What does a tilde underneath an inequality mean? I've recently come across an expression of the form
$$\large x \lesssim y$$
What does this expression mean?
 A: In the context of partial differential equations and harmonic analysis, this notation is often used to mean 
$$ A \lesssim B \iff \exists C > 0 \text{ s.t. } A \leq C B $$
As written this expression is pretty damn useless when $A$ and $B$ are just two real numbers. What's more useful is supposing $A(\lambda), B(\lambda)$ are two families of objects parametrised by $\lambda\in \Lambda$ for some set. Then 
$$ A \lesssim B \iff \exists C > 0 \text{ s.t. } \forall \lambda\in \Lambda~,~ A(\lambda) \leq C B(\lambda) $$
Sometimes you will see when $A(\lambda,\pi), B(\lambda,\pi)$ are two families of objects parametrised by $(\lambda,\pi) \in \Lambda\times\Pi$, the notation 
$$ A\lesssim_\pi B \iff \forall \pi\in \Pi \exists C = C(\pi)>0 \text{ s.t. } \forall \lambda\in \Lambda ~,~ A(\lambda,\pi) \leq C(\pi) B(\lambda,\pi) $$
that is, the constant in the inequality is universal over $\lambda$ but may depend on $\pi$. 

Similarly one sees the notation 
$$ A\approx B \iff A\lesssim B \text{ and } B\lesssim A $$
which also has the $\approx_\pi$ variant in the same way. 
A: One possible interpretation of $x \lesssim y$ is,
$$x < y \ \mbox{ or }\ x \approx y.$$
This is analogous to the way $x \le y$ signifies "$x < y$ or $x = y$."
But from the comments, clearly this is not the only way this symbol might be used.
One would hope that prior to its first use in a publication, the same publication would define the symbol.
A: Maybe $x \lesssim y$ means $y-x<\epsilon$ fro some small $\epsilon$. In other words, the difference is small but x IS less than y
A: The symbols are used frequently in high-energy physics by theorists and phenomenologists. 
In that context, they mean less than about or greater than about, i.e., $x \lesssim 10$ means that $x$ is less than about 10, so as @DavidK suggests, either $x<10$ or $x\approx 10$.
E.g., there are experimental constraints on the mass of a hypothetical particle called the gluino. The limits are somewhat complicated, depending on many other assumptions. Rather than write something technical with many caveats and explanations, the situation is summarized by just saying $m_\text{gluino} \gtrsim 1\,\text{TeV}$, meaning that the gluino mass must be greater than about $1\,\text{TeV}$.
A: One of several possible meanings is a total preorder.
