Finite Number Of Zeros Research Articles

Existence and Multiplicity Results for an Elliptic Problem Involving Mixed Local and Nonlocal Operator

In this work, we consider the following mixed local-nonlocal quasilinear ellipic problem$$(P_\lambda)\left\{\begin{array}{rcll} -\Delta_p u+(-\Delta)_p^s u = \lambda f(u) \text{in}\Omega,\\ u 0 \text{in }\Omega,\\ u = 0 \text{in }\mathbb{R}^N \setminus\Omega,\end{array}\right.$$where Ω⊂RN is a bounded regular domain in RN with 0s1pN and f: R→R is a continuous function, that have a finite number of zeroes, changing sign between them. The main goal of this paper is to prove the existence and multiplicity of positive solutions for such problems by using variational methods.

Об одном подходе к нахождению сумм кратных числовых рядов

An approach to calculating sums of some types of multiple numerical series is presented. This approach is based on using the formula for the resultant of a polynomial (or an entire function with a finite number of zeros) and an entire function obtained earlier by A.M. Kytmanov and E.K. Myshkina. This formula does not require values of the roots of the functions under study and is a combinatorial expression. By calculating the resultant of a polynomial and an entire function in two different ways, it is possible to obtain a relation for multiple numerical series. For the second way to find the resultant, we use the product of one function at the roots of another. In this article, the sums of some types of multiple numerical series that were previously absent in known reference books are found. They are expressed in terms of well-known special functions such as the Bessel function. This approach to calculating the sums of multiple numerical series differs significantly from the method based on the use of residue integrals associated with a system of equations. The relevance of this problem is determined by the fact that in applied problems, for example, in the equations of chemical kinetics, there are functions and systems of equations consisting of exponential polynomials.

Asymptotic vectors of entire curves

We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate a relationship between the asymptotic vectors and the Picard exceptional vectors.
 A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called an asymptotic vector for the entire curve $\vec{G}(z)=(g_1(z),g_2(z),\ldots,g_p(z))$ if there exists a continuous curve $L: \mathbb{R}_+\to \mathbb{C}$ given by an equation $z=z\left(t\right)$, $0\le t<\infty $, $\left|z\left(t\right)\right|<\infty $, $z\left(t\right)\to \infty $ as $t\to \infty $ such that$$\lim\limits_{\stackrel{z\to\infty}{z\in L}} \frac{\vec{G}(z)\vec{a} }{\big\|\vec{G}(z)\big\|}=\lim\limits_{t\to\infty} \frac{\vec{G}(z(t))\vec{a} }{\big\|\vec{G}(z(t))\big\|} =0,$$ where $\big\|\vec{G}(z)\big\|=\big(|g_1(z)|^2+\ldots +|g_p(z)|^2\big)^{1/2}$, $\vec{G}(z)\vec{a}=g_1(z)\cdot\bar{a}_1+g_2(z)\cdot\bar{a}_2+\ldots+g_p(z)\cdot\bar{a}_p$. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called a Picard exceptional vector of an entire curve $\vec{G}(z)$ if the function $\vec{G}(z)\vec{a}$ has a finite number of zeros in $\left\{\left|z\right|<\infty \right\}$.
 We prove that any Picard exceptional vector of transcendental entire curve with linearly independent com\-po\-nents and without common zeros is an asymptotic vector.Here we de\-mon\-stra\-te that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which $\infty $ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph``Value Distribution of Meromorphic Functions''.Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve.At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve.

A KAM Theorem for the Hamiltonian with Finite Zero Normal Frequencies and Its Applications (In Memory of Professor Walter Craig)

we investigate the existences of KAM tori for the infinite dimensional Hamiltonian system with finite number of zeros among normal frequencies. By constructing a constant vector we show that, for “most” tangent frequencies in the sense of Lebesgue measure, either if the vector is zero, there is a KAM torus or if the vector is not zero, there is no KAM torus in some domain. As an application, we show that the nonlinear Schrodinger equation with a zero among normal frequencies possesses many quasi-periodic solutions.

An inverse problem of the theory of the electromagnetic radiation for a function spectrum with a finite number of zeros

The radiation energy of an electromagnetic field in a certain frequency interval is proportional to the square of the modulus of the Fourier transform of this field. The determination of the radiation energy is a direct problem and has an independent value in solving many problems of radiation theory. It is of great
 interest, according to the known spectrum of the energy distribution of radiation over frequencies, to find the function of the electromagnetic field that created this spectrum, i.e. solve the inverse problem of radiation theory. In this paper, the above inverse problem is solved, namely, the function of the electromagnetic field is sought from a known function of the spectrum having a finite number of zeros on the frequency interval.

