Location Of Zeros Research Articles

Tree/link method for transfer function and stability analysis of switched‐capacitor circuits

SummaryThe wide spread application of switched‐capacitor circuits has resulted in complex structures wherein the derivation of transfer function in discrete form is not quite straight forward. A computer‐aided tool for direct transfer function analysis of switched‐capacitor circuits using state formulation method is proposed in this paper. The state equation matrices are directly constructed from the circuit topology using tree/link analysis. The entire topology is formulated only using cutset equations to obtain the state matrices without any modifications. The discretized topological state matrices are used to generate the transfer function in z form by forward substitution of state equation in successive clock phase. Lyapunov criteria‐based matrix equation constructed using the same state matrix is used to investigate the circuit stability. The formulation is also suitable in addressing the effect of non‐idealities like switch resistance and finite gain of operational amplifiers, on the location of poles and zeros. Various numerical examples have been presented to demonstrate the performance of the simulator.

Riemann zeros from Floquet engineering a trapped-ion qubit

The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in random matrix theory and quantum chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped-ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.

Prime zeta function statistics and Riemann zero-difference repulsion

We present a derivation of the numerical phenomenon in which differences between the Riemann zeta function’s nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical ‘repulsion’ between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function’s magnitude on the one-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related L-functions.

Miniaturized dual‐mode ultra‐wideband filter using sector substrate integrated waveguide

AbstractA miniaturized dual‐mode ultra‐wideband (UWB) bandpass filter employing a single sector substrate integrated waveguide (SSIW) is proposed and designed. In this design, a SSIW cavity with a significantly small angle 2.2° introduces the filter to be realizable for UWB characteristics, and further miniaturized size. Moreover, this structure permits the locations of the transmission zeros to be controlled independently, thus enhancing the proposed filter selectivity and flexibility. This filter with great performances has been designed and fabricated, and it possesses obvious advantages of low loss, compact size, and wide bandwidth.

Millimeter-Wave Wide-Band Bandpass Filter in CMOS Technology Using a Two-Layered Highpass-Type Approach With Embedded Upper Stopband

An on-chip millimeter-wave (mm-wave) broad-band bandpass filter (BPF) developed in bulk CMOS technology is reported. It is based on a two-layered implementation in which the circuit structure patterned in the top layer exhibits a highpass-type filtering response, whereas an upper stopband is created by the bottom-layer cell when coupled to the top-layer one to obtain a composite overall quasi-elliptic-type wide-band BPF functionality. The locations of the transmission zeros (TZs), which confer sharp-rejection capabilities to the overall frequency response, can be flexibly adjusted with the values of the capacitors that are employed in both layers. A simplified equivalent lumped-element circuit model of the proposed wide-band BPF approach is provided and applied to multi-stage BPF arrangements for higher-selectivity designs. Furthermore, for practical-demonstration purposes, an on-chip passive-integrated single-stage broad-band BPF prototype on silicon with 34.5-GHz center frequency and 61.2% 3-dB relative bandwidth is designed, manufactured, and characterized.

Lee-Yang zeros of the antiferromagnetic Ising model

AbstractWe investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

Some Remarks and Propositions on Riemann Hypothesis

In 1859, Bernhard Riemann, a German mathematician, published a paper to the Berlin Academy that would change mathematics forever. The mystery of prime numbers was the focus. At the core of the presentation was indeed a concept that had not yet been proven by Riemann, one that to this day baffles mathematicians. The way we do business could have been changed if the Riemann hypothesis holds true, which is because prime numbers are the key element for banking and e-commerce security. It will also have a significant influence, impacting quantum mechanics, chaos theory, and the future of computation, on the cutting edge of science. In this article, we look at some well-known results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. Initially, we observe omitting a logical undefined term in the integral representation of Zeta function by the means of Gamma function. For that we propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: s= 1/2 by assuming that ζ(s) and ζ(1-s) simultaneously equal zero. Consequently, we conditionally prove Riemann Hypothesis.

Directional pattern of an antenna array with non-equidistant arrangement of receiving elements

This article presents an analysis of the directional pattern of an antenna array with a non-equidistant arrangement of receiving elements. This problem is relevant for the synthesis of antenna arrays with the required location of zeros and maxima in the radiation pattern. The analysis is suitable for obtaining an analytical expression describing the directional pattern of a flat phased antenna array, the receiving elements of which are located on the antenna curtain.

