n\geqslant 1$, and its
basic properties. In Calculus we had to define the concept of absolute value
in order to study distance in real line. The same can happen in the vector
space $\mathbb{R}^{n}$. We know that the $n$ vectors
$e_{1}=\left({1,0,...,0}\right),\,e_{2}=\left({0,1,...,0}\right),\,...,e_{n}=\left({0,0,...,1}\right)$
of $\mathbb{R}^{n}$ consist a basis of it and in fact the normal basis (that
is, it is unit and perpendicular to two) and they generalize its usual
orthogonal basis
$B=\left\\{{i,\,j,\,k}\right\\}\,,\,\,\,\left({i=e_{1},\,j=e_{2},\,k=e_{3}}\right)$
of $\mathbb{R}^{3}$.
We know that for two possible vectors $x=\left({x_{1},\,x_{2},x_{3}}\right)$
and $y=\left({y_{1},\,y_{2},y_{3}}\right)$ of $\mathbb{R}^{3}$ their inner
product is defined as

$x\cdot y=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\in\mathbb{R}.$

We can now define in a similar way the inner product of two vectors of
$\mathbb{R}^{n}$ for any $n$. So if
$x=\left({x_{1},\,...,x_{n}}\right)\in\mathbb{R}^{n}$ and
$y=\left({y_{1},\,...,y_{n}}\right)\in\mathbb{R}^{n}$ extending the definition
we gave for the inner product of two vectors of $\mathbb{R}^{3}$ for any $n$
we can define the inner product of the vectors of $\mathbb{R}^{n}$ in a
similar way via

$x\cdot
y=x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}=\sum\limits_{i=1}^{n}{x_{i}y_{i}}~{}.$

The length or Euclidean norm of $x=\left({x_{1},\,...,x_{n}}\right)$ is
defined to be the non-negative real

$\left\|x\right\|=\sqrt{x\cdot x}=\sqrt{x_{1}^{2}+...+x_{n}^{2}}~{}.$

The pair $\left({\mathbb{R}^{n},\left\|{\,\cdot\,}\right\|}\right)$ is called
the Euclidean space of dimension $n$. We often use the notation $\langle
x,y\rangle$ instead $x\cdot y$ and the notation $|x|$ instead $\|x\|$.

The basic properties of the inner product are described in the following
proposition.

######  Proposition 2.

For all $x,y,z\,\in\mathbb{R}^{n}$, the following properties hold

$\begin{gathered}(\rm i)\,\,\langle x,y\rangle=\langle y,x\rangle\hfill\\\
(\rm{ii})\,\,\langle x+z,y\rangle=\langle x,y\rangle+\langle
z,y\rangle\hfill\\\ (\rm{iii})\,\,\langle x,x\rangle\geqslant 0\hfill\\\
(\rm{iv})\,\,\langle x,x\rangle=0\Leftrightarrow x=0\Leftrightarrow
x_{1}=x_{2}=...=x_{n}=0\hfill\\\ \end{gathered}$

The proof of properties is achieved by simply applying the definition of the
inner product.

###  2.2 Cauchy-Bunyakovsky-Schwarz Inequality

The $\rm CBS$ inequality is expressed as follows.

######  Theorem 1.

Let $x,\,y\in\mathbb{R}^{n}$. Then

$\left|{x\cdot y}\right|\leqslant\left\|x\right\|\cdot\left\|y\right\|~{}.$
(3)

The equality in $\rm CBS$ inequality holds if and only if the vectors $x$ and
$y$ are collinear.

_Proof._ If $x=0$ or $y=0$ then inequality applies trivially as equality.
Assume that $x\neq 0$ and $y\neq 0$.

$\rm(i)$ Let’s say that $\left\|x\right\|$=1= $\left\|y\right\|$. Then from
the obvious inequality
$\left({\left|{x_{i}}\right|-\left|{y_{i}}\right|}\right)^{2}\geqslant 0$ we
get that

$x_{i}^{2}+y_{i}^{2}-2\left|{x_{i}y_{i}}\right|\geqslant 0$

from which it follows that

$|x_{i}y_{i}|\leq\frac{x_{i}^{2}+y_{i}^{2}}{2},\,i=1,2,...,n.$

Therefore,

$\left|{\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right|\leqslant\sum\limits_{i=1}^{n}{\left|{x_{i}y_{i}}\right|}\leqslant\frac{1}{2}\sum\limits_{i=1}^{n}{\left({x_{i}^{2}+y_{i}^{2}}\right)}=\frac{1}{2}\sum\limits_{i=1}^{n}{x_{i}^{2}}+\frac{1}{2}\sum\limits_{i=1}^{n}{y_{i}^{2}}=\frac{1}{2}+\frac{1}{2}=1~{},$

that is

$\left|{x\cdot
y}\right|=\left|{\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right|\leqslant 1=1\cdot
1=\left|x\right|\left|y\right|.$

Thus, in this case inequality applies.

$\rm(ii)$ General case: Let
$x=\left({x_{1},\,...,x_{n}}\right)\in\mathbb{R}^{n}$ and
$y=\left({y_{1},\,...,y_{n}}\right)\in\mathbb{R}^{n}$ two possible vectors. We
set

$X_{i}=\frac{{x_{i}}}{{\left\|x\right\|}},\,\,Y_{i}=\frac{{y_{i}}}{{\left\|y\right\|}},i=1,2,..,n.$

It follows from the case $\rm(i)$ that

$\left|{\sum\limits_{i=1}^{n}{X_{i}Y_{i}}}\right|=\left|{\sum\limits_{i=1}^{n}{\left({\frac{{x_{i}}}{{\left\|x\right\|}}\cdot\frac{{y_{i}}}{{\left\|y\right\|}}}\right)}}\right|\leqslant\sum\limits_{i=1}^{n}{\frac{{\left|{x_{i}}\right|}}{{\left\|x\right\|}}\cdot\frac{{\left|{y_{i}}\right|}}{{\left\|y\right\|}}}\leqslant
1~{}.$

Because

$\sum\limits_{i=1}^{n}{X_{i}^{2}}=\sum\limits_{i=1}^{n}{\left({\frac{{x_{i}}}{{\left\|x\right\|}}}\right)^{2}=\sum\limits_{i=1}^{n}{\frac{{x_{i}^{2}}}{{\left\|x\right\|^{2}}}=}}\frac{1}{{\left\|x\right\|^{2}}}\sum\limits_{i=1}^{n}{x_{i}^{2}=}\frac{1}{{\left\|x\right\|^{2}}}\left\|x\right\|^{2}=1$

and similarly

$\sum\limits_{i=1}^{n}{Y_{i}^{{}^{\prime}2}}=1~{}.$

It follows that

$\left|{x\cdot
y}\right|=\left|{\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right|\leqslant 1\cdot
1=\left\|x\right\|\left\|y\right\|.$

If the vectors $x$ and $y$ are linear, that is, if we have for someone then we
easily find the equality in the $\rm CBS$ inequality.

We now assume that $\left|{x\cdot y}\right|=\left\|x\right\|\left\|y\right\|$.
If one of $x$ and $y$ is $0$, let $x$ be, then we set $\lambda=0$ and observe
that $x=\lambda y$. We now assume that $x=\left({x_{1},\,...,x_{n}}\right)\neq
0$ and $y=\left({y_{1},\,...,y_{n}}\right)\neq 0$. Then the equality
$\left|{x\cdot y}\right|=\left\|x\right\|\left\|y\right\|\,$ is written as

$\left|{\sum\limits_{i=1}^{n}{\frac{{\left|{x_{i}}\right|}}{{\left\|x\right\|}}\cdot\frac{{\left|{y_{i}}\right|}}{{\left\|y\right\|}}}}\right|=1$

and posing $X_{i}=\dfrac{{x_{i}}}{{\left\|x\right\|}}$ and
$Y_{i}=\dfrac{{y_{i}}}{{\left\|y\right\|}}$, yields

$\left|{\sum\limits_{i=1}^{n}{X_{i}\cdot Y_{i}}}\right|=1,$

where as we proved above
$\sum\limits_{i=1}^{n}{X_{i}^{2}}=1=\sum\limits_{i=1}^{n}{Y_{i}^{2}}$.
Therefore

$1=\left|{\sum\limits_{i=1}^{n}{X_{i}\cdot
Y_{i}}}\right|\leqslant\sum\limits_{i=1}^{n}{\left|{X_{i}\cdot
Y}\right|}\leqslant\frac{1}{2}\sum\limits_{i=1}^{n}{X_{i}^{2}}+\frac{1}{2}\sum\limits_{i=1}^{n}{Y_{i}^{2}}=1.$
(4)

We conclude that the numbers $X_{i},Y_{i},\,i=1,2,...,n$ are identical, since
the equation holds in the triangular inequality, that is

$\left|{\sum\limits_{i=1}^{n}{X_{i}\cdot
Y_{i}}}\right|=\sum\limits_{i=1}^{n}{\left|{X_{i}\cdot Y_{i}}\right|}~{}.$

By (4), also, it follows that necessary

$|X_{i}Y_{i}|=\frac{1}{2}X_{i}^{2}+\frac{1}{2}Y_{i}^{2},\,\,i=1,\,2,...,n.$
(5)

$\rm(a)$ If the common sign of the numbers $X_{i},Y_{i},\,i=1,2,...,n$ is
positive then from the equalities (5) we conclude that

$X_{i}Y_{i}=\frac{1}{2}\left({X_{i}^{2}+Y_{i}^{2}}\right)\Leftrightarrow\left({X_{i}-Y_{i}}\right)^{2}=0\Leftrightarrow
X_{i}=Y_{i},\,\,\,\forall\,\,i=1,\,2,...,n\,$

or

$\frac{{x_{i}}}{{\left\|x\right\|}}=\frac{{y_{i}}}{{\left\|y\right\|}}\Leftrightarrow
x_{i}=\frac{{\left\|x\right\|}}{{\left\|y\right\|}}\cdot y_{i}\Leftrightarrow
x=\frac{{\left\|x\right\|}}{{\left\|y\right\|}}\cdot y$

and

$\lambda=\frac{{\left\|x\right\|}}{{\left\|y\right\|}}.$

$\rm(b)$ If the common sign of the numbers $X_{i},Y_{i},\,i=1,2,...,n$ is
negative then from the equalities (5) we find that

$-X_{i}Y_{i}=\frac{1}{2}\left({X_{i}^{2}+Y_{i}^{2}}\right)\Leftrightarrow\left({X_{i}+Y_{i}}\right)^{2}=0\Leftrightarrow
X_{i}=-Y_{i},\,\,\,\forall\,\,i=1,\,2,...,n\,$

and

$\lambda=-\frac{{\left\|x\right\|}}{{\left\|y\right\|}}.$

$\Box$

Historical Notation 3. _Karl Hermann Amandus Schwarz, a German mathematician
known for his work on complex analysis, was born on January 25, 1843 in
Hermsnstorf, Silesia (southern Poland) and died on November 30, 1921 in
Berlin. Schwarz initially studied chemistry in Berlin, but Kummer and Karl
Weierstrass encouraged him to switch to mathematics. He received his Ph.D.
from the University of Berlin in 1864 under the supervision of Kummer and
Weierstrass. Between 1867 and 1869, he worked at the University of Halle and
then at the Swiss Federal Polytechnic. From 1875 he worked at the University
of Gottingen and was involved in complex analysis, differential geometry and
calculus of change._

Historical Notation 4. _Viktor Yakovlevich Bunyakovsky, a Russian
mathematician known for his work in theoretical engineering and number theory,
was born on December 16, 1804 in Bar, Russia, and died on November 12, 1889 in
St. Petersburg. He was a member and later vice-president of the Russian
Academy of Sciences. He graduated from the Sorbonne University in 1824 and
received his doctorate under Cauchy. He proved the $\rm CBS$ inequality for
the infinite case in 1859, long before Schwarz dealt with it._

###  2.3 Geometrical Interpretation of the $\rm CBS$ Inequality

From the $\rm CBS$ inequality it follows that for non-zero vectors $x$ and $y$
in $\mathbb{R}^{n}$ the quotient

$\frac{{x\cdot y}}{{\left\|x\right\|\left\|y\right\|}}$

belongs to the interval $[-1,1]$, i.e.

