Location Of Zeros Research Articles

A planar three-section coupled-line high-selectivity wideband bandstop filter

A planar three-section coupled-line high-selectivity wideband bandstop filter (BSF) is proposed. The proposed filter topology contains three coupled-line sections (CLs) and two transmission-line sections (TLs). The four locations of transmission zeros (TZs) are obtained through the deduction of the theoretical equations, which validates the proposed topology. During circuit simulation, the effect of the Z e1 on the skirt roll-off is investigated. A highly selective wideband BSF with center frequency at 2.4 GHz, rejection level below 20 dB and attenuation rate at 278.69 dB/GHz is designed and manufactured. The theoretical and simulation results are in good agreement. Indeed, it illustrates the correctness of the theoretical analysis.

Asymptotic Zeros of Poincaré Series

Abstract We study the zeros of Poincaré series $P_{k,m}$ for the full modular group. We consider the case where $m \sim \alpha k$ for some constant $\alpha> 0$. We show that in this case a positive proportion of the zeros lie on the line $\frac{1}{2} + it$. We further show that if $\alpha> \frac{\log (2)}{2\pi }$ then the imaginary axis also contains a positive proportion of zeros. We also give a description for the location of the non-real zeros when $\alpha $ is small.

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.

Brauer and Geršgorin Locations of Zeros of Perturbed Chebyshev Polynomials of the Second Kind by Dilation

We consider some perturbations of Chebyshev polynomials of the second kind obtained by modifying by dilation one of its recurrence coefficients at an arbitrary order. By applying Brauer and Geršgorin theorems to Jacobi matrices associated with such perturbed sequences we obtain some locations of their zeros.

When D-companion matrix meets incomplete polynomials

In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we obtain a majorization relationship between the zeros of convex combinations of incomplete polynomials and the original polynomial. Moreover, we prove that the set of zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. Location of zeros of convex combinations of incomplete polynomials is determined.

Location and double interlacing of zeros of certain combination of the Eisenstein series for [formula omitted]

We show that the zeros of aE2k+(z)−(Ek+(z))2 and bE2k+(z)+(Ek+(z))2 lie on the arc of the fundamental domain for the Fricke group Γ0+(2) of level 2, where Ek+(z) is the Eisenstein series for Γ0+(2), and we investigate the doubly interlacing property between non-elliptic zeros of aE2k+(z)−(Ek+(z))2 and bE2k+(z)+(Ek+(z))2.

On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

In this paper we study the following hypergeometric polynomials: $$ \mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1, \dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1, \dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1, \dots,\kappa_\rho+\delta_\rho+1;x), $$ and $$ \mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots, \delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1; \alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x), \qquad n\in\mathbb{Z}_+, $$ where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters. The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. Of course, such polynomials and more general ones appeared in the literature earlier. Our aim here is to show that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrices of measures. The importance of the orthogonality property was our main reason to concentrate our attention on polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$. Here we shall use some our tools developed earlier. In particular, it was shown recently that Sobolev orthogonal polynomials are related by a differential equation with orthogonal systems $\mathcal{A}$ of functions acting in the direct sums of usual $L^2_\mu$ spaces of square-summable (classes of the equivalence of) functions with respect to a positive measure $\mu$. The case of a unique $L^2_\mu$ is of a special interest, since it allows to use OPRL to obtain explicit systems of Sobolev orthogonal polynomials. The main problem here is \textit{to choose a suitable linear differential operator in order to get explicit representations for Sobolev orthogonal polynomials}. The proof of the orthogonality relations is then a verification of such a choice and it goes in another direction: we start from the already known polynomials to their properties. We also study briefly such properties of the above polynomials: integral representations, differential equations and location of zeros. A system of such polynomials with a kind of the bispectrality property is constructed.

On the zeroes of hypergraph independence polynomials

AbstractWe study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $\Delta$ have a zero-free disc almost as large as the optimal disc for graphs of maximum degree $\Delta$ established by Shearer (of radius $\sim 1/(e \Delta )$ ). Up to logarithmic factors in $\Delta$ this is optimal, even for hypergraphs with all edge sizes strictly greater than $2$ . We conjecture that for $k\ge 3$ , $k$ -uniform linear hypergraphs have a much larger zero-free disc of radius $\Omega (\Delta ^{- \frac{1}{k-1}} )$ . We establish this in the case of linear hypertrees.

