Phase-Change Memory

The Noether Charge is a fundamental concept in theoretical physics that arises from Noether's theorem, which links symmetries and conservation laws. Specifically, for every continuous symmetry of the action of a physical system, there is a corresponding conserved quantity. This conserved quantity is referred to as the Noether Charge. For instance, if a system exhibits time translation symmetry, the associated Noether Charge is the energy of the system, which remains constant over time. Mathematically, if a symmetry transformation can be expressed as a change in the fields of the system, the Noether Charge $Q$ can be computed from the Lagrangian density $\mathcal{L}$ using the formula:

where $\phi$ represents the fields of the system and $\delta \phi$ denotes the variation due to the symmetry transformation. The importance of Noether Charges lies in their role in understanding the conservation laws that govern physical systems, thereby providing profound insights into the nature of fundamental interactions.

Bayes' Theorem is a fundamental concept in probability theory that describes how to update the probability of a hypothesis based on new evidence. It mathematically expresses the idea of conditional probability, showing how the probability $P(H | E)$ of a hypothesis $H$ given an event $E$ can be calculated using the formula:

In this equation:

• $P(H | E)$ is the posterior probability, the updated probability of the hypothesis after considering the evidence.
• $P(E | H)$ is the likelihood, the probability of observing the evidence given that the hypothesis is true.
• $P(H)$ is the prior probability, the initial probability of the hypothesis before considering the evidence.
• $P(E)$ is the marginal likelihood, the total probability of the evidence under all possible hypotheses.

Bayes' Theorem is widely used in various fields such as statistics, machine learning, and medical diagnosis, allowing for a rigorous method to refine predictions as new data becomes available.

The Schrödinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a key result that encapsulates the principles of wave-particle duality and the probabilistic nature of quantum systems. The equation can be expressed in two main forms: the time-dependent Schrödinger equation and the time-independent Schrödinger equation.

The time-dependent form is given by:

where $\Psi(x, t)$ is the wave function of the system, $i$ is the imaginary unit, $\hbar$ is the reduced Planck's constant, and $\hat{H}$ is the Hamiltonian operator representing the total energy of the system. The wave function $\Psi$ provides all the information about the system, including the probabilities of finding a particle in various positions and states. The time-independent form is often used for systems in a stationary state and is expressed as:

where $E$ represents the energy eigenvalues. Overall, the Schrödinger Equation is crucial for predicting the behavior of quantum systems and has profound implications in fields ranging from chemistry to quantum computing.

The PageRank algorithm, developed by Larry Page and Sergey Brin, assigns a ranking to web pages based on their importance, which is determined by the links between them. The convergence of the PageRank vector $\mathbf{p}$ is proven through the properties of Markov chains and the Perron-Frobenius theorem. Specifically, the PageRank matrix $M$, representing the probabilities of transitioning from one page to another, is a stochastic matrix, meaning that its columns sum to one.

To demonstrate convergence, we show that as the number of iterations $n$ approaches infinity, the PageRank vector $\mathbf{p}^{(n)}$ approaches a unique stationary distribution $\mathbf{p}$. This is expressed mathematically as:

where $M$ is the transition matrix. The proof hinges on the fact that $M$ is irreducible and aperiodic, ensuring that any initial distribution converges to the same stationary distribution regardless of the starting point, thus confirming the robustness of the PageRank algorithm in ranking web pages.

The Brayton Cycle, also known as the gas turbine cycle, is a thermodynamic cycle that describes the operation of a gas turbine engine. It consists of four main processes: adiabatic compression, constant-pressure heat addition, adiabatic expansion, and constant-pressure heat rejection. In the first process, air is compressed, increasing its pressure and temperature. The compressed air then undergoes heat addition at constant pressure, usually through combustion with fuel, resulting in a high-energy exhaust gas. This gas expands through a turbine, performing work and generating power, before being cooled at constant pressure, completing the cycle. Mathematically, the efficiency of the Brayton Cycle can be expressed as:

where $T_1$ is the inlet temperature and $T_2$ is the maximum temperature in the cycle. This cycle is widely used in jet engines and power generation due to its high efficiency and power-to-weight ratio.

Jensen’s Alpha is a performance metric used to evaluate the excess return of an investment portfolio compared to the expected return predicted by the Capital Asset Pricing Model (CAPM). It is calculated using the formula:

where:

• $\alpha$ is Jensen's Alpha,
• $R_p$ is the actual return of the portfolio,
• $R_f$ is the risk-free rate,
• $\beta$ is the portfolio's beta (a measure of its volatility relative to the market),
• $R_m$ is the expected return of the market.

A positive Jensen’s Alpha indicates that the portfolio has outperformed its expected return, suggesting that the manager has added value beyond what would be expected based on the portfolio's risk. Conversely, a negative alpha implies underperformance. Thus, Jensen’s Alpha is a crucial tool for investors seeking to assess the skill of portfolio managers and the effectiveness of investment strategies.