$-1\leqslant\frac{{x\cdot y}}{{\left\|x\right\|\left\|y\right\|}}\leqslant
1~{}.$

This defines the angle $\theta$ between $x$ and $y$ by the formula

$\cos\theta=\frac{{x\cdot
y}}{{\left\|x\right\|\cdot\left\|y\right\|}},\,\theta\in\left[{0,\pi}\right].$
(6)

Moreover, if the vectors $x$ and $y$ belong to $\mathbb{R}^{3}$ then $\rm CBS$
inequality follows from the law of the cosine of trigonometry.

Applying the law of cosines to the triangle with vertice $O(0,0,0)$ and sides
the vectors $x$ and $y$ it will result that:

$\left\|{x-y}\right\|^{2}=\left\|x\right\|^{2}+\left\|y\right\|^{2}-2\left\|x\right\|\left\|y\right\|\cos\theta,$
(7)

where $\theta$ the angle of $x$ and $y$. From (7) using the properties of the
inner product we get successively

$\displaystyle\left\|{x-y}\right\|^{2}=\left\|x\right\|^{2}+\left\|y\right\|^{2}-2\left\|x\right\|\cdot\left\|y\right\|\cdot\cos\theta\,\hfill$
$\displaystyle\Leftrightarrow{}$
$\displaystyle\left({x-y}\right)\cdot\left({x-y}\right)=\left\|x\right\|^{2}+\left\|y\right\|^{2}-2\left\|x\right\|\left\|y\right\|\cos\theta\,\hfill$
$\displaystyle\Leftrightarrow{}$ $\displaystyle x^{2}-2x\cdot
y+y^{2}=x^{2}+y^{2}-2\left\|x\right\|\left\|y\right\|\cos\theta\,\hfill$
$\displaystyle\Leftrightarrow{}$ $\displaystyle-2x\cdot
y=-2\left\|x\right\|\left\|y\right\|\cos\theta\,\hfill$
$\displaystyle\Leftrightarrow{}$ $\displaystyle x\cdot
y=\left\|x\right\|\left\|y\right\|\cos\theta\,.\hfill$

Therefore,

$\left|{\cos\theta}\right|=\frac{{\left|{x\cdot
y}\right|}}{{\left\|x\right\|\left\|y\right\|}}\leqslant
1\Leftrightarrow\left|{x\cdot
y}\right|\leqslant\left\|x\right\|\left\|y\right\|.$

If the vectors $x$ and $y$ are collinear, that is $x=\lambda y$, for some
$\lambda\in\mathbb{R}$, then $\theta=0$ or $\theta=\pi$, therefore
$\cos\theta=1$ and so $\left|{x\cdot
y}\right|=\left\|x\right\|\left\|y\right\|$. If the vectors $x$ and $y$ are
not collinear, that is, linearly independent, then
$\theta\in\left({0,\pi}\right)$ therefore $\left|{\cos\theta}\right|<1$ and so
$\left|{x\cdot y}\right|<\left\|x\right\|\left\|y\right\|$. So, we come to the
part where we talk about the inner product of non-zeros vectors

$x\cdot
y=\left\|x\right\|\left\|y\right\|\cos\theta,\,\,\,x,y\in\mathbb{R}^{n}\,,$

where $\theta\in\left[{0,\pi}\right]$ is the unique angle between $x$ and $y$.
Two vectors $x,y\in\mathbb{R}^{n}\backslash\\{0\\}$ are called _orthogonal_ if
$x\cdot y=0\Leftrightarrow\theta=\dfrac{\pi}{2}$.

###  2.4 Triangular Inequality

We can prove our well-known triangular inequality in $\mathbb{R}^{n}$ on the
basis of the $\rm CBS$ inequality. That is, we will prove that if
$x,y\in\mathbb{R}^{n}$, then we have

$\left\|{x+y}\right\|\leqslant\left\|x\right\|+\left\|y\right\|.$ (8)

Indeed using the $\rm CBS$ inequality (3) we have

$\displaystyle\left\|{x+y}\right\|^{2}$ $\displaystyle=$
$\displaystyle\left({x+y}\right)\cdot\left({x+y}\right)\hfill$
$\displaystyle=$ $\displaystyle x\cdot x+x\cdot y+y\cdot x+y\cdot y\hfill$
$\displaystyle=$ $\displaystyle\left\|x\right\|^{2}+2x\cdot
y+\left\|y\right\|^{2}\hfill$ $\displaystyle\leqslant$
$\displaystyle\left\|x\right\|^{2}+2\left\|x\right\|\left\|y\right\|+\left\|y\right\|^{2}\hfill$
$\displaystyle=$
$\displaystyle\left({\left\|x\right\|+\left\|y\right\|}\right)^{2}.$

Taking square roots in each member results in the required inequality.

Based on the $\rm CBS$ inequality we can get the following very useful
proposition.

######  Proposition 3.

If $(x_{1},,...,x_{n})$ and $(y_{1},...,y_{n})$ are two sequences of real
numbers, then the following inequality holds

$\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)\left({\sum\limits_{i=1}^{n}{y_{i}^{2}}}\right)\geqslant\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)^{2},$
(9)

with the equality to be valid if and only if the sequences
$(x_{1},,...,x_{n})$ and $(y_{1},...,y_{n})$ are proportional term by term,
that is, if there exists a fixed number $\lambda$ such that $x_{k}=\lambda
y_{\kappa}\,\forall\,k\in\left\\{{1,2,...,n,...}\right\\}$.

$\rm CBS$ inequality is known to be one of the most basic and useful
inequalities in algebra, and because of its particular weight it has greatly
preoccupied mathematicians. This has resulted in a very large number of proofs
of great interest. Then we will present some of the ones we have chosen based
on the diversity, the brevity and the beauty that they present and of course
these criteria are subjective.

In the Appendix we present some of the most beautiful and useful proofs of the
$\rm CBS$ Inequality.

$\rm CBS$ inequality, as mentioned and above, without any doubt, is one of the
most widely used and important inequalities in all mathematics. If, for
example, we write it in the form

$x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}\leqslant\sqrt{x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}}\sqrt{y_{1}^{2}+y_{2}^{2}+...+y_{n}^{2}},$

one of the most important and immediate conclusions that emerges from this is
that from $\sum\limits_{i=1}^{\infty}{x_{i}^{2}}<\infty$ and
$\sum\limits_{i=1}^{\infty}{y_{i}^{2}}<\infty$ implies that
$\sum\limits_{i=1}^{\infty}{\left|{x_{i}y_{i}}\right|}<\infty$.

We can generalize the above CBS inequality in arbitrary linear spaces. Let for
example $V$ be an arbitrary real vector space. If we can define a function in
$V\times V$ which is defined by the map $(x,y)\mapsto\langle x,y\rangle$ in a
such way that the $\langle x,y\rangle$ satisfies the properties in Proposition
2 then the $\langle x,y\rangle$ is an inner product and
$({V,\langle{\cdot,\cdot}\rangle})$ is a real space with inner product.

If we go one step further and think that the integration of a function in a
space is nothing more than a sum in the continuous case, the question arises
whether the $\rm CBS$ inequality holds in this continuous case. That is, if we
replace the sums with integrals arises the following inequality

$\int_{a}^{b}{f\left(x\right)g\left(x\right)dx}\leqslant\left({\int_{a}^{b}{f^{2}\left(x\right)dx}}\right)^{\frac{1}{2}}\left({\int_{a}^{b}{g^{2}\left(x\right)dx}}\right)^{\frac{1}{2}}.$

The question therefore is whether such inequality applies. The answer to this
question is affirmative and in fact we can get a more general inequality the
so-called Hölder inequality. In other words, the following proposition is
valid.

######  Proposition 4.

(_Hölder’s inequality_) Let $p,q$ be two real numbers such that $1<p<+\infty$
and $\dfrac{1}{p}+\dfrac{1}{q}=1$. Let, also, the integrable functions
$f,g\,:\left[{\alpha,\,\beta}\right]\to\left[{0,\,+\infty}\right]$. Then the
following inequality holds

$\int_{a}^{b}{f\left(x\right)g\left(x\right)dx}\leqslant\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}\left({\int_{a}^{b}{g^{q}\left(x\right)dx}}\right)^{\frac{1}{q}},$

A special case of the latter is taken for $p=q=\dfrac{1}{2}$. The positive
exponents $p$ and $q$ are called _conjugate_ exponents if
$\dfrac{1}{p}+\dfrac{1}{q}=1$ or equivalent $p+q=pq$.

Historical Notation 5. _We mention that this inequality first appeared in
print in Victor Yacovlevich Bunyakovsky’s Memoire which was published by the
Imperial Academy of Sciences of St. Petersburg in 1859._

In aim to prove Hölder’s inequality we will assume that

$0<\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}<+\infty,\,\,\,0<\left({\int_{a}^{b}{g^{q}\left(x\right)dx}}\right)^{\frac{1}{q}}<+\infty,$

and that

$\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}\neq
0,\,\,\,\left({\int_{a}^{b}{g^{q}\left(x\right)dx}}\right)^{\frac{1}{q}}\neq
0,$

that is, that the functions $f^{p}$ and $g^{q}$ are integrable with non-zero
integrals because these cases are trivial.

For the proof we will need the following proposition.

