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In this paper, we show that the Central Limit Theorem is deeply ingrained in the mathematical and physical structure of Quantum Mechanics. We show, furthermore, that the Central Limit Theorem provides us with a clarification of the assumptions made by other quantization processes – in particular those using the notion of expansion of some variable up to second order.

In a previous paper, we have mathematically derived the Schrödinger equation using the construct of a Characteristic Function. We have shown that this derivation has a great number of consequences and may help to understand what Quantum Mechanics is really about. In this paper, we present another axiomatic mathematical derivation based on the construct of Boltzmann’s entropy. We also show how these two derivations are mathematically connected and obtain, from this, the positive definite phase space probability distribution function. This probability distribution function is shown to be the only one that reproduces the Schrödinger equation and maximizes entropy, while minimizing the energy. Bohmian Mechanics is considered and reinterpreted from the perspective of the mathematical results of the present approach. Some examples are worked out to give teachers interested in using this material in their classrooms some concreteness.

In this paper, we present a mathematical derivation of the Schrödinger equation departing from only two axioms. We also show that, using this formal derivation process, it is possible to directly derive the Schrödinger equation in generalized curvilinear coordinate systems. This derivation is also shown to be equivalent to Feynman’s path integral approach, but goes further, allowing us to mathematically derive the Bohr-Sommerfeld quantization rules. The use of a small parameter, both in the present derivation, where it is δr, and Feynman’s derivation, where it is ϵ = δt, is also clarified in terms of the Central Limit Theorem. Therefore, the article makes a didactic transposition of the topic of quantization, allowing it to be addressed in the context of teaching Quantum Mechanics. The epistemological importance of axiomatic approaches for the mathematical derivation and the interpretation of the symbols of the theory is also considered.

This paper presents an adapted technique to calibrate the AC voltage divider of AC high-voltage measuring systems up to 200 kV at the Egyptian National Institute of Standards. Two identical capacitors have been used as two similar AC high-voltage capacitive dividers. Firstly, the dividing ratios of two capacitors have been achieved by calibrating each capacitor via a traceable reference standard AC high-voltage divider up to 100 kV. Then, both capacitors have been connected in series to perform as one 200-kV AC voltage divider. They have been calibrated via the same 100-kV reference standard divider to experimentally get their actual dividing ratios up to 100 kV. In order to get their dividing ratios from 100 to 200 kV, a mathematical derivation has been determined. The concept of this derivation is when applying the doubled voltage to two identical pre-calibrated voltage dividers connected in series, their input voltage is almost equally divided across these two dividers. Finally, these calibrated two series capacitors have been used to calibrate a voltage divider of a 200-kV AC HV measuring system. The uncertainty contributions for all the results have been estimated. Enhanced uncertainties have been acquired using this proposed adapted calibration technique. [ABSTRACT FROM AUTHOR]

The Doppler sensitivity issue is an urgent problem to be solved for the phase-coded radar although it can give the promise of improved range resolution and resistance against various kinds of distortion. Due to the ideal autocorrelation and cross-correlation properties, the complementary sequence is utilised in high-frequency surface wave radar and MIMO radar. However, the Doppler frequency shift affects the autocorrelation property of the complementary sequence. This study aims to solve this problem. Based on the signal model and mathematical derivation, a new algorithm is proposed and the simulation results indicate the effectiveness of this method. This study also puts forward the signal processing diagram and the method to obtain the new code. [ABSTRACT FROM AUTHOR]

The Doppler sensitivity issue is an urgent problem to be solved for the phase-coded radar although it can give the promise of improved range resolution and resistance against various kinds of distortion. Due to the ideal autocorrelation and cross-correlation properties, the complementary sequence is utilised in high-frequency surface wave radar and MIMO radar. However, the Doppler frequency shift affects the autocorrelation property of the complementary sequence. This study aims to solve this problem. Based on the signal model and mathematical derivation, a new algorithm is proposed and the simulation results indicate the effectiveness of this method. This study also puts forward the signal processing diagram and the method to obtain the new code.

Here, the response characteristics of high-voltage DC (HVDC) control system are studied in the case that fault occurs in the sending-end AC system. Then the characteristics of DC system which is equivalent to power-frequency variation impedance in the fault super-imposed network are analysed by mathematical derivation and verified by simulations on PSCAD/EMTDC. The result shows that the equivalent power-frequency variation impedance of HVDC system is affected by the rated capacity and the parameters of HVDC control system. The power-frequency variation impedance becomes capacitive under certain fault conditions, which reduces the zone of distance protection based on power-frequency variation and even leads to the operation failure.

Here, the response characteristics of high-voltage DC (HVDC) control system are studied in the case that fault occurs in the sending-end AC system. Then the characteristics of DC system which is equivalent to power-frequency variation impedance in the fault super-imposed network are analysed by mathematical derivation and verified by simulations on PSCAD/EMTDC. The result shows that the equivalent power-frequency variation impedance of HVDC system is affected by the rated capacity and the parameters of HVDC control system. The power-frequency variation impedance becomes capacitive under certain fault conditions, which reduces the zone of distance protection based on power-frequency variation and even leads to the operation failure. [ABSTRACT FROM AUTHOR]

We demonstrate analytically a novel scheme to obtain polarization-independent operation in low-cost coherent OOK receivers, suitable for access networks. We move from a well-known phase-diversity receiver, exploiting a 3 × 3 coupler, three photodiodes, and basic analogue processing. If that receiver is modified to inject the local oscillator (or the signal) into two inputs with proper polarization states, we show, by both mathematical derivation and numerical simulations, that the resulting electrical signal can be polarization-independent; this is attained only if the frequency-detuning between the signal carrier and the local oscillator is high enough, so that intrinsic second order distortions can be spectrally isolated and suppressed by the low-pass filtering of the receiver. [ABSTRACT FROM PUBLISHER]

The electronic power transformer (EPT) is a novel transmission and transformation device, which consists of a series connection of two voltage‐source H‐bridge converters and a DC–DC converter with high‐frequency isolation transformer. When applied to microgrids, EPT will not only deliver energy to loads from the utility grid but also inject some amount of excess power into the utility grid. Hence, the capability of bidirectional power flow is important for EPT, which depends on the DC‐link stage. The traditional synchronous chopper control for a dual active bridge (DAB) converter has some limitations in the application of bidirectional power flow area. This study proposes a novel synchronous chopper control strategy for the DAB converter to implement bi‐direction power flow, details the basic principle and steady‐state operation and presents the mathematical derivations. A three‐phase three‐stage circuit configuration of 10 kV/400 V bi‐direction EPT system based on the novel control DAB converter is designed, and corresponding control schemes for the system are discussed. The performance of this EPT system is validated by the MATLAB/Simulink‐based simulations and the laboratory prototype experiments. [ABSTRACT FROM AUTHOR]

The mathematical derivation and experimental study on thermoacoustic vibration frequency of a Rijke combustor were carried out by adopting the method of Separation of Variables and HP E1432A High-Speed Data Acquisition System. The acoustic vibration frequency formula was derived by Newton's Second Law, Mass Conservation Law and State Equation, and the analytical solution is obtained accordingly. Stainless steel tubes which were closed in one end and open in the other were used in the experimental setup, the porous flame holder made of dense cordierite was installed inside the square place from the bottom of the combustor, over which the premixed methane/air mixture were ignited to form a flat flame, acting as a heat source. Pressure signals along the tube were measured with pressure sensors and HP E1432A, and their corresponding spectrums were obtained by Fast Fourier Transform. The results showed that the analytical solutions of thermoacoustic vibration frequency agree well with the experimental data. [ABSTRACT FROM AUTHOR]

This letter treats the basic problem of economic operation of power systems and presents a mathematical derivation that proves that the classic economic dispatch (ED) problem, with quadratic-convex cost functions, can be solved analytically, i.e., without any approximations or need for numerical iterative optimization algorithms. Duality theory is employed to determine both the exact primal and exact dual solutions. All this requires, at most, are 2n function evaluations. It is stated, therefore, that the use of an ED model as an optimization-based electricity auction does not cause any conflict of interest [ABSTRACT FROM PUBLISHER]

Signal‐to‐interference ratio (SIR) performance of a multiband orthogonal frequency division multiplexing ultra‐wideband system with residual timing offset is investigated. To do so, an exact mathematical derivation of the SIR of this system is derived. It becomes obvious that, unlike a cyclic prefixing based system, a zero padding based system is sensitive to residual timing offset. [ABSTRACT FROM AUTHOR]

This Letter reveals the underlying relations of three state‐of‐the‐art interval arithmetic (IA)‐based algorithms for antenna array pattern upper bound estimation with excitation amplitude errors. Theoretical analysis proves that the matrix IA algorithm has better performance than both the rIA and circular IA algorithms. Mathematical derivations are presented in detail, and representative numerical examples are given for validating this conclusion. [ABSTRACT FROM AUTHOR]