Mod 3

Lecture 24: Pumping Lemma

• The pumping lemma is used to prove that a language is not regular.
• A five-step process is used: assume the language L is regular, let p be the pumping length, choose string s ∈ L where |s| ≥ p, analyze all ways to split s into s = xyz, such that |xy| ≤ p and y ≠ ∈, demonstrate that there exists i ≥ 0 such that xyⁱz ∉ L.

Example 1: L = {0ⁿ1ⁿ | n ≥ 0}

• Assume L is regular, let p be the pumping length, and let s = 0^p1^p where |s| ≥ p, such that s = xyz, where |xy| ≤ p and y ≠ ∈.
• Then for all i ≥ 0, xyⁱz ∈ L.
• y comprises only 0s, so, y = 0^k for some k > 0.
• When i = 0, xy⁰z = xz = 0^p-k1^p, but p - k < p, so xz ∉ L, a contradiction; therefore, L is not regular.

Example 2: L = {w | w has an equal number of 0s and 1s}

• Assume L is regular, with p as pumping length and s = 0^p1^p.
• s can be split into s = xyz, and with |xy| ≤ p and y ≠ ε, y consists only of 0s (y = 0^k, k > 0).
• For i = 2, xy²z = 0^p+k1^p.
• Since k > 0, p + k > p, meaning xy²z ∉ L, which is a contradiction.
• Therefore, L is not regular.

Example 3: L = {0ⁿ1^m | n > m}

• Let s = 0^p+11^p, split into s = xyz.
• with |xy| ≤ p, y ≠ ε, y = 0^k for some k > 0.
• When i = 0, xy⁰z = xz = 0^p+1-k1^p, but k > 0, so p + 1 - k < p + 1, thus xz ∉ L.

Regulation of Gene Expression: Introduction

• Cells must conserve energy, respond to their environment, and differentiate, making gene regulation vital.

Why Regulate Gene Expression?

• Producing proteins requires energy, so cells regulate genes to conserve energy.
• Regulation also allows cells to produce proteins at the correct time, enabling adaptation to change.
• Gene regulation is essential for cell specialization and the creation of diverse tissues in multicellular organisms.

Mechanisms of Gene Regulation: Bacteria

• Bacterial gene regulation often involves operons - promoter: initiates transcription by binding RNA polymerase
• Operator: DNA sequence where a repressor protein binds
• Repressor: halts transcription by blocking RNA polymerase

Types of Operons

• Inducible operons are typically "off" but can be activated by an inducer molecule, for example, the lac operon controls lactose breakdown and is activated in the presence of lactose.
• Repressible operons are normally "on" but can be turned "off" by a corepressor molecule, for example, the trp operon controls tryptophan synthesis and is deactivated when tryptophan is abundant.

Eukaryotes: Levels of Regulation

• DNA is packaged with proteins to form chromatin, in which histone acetylation loosens chromatin to promote transcription and DNA methylation condenses chromatin to reduce transcription.
• Transcriptional control:Transcription factors (activators enhance, repressors inhibit), enhancers increase transcription, and silencers decrease transcription.
• Post-transcriptional control includes alternative splicing, mRNA degradation, and RNA interference.
• Translational control: Involves factors impacting translation initiation.
• Post-translational control: Protein modification (phosphorylation, glycosylation) and degradation impacts protein activity.

Teorema de Bayes

Teorema de Bayes descreve a probabilidade de um evento baseado em conhecimento prévio de condições relacionadas ao evento.

• A fórmula é:

$$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} $$

• P(A|B): probabilidade condicional de A dado que B ocorre
• P(B|A): probabilidade condicional de B dado que A ocorre
• P(A) e P(B): probabilidades de observar A e B independentemente

Dedução do Teorema

• Probabilidade Condicional:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) \neq 0$$

$$P(B|A) = \frac{P(B \cap A)}{P(A)}, \quad P(A) \neq 0$$

Etapas da Dedução

• Comece com $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
• Multiplique ambos os lados por $$P(B): P(A|B)P(B) = P(A \cap B)$$
• Similarmente, $$P(B|A) = \frac{P(B \cap A)}{P(A)}$$
• Multiplique ambos os lados por $$P(A): P(B|A)P(A) = P(B \cap A)$$
• Igualando as expressões (já que$$P(A \cap B) = P(B \cap A)): P(A|B)P(B) = P(B|A)P(A)$$
• Divida ambos os lados por $$P(B): P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

