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Klee Irwin

  • An author, researcher and entrepreneur, Klee Irwin is the director of Quantum Gravity Research, leading a dedicated t... moreedit
In this work, we define quasicrystalline spin networks as a subspace within the standard Hilbert space of loop quantum gravity, effectively constraining the states to coherent states that align with quasicrystal geometry structures. We... more
In this work, we define quasicrystalline spin networks as a subspace within the standard Hilbert space of loop quantum gravity, effectively constraining the states to coherent states that align with quasicrystal geometry structures. We introduce quasicrystalline spin foam amplitudes, a variation of the EPRL spin foam model, in which the internal spin labels are constrained to correspond to the boundary data of quasicrystalline spin networks. Within this framework, the quasicrystalline spin foam amplitudes encode the dynamics of quantum geometries that exhibit aperiodic structures. Additionally, we investigate the coupling of fermions within the quasicrystalline spin foam amplitudes. We present calculations for three-dimensional examples and then explore the 600-cell construction, which is a fundamental component of the four-dimensional Elser-Sloane quasicrystal derived from the E8 root lattice.
Aperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered... more
Aperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered structures and in the construction of some integrable models in quantum mechanics. Starting from the works of Patera and Twarock we present three aperiodic algebras based on Fibonacci-chain quasicrystals: a quasicrystal Lie algebra, an aperiodic Witt algebra and, finally, an aperiodic Jordan algebra. While a quasicrystal Lie algebra was already constructed from a modification of the Fibonacci chain, we here present an aperiodic algebra that matches exactly the original quasicrystal. Moreover, this is the first time to our knowledge, that an aperiodic Jordan algebra is presented leaving room for both theoretical and applicative developments.
Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing like the fast Fourier transform, Ramanujan sum signal processing and many other techniques. For space-time... more
Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing like the fast Fourier transform, Ramanujan sum signal processing and many other techniques. For space-time topological objects in physics and biology, we propose a type of algebraic processing based on schemes in which the discrimination of singularities within objects is based on the space-time-spin group SL(2, C). Such topological objects possess an homotopy structure encoded in their fundamental group and the related SL(2, C) multivariate polynomial character variety contains a plethora of singularities somehow analogous to the frequency spectrum in time structures. Our approach is applied to an Akbulut cork in exotic R 4 , to an hyperbolic model of topological quantum computing based on algebraic surfaces and to microRNAs in genetics. Such diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum.
In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz... more
In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find 7 pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld's framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation à la Rosenfeld.
Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. In this paper, we describe the algebraic geometry of both TFs and miRNAs thanks to group... more
Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. In this paper, we describe the algebraic geometry of both TFs and miRNAs thanks to group theory. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group π, we compute the SL(2, C) character variety, where SL(2, C) is simultaneously a 'space-time' (a Lorentz group) and a 'quantum' (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when π looks like a free group F r (r = 1 to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis G of the variety. A surface with simple singularities (like the well known Cayley cubic) within G is a signature of a potential disease even when G looks like a free group F r in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in {A, T/U, G, C}. Several human TFs and miRNAs are investigated in detail thanks to our approach.
We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C)... more
We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd=xyz−x2−y2−z2+d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 521, the Berge link 622 or the double-eight link 623) or the link 723, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d<4), with |x|,|y|,|z|≤2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation.
Taking a DNA sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group π, one can discriminate between two important families: (i) the cardinality structure for conjugacy classes of... more
Taking a DNA sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group π, one can discriminate between two important families: (i) the cardinality structure for conjugacy classes of subgroups of π is that of a free group on one to four bases, and the DNA word, viewed as a substitution sequence, is aperiodic; (ii) the cardinality structure for conjugacy classes of subgroups of π is not that of a free group, the sequence is generally not aperiodic and topological properties of π have to be determined differently. The two cases rely on DNA conformations such as A-DNA, B-DNA, Z-DNA, G-quadruplexes, etc. We found a few salient results: Z-DNA, when involved in transcription, replication and regulation in a healthy situation, implies (i). The sequence of telomeric repeats comprising three distinct bases most of the time satisfies (i). For two-base sequences in the free case (i) or non-free case (ii), the topology of π may be found in terms of the SL(2,C) character variety of π and the attached algebraic surfaces. The linking of two unknotted curves—the Hopf link—may occur in the topology of π in cases of biological importance, in telomeres, G-quadruplexes, hairpins and junctions, a feature that we already found in the context of models of topological quantum computing. For three- and four-base sequences, other knotting configurations are noticed and a building block of the topology is the four-punctured sphere. Our methods have the potential to discriminate between potential diseases associated to the sequences.
The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver... more
The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. Different from anyons, quasicrystals are already implemented in laboratories. In particular, we study the correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of the one-dimensional Fibonacci chain and the two-dimensional Penrose tiling quasicrystals. A concrete encoding on these tiling spaces of topological quantum information processing is also presented by making use of inflation and deflation of such tiling spaces. While we outline the theoretical basis for such a platform, details on the physical implementation remain open.