On the regions containing all the zeros of polynomials and related analytic functions

In this paper, by using standard techniques we shall obtain results with relaxed hypothesis which give zero bounds for the larger class of polynomials. Our results not only generalizes several well-known results but also provide better information about the location of zeros. We also obtain a similar result for analytic functions. In addition to this, we show by examples that our result gives better information on the zero bounds of polynomials than some known results.

Location of zeros of the polar derivative of a polynomial

Location of zeros of the polar derivative of a polynomial

On the location of zeros of quasi-orthogonal polynomials with applications to some real self-reciprocal polynomials

On the location of zeros of quasi-orthogonal polynomials with applications to some real self-reciprocal polynomials

Location of zeros of polynomials with complex coefficients

In this paper we extended Enestrom-Kakeya theorem (Let P(z)=∑aizi be a polynomial of degree n such that 0<a0≤a1≤…≤an then all the zeros of P(z) lie in |z|≤1) and our results [1] by relaxing the hypothesis in different ways by considering complex coefficients we get various other results which in term generalizes.

Triple-Mode Microwave Filters With Arbitrary Prescribed Transmission Zeros

This article presents a new filter design for arbitrary placement of real frequency transmission zeros in triple-mode filters, allowing for either asymmetric or symmetric responses. Triple-mode filters can be viewed, in general, as a series combination of coupled resonators with non-adjacent couplings, allowing for the realisation of transmission zeros. The strength and phase of couplings dictate the transmission zeros locations. The arbitrary placement of transmission zeros requires controlling all amplitude and phase rations, i.e., the ability to realise both negative and positive couplings. In general, this necessitates the use of capacitive and inductive probe structures complicating the construction and hence increasing the filter cost. The method presented in this article enables arbitrary placement of transmission zeros using only inductive or only capacitive couplings structures. In addition, broadband filtering characteristics can be realised using triple-mode filters. Examples of several narrow and broadband bandpass filters are given indicating that this method is valid for most triple-mode structures.

Synthesis of Linear-Phase FIR Filters With a Complex Exponential Impulse Response

In this paper, a novel recursive realization method with word length reduced is proposed to synthesize linear-phase FIR filters for arbitrary bandwidth. By exploring the sparsity in the frequency domain, it constructs the desired impulse response with few complex exponential series (CES) and very sparse additional coefficients (AC) with equal length. The design problem is formulated as a combinatorial optimization problem by confining the frequencies of CES to a finite dictionary. To reduce the computational complexity at optimization, an iterative algorithm is presented, where appropriate frequencies of CES and zero locations of AC can be quickly determined. Thereafter, an efficient parallel structure is introduced to implement the proposed filter. To verify the performance of the proposed method, MATLAB simulation and Field-programmable gate array (FPGA) implementation are made. It shows that the proposed method requires 18bits23bits word length while the existing advanced recursive realization method the piecewise-polynomial-sinusoidal recursiveneeds 59bits59bits for the same filter specifications. Meanwhile, the method can achieve comparable multiplier reduction compared with the state-of-the-art low-complexity techniques but produce a low extra group delay. Moreover, it can realize variable bandedges with a fixed filter structure.

Stabilization with Zero Location Constraints for Delay-Based Non-collocated Vibration Suppression

Stabilization with Zero Location Constraints for Delay-Based Non-collocated Vibration Suppression

Özel Yeteneklilerin Eğitiminde Lego Ev3 Robotlarla Tam Sayılarda İşlemlere Yönelik Bir Etkinlik Çalışması

The study aims to overcome the distress of the gifted students by concretizing abstract mathematical terms and to remove the difficulties the students face when teaching addition and subtraction with whole numbers. In this study, it has been observed that gifted students have reached informal information about negative whole number operations when implementing the lesson plan about whole number operations via robots. The study group of the survey included two gifted sixth grade students who receive education at Sahinbey Science and Arts Centre in Gaziantep. This study designed as a qualitative study. In this study, the case study method was applied, in order to understand how the lesson plan designed with robots affects the data analysis process and prediction skills of the students. In this way, it was aimed to examine the development of students in detail. The lesson plan were specified according to the students. The lesson plan consisted of two parts. In the first part, we tried to implicate the students the location of zero in the number line and the existence of the negative numbers. In the second part, we made the students gain the mathematical logic of the addition and subtraction operations with whole numbers thanks to the activities and modellings by using robots. In conclusion that the gifted students were able to show the locations of the negative whole numbers in the number line thanks to the activities, and it was observed that the students were able to make meaning of the logic about operations with negative whole numbers. The movements of the robots were associated with the whole number of operations and the locations of the numbers. It was observed that the students were able to express negative numbers more easily thanks to the robots.

