Search Results (15)

If spacetime is a physical object, it is conceivable that it loses its integrity or is destroyed in some way as a continuum in an abrupt process initiated in spacetime itself. An example is a gravitational collapse leading to a spacetime singularity, as in the interior of a black hole. We find a conservative extension of quantum mechanics by quantum set theory over the singular domain and show that it is reconcilable with the special extension of spacetime 4-diffeomorphisms by automorphisms of Boolean models of set theory. The extension of quantum mechanics supports the random sequences of the quantum mechanical outcomes that can negate Tsirelson’s conjecture, whereas the extension of 4-diffeomorphisms indicates the role of exotic smooth 4-spheres as gravitational instantons. This leads to the negation of the smooth 4-dimensional Poincaré conjecture before its final resolution by mathematicians. We also discuss the case where the Poincaré conjecture would remain true. Full article

Finite difference methods are commonly used in the pricing of discretely monitored exotic options in the Black–Scholes framework, but they tend to converge slowly due to discontinuities contained in terminal conditions. We present an effective analytical modification to existing finite difference methods that greatly enhances their performance on discretely monitored options with non-smooth terminal conditions. We apply this modification to the popular Crank–Nicolson method and obtain highly accurate option pricing results with significantly reduced CPU cost. We also introduce an adaptive mesh refinement technique that further improves the computational speed of the modified finite difference method. The proposed method is especially useful for options with high monitoring frequencies, which are difficult to price using other existing methods. Full article

The deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) has garnered significant attention for its ability to describe the properties of nuclei across the entire nuclear chart, from light to heavy nuclei, including both stable and exotic ones. As part of ongoing efforts to construct a mass table using the DRHBc theory, determining the ground states of nuclei is a crucial task in the systematic studies of deformed nuclei. In this work, a strategy for identifying the ground state in the superheavy nuclei region is proposed and evaluated, by taking $Z=134$ and 135 isotopes as examples. First, we examine how the step size of the initial quadrupole deformation parameter, $\Delta {\beta }_2$, affects the pattern of the potential energy curves (PECs) and the determination of the ground state. Our findings indicate that $\Delta {\beta }_2=0.05$ producing smooth and well-defined PECs while maintaining an acceptable numerical cost. Next, we explore the convergence of PECs with respect to the angular momentum cutoff, $J_{\max }$. Based on the results, we recommend using $J_{\max }=31/2\hslash$, especially for nuclei with competing oblate and prolate minima. Finally, we conclude that the accurate identification of the ground state can be achieved by performing unconstrained calculations around the minima of the PECs. Full article

The structure of skyrmion and spiral solutions, investigated within the phenomenological Dzyaloshinskii model of chiral magnets near the ordering temperatures, is characterized by the strong interplay between longitudinal and angular order parameters, which may be responsible for experimentally observed precursor effects. Within the precursor regions, additional effects, such as pressure, electric fields, chemical doping, uniaxial strains and/or magnetocrystalline anisotropies, modify the energetic landscape and may even lead to the stability of such exotic phases as a square staggered lattice of half-skyrmions, the internal structure of which employs the concept of the “soft” modulus and contains points with zero modulus value. Here, we additionally alter the stiffness of the magnetization modulus to favor one- and two-dimensional modulated states with large modulations of the order parameter magnitude. The computed phase diagram, which omits any additional effects, exhibits stability pockets with a square half-skyrmion lattice, a hexagonal skyrmion lattice with the magnetization in the center of the cells parallel to the applied magnetic field, and helicoids with propagation transverse to the field, i.e., those phases in which the notion of localized defects is replaced by the picture of a smooth but more complex tiling of space. We note that the results can be adapted to metallic glasses, in which the energy contributions are the same and originate from the inherent frustration in the models, and chiral liquid crystals with a different ratio of elastic constants. Full article

Recent measurements on heavy-flavour production as a function of charged-particle multiplicity at the LHC are discussed. Focus is given to quarkonium results in small (pp or pPb) collision systems. The measurements of relative yields, i.e., the ratio of the particle yields in given multiplicity intervals to the multiplicity integrated yield are presented and compared to model calculations from Monte Carlo event generators as well as to models considering effects at play in the initial and/or final state of the collision. The absolute inclusive J/$\psi$ yield as a function of the absolute charged-particle multiplicity is evaluated; a smooth behaviour of the absolute yield is observed across collision systems, from pp to pPb and PbPb collisions. Analogous measurements of the excited-to-ground state quarkonium ratios as a function of charged-particle multiplicity are also reviewed. Finally, the study of exotic particle production as a function of charged-particle multiplicity is introduced as a complementary tool to investigate the nature of the ${\chi }_{\mathrm{c}1}\left(3872\right)$ hadron. Full article