Nonstandard Transmission Responses Enabling Feasible Microwave-Acoustic Ladder Filters

Microwave-acoustic ladder filter devices are enabled by piezoelectricity, but achievable filter responses are harshly constrained by material properties and complex resonator geometries. While a response depicting an equiripple return loss (RL) is known to provide the highest out-of-band (OoB) rejection, such response might not be feasible after considering material-related constraints, due to footprint limitations or to an excess of helper reactive elements to be implemented in the laminate. This article explores how the ladder filter structure composed of acoustic wave (AW) resonators can trade the location of reflection zeros (RZs) of the transfer function in exchange for greater flexibility to accommodate technological constraints, such as the electromechanical coupling coefficient. This work proposes to move away from the classical generalized Chebyshev (GC) response toward a solution space defined by nonstandard transmission responses. For this purpose, two methods based on a rigorous and accurate filter synthesis methodology are proposed to deal with the arrangement of RZs. The first proposes detuning the RZs along imaginary axis producing a non-equiripple (NE) response, while the second proposes arranging the RZs in the complex plane. This arrangement on the complex plane is carried out in such a way the device shows an equiripple RL but depicting shallow RZs in comparison with the deep ones of the Chebyshev response. Two synthesis examples validate the proposal reaching filters where all resonators require a common electromechanical coupling coefficient, while the ideal equiripple response would require resonators made of different materials, and so, the use of external reactive elements. The solution that in which RZs are arranged into the complex plane depicts an OoB rejection close to that achievable with an ideal equiripple response.

Functional flows for complex effective actions

In the present work we set up a general functional renormalisation group framework for the computation of complex effective actions. For explicit computations we consider both flows of the Wilsonian effective action and the one-particle irreducible (1PI) effective action. The latter is based on an appropriate definition of a Legendre transform for complex actions, and we show its validity by comparison to exact results in zero dimensions, as well as a comparison to results for the Wilsonian effective action. In the present implementations of the general approaches, the flow of the Wilsonian effective action has a wider range of applicability and we obtain results for the effective potential of complex fields in \phi^4ϕ4-theories from zero up to four dimensions. These results are also compared with results from the 1PI effective action within its range of applicability. The complex effective action also allows us to determine the location of the Lee-Yang zeros for general parameter values. We also discuss the extension of the present results to general theories including QCD.

Routh Zero Location Tests Unhampered by Nonessential Singularities

The generalization of the Routh stability test to the corresponding zero location problem of determining how many of the zeros of a polynomial have negative real parts, positive real parts, or are purely imaginary encountered intensive and controversial activity about overcoming singular cases. This paper revisits this problem and presents two (MaxQ and LinQ) ZL methods for an arbitrary complex polynomial. The first method produces for an investigated polynomial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P(s)$</tex-math> </inline-formula> of degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$</tex-math> </inline-formula> a sequence of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\leq $</tex-math> </inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n+1$</tex-math> </inline-formula> of para-even or para-odd (para-paritic) polynomials of descending degrees obtained by a polynomial remainder routine with quotients of maximal degrees. The second method uses successively linear quotients for the reduction of degrees and thus assign to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P(s)$</tex-math> </inline-formula> always a sequence of exactly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n+1$</tex-math> </inline-formula> para-paritic polynomials. Previous so-called first-type singularities become “non-essential singularities” in the sense that they are now absorbed into modified forms of the polynomial recursions. The distribution of zeros with respect to the imaginary axis is extracted in both methods by sign variation rules posed on the leading coefficients (that stay real when testing a complex polynomial as well) of the polynomials in the sequence.

On the zeros of solutions of ordinary and fractional differential equations

This paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second‐order Sturm–Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.

An Analytical Approach to Locate Short Circuit Fault in a Cable Using Sweep Frequency Response Analysis

In this paper, analytical methods are proposed for the location of a short circuit fault in a power cable using sweep frequency response analysis (SFRA). The impedance spectroscopy is estimated using classical transmission line (TL) theory whereas the driving point impedance function (DPIF) of the cable is proposed to be constructed either by using experimental SFRA data or analytically from the knowledge of the frequency-dependent parameters, material properties, and dimensions of the cable. Finally, an analytical formula is proposed for the estimation of the location of a short-circuit fault using the location of poles and zeros of the impedance function before and after the fault. The formulae are validated using FEM software simulations as well as experimentally on a test cable.

Location of Zeros of Lacunary-Type Polynomials

Location of Zeros of Lacunary-Type Polynomials

Location of zeros of lacunary-type polynomials

Although recently proved generalization of Enestr¨om-Kakey theorem by N.A.Rather et al. are applicable to the larger class of polynomials as compared to all other Enestr¨om- Kakeya type results, but are not applicable to the polynomials whose one or two coefficients are zero. For instance, if we consider the polynomial whose one or two coefficients are zero, then one can note that all Enestr¨om-Kakey type results including the recent generalizations by N.A.Rather et al. are not applicable to this polynomial. So it is interesting to look for the results applicable to such class of polynomials. Motivated by this, here we establish the following results applicable to such class of polynomials called the lacunary-type polynomials. In this paper, by relaxing the hypothesis of well-known Enestr¨om-Kakey theorem, we obtain a result which is applicable to the lacunary-type of polynomials and generalizes several well-known results concerning the location of zeros of polynomials. In addition to this, we show by examples that our results presents better information about the bounds of zeros of polynomials than some known results.