It is shown that the representation theory of some finitely presented groups thanks to their SL 2 (C) character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the... more
It is shown that the representation theory of some finitely presented groups thanks to their SL 2 (C) character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. We study the connection of SL 2 (C) character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface f H (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups as well as that of the fundamental group for the singular fibersẼ 6 andD 4 contain f H. A surface birationally equivalent to a K 3 surface is another compound of their character varieties.
A comprehensive study of the technical parameters and conditions for the synthesis of ternary alloys in the Ti-Zr-Ni system by the "hydride cycle" method was carried out. The influence on the synthesis process of such parameters as:... more
A comprehensive study of the technical parameters and conditions for the synthesis of ternary alloys in the Ti-Zr-Ni system by the "hydride cycle" method was carried out. The influence on the synthesis process of such parameters as: temperature and annealing time, heating rate, cooling conditions, material composition, dispersion, hydrogen content in the hydrides used, the presence of impurities, mixing and pressing methods, and the degree of pressing of the starting components was determined. The alloys of the following compositions were synthesized and investigated: Ti 40.5 Zr 31.9 Ni 27.6 , Ti 41.5 Zr 41.5 Ni 17 , Ti 40 Zr 40 Ni 20 , Ti 44 Zr 40 Ni 16. The optimal technological parameters and conditions for the synthesis of ternary alloys are determined. It has been established that the key factors in the process of compound formation during hydride dissociation are the dispersion and homogeneity of the initial compacted components. It was found that the synthesis of ternary alloys in the Ti-Zr-Ni system occurs during a short-term exothermic reaction in the "thermal explosion" mode, which begins in the temperature region corresponding to the α↔β polymorphic transformation of zirconium and titanium.
We consider partition functions, in the form of state sums, and associated probabilistic measures for aperiodic substrates described by model sets and their associated tiling spaces. We propose model set tiling spaces as microscopic... more
We consider partition functions, in the form of state sums, and associated probabilistic measures for aperiodic substrates described by model sets and their associated tiling spaces. We propose model set tiling spaces as microscopic models for small scales in the context of quantum gravity. Model sets possess special self-similarity properties that allow us to consider implications on large and observable scales from the underlying (non-ergodic) dynamics. In particular we consider the implication of the underlying aperiodic substrate for the well known problem of time in quantum gravity, and propose a correspondence between small and large scales, the so-called ergodic correspondence, that addresses the emergence of matter properties and spacetime structure. In the process we find a possible bound in the mass spectrum of fundamental particles.
Transcription factors (TFs) are proteins that recognize specific DNA fragments in order to decode the genome and ensure its optimal functioning. TFs work at the local and global scales by specifying cell type, cell growth and death, cell... more
Transcription factors (TFs) are proteins that recognize specific DNA fragments in order to decode the genome and ensure its optimal functioning. TFs work at the local and global scales by specifying cell type, cell growth and death, cell migration, organization and timely tasks. We investigate the structure of DNA-binding motifs with the theory of finitely generated groups. The DNA 'word' in the binding domain-the motif-may be seen as the generator of a finitely generated group F dna on four letters, the bases A, T, G and C. It is shown that, most of the time, the DNA-binding motifs have subgroup structure close to free groups of rank three or less, a property that we call 'syntactical freedom'. Such a property is associated to the aperiodicity of the motif when it is seen as a substitution sequence. Examples are provided for the major families of TFs such as leucine zipper factors, zinc finger factors, homeo-domain factors, etc. We also discuss the exceptions to the existence of such a DNA syntactical rule and their functional role. This includes the TATA box in the promoter region of some genes, the single nucleotide markers (SNP) and the motifs of some genes of ubiquitous role in transcription and regulation.
We explore the structural similarities in three different languages, first in the protein language whose primary letters are the amino acids, second in the musical language whose primary letters are the notes, and third in the poetry... more
We explore the structural similarities in three different languages, first in the protein language whose primary letters are the amino acids, second in the musical language whose primary letters are the notes, and third in the poetry language whose primary letters are the alphabet. For proteins, the non local (secondary) letters are the types of foldings in space (α-helices, β-sheets, etc.); for music, one is dealing with clear-cut repetition units called musical forms and for poems the structure consists of grammatical forms (names, verbs, etc.). We show in this paper that the mathematics of such secondary structures relies on finitely presented groups fp on r letters, where r counts the number of types of such secondary non local segments. The number of conjugacy classes of a given index (also the number of graph coverings over a base graph) of a group fp is found to be close to the number of conjugacy classes of the same index in the free group Fr−1 on r−1 generators. In a concrete way, we explore the group structure of a variant of the SARS-Cov-2 spike protein and the group structure of apolipoprotein-H, passing from the primary code with amino acids to the secondary structure organizing the foldings. Then, we look at the musical forms employed in the classical and contemporary periods. Finally, we investigate in much detail the group structure of a small poem in prose by Charles Baudelaire and that of the Bateau Ivre by Arthur Rimbaud.
Most quasicrystals can be generated by the cut-and-project method from higher dimensional parent lattices. In doing so they lose the periodic order their parent lattice possess, replaced with aperiodic order, due to the irrationality of... more
Most quasicrystals can be generated by the cut-and-project method from higher dimensional parent lattices. In doing so they lose the periodic order their parent lattice possess, replaced with aperiodic order, due to the irrationality of the projection. However, perfect periodic order is discovered in the perpendicular space when gluing the cut window boundaries together to form a curved loop. In the case of a 1D quasicrystal projected from a 2D lattice, the irrationally sloped cut region is bounded by two parallel lines. When it is extrinsically curved into a cylinder, a line defect is found on the cylinder. Resolving this geometrical frustration removes the line defect to preserve helical paths on the cylinder. The degree of frustration is determined by the thickness of the cut window or the selected pitch of the helical paths. The frustration can be resolved by applying a shear strain to the cut-region before curving into a cylinder. This demonstrates that resolving the geometrical frustration of a topological change to a cut window can lead to preserved periodic order.
Considering the predictions from the standard model of particle physics coupled with experimental results from particle accelerators, we discuss a scenario in which from the infinite possibilities in the Lie groups we use to describe... more
Considering the predictions from the standard model of particle physics coupled with experimental results from particle accelerators, we discuss a scenario in which from the infinite possibilities in the Lie groups we use to describe particle physics, nature needs only the lower dimensional representations - an important phenomenology that we argue indicates nature is code theoretic. We show that the quantum deformation of the SU(2) Lie algebra at the fifth root of unity can be used to address the quantum Lorentz group representation theory through its universal covering group and gives the right low dimensional physical realistic spin quantum numbers confirmed by experiments. In this manner we can describe the spacetime symmetry content of relativistic quantum fields in accordance with the well known Wigner classification. Further connections of the fifth root of unity quantization with the mass quantum number associated with the Poincaré Group and the SU(N ) charge. Quantum numbers are discussed as well as their implication for quantum gravity.
Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The sequence unfolds in the 3-dimensional space through secondary (local foldings), tertiary (bonds) and qua-ternary (disjoint multiple) structures.... more
Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The sequence unfolds in the 3-dimensional space through secondary (local foldings), tertiary (bonds) and qua-ternary (disjoint multiple) structures. The mere existence of the genetic code for the 20 letters of the linear chain could be predicted with the (informationally complete) irreducible characters of the finite group Gn := Zn ⋊ 2O (with n = 5 or 7 and 2O the binary octahedral group) in our previous two papers. It turns out that some quaternary structures of protein complexes display n-fold symmetries. We propose an approach of secondary structures based on free group theory. Our results are compared to other approaches of predicting secondary structures of proteins in terms of α helices, β sheets and coils, or more refined techniques. It is shown that the secondary structure of proteins shows similarities to the structure of some hyperbolic 3-manifolds. The hyperbolic 3-manifold of smallest volume-Gieseking manifold-, some other 3 manifolds and Grothendieck's cartographic group are singled out as tentative models of such secondary structures. For the quaternary structure, there are links to the Kummer surface. Arxiv: quant-ph, math.GR, math.AG, q-bio.OT
In light of the self-simulation hypothesis, a simple form implementation of the principle of efficient language is discussed in a self-referential geometric quasicrystalline state sum model in three dimensions. Emergence is discussed in... more
In light of the self-simulation hypothesis, a simple form implementation of the principle of efficient language is discussed in a self-referential geometric quasicrystalline state sum model in three dimensions. Emergence is discussed in context of geometric state sum models.
The Kummer surface was constructed in 1864. It corresponds to the desingularisation of 1 the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in 2 some models of quantum gravity. Following our... more
The Kummer surface was constructed in 1864. It corresponds to the desingularisation of 1 the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in 2 some models of quantum gravity. Following our recent model of the DNA genetic code based on the 3 irreducible characters of the finite group G 5 := (240, 105) ∼ = Z 5 2O (with 2O the binary octahedral 4 group), we now find that groups G 6 := (288, 69) ∼ = Z 6 2O and G 7 := (336, 118) ∼ = Z 7 2O can be 5 used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some 6 biological functions. Groups G 6 and G 7 are found to involve the Kummer surface in the structure of 7 their character table. An analogy between quantum gravity and DNA/RNA packings is suggested. 8
We introduce a quantum model for the Universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of... more
We introduce a quantum model for the Universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac-Moody Lie algebra e 9. We investigate Kac-Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.
In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra g u that extends e 9. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn g u into a Lie superal-gebra... more
In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra g u that extends e 9. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn g u into a Lie superal-gebra sg u with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincaré group on sg u , which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular resonant scattering. Finally, we complete the model by merging the local sg u into a vertex-type algebra.