Lecture 14: Hashing II

• Collision Resolution's two basic approaches
  • Chaining: hash table entries are linked lists that store keys that hash to that entry.
  • Open addressing: table is probed until an empty slot is discovered after a collision

Chaining: Implementation

• insert(x): inserts x at the front of list H[h(x)].
• search(x): searches for x in list H[h(x)].
• delete(x): deletes x from list H[h(x)].

Simple Uniform Hashing

• Each key has equal likelihood of hashing to each of the m slots, independently of other keys.
• Load factor α = n/m = average number of keys per slot, where n is the number of keys in the table.

Anaylsis Simple Uniform Hashing

• Assuming simple uniform hashing
  • E[time to search for x] = Θ(1 + α)
  • E[time to insert x] = Θ(1)
  • E[time to delete x] = Θ(1 + α)

Choosing a Good Hash Function

• Should distribute keys uniformly into slots, requires accounting for non-uniform keys.

Division Method:

h(k) = k mod m

• Advantage: fast
• Disadvantage: have to avoid some values of m
• When m = 2^p, h(k) is just the p lowest order bits of k
• Select m to be a prime that is not too close to an exact power of 2.

Multiplication Method:

h(k) = ⌊m(kA mod 1)⌋, where A is a constant, 0 < A < 1

• Advantage: value of m is non-critical
• Disadvantage: slower than division method
• Some suggest optimal A = (**√**5 − 1) / 2 = 0.6180339887 …

Universal Hashing

• A malicious adversary can choose keys that all hash to the same location, leading to Θ(n) average search time.
• Randomly select a hash function from a collection of hash function
• A set of hash functions H is universal if, for each pair of distinct keys, x, y ∈ U, the number of hash functions h ∈ H for which h(x) = h(y) is |H|/m

Universal Hashing Analysis

$\Rightarrow$ The expected time for an unsuccessful search is Θ(1 + α)

Constructing a Universal Hash Function

• Choose prime number (p), is large enough that every key( k) is in range [0, p-1], or p > k
• Let a ∈ {1, 2, …, p-1} and b ∈ {0, 1, …, p-1}
• Define h_{a, b}(k) = ((ak + b) mod p) mod m
• Then H = {h_{a, b} | a ∈ {1, 2, …, p-1} and b ∈ {0, 1, …, p-1}} is a universal set of hash functions.

Open Addressing

• All elements are stored in the hash table itself.
• Each table entry contains either a key or NIL.
• Probe consecutive slots until an empty slot is found when inserting.

Algorithm to Insert

• Probes until empty slot is found

insert(x)
    i = 0
    repeat
        j = h(x, i)
        if H[j] == NIL
            H[j] = x
            return
        else i = i + 1
    until i == m
    error "hash table overflow"

• The probe sequence < h(k, 0), h(k, 1), ..., h(k, m-1) > should be a permutation of < 0, 1, ..., m-1 > to consider every table position.

Algorithm to Search

search(x)
    i = 0
    repeat
        j = h(x, i)
        if H[j] == x
            return j
        i = i + 1
    until H[j] == NIL or i == m
    return NIL

Deletion is difficult (why?)

Three Common Techniques for Computing Probe Sequences

• Linear probing
• Quadratic probing
• Double hashing

Linear Probing

• The probe sequence is given by h(k, i) = (h'(k) + i) mod m for i = 0, 1, …, m-1. This utilizes an ordinary hash function h'(k).
• A drawback is primary clustering, where long occupied slots accumulate, increasing the average search time.

Quadratic Probing

• Here, the probe sequence is given by h(k, i) = (h'(k) + c1i + c2i2) mod m, where c1 and c2 are auxiliary constants with c2 ≠ 0.
• It is more effective than linear probing, but attention must be paid to the values of c1, c2, and m.
• Primary Clustering becomes Secondary Clustering as the algorithm can have identical probe sequences if two keys have the same initial probe point

Double Hashing

• Here, the probe sequence is given by h(k, i) = (h1(k) + ih2(k)) mod m, where h1(k) and h2(k) are auxiliary hash functions.
• Different from single and quadratic in that it produces m² different prove sequences compared to m different proving sequences.

