Existence Of Zeros Research Articles

An infinite dimensional version of the intermediate value theorem

Let mathfrak {f}= I-k be a compact vector field of class C^1 on a real Hilbert space mathbb {H}. In the spirit of Bolzano’s Theorem on the existence of zeros in a bounded real interval, as well as the extensions due to Cauchy (in mathbb {R}^2) and Kronecker (in mathbb {R}^k), we prove an existence result for the zeros of mathfrak {f} in the open unit ball mathbb {B} of mathbb {H}. Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction mathfrak {f}|_mathbb {S} of mathfrak {f} to the boundary mathbb {S} of mathbb {B}. As an extension of this, but not as a consequence, we obtain as well an Intermediate Value Theorem whose statement needs the topological degree. Such a result implies the following easily comprehensible, nontrivial, generalization of the classical Intermediate Value Theorem: If a half-line with extremeq notin mathfrak {f}(mathbb {S})intersects transversally the functionmathfrak {f}|_mathbb {S}for only one point of mathbb {S}, then any value of the connected component ofmathbb {H}{setminus }mathfrak {f}(mathbb {S})containingqis assumed bymathfrak {f}inmathbb {B}. In particular, such a component is bounded.

An active noise control filter design method based on machine learning algorithms

Filter design is the essential problem in active noise control. In the case of FIR filter design, the whole problem can be formulated as a convex optimization. In practice, limited by computation capability, IIR filters are preferred but more challenging to design. On the one hand, the existence of zeros could cause instablity; on the other hand, the problem can not be formulated as a convex optimization problem. This work introduces a new IIR filter design method, which stablizes the system by modeling the ANC controller as casade parametric biquadratic IIRs and optimizes their cofficients by using machine learning algorithms. A neural network with zero input but learnable bias and network weights is used to generate the filter coefficients; and the loss function consists of a reward term for reducing noise and penalty terms for breaking the constraints. Training the neural network with gradient based machine learning algorithm directly leads to a group of coefficients that achieves good ANC performance and is subject to given constraints. It turned out that no big and deep neurual network is required for this method to perform well and therefore it is computationally efficient. Experiments showed that the proposed method was able to consistently yield good ANC filters for a variety of given acoustical environments.

Local exponential stability of four almost-periodic positive solutions for a classic Ayala-Gilpin competitive ecosystem provided with varying-lags and control terms

Ayala-Gilpin ecosystem is one of the most famous differential equation models based on experimental and theoretical analysis. Most of the previous works on the almost periodic solution and its stability are mainly about the existence of an almost-periodic solution and its global stability. In fact, the real ecosystem presents multiple stable states due to the influence of various external factors. However, there are few researches on multiple almost-periodic solutions and local stability of Ayala-Gilpin ecosystem. Therefore, we focus on the existence and local exponential stability of multiple almost-periodic solutions for a classical nonlinear competitive Ayala-Gilpin ecosystem provided with varying-lags and control terms in this paper. Firstly, a class of functions having only two zeros in is investigated. Next, based on the existence of zeros of these functions, we obtain some sufficient conditions to ensure the existence of at least four almost-periodic positive solutions by utilising theorem of coincidence degree and some inequality techniques. Furthermore, we also conclude that this ecosystem exists at least two almost-periodic positive solutions under relatively weak conditions. Finally, we build some Lyapunov functionals to discuss the local exponential stability of each almost-periodic positive solution in its own region. As an application, a numerical simulation inspect the effectiveness of our major results.

Water wave scattering by three thin vertical barriers with middle one partially immersed and outer two submerged

The problem of oblique water wave scattering by three thin vertical barriers in deep water is investigated here assuming linear theory. The geometrical configurations of the three barriers are such the inner(middle) barrier is partially immersed and the two outer barriers are completely submerged and extend infinitely downwards. A system of three simultaneous integral equations of first kind involving differences of velocity potentials across the barriers has been obtained in the mathematical analysis by employing Havelock’s expansion of water wave potential alongwith Havelock’s inversion formulae. Approximate numerical solutions of these integral equations are obtained by using single-term Galerkin approximations where the single term is chosen to be the exact solution of the integral equation obtained for a single vertical barrier in deep water for normal incidence of surface waves. Fairely accurate numerical estimates for the reflection and transmission coefficients are obtained by solving this system of linear equations. These estimates satisfy the energy identity, thereby justifying the validity of the present method. Due to energy identity, the behaviour of reflection coefficient is discussed by depicting it graphically against wavenumber. Existence of zeros of reflection coefficient is observed only when the two outer barriers are identical i.e. they are submerged from the same depth below the mean free surface. Some earlier results for two completely submerged barriers and a single barrier are recovered as special cases for a particular set of values of the parameters involved in the problem. This provides another check on the correctness of the present method.

