Real Roots Research Articles

Properties of the Wrońskian determinant of a system of solutions to a linear homogeneous equation: The case when the number of solutions is less than the order of the equation

Objectives. The work sets out to study the properties of the Wrońskian determinant of the system of solutions to a linear homogeneous equation in cases when the number of solutions is less than the order of the equation, comparing them with the known properties of the same determinant when the number of solutions is equal to the order of the equation.Methods. The work uses the methods of linear algebra according to the theory of ordinary differential equations, as well as mathematical and complex analysis.Results. It is shown that the vanishing of a considered determinant on an arbitrarily small interval implies its vanishing on the entire domain of definition; the solutions turn out to be linearly dependent. A stronger result is obtained in three cases: (1) if the coefficients of the equation are analytic functions; (2) if the number of solutions is equal to one; (3) if the number of solutions is one less than the order of the equation. Namely, if the set of zeros of the considered Wrońskian has a limit point belonging to the domain of definition of solutions, then the determinant is identically equal to zero and the solutions are linearly dependent.Conclusions. According to the obtained results, the Wrońskian of a system of solutions of a linear homogeneous equation can serve as an indicator of the linear dependence or independence of this system in cases where the number of solutions is lower than the order of the equation; here, the solutions are linearly dependent if and only if their Wrońskian is identically equal to zero. In this case, there is no need to check whether the determinant vanishes over the entire domain of definition, since it is sufficient to do this on an arbitrarily chosen interval or even (in the special cases listed above) on an arbitrarily chosen set having a limit point.

Real polynomials with constrained real divisors. I. Fundamental groups

In the late 80s, V. Arnold and V. Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree [Formula: see text] and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces [Formula: see text] of real monic univariate polynomials of degree [Formula: see text] whose real divisors avoid sequences of root multiplicities, taken from a given poset [Formula: see text] of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of [Formula: see text] and of some related topological spaces. We find explicit presentations for the groups [Formula: see text] in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in [Formula: see text]. We also show that [Formula: see text] stabilizes for [Formula: see text] large. The mechanism that generates [Formula: see text] has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree [Formula: see text] polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups [Formula: see text] admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder [Formula: see text], whose images avoid the tangency patterns from [Formula: see text] with respect to the generators of the cylinder.

Spectrality of random convolutions generated by finitely many Hadamard triples

Let {(Nj,Bj,Lj):1⩽j⩽m} be finitely many Hadamard triples in R . Given a sequence of positive integers {nk}k=1∞ and ω=(ωk)k=1∞∈{1,2,…,m}N , let μω,{nk} be the infinite convolution given by μω,nk=δNω1−n1Bω1∗δNω1−n1Nω2−n2Bω2∗⋯∗δNω1−n1Nω2−n2⋯Nωk−nkBωk∗⋯. In order to study the spectrality of μω,{nk} , we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if gcd(Bj−Bj)=1 for 1⩽j⩽m , then all infinite convolutions μω,{nk} are spectral measures. This implies that we may find a subset Λω,{nk}⊆R such that {eλ(x)=e2πiλx:λ∈Λω,{nk}} forms an orthonormal basis for L2(μω,{nk}) .

Simulation and Modelling of Tree Roots using GPR and Numerical Analysis

Traditional methods to check tree roots are complicated due to their destructive nature and limited quantitative assessments in long-term research. Therefore, this study aims to understand the synergistic use of the Ground Penetrating Radar (GPR) to obtain accurate information on tree root health. For this purpose, geophysical and surveying techniques are used. In the study, image data of tree roots were obtained using IDS Duo GPR under normal conditions. To obtain the most accurate results in GPRMax, the simulation used the greatest contrast dielectric value of soil and roots. Then, an analysis is conducted to compare synthetic and real data. Modelling is crucial to understand electromagnetic wave propagation and interaction with tree roots. First, the synthetic hyperbola’s shape is compared with the real root’s hyperbola. Second, roots with increasing diameters were simulated and the time interval associated with each diameter was determined to produce a regression line model. Finally, depending on the real-time interval and the collected data, the regression model is utilised to estimate the true diameter. The study found the following: (1) the results show that the high dielectric value of the detected roots, real and synthetic hyperbola, have similar amplitude and tail; (2) the findings demonstrate that the estimation model is good with an average error of ±8 mm under ideal conditions and ±20 mm under normal conditions. The estimation variation is strongly influenced by soil moisture. The GPR resolution and signal deteriorate when the high soil moisture content is high. As a result, this study could provide vital insight for more effective assessment of tree roots and serve as an important reference for researchers seeking to expand on present findings.

