Spectral Theorem, Projections & Operators

Spectral Theorem

• States that a self-adjoint operator on a finite-dimensional inner product space has an orthonormal basis of eigenvectors.

Projections

• Defined as a linear transformation where applying it twice is the same as applying it once ($P^2 = P$).

Orthogonal Projection

• For a subspace $W$ of $V$, the orthogonal projection $P_W$ maps a vector $v$ in $V$ to a vector in $W$ such that $v - P_W(v)$ is in $W^\perp$.

Key Theorems Related to Self-Adjoint Operators

• If $A$ is a self-adjoint operator with distinct eigenvalues $\lambda_1, \ldots, \lambda_m$, eigenspaces $W_i = \ker(A - \lambda_i I)$, and orthogonal projections $P_i$ onto $W_i$:
  • The vector space $V$ can be decomposed into the direct sum of eigenspaces: $V = W_1 \oplus \cdots \oplus W_m$.
  • Different orthogonal projections are mutually exclusive: $P_i P_j = 0$ for $i \neq j$.
  • The operator $A$ can be expressed as a linear combination of projections weighted by their eigenvalues: $A = \lambda_1 P_1 + \cdots + \lambda_m P_m$.
  • The identity operator $I$ can be expressed as the sum of all orthogonal projections: $I = P_1 + \cdots + P_m$.

Positive Operators

• A self-adjoint operator $A$ is positive if for all vectors $v$ in $V$, $\langle A v, v\rangle \geq 0$.
• For any operator $B$, the operator $A = B^* B$ is positive since $\langle B^* B v, v\rangle = |B v|^2 \geq 0$.
• A self-adjoint operator is positive if and only if all of its eigenvalues are non-negative.
• If $A$ is positive, there is a positive operator $B$ such that $B^2 = A$, where B the square root of A, is equal to $\sqrt{A}$.
• If $A$ has distinct eigenvalues $\lambda_1, \ldots, \lambda_m$ and orthogonal projections $P_1, \ldots, P_m$, then $B = \sqrt{\lambda_1} P_1 + \cdots + \sqrt{\lambda_m} P_m$ is a square root of $A$.

Planck's Constant

• The quantum of action in quantum mechanics, symbolized as $h$.
• Frequently, the reduced Planck's constant, $\hbar$, is used, and is calculated as $\frac{h}{2\pi}$.
• Possesses dimensions of physical action (energy × time, or momentum × distance).
• It has a value of $6.62607015 \times 10^{-34}$ J⋅s in SI units.
• Establishes a proportional relationship between the energy ($E$) of a photon and its frequency ($v$): $E = h\nu$.
• Can be related to wavelength $\lambda$ by $E = h\frac{c}{\lambda}$, where $c$ is the speed of light.
• Defines a proportional relationship between the de Broglie wavelength $\lambda$ and momentum $p$: $\lambda = \frac{h}{p}$.

Central Dogmas of Molecular Biology

• Describes the flow of genetic information within biological systems.

Original Dogma (1958)

• DNA -> RNA -> Protein

Current Dogma

• Includes processes like replication, transcription, and translation.
  • Replication: DNA -> DNA
  • Transcription: DNA -> RNA
  • Translation: RNA -> Protein

Exceptions to the Central Dogma

• Reverse transcription: RNA -> DNA (occurs in retroviruses)
• Direct replication: RNA -> RNA (occurs in RNA viruses)
• Prions: Protein -> Protein (conformational change)

DNA Replication

• Process of copying a DNA molecule.
  • Enzymes: DNA polymerases.
  • Process: Semiconservative (one original strand and one new strand).
  • Direction: 5' -> 3' (the new strand is synthesized in this direction).
  • Origin of replication: Specific regions of DNA where replication begins.
  • Replication fork: Y-shaped structure where replication is occurring.
  • Leading strand: Synthesized continuously.
  • Lagging strand: Synthesized in fragments (Okazaki fragments).
  • Primers: Short sequences of RNA needed to initiate synthesis.
  • Ligase: An enzyme that joins Okazaki fragments.

DNA Polymerase

• Main enzyme in replication.
  • Function: Synthesizes new DNA strands from a template strand.
  • Error correction: Some DNA polymerases can correct errors.

DNA Transcription

• Synthesis of RNA from a DNA strand, and done by RNA polymerases.
  • Process: RNA polymerase binds to DNA and synthesizes a complementary RNA molecule.
  • Direction: 5' -> 3' (the new RNA strand is synthesized in this direction).
  • Promoter: Region of DNA where RNA polymerase binds to begin transcription.
  • Terminator: Signal in DNA that indicates the end of transcription.
  • Types of RNA:
    • Messenger RNA (mRNA): Contains the information for protein synthesis.
    • Transfer RNA (tRNA): Transports amino acids to ribosomes.
    • Ribosomal RNA (rRNA): Forms part of ribosomes.

RNA Polymerase

• Main enzyme in transcription.
  • Function: Synthesizes RNA from a DNA template strand.
  • No primer needed: Unlike DNA polymerase, it does not need a primer to begin synthesis.

RNA Translation

• Synthesis of proteins from the information contained in mRNA.
  • Location: Ribosomes.
  • Genetic code: Set of rules that define how a sequence of nucleotides is translated into a sequence of amino acids.
  • Codon: Sequence of three nucleotides in mRNA that codes for a specific amino acid.
  • Anticodon: Sequence of three nucleotides in tRNA that binds to the codon in mRNA.
  • Stages:
    • Initiation: The ribosome binds to the mRNA and the initiator tRNA.
    • Elongation: The ribosome moves along the mRNA, adding amino acids to the polypeptide chain.
    • Termination: The ribosome encounters a stop codon, and protein synthesis stops.

Ribosomes

• Cellular structures where protein synthesis occurs.
  • Subunits: Two subunits (large and small) that join during translation.
  • Ribosomal RNA (rRNA): RNA that catalyzes the formation of peptide bonds between amino acids.

The Genetic Code Table

• Includes information on the codons, amino acids they code for, and the start and stop signals for translation.

Fourier Transform Usage Examples

• To solve partial differential equations on the whole line.

Wave Equation Example

• Fourier transform can be applied to solve the wave equation: $u_{tt} = c^2 u_{xx}$, with initial conditions $u(x, 0) = f(x)$ and $u_t(x, 0) = g(x)$.

Heat Equation Example

• Fourier transform can also be employed to solve the heat equation: $u_t = k u_{xx}$, with initial condition $u(x, 0) = f(x)$.

Definition of Fourier Transform

• The Fourier Transform of a function $f(x)$ is $\hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} dx$
• The Inverse Fourier Transform of a function $\hat{f}(k)$ is $f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(k) e^{ikx} dk$

Properties of Fourier Transform

• Linearity: $\widehat{af + bg} = a \hat{f} + b \hat{g}$
• Scaling: $\widehat{f(ax)} = \frac{1}{|a|} \hat{f}(\frac{k}{a})$
• Translation: $\widehat{f(x - x_0)} = e^{-ikx_0} \hat{f}(k)$
• Differentiation: $\widehat{f'(x)} = ik \hat{f}(k)$
• Convolution: $\widehat{f * g} = \hat{f} \cdot \hat{g}$, where $f * g (x) = \int_{-\infty}^{\infty} f(y) g(x - y) dy$

Static Electricity Basics

• All matter consists of atoms.
• Atoms are composed of positively charged protons, neutrally charged neutrons, and negatively charged electrons.
• Opposite charges exhibit attraction, while like charges exhibit repulsion.

Electric Charge

• A physical property (q) that triggers an object to experience a force when it is close to other charged objects.
• Unit is the Coulomb (C)
• The electric charge is connected to the quantity of electrons and protons present in an object.
• The charge of a single electron is $-1.602 \times 10^{-19} \text{C}$

Charging Objects

• Friction: Electrons can move between objects when they are rubbed together, leading one to become positive and the other negative.
• Conduction: Requires direct contact between a charged object and a neutral or charged object.
• Induction: No contact needed; it entails the redistribution of charge within an object situated near a charged object.

Conductors vs Insulators

• Conductors: Materials allowing free electron movement, and includes metals (copper, aluminum, gold, silver), electrolytes (salt water), and plasma (ionized gas).
• Insulators: Materials resisting electron movements, for example: rubber, glass, plastic, wood.

Electric Force

• Defined as attraction or repulsion force between two charged objects.

Coulomb's Law

• Describes the magnitude of the electric force ($F_e$) between two charged objects as follows:
  • Directly proportional to the magnitude of the charges ($q_1, q_2$).
  • Inversely proportional to the square of the distance between them ($r^2$).
  • With the equation: $F_e = k \frac{q_1 q_2}{r^2}$
    • $F_e$: electric force (N).
    • $q_1$ and $q_2$: electric charges (C).
    • $r$: distance between the charges.
    • $k$: Coulomb's constant ($8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2$).

Difference between electrical force and gravity.

• Electric force acts only on charged objects and causes attraction or repulsion.
• Whereas gravity acts on all objects and is always is attractive.

Physics 1 Equations: Mechanics

• Kinematic equations for constant acceleration.
• Formulas for net force and its components.
• Friction force formula.
• Momentum, impulse, and kinetic energy equations.
• Power formulas.
• Gravitational potential energy.
• Center of mass formula.
• Torque and moment of inertia equations.
• Angular velocity and acceleration equations.
• Angular momentum and rotational kinetic energy formulas.
• Newton's law of gravitation and gravitational potential energy.

Physics 1 Equations: Waves and Optics

• Wave speed, frequency, and wavelength relationship.
• Equations for wave speed in strings and bulk materials.
• Wavelength and frequency for standing waves in strings and open/closed pipes.

Physics 1 Equations: Electricity and Magnetism

• Coulomb's law for electric force.
• Electric field strength.
• Electric potential energy and voltage formulas.
• Electric field in terms of potential.
• Capacitance and energy stored in a capacitor.
• Current, resistance, and resistivity equations.
• Ohm's law.
• Power in a circuit.
• Equivalent resistance and capacitance for series and parallel circuits.
• Magnetic force on a moving charge.
• Magnetic field due to a long straight wire and solenoid.
• Magnetic flux and Faraday's law of induction.

Physics 1 Equations: Thermodynamics

• First law of thermodynamics.
• Efficiency of a heat engine.
• Carnot efficiency.
• Heat transfer formula.

Physics 1 Equations: Modern Physics

• Energy of a photon.
• De Broglie wavelength.
• Mass-energy equivalence.

Physics 1 Equations: Fluids

• Density formula.
• Pressure formula.
• Pressure at a depth in a fluid.
• Continuity equation.
• Bernoulli's equation.

Physics 1 Equations: Constants and Conversions

• Values for fundamental constants like proton mass, electron charge, gravitational constant, speed of light, etc.
• Conversion factors for atomic mass unit.

Physics 1 Equations: Unit Prefixes

• Table of prefixes from giga to pico.

Physics 1 Equations: Trigonometry

• Trigonometric relationships for right triangles including Sine, Cosine, and Tangent.
• Pythagorean theorem.

MDI Basics: Overview

• Collection of questions and diagrams to test understanding of Model Driven Interoperability (MDI).

MDI Basics: Terminology

• Matching terms to definitions:
  • Terms: Model, Metamodel, DSL, Transformation, Model validation, Model verification.
  • Definitions:
    • A description or representation of a system, component, or phenomenon.
    • A model that defines the language for creating models.
    • A standard way of modeling a particular domain.
    • The process of automatically transforming a model into code or another model.
    • Checking a model against the rules defined in its metamodel (Well-formedness).
    • Checking that a model meets the requirements of the system it represents (Correctness).

MDI Basics: Diagram Identification

• Three diagrams are identified:
  • Model-to-Code Transformation
  • Model-to-Model Transformation (Platform Specific Model to Platform Independent Model)
  • Model Driven Development (MDD)

MDI Basics: Short Answer

• Common MDI benefits are; increased productivity, improved quality, better interoperability, and reduced development costs.
• Examples of challenges applying MDI are; learning curve with new modeling languages and tools, the complexity of defining and maintaining metamodels, and the risk of vendor lock-in.
• A metamodel defines the language for creating models, while a model is a specific instance of a metamodel that describes a particular system, component, or phenomenon.

MDI Basics: UML Modeling

• The simple UML class diagram for a library system including classes for Book, Author, and Library shows the relationships between the classes, including attributes and methods.

MDI Basics: Model Transformation

• Transformation Steps:
  • Define a metamodel for UML and the target database schema.
  • Create a transformation rule.
  • Apply the transformation.
  • Validate the generated schema.

PETIC Overview

• Consists of establishing the strategies of the DTI that allow to support the fulfillment of the institutional objectives through IT projects, optimization of the IT services and the technologic innovation.
• Covers strategic lines such as; technologically based infrastructure, systems development, IT innovation, information security and TI Service management.

PETIC Methodology

• Based on the Federal Government methodology as purposed by the Digital Federal Government Unit, with a base on the Deming Cycle (Plan, Do, Verify and Act).

IT Strategic Design Model

• Based on the cycle of Deming, with four steps:
  • Plan: Definition of DTI's objectives and the TI projects oriented to such fulfilment
  • Do: Execution of the designed TI projects on the planning stage
  • Verify: Evaluation of results related to the TI projects being compared to the designed objectives.
  • Act: Apply the correct measuring process on the fulfilment of the TI projects to achieve the designed objectives.

PETIC Justification

• Aligns the DTI strategies with the institutional strategies.
• Optimizes TI resources.
• Improves the quality of TI services.
• Promotes technological innovation.

DTI's Main Challenges

• Aligning with institutional strategies.
• Optimizing resources.
• Improving service quality.
• Promoting innovation.

DTI's Main Opportunities

• Leveraging new technologies.
• Collaboration with other institutions.
• Becoming a benchmark in ICT.