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Stochastic Gradient Descent Proofs

Stochastic Gradient Descent (SGD) is an optimization algorithm used to minimize an objective function, typically in the context of machine learning. The fundamental idea behind SGD is to update the model parameters iteratively based on a randomly selected subset of the training data, rather than the entire dataset. This leads to faster convergence and allows the model to escape local minima more effectively.

Mathematically, at each iteration ttt, the parameters θ\thetaθ are updated as follows:

θt+1=θt−η∇L(θt;x(i),y(i))\theta_{t+1} = \theta_t - \eta \nabla L(\theta_t; x^{(i)}, y^{(i)})θt+1​=θt​−η∇L(θt​;x(i),y(i))

where η\etaη is the learning rate, and (x(i),y(i))(x^{(i)}, y^{(i)})(x(i),y(i)) is a randomly chosen training example. Proofs of convergence for SGD typically involve demonstrating that, under certain conditions (like a diminishing learning rate), the expected value of the loss function will converge to a minimum as the number of iterations approaches infinity. This is crucial for ensuring that the algorithm is both efficient and effective in practice.

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Cantor Function

The Cantor function, also known as the Cantor staircase function, is a classic example of a function that is continuous everywhere but not absolutely continuous. It is defined on the interval [0,1][0, 1][0,1] and maps to [0,1][0, 1][0,1]. The function is constructed using the Cantor set, which is created by repeatedly removing the middle third of intervals.

The Cantor function is defined piecewise and has the following properties:

  • It is non-decreasing.
  • It is constant on the intervals removed during the construction of the Cantor set.
  • It takes the value 0 at x=0x = 0x=0 and approaches 1 at x=1x = 1x=1.

Mathematically, if you let C(x)C(x)C(x) denote the Cantor function, it has the property that it increases on intervals of the Cantor set and remains flat on the intervals that have been removed. The Cantor function is notable for being an example of a continuous function that is not absolutely continuous, as it has a derivative of 0 almost everywhere, yet it increases from 0 to 1.

Hotelling’S Law

Hotelling's Law is a principle in economics that explains how competing firms tend to locate themselves in close proximity to each other in a given market. This phenomenon occurs because businesses aim to maximize their market share by positioning themselves where they can attract the largest number of customers. For example, if two ice cream vendors set up their stalls at opposite ends of a beach, they would each capture a portion of the customers. However, if one vendor moves closer to the other, they can capture more customers, leading the other vendor to follow suit. This results in both vendors clustering together at a central location, minimizing the distance customers must travel, which can be expressed mathematically as:

Distance=1n∑i=1ndi\text{Distance} = \frac{1}{n} \sum_{i=1}^{n} d_iDistance=n1​i=1∑n​di​

where did_idi​ represents the distance each customer travels to the vendors. In essence, Hotelling's Law illustrates the balance between competition and consumer convenience, highlighting how spatial competition can lead to a concentration of firms in certain areas.

Floyd-Warshall

The Floyd-Warshall algorithm is a dynamic programming technique used to find the shortest paths between all pairs of vertices in a weighted graph. It works on both directed and undirected graphs and can handle graphs with negative weights, but it does not work with graphs that contain negative cycles. The algorithm iteratively updates a distance matrix DDD, where D[i][j]D[i][j]D[i][j] represents the shortest distance from vertex iii to vertex jjj. The core of the algorithm is encapsulated in the following formula:

D[i][j]=min⁡(D[i][j],D[i][k]+D[k][j])D[i][j] = \min(D[i][j], D[i][k] + D[k][j])D[i][j]=min(D[i][j],D[i][k]+D[k][j])

for all vertices kkk. This process is repeated for each vertex kkk as an intermediate point, ultimately ensuring that the shortest paths between all pairs of vertices are found. The time complexity of the Floyd-Warshall algorithm is O(V3)O(V^3)O(V3), where VVV is the number of vertices in the graph, making it less efficient for very large graphs compared to other shortest-path algorithms.

Vco Modulation

VCO modulation, or Voltage-Controlled Oscillator modulation, is a technique used in various electronic circuits to generate oscillating signals whose frequency can be varied based on an input voltage. The core principle revolves around the VCO, which produces an output frequency that is directly proportional to its input voltage. This allows for precise control over the frequency of the generated signal, making it ideal for applications like phase-locked loops, frequency modulation, and signal synthesis.

In mathematical terms, the relationship can be expressed as:

fout=k⋅Vin+f0f_{\text{out}} = k \cdot V_{\text{in}} + f_0fout​=k⋅Vin​+f0​

where foutf_{\text{out}}fout​ is the output frequency, kkk is a constant that defines the sensitivity of the VCO, VinV_{\text{in}}Vin​ is the input voltage, and f0f_0f0​ is the base frequency of the oscillator.

VCO modulation is crucial in communication systems, enabling the encoding of information onto carrier waves through frequency variations, thus facilitating effective data transmission.

Pell Equation

The Pell Equation is a classic equation in number theory, expressed in the form:

x2−Dy2=1x^2 - Dy^2 = 1x2−Dy2=1

where DDD is a non-square positive integer, and xxx and yyy are integers. The equation seeks integer solutions, meaning pairs (x,y)(x, y)(x,y) that satisfy this relationship. The Pell Equation is notable for its deep connections to various areas of mathematics, including continued fractions and the theory of quadratic fields. One of the most famous solutions arises from the fundamental solution, which can often be found using methods like the continued fraction expansion of D\sqrt{D}D​. The solutions can be generated from this fundamental solution through a recursive process, leading to an infinite series of integer pairs (xn,yn)(x_n, y_n)(xn​,yn​).

Transcendence Of Pi And E

The transcendence of the numbers π\piπ and eee refers to their property of not being the root of any non-zero polynomial equation with rational coefficients. This means that they cannot be expressed as solutions to algebraic equations like axn+bxn−1+...+k=0ax^n + bx^{n-1} + ... + k = 0axn+bxn−1+...+k=0, where a,b,...,ka, b, ..., ka,b,...,k are rational numbers. Both π\piπ and eee are classified as transcendental numbers, which places them in a special category of real numbers that also includes other numbers like eπe^{\pi}eπ and ln⁡(2)\ln(2)ln(2). The transcendence of these numbers has profound implications in mathematics, particularly in fields like geometry, calculus, and number theory, as it implies that certain constructions, such as squaring the circle or duplicating the cube using just a compass and straightedge, are impossible. Thus, the transcendence of π\piπ and eee not only highlights their unique properties but also serves to deepen our understanding of the limitations of classical geometric constructions.