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In this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric metric connection, i.e., relations between the mean curvature associated with the semi-symmetric metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered.

In this study, a novel water pumping module fed by grid interactive Photo-Voltaic with a bidirectional Power Flow Control was proposed. In addition to improving the pumping system’s reliability, a water pump is powered by a brushless DC motor drive. This method enables the pump to work at its maximum capacity for the entirety of that day, regardless of the weather. The entire system becomes more reliable as a result of the motor’s increased use of photovoltaic (PV) generated power for pumping applications. Maximum Power Point Tracking (MPPT) controller incorporating Machine Learning algorithm drives bridgeless greater static gain DCDC converter to achieve higher power generation point and increment PV efficiency. The PV array’s operation would be managed using the ML back propagation technology to capture the most electricity under any ecological circumstance. A BLDC motor is fed by a Voltage Source Inverter (VSI) that includes a DC bus controlled in both directions by a unit vector template (UVT) approach incorporated in a single-phase voltage source converter (VSC). Additionally, utilizing a PI controller to manage the DC capacitor voltage in the UVT controller at a particular level is not appropriate for the increased PQ capabilities. However, due to tuning problems with the current controller, this controller is unpopular. The aforementioned problems are resolved by employing a unique intelligent-based fuzzy logic controller that achieves good performance features. In this technique, the function of a PV array at its Maximum Power Point (MPP), as well as power quality enhancements and a decrease in Total Harmonic Distortion (THD) of the grid are accomplished. The proposed PI controller attains a significant voltage THD of 3.736. The PI controller, on the other hand, managed to achieve a load voltage THD of 2.629%. The ANFIS method, whose value is 1.739%, is discovered to have a lower THD than all remotes with improved features, it lessens abrupt swings while maintaining steady DC-link voltage.

Heisenberg’s uncertainty principle is one of the main tenets of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum foundations, there has been some confusion in its interpretation: Although Heisenberg’s first argument was that the measurement of one observable on a quantum state necessarily disturbs another incompatible observable, standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other observable is measured. In this paper, we quantify precisely Heisenberg’s intuition. Even if two incompatible observables cannot be measured together, one can still approximate their joint measurement, at the price of introducing some errors with respect to the ideal measurement of each of them. We present a tight relation characterizing the optimal tradeoff between the error on one observable vs. the error on the other. As a particular case, our approach allows us to characterize the disturbance of an observable induced by the approximate measurement of another one; we also derive a stronger error-disturbance relation for this scenario.

We perform a finite sample analysis of the detection levels for sparse principal components of a high-dimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known to be NP-complete in general, and we describe a computationally efficient alternative test using convex relaxations. Our relaxation is also proved to detect sparse principal components at near optimal detection levels, and it performs well on simulated datasets. Moreover, using polynomial time reductions from theoretical computer science, we bring significant evidence that our results cannot be improved, thus revealing an inherent trade off between statistical and computational performance.

Recently, W. M. Schmidt and L. Summerer introduced a new theory that allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in ℝn and to discover new ones. They first note that these exponents can be computed in terms of the successive minima of a parametric family of convex bodies attached to the given point. Then they prove that the n-tuple of these successive minima can in turn be approximated up to bounded difference by a function from a certain class. In this paper, we show that the same is true within a smaller and simpler class of functions, which we call rigid systems. We also show that conversely, given a rigid system, there exists a point in ℝn whose associated family of convex bodies has successive minima that approximate that rigid system up to bounded difference. As a consequence, the problem of describing the joint spectrum of a family of exponents of Diophantine approximation is reduced to combinatorial analysis.

A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall.

We consider the problem of testing multiple quantum hypotheses $\\left\\{ {\\rho _1^{ \\otimes n},...,\\rho _r^{ \\otimes n}} \\right\\}$, where an arbitrary prior distribution is given and each of the r hypotheses is n copies of a quantum state. It is known that the minimal average error probability Pe decays exponentially to zero, that is, Pe = exp{–ξn + 0(n)}. However, this error exponent ξ is generally unknown, except for the case that r = 2. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkola's conjecture that ξ = mini≠j C(ρi, ρj). The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C\\left( {{\\rho _i},{\\rho _j}} \\right): = \\max {}_{0 \\leqslant s \\leqslant 1}\\left\\{ { - \\log Tr\\rho _i^s\\rho _j^{1 - s}} \\right\\}$ has been previously identified as the optimal error exponent for testing two hypotheses, $\\rho _i^{ \\otimes n}$ versus $\\rho _j^{ \\otimes n}$. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szkola's lower bound. Specialized to the case r = 2, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.

We say that an operator T∈B(H)T \\in B(\\mathcal {H}) is complex symmetric if there exists a conjugate-linear, isometric involution C:H→HC:\\mathcal {H}\\rightarrow \\mathcal {H} so that T=CT∗CT = CT^*C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim⁡ker⁡T,dim⁡ker⁡T∗)(\\dim \\ker T, \\dim \\ker T^*).

We introduce a mutation algorithm for puzzles that is a three-direction analogue of the classical jeu de taquin algorithm for semistandard tableaux. We apply this algorithm to prove our conjectured puzzle formula for the equivariant Schubert structure constants of two-step flag varieties. This formula gives an expression for the structure constants that is positive in the sense of Graham. Thanks to the equivariant version of the 'quantum equals classical' result, our formula specializes to a Littlewood-Richardson rule for the equivariant quantum cohomology of Grassmannians.

We present a preconditioned nullspace method for the numerical solution of large sparse linear systems that arise from discretizations of continuum models for the orientational properties of liquid crystals. The approach effectively deals with pointwise unit-vector constraints, which are prevalent in such models. The indefinite, saddle-point nature of such problems, which can arise from either or both of two sources (pointwise unit-vector constraints, coupled electric fields), is illustrated. Both analytical and numerical results are given for a model problem. [PUBLICATION ABSTRACT]