Efficient Inverse Fractional Neural Network-Based Simultaneous Schemes for Nonlinear Engineering Applications

Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders ς+2 and 2ς+4, respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time.

Fast Optimistic Gradient Descent Ascent (OGDA) Method in Continuous and Discrete Time

AbstractIn the framework of real Hilbert spaces, we study continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V. The starting point of our investigations is a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. Our method exhibits fast convergence rates of order $$o \left( \frac{1}{t\beta (t)} \right) $$ o 1 t β ( t ) for $$\Vert V(z(t))\Vert $$ ‖ V ( z ( t ) ) ‖ , where $$z(\cdot )$$ z ( · ) denotes the generated trajectory and $$\beta (\cdot )$$ β ( · ) is a positive nondecreasing function satisfying a growth condition, and also for the restricted gap function, which is a measure of optimality for variational inequalities. We also prove the weak convergence of the trajectory to a zero of V. Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method for monotone operators, for which we prove that the generated sequence of iterates $$(z_k)_{k \ge 0}$$ ( z k ) k ≥ 0 shares the asymptotic features of the continuous dynamics. In particular we show for the implicit numerical algorithm convergence rates of order $$o \left( \frac{1}{k\beta _k} \right) $$ o 1 k β k for $$\Vert V(z^k)\Vert $$ ‖ V ( z k ) ‖ and the restricted gap function, where $$(\beta _k)_{k \ge 0}$$ ( β k ) k ≥ 0 is a positive nondecreasing sequence satisfying a growth condition. For the explicit numerical algorithm, we show by additionally assuming that the operator V is Lipschitz continuous convergence rates of order $$o \left( \frac{1}{k} \right) $$ o 1 k for $$\Vert V(z^k)\Vert $$ ‖ V ( z k ) ‖ and the restricted gap function. All convergence rate statements are last iterate convergence results; in addition to these, we prove for both algorithms the convergence of the iterates to a zero of V. To our knowledge, our study exhibits the best-known convergence rate results for monotone equations. Numerical experiments indicate the overwhelming superiority of our explicit numerical algorithm over other methods designed to solve monotone equations governed by monotone and Lipschitz continuous operators.

Stability optimization of time-delay systems with zero-location constraints applied to non-collocated vibration suppression

We present an active control method for achieving non-collocated vibration absorption, implemented on a lumped mass system. The task of suppressing vibrations at a number of frequencies, at a target mass is recast into a problem of assigning zeros of the transfer function from the external excitation force to the position of the target mass. The controller design involves computing the controller parameters by utilizing the available output feedback to directly assign a set of zeros on the imaginary axis while simultaneously preserving the overall stability with a sufficient margin. The design also takes into account delays in the feedback path. The presence of delays leads to an infinite dimensional system for which, there exists in general, infinitely many characteristic roots. The requirement of achieving sufficient damping in the resulting closed-loop system together with suppressing vibrations of the desired frequency leads to a constrained optimization problem of minimizing the spectral abscissa function subject to zero-location constraints. These constraints exhibit polynomial dependence on the controller parameters. We present two approaches for controller parameterization for the single and multiple input case. Both approaches are based on constraint elimination and exploit the property that the constraints are affine in a selection of the gain parameters. Finally, the methodology is verified by simulating on a spring–mass–damper system following which, it is then validated experimentally on a laboratory setup.

