Fixed point results for the complex fractal generation in the S -iteration orbit with s -convexity

Since the introduction of complex fractals by Mandelbrot they gained much attention by the researchers. One of the most studied complex fractals are Mandelbrot and Julia sets. In the literature one can find many generalizations of those sets. One of such generalizations is the use of the results from fixed point theory. In this paper we introduce in the generation process of Mandelbrot and Julia sets a combination of the S-iteration, known from the fixed point theory, and the s-convex combination. We derive the escape criteria needed in the generation process of those fractals and present some graphical examples

Relationship between the Mandelbrot Algorithm and the Platonic Solids

This paper focuses on the dynamics of the eight tridimensional principal slices of the tricomplex Mandelbrot set for the power 2: the Tetrabrot, the Arrowheadbrot, the Mousebrot, the Turtlebrot, the Hourglassbrot, the Metabrot, the Airbrot (octahedron) and the Firebrot (tetrahedron). In particular, we establish a geometrical classification of these 3D slices using the properties of some specific sets that correspond to projections of the bicomplex Mandelbrot set on various two-dimensional vector subspaces, and we prove that the Firebrot is a regular tetrahedron. Finally, we construct the so-called "Stella octangula" as a tricomplex dynamical system composed of the union of the Firebrot and its dual, and after defining the idempotent 3D slices of $\mathcal{M}_{3}$, we show that one of them corresponds to a third Platonic solid: the cube

The role of supersymmetry in the black hole/qubit correspondence

This thesis explores the numerous relationships between the entropy of black hole solutions in supergravity and the entanglement of multipartite systems in quantum information theory: the so-called black hole/qubit correspondence. We examine how, through the correspondence, the dyonic charges in the entropy of supersymmetric black hole solutions are directly matched to the state vector coefficients in the entanglement measures of their quantum information analogues. Moreover the Uduality invariance of the black hole entropy translates to the stochastic local operations and classical communication (SLOCC) invariance of the entanglement measures. Several examples are discussed, with the correspondence broadening when the supersymmetric classification of black holes is shown to match the entanglement classification of the qubit/qutrit analogues. On the microscopic front, we study the interpretation of D-brane wrapping configurations as real qubits/qutrits, including the matching of generating solutions on black hole and qubit sides. Tentative generalisations to other dimensions and qubit systems are considered. This is almost eclipsed by more recent developments linking the nilpotent U-duality orbit classi cation of black holes to the nilpotent classi cation of complex qubits. We provide preliminary results on the corresponding covariant classi cation. We explore the interesting parallel development of supersymmetric generalisations of qubits and entanglement, complete with two- and three-superqubit entanglement measures. Lastly, we briefly mention the supergravity technology of cubic Jordan algebras and Freudenthal triple systems (FTS), which are used to: 1) Relate FTS ranks to threequbit entanglement and compute SLOCC orbits. 2) Define new black hole dualities distinct from U-duality and related by a 4D/5D lift. 3) Clarify the state of knowledge of integral U-duality orbits in maximally extended supergravity in four, five, and six dimensions

Twisted Covariant Form Hierarchies: From Hidden Symmetries in M-theory to Anomalies in Heterotic Backgrounds.

In this thesis, we compute the twisted covariant form hierarchies (TCFHs) of minimal supergravity theories in four and five dimensions, as well as, eleven-dimensional super-gravity and the internal space of its warped AdS backgrounds. As a consequence, the form bilinears satisfy a generalised conformal Killing-Yano equation with respect to the TCFH connection. Then, we find the (hidden) symmetries generated by the form bilin-ears in spinning particle actions propagating in certain supersymmetric backgrounds of D = 4, N = 2 and D = 5, N = 1 minimal supergravities, M-brane backgrounds which include the M2-brane, M5-brane, pp-wave and KK-monopole, maximal supersymmetric AdS backgrounds and some AdS backgrounds that arise as near horizon geometries of intersecting M-branes. In addition, we explore whether the form bilinears are sufficient to prove the integrability of particle probe dynamics on M-brane backgrounds. More-over, we show that the covariantly constant forms of heterotic backgrounds with SU(2) and SU(3) holonomy generate a W-symmetry algebra in two-dimensional non-linear su-persymmetric sigma models with the previous backgrounds as target spaces and analyse the consistency conditions of the chiral anomalies arising from all symmetry generators required for the closure of the algebra