The Zeros of the Bergman Kernel for Some Reinhardt Domains

We consider the Reinhardt domainDn={(ζ,z)∈C×Cn:|ζ|2<(1-|z1|2)⋯(1-|zn|2)}.We express the explicit closed form of the Bergman kernel forDnusing the exponential generating function for the Stirling number of the second kind. As an application, we show that the Bergman kernelKnforDnhas zeros if and only ifn≥3. The study of the zeros ofKnis reduced to some real polynomial with coefficients which are related to Bernoulli numbers. This result is a complete characterization of the existence of zeros of the Bergman kernel forDnfor all positive integersn.

Front instability in a condensed phase combustion model

AbstractWe consider a condensed phase (or solid) combustion model and its linearization around the travelling front solution. We construct an Evans function to characterize the eigenvalues of the linearized problem. We estimate this functional in the high activation energy limit. We deduce the existence of zeros with nonnegative real part for high activation energy, which proves the linear instability of the travelling front solution.

The existence of zeros of set-valued mappings in reflexive Banach spaces

In this paper, we establish the equivalence between the existence of zeros for set-valued mappings and the solvability of variational inequality, under some conditions involving the generalized projection operator. Basing on this result, we obtain some existence theorems of zeros for quasimonotone set-valued mappings. As application, we derive several fixed point theorems for generalized inward mappings in Hilbert spaces.

Convergent Series for Riemann's Hypothesis

Riemann's hypothesis affirms that the existence of zeros for zed function have as the royal part (a = 1 ) as the only solution. In this article I analyzed and in turn demonstrates the existence of infinitys royal numbers for his royal part. To define all zeros, we will apply the method of progressive substitution.

Existence Results for Variational Inequalities with Surjectivity Consequences Related to Generalized Monotone Operators

We present existence results for variational inequalities given by generalized monotone operators. As a consequence, we deduce the existence of zeros, or even more, the surjectivity of some classes of set-valued operators. We show that by strengthening the continuity assumptions, similar surjectivity results can be obtained without any monotonicity assumption. In the framework of reflexive Banach spaces, we extend a related result due to Inoan and Kolumban (Nonlinear Anal. 68:47–53, 2008).

The Bergman kernel of the Fock–Bargmann–Hartogs domain and the polylogarithm function

We consider the Fock–Bargmann–Hartogs domain D n,m which is defined by the inequality where (z, ζ) ∈ ℂ n × ℂ m and μ > 0. We give an explicit formula for the Bergman kernel of the domain in terms of the polylogarithm functions. Moreover, using the interlacing property, we describe how the existence of zeros of the Bergman kernel depends on the integers m and n.

An Index Theory for Positive Dynamical Systems with an Application to Lotka–Voterra Systems

In this paper we develop an index theory of positive dynamical systems in planar case. We present a formula of index sum of isolated zeros a positive dynamical system on the positive orthant of R2. As an application, we present some sufficient conditions on existence of zeros of Lotka–Voterra systems in the positive orthant of R2.

Centers of graded fusion categories

Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center ZD(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara–Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara–Yamagami categories. Finally, we prove a general result about the existence of zeroes in S-matrices of weakly integral modular categories.

A framework for existence tests based on the topological degree and homotopy

The invariance of the topological degree under certain homotopies is used to derive a framework for tests to computationally prove the existence of zeros of nonlinear mappings in $${\mathbb {R}^{n}}$$. These tests use interval arithmetic to enclose the range of a function over a box and are provably more general than many other tests like the Moore–Kioustelidis test, a test based on the Krawczyk operator, and another degree–based test published recently. A numerical example is included.

The fixed point property for mappings admitting a center

We introduce a class of nonlinear continuous mappings in Banach spaces which allow us to characterize the Banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping. Later on, we give an application to the existence of zeroes for certain kinds of accretive operators.

On the existence of zeros of monotone operators in reflexive Banach spaces

This paper is devoted to the investigation on the existence of zeros of monotone operators in reflexive Banach spaces. We first present a sufficient condition under which single-valued monotone operators have zeros. The obtained theorem includes a previous result as a special case. A necessary and sufficient condition for the existence of zeros of maximal monotone operators is presented.

Surjectivity and fixed points of relaxed dissipative multifunctions. Differential inclusions approach

One from the most important properties of accretive and monotone operators is the existence of zeros and surjectivity. In the paper we introduce relaxed variants of dissipative, accretive and monotone operators. Using essentially the properties of the solution set of appropriate differential inclusions we study the existence of zeros of such operators. As corollaries the existence of fixed points of relaxed contractive and relaxed nonexpansive multifunctions are obtained.

