Information Retrival is trying to find material within large collections which is unstructured which satisfies an information need (of structured info).

Unstructured information has had a massive outburst after the millennium.

IMPORTANTLY: evaluating Information Retrival is based on Precision/Recall/F on information need and not the query.

For ranked system, we can come up with a curve of precision-recall curve by selecting increasing \(k\), or mean average precision.

Basic Terminology

collection

a set of documents—could by static, or dynamically added

Information Units are unique entities mentioned during an utterance; for a sentence like “There is a boy. The boy is a brother. He is stealing a cookie. The sister is watching.”, “boy, cookie, sister” are possible IUs.

First order IVP

The class of problems described as:

\begin{equation} \dv{y}{t} = f(t, y) \end{equation}

and:

\begin{equation} y(t_0) = y_0 \end{equation}

we need to figure “which of the general solutions of the DiffEqu satisfy the general value.

To do this, we simply have to plug in the initial value and solve for our constant \(K\).

Second order IVP

\begin{equation} \dv[2]{d}{t} = f(t,y,y’) \end{equation}

this requires two initial conditions to fully specify (because two variables becomes constant and goes away).

An injective function is one which is one-to-one: that it maps distinct inputs to distinct outputs.

constituents

• A function \(T: V \to W\)

requirements

\(T\) is injective if \(Tu = Tv\) implies \(u=v\).

additional information

injectivity implies that null space is \(\{0\}\)

Proof: let \(T \in \mathcal{L}(V,W)\); \(T\) is injective IFF \(null\ T = \{0\}\).

given injectivity

Suppose \(T\) is injective.

Now, we know that \(0\), because it indeed gets mapped by \(T\) to \(0\), is in the null space of \(T\).