Analysis of Open Addressing

Assumptions:

• Uniform hashing: each key is equally likely to have any one of the m! permutations of < 0, 1, ..., m-1 > as its probe sequence.
• The load factor α = n/m < 1.

Theorem

Given an open-address hash table with load factor α = n/m < 1, the expected number of probes in an unsuccessful search is at most 1/(1 - α).

Proof

• $E[X] \le \frac{1}{1} + \frac{n}{m} + \frac{n}{m} \cdot \frac{n-1}{m-1} + \frac{n}{m} \cdot \frac{n-1}{m-1} \cdot \frac{n-2}{m-2} + \dots$
• $E[X] \le \frac{1}{1} + \frac{n}{m} + \left(\frac{n}{m}\right)^2 + \left(\frac{n}{m}\right)^3 + \dots$
• $E[X] \le \frac{1}{1} + \alpha + \alpha^2 + \alpha^3 + \dots$
• $E[X] \le \sum_{i = 0}^{\infty} \alpha^i = \frac{1}{1 - \alpha}$

Corollary

The expected number of probes in a successful search is at most $\frac{1}{\alpha} \ln \frac{1}{1 - \alpha}$.

Summary

Hashing

• an efficient technique for storing and retrieving data
• Chaining and open addressing
  • two approaches to collision resolution.
    • Chaining simpler, while open addressing can be more space-efficient. Choice of hash function - critical.
      • Universal hashing - way to choose a hash function randomly from a set.
        • Open addressing suffers from clustering, but double hashing can reduce this effect.
      • Load factor α - is a key parameter in the analysis of hashing.

MPSI

• Presented is a guide to linear algebra, designed for students in their first year of scientific preparatory classes (MPSI).
• The objective is to present essential linear algebra concepts concisely, illustrating their relevance through numerous examples and exercises.

Table of Contents:

1. The System $\mathbb{K}$
   • 1.1 Axiomatics of fields
   • 1.2 Examples of fields
   • 1.3 Characteristic of a field
2. Vector Spaces
   • 2. 1 Definitions and examples
   • 2. 2 Vector subspaces
   • 2. 3 Sum of vector subspaces
3. Linear Applications
   • 3. 1 Definitions and properties
   • 3. 2 Image and kernel
   • 3. 3 Rank theorem
4. Vector Spaces of Finite Dimension
   • 4.1 Free families, generating families, bases
   • 4.2 Existence of bases in finite dimension
   • 4.3 Rank of a family of vectors
5. Matrices
   • 5.1 Generalities
   • 5.2 Matrices and linear applications
   • 5.3 Operations on matrices
   • 5.4 Invertible matrices
   • 5.5 Trace of a square matrix
6. Determinants
   • 6.1 Alternating n-linear forms
   • 6.2 Definition and properties of the determinant
   • 6.3 Calculation of determinants
   • 6.4 Applications of determinants

Lecture 16: The Simplex Method

• LP in standard form:

$$\begin{aligned} \text{minimize } & c^T x \
\text{subject to } & Ax = b \
& x \ge 0 \end{aligned}$$

Optimal solution = located at BFS, thus the method traverses the BFSs while improving the function

Recap: Basic Feasible Solutions

• vector x ∈ Rⁿ is a basic solution of Ax = b if: We choose m columns of A, which are linearly independent. Let B be the m×m matrix formed by these columns (the basis); Set all variables not corresponding to columns of B to 0 (non-basic); Solve Bx = b to obtain the values of the basic variables.
• A basic solution x is a basic feasible solution (BFS) if x ≥ 0.

