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Investigating the Tensorial Cross Product Modulation (TCPM): Applications in Engineering and Finance

In this series, our primary aim is to demystify and popularize intricate mathematical structures, making them accessible to the everyday individual, both aspirational learners and mindful engineers. Even though mathematics is seen as mysterious and abstract, it is full of deep symmetry, profound truths, and beautiful things. Gaining mathematics literacy improves our ability to solve problems and creates new opportunities across many industries. A mathematically literate society is more powerful and capable of making the most use of equations and numbers to improve living standards and get a better understanding of the universe.

In mathematics, manipulating and comprehending complicated structures is crucial to addressing real-world issues, especially in the domains of engineering and finance. The Tensorial Cross Product Modulation (TCPM) is one such mathematical procedure that shows how abstract mathematical ideas may be used in real-world situations. This operation provides a sophisticated method for analysing and interpreting multidimensional data and interactions by extending the concept of the cross product into the tensorial domain and incorporating modulation by the Levi-Civita symbol.

Mathematical Foundation of TCPM

The TCPM operation involves the calculation of a matrix that effectively represents a generalized cross product between a vector $\vec{A}$ and a dyadic tensor $\overline{\overline{D}}$ . This matrix is modulated by the Levi-Civita symbol, which introduces the antisymmetric property characteristic of cross products, and is further articulated through the directional components provided by the unit vectors of the standard basis in $R^3$ .

The mathematical definition of TCPM is given by the equation:

$\overline{\overline{B}}=\sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \delta_i (\sum_{l=1}^3 \epsilon_{ljk} \delta_l) D_{ij} A_k$
where $i,j,k=1,2,3$ in cyclic order and $\epsilon_{ljk}$ is the Levi-Civitta tensor and the $\delta_i$ are the unit vectors.

import sympy as sp

# Define symbols
x, y, z = sp.symbols('x y z')
D_11, D_12, D_13, D_21, D_22, D_23, D_31, D_32, D_33 = sp.symbols('D_11 D_12 D_13 D_21 D_22 D_23 D_31 D_32 D_33')

# Symbolic representation of D (as p)
D = sp.Matrix([
    [D_11, D_12, D_13],
    [D_21, D_22, D_23],
    [D_31, D_32, D_33]
])

# Levi-Civita tensor (epsilon_ijk) for three dimensions
def epsilon(i, j, k):
    if (i, j, k) in [(1, 2, 3), (2, 3, 1), (3, 1, 2)]:
        return 1
    elif (i, j, k) in [(3, 2, 1), (1, 3, 2), (2, 1, 3)]:
        return -1
    else:
        return 0

# Symbolic representation of D and A
D = sp.Matrix([
    [D_11, D_12, D_13],
    [D_21, D_22, D_23],
    [D_31, D_32, D_33]
    ])

A = sp.Matrix([
    [x],
    [y],
    [z]
    ])

# Unit vectors (standard basis in R^3) as column matrices
delta = [sp.Matrix([1, 0, 0]), sp.Matrix([0, 1, 0]), sp.Matrix([0, 0, 1])]

# Initialize the resulting matrix B_bar_bar as a zero matrix
B_bar_bar = sp.zeros(3, 3)

# Perform the operation to calculate B_bar_bar according to the formula
for i in range(3):
    for j in range(3):
        for k in range(3):
            # Reset sum_over_l for each new combination of i, j, k
            sum_over_l = sp.zeros(3, 1)  # Transpose delta[l] here, initializing as a row vector
            for l in range(3):
                # Apply transpose to delta[l] directly
                sum_over_l += epsilon(l+1, k+1, i+1) * delta[l] * D[i, j] * A[k]
            # Multiply by delta_i on the left without transposing the sum_over_l
            B_bar_bar += sum_over_l * delta[j].transpose() # delta[i] as a column vector, sum_over_l already a row vector

B_bar_bar

Notice that $\vec{A} \times \overline{\overline{D}} \neq \overline{\overline{D}} \times \vec{A}$ .

The result is

Applications in Engineering

The TCPM is useful in engineering applications where it is necessary to analyse forces, moments, and field characteristics in three-dimensional environments. For example, in fluid dynamics, the procedure may be used to examine the circulation and vorticity in a fluid flow, assisting engineers in creating fluid handling systems that are more effective. Analogously, in the field of structural engineering, the TCPM can facilitate the assessment of stress and strain in materials, providing critical information for determining the safety and integrity of the structure.