Location of zeros of a Lacunary type polynomial

For a given polynomial p(z) of degree n with real or complex coefficients, our basic aim has been to determine the smallest region in which all the zeros of p(z) lie. In the present paper, we have obtained a result by using Lacunary type polynomial which gives the region of zeros neither circular nor annular except in some particular cases. Our result plays an important role to reduce the region of polynomial zeros.

Lattice scalar field theory at complex coupling

Lattice scalar field theories encounter a sign problem when the coupling constant is complex. This is a close cousin of the real-time sign problems that afflict the lattice Schwinger-Keldysh formalism, and a more distant relative of the fermion sign problem that plagues calculations of QCD at finite density. We demonstrate the methods of complex normalizing flows and contour deformations on scalar fields in $0+1$ and $1+1$ dimensions, respectively. In both cases, intractable sign problems are readily bypassed. These methods extend to negative couplings, where the partition function can be defined only by analytic continuation. Finally, we examine the location of partition function zeros, and discuss their relation to the performance of these algorithms.

A Filtering Antenna Using Transversal Coupling Topology

In this letter, a coupling scheme for the realization of filtering antennas is introduced. This coupling scheme allows the filtering antenna to realize a filter-like reflection coefficient using an all-radiator structure. Instead of cascading resonators one by one in a ladder manner, in a transversal filtering antenna (TFA), all the radiative resonators are coupled to the source simultaneously. The impedance and radiation behaviors of these subantennas (SAs) can be designed individually. So, the performance of these filtering antennas has much lower sensitivity to their physical structure than conventional cascaded designs, which makes the design of such a filtering antenna much easier. Besides, this topology can realize out-of-band gain zeros without using any parasitic elements or cross-coupling structures. The locations of these gain zeros can be controlled by simply changing the lengths of each SA. To the authors’ knowledge, this is the first time that a transversal filtering antenna with all resonators radiative is presented. For validation, some transversal filtering antenna designs with different coupling topologies are introduced to prove the concept and show their advantages. A third-order prototype is fabricated and measured for demonstration, showing good agreement between the simulated and measured results.

Substrate Integrated Waveguide Quasi-Elliptic Filter with Arbitrary Termination Impedances

This paper presents a quasi-elliptic filter (QEF) with arbitrary termination impedances (ATI). The proposed QEF is designed by adding cross-coupling between the first and last resonators of an ATI bandpass filter (BPF) with the Chebyshev response. The proposed QEFs with ATI can be designed in even-order resonators and the location of the pair transmission zeros (TZs) is controllable. To prove the validity of the proposed design, the fourth-order QEFs with ATI were implemented on a single-layer substrate-integrated waveguide (SIW) cavity at a center frequency (&lt;i&gt;f&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;) of 10 GHz with the pair TZs at 10 ± 1.4 GHz. These SIW QEFs with ATI improve frequency selectivity and effectively suppress the out-of-band signal with high power handling. The measured maximum insertion loss (|&lt;i&gt;S&lt;/i&gt;&lt;sub&gt;21&lt;/sub&gt;|) and minimum return loss (|&lt;i&gt;S&lt;/i&gt;&lt;sub&gt;11&lt;/sub&gt;|) of the SIW QEF with unequal real-to-real ATI are 0.93 dB and 17.4 dB, respectively, in the passband. Similarly, the maximum |&lt;i&gt;S&lt;/i&gt;&lt;sub&gt;21&lt;/sub&gt;| and minimum |S&lt;i&gt;S&lt;/i&gt;&lt;sub&gt;11&lt;/sub&gt;| of the SIW QEF with complex-to-real ATI are 1.2 dB and 18 dB, respectively.yer substrate-integrated waveguide (SIW) cavity at a center frequency (

Correlation Decay and Partition Function Zeros: Algorithms and Phase Transitions

We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite $\Delta$-regular tree) implies strong spatial mixing on all graphs of maximum degree $\Delta$ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing. This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard core lattice gas. It also allows us to prove new results on the location of Lee-Yang zeros of the anti-ferromagnetic Ising model. We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs. In particular, we show that results on strong spatial mixing in the anti-ferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function. This extension allows us to give the first deterministic FPTAS for counting the number of $q$-colorings of a graph of maximum degree $\Delta$ provided only that $q\ge 2\Delta$. This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. We also give an improved version of this result for triangle-free graphs.