We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature G = Z5 H where H = Z2.S4 ∼ = 2O is isomorphic to... more
We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature G = Z5 H where H = Z2.S4 ∼ = 2O is isomorphic to the binary octahedral group 2O and S4 is the symmetric group on four letters/bases. The second group has signature G = Z5 GL(2, 3) and points out a threefold symmetry of base pairings. For those groups, the representations for the 22 conjugacy classes of G are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the 22 characters of G attached to those representations are informationally complete. The biological meaning of these coincidences is discussed. Arxiv: q-bio.OT, quant-ph, math.GR, math.AG
A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters κ of a finite group G. Here we introduce a d-dimensional Hilbert space with d = cc(G), the number of conjugacy classes of... more
A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters κ of a finite group G. Here we introduce a d-dimensional Hilbert space with d = cc(G), the number of conjugacy classes of G. Groups under consideration should follow two rules, (a) the character table contains both two-and three-dimensional representations with at least one of them faithful and (b) there are minimal informationally complete measurements under the action of a d-dimensional Pauli group over the characters of these representations. Groups with small d that satisfy these rules coincide in a large part with viable ones derived so far for reproducing simultaneously the CKM (quark) and PNMS (lepton) mixing matrices. Groups leading to physical CP violation are singled out.
In this work we explore how the heat kernel, which gives the solution to the diffusion equation and the Brownian motion, would change when we introduce quasiperiodicity in the scenario. We also study the random walk in the Fibonacci... more
In this work we explore how the heat kernel, which gives the solution to the diffusion equation and the Brownian motion, would change when we introduce quasiperiodicity in the scenario. We also study the random walk in the Fibonacci sequence. We discuss how these ideas would change the discrete approaches to quantum gravity and the construction of quantum geometry.
In this work, the structural transformation from a crystalline to quasicrystalline symmetry in palladium (Pd) and palladium-hydrogen (Pd-H) atomic clusters upon thermal annealing and hydrogenation has been addressed by means of atomistic... more
In this work, the structural transformation from a crystalline to quasicrystalline symmetry in palladium (Pd) and palladium-hydrogen (Pd-H) atomic clusters upon thermal annealing and hydrogenation has been addressed by means of atomistic simulations. A structural analysis of the clusters was performed during the heating up to the melting point to identify the temperature for the phase transformation. It has been demonstrated that nanometric pure Pd clusters transform from cuboctahedral to icosahedral structures under heating. This transformation is thermally activated process and the activation barrier depends on the cluster size. The activation energy of the cubo-ico symmetry transformation was measured using the variable heating rate method and was found to increase with the cluster size from 0.05 eV for 55 atomic cluster up to 0.66 eV for 147 atomic cluster. Hydrogenation of the nanometric Pd clusters yields to the modification of the transformation barrier in a non-monotonic form. At low H concentration, the transformation barrier decreases, while by increasing H concentration above a certain threshold, the barrier grows again thus making a minimum around a specific hydrogen concentration. This behaviour was rationalized as a competition between two processes, namely: the structure symmetry breaking at low H concentrations and stabilization of cuboctahedral phase of the clusters at high H concentration. The obtained results provide an estimation of the temperature range at which the symmetry transformation should occur under thermal annealing with experimentally achievable heating rates.
The synthesis of intermetallic material was carried out by means of dehydrogenating annealing of a (TiH 2) 30 Zr 45 Ni 25 sample in vacuum by an electron beam. The properties of the obtained material were studied for establishing the... more
The synthesis of intermetallic material was carried out by means of dehydrogenating annealing of a (TiH 2) 30 Zr 45 Ni 25 sample in vacuum by an electron beam. The properties of the obtained material were studied for establishing the structural phase composition by scanning electron microscopy and X-ray structural analysis. It was found that prolonged exposure of an electron beam to a sample containing titanium hydride leads to a number of structural transformations in the material, accompanied by a redistribution of hydrogen from titanium to zirconium and culminating in the synthesis of a ternary alloy with characteristic growth structures. The processes of hydrogen sorption-desorption by a synthesized sample were studied, the temperature ranges of these processes and the absorption capacity of the obtained material were established. It was shown that the structure of the sample formed upon heating by an electron beam promotes the absorption of hydrogen at room temperature up to 1.41 wt.%.
Research Interests:
The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a 'magic' state |ψ in d-dimensional... more
The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a 'magic' state |ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite 'contextual' geometry. In the present work, we choose G as the fundamental group π1(V) of an exotic 4-manifold V , more precisely a 'small exotic' (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4). Our selected example, due to to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurence of standard contextual geometries such as the Fano plane (at index 7), Mermin's pentagram (at index 10), the two-qubit commutation picture GQ(2, 2) (at index 15) as well as the combinatorial Grassmannian Gr(2, 8) (at index 28) , (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) R 4 's. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of 'quantum gravity'.
We modify the simulation hypothesis to a self-simulation hypothesis, where the physical universe, as a strange loop, is a mental self-simulation that might exist as one of a broad class of possible code theoretic quantum gravity models of... more
We modify the simulation hypothesis to a self-simulation hypothesis, where the physical universe, as a strange loop, is a mental self-simulation that might exist as one of a broad class of possible code theoretic quantum gravity models of reality obeying the principle of efficient language axiom. This leads to ontological interpretations about quantum mechanics. We also discuss some implications of the self-simulation hypothesis such as an informational arrow of time.