On sum of squares certificates of non-negativity on a strip

In [6] , Murray Marshall proved that every f ∈ R [ X , Y ] non-negative on the strip [ 0 , 1 ] × R can be written as f = σ 0 + σ 1 X ( 1 − X ) with σ 0 , σ 1 sums of squares in R [ X , Y ] . In this work, we present a few results concerning this representation in particular cases. First, under the assumption deg Y ⁡ f ≤ 2 , by characterizing the extreme rays of a suitable cone, we obtain a degree bound for each term. Then, we consider the case of f positive on [ 0 , 1 ] × R and non-vanishing at infinity , and we show again a degree bound for each term, coming from a constructive method to obtain the sum of squares representation. Finally, we show that this constructive method also works in the case of f having only a finite number of zeros, all of them lying on the boundary of the strip, and such that ∂ f ∂ X does not vanish at any of them.

The long-time behavior of weighted p-Laplacian equations

In this work we study weighted $p$-Laplacian equations in abounded domain with a variable and generally non-smooth diffusion coefficient having at most a finite number of zeroes. The main attention is focused on the case that the diffusion coefficient $a(x)$ in such equations satisfies the inequality $\liminf\limits_{x\to z}|x-z|^{-p}a(x)> 0$ for every $ z\in \overline\Omega$. We show the existence of weak solutions and global attractors in $L^2(\Omega)$, $L^q(\Omega)(q\geq 2)$ and $D_0^{1,p}(\Omega)$, respectively.

On the Cubic Lowest Landau Level Equation

We study dynamical properties of the cubic lowest Landau level equation, which is used in the modeling of fast rotating Bose-Einstein condensates. We obtain bounds on the decay of general stationary solutions. We then provide a classification of stationary waves with a finite number of zeros. Finally, we are able to establish which of these stationary waves are stable, through a variational analysis.

Weighted Bergman kernel functions associated to meromorphic functions

We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with modulus squared weight of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng problem.

Functions and Vector Fields on C(CP^n)-singular manifolds

In this paper we study functions and vector fields with isolated singularities on a $C(\mathbb{C}P^n)$-singular manifold. In general, a$C(\mathbb{C}P^n)$-singular manifold is obtained from a~smooth $(2n+1)$-manifold with boundary which is a disjoint union of complex projective spaces $\mathbb{C}P^n \cup\ldots \cup\mathbb{C}P^n$ and subsequent capture of the cone over each component $\mathbb{C}P^n$ of the boundary. We calculate the Euler characteristic of a compact $C(\mathbb{C}P^n)$-singular manifold $M^{2n+1}$ with finite isolated singular points. We also prove a version of the Poincare-Hopf Index Theorem for an almost smooth vector field with finite number of zeros on a~$C(\mathbb{C}P^n)$-singular manifold.

Proton scattering by a hydrogen atom in an effectively two-body model

It is assumed that the total potential of proton interaction with a hydrogen atom is the sum of the short-range nuclear soft-core Reid potential and the long-range Thomas-Fermi potential. A quantum mechanical analysis of low-energy features of the phase shift and cross section for elastic proton scattering on a hydrogen atom is given for the case of zero total angular momentum. The calculations performed in the present study within a nonlinear version of the variable-phase approach ultimately revealed that, because of a long-range character of the asymptotic behavior of the Thomas-Fermi potential, the respective cross section at low energies oscillates but has a finite number of zeros.

DUAL EQUATION AND INVERSE PROBLEM FOR AN INDEFINITE STURM–LIOUVILLE PROBLEM WITH M TURNING POINTS OF EVEN ORDER

In this paper the differential equation y″ + (ρ 2 φ 2 (x) –q(x))y = 0 is considered on a finite interval I, say I = [0, 1], where q is a positive sufficiently smooth function and ρ 2 is a real parameter. Also, [0, 1] contains a finite number of zeros of φ 2 , the so called turning points, 0 < x 1 < x 2 < … < x m < 1. First we obtain the infinite product representation of the solution. The product representation, satisfies in the original equation. As a result the associated dual equation is derived and then we proceed with the solution of the inverse problem.