######  Proposition 5.

Let $p,q$ be two real numbers such that $1<p<+\infty$ and
$\dfrac{1}{p}+\dfrac{1}{q}=1$. Then, for all $x,y\geq 0$, the following
inequality holds

$xy\leqslant\frac{{x^{p}}}{p}+\frac{{y^{q}}}{q}.$

_Proof._ We observe that if $x=0$ or $y=0$ the inequality holds, then we
assume that $x>0$ and $y>0$. Dividing inequality by members with $y^{q}$ we
obtain the inequality

$\frac{x}{{y^{q-1}}}\leqslant\frac{{x^{p}}}{{py^{q}}}+\frac{1}{q}$

and if we put in the latter $t=\dfrac{{x^{p}}}{{y^{q}}}$, because of
$\dfrac{1}{p}+\dfrac{1}{q}=1$, we have $(q-1)p=q$ and so it turns out that

$\frac{x}{{y^{q-1}}}=\left({\frac{{x^{p}}}{{y^{\left({q-1}\right)p}}}}\right)^{\frac{1}{p}}=\left({\frac{{x^{p}}}{{y^{pq-p}}}}\right)^{\frac{1}{p}}=\left({\frac{{x^{p}}}{{y^{q}}}}\right)^{\frac{1}{p}}=t^{\frac{1}{p}}~{},$

so we get the inequality

$t^{\frac{1}{p}}\leqslant\frac{1}{p}t+\frac{1}{q},\,\,t>0.$

The function
$f\left(t\right)=t^{\frac{1}{p}}-\frac{1}{p}t-\frac{1}{q},\,\,t>0$ is
differentiable with
$f^{\prime}\left(t\right)=\frac{1}{p}\left({t^{-\frac{1}{q}}-1}\right)$.
We observe that $f^{\prime}(1)=0$,
$f^{\prime}\left(t\right)<0,\,\,\,\forall\,\,t>1$ and
$f^{\prime}\left(t\right)>0,\,\,\,\forall\,\,t\in(0,1)$ so
$f\left(t\right)\leqslant f(1)\,=0\,\,\forall\,\,t>0$, and then

$t^{\frac{1}{p}}-\frac{1}{p}t-\frac{1}{q}\leqslant 0,\,\,\forall\,\,t>0,$

where the equality holds only when $t=1$.  $\Box$

_Proof of Hölder’s inequality_. We set
$A=\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}$ and
$B=\left({\int_{a}^{b}{g^{q}\left(x\right)dx}}\right)^{\frac{1}{q}}$. So from
the above proposition for the numbers $f(x)/A$ and $\ g(x)/B$ we get

$\frac{{f\left(x\right)}}{A}\frac{{g\left(x\right)}}{B}\leqslant\frac{(f(x)/A)^{p}}{p}+\frac{(g(x)/B)^{q}}{q}=\frac{{\left({f\left(x\right)}\right)^{p}}}{{pA^{p}}}+\frac{{\left({g\left(x\right)}\right)^{q}}}{{qB^{q}}}~{}.$

By integration by parts from the last we obtain

$\begin{gathered}\,\frac{1}{{AB}}\int\limits_{\alpha}^{\beta}{f\left(x\right)}g\left(x\right)dx\leqslant\frac{1}{{pA^{p}}}\int\limits_{\alpha}^{\beta}{\left({f\left(x\right)}\right)^{p}dx}+\frac{1}{{qB^{q}}}\int\limits_{\alpha}^{\beta}{\left({g\left(x\right)}\right)^{q}dx}\hfill\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{1}{{pA^{p}}}A^{p}+\frac{1}{{qB^{q}}}B^{q}=\frac{1}{p}+\frac{1}{q}=1,\hfill\\\
\end{gathered}$

so

$\int\limits_{\alpha}^{\beta}{f\left(x\right)}g\left(x\right)dx\leqslant
AB\Leftrightarrow\,\int\limits_{\alpha}^{\beta}{f\left(x\right)}g\left(x\right)dx\leqslant\left({\int\limits_{\alpha}^{\beta}{f^{p}\left(x\right)}}\right)^{\frac{1}{p}}\left({\int\limits_{\alpha}^{\beta}{g^{q}\left(x\right)}}\right)^{\frac{1}{q}}.$

If we use the classic notation for the norm in $L^{p}$ spaces, that is, if we
set

$\left\|f\right\|_{p}=\left({\int\limits_{\alpha}^{\beta}{f^{p}\left(x\right)}}\right)^{\frac{1}{p}},\,\,\,1\leqslant
p<+\infty,$

the last inequality takes the form

$\left\|{fg}\right\|_{1}=\left\|f\right\|_{p}\left\|g\right\|_{q}.$

$\Box$

Historical Notation 6. _Ludwig Otto Holder, German mathematician, was born on
December 22, 1859 in Stuttgart and died on August 29, 1937. He studied at the
present University of Stuttgart where he was a student of Kronecker,
Weirstrass and Kummer. He received his doctorate from the University of
Tubingen in 1882. He is known for many theorems such as the Holder Inequality,
the Jordan-Holder theorem and the abnormal external automorphism of the
symmetric group._

Completing this generalization in the continuous case, that is, in spaces, we
can get the generalization of the triangular inequality called Minkowski
inequality.

######  Proposition 6.

Let $1<p<+\infty$ and the integrable functions
$f,\,g\,:\,\left[{\alpha,\,\beta}\right]\to\left[{0,\,+\infty}\right]$. Then,

$\left({\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}\leqslant\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}+\left({\int_{a}^{b}{g^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}$

_Proof._ We notice that

$\displaystyle\left[{f\left(x\right)+g\left(x\right)}\right]^{p}$
$\displaystyle\leqslant$
$\displaystyle\left[{2\max\left\\{{f\left(x\right),g\left(x\right)}\right\\}}\right]^{p}\hfill$
$\displaystyle=$ $\displaystyle
2^{p}\max\left\\{{f^{p}\left(x\right),g^{p}\left(x\right)}\right\\}\hfill$
$\displaystyle\leqslant$ $\displaystyle
2^{p}\left[{f^{p}\left(x\right)+g^{p}\left(x\right)}\right]$

or

$\left({f\left(x\right)+g\left(x\right)}\right)^{p}\leqslant
2^{p}f^{p}\left(x\right)+2^{p}g^{p}\left(x\right),$

so with integration by parts we get this inequality

$\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}\leqslant
2^{p}\int_{a}^{b}{f^{p}\left(x\right)dx}+2^{p}\int_{a}^{b}{g^{p}\left(x\right)dx}.$

We can assume that $\int_{a}^{b}{f^{p}\left(x\right)dx}<+\infty$,
$\int_{a}^{b}{g^{p}\left(x\right)dx}\leqslant+\infty$, and then due to the
last inequality arises
$\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}<+\infty$ and so in this
case the inequality we want to prove is obvious.

Let now be the conjugate exponent $q$ of $p$. Then, $\left({p-1}\right)q=p$,
so from Hölder’s inequality we get

$\int_{a}^{b}{f\left({f+g}\right)^{p-1}\left(x\right)dx}\leqslant\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}\left({\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}}\right)^{\frac{1}{q}}$

and

$\int_{a}^{b}{g\left({f+g}\right)^{p-1}\left(x\right)dx}\leqslant\left({\int_{a}^{b}{g^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}\left({\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}}\right)^{\frac{1}{q}}.$

Adding by members the last two inequalities, we reach

$\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}\leqslant\left[{\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}+\left({\int_{a}^{b}{g^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}}\right]\left({\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}}\right)^{\frac{1}{q}}$

and from the latter, by division by members, we obtain the inequality

$\left({\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}}\right)^{1-\frac{1}{q}}\leqslant\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}+\left({\int_{a}^{b}{g^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}$

and because $1-\dfrac{1}{q}=\dfrac{1}{p}$ we finally arrive to

$\left({\int_{a}^{b}{\left({f+g}\right)^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}\leqslant\left({\int_{a}^{b}{f^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}+\left({\int_{a}^{b}{g^{p}\left(x\right)dx}}\right)^{\frac{1}{p}}.$

The last inequality, again using the classic notation of $L^{p}$-norm, can be
written in the form

$\left\|{f+g}\right\|_{p}\leqslant\left\|f\right\|_{p}+\left\|g\right\|_{p},$

which is the generalization of triangular inequality in spaces.  $\Box$

Historical Notation 7. _Herman Minkowski, German mathematician, was born on 22
June 1864 in Lithuania and died on 12 January 1909 in Gottingen. He received
his PhD from the University of Konigsbgerg at the age of 21\. He taught at the
Federal Institute of Technology (ETH) in Zurich and in the last years of his
life at the University of Gottingen, where he had the mathematics course
created for him by David Hilbert. But Schwarz needed a two-dimensional form of
Cauchy inequality. In particular, he had to show that if
$S\subset\mathbb{R}^{2}$ and $f,g:S\to\mathbb{R}$, then the double integrals
$A=\iint_{S}{f^{2}dxdy}$, $B=\iint_{S}{fgdxdy}$, and $A=\iint_{S}{g^{2}dxdy}$
must satisfy the inequality $\left|B\right|\leqslant\sqrt{A}\cdot\sqrt{C}$ and
he had to know that the inequality is strict unless the functions are
proportional. Schwarz would face a problem for a number of reasons, including
the fact that the severity of the inequality can be lost using integrals. So
he had to find a different solution, which he did as he discovered a proof
that went down in history. He based his proof on an impressive observation. He
observed that the real polynomial
$p\left(t\right)=\iint_{S}{\left({t\,f\left({x,y}\right)+g\left({x,y}\right)}\right)}^{2}dxdy=At^{2}+2Bt+C$
is always non-negative and that $p(t)$ it is strictly positive unless $f$ and
and $g$ are proportional. The binomial formula tells us that the coefficients
must satisfy $B^{2}\leqslant AC$ and unless $f$ and $g$ are proportional then
we have strict inequality $B^{2}<AC$. So from a simple algebraic observation
Schwarz discovered everything he needed to know._

##  3 A Cauchy-Bunyakovsky-Schwarz Type Inequality

In this section we present a Cauchy-Schwarz type inequality, which arose from
the study of mechanical properties of elastic rod (see in Birsan and Birsan
(2011)). Firstly, we present the discreet case and secondly its integral
version.

######  Theorem 2.

Let $x=(x_{1},...,x_{n})$ and $y=(y_{1},...,y_{n})$ be arbitrary elements of
$\mathbb{R}^{n}$. Then the following holds

$\displaystyle\left({\sum\limits_{i=1}^{n}{p_{i}}}\right)\left({\sum\limits_{i=1}^{n}{p_{i}x_{i}}y_{i}}\right)-\left({\sum\limits_{i=1}^{n}{p_{i}x_{i}}}\right)\left({\sum\limits_{i=1}^{n}{p_{i}}y_{i}}\right)$
$\displaystyle\leqslant\sqrt{\left({\sum\limits_{i=1}^{n}{p_{i}}}\right)\left({\sum\limits_{i=1}^{n}{p_{i}x_{i}^{2}}}\right)-\left({\sum\limits_{i=1}^{n}{p_{i}x_{i}}}\right)^{2}}$
(10)
$\displaystyle\times\sqrt{\left({\sum\limits_{i=1}^{n}{p_{i}}}\right)\left({\sum\limits_{i=1}^{n}{p_{i}y_{i}^{2}}}\right)-\left({\sum\limits_{i=1}^{n}{p_{i}y_{i}}}\right)^{2}},$

for any positive real numbers $p_{1},p_{2},...,p_{n}$ (weights). The above
relation (2) becomes an equality if and only if there exist $a,b\in\mathbb{R}$
such that $ax+by=c$, a constant vector, i.e. if there exists a linear
combination of the vectors $x$ and $y$ which is a constant vector.

_Sketch of the proof_. We present the basic steps of the proof. (A proof in
detail and some important remarks are available in Birsan and Birsan (2011)).