The Simplex Algorithm Process

0. Have 𝐴 matrix, b solution, and minimize statement for linear program
1. Construct the simplex tableau
2. Find a BFS.
3. Check if current BFS is optimal; if yes, end
4. Else, find non-basic variable with negative reduced cost to use as the entering variable
5. Find basic variable that will become non-basic when the entering variable increases and use as the leaving variable
6. Pivot to repeat

The Simplex Tableau

• LP in standard form:

$$\begin{aligned} \text{minimize } & c^T x \
\text{subject to } & Ax = b \
& x \ge 0 \end{aligned}$$

• Construct the simplex tableau using shown matrix

$$\begin{bmatrix} A & b \ c^T & 0 \end{bmatrix}$$

Finding A BFS

• Finding a BFS isn't always easy as it depends on the LP form - If the form displays 𝐴𝑥 ≤ b, with non-negative b, we can add slack variables to show new standard form in, 𝐴𝑥+𝑠=𝑏 , x,s≥0; if so set x=0 and s=b to find BFS - If no, must use two-phase simplex example - Phase 1: BFS finding using second LP algorithm Phase 2: Simplex method used to solve

Checking of Optimality

• The current BFS is optimal if all reduced costs are non-negative.
• The reduced cost of a non-basic variable is the amount by which the objective function will increase if we increase the value of the non-basic variable by 1.

The Ratio Test

• When the value of the entering variable is increased, some basic variables will decrease.
• Determine the basic variable that will reach 0 first
• Perform as follows -For each basic variable, calculate the ratio where is right hand side of constraint, is the coefficient of the entering valuable and constraint -Smallest Ratio to pick as the leaving variable

Unboundeness

• If all ratios are negative/ infinite → unbounded LP -The value of entering variable can increase indefinitely without violating constraints -Objective function can become arbitrarily low

Degeneracy

If two ore more basic variables hold similar ratios → degenerate LP The simplex method repeats cycles without getting anywhere, but may try Bland's Rule: Pick Entering And Leaving variables with the smallest indices to prevent cycling

Summary

• powerful algorithm for solving LPs
• traverses the BFSs of the feasible region, while improving the function

Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

How wings generate lift

• The wing is shaped so that the air flows faster over the top of the wing than under the wing.
  • Faster airflow on top of the wing creates less pressure which pushes the wind up,
  • slower airflow on the bottom of the wing creates more pressure

Bernoulli's Equation

$P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$

• regardes conservation of energy principle for flowing fluids.
  • Pressure energy
• Kinetic energy
  • Potential energy

Variables

• P = Pressure
• ρ = Density
• v = Velocity
• g = Gravity
• h = Height

Guía para la detección en fase temprana del cáncer infantil

La detección es importante y puede incluir:

• Estar atento a los signos y síntomas y/o realizarse exámenes médicos periódicos.

¿Por qué es importante la detección temprana?

• Conlleva a mejor resultado, menos tratamientos y menos efectos secundarios

¿Cuáles son los signos y síntomas del cáncer infantil?

• Fatiga, pérdida de peso, fiebre, dolor, bulto o hinchazón, moretones o sangrado, dolores de cabeza cambios en la visión, ganglios linfáticos inflamados y erupción cutánea,

¿Qué hacer si nota signos o síntomas de cáncer infantil?

• Consultar a un médico Inmediatamente,

¿Cómo se diagnostica el cáncer infantil?

• Examen físico, revisión de historial clínico pruebas, análisis de sangre, estudios por imágenes como radiografías tomografías computarizadas o resonancias magnéticas para observar el interior del cuerpo y biopsia.

¿Cómo se trata el cáncer infantil?

• Quimioterapia, radioterapia , cirugía , trasplante de células madre, terapia dirigida e Inmunoterapia

Funciones Vectoriales de Variable Real

• Funciones vectoriales map real numbers to vectors in Rⁿ

Funciones Vectoriales

• A function whose domain is a subset of real numbers and whose range is a set of vectors.

$\qquad \vec{r}: \mathbb{R} \rightarrow \mathbb{R}^{n}$ $\qquad t \mapsto \vec{r}(t)=\left(f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right)$ where the component functions are real functions of the variable t.

LImits

Take the limit of a vector-valued function by taking the limit of each of its components.

$\qquad \lim _{t \rightarrow a} \vec{r}(t)=\left\langle\lim {t \rightarrow a} f{1}(t), \lim {t \rightarrow a} f{2}(t), \ldots, \lim {t \rightarrow a} f{n}(t)\right\rangle$ provided the limits of the component functions exist.