The TCPM has another interesting use in electromagnetism, where it may be used to calculate the distributions of magnetic and electric fields more easily. This is especially helpful for the design and optimisation of electrical equipment, where it’s important to comprehend how magnetic fields and electric currents interact, such transformers and motors.

Example of application: Torque Calculation using Maxwell Stress Tensor

Consider a flat plate of area A lying in the xy-plane, centered at the origin for simplicity. Incident light is assumed to come in along the z-axis, exerting pressure on the plate due to its electromagnetic field. The Maxwell stress tensor for electromagnetic fields in a vacuum is defined as:

$T_{ij} = \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right) + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right)$ . Focusing on the electric field component, we simplify our consideration by assuming that the light’s pressure effect is primarily due to its electric field component perpendicular to the surface. The force differential $d\mathbf{F}$ on an elemental area dA due to the stress tensor is:

$d\mathbf{F} = \mathbf{T} \cdot d\mathbf{A}$ . For light incident normally on the xy-plane, $d\mathbf{A}$ can be considered as $dA \hat{z}$ , where $\hat{z}$ is the unit vector in the z-direction. The torque $\mathbf{\tau}$ about the origin is calculated by integrating the cross product of the position vector $\mathbf{r}$ with the force differential over the surface S:

$\mathbf{\tau} = \int_S (\mathbf{r} \times d\mathbf{F})$ , $\mathbf{\tau} = \int_S (\mathbf{r} \times (T_{zz} \hat{z} \cdot dA \hat{z})) = \int_S (\mathbf{r} \times (T_{zz} dA \hat{z}))$ .

Given $T_{zz}$ as a uniform pressure p across the surface, and $T_{zz} = p$ , we have:

$\mathbf{\tau} = \int_S (\mathbf{r} \times (p dA \hat{z}))$

Visual representation of the experimental setup involving a flat plate exposed to incident light, illustrating the electromagnetic interaction between the light and the plate amidst a laboratory environment, created by the author with DALL-E.

For a flat plate centered at the origin, the cross product becomes: $\mathbf{\tau} = \int_S \left( -py dA \hat{x} + px dA \hat{y} \right)$ . Due to the symmetry about the origin and the uniform pressure p, the integral expressions for $x dA$ and $y dA$ over the area S will cancel out, leading to no net torque for a perfectly symmetrical plate. For scenarios involving non-symmetrical plates or non-uniform pressure, detailed numerical integration over the actual geometry and electromagnetic field distribution is required for a precise torque calculation.

Applications in Finance

TCPM has potential in risk management and derivatives pricing, even if its use in finance may not be as straightforward as in engineering. Tensors are a useful tool in the financial markets to describe the behaviour of diverse financial instruments because they can capture the multidimensional character of market movements and correlations among different assets. In this case, modelling the dynamics of complicated derivatives—whose values rely on several underlying variables—can be greatly aided by the TCPM method. Financial analysts may improve risk assessment and hedging strategies by using TCPM to obtain greater insights into the sensitivities and exposures of these instruments.

Additionally, the TCPM can aid in the comprehension of the multifaceted risk-return environment during portfolio optimisation, allowing for the development of portfolios that are more closely linked with the return goals and risk tolerance of investors. The operation is an effective instrument for managing the numerous interdependencies of financial markets because of its capacity to manage complex, multidimensional interactions. In engineering and finance, the Tensorial Cross Product Modulation operation serves as a link between abstract mathematical theory and real-world applications. The TCPM provides a comprehensive method for analysing and understanding multidimensional interactions, such as physical pressures in engineering contexts or market dynamics in finance, by extending standard vector operations into the tensorial domain. It is certain that the applications of these mathematical structures will increase as we learn more about and comprehend their possibilities.

Although the Tensorial Cross Product Modulation (TCPM) may not be widely used or documented in the financial industry, its conceptual foundations offer a rich environment for creative financial modelling, especially when it comes to intricate, multifaceted financial instruments and risk management techniques. Even though the particular mathematical structure isn’t utilised directly, let’s examine how the concepts behind TCPM might be theoretically applied to the banking industry.

Portfolio Optimization and Risk Management

In finance, the complexity of relationships between assets in a portfolio can often mirror the multidimensional interactions captured by tensorial operations. Portfolio optimization, which seeks to balance the trade-off between risk and return across a basket of investments, can benefit from the high-dimensional analysis capabilities that TCPM suggests.

For example, the covariance matrix, a key component in modern portfolio theory, could be extended into a higher-dimensional tensor to capture not only pairwise asset volatilities and correlations but also the influence of external multidimensional factors such as macroeconomic indicators or global financial trends. The principles behind TCPM could inform the development of these tensorial models, providing a more nuanced view of portfolio dynamics and risks.

