Search Results (8)

The derivations presented in this paper suggest an intimate relationship between geometry and the electroweak sector at the Planck scale. A Lorentz-invariant maximally symmetric stochastically perturbed spacetime transformed to spherical coordinates reveals an emergent Schwarzschild metric, entirely a statistical structure of stochastic spacetime. Similarly, the transition from a maximally symmetric universe with a complex $SU\left(2\right)$ scalar doublet $\varphi$, comprising four independent real scalar fields with a zero vacuum expectation value (VEV), to spherical coordinates at the Planck scale reveals the spontaneously broken electroweak (EW) sector. Working in the unitarity gauge, the resulting EW potential can be simultaneously mapped in space at the Planck scale and across the EW sector. In space, the resulting EW potential includes a deep well within the Schwarzschild sphere and a shallow well just outside corresponding to an accretion disk. The same potential mapped in the EW space provides an entire family of possible sombrero hat potentials with fourth-order coupling specific to a point in space. At the minimum points of the potential in space, inside the Schwarzschild sphere and at the accretion disk, the $\lambda$ corresponding to the Standard Model (SM) fourth-order coupling is instead derived as $\frac{\lambda }{5}$. The factor of $\frac{1}{5}$ is a simple consequence of the conservation of the EW VEV and the fact that the SM formulation of the EW potential does not account for situations where the perturbations in $\varphi$ dominate. A more general formulation of the EW potential restores the SM quartic coupling and preserves $\lambda$ in space. An emergent Higgs field inside the Schwarzschild black hole is found to directly relate to the stochastic spacetime fields normalized by the Schwarzschild radius. The corresponding Higgs vacuum has both a ground and excited state and the possibility of both positive and negative vacuum entropy. Finally, the scalar-field VEV degeneracy in EW space of the metastable Higgs vacuum appears instead differentiated in space with possible probability, tunneling, and entropy implications. Full article

In this work, we define the multi-radical quadratic and multi-radical quartic mappings as two systems of radical functional equations and then represent them as two equations. Then, we establish some results concerning the stability of multi-radical quadratic and multi-radical quartic mappings by applying a fixed-point based on Brzdȩk. As a direct consequence, we prove the Rassias and Hyers stability of the mentioned mappings. Full article

In this article, we defined the generalized intuitionistic P-pseudo fuzzy 2-normed spaces and investigated the Hyers stability of m-mappings in this space. The m-mappings are interesting functional equations; these functional equations are additive for m = 1, quadratic for m = 2, cubic for m = 3, and quartic for m = 4. We have investigated the stability of four types of functional equations in generalized intuitionistic P-pseudo fuzzy 2-normed spaces by the fixed point method. Full article

Designing blades for efficient energy transfer by turning the flow and angular momentum change is both an art and iterative multidisciplinary engineering process. A robust parametric design tool with few inputs to create 3D blades for turbomachinery and rotating or non-rotating energy converters is described in this paper. The parameters include axial–radial coordinates of the leading/trailing edges, construction lines (streamlines), metal angles, thickness-to-chord ratio, standard, and user-defined airfoil type among others. Using these, 2D airfoils are created, conformally mapped to 3D stream surfaces, stacked radially with multiple options, and they are transformed to a 3D Cartesian coordinate system. Smooth changes in blade curvature are essential to ensure a smooth pressure distribution and attached flow. B-splines are used to control meanline curvature, thickness, leading edge shape, sweep-lean, and other parameters chordwise and spanwise, making the design iteration quick and easy. C2 curve continuity is achieved through parametric segments of cubic and quartic B-splines and is better than G2. New geometries using an efficient parametric scheme and minimal CAD interaction create watertight solid bodies and optional fluid domains. Several examples of ducted axial and radial turbomachinery with special airfoil shapes or otherwise, unducted rotors including propellers and wind and hydrokinetic turbines are presented to demonstrate versatility and robustness of the tool and can be easily tied to any automation chain and optimizer. Full article

An elliptical Hough transform (EHT) algorithm is proposed in the framework of track-before-detect (TBD) for joint detection and tracking of weak exoatmospheric targets. The new approach exploits the fact that when restricted to a two-body problem, the exoatmospheric target often follows an elliptical orbit, and thus the Hough transform integrated with orbital geometry information would have better detection performance. The relationship between the original radar measurements in data space and the elliptical parameters in parameter space is explicitly derived with multiple steps of coordinate transformation. It is found that the data points mapping into the parameter space essentially represent a quartic curve. An EHT-based algorithm is then designed, and orbit planarity is also taken into account to reduce the effect of noise accumulation. The influences of primary and secondary thresholds and the signal-to-noise ratio (SNR) on the detection performance are compared by simulations. Additionally, a real radar tracking dataset from a scientific satellite on 28 May 2017 is used to investigate the efficiency of the method. By adding some imaginary clutter to the raw orbit, the results indicate that it is very effective in detecting the real satellite trajectory in a low signal-to-noise ratio (SNR) environment. The advantage of the new method lies in it can not only simultaneously detect and track weak exoatmospheric targets but also can predict the trajectory by using these available detected parameters. Full article

In this article, we deal with stabilities of several functional equations in n-Banach spaces. For a surjective mapping f into a n-Banach space, we prove the generalized Hyers–Ulam stabilities of the cubic functional equation and the quartic functional equation for f in n-Banach spaces. Full article