We define the function of two variables
$\langle.,.\rangle:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ defined by

$\displaystyle\left\langle{x,y}\right\rangle=\left({\sum\limits_{i=1}^{n}{p_{i}}}\right)\left({\sum\limits_{i=1}^{n}{p_{i}x_{i}}y_{i}}\right)-\left({\sum\limits_{i=1}^{n}{p_{i}x_{i}}}\right)\left({\sum\limits_{i=1}^{n}{p_{i}}y_{i}}\right)~{}.$
(11)

It is easy to prove that the function
$\langle.,.\rangle:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ is a
bilinear form, i.e. it satisfies the properties of Proposition 2. These
properties can be easy verified by taking into account the definition (11).
Concerning the first property it can be proved using the CBS inequality (3),
that is

$\displaystyle\left\langle{x,x}\right\rangle$ $\displaystyle=$
$\displaystyle\left({\sum\limits_{i=1}^{n}{p_{i}}}\right)\left({\sum\limits_{i=1}^{n}{p_{i}x_{i}^{2}}_{i}}\right)-\left({\sum\limits_{i=1}^{n}{p_{i}x_{i}}}\right)\left({\sum\limits_{i=1}^{n}{p_{i}}x_{i}}\right)\hfill$
$\displaystyle=$
$\displaystyle\left[{\sum\limits_{i=1}^{n}{\left({\sqrt{p_{i}}}\right)^{2}}}\right]\left[{\sum\limits_{i=1}^{n}{\left({\sqrt{p_{i}}x_{i}}\right)^{2}}}\right]-\left({\sum\limits_{i=1}^{n}{\sqrt{p_{i}}\sqrt{p_{i}x_{i}}}}\right)^{2}\,\geqslant
0.$

The inequality becomes equality, i.e. $\left\langle{x,x}\right\rangle=0$, if
and only if the vectors $(\sqrt{p_{1}},\sqrt{p_{2}},...,\sqrt{p_{n}})$ and
$(\sqrt{p_{1}}x_{1},\sqrt{p_{2}}x_{2},...,\sqrt{p_{n}}x_{n})$ are linear
dependent, that is, $x=(x_{1},x_{2},...,x_{n})$ is a constant vector. With
these considerations inequality (11) takes the form

$\left\langle{x,y}\right\rangle\leqslant\sqrt{\left\langle{x,x}\right\rangle}\sqrt{\left\langle{y,y}\right\rangle},\,\,\,\forall\,x,y\in\mathbb{R}^{n}~{}.$

which is inequality (2).

The integral version of the CBS type inequality is presented by the following
proposition. (We will relax the notation $\int_{\Omega}f(x)dx$ and simply
write $\int_{\Omega}f$).

######  Proposition 7.

Let the functions $f,g,p:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}$ be
continuous on a compact domain $\Omega$, and let $p$ be positive. The
following inequality holds

$\displaystyle\int\limits_{\Omega}p\int\limits_{\Omega}{pfg}-\int\limits_{\Omega}pf\int\limits_{\Omega}pg\leqslant\sqrt{\int\limits_{\Omega}p\int\limits_{\Omega}pf^{2}-\left({\int\limits_{\Omega}pf}\right)^{2}}\sqrt{\int\limits_{\Omega}p\int\limits_{\Omega}pg^{2}-\left({\int\limits_{\Omega}pg}\right)^{2}},$
(12)

where the inequality becomes equality if and only if there exist
$\alpha,\beta\in\mathbb{R}$ such that $\alpha f+\beta g=c$, a constant.

Notation. We can imagine that inequality (12) is obtained directly from (2) by
replacing the discrete “sums" by continuous ones, i.e. by integrals, and what
remains to be shown is that the steps in the proof of the continuous case are
exactly the same as those of the discrete one.
_Sketch of the proof_. We set

$\displaystyle\left\langle{f,g}\right\rangle=\int\limits_{\Omega}p\int\limits_{\Omega}pfg-\int\limits_{\Omega}pf-\int\limits_{\Omega}pg$
(13)

It easy to verify that the properties $\rm(i)-(v)$ in Proposition 2 are
satisfied and inequality (12) can be written

$\displaystyle\left\langle{f,g}\right\rangle\leqslant\sqrt{\left\langle{f,f}\right\rangle}\sqrt{\left\langle{g,g}\right\rangle},\,\,\,\forall\,f,g\in
C(\Omega),$ (14)

which leads to the proof of the result.

##  4 Strengthened Cauchy-Bunyakovsky-Schwarz Inequality for Elasticity

In this section we present the strengthened CBS inequality,

$|\langle u,v\rangle|\leq\gamma\sqrt{\langle u,u\rangle}\sqrt{\langle
v,v\rangle},\,\,u\in U,\,\,v\in V,\,\,U\cap V=\\{0\\},$ (15)

where $U$ and $V$ are two linear subspaces $\langle.,.\rangle$ is the bilinear
form corresponding to the variational formulation of the problem and $\gamma$
is a constant between 0 and 1 depending only on the spaces $U$ and $V$. (In
fact the constant $\gamma$ could take the values 0 and 1, but if $\gamma=0$
then the vectors $u$ and $v$ are orthogonal and for $\gamma=1$ arises the
classical CBS inequality). The estimate of the constant that appears in the
second term of this inequality and in particular its optimal value (that is,
the smallest value it can taken to hold for all $u\in U$ and $v\in V$) is of
fundamental practical importance in solving the problems that appear and
especially in solving 2D and 3D elasticity problems, which are the ones that
interest us. For example, Margenov in Margenov (1994) gave estimates on the 2D
elasticity problem on a triangular mesh and proved for a uniform mesh of right
isosceles triangles, that the constant $\gamma^{2}$ is bounded above by $3/4$
uniformly on the mesh. Achchab and Maitre in Achchab and Maitre (1996) proved
that this result remains true for every triangular mesh. In the case of a 3D
problem, Jung and Maire in Jung and Maitre (1999) reported that numerical
experiments on a particular triangulation and specifically in the refinement
of a standard tetrahedron showed that the constant $\gamma^{2}$ is bounded by
$9/10$. Achchab et al. in Achchab et al. (2001) generalized this result and
proved that it remains valid for an arbitrary tetrahedron mesh. In addition,
Achchab et al. proves that the constant $\gamma$ plays significant role in
posteriori error estimates for elasticity problems.

In this section we present the basic theorem needed for the establishment of
the strengthened CBS inequality, which is needed for the estimation of the
constant $\gamma$. (For more details and for the proofs see also the work of
Achchab et al.Achchab et al. (2001))

######  Theorem 3.

Given a finite-dimensional Hilbert space $H$, an inner product
$\langle,.,\rangle$ on it and the subspaces $U$ and $V$ of $H$ with $U\cap
V=\\{0\\}$, there exist a constant $\gamma\in(0,1)$ depending only on $U$ and
$V$ such that for all $u\in U$ and $v\in V$ the following strengthened CBS
inequality holds

$|\langle u,v\rangle|\leq\gamma\|u\|\|v\|,$ (16)

where the norm induced by the inner product.

######  Theorem 4.

Let $A$ be a symmetric, positive semidefine $2\times 2$ block matrix

$A=\left({\begin{array}[]{*{20}c}{A_{11}}&{A_{12}}\\\
{A_{21}}&{{\text{A}}_{{\text{22}}}}\\\ \end{array}}\right),$

where $A_{11}$ is invertible and where the partitioning components to $U$ ,
$V$, which are the spaces of vectors with only non-zero first, and second
components $u\in U$, $v\in V$ of the form
$u=\left({\begin{array}[]{*{20}c}{u_{1}}\\\ 0\\\ \end{array}}\right)$ and
$v=\left({\begin{array}[]{*{20}c}0\\\ {v_{2}}\\\ \end{array}}\right),$
respectively. Assuming that the kernel of $A$, N(A), is included in $V$, the
optimal value of the constant $\gamma$ is given by

$\gamma^{2}=\mathop{\mathop{\sup}\limits_{u\in U}}\limits_{v\in V\backslash
N\left(A\right)}\frac{{\left({u^{T}Av}\right)^{2}}}{{\left({u^{T}Au}\right)\left({v^{T}Av}\right)}}~{}.$

Let $\Omega$ be a bounded open connected subset of $\mathbb{R}^{3}$ with a
Lipschitz continuous boundary $\Gamma$. Let, also, $\Gamma_{0}$ a measurable
subset of $\Gamma$ and $\Gamma_{1}$ its complement on $\Gamma$. We consider
the following 3-dimensional boundary value problem of linear elasticity

$\begin{gathered}\sum\limits_{j=1}^{3}{\frac{{\partial\sigma_{ij}\left(u\right)}}{{\partial
x_{j}}}}+f_{i}\,\,\,\,\,\mathrm{in}\,\,\,\Omega,\,\,\,i=1,2,3\hfill\\\
u_{i}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{on}\,\,\Gamma_{0},\,\,i=1,2,3\hfill\\\
\sum\limits_{j=1}^{3}{\sigma_{ij}\left(u\right)v_{j}}=g_{i}\,\,\,\,\mathrm{on}\,\,\Gamma_{1},\,\,i=1,2,3.\hfill\\\
\end{gathered}$

$v=(v_{1},v_{2},v_{3})$ is the external normal to $\Omega$, with the classical
constitutive law for isotropic material with Láme modula $\lambda$ and $\mu$

$c_{ij}\left(u\right)=\lambda\left({\sum\limits_{k=1}^{3}{\varepsilon_{kk}}\left(u\right)}\right)\delta_{ij}+2\mu\varepsilon_{ij}\left(u\right),$

where

$\varepsilon_{ij}\left(u\right)=\frac{1}{2}\left({\frac{{\partial
u_{i}}}{{\partial x_{j}}}+\frac{{\partial u_{j}}}{{\partial x_{i}}}}\right).$

We introduce the space

$V\left(\Omega\right)=\left\\{{v\in\left[{H^{1}\left(\Omega\right)}\right]^{3};\,\,v=0\,\,\mathrm{on}\,\,\Gamma_{0}}\right\\}$

For $f\in\left[{L^{2}\left(\Omega\right)}\right]^{3}$ and
$\,g\in\left[{L^{2}\left({\Gamma_{0}}\right)}\right]^{3}$ the variational
formulation of the problem is given by

$\mathrm{Find}\,\,u\in
V(\Omega)\,\,\mathrm{such}\,\mathrm{that}\,\,\,a(u,v)=L(v),\,\,\forall\,v\in\Omega,$

where

$a\left({u,v}\right)=\lambda\int_{\Omega}{div\left(u\right)}div\left(v\right)dx+2\mu\sum\limits_{i,j=1}^{3}{\varepsilon_{ij}\left(u\right)\varepsilon_{ij}\left(v\right)}dx$

and

$L\left(v\right)=\int_{\Omega}{fv}dx+\int_{\Gamma_{1}}{gv}d\sigma.$

Finally, we define the two forms $a_{1}(,.,)$ and $a_{2}(,.,)$ by

$a_{1}(u,v)=\int_{\Omega}\rm{div}(u)\rm{div}(v)dx$

and

$\;\;\qquad
a_{2}\left({u,v}\right)=\sum\limits_{i,j=1}^{3}{\int_{\Omega}{c_{ij}}}\left(u\right)c_{ij}\left(v\right)dx.$

Under the above considerations the following theorem holds.

