I'm referring to this symbol: $\geqslant$.
No symbol list I've come across mentions it. I saw it in the book Pattern Recognition and Machine Learning by Christopher M. Bishop.
[Math] What does the sign that looks like $\geq$ except with the bottom line being sloped mean
notation
Related Solutions
The symbol $\Pi$ is the pi-product. It is like the summation symbol $\sum$ but rather than addition its operation is multiplication. For example, $$ \prod_{i=1}^5i=1\cdot2\cdot3\cdot4\cdot5=120 $$ The other symbol is the coproduct.
According to the digital copy of the text which was linked by the user Chappers, this notation is the Cauchy principle value. This use is listed in the rather extensive index of notation at the end of the text—specifically, in the section labeled Miscellaneous Notation.
The Cauchy Principal Value is a way of assigning a value to certain "improper" integrals which would otherwise be undefined. If $f : \mathbb{R} \to \mathbb{R}$ has a singularity at $c \in [a,b]$, then the Cauchy Principal value is given by
$$ -\kern-9pt\int_{a}^{b} f(x)\,\mathrm{d}x := \lim_{\varepsilon\searrow 0} \left[ \int_{a}^{c-\varepsilon} f(x)\,\mathrm{d}x + \int_{c+\varepsilon}^b f(x)\,\mathrm{d}x\right].$$
A similar definition applies if $f$ has a singularity at infinity:
$$ -\kern-9pt\int_{-\infty}^{\infty} f(x) = \lim_{R\to\infty} \int_{-R}^{R} f(x)\,\mathrm{d}x. $$
In this second case, it is easier to see how Cauchy principal value differs from the "usual" method of assigning a value to an improper integral. In the usual setting, we define $$ \int_{-\infty}^{\infty} f(x)\,\mathrm{d}x := \lim_{a\to-\infty} \int_{a}^{c} f(x)\,\mathrm{d}x + \lim_{b\to\infty} \int_{c}^{b} f(x),\mathrm{d}x, $$ where $c$ is any real number. Using this standard definition, the sine function cannot be integrated over the entire real line. However, the Cauchy principle value does exist:
$$ -\kern-9pt\int_{-\infty}^{\infty} \sin(x)\,\mathrm{d}x = \lim_{R\to\infty} \int_{-R}^{R} \sin(x) \,\mathrm{d}x = 0, $$ since sine is an odd function.
It is also well worth noting that $-\kern-7.5pt\int$ is not standardized notation for the Cauchy principal value. Most authors will, instead, use the notation $PV\kern-4pt\int$, or something similar. Also, the notation $-\kern-7.5pt\int$ is used by other authors to mean something different. For example, in his text on PDEs, Evans uses $-\kern-7.5pt\int$ to denote the average integral over a ball, i.e. $$ -\kern-9pt\int_{B(x,r)} f(y)\,\mathrm{d}y = \frac{1}{\mu(B(x,r))} \int_{B(x,r)} f(y)\,\mathrm{d}y,$$ where $B(x,r)$ denotes a ball in $n$-dimensional Euclidean space with center $x$ and radius $r$, and $\mu(B(x,r))$ denotes the $n$-dimensional volume (Lebesgue measure) of that ball.
Best Answer
An example of the symbol's context can be found on p19 of Bishop - Pattern Recognition and Machine Learning:
$\qquad\begin{align*} p(x) &\geqslant 0 \\ \int p(x)\,\mathrm{d}x &= 1 \end{align*}$
The symbols $\geq$, $\geqslant$ and
>=, all mean the same thing: Greater than or equal to.The same applies to its converse: $\leq$, $\leqslant$ and
<=, all mean: Less than or equal to.Context for the latter symbol:
$\qquad\displaystyle f\left(\sum_{i=1}^M \lambda_i x_i\right) \leqslant \sum_{i=1}^M \lambda_i f(x_i)$
The differences can be attributed to the character set available, and/or the particular font that is used. In this particular book, the author specifies, on page viii, than they have used $\LaTeX$.