In quasicrystals, any given local patch-called an emperor-forces at all distances the existence of accompanying tiles-called the empire-revealing thus their inherent nonlocality. In this chapter, we review and compare the methods... more
In quasicrystals, any given local patch-called an emperor-forces at all distances the existence of accompanying tiles-called the empire-revealing thus their inherent nonlocality. In this chapter, we review and compare the methods currently used for generating the empires, with a focus on the cut-and-project method, which can be generalized to calculate empires for any quasicrystals that are projections of cubic lattices. Projections of non-cubic lattices are more restrictive and some modifications to the cut-and-project method must be made in order to correctly compute the tilings and their empires. Interactions between empires have been modeled in a game-of-life approach governed by nonlocal rules and will be discussed in 2D and 3D quasicrystals. These nonlocal properties and the consequent dynamical evolution have many applications in quasicrystals research, and we will explore the connections with current material science experimental research.
Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inflation rule protect quantum information stored in the bulk from errors on the boundary provided the code rate is less than one. Hyperbolic... more
Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inflation rule protect quantum information stored in the bulk from errors on the boundary provided the code rate is less than one. Hyperbolic geometry bounds the holographic code rate and guarantees quantum error correction for codes grown with any inflation rule on all regular hyperbolic tessellations in a class whose size grows exponentially with the rank of the perfect tensors for rank five and higher. For the tile completion inflation rule, holographic triangle codes have code rate more than one but all others perform quantum error correction.
Let H be a non trivial subgroup of index d of a free group G and N the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of... more
Let H be a non trivial subgroup of index d of a free group G and N the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of MIC states associated to minimal informationally complete measurements. It is shown that, in most cases, the existence of a MIC state entails that the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups), or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π1(M) of some manifolds encountered in our recent papers, e.g. the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).
Research Interests:
We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within SU (2) quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering... more
We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within SU (2) quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering "fermionic" cycles through the foam we couple this SU (2) quantum group with the same deformation of SU (3), so that we have quantum numbers linked with spacetime symmetry and charge gauge symmetry in the computation of observables. The generalization to higher-dimensional Lie groups SU (N), G 2 and E 8 is suggested. On this basis we discuss a unifying framework for quantum gravity. Inside the transition amplitude or partition function for geometries, we have the quantum numbers of particles and fields interacting in the form of a spin foam network − in the framework of state sum models, we have a sum over quantum computations driven by the interplay between aperiodic order and topological order.
Based on the Cayley-Dickson process, a sequence of multidimensional structured natural numbers (infinions) creates a path from quantum information to quantum gravity. Octonionic structure, exceptional Jordan algebra, and e8 Lie algebra... more
Based on the Cayley-Dickson process, a sequence of multidimensional structured natural numbers (infinions) creates a path from quantum information to quantum gravity. Octonionic structure, exceptional Jordan algebra, and e8 Lie algebra are encoded on a graph with E9 connectivity, decorated by integral matrices. With the magic star, a toy model for a quantum gravity is presented with its naturally emergent quasicrystalline projective compactification.
The fundamental group π1(L) of a knot or link L may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In... more
The fundamental group π1(L) of a knot or link L may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the L of such a quantum computer model and computes their Seifert surfaces and the corresponding Alexander polynomial.
In particular, some d-fold coverings of the trefoil knot, with d=3, 4, 6 or 12, define appropriate links L and the latter two cases connect to the Dynkin diagrams of E6 and D4, respectively. In this new context, one finds that this correspondence continues with the Kodaira’s classification of elliptic singular fibers. The Seifert fibered toroidal manifold Σ′, at the boundary of the singular fiber E8~, allows possible models of quantum computing.
Computational assessment of the discrete breathers (also known as intrinsic localised modes) is performed in nickel and palladium hydrides with an even stoichiometry by means of molecular dynamics simulations. The breathers consisting of... more
Computational assessment of the discrete breathers (also known as intrinsic localised modes) is performed in nickel and palladium hydrides with an even stoichiometry by means of molecular dynamics simulations. The breathers consisting of hydrogen and metallic atoms were excited following the experience obtained earlier by modelling the breathers in pure metallic systems. Stable breathers were only found in the nickel hydride system and only for the hydrogen atoms oscillating along 〈1 0 0〉 and 〈1 1 1〉 polarization axes. At this, two types of the stable breathers involving single oscillating hydrogen and a pair of hydrogen atoms beating in antiphase mode were discovered. Analysis of the breather characteristics reveals that its frequency is located in the phonon gap or lying in the optical phonon band of phonon spectrum near the upper boundary. Analysis of the movement of atoms constituting the breather was performed to understand the mechanism that enables the breather stabilization and long-term oscillation without dissipation its energy to the surrounding atoms. It has been demonstrated that, while in palladium hydride, the dissipation of the intrinsic breather energy due to hydrogen-hydrogen attractive interaction occurs, the stable oscillation in the nickel hydride system is ensured by the negligibly weak hydrogen-hydrogen interaction acting within a distance of the breather oscillation amplitude. Thus, our analysis provides an explanation for the existence of the long-living stable breathers in metallic hy-dride systems. Finally, the high energy oscillating states of hydrogen atoms have been observed for the NiH and PdH lattices at finite temperatures which can be interpreted as a fingerprint of the finite-temperature analogues of the discrete breathers.