######  Theorem 5.

The constants $\gamma_{1}$, $\gamma_{2}$ of the strengthened CBS inequality
associated the forms $a_{1}(u,v)$ and $a_{2}\left({u,v}\right)$ satisfies

$\gamma_{i}^{2}\leq\frac{9}{10},\,\,i=1,2.$

## Appendix

_Proof 1._ Consider the double sum
$\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}}(x_{i}y_{j}-x_{j}y_{i})^{2}.$ By
deleting the parentheses and grouping the same terms we get successively

$\displaystyle\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{\left({x_{i}y_{j}-x_{j}y_{i}}\right)}^{2}}$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{\left({x_{i}^{2}y_{j}^{2}+x_{j}^{2}y_{i}^{2}-2x_{i}x_{j}y_{i}y_{j}}\right)}}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{\left({\sum\limits_{j=1}^{n}{x_{i}^{2}y_{j}^{2}}+\sum\limits_{j=1}^{n}{x_{j}^{2}y_{i}^{2}}-\sum\limits_{j=1}^{n}{2x_{i}x_{j}y_{i}y_{j}}}\right)}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{\left({x_{i}^{2}\sum\limits_{j=1}^{n}{y_{j}^{2}}+y_{i}^{2}\sum\limits_{j=1}^{n}{x_{j}^{2}}-2x_{i}y_{i}\sum\limits_{j=1}^{n}{x_{j}y_{j}}}\right)}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{\left({x_{i}^{2}\sum\limits_{j=1}^{n}{y_{j}^{2}}}\right)}+\sum\limits_{i=1}^{n}{\left({y_{i}^{2}\sum\limits_{j=1}^{n}{x_{j}^{2}}}\right)\,-2\sum\limits_{i=1}^{n}{\left({x_{i}y_{i}\sum\limits_{j=1}^{n}{x_{j}y_{j}}}\right)}}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{n}{y_{j}^{2}}\sum\limits_{i=1}^{n}{x_{i}^{2}}+\sum\limits_{j=1}^{n}{x_{j}^{2}}\sum\limits_{i=1}^{n}{y_{i}^{2}\,-2\sum\limits_{j=1}^{n}{x_{j}y_{j}}\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{x_{i}^{2}\sum\limits_{j=1}^{n}{y_{j}^{2}}}+\sum\limits_{i=1}^{n}{y_{i}^{2}\sum\limits_{j=1}^{n}{x_{j}^{2}}}-2\sum\limits_{j=1}^{n}{x_{i}y_{i}}\sum\limits_{j=1}^{n}{x_{j}y_{j}}\hfill$
$\displaystyle=$
$\displaystyle\,\,2\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)\left({\sum\limits_{i=1}^{n}{y_{i}^{2}}}\right)-2\left({\sum\limits_{j=1}^{n}{x_{i}y_{i}}}\right)^{2}$

Because the left-hand side of the equation is a sum of squares of real numbers
is greater or equal to zero, so

$\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)\left({\sum\limits_{i=1}^{n}{y_{i}^{2}}}\right)\geqslant\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)^{2},$

that is, the inequality (9).  $\Box$

_Proof 2._ Consider the next trinomial with respect to $t$

$f\left(t\right)=\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)t^{2}-2\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)t+\sum\limits_{i=1}^{n}{y_{i}^{2}}.$

Doing the operations in the second member we notice that

$\displaystyle f\left(t\right)$ $\displaystyle=$
$\displaystyle\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)t^{2}-2\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)t+\sum\limits_{i=1}^{n}{y_{i}^{2}}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{x_{i}^{2}}t^{2}-2\sum\limits_{i=1}^{n}{x_{i}y_{i}}t+\sum\limits_{i=1}^{n}{y_{i}^{2}}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{\left({x_{i}t}\right)^{2}}-2\sum\limits_{i=1}^{n}{\left({x_{i}t}\right)y_{i}}+\sum\limits_{i=1}^{n}{y_{i}^{2}}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{\left[{\left({x_{i}t}\right)^{2}-2\left({x_{i}t}\right)y_{i}+y_{i}^{2}}\right]}\hfill$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}{\left({x_{i}t-y_{i}}\right)^{2}}\geqslant
0.$

Since $f\left(t\right)\geqslant 0$ for every $t\in\mathbb{R}$ the discriminant
of $f(t)$ is negative so it holds that

$\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)^{2}-\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)\left({\sum\limits_{i=1}^{n}{y_{i}^{2}}}\right)\leqslant
0\Leftrightarrow\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)^{2}\leqslant\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)\left({\sum\limits_{i=1}^{n}{y_{i}^{2}}}\right),$

that is, the inequality (9) holds.  $\Box$

_Proof 3._ We assume that the $\rm CBS$ inequality (4) is true for every
$\left({x_{1},\,...,x_{n}}\right)$ and $\left({y_{1},\,...,y_{n}}\right)$.
Based on this hypothesis we will come to the true. If
$\sum\limits_{i=1}^{n}{x_{i}^{2}}=0$ or $\sum\limits_{i=1}^{n}{y_{i}^{2}}=0$,
then $x_{1}=x_{2}=...=x_{n}=0$ or $y_{1}=y_{2}=...=y_{n}=0$ and then the
conclusion is obvious. Now we assume that
$X_{n}=\sum\limits_{i=1}^{n}{x_{i}^{2}}\neq 0$ and
$Y_{n}=\sum\limits_{i=1}^{n}{y_{i}^{2}}\neq 0$ and we set
$a_{i}=\dfrac{{x_{i}}}{{\sqrt{X_{n}}}}$ and
$\,b_{i}=\dfrac{{y_{i}}}{{\sqrt{Y_{n}}}}$, for each $i=1,2,...,n$. Then we
will have

$\sum\limits_{i=1}^{n}{a_{i}^{2}}=\sum\limits_{i=1}^{n}{\left({\frac{{x_{i}}}{{\sqrt{X_{n}}}}}\right)^{2}}=\sum\limits_{i=1}^{n}{\frac{{x_{i}^{2}}}{{X_{n}}}=}\frac{1}{{X_{n}}}\sum\limits_{i=1}^{n}{x_{i}^{2}=}\frac{1}{{X_{n}}}X_{n}=1$

and

$\sum\limits_{i=1}^{n}{b_{i}^{2}}=1.$

Since we have assumed that the initial inequality (4) is true for all
$n$-groups we apply it to $\left({a_{1},\,...,a_{n}}\right)$ and
$\left({b_{1},\,...,b_{n}}\right)$ and then we get

$1\cdot 1\geqslant\left({\sum\limits_{i=1}^{n}{a_{i}b_{i}}}\right)^{2}$

from which inequality arises

$a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n}\leqslant 1.$

From the latter we take successively

$\displaystyle 2\left({a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n}}\right)$
$\displaystyle\leqslant$ $\displaystyle 2\,\,\,\Leftrightarrow\hfill$
$\displaystyle 2\left({a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n}}\right)$
$\displaystyle\leqslant$ $\displaystyle 1+1\Leftrightarrow\hfill$
$\displaystyle 2\left({a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n}}\right)$
$\displaystyle\leqslant$
$\displaystyle\left({a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}}\right)+\left({b_{1}^{2}+b_{2}^{2}+...+b_{n}^{2}}\right),$

thus

$\left({a_{1}-b_{1}}\right)^{2}+\left({a_{2}-b_{2}}\right)^{2}+...+\left({a_{n}-b_{n}}\right)^{2}\geqslant
0,$

which is obviously true.  $\Box$

_Proof 4._ Let

$X=\sqrt{x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}},\,\,\,Y=\sqrt{y_{1}^{2}+y_{2}^{2}+...+y_{n}^{2}}$

From the inequality of the arithmetic-geometric mean, we have

$\displaystyle\sum\limits_{i=1}^{n}{\frac{{x_{i}y_{i}}}{{XY}}}$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{n}{\left({\frac{{x_{i}^{2}}}{{X^{2}}}+\frac{{y_{i}^{2}}}{{Y^{2}}}}\right)}\hfill$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left({\sum\limits_{i=1}^{n}{\frac{{x_{i}^{2}}}{{X^{2}}}}+\sum\limits_{i=1}^{n}{\frac{{y_{i}^{2}}}{{Y^{2}}}}}\right)\hfill$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left({\frac{1}{{X^{2}}}\sum\limits_{i=1}^{n}{x_{i}^{2}}+\frac{1}{{Y^{2}}}\sum\limits_{i=1}^{n}{y_{i}^{2}}}\right)\hfill$
$\displaystyle=$ $\displaystyle 1,$

so it will be

$\sum\limits_{i=1}^{n}{x_{i}y_{i}}\leqslant{\rm
X}\Upsilon=\sqrt{x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}}\sqrt{y_{1}^{2}+y_{2}^{2}+...+y_{n}^{2}}.$

Therefore

$\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)^{2}\leqslant\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)\left({\sum\limits_{i=1}^{n}{y_{i}^{2}}}\right).$

$\Box$

_Proof 5._ Let

| $A_{n}$ | = |  $x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}$,
---|---|---|---
| $B_{n}$ | = | $y_{1}^{2}+y_{2}^{2}+...+y_{n}^{2}$
and  |  |  |
| $C_{n}$ | = |  $x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}$.

It follows from the inequality of the arithmetic-geometric mean that

$\displaystyle\frac{{\sum\limits_{i=1}^{n}{x_{i}^{2}}\sum\limits_{i=1}^{n}{y_{i}^{2}}}}{{\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)^{2}}}+1$
$\displaystyle=$
$\displaystyle\frac{{A_{n}B_{n}}}{{C_{n}^{2}}}+1=\frac{{\sum\limits_{i=1}^{n}{x_{i}^{2}}B_{n}}}{{C_{n}^{2}}}+\frac{{\sum\limits_{i=1}^{n}{y_{i}^{2}}}}{{B_{n}}}=\sum\limits_{i=1}^{n}{\left({\frac{{x_{i}^{2}B_{n}}}{{C_{n}^{2}}}+\frac{{y_{i}^{2}}}{{B_{n}}}}\right)}a\hfill$
$\displaystyle\geqslant$
$\displaystyle\sum\limits_{i=1}^{n}{\left({2\frac{{x_{i}^{2}B_{n}}}{{C_{n}^{2}}}\cdot\frac{{y_{i}^{2}}}{{B_{n}}}}\right)}=2\frac{1}{{C_{n}}}\sum\limits_{i=1}^{n}{x_{i}y_{i}}=2\frac{1}{{C_{n}}}C_{n}=2~{}.\hfill$

Therefore

$A_{n}B_{n}\geqslant C_{n}^{2},$

that is

$\left({x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}}\right)\left({y_{1}^{2}+y_{2}^{2}+...+y_{n}^{2}}\right)\geqslant\left({x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}}\right)^{2}.$

$\Box$

_Proof 6._ We will prove the $\rm CBS$ inequality (9) by mathematical
induction. Starting the induction for 1, the hypothesis is trivial. Note that

$\displaystyle\left({x_{1}y_{1}+x_{2}y_{2}}\right)^{2}$
$\displaystyle=x_{1}^{2}y_{1}^{2}+2x_{1}y_{1}x_{2}y_{2}+x_{2}^{2}y_{2}^{2}$
$\displaystyle\leqslant
x_{1}^{2}y_{1}^{2}+x_{1}^{2}y_{2}^{2}+x_{2}^{2}y_{1}^{2}+x_{2}^{2}y_{2}^{2}$
$\displaystyle=\left({x_{1}^{2}+x_{2}^{2}}\right)\left({y_{1}^{2}+y_{2}^{2}}\right),$

which shows that the inequality (9) holds for $n=2$. Assume that the
inequality (9) holds for $n=k$, that is,