Three-dimensional topological insulators (3D-TIs) are a new generation of materials with insulating bulk and exotic metallic surface states that facilitate a wide variety of ground-breaking applications. However, utilization of the surface channels is often hampered by the presence of crystal defects, such as antisites, vacancies, and twin domains. For terahertz device applications, twinning is shown to be highly deleterious. Previous attempts to reduce twins using technologically important InP(111) substrates have been promising, but have failed to completely suppress twin domains while preserving high structural quality. Here we report growth of twin-free molecular beam epitaxial Bi₂Se₃ and Sb₂Te₃ structures on ultra-thin In₂Se₃ layers formed by a novel selenium passivation technique during the oxide desorption of smooth, non-vicinal InP(111)B substrates, without the use of an indium source. The formation of un-twinned In₂Se₃ provides a favorable template to fully suppress twin domains in 3D-TIs, greatly broadening novel device applications in the terahertz regime. Full article

The study of the exotic properties of the surface states of topological insulators requires defect-free and smooth surfaces. This work aims to study the enhancement of the surface morphology of optimally doped, high-crystalline (Bi_0.4Sb_0.6)₂Te₃ films deposited by molecular beam epitaxy on Al₂O₃ (001) substrates. Atomic force microscopy shows that by employing an in situ thermal post anneal, the surface roughness is reduced significantly, and transmission electron microscopy reveals that structural defects are diminished substantially. Thence, these films provide a great platform for the research on the thickness-dependent properties of topological insulators. Full article

In this paper, we discuss the initial state of the universe at the Big Bang. By using the ideas of Freedman in the proof of the disk embedding theorem for 4-manifolds, we describe the corresponding spacetime as a gravitational instanton. The spatial space is a fractal space (wild embedded 3-sphere). Then, we construct the quantum state from this fractal space. This quantum state is part of the string algebra of Ocneanu. There is a link between the Jones polynomial and Witten’s topological field theory. Using this link, we are able to determine the physical theory (action) as the Chern–Simons functional. The gauge fixing of this action determines the foliation of the spacetime and the smoothness properties. Finally, we determine the quantum symmetry of the quantum state to be the enveloped Lie algebra $U_q\left(s{l}_2\left(\mathbb{C}\right)\right)$, where q is the fourth root of unity. Full article

Smooth brome (Bromus inermis Leyss.) is an invasive cool-season grass that has spread throughout the Great Plains of North America. The species is considered one of the most widespread exotic grasses that has successfully invaded both cool-season and warm-season native prairies. In the prairies where it has invaded, there has often been a total elimination of native species and an overall homogenization of ecosystems. Smooth brome has greater competitive abilities compared to many native grasses and can foster their total elimination in many instances. The greater competitiveness can be partially attributed to its ability to alter the soil and hydrological properties of a site. It is a deep-rooted rhizomatous grass species that thrives in nitrogen-enriched soil, and since its leaf tissue decomposes faster than native species, it in turn increases the soil nitrogen level, causing positive plant-soil feedback. Moreover, smooth brome is able to transport the required nutrients from older plants to the newer progenies invading new nutrient-depleted areas, making it a potent invader. However, the impact of smooth brome is not limited to soil biochemistry alone; it also affects other ecosystem components such as the movement and behavior of many native arthropods, thereby altering the overall population dynamics of such species. Thus, smooth brome invasion poses a serious threat to the remnant prairies of the Great Plains, and efficient management strategies are urgently needed to control its invasion. Control measures such as mowing, grazing, burning, and herbicide application have been effectively used to manage this species. However, due to the widespread distribution of smooth brome across North America and its adaptability to a wide range of environmental conditions, it is challenging to translate the management strategies from one area to another. Full article

In this paper, we focus on some aspects of the relation of spacetime and quantum mechanics and the study counterparts (in Set) of the categorical local symmetries of smooth 4-manifolds. In the set-theoretic limit, there emerge some exotic smoothness structures on ${\mathbb{R}}^4$ (hence the Riemannian nonvanishing curvature), which fit well with the quantum mechanical lattice of projections on infinite-dimensional Hilbert spaces. The method we follow is formalization localized on the open covers of the spacetime manifold. We discuss our findings in the context of the information paradox assigned to evaporating black holes. A black hole can evaporate entirely, but the smoothness structure of spacetime will be altered and, in this way, the missing information about the initial states of matter forming the black hole will be encoded. Thus, the possible global geometric remnant of black holes in spacetime is recognized as exotic 4-smoothness. The full-fledged verification of this proposal will presumably be possible within the scope of future quantum gravity theory research. Full article

