Zeros Of Function Research Articles

Aggregating Composite Indicators through the Geometric Mean: A Penalization Approach

In this paper, we introduce a penalized version of the geometric mean. In analogy with the Mazziotta Pareto Index, this composite indicator is derived as a product between the geometric mean and a penalization term to account for the unbalance among indicators. The unbalance is measured in terms of the (horizontal) variability of the normalized indicators opportunely scaled and transformed via the Box–Cox function of order zero. The penalized geometric mean is used to compute the penalized Human Development Index (HDI), and a comparison with the geometric mean approach is presented. Data come from the Human Development Data Center for 2019 and refer to the classical three dimensions of HDI. The results show that the new method does not upset the original ranking produced by the HDI but it impacts more on countries with poor performances. The paper has the merit of proposing a new reading of the Mazziotta Pareto Index in terms of the reliability of the arithmetic mean as well as of generalizing this reading to the geometric mean approach.

Perturbed Fourier uniqueness and interpolation results in higher dimensions

We obtain new Fourier interpolation and uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa [11] and the second author [12]. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radii are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an application, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Viazovska [1].

On a property of the ideals of the polynomial ring $R[x

Let RR be a commutative ring with unity 1≠01≠0. In this paper we introduce the definition of the first derivative property on the ideals of the polynomial ring R[x]R[x]. In particular, when RR is a finite local ring with principal maximal ideal m≠{0}m≠{0} of index of nilpotency ee, where 1<e≤|R/m|+11≤e≤|R/m|+1, we show that the null ideal consisting of polynomials inducing the zero function on RR satisfies this property. As an application, when RR is a finite local ring with null ideal satisfying this property, we prove that the stabilizer group of RR in the group of polynomial permutations on the ring R[x]/(x2)R[x]/(x2), is isomorphic to a certain factor group of the null ideal.

First and second countable generators

A subset G of Cp(X) - the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence - is a generator (resp., (0,≠0)-generator, or (0,1)-generator) if whenever a point x is not in a closed set C of X, there is a g in G such that g(x)∉g(C)‾ (resp., g(x)≠0 and g(C)⊆{0}, or g(x)=1 and g(C)⊆{0}).The motivation behind the questions: Which spaces have a first countable(0,≠0)-generator containing the zero function,0? Are they precisely the separable spaces? Which spaces have a second countable(0,≠0)-generator containing0? Are they precisely the cosmic spaces? is elucidated, and partial solutions presented.

Isoparametric foliations and the Pompeiu property

<abstract><p>A bounded domain $ \Omega $ in a Riemannian manifold $ M $ is said to have the Pompeiu property if the only continuous function which integrates to zero on $ \Omega $ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails to have the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $ M $ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $ M $ under which the level domains of an isoparametric function fail to have the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.</p></abstract>

Non-trivial zeros of riemann zeta function and riemann hypothesis

Non-trivial zeros of riemann zeta function and riemann hypothesis

Koopman Operator Based Modeling for Quadrotor Control on SE(3)

In this letter, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on SE(3) in terms of an infinite-dimensional linear system which evolves in the space of observables (lifted space) and which is more appropriate for control design purposes. The major challenge when using the Koopman operator is the characterization of a set of observables that can span the lifted space. Most of the existing methods either start from a set of dictionary functions and then search for a subset that best fits the underlying nonlinear dynamics or they rely on machine learning algorithms to learn these observables. Instead of guessing or learning the observables, in this work we derive them in a systematic way for the quadrotor dynamics on SE(3). In addition, we prove that the proposed sequence of observables converges pointwise to the zero function, which allows us to select only a finite set of observables to form (an approximation of) the lifted space. Our theoretical analysis is also confirmed by numerical simulations which demonstrate that by increasing the dimension of the lifted space, the derived linear state space model can approximate the nonlinear quadrotor dynamics more accurately.

Riemann Hypothesis and Random Walks: The Zeta Case

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

On the Zeta-Function of Zeros of Entire Function

This article is devoted to the study of the properties of the zeta-function of zeros of entire function. We obtained an explicit expression for the kernel of the integral representation of the zetafunction in one case

Partial integral operators of Fredholm type on Kaplansky-Hilbert module over $L_0$

The article studies some characteristic properties of self-adjoint partially integral operators of Fredholm type in the Kaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $L_{0}\left(\Omega_{2}\right)$. Some mathematical tools from the theory of Kaplansky-Hilbert module are used. In the Kaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $ L_{0} \left (\Omega _ {2} \right)$ we consider the partially integral operator of Fredholm type $T_{1}$ ($ \Omega_{1} $ and $\Omega_{2} $ are closed bounded sets in $ {\mathbb R}^{\nu_{1}}$ and $ {\mathbb R}^{\nu_{2}},$ $\nu_{1}, \nu_{2} \in {\mathbb N} $, respectively). The existence of $ L_{0} \left (\Omega _ {2} \right) $ nonzero eigenvalues for any self-adjoint partially integral operator $T_{1}$ is proved; moreover, it is shown that $T_{1}$ has finite and countable number of real $L_{0}(\Omega_{2})$-eigenvalues. In the latter case, the sequence $ L_{0}(\Omega_{2})$-eigenvalues is order convergent to the zero function. It is also established that the operator $T_{1}$ admits an expansion into a series of $\nabla_{1}$-one-dimensional operators.

The origin of the surface barrier in nanoporous materials

Surface barriers are influencing the mass transfer in nanopores, but their origin is unclear and can be quite different in different materials. For MFI and CHA membranes studied here, we show that the surface barrier may be a surface diffusion process with higher activation energy than the surface diffusion process in the pores, but other possible mechanisms such as pore blocking and pore narrowing has not been ruled out. The higher activation energy is probably a result of less interaction between adsorbed molecules at the pore mouth than inside the pores, i.e. the barrier is simply a geometrical effect in these materials. For pure components at low concentration in MFI zeolite, we found that barrier is proportional to the product of the molecular weight and heat of desorption. For MFI and CHA zeolite, we observed that the barrier is a function of concentration and approach zero at high concentration and that the barriers of the components become more similar due to interaction between the components in mixtures, which explains the high and selective mass transfer displayed by these nanoporous materials at high concentration.

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.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

A Principle Related to Zeros of Real Functions

A Principle Related to Zeros of Real Functions

The Fourier transform of a function related to cylinder functions and asymptotic expansions with logarithmic terms

We consider a function that is obtained by the first kind Hankel function of order zero by removing its logarithmic singularity. This function, apart from a multiple of the zero-order Bessel function as an additive term, can be considered as the ‘entire part’ of the Hankel function. We constitute the Fourier transform of this function by considering a fifth-order ordinary differential equation which can be considered as a generalization of the differential equation for the zero-order cylinder functions. Since the asymptotics of our entire part of the Hankel function contains logarithmic terms, this new function can be used to derive asymptotic expansions of some other functions whose asymptotics contain logarithmic terms too. In the Fourier image it can be shown that these expansions are not merely asymptotic but actually convergent.

Zeros of the Riemann Zeta Function on the Line z = 1/2 + it 0 II

It is shown that all nontrivial zeros of the Riemann zeta function lie on the line z = 1/2 + it0 and can be classified into two sets: normal zeros, whose numbers are uniquely restored by the value of the root, and anomalous zeros, the unique restoration of whose numbers requires the knowledge of the values of two neighboring zeros (the left and right ones). The methods of analysis applied can be useful for the study of physical phenomena related to the phase slip.

UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.

A Speedy New Proof of the Riemann's Hypothesis

In this paper we show that Riemann's function <i>(xi)</i>, involving the Riemann’s (<i>zeta)</i> function, is holomorphic and is expressed as a convergent infinite polynomial product in relation to their zeros and their conjugates. Our work will be done on the critical band in which non-trivial zeros exist. Our approach is to use the properties of power series and infinite product decomposition of holomorphic functions. We take inspiration from the Weierstrass method to construct an infinite product model which is convergent and whose zeros are the zeros of the zeta function. By applying the symetric functional equation of the <i>xi</i> function we deduce a relation between each zero of the function <i>xi</i> and its conjugate. Because of the convergence of the infinite product, and that the elementary polynomials of the second degree of this same product are irreducible into the complex set, then this relation is well determined. The apparent simplicity of the reasoning is based on the fundamental theorems of Hadamard and Mittag-Leffler. We obtain the sought result: the real part of all zeros is equal to ½. This article proves that the Riemann’ hypothesis is true. Our perspectives for a next article are to apply this method to Dirichlet series, as a generalization of the Riemann function.

非均匀复变解析函数零点的孤立性及惟一性

非均匀复变解析函数零点的孤立性及惟一性

Fourier Transform Associated with Riemann Zeta Function

The Fourier transform associated with the normalized logarithm of the modulus of the Riemann Zeta Function is considered. The formulas linking the Fourier transform and the zeros of the Riemann function are established that lead to the necessary and sufficient condition of satisfaction of the Riemann hypothesis.