User:Adriano Ferrari

From McGill University Physics Department Technical Services Wiki

Office: Room 228 Rutherford Physics Building
Office Hours: W 12:00 - 14:00
E-mail: $\mbox{adriano.ferrari}\otimes\mbox{gmail.com}$

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Project 1

link

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Physiology 610- "Hydrophobic Collapse in Proteins"

This has been moved here

$V=k\int\frac{\rho}{r}\,d\tau$

$\nabla^2 V=-\rho/\epsilon_0$

$\nabla\cdot\mathbf{E}=\rho/\epsilon_0$

$\nabla\times\mathbf{E}=0$

$\mathbf{E}=k\int\frac{\hat{r}}{r^2}\rho \,d\tau$

$\mathbf{E}=-\nabla V$

$V=-\int\mathbf{E}\cdot d\mathbf{l}$

$d\tau=dx\,dy\,dz$ $d\tau=r\,dr\,dh\,d\phi$

$d\tau=r^2\sin\theta dr\,d\theta\,d\phi$

$\int \frac{x}{\sqrt{x^2\pm a^2}}\,dx=\sqrt{x^2\pm a^2}$

$\int \frac{dx}{\sqrt{x^2\pm a^2}}=\ln\left|x+\sqrt{x^2\pm a^2}\right|$

$\int \sqrt{x^2\pm a^2}=\frac{x\sqrt{x^2\pm a^2}}{2}\pm \frac{a^2}{2}\ln\left|x+\sqrt{x^2\pm a^2}\right|$

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Solids Project

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Random Notes

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Proteomics

Unknown terms:

"Steric" hindrances/constraints etc...

In context: "An even more fundamental structural principle pertains to steric constraints related to the diversity of residue sizes, the prohibition of overlaps of atoms and a close packing of the residues leading to small cavity volumes."

"Co-operativity"
"Specificity"
"Associate" as in "the tendency of nonpolar amino acids to associate in water."
"Water-to-oil transfer free energy"... is "a measure of the interactions among monomer contacts."
"solvation"
"site-directed mutagenesis"
"amphipathic"

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Math

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Definitions

Open set:

English: a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U; it can't be on the edge of U.
MathSpeak: A subset U of the Euclidean n-space Rⁿ is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rⁿ whose Euclidean distance from x is smaller than ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U)
Examples: As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. Here, the topology is the usual topology on the real line. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and move even the tiniest bit in the positive direction, you will be outside of (0,1].

Complement:

If a universal set U is defined, then the relative complement of A in U is called the complement of A, and is denoted by A^C (or sometimes A′), that is:

A^C = U − A

Example: If the universal set is the set of natural numbers, then the complement of the set of odd numbers is the set of even numbers.

De Morgan's laws:

(A ∪ B)^C = A^C ∩ B^C
(A ∩ B)^C = A^C ∪ B^C

Complement laws:

A ∪ A^C = U
A ∩ A^C = Ø
Ø^C = U
U^C = Ø
If A⊆B, then B^C⊆A^C (this follows from the equivalence of a conditional with its contrapositive)

Involution or double complement law:

A^C^C = A.

Closed set:

English: A closed set contains its own boundary. In other words, if you are "outside" a closed set and you "wiggle" a little bit, you will stay outside the set. A closed set is the complement of an open set.

Compact:

MathSpeak: A subset of Euclidean space Rⁿ is called compact if it is closed and bounded.

Compact-support:

The support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. Functions with compact support in X are those with support that is a compact subset of X.
Example: If X is the real line, they are examples of functions that vanish at infinity (and negative infinity).

Supremum:

The supremum of a set S is the least element that is greater than or equal to each element of S. a.k.a. the Least Upper Bound.
Example: $S_1=\{x\in\mathbb{Q} | x^2 <2\}$ . e.g. $1414/1000 \in S_1$ , $14142/10000 \in S_1$ , but $\sup(S_1)=\sqrt{2}$ isn't in S₁. i.e. $x \le \sup(S_1)=\sqrt{2} \quad \forall\, x\in S_1$ .

Infimum:

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Formulae

Green's Formula I:

$\int_\Omega u\,\mbox{div}\!\left(\mathbf{v}\right)\,d\mathbf{x} = -\int_\Omega\nabla u \cdot \mathbf{v}\,d\mathbf{x}+\int_{\partial \Omega}u\left(\mathbf{v}\cdot\mathbf{n}\right)dS_\mathbf{x}$

(note $u=u(\mathbf{x})$ , $\mathbf{v}=\mathbf{v}(\mathbf{x})$ , etc.)

Green's Formula II:

$\int_\Omega u\Delta w\,d\mathbf{x} = -\int_\Omega\nabla u \cdot \nabla w\,d\mathbf{x}+\int_{\partial \Omega}u\left(\nabla w\cdot\mathbf{n}\right)dS_\mathbf{x}$