Geometry of strings and branes

De elementaire-deeltjesfysica probeert de fundamentele bouwstenen van de Natuur en hun onderlinge wisselwerkingen te beschrijven. Uit experimenten is gebleken dat de elementaire deeltjes in twee klassen zijn onder te brengen: de leptonen, waaronder het elektron en het neutron; en de quarks, de bouwstenen van protonen en neutronen. ... Zie: Samenvatting

Multispace & Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of Sciences), Vol. IV

The fourth volume, in my book series of “Collected Papers”, includes 100 published and unpublished articles, notes, (preliminary) drafts containing just ideas to be further investigated, scientific souvenirs, scientific blogs, project proposals, small experiments, solved and unsolved problems and conjectures, updated or alternative versions of previous papers, short or long humanistic essays, letters to the editors - all collected in the previous three decades (1980-2010) – but most of them are from the last decade (2000-2010), some of them being lost and found, yet others are extended, diversified, improved versions. This is an eclectic tome of 800 pages with papers in various fields of sciences, alphabetically listed, such as: astronomy, biology, calculus, chemistry, computer programming codification, economics and business and politics, education and administration, game theory, geometry, graph theory, information fusion, neutrosophic logic and set, non-Euclidean geometry, number theory, paradoxes, philosophy of science, psychology, quantum physics, scientific research methods, and statistics. It was my preoccupation and collaboration as author, co-author, translator, or cotranslator, and editor with many scientists from around the world for long time. Many topics from this book are incipient and need to be expanded in future explorations

Index relations and fusion rules: Explorations of Supersymmetric, Conformal, and Topological Field Theories.

PhD Theses.This thesis explores the world of quantum eld theories through an analytic approach. It focuses on three special types of quantum eld theories: supersymmetric, conformal and topological ones. The necessary background knowledge is introduced in chapter one, then two types of problems are studied in the next three chapters: index relations and fusion rules.1 For index relations we study certain exactly marginal gaugings involving arbitrary numbers of Argyres-Douglas (AD) theories and show that the resulting Schur indices are related to those of certain Lagrangian theories of class S via simple transformations. By writing these quantities in the language of 2D topological quantum eld theory (TQFT), we easily read o the S-duality action on the avor symmetries of the AD quivers and also nd expressions for the Schur indices of various classes of exotic AD theories appearing in di erent decoupling limits. The TQFT expressions for these latter theories are related by simple transformations to the corresponding quantities for certain well-known isolated theories with regular punctures (e.g., the Minahan-Nemeschansky E6 theory and various generalizations). We then reinterpret the TQFT expressions for the indices of our AD theories in terms of the topology of the corresponding 3D mirror quivers, and we show that our isolated AD theories generically admit renormalization group (RG) ows to interacting superconformal eld theories (SCFTs) with thirty-two (Poincar e plus special) supercharges. Motivated by these examples, we argue that, in a sense we make precise, the existence of RG ows to interacting SCFTs with thirty-two supercharges is generic in a far larger class of 4D N = 2 SCFTs arising from compacti cations of the 6D (2; 0) theory on surfaces with irregular singularities. Then we study fusion rules in modular tensor categories. We rst relate fusion rules to the mathematical conjecture of Arad and Herzog (AH) in group theory: in nite simple groups, the product of two conjugacy classes of length greater than one is never a single conjugacy class. We discuss implications of this conjecture for nonabelian anyons in 2 + 1-dimensional discrete gauge theories. Thinking in this way suggests closely related statements about nite simple groups and their associated discrete gauge theories. We prove these statements and give physical intuition for their validity. Finally, we explain that the lack of certain dualities in theories with nonabelian nite simple gauge groups provides a non-trivial check of the AH conjecture. We also study the implications of the anyon fusion equation a b = c on global properties of 2 + 1D topological quantum eld theories (TQFTs). Here a and b are anyons that fuse together to give a unique anyon, c. As is well known, when at least one of a and b is abelian, such equations describe aspects of the one-form symmetry of the theory. When a and b are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and a and b lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian a and b can also be an indication of zero-form symmetries in a TQFT, of what we term \quasi-zero-form symmetries" (as in the case of discrete gauge 1Chapter two, three , four are based on the papers [34],[35],[36] respectively. 2 theories based on the largest Mathieu group, M24), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition