81 research outputs found
p-Adic Mathematical Physics
A brief review of some selected topics in p-adic mathematical physics is
presented.Comment: 36 page
Emergent quantum mechanics of the event-universe, quantization of events via Denrographic Hologram Theory
Quantum mechanics (QM) is derived on the basis of a universe composed solely
of events, for example, outcomes of observables. Such an event universe is
represented by a dendrogram (a finite tree) and in the limit of infinitely many
events by the p-adic tree. The trees are endowed with an ultrametric expressing
hierarchical relationships between events. All events are coupled through the
tree structure. Such a holistic picture of event-processes was formalized
within the Dendrographic Hologram Theory (DHT). The present paper is devoted to
the emergence of QM from DHT. We used the generalization of the QM-emergence
scheme developed by Smolin. Following this scheme, we did not quantize events
but rather the differences between them and through analytic derivation arrived
at Bohmian mechanics. Previously, we were able to embed the basic elements of
general relativity (GR) into DHT, and now after Smolin-like quantization of
DHT, we can take a step toward quantization of GR. Finally, we remark that DHT
is nonlocal in the treelike geometry, but this nonlocality refers to relational
nonlocality in the space of events and not Einstein's spatial nonlocality.Comment: The paper was presented as an invited talk at the conference DICE202
Discrete scale invariance and complex dimensions
We discuss the concept of discrete scale invariance and how it leads to
complex critical exponents (or dimensions), i.e. to the log-periodic
corrections to scaling. After their initial suggestion as formal solutions of
renormalization group equations in the seventies, complex exponents have been
studied in the eighties in relation to various problems of physics embedded in
hierarchical systems. Only recently has it been realized that discrete scale
invariance and its associated complex exponents may appear ``spontaneously'' in
euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples
are diffusion-limited-aggregation clusters, rupture in heterogeneous systems,
earthquakes, animals (a generalization of percolation) among many other
systems. We review the known mechanisms for the spontaneous generation of
discrete scale invariance and provide an extensive list of situations where
complex exponents have been found. This is done in order to provide a basis for
a better fundamental understanding of discrete scale invariance. The main
motivation to study discrete scale invariance and its signatures is that it
provides new insights in the underlying mechanisms of scale invariance. It may
also be very interesting for prediction purposes.Comment: significantly extended version (Oct. 27, 1998) with new examples in
several domains of the review paper with the same title published in Physics
Reports 297, 239-270 (1998
Aspects of emergent cyclicity in language and computation
This thesis has four parts, which correspond to the presentation and development of a theoretical
framework for the study of cognitive capacities qua physical phenomena, and a case study of locality conditions over natural languages.
Part I deals with computational considerations, setting the tone of the rest of the thesis, and introducing and defining critical concepts like ‘grammar’, ‘automaton’, and the relations between them
. Fundamental questions concerning the place of formal language theory in
linguistic inquiry, as well as the expressibility of linguistic and computational concepts in
common terms, are raised in this part.
Part II further explores the issues addressed in Part I with particular emphasis on how
grammars are implemented by means of automata, and the properties of the formal languages
that these automata generate. We will argue against the equation between effective computation
and function-based computation, and introduce examples of computable procedures which are
nevertheless impossible to capture using traditional function-based theories. The connection
with cognition will be made in the light of dynamical frustrations: the irreconciliable tension
between mutually incompatible tendencies that hold for a given dynamical system. We will
provide arguments in favour of analyzing natural language as emerging from a tension between
different systems (essentially, semantics and morpho-phonology) which impose orthogonal
requirements over admissible outputs. The concept of level of organization or scale comes to
the foreground here; and apparent contradictions and incommensurabilities between concepts
and theories are revisited in a new light: that of dynamical nonlinear systems which are
fundamentally frustrated. We will also characterize the computational system that emerges from
such an architecture: the goal is to get a syntactic component which assigns the simplest
possible structural description to sub-strings, in terms of its computational complexity. A
system which can oscillate back and forth in the hierarchy of formal languages in assigning
structural representations to local domains will be referred to as a computationally mixed
system.