In this paper we will detail an approach to generate metrics and matter models on end-periodic manifolds, which are used extensively in the study of the exotic smooth structures of ${\mathbb{R}}^4$. After an overview of the technique, we will present two specific examples, discuss the associated matter models by solving the Einstein equations, and determine the physical viability by examining the energy conditions. We compare the resulting model directly with existing models of matter distributions in extragalactic systems, to highlight the viability of utilizing exotic smooth structures to understand the existence and distribution of dark matter. Full article

In this paper, we will describe the idea that dark matter partly consists of gravitational solitons (gravisolitons). The corresponding solution is valid for weak gravitational fields (weak field limit) with respect to a background metric. The stability of this soliton is connected with the existence of a special foliation and amazingly with the smoothness properties of spacetime. Gravisolitons have many properties of dark matter, such as no interaction with light but act on matter via gravitation. In this paper, we showed that the gravitational lensing effect of gravisolitons agreed with the lensing effect of usual matter. Furthermore, we obtained the same equation of state $w=0$ as matter. Full article

Various experimentally verified values of physical parameters indicate that the universe evolves close to the topological phase of exotic smoothness structures on ${\mathbb{R}}^4$ and K3 surface. The structures determine the $\alpha$ parameter of the Starobinski model, the number of e-folds, the spectral tilt, the scalar-to-tensor ratio and the GUT and electroweak energy scales, as topologically supported quantities. Neglecting exotic $R^4$ and K3 leaves these free parameters undetermined. We present general physical and mathematical reasons for such preference of exotic smoothness. It appears that the spacetime should be formed on open domains of smooth $\mathrm{K}3\#\overline{C{P}^2}$ at extra-large scales possibly exceeding our direct observational capacities. Such potent explanatory power of the formalism is not that surprising since there exist natural physical conditions, which we state explicitly, that allow for the unique determination of a spacetime within the exotic K3. Full article

Category theory allows one to treat logic and set theory as internal to certain categories. What is internal to SET is 2-valued logic with classical Zermelo–Fraenkel set theory, while for general toposes it is typically intuitionistic logic and set theory. We extend symmetries of smooth manifolds with atlases defined in Set towards atlases with some of their local maps in a topos $\mathcal{T}$ . In the case of the Basel topos and ${\mathbb{R}}^4$ , the local invariance with respect to the corresponding atlases implies exotic smoothness on ${\mathbb{R}}^4$ . The smoothness structures do not refer directly to Casson handless or handle decompositions, which may be potentially useful for describing the so far merely putative exotic ${\mathbb{R}}^4$ underlying an exotic $S^4$ (should it exist). The tovariance principle claims that (physical) theories should be invariant with respect to the choice of topos with natural numbers object and geometric morphisms changing the toposes. We show that the local $\mathcal{T}$ -invariance breaks tovariance even in the weaker sense. Full article

Recently, a cosmological model based on smooth open 4-manifolds admitting non-standard smoothness structures was proposed. The manifolds are exotic versions of ${\mathbb{R}}^4$ and $S^3\times \mathbb{R}$ . The model has been developed further and proven to be capable of obtaining some realistic cosmological parameters from these exotic smoothings. The important problem of the quantum origins of the exotic smoothness of space-time is addressed here. It is shown that the algebraic structure of the quantum-mechanical lattice of projections enforces exotic smoothness on ${\mathbb{R}}^n$ . Since the only possibility for such a structure is exotic $R^4$ , it is found to be a reasonable explanation of the large-scale four-dimensionality of space-time. This is based on our recent research indicating the role of set-theoretic forcing in quantum mechanics. In particular, it is shown that a distributive lattice of projections implies the standard smooth structure on ${\mathbb{R}}^4$ . Two examples of models valid for cosmology are discussed. The important result that the cosmological constant can be identified with the constant curvature of the embedding $\left(\mathrm{exotic}{R}^4\right)\to {\mathbb{R}}^4$ is referred. . The calculations are in good agreement with the observed small value of the dark energy density. Full article