Gram-Schmidt Orthogonalization

Financial derivatives pricing refers to the process of determining the fair value of financial instruments whose value is derived from the performance of underlying assets, such as stocks, bonds, or commodities. The pricing of these derivatives, including options, futures, and swaps, is often based on models that account for various factors, such as the time to expiration, volatility of the underlying asset, and interest rates. One widely used method is the Black-Scholes model, which provides a mathematical framework for pricing European options. The formula is given by:

where $C$ is the call option price, $S_0$ is the current stock price, $X$ is the strike price, $r$ is the risk-free interest rate, $T$ is the time until expiration, and $N(d)$ is the cumulative distribution function of the standard normal distribution. Understanding these pricing models is crucial for traders and risk managers as they help in making informed decisions and managing financial risk effectively.

The Riemann Zeta function is a complex function denoted as $\zeta(s)$, where $s$ is a complex number. It is defined for $s > 1$ by the infinite series:

This function converges to a finite value in that domain. The significance of the Riemann Zeta function extends beyond pure mathematics; it is closely linked to the distribution of prime numbers through the Riemann Hypothesis, which posits that all non-trivial zeros of this function lie on the critical line where the real part of $s$ is $\frac{1}{2}$. Additionally, the Zeta function can be analytically continued to other values of $s$ (except for $s = 1$, where it has a simple pole), making it a pivotal tool in number theory and complex analysis. Its applications reach into quantum physics, statistical mechanics, and even in areas of cryptography.

RSA encryption is a widely used asymmetric cryptographic algorithm that secures data transmission. It relies on the mathematical properties of prime numbers and modular arithmetic. The process involves generating a pair of keys: a public key for encryption and a private key for decryption. To encrypt a message $m$, the sender uses the recipient's public key $(e, n)$ to compute the ciphertext $c$ using the formula:

where $n$ is the product of two large prime numbers $p$ and $q$. The recipient then uses their private key $(d, n)$ to decrypt the ciphertext, recovering the original message $m$ with the formula:

The security of RSA is based on the difficulty of factoring the large number $n$ back into its prime components, making unauthorized decryption practically infeasible.

Dynamic Programming (DP) is an algorithmic paradigm used to solve complex problems by breaking them down into simpler subproblems. It is particularly effective for optimization problems and is characterized by its use of overlapping subproblems and optimal substructure. In DP, each subproblem is solved only once, and its solution is stored, usually in a table, to avoid redundant calculations. This approach significantly reduces the time complexity from exponential to polynomial in many cases. Common applications of dynamic programming include problems like the Fibonacci sequence, shortest path algorithms, and knapsack problems. By employing techniques such as memoization or tabulation, DP ensures efficient computation and resource management.

Van der Waals heterostructures are engineered materials composed of two or more different two-dimensional (2D) materials stacked together, relying on van der Waals forces for adhesion rather than covalent bonds. These heterostructures enable the combination of distinct electronic, optical, and mechanical properties, allowing for novel functionalities that cannot be achieved with individual materials. For instance, by stacking transition metal dichalcogenides (TMDs) with graphene, researchers can create devices with tunable band gaps and enhanced carrier mobility. The alignment of the layers can be precisely controlled, leading to the emergence of phenomena such as interlayer excitons and superconductivity. The versatility of van der Waals heterostructures makes them promising candidates for applications in next-generation electronics, photonics, and quantum computing.

The Kolmogorov Axioms form the foundational framework for probability theory, established by the Russian mathematician Andrey Kolmogorov in the 1930s. These axioms define a probability space $(S, \mathcal{F}, P)$, where $S$ is the sample space, $\mathcal{F}$ is a σ-algebra of events, and $P$ is the probability measure. The three main axioms are:

1. Non-negativity: For any event $A \in \mathcal{F}$, the probability $P(A)$ is always non-negative:

$P(A) \geq 0$

2. Normalization: The probability of the entire sample space equals 1:

$P(S) = 1$

3. Countable Additivity: For any countable collection of mutually exclusive events $A_1, A_2, \ldots \in \mathcal{F}$, the probability of their union is equal to the sum of their probabilities:

$P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i)$

These axioms provide the basis for further developments in probability theory and allow for rigorous manipulation of probabilities