Trigonometric functions
View article: The Fundamental Asymmetry Tensor Master Monograph
The Fundamental Asymmetry Tensor Master Monograph Open
This monograph presents a unified framework for physical emergence based on a single geometric axiom: that existence is the breaking of symmetry. We identify the Fundamental Asymmetry Tensor (\(\mathcal{A}_{\mu\nu}\)) as t…
View article: The Fundamental Asymmetry Tensor Master Monograph
The Fundamental Asymmetry Tensor Master Monograph Open
This monograph presents a unified framework for physical emergence based on a single geometric axiom: that existence is the breaking of symmetry. We identify the Fundamental Asymmetry Tensor (\(\mathcal{A}_{\mu\nu}\)) as t…
View article: The Cotangent Geometry of Distributional Regularity
The Cotangent Geometry of Distributional Regularity Open
This paper explores the intricate relationship between distributional regularity and the geometric structures provided by cotangent bundles. Distributional regularity extends classical notions of smoothness to generalized functions, crucia…
View article: From Ancient to Modern Space Study with Architectural Point of View
From Ancient to Modern Space Study with Architectural Point of View Open
India’s astronomical heritage—from the Āryabhaṭīya to the Sūrya Siddhānta—lays the mathematical foundation for many principles used in modern orbital science. This paper compares ancient models of planetary motion, trigonometry, and eclips…
View article: The Angular Dilemma: Rethinking Cosine Similarity for Bias Mitigation in Latent Spaces
The Angular Dilemma: Rethinking Cosine Similarity for Bias Mitigation in Latent Spaces Open
The pervasive deployment of machine learning models across critical domains necessitates a deep scrutiny of inherent biases embedded within their learned representations, particularly in latent spaces. Cosine similarity, a ubiquitous metri…
View article: The Angular Dilemma: Rethinking Cosine Similarity for Bias Mitigation in Latent Spaces
The Angular Dilemma: Rethinking Cosine Similarity for Bias Mitigation in Latent Spaces Open
The pervasive deployment of machine learning models across critical domains necessitates a deep scrutiny of inherent biases embedded within their learned representations, particularly in latent spaces. Cosine similarity, a ubiquitous metri…
View article: The Cotangent Geometry of Distributional Regularity
The Cotangent Geometry of Distributional Regularity Open
This paper explores the intricate relationship between distributional regularity and the geometric structures provided by cotangent bundles. Distributional regularity extends classical notions of smoothness to generalized functions, crucia…
View article: From Ancient to Modern Space Study with Architectural Point of View
From Ancient to Modern Space Study with Architectural Point of View Open
India’s astronomical heritage—from the Āryabhaṭīya to the Sūrya Siddhānta—lays the mathematical foundation for many principles used in modern orbital science. This paper compares ancient models of planetary motion, trigonometry, and eclips…
View article: The Implicit Regularization of Cosine Similarity in Large Language Model Alignment
The Implicit Regularization of Cosine Similarity in Large Language Model Alignment Open
Large Language Models (LLMs) have demonstrated remarkable capabilities across various natural language processing tasks, yet their practical deployment necessitates robust alignment with human values and intentions. Reinforcement Learning …
View article: The Implicit Regularization of Cosine Similarity in Large Language Model Alignment
The Implicit Regularization of Cosine Similarity in Large Language Model Alignment Open
Large Language Models (LLMs) have demonstrated remarkable capabilities across various natural language processing tasks, yet their practical deployment necessitates robust alignment with human values and intentions. Reinforcement Learning …
View article: Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics
Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics Open
We introduce a \emph{point--spread--function} (PSF)--aware \emph{moving--wall Womersley}(WMW) inversion that turns retinal Doppler holography (DH) into compact,phase-referenced readouts of flow, shear, wall motion, and impedance-like behav…
View article: The Calibrated Dot Product: Decoupling Magnitude and Direction for Robust AI Similarity
The Calibrated Dot Product: Decoupling Magnitude and Direction for Robust AI Similarity Open
The efficacy of artificial intelligence systems, particularly in tasks reliant on vector representations, hinges critically on robust and meaningful similarity measures. Traditional methods such as the dot product and cosine similarity, wh…
View article: Beyond Cosine: Adaptive Embedding Distances for Semantic Search Across Modalities
Beyond Cosine: Adaptive Embedding Distances for Semantic Search Across Modalities Open
Semantic search across modalities, such as text and images, presents a significant challenge due to the inherent heterogeneity of data representations. Traditional methods often rely on cosine similarity applied to pre-computed embeddings,…
View article: Unveiling Hidden Manifolds: A Causal Geometric Approach to Debiasing Cosine Similarity in AI Embeddings
Unveiling Hidden Manifolds: A Causal Geometric Approach to Debiasing Cosine Similarity in AI Embeddings Open
AI embeddings have become foundational to numerous artificial intelligence applications, encoding complex data into dense vector representations. Cosine similarity is a widely adopted metric for measuring the proximity of these embeddings,…
View article: Unveiling Hidden Manifolds: A Causal Geometric Approach to Debiasing Cosine Similarity in AI Embeddings
Unveiling Hidden Manifolds: A Causal Geometric Approach to Debiasing Cosine Similarity in AI Embeddings Open
AI embeddings have become foundational to numerous artificial intelligence applications, encoding complex data into dense vector representations. Cosine similarity is a widely adopted metric for measuring the proximity of these embeddings,…
View article: Beyond Cosine Similarity: Rethinking Sentence Embedding Evaluation with Distributional Geometry