Representation of analytic functions in bounded convex domains on the complex plane

The paper is devoted to entire functions of exponential type and regular growth. Exceptional sets are investigated outside of which these functions have estimates from below that asymptotically coincide with their estimates from above. An explicit construction of an exceptional set, which consists of disks with centers at zeros of the entire function, is indicated. The concept of a properly balanced set is introduced, which is a natural generalization of the concept of a regular set by B. Ya. Levin. It is proved that the zero set of an entire function is properly balanced if and only if each function analytic in the interior of the conjugate diagram of the entire function in question and continuous up to the boundary is represented by a series of exponential monomials whose exponents are zeros of this entire function. This result generalizes the classical result of A. F. Leont′ev on the representation of analytic functions in a convex domain to the case of a multiple zero set.

Closure condition of a function by the Newton-Raphson method

The work was developed with the purpose of briefly presenting the Newton-Raphson method for the calculation of real roots of equations. The objective was to present an experiment that would consolidate with the study in simulation format to apply the concepts developed through the root isolation procedures, definition of the initial value by the appropriate interval, as well as in the stopping criteria, by the ge-ometric observation of the tangent line of the graphs and the iteration processes, approximating the values of the reality of the zeros of the function. We also discuss the possible flaws of the simulation system presented, given the rules of the method used.

The Singlet: Direct Synthesis of Pseudo-Elliptic Inline Filters With Frequency Variant Couplings

A general and direct synthesis of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{p}$</tex-math> </inline-formula> poles pseudo-elliptic inline filters having <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{p}$</tex-math> </inline-formula> purely real and/or imaginary frequency transmission zeros is presented. Using an appropriate model of the resonators (bandpass distributed or lumped element) and shifting the reference planes at the input and output of the filter network, a novel direct synthesis and realization of filters based on frequency variant coupling (inductive or capacitive) in the distributed domain is demonstrated for accurate modeling of narrow and broadband filters. First, a complete general synthesis procedure is described that allows for flexible or mixed topologies realizing transmission zeros anywhere on the frequency plane (including complex and infinite frequencies). Furthermore, by exploiting the power of modular design, a synthesis procedure that enables simultaneous extraction of reflection zero (a pole) and a transmission zero in the form of a symmetrical and physically realizable singlet as an elementary unit is showcased. The physical dimensions of the extracted singlets are independently determined in any chosen technology, and the overall filter is obtained by directly cascading the physical singlets. To validate this novel synthesis technique, three design examples are shown: a narrowband four-pole, four transmission zeros (fully canonical) filter from literature is synthesized and physically dimensioned, a wideband three-pole, three transmission zero (fully canonical) filter, and a broadband three-pole, one transmission zero, filter. In the first two examples, each of the singlet blocks was implemented in an overmoded rectangular waveguide cavity operating at the TE <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$_{201}$</tex-math> </inline-formula> mode coupled at the input and output to the fundamental nonresonating TE <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$_{10}$</tex-math> </inline-formula> mode. The third example was implemented in a re-entrant coaxial resonator operating at the fundamental mode. The simulation of the electromagnetic (EM) high-frequency structure simulator (HFSS) models of the overall example filters was in excellent agreement with the synthesized models, thereby validating the effectiveness of the proposed synthesis technique. Furthermore, the measured results of the fabricated wide bandpass filter showed close agreement with synthesis and EM model.

Calibration of the Mechanical Boundary Conditions for a Patient-Specific Thoracic Aorta Model Including the Heart Motion Effect.

We propose a procedure for calibrating 4 parameters governing the mechanical boundary conditions (BCs) of a thoracic aorta (TA) model derived from one patient with ascending aortic aneurysm. The BCs reproduce the visco-elastic structural support provided by the soft tissue and the spine and allow for the inclusion of the heart motion effect. We first segment the TA from magnetic resonance imaging (MRI) angiography and derive the heart motion by tracking the aortic annulus from cine-MRI. A rigid-wall fluid-dynamic simulation is performed to derive the time-varying wall pressure field. We build the finite element model considering patient-specific material properties and imposing the derived pressure field and the motion at the annulus boundary. The calibration, which involves the zero-pressure state computation, is based on purely structural simulations. After obtaining the vessel boundaries from the cine-MRI sequences, an iterative procedure is performed to minimize the distance between them and the corresponding boundaries derived from the deformed structural model. A strongly-coupled fluid-structure interaction (FSI) analysis is finally performed with the tuned parameters and compared to the purely structural simulation. The calibration with structural simulations allows to reduce maximum and mean distances between image-derived and simulation-derived boundaries from 8.64 mm to 6.37 mm and from 2.24 mm to 1.83 mm, respectively. The maximum root mean square error between the deformed structural and FSI surface meshes is 0.19 mm. This procedure may prove crucial for increasing the model fidelity in replicating the real aortic root kinematics.