Emergence theory is a code-theoretic first-principles based discretized quantum field theoretic approach to quantum gravity and particle physics. This overview covers the primary set of ideas being assembled by Quantum Gravity Research.
On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game of life dynamics governed by non-local rules. Quasicrystals have inherently non-local order since any local patch, the emperor, forces the... more
On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game of life dynamics governed by non-local rules. Quasicrystals have inherently non-local order since any local patch, the emperor, forces the existence of a large number of tiles at all distances, the empires. Considering the emperor and its local patch as a quasiparticle, in this case a glider, its empire represents its field and the interaction between quasiparticles can be modeled as the interaction between their empires. Following a set of rules, we model the walk of life in different setups and we present examples of self-interaction and two-particle interactions in several scenarios. This dynamic is influenced by both higher dimensional representations and local choice of hinge variables. We discuss our results in the broader context of particle physics and quantum field theory, as a first step in building a geometrical model that bridges together higher dimensional representations, quasicrystals and fundamental particles interactions.
It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing d-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs.
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The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal's vertex configurations, frequencies, and empires (forced tiles). We review the... more
The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal's vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.
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In geometrically frustrated clusters of polyhedra, gaps between faces can be closed without distorting the polyhedra by the long established method of discrete curvature, which consists of curving the space into a fourth dimension,... more
In geometrically frustrated clusters of polyhedra, gaps between faces can be closed without distorting the polyhedra by the long established method of discrete curvature, which consists of curving the space into a fourth dimension, resulting in a dihedral angle at the joint between polyhedra in 4D. An alternative method—the twist method—has been recently suggested for a particular case, whereby the gaps are closed by twisting the cluster in 3D, resulting in an angular offset of the faces at the joint between adjacent polyhedral. In this paper, we show the general applicability of the twist method, for local clusters, and present the surprising result that both the required angle of the twist transformation and the consequent angle at the joint are the same, respectively, as the angle of bending to 4D in the discrete curvature and its resulting dihedral angle. The twist is therefore not only isomorphic, but isogonic (in terms of the rotation angles) to discrete curvature. Our results apply to local clusters, but in the discussion we offer some justification for the conjecture that the isomorphism between twist and discrete curvature can be extended globally. Furthermore, we present examples for tetrahedral clusters with three-, four-, and fivefold symmetry.
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A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of 3-manifolds. A magic state and the... more
A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of uqc as a POVM that one recognizes to be a 3-manifold M 3. E. g., the d-dimensional POVMs defined from subgroups of finite index of the modular group P SL(2, Z) in [2] correspond to d-fold M 3-coverings of the trefoil knot. In this paper, one also investigates quantum information on a few 'universal' knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings [3], making use of the catalog of platonic manifolds available on SnapPy [4]. Further connections between POVMs based uqc and M 3 's obtained from Dehn fillings are explored.
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This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the... more
This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth.
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We apply a discrete quantum walk from a quantum particle on a discrete quantum spacetime from loop quantum gravity and show that the related entanglement entropy drives an entropic force. We apply these concepts in a model where walker... more
We apply a discrete quantum walk from a quantum particle on a discrete quantum spacetime from loop quantum gravity and show that the related entanglement entropy drives an entropic force. We apply these concepts in a model where walker positions are topologically encoded on a spin network. Then, we discuss the role of the golden ratio in fundamental physics by addressing charge and length quantization and by analyzing the ratios of fundamental constants−the limits of nature. The limit of minimal length and volume arising in quantum gravity theory indicates an underlying principle that we develop herein.
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Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one... more
Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N) region. The ordinates λn are the positive imaginary parts of the nontrivial zeta zeros in the critical line : sn = 1 2 + iλn. The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.