Part III is where the really fun stuff starts. Field theory is introduced, and its applicability to
neurocognitive phenomena is made explicit, with all due scale considerations. Physical and
mathematical concepts are permanently interacting as we analyze phrase structure in terms of
pseudo-fractals (in Mandelbrot’s sense) and define syntax as a (possibly unary) set of
topological operations over completely Hausdorff (CH) ultrametric spaces. These operations, which makes field perturbations interfere, transform that initial completely Hausdorff
ultrametric space into a metric, Hausdorff space with a weaker separation axiom. Syntax, in this
proposal, is not ‘generative’ in any traditional sense –except the ‘fully explicit theory’ one-:
rather, it partitions (technically, ‘parametrizes’) a topological space. Syntactic dependencies are
defined as interferences between perturbations over a field, which reduce the total entropy of
the system per cycles, at the cost of introducing further dimensions where attractors
corresponding to interpretations for a phrase marker can be found.
Part IV is a sample of what we can gain by further pursuing the physics of language approach,
both in terms of empirical adequacy and theoretical elegance, not to mention the unlimited
possibilities of interdisciplinary collaboration. In this section we set our focus on island
phenomena as defined by Ross (1967), critically revisiting the most relevant literature on this
topic, and establishing a typology of constructions that are strong islands, which cannot be
violated. These constructions are particularly interesting because they limit the phase space of
what is expressible via natural language, and thus reveal crucial aspects of its underlying
dynamics. We will argue that a dynamically frustrated system which is characterized by
displaying mixed computational dependencies can provide straightforward characterizations of
cyclicity in terms of changes in dependencies in local domains
Neural Distributed Autoassociative Memories: A Survey
Introduction. Neural network models of autoassociative, distributed memory
allow storage and retrieval of many items (vectors) where the number of stored
items can exceed the vector dimension (the number of neurons in the network).
This opens the possibility of a sublinear time search (in the number of stored
items) for approximate nearest neighbors among vectors of high dimension. The
purpose of this paper is to review models of autoassociative, distributed
memory that can be naturally implemented by neural networks (mainly with local
learning rules and iterative dynamics based on information locally available to
neurons). Scope. The survey is focused mainly on the networks of Hopfield,
Willshaw and Potts, that have connections between pairs of neurons and operate
on sparse binary vectors. We discuss not only autoassociative memory, but also
the generalization properties of these networks. We also consider neural
networks with higher-order connections and networks with a bipartite graph
structure for non-binary data with linear constraints. Conclusions. In
conclusion we discuss the relations to similarity search, advantages and
drawbacks of these techniques, and topics for further research. An interesting
and still not completely resolved question is whether neural autoassociative
memories can search for approximate nearest neighbors faster than other index
structures for similarity search, in particular for the case of very high
dimensional vectors.Comment: 31 page
Small Transformers Compute Universal Metric Embeddings
We study representations of data from an arbitrary metric space
in the space of univariate Gaussian mixtures with a transport metric (Delon and
Desolneux 2020). We derive embedding guarantees for feature maps implemented by
small neural networks called \emph{probabilistic transformers}. Our guarantees
are of memorization type: we prove that a probabilistic transformer of depth
about and width about can bi-H\"{o}lder embed any -point
dataset from with low metric distortion, thus avoiding the curse
of dimensionality. We further derive probabilistic bi-Lipschitz guarantees,
which trade off the amount of distortion and the probability that a randomly
chosen pair of points embeds with that distortion. If 's geometry
is sufficiently regular, we obtain stronger, bi-Lipschitz guarantees for all
points in the dataset. As applications, we derive neural embedding guarantees
for datasets from Riemannian manifolds, metric trees, and certain types of
combinatorial graphs. When instead embedding into multivariate Gaussian
mixtures, we show that probabilistic transformers can compute bi-H\"{o}lder
embeddings with arbitrarily small distortion.Comment: 42 pages, 10 Figures, 3 Table
Quantum walks: a comprehensive review
Quantum walks, the quantum mechanical counterpart of classical random walks,
is an advanced tool for building quantum algorithms that has been recently
shown to constitute a universal model of quantum computation. Quantum walks is
now a solid field of research of quantum computation full of exciting open
problems for physicists, computer scientists, mathematicians and engineers.
In this paper we review theoretical advances on the foundations of both
discrete- and continuous-time quantum walks, together with the role that
randomness plays in quantum walks, the connections between the mathematical
models of coined discrete quantum walks and continuous quantum walks, the
quantumness of quantum walks, a summary of papers published on discrete quantum
walks and entanglement as well as a succinct review of experimental proposals
and realizations of discrete-time quantum walks. Furthermore, we have reviewed
several algorithms based on both discrete- and continuous-time quantum walks as
well as a most important result: the computational universality of both
continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing
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