Plasmonic Waveguides

The Cournot Oligopoly model describes a market structure in which a small number of firms compete by choosing quantities to produce, rather than prices. Each firm decides how much to produce with the assumption that the output levels of the other firms remain constant. This interdependence leads to a Nash Equilibrium, where no firm can benefit by changing its output level while the others keep theirs unchanged. In this setting, the total quantity produced in the market determines the market price, typically resulting in a price that is above marginal costs, allowing firms to earn positive economic profits. The model is named after the French economist Antoine Augustin Cournot, and it highlights the balance between competition and cooperation among firms in an oligopolistic market.

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.

A brushless motor is an electric motor that operates without the use of brushes, which are commonly found in traditional brushed motors. Instead, it uses electronic controllers to switch the direction of current in the motor windings, allowing for efficient rotation of the rotor. The main components of a brushless motor include the stator (the stationary part), the rotor (the rotating part), and the electronic control unit.

One of the primary advantages of brushless motors is their higher efficiency and longer lifespan compared to brushed motors, as they experience less wear and tear due to the absence of brushes. Additionally, they provide higher torque-to-weight ratios, making them ideal for a variety of applications, including drones, electric vehicles, and industrial machinery. The typical operation of a brushless motor can be described by the relationship between voltage ($V$), current ($I$), and resistance ($R$) in Ohm's law, represented as:

This relationship is essential for understanding how power is delivered and managed in brushless motor systems.

A Minsky Moment refers to a sudden and dramatic collapse of asset prices following a prolonged period of speculation and increasing debt levels, named after the economist Hyman Minsky. According to Minsky's financial instability hypothesis, economies go through cycles of boom and bust driven by investors' changing risk appetites. During the boom phase, optimism leads to increased borrowing and speculative investments, creating an illusion of stability. However, as debts accumulate and asset prices become overvalued, even a minor negative event can trigger a panic, leading to a rapid decline in asset prices and a financial crisis. This phenomenon highlights the inherent instability of financial markets and the tendency for economic systems to oscillate between periods of euphoria and despair.

PWM (Pulse Width Modulation) is a technique used to control the amount of power delivered to electrical devices, particularly in applications involving motors, lights, and heating elements. It works by varying the duty cycle of a square wave signal, which is defined as the percentage of one period in which a signal is active. For instance, a 50% duty cycle means the signal is on for half the time and off for the other half, effectively providing half the power. This can be mathematically represented as:

By adjusting the duty cycle, PWM can control the speed of a motor or the brightness of a light with great precision and efficiency. Additionally, PWM is beneficial because it minimizes energy loss compared to linear control methods, making it a popular choice in modern electronic applications.

The Rayleigh Criterion is a fundamental principle in optics that defines the limit of resolution for optical systems, such as telescopes and microscopes. It states that two point sources of light are considered to be just resolvable when the central maximum of the diffraction pattern of one source coincides with the first minimum of the diffraction pattern of the other. Mathematically, this can be expressed as:

where $\theta$ is the minimum angular separation between two point sources, $\lambda$ is the wavelength of light, and $D$ is the diameter of the aperture (lens or mirror). The factor 1.22 arises from the circular aperture's diffraction pattern. This criterion is critical in various applications, including astronomy, where resolving distant celestial objects is essential, and in microscopy, where it determines the clarity of the observed specimens. Understanding the Rayleigh Criterion helps in designing optical instruments to achieve the desired resolution.