On the Value Distribution of Hurwitz Zeta-Functions at the Nontrivial Zeros of the Riemann Zeta-Function

We consider the value distribution of Hurwitz zeta-functions\(\zeta (s,\alpha ): = \sum\nolimits_{n = 0}^\infty {\frac{1}{{(n + \alpha )^s }}} \) at the nontrivial zeros ϱ= β + iγ of the Riemann zeta-function ζ (s):= ζ (s, 1). Using the method of Conrey, Ghosh and Gonek we prove for fixed 0< α< 1 andH ≤T that $$\begin{gathered} \sum\limits_{T< \gamma \leqslant T + H} {\zeta (\varrho ,\alpha ) = - \left( {\Lambda \frac{1}{\alpha } + \sum\limits_{n = 1}^\infty {\frac{{\exp ( - 2\pi i\alpha n}}{n}} } \right)\frac{H}{{2\pi }}} \hfill \\ + O(H exp( - C(log T)^{1/3} ) + T^{1/2 + \varepsilon } ) \hfill \\ \end{gathered} $$ with some absolute constantC > 0 (a similar result was first proved by Fujii [4] under assumption of the Riemann hypothesis). It follows that\(\frac{{\zeta (s,\alpha )}}{{\zeta (s)}}\) is an entire function if and only if α = 1/2 or α = l. Further, we prove for α ≠ 1/2, 1 the existence of zeros ϱ = β +iγ withT < γ ≤T + T3/4, 1/2 β ≤ 9/10+ e and ζ(ϱ,α)≠0.

INEXACT VERSIONS OF PROXIMAL POINT AND AUGMENTED LAGRANGIAN ALGORITHMS IN BANACH SPACES

We generalize a proximal-like method for finding zeroes of maximal monotone operators in Hilbert spaces with quadratic regularization due to Solodov and Svaiter, making it possible the use of other kind of regularizations and extending it to Banach spaces. In particular, we introduce an appropriate error criterium to obtain an inexact proximal iteration based on Bregman functions and construct a hyperplane which strictly separates the current iterate from the solution set. A Bregman projection onto this hyperplane is then used to obtain the next iterate. Boundedness of the sequence and optimality of the weak accumulation points are established under suitable assumptions on the regularizing function, which hold for any power greater than one of the norm of any uniformly smooth and uniformly convex Banach space, without any assumption on the operator other than existence of zeroes. These assumptions let us, also, obtain similar results in Banach spaces for the Hybrid Extragradient-Generalized Proximal Point method, proposed by Solodov and Svaiter for finite dimensional spaces. We then transpose such methods to generate augmented Lagrangian methods for L P -constrained convex optimization problems in Banach spaces, obtaining two alternative procedures which allow for sequences, and optimality of primal and dual weak accumulation points, are then established, assuming only existence of Karush-Kuhn-Tucker pairs. *The work of this author was partially supported by CNPq grant no. 301280/86 and by Marcel Dassault chair, Centro de Modelamiento Matemático, Universidad de Chile.

Infrequency of Purchase: A Model for Clothing Consumption with Panel Data

Microeconomic survey data sets offer important advantages for the analysis of consumer demand, although one of the problems associated with these data is the existence of zeros in the expenditure records of households. One of the reasons behind a zero record is infrequency of purchase. This paper focuses on the analysis of Infrequency of Purchase Models (IPMs). We develop an IPM for panel data and compare it with the IPM for cross-sectional data. The estimation of these models using panel data might shed some light on the restrictions that one has to make in order to estimate them using cross-sectional data. Finally, we test for individual heterogeneity and compare the two sets of estimation results. The data are drawn from the Spanish Family Expenditure Survey (ECPF) for the period 1985 to 1991.

Implicit function theorems for m-accretive and locally accretive set-valued mappings

In this paper we elaborate new methods to the theory of parametric generalized equations governed by accretive mappings. Two implicit function theorems will be proved, one for m-accretive and another for locally accretive and continuous set-valued mappings. Making use of the theorem for m-accretive mappings we will study parametric nonlinear evolution problems. It will be also shown that there is possible a unified approach to the theory of implicit function theorems, namely many results on the stability of the solutions of perturbed nonsmooth or monotone generalized equations, variational inequalities [?, ?, 1, 5, 6, ?], as well the standard implicit function theorem (Theorem 15.1, [?]) are consequences of the present theorem for locally-accretive and continuous set-valued mappings. In the following investigations the geometrical properties of Banach spaces, the extension problems for accretive mappings and the existence of zeros for accretive mappings play an important role. The two implicit function theorems are independent because of the above mentioned extension problems. The possibility of extending an accretive mapping to an m-accretive one was studied in many papers [?, ?, 4]. The most important result is that if C is a closed, convex subset of a reflexive, strictly-convex Banach space X, which is not a nonexpansive retract of X, then no accretive mappings A : X ; X, with co (Dom (A)) = C, can be extended to an m-accretive mapping B : X ; X, with co(Dom (B)) = C. The closed balls in a non-Hilbert Banach space are not nonexpansive retracts of the whole space. A consistency condition appears in the following theorems. Such conditions are frequently used in implicit function theorems, see [?, ?, ?, ?], because they give ”almost” solutions for each parameter close enough to a fixed parameter and replace the continuity with respect to parameters. Our condition is that from the papers [?, ?] and as will be proved it is a weaker property than the lower-semicontinuity or the pseudo-continuity. The consistency will be used toghether with results on the existence of zeros for accretive mappings, see for example [?, ?, ?, ?, ?, 3, ?]. In this paper we do not use compactness