$\left({\sum\limits_{i=1}^{k}{x_{i}y_{i}}}\right)^{2}\leqslant\left({\sum\limits_{i=1}^{k}{x_{i}^{2}}}\right)\left({\sum\limits_{i=1}^{k}{y_{i}^{2}}}\right)$

and we will prove that it is true for $n=k+1$. We have

$\displaystyle\sqrt{\sum\limits_{i=1}^{k+1}{x_{i}^{2}}}\cdot\sqrt{\sum\limits_{i=1}^{k+1}{y_{i}^{2}}}$
$\displaystyle=$
$\displaystyle\sqrt{\sum\limits_{i=1}^{k}{x_{i}^{2}+x_{k+1}^{2}}}\cdot\sqrt{\sum\limits_{i=1}^{k}{y_{i}^{2}+y_{k+1}^{2}}}\hfill$
$\displaystyle\geqslant$
$\displaystyle\sqrt{\sum\limits_{i=1}^{k}{x_{i}^{2}}}\cdot\sqrt{\sum\limits_{i=1}^{k}{y_{i}^{2}}}+\left|{x_{k+1}y_{k+1}}\right|\hfill$
$\displaystyle\geqslant$
$\displaystyle\sum\limits_{i=1}^{k}{\left|{x_{i}y_{i}}\right|}+\left|{x_{k+1}y_{k+1}}\right|=\sum\limits_{i=1}^{k+1}{\left|{x_{i}y_{i}}\right|}.\hfill$

This means that the inequality (9) holds for $n=k+1$, and so we conclude that
$\rm CBS$ holds for all natural numbers. This completes the proof.  $\Box$

_Proof 7._ Let

$\displaystyle X$ $\displaystyle=$
$\displaystyle\left\\{{x_{1}y_{1},...,x_{1}y_{n},x_{2}y_{1},...,x_{2}y_{n},...,x_{n}y_{1},...,x_{n}y_{n}}\right\\},\hfill$
$\displaystyle Y$ $\displaystyle=$
$\displaystyle\left\\{{x_{1}y_{1},...,x_{1}y_{n},x_{2}y_{1},...,x_{2}y_{n},...,x_{n}y_{1},...,x_{n}y_{n}}\right\\},\hfill$
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\left\\{{x_{1}y_{1},...,x_{1}y_{n},x_{2}y_{1},...,x_{2}y_{n},...,x_{n}y_{1},...,x_{n}y_{n}}\right\\},\hfill$
$\displaystyle W$ $\displaystyle=$
$\displaystyle\left\\{{x_{1}y_{1},...,x_{n}y_{1},x_{1}y_{2},...,x_{n}y_{2},...,x_{1}y_{n},...,x_{n}y_{n}}\right\\}.\hfill$

It is easy to observe that the $X$ and $Y$ have the same classification, while
the $Z$ and $W$ have a mixed classification. Applying the rearrangement
inequality, we have

$\begin{gathered}\left({x_{1}y_{1}}\right)\left({x_{1}y_{1}}\right)+...+\left({x_{1}y_{n}}\right)\left({x_{1}y_{n}}\right)+\left({x_{2}y_{1}}\right)\left({x_{2}y_{1}}\right)+...+\left({x_{2}y_{n}}\right)\left({x_{2}y_{n}}\right)+...\hfill\\\
+\left({x_{n}y_{1}}\right)\left({x_{n}y_{1}}\right)+...+\left({x_{n}y_{n}}\right)\left({x_{n}y_{n}}\right)\hfill\\\
\geqslant\left({x_{1}y_{1}}\right)\left({x_{1}y_{1}}\right)+...+\left({x_{1}y_{n}}\right)\left({x_{n}y_{1}}\right)+\left({x_{2}y_{1}}\right)\left({x_{1}y_{2}}\right)+...+\left({x_{2}y_{n}}\right)\left({x_{n}y_{2}}\right)+...\hfill\\\
+\left({x_{n}y_{1}}\right)\left({x_{n}y_{1}}\right)+...+\left({x_{n}y_{n}}\right)\left({x_{n}y_{n}}\right),\hfill\\\
\end{gathered}$

which can be simplified and written as follows

$\left({x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}}\right)\left({y_{1}^{2}+y_{2}^{2}+...+y_{n}^{2}}\right)\geqslant\left({x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}}\right)^{2}$

The last inequality is the desired one.  $\Box$

_Proof 8._ Because if $x=\left({0,0,...,0}\right)$ or
$y=\left({0,0,...,0}\right)$ inequality (4) is always valid, we consider the
non-zero vectors $x=\left({x_{1},x_{2},...,x_{n}}\right)$ and
$y=\left({y_{1},y_{2},...,y_{n}}\right)$. Then for any real number $t$, the
following holds

$\left|{x+ty}\right|^{2}=\left({x+ty}\right)\cdot\left({x+ty}\right)=x\cdot
x+2\left({x\cdot y}\right)t+\left({y\cdot
y}\right)t^{2}=\left|x\right|^{2}+2\left({x\cdot
y}\right)t+\left|y\right|^{2}t^{2},$

so inequality

$\left|x\right|^{2}+2\left({x\cdot y}\right)t+\left|y\right|^{2}t^{2}\geqslant
0$

holds for every $t\,\in\,\mathbb{R}$. But the first term of the last
inequality is a trinomial with respect to $t$ and because it is greater than
or equal to $0$ for each $t\in\mathbb{R}$ and its coefficient is
$\left|y\right|^{2}>0$ (because we have assumed that
$y=\left({y_{1},y_{2},...,y_{n}}\right)\neq 0$ its discriminant will be less
than or equal to zero. Therefore,

$4\left({x\cdot y}\right)^{2}-4\left|x\right|^{2}\left|y\right|^{2}\leqslant
0$

so it will be

$\left({x\cdot y}\right)^{2}\leqslant\left|x\right|^{2}\left|y\right|^{2}.$

Using the notations

$x\cdot
y=x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n},\,\,\left|x\right|^{2}=\sum\limits_{i=1}^{k}{x_{i}^{2}},\,\,\left|y\right|^{2}=\sum\limits_{i=1}^{k}{y_{i}^{2}},$

we result the following

$\left({\sum\limits_{i=1}^{n}{x_{i}y_{i}}}\right)^{2}\leqslant\left({\sum\limits_{i=1}^{n}{x_{i}^{2}}}\right)\left({\sum\limits_{i=1}^{n}{y_{i}^{2}}}\right),$

which is the inequality (9).  $\Box$

_Proof 9._ Consider the vectors

$x=\left({x_{1},x_{2},...,x_{n}}\right),\,\,y=\left({y_{1},y_{2},...,y_{n}}\right).$

From the formula of inner product $x\cdot
y=\left|x\right|\left|y\right|\cos\left({x,y}\right)$ we conclude that $x\cdot
y\leqslant\left|x\right|\left|y\right|$. Using the notations

$x\cdot
y=x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n},\,\,\left|x\right|^{2}=\sum\limits_{i=1}^{k}{x_{i}^{2}},\,\,\left|y\right|^{2}=\sum\limits_{i=1}^{k}{y_{i}^{2}},$

the $\rm CBS$ inequality arises.

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# The High-Frequency and Rare Events Barriers to Neural Closures of
Atmospheric Dynamics

Mickaël D. Chekroun<EMAIL_ADDRESS>Department of Atmospheric and
Oceanic Sciences, University of California, Los Angeles, CA  Department of
Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot 76100,
Israel  Honghu Liu  Department of Mathematics, Virginia Tech, Blacksburg, VA
24061, USA  Kaushik Srinivasan  Department of Atmospheric and Oceanic
Sciences, University of California, Los Angeles, CA 90095-1565, USA  James C.
McWilliams  Department of Atmospheric and Oceanic Sciences and Institute of
Geophysics and Planetary Physics, University of California, Los Angeles, CA
90095-1565, USA

###### Abstract

Neural parameterizations and closures of climate and turbulent models have
raised a lot of interest in recent years. In this short paper, we point out
two fundamental problems in this endeavour, one tied to sampling issues due to
rare events, and the other one tied to the high-frequency content of slow-fast
solutions which constitute an intrinsic barrier to neural closure of such
multiscale systems. We argue that the atmospheric 1980 Lorenz model, a
truncated model of the Primitive Equations—the fuel engine of climate
models—serves as a remarkable metaphor to illustrate these fundamental issues.

Atmospheric and oceanic flows constrained by Earth’s rotation satisfy an
approximately geostrophic momentum balance on larger scales, associated with
slow evolution on time scales of days, but they also exhibit fast inertia-
gravity wave oscillations. The problems of identifying the slow component
(e.g., for weather forecast initialization [1, 2, 3, 4]) and of characterizing
slow-fast interactions are central to geophysical fluid dynamics, and the
former was first coined as a slow manifold problem by Leith [5]. The L63 model
[6] famous for its chaotic strange attractor is a paradigm for the geostrophic
component, while the L80 model [7] is its paradigmatic successor both for the
generalization of slow balance and for slow-fast coupling.

The explosion of machine learning (ML) methods provides an unprecedented
opportunity to analyze data and accelerate scientific progress. A variety of
ML methods have emerged for solving dynamical systems [8, 9, 10], predicting
[11] or discovering [12] them from data. For larger scale problems, much
effort has been devoted lately to the learning of neural subgrid-scale
parameterizations in coarse-resolution climate models [13] but yet lack of
interpretability and reliability prevents a widespread adoption [14, 15].

In parallel, the learning of stable neural parameterizations (not yet
interpretable) has progressed remarkably for the closure of fluid models in
turbulent regimes such as the forced Navier-Stokes equations or quasi-
geostrophic flow models on a $\beta$-plane; see [16, 17, 18, 19, 20].

Nevertheless, the case of the Primitive Equations (PE) has its mysteries yet
untouched by neural parameterizations. This study is aimed at pointing out
potential challenges that one may have to face for the efficient neural
closures of the PE. To do so, the L80 model (recalled in Methods), a truncated
model of the PE, serves as a remarkable metaphor to illustrate some
fundamental issues in this task, as explained in this Letter.

At small Rossby numbers, the solutions to the L80 model remain entirely slow
for all time (i.e. dominated by Rossby waves) whereas fast oscillations get
superimposed to such slow solutions as the Rossby number is further increased.
Such a spontaneous emergence of fast oscillations, tied to inertia gravity
waves (IGWs) evolving on top of the slow geostrophic motion, complicate
severely the closure problem [21, 22].

Regimes with a multiscale mixture of slow and fast dynamics without timescale
separation are not only intimate to the L80 model. They have been observed in
PE models accounting for a greater amount of multiscale interactions as
conspicuously generated by fronts and jets [23, 24], and in cloud-resolving
models in which large-scale convectively coupled gravity waves spontaneously
develop [25]. Regions of organized convective activity in the tropics generate
also gravity waves leading to a spectrum that contains notable contributions
from horizontal wavelengths of 10 km through to scales beyond 1000 km [26] and
such IGWs have been also identified from satellite observation of continental
shallow convective cumulus forming organized mesoscale patterns over vegetated
areas [27].

In fact, IGWs can be energetic on surprisingly large scales. For instance, in
certain regions of the oceans too, Rocha et al. have shown in [28] that IGWs
can account for roughly half of the near-surface kinetic energy at scales
between 10 and 40 km. Thus, geophysical kinetic energy spectra can exhibit a
band of wavenumbers within which waves and turbulence are equally energetic. A
painful consequence of this length-scale overlap is that perturbation methods
such as that of Wentzel, Kramers, and Brillouin (WKB) [29] do not apply to all
scales [30].

