Mapping the Geometry of Law using Document Embeddings

Recent work in natural language processing represents language objects (words and documents) as dense vectors that encode the relations between those objects. This paper explores the application of these methods to legal language, with the goal of understanding judicial reasoning and the relations between judges. In an application to federal appellate courts, we show that these vectors encode information that distinguishes courts, time, and legal topics. The vectors do not reveal spatial distinctions in terms of political party or law school attended, but they do highlight generational differences across judges. We conclude the paper by outlining a range of promising future applications of these methods

Categorical Ontology of Complex Systems, Meta-Systems and Theory of Levels: The Emergence of Life, Human Consciousness and Society

Single cell interactomics in simpler organisms, as well as somatic cell interactomics in multicellular organisms, involve biomolecular interactions in complex signalling pathways that were recently represented in modular terms by quantum automata with ‘reversible behavior’ representing normal cell cycling and division. Other implications of such quantum automata, modular modeling of signaling pathways and cell differentiation during development are in the fields of neural plasticity and brain development leading to quantum-weave dynamic patterns and specific molecular processes underlying extensive memory, learning, anticipation mechanisms and the emergence of human consciousness during the early brain development in children. Cell interactomics is here represented for the first time as a mixture of ‘classical’ states that determine molecular dynamics subject to Boltzmann statistics and ‘steady-state’, metabolic (multi-stable) manifolds, together with ‘configuration’ spaces of metastable quantum states emerging from complex quantum dynamics of interacting networks of biomolecules, such as proteins and nucleic acids that are now collectively defined as quantum interactomics. On the other hand, the time dependent evolution over several generations of cancer cells --that are generally known to undergo frequent and extensive genetic mutations and, indeed, suffer genomic transformations at the chromosome level (such as extensive chromosomal aberrations found in many colon cancers)-- cannot be correctly represented in the ‘standard’ terms of quantum automaton modules, as the normal somatic cells can. This significant difference at the cancer cell genomic level is therefore reflected in major changes in cancer cell interactomics often from one cancer cell ‘cycle’ to the next, and thus it requires substantial changes in the modeling strategies, mathematical tools and experimental designs aimed at understanding cancer mechanisms. Novel solutions to this important problem in carcinogenesis are proposed and experimental validation procedures are suggested. From a medical research and clinical standpoint, this approach has important consequences for addressing and preventing the development of cancer resistance to medical therapy in ongoing clinical trials involving stage III cancer patients, as well as improving the designs of future clinical trials for cancer treatments.\ud \ud \ud KEYWORDS: Emergence of Life and Human Consciousness;\ud Proteomics; Artificial Intelligence; Complex Systems Dynamics; Quantum Automata models and Quantum Interactomics; quantum-weave dynamic patterns underlying human consciousness; specific molecular processes underlying extensive memory, learning, anticipation mechanisms and human consciousness; emergence of human consciousness during the early brain development in children; Cancer cell ‘cycling’; interacting networks of proteins and nucleic acids; genetic mutations and chromosomal aberrations in cancers, such as colon cancer; development of cancer resistance to therapy; ongoing clinical trials involving stage III cancer patients’ possible improvements of the designs for future clinical trials and cancer treatments. \ud \u

Response to `Comment on "Quantum correlations are weaved by the spinors of the Euclidean primitives"'

In this paper I respond to a critique of one of my papers previously published in the Royal Society Open Science entitled "Quantum correlations are weaved by the spinors of the Euclidean primitives." Without engaging with the geometrical framework presented in my paper, the critique incorrectly claims that there are mathematical errors in it. I demonstrate that the critique is based on a series of misunderstandings, and refute each of its claims of error. I also bring out a number of logical, mathematical, and conceptual errors from the critique and the critiques it relies on.Comment: 49 pages, Invited reply to comment arXiv:2106.03169 [quant-ph] on the earlier paper arXiv:1806.02392 [quant-ph], published by invitation of the editors of `Royal Society Open Science.' For complete transparency of the peer review process, the Review History of the paper (both rounds) is included, which is also accessible from the journal's webpage for the pape