Beyond Cosine Similarity: Rethinking Sentence Embedding Evaluation with Distributional Geometry Open
Sentence embeddings have become a cornerstone of modern natural language processing, enabling machines to understand and manipulate text with increasing sophistication. Evaluating the quality of these embeddings, however, remains a persist…
View article: Beyond Cosine Similarity: Rethinking Sentence Embedding Evaluation with Distributional Geometry
Beyond Cosine Similarity: Rethinking Sentence Embedding Evaluation with Distributional Geometry Open
Sentence embeddings have become a cornerstone of modern natural language processing, enabling machines to understand and manipulate text with increasing sophistication. Evaluating the quality of these embeddings, however, remains a persist…
View article: Beyond Cosine Similarity: Rethinking Sentence Embedding Evaluation for Robust Semantic Understanding
Beyond Cosine Similarity: Rethinking Sentence Embedding Evaluation for Robust Semantic Understanding Open
Sentence embeddings are vector representations of sentences that aim to capture their semantic meaning. These embeddings are crucial for various natural language processing (NLP) tasks, including semantic similarity, paraphrase detection, …
View article: Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics
Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics Open
We introduce a \emph{point--spread--function} (PSF)--aware \emph{moving--wall Womersley}(WMW) inversion that turns retinal Doppler holography (DH) into compact,phase-referenced readouts of flow, shear, wall motion, and impedance-like behav…
View article: Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics
Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics Open
We introduce a \emph{point--spread--function} (PSF)--aware \emph{moving--wall Womersley}(WMW) inversion that turns retinal Doppler holography (DH) into compact,phase-referenced readouts of flow, shear, wall motion, and impedance-like behav…
View article: Beyond Cosine: Adaptive Embedding Distances for Semantic Search Across Modalities
Beyond Cosine: Adaptive Embedding Distances for Semantic Search Across Modalities Open
Semantic search across modalities, such as text and images, presents a significant challenge due to the inherent heterogeneity of data representations. Traditional methods often rely on cosine similarity applied to pre-computed embeddings,…
View article: Beyond Cosine Similarity: Rethinking Sentence Embedding Evaluation for Robust Semantic Understanding
Beyond Cosine Similarity: Rethinking Sentence Embedding Evaluation for Robust Semantic Understanding Open
Sentence embeddings are vector representations of sentences that aim to capture their semantic meaning. These embeddings are crucial for various natural language processing (NLP) tasks, including semantic similarity, paraphrase detection, …
View article: Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics
Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics Open
We introduce a \emph{point--spread--function} (PSF)--aware \emph{moving--wall Womersley}(WMW) inversion that turns retinal Doppler holography (DH) into compact,phase-referenced readouts of flow, shear, wall motion, and impedance-like behav…
View article: The Calibrated Dot Product: Decoupling Magnitude and Direction for Robust AI Similarity
The Calibrated Dot Product: Decoupling Magnitude and Direction for Robust AI Similarity Open
The efficacy of artificial intelligence systems, particularly in tasks reliant on vector representations, hinges critically on robust and meaningful similarity measures. Traditional methods such as the dot product and cosine similarity, wh…
View article: Mathematical Foundations of Emotional Turbulence: A Theoretical Framework Based on the Model of Internal States Dynamics (MDEI) and its MESN Extension
Mathematical Foundations of Emotional Turbulence: A Theoretical Framework Based on the Model of Internal States Dynamics (MDEI) and its MESN Extension Open
This work presents the mathematical for- malization of the Model of Internal States Dynamics (MDEI) and its MESN extension, integrating emo- tional vectorial geometry, a normalized trigonometric matrix, and an angular bifurcation criterion…
View article: Mathematical Foundations of Emotional Turbulence: A Theoretical Framework Based on the Model of Internal States Dynamics (MDEI) and its MESN Extension
Mathematical Foundations of Emotional Turbulence: A Theoretical Framework Based on the Model of Internal States Dynamics (MDEI) and its MESN Extension Open
This work presents the mathematical for- malization of the Model of Internal States Dynamics (MDEI) and its MESN extension, integrating emo- tional vectorial geometry, a normalized trigonometric matrix, and an angular bifurcation criterion…
View article: Reconstruction of Schrödinger Dynamics from C3 Phase Geometry
Reconstruction of Schrödinger Dynamics from C3 Phase Geometry Open
This monograph develops a consistent formulation of quantum mechanics on the cyclic algebra C3C_3C3, replacing the traditional complex imaginary unit iii with the unique anti-Hermitian axis j+j2j + j^2j+j2. This replacement yields a two-ch…
View article: Reconstruction of Schrödinger Dynamics from C3 Phase Geometry
Reconstruction of Schrödinger Dynamics from C3 Phase Geometry Open
This monograph develops a consistent formulation of quantum mechanics on the cyclic algebra C3C_3C3, replacing the traditional complex imaginary unit iii with the unique anti-Hermitian axis j+j2j + j^2j+j2. This replacement yields a two-ch…
View article: Reconstruction of Schrödinger Dynamics from C3 Phase Geometry
Reconstruction of Schrödinger Dynamics from C3 Phase Geometry Open
This monograph develops a consistent formulation of quantum mechanics on the cyclic algebra C3C_3C3, replacing the traditional complex imaginary unit iii with the unique anti-Hermitian axis j+j2j + j^2j+j2. This replacement yields a two-ch…
View article: Catalog of Arctangent Identities Generated by f (x) = (1 − x)/(1 + x)
Catalog of Arctangent Identities Generated by f (x) = (1 − x)/(1 + x) Open
Collection of all identities from f(x)=1−x1+xf(x)=1+x1−x: rational, algebraic (golden ratio), trigonometric, and transcendental families. Systematic reference.