Сompactness of one class of fractional integrating operator with variable upper limit

The paper establishes a characterization of the compactness for fractional operators of a general class, including the Riemann-Liouville, Hadamard and Erdelyi-Kober operators. The paper considers an integral fractional integration operator of Hardy type with nonnegative kernels and a variable limit of integration (a function as the upper limit of integration) and under certain conditions on the kernel, a criterion of the compactness in weighted Lebesgue spaces is obtained for this operator, when the parameters of the spaces satisfy the conditions Moreover, more general results are obtained for the weighted differential inequality of Hardy type on the set of locally absolutely continuous functions that vanish and infinity at the ends of the interval, covering the previously known results, and more precise estimates for the best constant are given. The localization method, Schauder’s theorem, the Kantorovich test, and the theorem on the uniform limit of compact operators were used in the proof of the main theorem. The obtained results of the study the compactness of fractional integration operators can be used in the estimation of solutions of differential equations that model various processes in mathematics. In particular, these results yield new results in the theory of Hardy-type inequalities.

Optimal strategy to certify quantum nonlocality

Certification of quantum nonlocality plays a central role in practical applications like device-independent quantum cryptography and random number generation protocols. These applications entail the challenging problem of certifying quantum nonlocality, something that is hard to achieve when the target quantum state is only weakly entangled, or when the source of errors is high, e.g. when photons propagate through the atmosphere or a long optical fiber. Here we introduce a technique to find a Bell inequality with the largest possible gap between the quantum prediction and the classical local hidden variable limit for a given set of measurement frequencies. Our method represents an efficient strategy to certify quantum nonlocal correlations from experimental data without requiring extra measurements, in the sense that there is no Bell inequality with a larger gap than the one provided. Furthermore, we also reduce the photodetector efficiency required to close the detection loophole. We illustrate our technique by improving the detection of quantum nonlocality from experimental data obtained with weakly entangled photons.

Optimal Strategy to Certify Quantum Nonlocality

Certification of quantum nonlocality plays a central role in practical applications like device-independent quantum cryptography and random number generation protocols. These applications entail the challenging problem of certifying quantum nonlocality, something that is hard to achieve when the target quantum state is only weakly entangled, or when the source of errors is high, e.g. when photons propagate through the atmosphere or a long optical fiber. Here we introduce a technique to find a Bell inequality with the largest possible gap between the quantum prediction and the classical local hidden variable limit for a given set of measurement frequencies. Our method represent an efficient strategy to certify quantum nonlocal correlations from experimental data without requiring extra measurements, in the sense that there is no Bell inequality with a larger gap than the one provided. Furthermore, we also reduce the photodetector efficiency required to close the detection loophole. We illustrate our technique by improving the detection of quantum nonlocality from experimental data obtained with weakly entangled photons.

Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters, are often of main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.

Probabilistic Safety Pin Coupling Failure Analysis to Explore

Material properties and geometry of the safety coupling of the safety pin has a random uncertainty. Limit torque passed by coupling is a random variable. Limit torque variation within the interval Tmax and Tmin Reduced Mechanical Propertiematerial of material and the size dispersion characteristics in the design of the safety pin is a question that can be considered in the design principle of probability. It can make the technical requirements of safety and limit the scope of changes in the coupling strength of the material limits. Probabilistic design principles proposed design safety couplings used safety pin number

A variable limit for the instability parameter of wall-frame or core-frame bracing structures

This work aims to investigate the viability and convenience of adopting a variable limit α1 for the instability parameter of buildings with reinforced concrete wall-frame or core-frame structures. Initially, the evolution of tall buildings global stability theory is summarized, giving emphasis to define when a second order analysis is needed. The treatment given to this subject by the present Brazilian code for concrete structures design (NBR 6118:2007) is also showed. It follows a detailed analytical study that led to the derivation of an equation for the variable limit α1; a series of examples is presented to check its accuracy. Results are analyzed, showing the validity bounds of the equation and research directions are suggested, in order to improve it.