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Since antiquity, the packing of convex shapes has been of great interest to many scientists and mathematicians [1-7]. Recently, particular interest has been given to packings of three-dimensional tetrahedra [8-20]. Dense packings of both... more
Since antiquity, the packing of convex shapes has been of great interest to many scientists and mathematicians [1-7]. Recently, particular interest has been given to packings of three-dimensional tetrahedra [8-20]. Dense packings of both crystalline [8, 10, 15, 17, 19] and semi-quasicrystalline [14] have been reported. It is interesting that a semi-quasicrystalline packing of tetrahedra can emerge naturally within a thermodynamic simulation approach [14]. However, this packing is not perfectly quasicrystalline and the packing density, while dense, is not maximal. Here we suggest that a "golden rotation" between tetrahedral facial junctions can arrange tetrahedra into a perfect quasicrystalline packing. Using this golden rotation, tetrahedra can be organized into "triangular", "pentagonal", and "spherical" locally dense aggregates. Additionally, the aperiodic Boerdijk-Coxeter helix [23, 24] (tetrahelix) is transformed into a structure of 3-or 5-fold periodicity—depending on the relative chiralities of the helix and rotation—herein referred to as the "philix". Further, using this same rotation, we build (1) a shell structure which resembles a Penrose tiling upon projection into two dimensions, and (2) a "tetragrid" structure assembled of golden rhombohedral unit cells. Our results indicate that this rotation is closely associated with Fuller's "jitterbug transformation" [21] and that the total number of face-plane classes (defined below) is significantly reduced in comparison with general tetrahedral aggregations, suggesting a quasicrystalline packing of tetrahedra which is both dynamic and dense. The golden rotation that we report presents a novel tool for arranging tetrahedra into perfect quasicrystalline, dense packings.
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The hard problem of consciousness must be approached through the ontological lens of 20th century physics, which tells us that reality is information theoretic [1,2] and quantized at the level of Planck scale spacetime[3]. Through careful... more
The hard problem of consciousness must be approached through the ontological lens of 20th century physics, which tells us that reality is information theoretic [1,2] and quantized at the level of Planck scale spacetime[3]. Through careful deduction, it becomes clear that information cannot exist without consciousness – the awareness of things. And to be aware is to hold the meaning of relationships of objects within consciousness – perceiving abstract objects, while enjoying degrees of freedom within the structuring of those relationships. This defines consciousness as language – (1) a set of objects and (2) an ordering scheme with (3) degrees of freedom used for (4) expressing meaning. And since even information at the Planck scale cannot exist without consciousness, we propose an entity called a " primitive unit of consciousness " , which acts as a mathematical operator in a quantized spacetime language. Quasicrystal mathematics based on E8 geometry [4] seems to be a candidate for the language of reality, possessing several qualities corresponding to recent physical discoveries and various physically realistic unification models.
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Geometric methods for constructing exact solutions of motion equations with first order α ′ corrections to the heterotic supergravity action implying a non-trivial Yang-Mills sector and six dimensional, 6-d, almost-Kähler internal spaces... more
Geometric methods for constructing exact solutions of motion equations with first order α ′ corrections to the heterotic supergravity action implying a non-trivial Yang-Mills sector and six dimensional, 6-d, almost-Kähler internal spaces are studied. In 10-d spacetimes, general parametrizations for generic off–diagonal metrics, nonlinear and linear connections and matter sources, when the equations of motion decouple in very general forms are considered. This allows us to construct a variety of exact solutions when the coefficients of fundamental geometric/physical objects depend on all higher dimensional spacetime coordinates via corresponding classes of generating and integration functions, generalized effective sources and integration constants. Such generalized solutions are determined by generic off–diagonal metrics and nonlinear and/or linear connections. In particular, as configurations which are warped/compactified to lower dimensions and for Levi–Civita connections. The corresponding metrics can have (non) Killing and/or Lie algebra symmetries and/or describing (1+2)-d and/or (1+3)-d domain wall configurations, with possible warping nearly almost-Kähler manifolds, with gravitational and gauge instantons for nonlinear vacuum configurations and effective polarizations of cosmological and interaction constants encoding string gravity effects. A series of examples of exact solutions describing generic off-diagonal supergravity modifications to black hole/ ellipsoid and solitonic configurations are provided and analyzed. We prove that it is possible to reproduce the Kerr and other type black solutions in general relativity (with certain types of string corrections) in 4-d and to generalize the solutions to non-vacuum configurations in (super) gravity/ string theories.
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Quasicrystals (QCs) are a novel form of matter, which are neither crystalline nor amorphous. Among many surprising properties of QCs is their high catalytic activity. We propose a mechanism explaining this peculiarity based on unusual... more
Quasicrystals (QCs) are a novel form of matter, which are neither crystalline nor amorphous. Among many surprising properties of QCs is their high catalytic activity. We propose a mechanism explaining this peculiarity based on unusual dynamics of atoms at special sites in QCs, namely, localized anharmonic vibrations (LAVs) and phasons. In the former case, one deals with a large amplitude (~ fractions of an angstrom) time-periodic oscillations of a small group of atoms around their stable positions in the lattice, known also as discrete breathers, which can be excited in regular crystals as well as in QCs. On the other hand, phasons are a specific property of QCs, which are represented by very large amplitude (~angstrom) oscillations of atoms between two quasi-stable positions determined by the geometry of a QC. Large amplitude atomic motion in LAVs and phasons results in time-periodic driving of adjacent potential wells occupied by hydrogen ions (protons or deuterons) in case of hydrogenated QCs. This driving may result in the increase of amplitude and energy of zero-point vibrations (ZPV). Based on that, we demonstrate a drastic increase of the D-D or D-H fusion rate with increasing number of modulation periods evaluated in the framework of Schwinger model, which takes into account suppression of the Coulomb barrier due to lattice vibrations. In this context, we present numerical solution of Schrodinger equation for a particle in a non-stationary double well potential, which is driven time-periodically imitating the action of a LAV or phason. We show that the rate of tunneling of the particle through the potential barrier separating the wells is enhanced drastically by the driving, and it increases strongly with increasing amplitude of the driving. These results support the concept of nuclear catalysis in QCs that can take place at special sites provided by their inherent topology. Experimental verification of this hypothesis can lead to new ways of engineering materials containing nuclear active environments based on QC catalytic properties.