Validation of public stigma of tuberculosis scale during the COVID-19 pandemic using the Rasch model.

Tuberculosis (TB) is still a major global health problem, and it has been particularly concerning during the COVID-19 pandemic. Non-compliance with anti-TB treatment increases the number of multidrug-resistant cases, causing ongoing transmission and increased morbidity and mortality. The main factors causing TB patients' non-compliance are stigma and lack of financial resources. Stigma harms patients and may cause them to delay seeking and adhering to treatment. Thus, it is important to measure the public stigma surrounding TB. However, few scales are available to measure this stigma as it developed during the COVID-19 pandemic. This study, therefore, aimed to develop and validate such a scale. Mixed methods were employed in this study, consisting of a qualitative phase using in-depth interviews with 26 community leaders and a descriptive quantitative survey of 37 people in the Sumedang District to validate the public stigma of tuberculosis scale during the COVID-19 pandemic. The qualitative data were analyzed using thematic analysis, and the quantitative data were analyzed using the Rasch model. The 21 items yielded by an initial qualitative analysis of the data gathered were validated using the RASCH model, yielding 17 valid items with a Cronbach's Alpha of 0.95, person separation of 3.61, real root mean square deviation (RMSE) of 0.37, infit mean square (INFIT MNSQ) of > +1.25, differential item functioning (DIF) of 1.000, the raw variance of 52.4%, and an unexplained variance ranging from 3.4% to 6.9%. The scale developed to measure the public stigma surrounding TB during the COVID-19 pandemic is valid and reliable to measure stigma surrounding TB in the community, especially the pandemic. Further research is needed to apply the scale to bigger and broader populations to evaluate its measurement consistency.

Spectra of quotient modules

We determine the Taylor spectra of quotient tuples of the d-shift on Drury-Arveson spaces with finite-dimensional coefficient spaces. We show that the Taylor spectrum can be described in terms of the approximate zero set of the annihilator ideal, and in terms of the pointwise behavior of the inner multiplier associated with the quotient tuple.

Asymptotic analysis of fundamental solutions of hypoelliptic operators

Abstract Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator 𝐏 ⁢ ( i ⁢ ∂ x ) = ( P 1 ⁢ ( i ⁢ ∂ x ) ) m 1 ⁢ ⋯ ⁢ ( P l ⁢ ( i ⁢ ∂ x ) ) m l \mathbf{P}(i\partial_{x})=(P_{1}(i\partial_{x}))^{m_{1}}\cdots(P_{l}(i\partial% _{x}))^{m_{l}} with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation 𝐏 ⁢ ( i ⁢ ∂ x ) ⁢ u = f {\mathbf{P}(i\partial_{x})u=f} in ℝ n {\mathbb{R}^{n}} . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation.

Braid group action and quasi-split affine 𝚤quantum groups I

This is the first of our papers on quasi-split affine quantum symmetric pairs ( U ~ ( g ^ ) , U ~ ı ) \big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big ) , focusing on the real rank one case, i.e., g = s l 3 \mathfrak g = \mathfrak {sl}_3 equipped with a diagram involution. We construct explicitly a relative braid group action of type A 2 ( 2 ) A_2^{(2)} on the affine ı \imath quantum group U ~ ı \widetilde {{\mathbf U}}^\imath . Real and imaginary root vectors for U ~ ı \widetilde {{\mathbf U}}^\imath are constructed, and a Drinfeld type presentation of U ~ ı \widetilde {{\mathbf U}}^\imath is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine ı \imath quantum groups in the sequels.