Gravitation as a Plastic Distortion of the Lorentz Vacuum

In this paper we present a theory of the gravitational field where this field (a kind of square root of g) is represented by a (1,1)-extensor field h describing a plastic distortion of the Lorentz vacuum (a real substance that lives in a Minkowski spacetime) due to the presence of matter. The field h distorts the Minkowski metric extensor in an appropriate way (see below) generating what may be interpreted as an effective Lorentzian metric extensor g and also it permits the introduction of different kinds of parallelism rules on the world manifold, which may be interpreted as distortions of the parallelism structure of Minkowski spacetime and which may have non null curvature and/or torsion and/or nonmetricity tensors. We thus have different possible effective geometries which may be associated to the gravitational field and thus its description by a Lorentzian geometry is only a possibility, not an imposition from Nature. Moreover, we developed with enough details the theory of multiform functions and multiform functionals that permitted us to successfully write a Lagrangian for h and to obtain its equations of motion, that results equivalent to Einstein field equations of General Relativity (for all those solutions where the manifold M is diffeomorphic to R^4. However, in our theory, differently from the case of General Relativity, trustful energy-momentum and angular momentum conservation laws exist. We express also the results of our theory in terms of the gravitational potential 1-form fields (living in Minkowski spacetime) in order to have results which may be easily expressed with the theory of differential forms. The Hamiltonian formalism for our theory (formulated in terms of the potentials) is also discussed. The paper contains also several important Appendices that complete the material in the main text.Comment: Misprints and typos have been corrected, Chapter 7 have been improved. Appendix E has been reformulated and Appendix F contains new remarks which resulted from a discussion with A. Lasenby. A somewhat modified version has been published in the Springer Series: Fundamental Theories of Physics vol. 168, 2010. http://www.ime.unicamp.br/~walrod/plastic2014.pd

Information Processing by Neuron Populations in the Central Nervous System: Mathematical Structure of Data and Operations

In the intricate architecture of the mammalian central nervous system, neurons form populations. Axonal bundles communicate between these clusters using spike trains as their medium. However, these neuron populations' precise encoding and operations have yet to be discovered. In our analysis, the starting point is a state-of-the-art mechanistic model of a generic neuron endowed with plasticity. From this simple framework emerges a profound mathematical construct: The representation and manipulation of information can be precisely characterized by an algebra of finite convex cones. Furthermore, these neuron populations are not merely passive transmitters. They act as operators within this algebraic structure, mirroring the functionality of a low-level programming language. When these populations interconnect, they embody succinct yet potent algebraic expressions. These networks allow them to implement many operations, such as specialization, generalization, novelty detection, dimensionality reduction, inverse modeling, prediction, and associative memory. In broader terms, this work illuminates the potential of matrix embeddings in advancing our understanding in fields like cognitive science and AI. These embeddings enhance the capacity for concept processing and hierarchical description over their vector counterparts.Comment: 34 pages, 12 figure

An Algebraic Method to Fidelity-based Model Checking over Quantum Markov Chains

Fidelity is one of the most widely used quantities in quantum information that measure the distance of quantum states through a noisy channel. In this paper, we introduce a quantum analogy of computation tree logic (CTL) called QCTL, which concerns fidelity instead of probability in probabilistic CTL, over quantum Markov chains (QMCs). Noisy channels are modelled by super-operators, which are specified by QCTL formulas; the initial quantum states are modelled by density operators, which are left parametric in the given QMC. The problem is to compute the minimumfidelity over all initial states for conservation. We achieve it by a reduction to quantifier elimination in the existential theory of the reals. The method is absolutely exact, so that QCTL formulas are proven to be decidable in exponential time. Finally, we implement the proposed method and demonstrate its effectiveness via a quantum IPv4 protocol

Case Vectors: Spatial Representations of the Law Using Document Embeddings

Recent work in natural language processing represents language objects (words and documents) as dense vectors that encode the relations between those objects. This paper explores the application of these methods to legal language, with the goal of understanding judicial reasoning and the relations between judges. In an application to federal appellate courts, we show that these vectors encode information that distinguishes courts, time, and legal topics. The vectors do not reveal spatial distinctions in terms of political party or law school attended, but they do highlight generational differences across judges. We conclude the paper by outlining a range of promising future applications of these methods

Mapping the Geometry of Law using Document Embeddings

Recent work in natural language processing represents language objects (words and documents) as dense vectors that encode the relations between those objects. This paper explores the application of these methods to legal language, with the goal of understanding judicial reasoning and the relations between judges. In an application to federal appellate courts, we show that these vectors encode information that distinguishes courts, time, and legal topics. The vectors do not reveal spatial distinctions in terms of political party or law school attended, but they do highlight generational differences across judges. We conclude the paper by outlining a range of promising future applications of these methods