In the L80 model, regimes with time-scale overlap in which the solutions
exhibit a mixture of slow motion punctuated by bursts of fast IGWs containing
a large fraction of the total energy (referred to as high-low frequency (HLF)
solutions) were shown to be responsible for a severe breakdown of slaving
relationships [21], that occurs at large Rossby numbers.

Only recently, the generic elements for solving such hard closure problems
with a lack of timescale separation, have been identified [22]. Key to its
solution is the Balance Equation (BE) manifold [31, 32] as rooted in the works
of Monin [33], Charney and Bolin [34, 1], and Lorenz [35]. The BE manifold has
been shown to provide, even for large Rossby numbers, the slow trend motion of
HLF solutions that optimally averages out the fast oscillations [21, 36],
while the fast motions can be efficiently modeled by means of networks of
stochastic oscillators [22]. In this Letter we emphasize the fundamental
issues encountered by neural networks to parameterize unequivocally the slow
balance motion and restore the fast oscillations.

## Learning slow neural closure: Sensitivity

The HLF solutions considered in this study correspond to those of [22, Fig.
6]; see Supplementary Material (SM). We first analyze the ability of neural
parameterizations to learn the slow motion of the L80 dynamics in the HLF
regime. To do so, we preprocess the target variables $\bm{x}$ and $\bm{z}$ to
be parameterized by applying a low-pass filter in order to extract the slow
motion. In that respect, a simple moving average is adopted with a window size
equal to $T_{GW}$, the dominant period of the gravity waves. The results are
shown in Fig. 1A for the $z_{3}$-variable for which we observe that the low-
pass filtered solution almost coincides this way with the BE parameterization
$\bm{z}_{BE}(t)=G(\bm{y}(t))$ with $\bm{y}(t)$ denoting the $\bm{y}$-component
of the HLF solution to the L80 model.

To parallel the construction of the BE manifold in the NN realm (see Methods),
we first learn an MLP, $\bm{\mathcal{Z}}_{\bm{\theta}}$ for $\bm{z}$ as in
(7), with $\bm{y}$ as input and the filtered $\bm{z}$-variable as output. We
then learn an MLP for $\bm{x}$, denoted by $\bm{\mathcal{X}}_{\bm{\theta}}$
with $(\bm{y},\bm{\mathcal{Z}}_{\bm{\theta}}(\bm{y}))$, as input and the
filtered $\bm{x}$-variable as output. It turns out that NNs with one hidden
layer and 5 neurons are sufficient to obtain loss functions with a small
residual; see Suppl. Table 1.

These parameterizations are learned through two consecutive minimizations.
First, given a discrete set of time instants $t_{j}$, one minimizises

$\displaystyle\mathcal{L}_{\bm{\theta}}(\bm{z};\bm{y})=\sum_{j}\left\lVert\bm{z}_{t_{j}}-\bm{\mathcal{Z}}_{\bm{\theta}}(\bm{y}_{t_{j}})\right\rVert^{2},$
(1)

in which $\bm{z}$ is filtered (in time) while $\bm{y}$ is not, followed by the
minimization of

$\displaystyle\mathcal{L}_{\bm{\theta}}(\bm{x};(\bm{y},\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}(\bm{y})))=\sum_{j}\left\lVert\bm{x}_{t_{j}}-\bm{\mathcal{X}}_{\bm{\theta}}(\bm{y}_{t_{j}},\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}(\bm{y}_{t_{j}}))\right\rVert^{2},$
(2)

with $\bm{x}$ filtered and where $\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}$
denotes the optimal parameterization obtained after minimization of (1). Note
that the use of the unfiltered $\bm{y}$-component of the HLF solution albeit
containing fast oscillations, is key for the discovery of a proper
parameterization of the slow motion. For instance, if one replaces it with a
filtered version of $\bm{y}$, the resulting closure leads to e.g. unrealistic
quasiperiodic solutions; see Supplementary Note 4.

To assess whether a neural parameterization is successful in capturing the
slow motion, we evaluate also the following high-frequency (HF) residual

$E_{NN}^{j}(t)=x_{j}(t)-\bm{\mathcal{X}}_{\bm{\theta}_{2}^{\ast}}^{j}(\bm{y}(t),\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}(\bm{y}(t))),$
(3)

in which the $x_{j}(t)$ and $\bm{y}(t)$ are both unfiltered. For an NN with
small residual, $E_{NN}^{j}(t)$ is typically void of slow oscillations (see
Fig. 1E) with mean $\langle E_{NN}^{j}\rangle\approx 0$ for each $1\leq j\leq
3$. Fig. 1 shows such a situation for two NNs, NN1 and NN2, learnt through
different modalities of selection of training, validation and testing sets.
The learnt neural parameterizations are thus able to capture, offline, the
slow motion as the BE manifold does, albeit with noticeable differences as
revealed by their geometric visualizations (see Methods) in Figs. 1B, C and D.

Figure 1: Panel A: Illustration (on $z_{3}$) of the ability of the BE manifold
derived analytically in [32] in capturing the L80’s slow motion. Panels (B)
and (C): Neural parameterizations $\bm{\mathcal{X}}_{\bm{\theta}}^{3}$ (of
$x_{3}$), as learnt through random selection (NN1)/predefined selection (NN2).
Visualized here as mappings from $(y_{1},y_{2})$ on the unit sphere in
$\mathbb{R}^{3}$. Panels (D): Same visualization adopted for the BE manifold.
Panels (E): High-frequency residual $E_{NN_{1}}(t)$ for $x_{3}$ (black) and
its difference with $E_{NN_{2}}(t)$ (red).

These geometric offline differences hide more profound consequences when the
neural parameterizations are used online, for closure. As explained below,
sensitivity of online predictions that are tied to sampling issues are indeed
observed. In that respect, recall that a common practice to train NNs is to
divide the dataset into three subsets. The first subset is the training set,
which is used for computing the loss function’s gradient and updating the
network weights and biases.

The second subset is the validation set. It corresponds to the second dataset
over which the prediction skills of the fitted model are assessed. The error
on the validation set is monitored during the training process to provide an
unbiased evaluation while tuning the model’s hyperparameters. When the network
begins to overfit the data, the error on the validation set typically begins
to rise after an initial decrease. The network parameters are saved at the
minimum. It gives then the ”final model” that is tested over the test set that
is typically a holdout dataset not used as a validation nor a training set.

The parameterization NN1 shown in Fig. 1A is learnt through a random selection
while NN2 through a predefined selection. In each case, ratios for training,
testing and validation are 0.7, 0.15 and 0.15, respectively. The total length
of the training is 700 days. Given the same input and target data, the minimal
values of the loss functions (1)-(2) for NN1 and NN2 are reported in
Supplementary Table 1, across epochs. Already after 500 epochs, one observes
that the loss function evaluations differ only by $1\%$ between the random or
predefined selection protocol of the training, validation, and testing sets.

We now discuss the sensitivity issue of online predictions driven by such
neural parameterizations close in terms of their loss function evaluation.
This point is illustrated in Fig. 2. There, we show online prediction
corresponding to a given slow NN-parameterization
$(\bm{\mathcal{X}}_{\bm{\theta}_{2}^{\ast}},\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}})$
learnt by minimization of the loss functions (1)-(2), namely the solution to
the slow neural closure

$\displaystyle a_{i}\frac{\,\mathrm{d}y_{i}}{\,\mathrm{d}t}$
$\displaystyle=-a_{k}b_{k}\bm{\mathcal{X}}_{\bm{\theta}_{2}^{\ast}}^{j}(\bm{y},\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}(\bm{y}))y_{k}-a_{j}b_{j}y_{j}\bm{\mathcal{X}}_{\bm{\theta}_{2}^{\ast}}^{k}(\bm{y},\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}(\bm{y}))+c(a_{k}-a_{j})y_{j}y_{k}-a_{i}\bm{\mathcal{X}}_{\bm{\theta}_{2}^{\ast}}^{i}(\bm{y},\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}(\bm{y}))-\nu_{0}a_{i}^{2}y_{i},$
(4)

which is obtained by replacing the $x_{\ell}$ in the $\bm{y}$-equation of the
L80 model by its neural parameterization, either NN1 or NN2.

The attractor corresponding to the slow NN1-closure (with random selection)
differs clearly from that of slow NN2-closure (with predefined selection) in
spite of convergence and closeness of the loss functions at their respective
minimal value; see Fig. 2B. Both predict periodic orbits with different
attributes, one self-intersecting in the $(y_{2},y_{3})$-plane (NN1), the
other without intersection point (NN2).

A closer inspection at these topological differences reveals in the time
domain that the slow NN1-closure is able to capture more accurately the low-
frequency content of certain temporal patterns exhibited by the HLF solutions
of the L80 model compared to the slow NN2-closure; blue vs red curves in Fig.
2A. We argue next that such a sensitivity between online solutions takes its
root in the rare events tied to the irregular transitions exhibited by the HLF
solutions to the L80 model that spoils the offline learning.

Figure 2: Sensitivity of the slow neural closures. Here, NN1 and NN2 differ
only in their training modalities. NN1 is learnt from random selection of the
training, validation, and testing sets, and NN2 from a predefined selection
with the same aspect ratios; see Text. The corresponding loss functions differ
by 1$\%$, while the dynamical differences of the online predictions are
substantial.

In contrast, at lower Rossby numbers, for regimes devoid of fast oscillations
such as shown in Fig. 3D below corresponding to $F_{1}=6.97\times 10^{-2}$ in
the L80 model, neural closures of high-accuracy are easily accessible with
skills comparable to those obtained with the BE manifold; see Supplementary
Fig. 1. As explained next, the reasons for this success lie in the absence of
high-frequencies in the solutions to parameterize and in the absence of rare
events in the statistics of lobe transitions.

Figure 3: Panel A: Attractor in the HLF case. Panel B: The sojourn episodes
within one particular lobe are marked by different colours. Here, the
parameters are those used in Lorenz’s original paper [7] except $F_{1}=0.3027$
in Eq. (6). Panel C: Lobe sojourn time distributions. The exponential fit is
calculated over 500 yr-long simulation of Eq. (6) and is shown by the black
curve $f(t)=ae^{bt}$ with $a=2292$ and $b=-6.05\times 10^{-2}$ with $t$ in
day. The inset in panel C shows a magnification of the distribution for the
rare and large sojourn times. Panels E and F: Same as panels B and C except
that $F_{1}=6.97\times 10^{-2}$, corresponding to the slow chaotic regime
shown in panel D in which the solutions are void of fast oscillations. In this
regime, no rare event statistics emerge.

## Irregular transitions, rare events and learning consequences

The sensitivity in the online capture of the low-frequency content between two
nearby neural parameterizations (as measured through their loss functions),
calls for more understanding. Keeping in mind that the differences pointed out
in Fig. 2 are only tied to the way the training, validation, and test sets are
chosen, we perform a statistical analysis of key features of the L80 dynamics
in the HLF regime. In particular, we focus on the irregular lobes’ transitions
exhibited by HLF solutions, and for comparison, we analyze the lobes’
transitions in the slow chaotic regime of Fig. 3D for which neural
parameterizations do not suffer from sensitivity and have no difficulty in
learning the proper closure; see Supplementary Note 3.

