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In this paper, a trajectory tracking control system using a neural network estimator is proposed to control the proof mass of a MEMS gyroscope,. The proposed control system incorporates a fractional controller based on the terminal sliding mode and a recurrent Chebyshev fuzzy neural network using a self-evolving mechanism. The fractional order terminal sliding mode control (FOTSMC) can guarantee the tracking error is exponential stable, and self-evolving recurrent Chebyshev fuzzy neural network (SERCFNN) is introduced to relax the requirement of nonlinear functional certainty. In addition, the SERCFNN develops the advantages of the self-evolving fuzzy neural network (SEFNN), recurrent fuzzy neural network (RFNN) and Chebyshev function network (CFN). The SEFNN can adaptively update dynamic structure through generating and adjusting fuzzy logic rules. The RFNN improves the performance for coping with temporal problem with the capability to store prior information. The CFN is capable to enlarge the dimensionality of the input variables. Moreover, the asymptotic stability of the proposed control system can be proved by Lyapunov stability theory. The effectiveness and superiority of performance are exhibited with simulation studies and comprehensive comparisons.

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. In 2011, Solé and Planat stated that, the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \prod_{p\leq x} (1+\frac{1}{p}) > e^{\gamma} \cdot \log \theta(x)$ holds for all $x \geq 5$, where $\theta(x)$ is the first Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, $\zeta(x)$ is the Riemann zeta function and $\log$ is the natural logarithm. In this note, using Solé and Planat criterion, we prove that, when the Riemann hypothesis is false, then there are infinitely many natural numbers $x$ for which $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + \varepsilon \cdot \log x \leq 2.062$ could be satisfied for some $\varepsilon > 0$. Since the inequality $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + \varepsilon \cdot \log x \leq 2.062$ never holds for every $\varepsilon > 0$ and large enough $x$, then the Riemann hypothesis is true by principle of non-contradiction.

Chacha 20 is a stream cypher that is used as lightweight on many CPUs that ‎do not have dedicated AES instructions. As stated by Google, that is the reason why they use it on many ‎devices, such as mobile devices, for authentication in TLS protocol. This paper ‎proposes an improvement of chaha20 stream cypher algorithm based on tent and ‎Chebyshev functions (IChacha20). The main objectives of the proposed IChacha20 ‎algorithm are increasing security layer, designing a robust structure of the IChacha20 ‎to be enabled to resist various types of attacks, implementing the proposed ‎algorithm for encryption of colour images, and transiting it in a secure manner. The ‎ test results proved that the MSE, PSNR, UQI and NCC metrics of ‎IChacha20 are better than those of the original Chacha20. Also, the proposed method has ‎a faster execution time (01:26:4 sec) compared with the original algorithm (02:07:1 sec).

In this article, we provide explicit bounds for the prime counting functions θ ( x ) \theta (x) for all ranges of x x . The bounds for the error term for θ ( x ) − x \theta (x)- x are of the shape ε x \varepsilon x and c k x ( log ⁡ x ) k \frac {c_k x}{(\log x)^k} , for k = 1 , … , 5 k=1,\ldots ,5 . Tables of values for ε \varepsilon and c k c_k are provided.

Under the assumption that the Riemann Hypothesis is true, von Koch deduced the asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$. A precise version of this was given by Schoenfeld. He found under the assumption that the Riemann Hypothesis is true that $\left| \theta(x) - x \right| < \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. We prove that if the Riemann Hypothesis is true, then \[\log \theta(x) > \log x - \log \left(\frac{8 \times \pi \times \sqrt{x}}{8 \times \pi \times \sqrt{x} - \log^{2} x}\right) - \left(\frac{4+\gamma -\log 4 \times \pi}{\sqrt{x-\frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x}}\right)\] for every $x \geq p_{120569}$, where $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $p_{120569}$ is the 120569th prime number.

Under the assumption that the Riemann hypothesis is true, von Koch deduced the asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$. A precise version of this was given by Schoenfeld. He found under the assumption that the Riemann hypothesis is true that $\left| \theta(x) - x \right| < \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. Using this result, we prove that if the Riemann hypothesis is true, then $\prod_{q \leq x} \frac{q}{q-1} > \left(e^{\gamma} \times \log x\right) \times \left(1 + \frac{\log (1 - \frac{1}{8 \times \pi \times \sqrt{x}} \times \log^{2} x)}{\log x}\right)$ for every $x \geq 599$.

The Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. In 2011, Sol{\'e} and and Planat stated that the Riemann Hypothesis is true if and only if the inequality $\prod_{q\leq q_{n}}\left(1+\frac{1}{q} \right) >\frac{e^{\gamma}}{\zeta(2)}\times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 3$, where $\theta(x)$ is the Chebyshev function, $\gamma\approx 0.57721$ is the Euler-Mascheroni constant and $\zeta(x)$ is the Riemann zeta function. Using this result, we create a new criterion for the Riemann Hypothesis. We prove the Riemann Hypothesis is true using this new criterion., Last version of the author without the final revision of the referees.

Under the assumption that the Riemann hypothesis is true, von Koch deduced the asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$. A precise version of this was given by Schoenfeld. He found under the assumption that the Riemann hypothesis is true that $\left| \theta(x) - x \right| < \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. Using this result, we prove that if the Riemann hypothesis is true, then $\prod_{q \leq x} \frac{q}{q-1} < \left(e^{\gamma} \times \log x\right) \times \left(1 - \frac{\log x}{8 \times \pi \times \sqrt{x}}\right)$ for every $x \geq 599$. Hence, we obtain that if the Riemann hypothesis is true, then $x^{\left(1 - \frac{\log x}{8 \times \pi \times \sqrt{x}}\right)} > \theta(x)$ for every $x \geq 599$. However, this is false since $(\theta(x) - x)$ changes sign infinitely often. By contraposition, the Riemann hypothesis is false.

Under the assumption that the Riemann hypothesis is true, von Koch deduced the improved asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$, where $\theta(x)$ is the Chebyshev function. On the contrary, we prove if there exists some real number $x \geq 10^{8}$ such that $\theta(x) > x + \frac{1}{\log \log \log x} \times \sqrt{x} \times \log^{2} x$, then the Riemann hypothesis should be false. Note that, the von Koch asymptotic formula uses the Big $O$ notation, where $f(x) = O(g(x))$ means that there exists a positive real number $M$ and a real number $y$, such that $|f(x)| \leq M \times g(x)$ for all $x \geq y$. However, no matter how big we get the real number $y \geq 10^{8}$, the another positive real number $M$ could always prevail over the value of $\frac{1}{\log \log \log x}$ for sufficiently large numbers $x \geq y$.

The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros. The duality between primes and Riemann zeta function zeros suggests some new goals and aspects to be studied: {\em adelic duality} and the {\em POSet of prime numbers}. The article presents computational evidence of the structure of the imaginary parts $t$ of the non-trivial zeros of the Riemann zeta function $\rho=1/2+it$, called in this article the {\em Riemann Spectrum}, using the study of their distribution. The novelty represents in considering the associated characters $p^{it}$, towards an algebraic point of view, than rather in the sense of Analytic Number Theory. This structure is tentatively interpreted in terms of adelic characters, and the duality of the rationals. Second, the POSet structure of prime numbers studied, is tentatively mirrored via duality in the Riemann spectrum. A direct study of the convergence of their Fourier series, along Pratt trees, is proposed. Further considerations, relating the Riemann Spectrum, adelic characters and distributions, in terms of Hecke idelic characters, local zeta integrals (Mellin transform) and $\omega$-eigen-distributions, are explored following., Comment: 15 pages

Srinivasa Ramanujan studied the function $S_{1}(x) = \sum_{\rho} \frac{x^{\rho - 1}}{\rho \cdot (1 - \rho)}$ where $\rho$ runs over the nontrivial zeros of the Riemann $\zeta$ function. Under the Riemann hypothesis, we know that $\lvert S_{1}(x) \rvert \leq \frac{\tau}{\sqrt{x}}$ for $\tau = 2 + \gamma - \log(4 \cdot \pi) \approx 0.04619$. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. In 2011, Solé and Planat stated that, the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \prod_{p\leq x} (1+\frac{1}{p}) > e^{\gamma} \cdot \log \theta(x)$ holds for all $x \geq 5$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, $\zeta(x)$ is the Riemann zeta function and $\log$ is the natural logarithm. In this note, using Solé and Planat criterion, we prove that, when the Riemann hypothesis is false, then there are infinitely many natural numbers $x$ for which $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + S_{1}(x) \cdot \sqrt{x} \cdot \log x \leq 2.062$ could be satisfied. In addition, we show that the Riemann hypothesis is true when $S_{1}(x) \geq \frac{\varepsilon}{\sqrt{x}}$ for $\varepsilon \geq -1.9999999$ and large enough $x$.