When amenable groups have real rank zero C∗-algebras

We investigate when discrete, amenable groups have [Formula: see text]-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse is an open problem. We show that if [Formula: see text] has real rank zero, then all normal subgroups of [Formula: see text] that are elementary amenable and have finite Hirsch length must be locally finite.

Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

A real univariate polynomial of degree n is called hyperbolic if all of its n roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article is on families of hyperbolic polynomials which are determined through k linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a hyperbolic slice. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the local extreme points of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most k distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.

Identifying Suitable Reference Gene Candidates for Quantification of DNA Damage-Induced Cellular Responses in Human U2OS Cell Culture System.

DNA repair pathways trigger robust downstream responses, making it challenging to select suitable reference genes for comparative studies. In this study, our goal was to identify the most suitable housekeeping genes to perform comparable molecular analyses for DNA damage-related studies. Choosing the most applicable reference genes is important in any kind of target gene expression-related quantitative study, since using the housekeeping genes improperly may result in false data interpretation and inaccurate conclusions. We evaluated the expressional changes of eight well-known housekeeping genes (i.e., 18S rRNA, B2M, eEF1α1, GAPDH, GUSB, HPRT1, PPIA, and TBP) following treatment with the DNA-damaging agents that are most frequently used: ultraviolet B (UVB) non-ionizing irradiation, neocarzinostatin (NCS), and actinomycin D (ActD). To reveal the significant changes in the expression of each gene and to determine which appear to be the most acceptable ones for normalization of real-time quantitative polymerase chain reaction (RT-qPCR) data, comparative and statistical algorithms (such as absolute quantification, Wilcoxon Rank Sum Test, and independent samples T-test) were conducted. Our findings clearly demonstrate that the genes commonly employed as reference candidates exhibit substantial expression variability, and therefore, careful consideration must be taken when designing the experimental setup for an accurate and reproducible normalization of RT-qPCR data. We used the U2OS cell line since it is generally accepted and used in the field of DNA repair to study DNA damage-induced cellular responses. Based on our current data in U2OS cells, we suggest using 18S rRNA, eEF1α1, GAPDH, GUSB, and HPRT1 genes for UVB-induced DNA damage-related studies. B2M, HPRT1, and TBP genes are recommended for NCS treatment, while 18S rRNA, B2M, and PPIA genes can be used as suitable internal controls in RT-qPCR experiments for ActD treatment. In summary, this is the first systematic study using a U2OS cell culture system that offers convincing evidence for housekeeping gene selection following treatment with various DNA-damaging agents. Here, we unravel an indispensable issue for performing and assessing trustworthy DNA damage-related differential gene expressional analyses, and we create a "zero set" of potential reference gene candidates.

Hypercyclic Algebras for Convolution Operators with Long Arithmetic Progressions in their Zero Sets

Hypercyclic Algebras for Convolution Operators with Long Arithmetic Progressions in their Zero Sets

Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions

ABSTRACT In this paper, we provide Dirichlet series with periodic coefficients that have Riemann’s functional equation and real zeros of Dirichlet L-functions. The details are as follows. Let L(s, χ) be the Dirichlet L-function and G(χ) be the Gauss sum associated with a primitive Dirichlet character χ (mod q). We define f ( s , χ ) : = q s L ( s , χ ) + i − κ ( χ ) G ( χ ) L ( s , χ ¯ ) , where χ ¯ is the complex conjugate of χ and κ(χ) := (1 – χ(−1))/2. Then, we prove that f (s, χ) satisfies Riemann’s functional equation in Hamburger’s theorem if χ is even. In addition, we show that f (σ, χ) ≠ 0 for all σ ≥ 1. Moreover, we prove that f(σ, χ) ≠ 0 for all 1/2 ≤ σ &lt; 1 if and only if L(σ, χ) ≠ 0 for all 1/2 ≤ σ &lt; 1. When χ is real, all zeros of f (s, χ) with ℜ ( s ) &gt; 0 are on the line σ = 1/2 if and only if the generalized Riemann hypothesis for L(s, χ) is true. However, f (s, χ) has infinitely many zeros off the critical line σ = 1/2 if χ is non-real.