In that respect, we perform a high-resolution and long-term integration of the
L80 model corresponding to a 500-yr simulation ($\sim 730000\times T_{GW}$)
with a time stepping of 0.75 min, for both the HLF regime and the slow chaotic
regime.

In either case, the L80 attractor exhibits two lobes as visible in e.g. the
$(y_{2},y_{3})$-projection; see Figs. 3A and D. These lobes are essentially
separated by the vertical line $y_{3}=0$. Numerical integration of the L80
model reveals that the visit of the right lobe comes with $y_{3}(t)$ getting
greater than some threshold value $y_{b}$, while the visit of the left lobe
comes with $y_{3}(t)$ getting smaller than $y_{a}=-y_{b}.$ A close inspection
of the solution in the HLF case reveals that the choice of $y_{b}=0.2$
constitutes a good one to identify the sojourn of the dynamics within one lobe
from the other. This choice leads furthermore to an interval $(-y_{b},y_{b})$
that provides a good bound of the bursts of fast oscillations crossing the
vertical line $y_{3}=0$ in the $(y_{2},y_{3})$-plane (”grey” zone).

To count the transitions from one lobe to the other one thus proceeds as
follows. Given our 500-yr long simulation of $y_{3}(t)$ we first find the
local maxima and minima that are above $y_{b}$ and below $y_{a}$,
respectively. No transition occurs between consecutive such local maxima or
minima. A transition occurs only when a local maximum above $y_{b}$ is
immediately followed by a local minimum below $y_{a}$ or vice versa. If a
local maximum is immediately followed by a local minimum, the intermediate
time instant at which the trajectory goes below zero is identified as the
transition instant, and the other way around if a local minimum is immediately
followed by a local maximum. These transition times characterized this way
allow us to count the sojourn times in a lobe and display the distribution of
these sojourn times shown in Figs. 3C and F.

These lobe sojourn time distributions reveal a striking difference between the
HLF and slow chaotic regimes. In the HLF case, we observe indeed that the
solution can stay in one lobe for a period of time that can be arbitrarily
long (see solution’s segment between $t=763$ and $t=893$ shown in blue in Fig.
3B) albeit of probability of occurrence vanishing exponentially as shown in
Fig. 3C. As a comparison, the transitions between the attractor’s lobes occur
at a much more regular pace in the slow chaotic regime (see Fig. 3E) in which
the solutions to the L80 model are void of fast oscillations. In this case,
the distribution of sojourn times drops quickly below a 60-day duration
barrier (Fig. 3F).

One can thus argue that such rare events with an exponential distribution are
troublesome for the derivation of a reliable slow neural closure. They add
diversity in the temporal distribution of the time series patterns that
explain the sensitivity results shown in Fig. 2. Indeed, a random selection of
the training set may contain temporal episodes that are more skewed towards
one lobe than those in a predefined set, confusing this way the learning
procedure.

## The high-frequency barrier to neural closure

We address now the issue of direct parameterizations of HLF solutions
containing a multiscale mixture of slow and fast motions. To do so, we learn
an MLP for $\bm{x}(t)$, denoted by $\bm{\mathcal{V}}_{\bm{\theta}}$, with (the
unfiltered) $\bm{y}(t)$-variable of the L80 model (6), as input, and the
unfiltered $\bm{x}$-component, $\bm{x}(t)$, as output. Note that unlike the
slow NN-parameterizations above, the parameterization
$\bm{\mathcal{V}}_{\bm{\theta}}$ aims at parameterizing $\bm{x}(t)$ directly
as a nonlinear mapping of $\bm{y}(t)$ without conditioning on $\bm{z}(t)$ nor
filtering of any sort. The corresponding closure, called a vanilla NN-closure,
consists then of Eq. (4) in which
$\bm{\mathcal{X}}_{\bm{\theta}_{2}^{\ast}}(\bm{y},\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}(\bm{y}))$
is replaced by $\bm{\mathcal{V}}_{\bm{\theta}^{\ast}}(\bm{y})$, obtained after
minimization of the following $L^{2}$-loss function

$\displaystyle\mathcal{L}_{\bm{\theta}}(\bm{x};\bm{y})=\sum_{j}\left\lVert\bm{x}_{t_{j}}-\bm{\mathcal{V}}_{\bm{\theta}}(\bm{y}_{t_{j}})\right\rVert^{2},$
(5)

for which the target variable $\bm{x}(t)$ is unfiltered, i.e. containing a
mixture of fast and slow oscillations. To address this more challenging
problem we use MLPs with a larger capacity either with more neurons and/or
layers.

Our experiments reveal that an NN with one hidden layer and 20 neurons turns
out to provide the best closure results; see Supplementary Note 5. The
corresponding vanilla neural closure is able to apprehend to a certain extent
the complexity of the temporal patterns exhibited by the HLF solution (see
Fig. 4A-B), albeit completely failing to predict the high-frequency content
i.e. the physics of IGWs; see also Supplementary Fig. 3.

Figure 4: Vanilla NN-closure vs L80 dynamics. Failure to capture the high-
frequency content and symmetry of the L80 attractor.

The failure experienced by vanilla NN-closure in capturing the IGWs dynamics
mirrors the spectral bias problem exhibited by NNs for function fitting [37];
feedforward NNs prioritizing the learning of the low frequency features. We
emphasize though that here the difficulty is of next order than for function
fitting. The problem is indeed to properly learn offline the neglected
variables along with their high-frequency content, so that the online solution
via neural closure reproduces both the slow and fast motions of the original
dynamics. Even global geometric features are misrepresented by vanilla NN-
closures, with e.g. distortion of the attractor and breakdown of symmetry
compared to the L80 attractor; see Fig. 4C.

Finally, it is worth mentioning that even larger NNs does not necessarily
help. A vanilla NN, $\bm{\mathcal{V}}_{\bm{\theta}}$, with e.g. 5 hidden
layers and 20 neurons predicts an unrealistic periodic orbit (of small
amplitude) when iterated through time-stepping online in the neural closure
and tends to exaggerate the high-frequency content of the parameterized
solutions offline; see Suppl. Fig. 5.

These results, tied with the multiscale nature of the L80 regime analyzed with
its rare events, show that the L80 model provides a cautionary illustration
for the challenges of using machine learning for deriving accurate dynamical-
system closures for geophysical fluid dynamics. Because rare events and
extreme statistics are likely to play a more and more dominant role in a
changing climate [38, 39, 40, 41, 42], this study shows that in spite of the
recent promises of neural parameterizations, still a lot of fundamental issues
lie in front of us to derive reliable parameterizations accounting for rare
events in a robust way. In that respect, rare events algorithms [43, 44, 45]
could be of great use to simulate rare events offline in order to better
sample e.g. the distribution’s tails to train the neural networks.

## Methods

The L80 model. The L80 model, obtained as a nine-dimensional truncation of the
PE onto three Fourier modes with low wavenumbers [7], can be written as:

$\displaystyle\\!\\!a_{i}\frac{\,\mathrm{d}x_{i}}{\,\mathrm{d}t}$
$\displaystyle=-\nu_{0}a_{i}^{2}x_{i}-c(a_{i}-a_{k})x_{j}y_{k}+c(a_{i}-a_{j})y_{j}x_{k}$
(6)
$\displaystyle\quad+a_{i}b_{i}x_{j}x_{k}-2c^{2}y_{j}y_{k}+a_{i}(y_{i}-z_{i}),$
$\displaystyle\\!\\!a_{i}\frac{\,\mathrm{d}y_{i}}{\,\mathrm{d}t}$
$\displaystyle=-a_{k}b_{k}x_{j}y_{k}-a_{j}b_{j}y_{j}x_{k}+c(a_{k}-a_{j})y_{j}y_{k}$
$\displaystyle\quad-a_{i}x_{i}-\nu_{0}a_{i}^{2}y_{i},$
$\displaystyle\\!\\!\frac{\,\mathrm{d}z_{i}}{\,\mathrm{d}t}$
$\displaystyle=g_{0}a_{i}x_{i}-b_{k}x_{j}(z_{k}-h_{k})-b_{j}(z_{j}-h_{j})x_{k}$
$\displaystyle+cy_{j}(z_{k}-h_{k})-c(z_{j}-h_{j})y_{k}-\kappa_{0}a_{i}z_{i}+F_{i},$

with $(i,j,k)$ in $\\{(1,2,3),(2,3,1),(3,1,2)\\}.$

The neural parameterizations. We adopt multilayer perceptrons (MLP) that
respect the analytic construction of the BE manifold; see Supplementary Note
2. First, we determine a neural parameterization of $\bm{z}$ in terms of
$\bm{y}$ and then we determine a neural parameterization of $\bm{x}$ that is
conditioned on the former.

Each neural parameterization, e.g. $\bm{z}$ in terms of $\bm{y}$ is sought by
means of an MLP with $L$ hidden layers of $p$ neurons each, it boils down to
find

$\bm{\mathcal{Z}}_{\bm{\theta}}(\bm{y})={\mathcal{N}}_{\rm
out}\circ{\mathcal{N}}_{L}\circ\cdots\circ{\mathcal{N}}_{1}\circ{\mathcal{N}}_{\rm
in}(\bm{y}),$ (7)

in which ${\mathcal{N}}_{\rm in}$ (resp. ${\mathcal{N}}_{\rm out}$) constitute
the input (resp. output) layer, while ${\cal N}_{k}$ is a mapping from
$\mathbb{R}^{p}$ (the space of neurons) onto itself, given by ${\cal
N}_{k}(\xi)=\Psi_{k}({\bf W}_{k}\xi+{\bf b}_{k})$ ($\xi$ in $\mathbb{R}^{p}$)
where $\Psi_{k}$ is a $p$-dimensional elementwise function, i.e. a function
that applies a (scalar) activation function to each of its inputs
individually, and the ${\bf W}_{k}$ and ${\bf b}_{k}$ denote respectively the
weight matrices and bias vectors to be learned. In (7), the subscript
$\bm{\theta}$ denotes the collection of these parameters. In this work, the
nonlinear activation function is a simple $\tanh$ function, and the input and
output layers consist just of linear normalization and reversal operations.

Geometric visualization of neural parameterizations. Let
$\bm{\mathcal{X}}_{\bm{\theta}_{2}^{\ast}}^{j}$ (resp.
$\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}^{j}$) be the $j$th component of the
neural parameterization of $x_{j}$ (resp. $z_{j}$). For each $1\leq j\leq 3$,
they are mappings from $\mathbb{R}^{3}$ into $\mathbb{R}$, since they are
functions of the variable $\bm{y}$. For a given radius $r$, the level set of
$\bm{\mathcal{X}}_{\bm{\theta}_{2}^{\ast}}^{j}$ (resp.
$\bm{\mathcal{Z}}_{\bm{\theta}_{1}^{\ast}}^{j}$) on the sphere,
$y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=r^{2}$, can be visualized as a 2D surface that
maps e.g $(y_{1},y_{2})$ to $x_{j}$ (resp. $z_{j}$). The surfaces shown in
Fig. 1 have been computed according to this protocol.

###### Acknowledgements.

This work has been partially supported by the Office of Naval Research (ONR)
Multidisciplinary University Research Initiative (MURI) grant
N00014-20-1-2023, by the National Science Foundation grant DMS-2108856, and by
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (grant agreement no. 810370). We also
acknowledge the computational resources provided by Advanced Research
Computing at Virginia Tech.

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