Derivatives Pricing and Risk Sensitivity Analysis

Financial derivatives, whose values depend on the behavior of one or more underlying assets, can exhibit complex risk sensitivities, known as “Greeks” in finance. These sensitivities can be influenced by a range of factors, including changes in the underlying asset prices, volatility, time decay, and interest rates.

Conceptually, the application of a TCPM-like operation could enhance the modeling of such sensitivities, especially for exotic derivatives that depend on multiple underlying variables in complex ways. By considering the interactions between these variables in a multidimensional tensor framework, analysts could potentially uncover new insights into the derivatives’ behavior and risk profiles.

High-dimensional Financial Data Analysis

The explosion of big data in finance requires analytical methods that can handle multidimensional datasets. Examples of this type of data include sentiment analysis on social media, economic indicators, and data from high-frequency trading. Although tensor operations are not directly related to TCPM, their multidimensional nature and mathematical complexity provide inspiration for methods of analysing such complicated data.

For example, latent features from high-dimensional financial data may be extracted using tensor decomposition techniques, providing insights that may be lost in lower-dimensional analysis. These techniques may help uncover underlying trends and connections in market behaviour, which may result in better investing choices.

The abstract nature of Tensorial Cross Product Modulation (TCPM) and its speculative use in finance allow us to investigate the conceptual effects of tensor-based operations on financial models, especially in high-dimensional data analysis, derivatives pricing, and portfolio optimisation. In the sections that follow, we go deeper into each topic and provide examples of possible applications for tensor operations, where suitable utilising Python for demonstrations. Refer to this reference [1].

Portfolio Optimization

In portfolio optimization, the goal is to allocate assets in a way that maximizes return for a given level of risk, or equivalently, minimizes risk for a given level of return. The covariance matrix, which captures the volatilities of assets and their correlations, is central to this problem. Extending this to incorporate external factors leads to a more complex model that can be represented using tensors.

Python Demonstration:

import numpy as np

# Assuming 3 assets and 2 external factors
# Mock data for asset returns
asset_returns = np.random.rand(3, 10)  # 3 assets, 10 time periods
# Mock data for factor sensitivities of each asset
factor_sensitivities = np.random.rand(3, 2)  # 3 assets, 2 factors
# Mock factor returns
factor_returns = np.random.rand(2, 10)  # 2 factors, 10 time periods

# Calculate asset returns influenced by factors
# This simplifies the interaction but illustrates the concept
influenced_returns = asset_returns + factor_sensitivities @ factor_returns

# Portfolio optimization would then proceed using these influenced returns

Although it streamlines the procedure, this example shows how to include outside variables in asset return computations, much like a higher-dimensional tensor.

Derivatives Pricing

Modelling the “Greeks” or risk sensitivity for financial derivatives entails comprehending how shifts in market circumstances impact the derivative’s value. The interactions between several underlying variables and their cumulative effect on the derivative may be modelled using tensors.

Conceptual Framework:

Consider a derivative where the two underlying assets determine its value. A number of variables, such as interest rates and market volatility, may have an impact on its price sensitivity to changes in these assets (Delta) and volatility (Vega). Tensors might capture these interactions in a multidimensional representation of this intricate connection.

High-dimensional Financial Data Analysis

High-dimensional financial data, such as latent elements in market movements or investor sentiment from social media, may be analysed using tensor decomposition techniques.

Python Demonstration using Tensor Decomposition:

from tensorly.decomposition import parafac
import tensorly as tl
import numpy as np

# Mock high-dimensional data: 3 assets, 4 time periods, 2 external factors
data = np.random.rand(3, 4, 2)

# Decompose the tensor to identify latent factors
factors = parafac(tl.tensor(data), rank=2)

# factors now contains the decomposed tensor representing principal components
</code>

In order to examine high-dimensional data and maybe uncover underlying patterns or variables that affect financial markets, this example makes use of tensor decomposition.

Conclusion

Though its application in finance is currently limited, the Tensorial Cross Product Modulation (TCPM) operation and its theoretical underpinnings—the analysis of complex, multidimensional relationships through the use of advanced mathematical structures—have the potential to inspire innovative thinking in the field of financial modelling and analysis. The use of sophisticated mathematical models, impacted by TCPM activities, will be crucial in handling the opportunities and challenges brought about by the ongoing development and intricacy of financial markets.

REFERENCES:

[1] ACM Transactions on Information SystemsVolume 34Issue 2Article No.: 11pp 1–30https://doi.org/10.1145/2838731