Global attractors for degenerate semilinear parabolic equations

Let Ω be a smooth bounded domain in RN,(N≥2). We consider the long-time behavior of solutions to the following problem {ut−div(σ(x)∇u)+f(u)=gin Ω×R+,u=0on ∂Ω×R+,u(x,0)=u0(x)in Ω, where u0,g∈L1(Ω). The diffusion coefficient σ(x) is measurable, nonnegative and is allowed to have a finite number of zeroes at some points. We provide the existence and uniqueness results for the problem. Then we establish some asymptotic regularity results on the solution and consider its long-time behavior. The existence of a global attractor is obtained in the proper space.

Asymptotic form and infinite product representation of solution of a second order initial value problem with a complex parameter and a finite number of turning points

The paper studies the differential equation $$y'' + (\rho ^2 \varphi ^2 (x) - q(x))y = 0$$ (*) on the interval I = [0, 1], containing a finite number of zeros 0 < x1 < x2 < ... < xm < 1 of ϕ2, i.e. so-called turning points. Using asymptotic estimates from [6] for appropriate fundamental systems of solutions of (*) as |ρ| → ∞, it is proved that, if there is an asymptotic solution of the initial value problem generated by (*) in the interval [0, x1), then the asymptotic solutions in the remaining intervals can be obtained recursively. Furthermore, an infinite product representation of solutions of (*) is studied. The representations are useful in the study of inverse spectral problems for such equations.

Weakly coercive mappings sharing a value

Some sufficient conditions are provided that guarantee that the difference of a compact mapping and a proper mapping defined between any two Banach spaces over $$ \mathbb{K} $$ has at least one zero. When conditions are strengthened, this difference has at most a finite number of zeros throughout the entire space. The proof of the result is constructive and is based upon a continuation method.

Gabor windows supported on [formula omitted] and compactly supported dual windows

Consider a bounded function g supported on [ − 1 , 1 ] and a modulation parameter b ∈ ] 1 / 2 , 1 [ for which the Gabor system { E m b T n g } m , n ∈ Z is a frame. We show that such a frame always has a compactly supported dual window. More precisely, we show that if b < N N + 1 for some N ∈ N , it is possible to find a dual window supported on [ − N , N ] . Under the additional assumption that g is continuous and only has a finite number of zeros on ] − 1 , 1 [ , we characterize the frame property of { E m b T n g } m , n ∈ Z . As a consequence we obtain easily verifiable criteria for a function g to generate a Gabor frame with a dual window having compact support of prescribed size.

First Colonization of a Spectral Outpost in Random Matrix Theory

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N −2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature. The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate N −γ , whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.

Periodic solutions of a periodic scalar piecewise ode

We study the distributions of periodic solutions of scalar piecewise equations defined by $x'=f(t,x)$ if $x\geq 0$, and $x'=g(t,x)$ if $x<0$, where $f,g$ are time periodic $\mathcal C^1$-functions such that $f(t,0)=g(t,0)$. Thus, these are equations on the cylinder where the vector field is not necessarily smooth on one of the equatorial circles.<br> We find that the solutions are $\mathcal C^1$-functions when the equation restricted to the equatorial line has a finite number of zeroes. Moreover, if $f$ and $g$ are analytic functions and the zeroes on the equatorial line are finite and simple, the set of periodic solutions consists of isolated periodic solutions and a finite number (determined by the number of zeroes) of closed "bands" of periodic solutions.

On the {\L}ojasiewicz exponent

Let $\Bbb K$ be an algebraically closed field and let $X\subset \Bbb K^l$ be an $n-$dimensional affine variety of degree $D.$ We give a sharp estimation of the degree of the set of non-properness for generically-finite separable and dominant mapping $f=(f_1,...,f_n): X\to \Bbb K^n$. We show that such a mapping must be finite, provided it has a sufficiently large geometric degree. Moreover, we estimate the \L ojasiewicz exponent at infinity of a polynomial mapping $f: X\to \Bbb K^m$ with a finite number of zeroes.

Vortex distribution in the lowest Landau level

We study the vortex distribution of the wave functions minimizing the Gross-Pitaevskii energy for a fast rotating condensate in the lowest Landau level (LLL): we prove that the minimizer cannot have a finite number of zeroes, thus the lattice is infinite, but not uniform. This uses the explicit expression of the projector onto the LLL. We also show that any slow varying envelope function can be approximated in the LLL by distorting the lattice. This is used in particular to approximate the inverted parabola and understand the role of ``invisible'' vortices: the distortion of the lattice is very small in the Thomas-Fermi region but quite large outside, where the ``invisible'' vortices lie.