Dual mode filter with tunable cross Iris

ABSTRACT The aim of this work is to analyze the effect of tuning screws located at the cross iris of a dual-mode vacuum filled cylindrical waveguide filter. Dual-mode waveguide filters with and without cross iris screws are designed and manufactured. Their simulation and measurement results are compared. It is clear that the bandwidth of the filter (and also Finite Transmission Zero (FTZ) locations) can be adjusted with cross iris screws. So, design and mechanical production tolerances can be easily compensated depending on the technical requirements.

A unified power series method for vibration analysis of composite laminate conical, cylindrical shell and annular plate

In this article, the power series method is adopted to investigate the free vibration characteristics of the composite laminated conical, cylindrical shell, and annular plate with arbitrary boundary conditions by the wave based matrix. By the first-order shear deformation shell theory, the governing equation of the composite laminated structure is established. The displacement and moment variables are transformed as the power series function form. Through the introduction of the boundary conditions and the continuous condition between the adjacent segments, the total matrix of the composite structure is established. Searching the zero locations of the total matrix determinant by the bisection method, the natural frequencies of the composite laminated structure under arbitrary boundary conditions are obtained. To further verify the correctness of the solutions by the presented method, some numerical examples are established. From the comparison of the solutions by the presented method and the results in some reported literature and experimental studies, the calculation accuracy is verified. Furthermore, the effect of the geometric parameters and material constants, such are elastic restrained spring stiffness constants, the angle between the shell surface and axis, length to the radius ratios, and modulus ratios, on the free vibration characteristics of the composite laminated structure, have been discussed. The advantage of the presented method is high calculation accuracy, simple matrix assembly, and convenient replacement of boundary conditions. Furthermore, some new numerical solutions are presented in this paper to provide the basic foundation for subsequent research.

Design of a miniaturized planar microstrip Wilkinson power divider withharmonic cancellation

In this paper, a compact microstrip Wilkinson power divider is designed and proposed using rectangularshaped resonator cells. The presented resonator cells are used instead of quarter-wave length branches in the traditional structure to reduce the circuit size, increase the bandwidth, and eliminate the unwanted harmonics. The designed resonator behavior is studied analytically and the locations of transmission zeros are investigated using transfer function andLCequivalentcircuitmethods. The proposed power divider(PD)operates at 2GHz frequency and suppresses the 2nd to 14th unwanted harmonics. The proposed PD achieves approximately 50% size reduction and 40%fractional bandwidth (FBW). The abilities of desirable size reduction and harmonic suppression along with simple structure and wide FBW make the proposed divider a good choice for the modern communication systems, such as multistage power amplifier applications. The proposed divider is implemented and measured. The measurement results verify the simulation responses.

On loci of complex polynomials of degree three

A locus of a complex polynomial is a minimal, with respect to inclusion, set that contains at least one point of every solution of the polarization of the polynomial. The notion of a locus of a complex polynomial was introduced in [2], where several examples were shown for polynomials of degree three. The study of such objects allows one to improve upon known results about the location of zeros and critical points of complex polynomials, see for example [3] and [4]. One of the main results in [2] is Theorem 5.1, which constructs a locus of a polynomial of degree three inside the smallest disk containing its zeros. The proof of that theorem is very long and relies on intricate geometric constructions. It also relies on a specific positioning of the zeros of the polynomial. The goals of this paper are three-fold. First, we give a simpler and more transparent proof of that result. Then we extend it by constructing a locus holder of the polynomial, when its zeros are in arbitrary position. Finally, third, we answer an open question formulated at the end of Section 7 in [2] and by doing so discover a new locus of the polynomial z3+1.