A trustworthy proof for the Riemann hypothesis has been considered as the Holy Grail of Mathematics by several authors. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. Let $Q$ be the set of prime numbers $q_{n}$ satisfying the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1}>e^{\gamma} \cdot\log\theta(q_{n})$ with the product extending over all prime numbers $q$ that are less than or equal to $q_{n}$, where $\gamma\approx 0.57721$ is the Euler-Mascheroni constant, $\theta(x)$ is the Chebyshev function and $\log$ is the natural logarithm. If the Riemann hypothesis is false, then there are infinitely many prime numbers $q_{n}$ outside and inside of $Q$. In this note, we obtain a contradiction when we assume that there are infinitely many prime numbers $q_{n}$ outside of $Q$. By reductio ad absurdum, we prove that the Riemann hypothesis is true.

The Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. In 2011, Patrick Solé and Michel Planat stated a new criterion for the Riemann Hypothesis. We prove the Riemann Hypothesis is true using this criterion.

For every prime number $p_{n}$, we define the sequence $X_{n} = \prod_{q \leq p_{n}} \frac{q}{q-1} - e^{\gamma} \times \log \theta(p_{n})$, where $\theta(x)$ is the Chebyshev function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas theorem states that the Riemann hypothesis is true if and only if the $X_{n} > 0$ holds for all prime $p_{n} > 2$. For every prime number $p_{k}$, $X_{k} > 0$ is called the Nicolas inequality. We show if the sequence $X_{n}$ is strictly decreasing for $n$ big enough, then the Riemann hypothesis must be true. For every prime number $p_{n} > 2$, we define the sequence $Y_{n} = \frac{e^{\frac{1}{2 \times \log(p_{n})}}}{(1 - \frac{1}{\log(p_{n})})}$ and show that $Y_{n}$ is strictly decreasing for $p_{n} > 2$. For all prime $p_{n} > 286$, we demonstrate that the inequality $X_{n} < e^{\gamma} \times \log Y_{n}$ is always satisfied. We prove that $\lim_{{n\to \infty }}X_{n}=\lim_{{n\to \infty }}(\log Y_{n})=0$.

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. This problem has remained unsolved for many years. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show some arguments in favor of the Riemann hypothesis is true.

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. It is known that $S(x)$ changes sign infinitely often. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}-\log \log \theta(x)-B \right)$. We prove that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann hypothesis should be false. The Riemann hypothesis is also false when the inequalities $\delta(x) \leq 0$ and $S(x)\geq 0$ are satisfied for some number $x \geq 3$ or when $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log \theta(x)} \leq 1$ is satisfied for some number $x \geq 13.1$ or when there exists some number $y \geq 13.1$ such that for all $x \geq y$ the inequality $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log (x + C \times \sqrt{x} \times \log \log \log x)} \leq 1$ is always satisfied for some positive constant $C$ independent of $x$.

We define the function $\upsilon(x) = \frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log (x + C \times \sqrt{x} \times \log \log \log x)} - 1$ for some positive constant $C$ independent of $x$. We prove that the Riemann hypothesis is false when there exists some number $y \geq 13.1$ such that for all $x \geq y$ the inequality $\upsilon(x) \leq 0$ is always satisfied. We know that the function $\upsilon(x)$ is monotonically decreasing for all sufficiently large numbers $x \geq 13.1$. Hence, it is enough to find a value of $y \geq 13.1$ such that $\upsilon(y) \leq 0$ since for all $x \geq y$ we would have that $\upsilon(x) \leq \upsilon(y) \leq 0$. Using the tool $\textit{gp}$ from the project PARI/GP, we found the first zero $y$ of the function $\upsilon(y)$ in $y \approx 8.2639316883312400623766461031726662911 \ E5565708$ for $C \geq 1$. In this way, we claim that the Riemann hypothesis could be false.

This thesis presents the synthesis of a bandpass filter to achieve high selectivity and rejection properties using a new class of filter functions called chained-elliptic function filters. Chained-elliptic filters have higher selectivity compared to Chebyshev function filters and have the property of sensitivity to manufacturing tolerance reduction in chained-function filters. The proposed design has high selectivity and reduced sensitivity, enabling easier and faster filter fabrication, which greatly reduces the time and cost required to manufacture a millimeter-wave filter. The characteristic polynomials of chained-elliptic function filters are derived (through chaining elliptic filtering function) and extracted to form the coupling matrix of the bandpass filter.

A new class of compact microstrip bandpass filters in doublet topology based on a dual-mode loop microstrip resonator with controllable locations of input/output feeding lines is presented and researched. Filters with up to five poles are proposed and researched. Diagonal cross-coupling path in box-like topology can be introduced to produce a finite transmission zero (TZ) with controllable position. Some design examples are provided to illustrate the performances of the proposed two-, three-, four- and five-pole filters with controllable TZs. A three-pole and a five-pole filters were fabricated and measured. Good agreement is achieved between the simulated and the measured results.

A stopband-extended sixth-order balanced bandpass filter with switchable passband function and stopband response is proposed in this paper. Based on the use of asymmetric coupled stepped-impedance resonators, the upper differential-mode stopband can be extended, and high common-mode suppression can be performed within passband. By means of high-order structure and cross coupling, which introduce transmission zeros closely around the passband, high selectivity is attained. The configuration utilizing switchable cross coupling is eventually employed in the balanced filter to perform either Elliptic function with steep passband edge or Chebyshev function with flat and deep stopband. The circuit is fabricated in high-temperature superconducting technique on an MgO substrate with YBCO thin films, which can result in relatively low insertion losses. Experiment at the temperature of 77 K was carried out to validate the design concept. The results attain good agreement with the predicted ones in simulation.

Cardiac disease detection is a tedious process. Classification of electrocardiogram (ECG) signals plays an important role in the diagnosis of heart diseases. The most important factor that limits the detection of cardiac disease is the rare availability of instances of the abnormal condition collected using ECG sensors. And if the signals contain noise, then the classification might become a challenging task. In this work, we address the problem of cardiac disease detection when the dataset has less number of noisy ECG sensor signals. Here, Chebyshev Type II filter and Chebyshev function, which is termed as Chebfun, are used. The Chebyshev filter is used for high-frequency noise removal and Chebfun is used to approximate the signal with its coefficients. These coefficients known as Chebfun coefficients are used as the features. These features are used for classification. In the proposed work, machine learning algorithms, like SVM, logistic regression, decision tree, and AdaBoost, are used for classifying the features extracted from Chebfun.

The Riemann hypothesis has been considered to be the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis could be true.

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems and it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

In this paper, the flexibility of Kaiser function utilized on power weighted linear antenna array is presented. Some linear antenna arrays working at the X-band frequency with odd and even number of radiator elements are applied to implement the weighting coefficient based on Kaiser function. The involvement of Kaiser function is expected to give other useful preference in the design of antenna array. The Kaiser function utilization on power weighted linear antenna array shows a flexibility in terms of width of mainlobe (WML) and sidelobe level (SLL). For constant array elements number and inter element spacing, the Kaiser function could improve the WML and SLL of linear array antennas by varying $\beta$. In addition, Chebyshev function is also employed in the analysis of performance as comparison.

An improved Blackman Harris apodization function is studied for rectify the shortcomings of large main lobe width of Blackman Harris apodization function in apodization processing of Fourier transform spectrometer. The basic principle of apodized interferogram is analyzed. The Chebyshev function with tunable parameters is introduced to improve Blackman Harris function. Compared with the original function through Matlab simulation, the improved apodization function shows not only the characteristics of main lobe width is enhanced, but also the side lobe attenuation can be adjusted, so the function is more flexible in engineering application.

This paper uses the Barycentric Lagrange interpolation method to explore the free vibration of a plate with the regular and irregular domain using the Chebyshev function, allowing us to consider multiple dimensions. From our results, it can be shown that the Barycentric Lagrange interpolation method can solve three-dimensional problems. In the analysis, we can see that the Barycentric Lagrange interpolation method can solve the dynamic motion of the plate with regular domain, and the error of the simulation can be reduced to 0.15%. The effect of the geometric node number on the simulated error of the natural frequency of the plate is very profound. The Barycentric Lagrange interpolation method and the extrapolation difference method can solve the natural frequency of the plate with irregular domain. The error of the simulation on the natural frequency can be reduced to 1.084%. This allows us to understand the vibration of the plate with the regular and irregular domain under various boundary conditions quickly.

In mathematics, the Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all sufficiently large $n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all $n > 5040$ if and only if the Riemann Hypothesis is true. In 2002, Lagarias proved that if the inequality $\sigma(n) \leq H_{n} + exp(H_{n}) \times \log H_{n}$ holds for all $n \geq 1$, then the Riemann Hypothesis is true, where $H_{n}$ is the $n^{th}$ harmonic number. We show certain properties of these both inequalities that leave us to a verified proof of the Riemann Hypothesis. These results are supported by the claim that a numerical computer calculation verifies that the subtraction of \[\log (e^{\gamma} \times q_{m} \times r) + e^{\gamma} \times q_{m} \times r \times \log \log (e^{\gamma} \times q_{m} \times r)\] with \[(q_{m} + 1) \times \log (e^{\gamma} \times (r + 1)) + (q_{m} + 1) \times e^{\gamma} \times (r + 1) \times \log \log (e^{\gamma} \times (r + 1))\] is monotonically increasing as much as $q_{m}$ and $r$ become larger just starting with the initial values of $q_{m} = 47$ and $r = 1$, where $q_{m}$ is a prime number and $r$ is a natural number. In this way, we can confirm that the Riemann Hypothesis is true based on computational mathematics using a simple and naive computer assisted proof.

Purpose This paper aims to solve the Falkner–Skan problem over an isothermal moving wedge using the combination of the quasilinearization method and the fractional order of rational Chebyshev function (FRC) collocation method on a semi-infinite domain. Design/methodology/approach The quasilinearization method converts the equation into a sequence of linear equations, and then by using the FRC collocation method, these linear equations are solved. The governing nonlinear partial differential equations are reduced to the nonlinear ordinary differential equation by similarity transformations. Findings The entropy generation and the effects of the various parameters of the problem are investigated, and various graphs for them are plotted. Originality/value Very good approximation solutions to the system of equations in the problem are obtained, and the convergence of numerical results is shown by using plots and tables.

This paper presents a novel wideband filtering power divider. Instead of the T-junction in the traditional power divider design, a three coupled-line structure is utilized to provide the equal power division to every outputs. Together with the quarter-wavelength coupled line in each way, a wideband filtering response can be obtained. In order to enhance the frequency selectivity, a pair of half-wavelength open stubs are loaded between the three-line coupled structure and parallel coupled lines to produce a pair of transmission zeros on both sides of the passband. To enhance the upper stopband rejection, two additional open stubs are installed at output ports. A new three-port equivalent network for the three-line coupled structure is proposed, and the circuit model for the filtering power divider can be developed for illustrating the working mechanism and the design process. Furthermore, the generalized Chebyshev function is applied to synthesize and analyze the whole structure systematically. For demonstration, a prototype of the wideband filtering power divider operating at 3 GHz is designed and fabricated. The measured results show a 16-dB bandwidth of 62%, an isolation better than 16.5 dB, and a 29-dB wide stopband from 4.39 to 7.73 GHz (1.1f0).

In this paper, an extended Chebyshev function is proposed to adapt the matching condition of the power amplifier (PA) by introducing a new factor. A set of impedance functions can be directly calculated along with the variation of a new variable, and the first element extracted from the functions is distributed in a wide range. In addition, the impedance function whose first element is the closest to the output capacitance of the transistor can be easily read out and selected as the original matching network. The fundamental impedance of the selected function will be reached a good matching state, and the impedances out of band will be on the edge of Smith chart. To achieve better performances, the real frequency technique is applied to adjust the harmonic impedances preventing it from falling into the low-efficiency region. Two PAs with a relative bandwidth of 34% and 75% are implemented to validate the proposed approach.