Continuity

• A vector valued function $\vec{r}$ is continuous if $\qquad \lim _{t \rightarrow a} \vec{r}(t)=\vec{r}(a)$

Derivatives

$\qquad \vec{r}^{\prime}(t)=\left\langle f_{1}^{\prime}(t), f_{2}^{\prime}(t), \ldots, f_{n}^{\prime}(t)\right\rangle$

Chapter 4: Data Types

• Every value is assigned to a "data type".

Data Types

• Describe the characteristics of a value,.
• Indicate operations and storage.
• Example Data Types: Integers (int), floating point numbers (float), strings (str)

Why Data Types Matter

• Knowing what it is to prevent errors

Integers

Definition

• Whole numbers with no fractional part Examples: 0, 10, -5, 1000000, -34567

Common Operations

+Add

• Subtraction * Multiplication /Division //Floor Division %Modulo **Exponentiation

Division Result

Division (/) always returns a float.

• Integer size is limited by available memory

4.3 Floating-Point Numbers

Numbers with Decimal Point Examples: 3. 14, -2.5, 0.0, 1.0, -123.456 Represented in base 2 (binary) with limited precision. Limitation precision examples 0. 1 + 0. 2 = 0. 30000000000000004 Operations same Used for Real Numbers/ Fractional Points

4.4 Complex Numbers

Definition Complex numbers have a real and imaginary part. A complex number is presented as x + yj Where x is the real amount y is the imaginary amount

Function

z.real: Returns the real part of the complex number z z.imag: Returns the imaginary part of the complex number z Use to represent scientific models, physics, math Examples Addition (+): Adds corresponding real and imaginary parts. Subtraction (-): Subtracts corresponding real and imaginary parts. Multiplication (*): Multiplies complex numbers using distributive property. Division (/): Divides complex numbers.

4.5 Strings

Are sequences of characters in "" or '' Ex: "Hello" Multiline strings are defined by """ Ex: """Hello Multiple lines"""

Operations 1.Concatenation 2. Repetition 3. Indexing 4. String Slcing

4.6 Boolean

Represent truth values → True or False. Commonly used in conditional statements. Operators: and, or, not. Non Boolean values can be applied.

Type Conversion

Can convert implicitly or explicitly Examples

x = 5 # int
y = 2.0 # float
z = x + y
print(z) # 7.0 float

Functions
int - converts to integer
float - converts to float value
str - converts to astring value
bool - converts to boolean value from a value
complex - convers to complex value

Determining Value

Type() , will output the value type
Ex:
x = 5
print(type(x))
#output : int

Isinstance(), will tell you if the values are the same or not
Ex:
x = 5
print(isinstance(x , int ))
Output : True

Number Operations

• Order of operations still apply

String Operations

• + ↔ Joins together “hello” + “world” = helloworld
• * ↔ Repeats itself “Hello” * 3 = HelloHelloHello
• [] ↔ Outputs an individual character location A = [“hello’] Ex; A [ 0 ] == h
• Slice ↔ A slice of the string example [Start:end:step] A[0:3] ex outputs “hel” A[1:] ex outputs “ello” A[:4] “hell” A[::2] “hlo”

Function

• Len () tells you how many data points are in astring
• Lower()/Upper() Changes the words to respective upper and lower
• Strip() removes "" space before or after string
• Spit() divides into substrings based on the “ “
• Replace() → replaces () whats inside Ex replace (“ ‘, “ ”)
• Find / Count tells you either were is something located or how many are there . "Hello”.find (“e”) = one. “Hello”. Count (‘L’) = . 2
• Startswith() / Endwith checks is start or end Ex: String startswith or = endswith Check is start or end equals to the value. isdigit, alpha and isnumeric, check is the variable has either numbers letters or words

String Formatting .format() Method

Replaces placeholder’s with values. Ex “(0) is cool “.format, (words) = (Word) is cool

F{} is format strings

Print a function with (the values) inside “ “ ex. f(“ The area is (width)

• Common specifiers
  • d == will set data as integer
• f == will set data an float
• “.nf” ↔ to specify data n point of decimal
• b ↔ binary values
• X ↔ Hexadecimal
• O ↔ Octal
• % ↔ Percentage