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The goal of this work on mathematical cosmology and geometric methods in modifed gravity theories, MGTs, is to investigate Starobinsky-like infation scenarios determined by gravitational and scalar field configurations mimicking... more
The goal of this work on mathematical cosmology and geometric methods in modifed gravity theories, MGTs, is to investigate Starobinsky-like infation scenarios determined by gravitational and scalar field configurations mimicking quasicrystal, QC, like structures.
Such spacetime aperiodic QCs are different from those discovered and studied in solid state physics but described by similar geometric methods. We prove that inhomogeneous and locally anisotropic gravitational and matter field effective QC mixed continuous and
discrete "ether" can be modeled by exact cosmological solutions in MGTs and Einstein gravity. The coeffcients of corresponding generic off-diagonal metrics and generalized connections depend (in general) on all spacetime coordinates via generating and integration
functions and certain smooth and discrete parameters. Imposing additional nonholonomic constraints, prescribing symmetries for generating functions and solving the boundary conditions for integration functions and constants, we can model various nontrivial torsion QC structures or extract cosmological Levi-Civita configurations with diagonal metrics reproducing de Sitter (inflationary) like and other type homogeneous inflation and acceleration phases. Finally, we speculate how various dark energy and dark matter effects
can be modelled by off-diagonal interactions and deformations of a nontrivial QC like gravitational vacuum structure and analogous scalar matter fields.
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Heterotic supergravity with (1+3)–dimensional domain wall configurations and (warped) internal, six dimensional, almost-Kähler manifolds 6 X are studied. Considering on ten dimensional spacetime, nonholo-nomic distributions with... more
Heterotic supergravity with (1+3)–dimensional domain wall configurations and (warped) internal, six dimensional, almost-Kähler manifolds 6 X are studied. Considering on ten dimensional spacetime, nonholo-nomic distributions with conventional double fibrations, 2+2+...=2+2+3+3, and associated SU (3) structures on internal space, we generalize for real, internal, almost symplectic gravitational structures the constructions with gravitational and gauge instantons of tanh-kink type [1, 2]. They include the first α ′ corrections to the heterotic supergravity action, parameterized in a form to imply nonholonomic deformations of the Yang-Mills sector and corresponding Bianchi identities. We show how it is possible to construct a variety of solutions, depending on the type of nonholonomic distributions and deformations of 'prime' instanton configurations characterized by two real supercharges. This corresponds to N = 1/2 supersymmetric, nonholonomic manifolds from the four dimensional point of view. Our method provides a unified description of embedding nonholonomically deformed tanh-kink-type instantons into half-BPS solutions of heterotic supergravity. This allows us to elaborate new geometric methods of constructing exact solutions of motion equations, with first order α ′ corrections to the heterotic supergravity. Such a formalism is applied for general and/or warped almost-Kähler configurations, which allows us to generate nontrivial (1+3)-d domain walls. This formalism is utilized in our associated publication [3] in order to construct and study generic off-diagonal nonholonomic deformations of the Kerr metric, encoding contributions from heterotic supergravity.
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We present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal can also be thought as a golden composition of five sets of Fibonacci... more
We present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal can also be thought as a golden composition of five sets of Fibonacci tetragrids. We found that this quasicrystal embeds the quasicrystals that are golden compositions of the three-dimensional tetrahedral cross-sections of the Elser-Sloane quasicrystal, which is a four-dimensional cut-and-project of the E8 lattice. These compound quasicrystals are subsets of the Fibonacci icosagrid, and they can be enriched to form the Fibonacci icosagrid. This creates a mapping between the Fibonacci icosagrid and the E−8 lattice. It is known that the combined structure and dynamics of all gravitational and Standard Model particle fields, including fermions, are part of the E8 Lie algebra. Because of this, the Fibonacci icosagrid is a good candidates, for representing states and interactions between particles and fields in quantum mechanics. We coin the name Quasicrystalline Spin-Network (QSN) for this quasicrystalline structure.
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And 11 more

Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one... more
Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N) region. The ordinates λn are the positive imaginary parts of the nontrivial zeta zeros in the critical line : sn = 1 2 + iλn. The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.
Research Interests:
Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one... more
Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N) region. The ordinates λ_n are the positive imaginary parts of the nontrivial zeta zeros in the critical line : s_n = 1 2 + iλ_n. The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.
Research Interests: