Zero Density Research Articles

Density of Real Zeros of the Tutte Polynomial

The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.

Lane–Emden equation with inertial force and general polytropic dynamic model for molecular cloud cores

We study self-similar hydrodynamics of spherical symmetry using a general polytropic ( GP) equation of state and derive the GP dynamic Lane-Emden equation ( LEE) with a radial inertial force. In reference to Lou & Cao, we solve the GP dynamic LEE for both polytropic index gamma = 1 + 1/n and the isothermal case n -> +infinity; our formalism is more general than the conventional polytropic model with n = 3 or gamma = 4/3 of Goldreich & Weber. For proper boundary conditions, we obtain an exact constant solution for arbitrary n and analytic variable solutions for n = 0 and n = 1, respectively. Series expansion solutions are derived near the origin with the explicit recursion formulae for the series coefficients for both the GP and isothermal cases. By extensive numerical explorations, we find that there is no zero density at a finite radius for n >= 5. For 0 0 for monotonically decreasing density from the origin and vanishing at a finite radius for c being less than a critical value C-cr. As astrophysical applications, we invoke our solutions of the GP dynamic LEE with central finite boundary conditions to fit the molecular cloud core Barnard 68 in contrast to the static isothermal Bonnor-Ebert sphere by Alves et al. Our GP dynamic model fits appear to be sensibly consistent with several more observations and diagnostics for density, temperature and gas pressure profiles.

An explicit bound for the least prime ideal in the Chebotarev density theorem

We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. In order to accomplish this, we prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon for Hecke $L$-functions. As an application, we prove the first explicit nontrivial upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also present applications to the group of $\mathbb{F}_p$-rational points of an elliptic curve and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.

Distribution of zeros of the S-matrix of chaotic cavities with localized losses and coherent perfect absorption: non-perturbative results

We employ the random matrix theory framework to calculate the density of zeroes of an M-channel scattering matrix describing a chaotic cavity with a single localized absorber embedded in it. Our approach extends beyond the weak-coupling limit of the cavity with the channels and applies for any absorption strength. Importantly it provides an insight for the optimal amount of loss needed to realize a chaotic coherent perfect absorbing trap. Our predictions are tested against simulations for two types of traps: a complex network of resonators and quantum graphs.

Integral points of bounded degree on the projective line and in dynamical orbits

Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $\varphi(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $\varphi^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.

Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.

Experimental demonstration of one-dimensional active plate-type acoustic metamaterial with adaptive programmable density

A class of active acoustic metamaterials (AMMs) with a fully controllable effective density in real-time is introduced, modeled, and experimentally verified. The density of the developed AMM can be programmed to any value ranging from −100 kg/m3 to 100 kg/m3 passing by near zero density conditions. This is achievable for any frequency between 500 and 1500 Hz. The material consists of clamped piezoelectric diaphragms with air as the background fluid. The dynamics of the diaphragms are controlled by connecting a closed feedback control loop between the piezoelectric layers of the diaphragm. The density of the material is adjustable through an outer adaptive feedback loop that is implemented by the real-time evaluation of the density using the 4-microphone technique. Applications for the new material include programmable active acoustic filters, nonsymmetric acoustic transmission, and programmable acoustic superlens.

A Chebotarev Variant of the Brun–Titchmarsh Theorem and Bounds for the Lang-Trotter conjectures

We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms. We also obtain new results on the distribution of primes represented by positive-definite integral binary quadratic forms.

Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato–Tate Conjecture

AbstractLet $K/\mathbb{Q}$ be a number field. Let π and π′ be cuspidal automorphic representations of $\textrm{GL}_{d}(\mathbb{A}_{K})$ and $\textrm{GL}_{d^{\prime }}(\mathbb{A}_{K})$. We prove an unconditional and effective log-free zero density estimate for all automorphic L-functions L(s, π) and prove a similar estimate for Rankin–Selberg L-functions L(s, π × π′) when π or π′ satisfies the Ramanujan conjecture. As applications, we make effective Moreno’s analog of Hoheisel’s short interval prime number theorem and extend it to the context of the Sato–Tate conjecture; additionally, we bound the least prime in the Sato–Tate conjecture in analogy with Linnik’s theorem on the least prime in an arithmetic progression. We also prove effective log-free density estimates for automorphic L-functions averaged over twists by Dirichlet characters, which allows us to prove an “average Hoheisel” result for GLdL-functions.

Universal crossover from ground-state to excited-state quantum criticality

We study the nonequilibrium properties of a nonergodic random quantum chain in which highly excited eigenstates exhibit critical properties usually associated with quantum critical ground states. The ground state and excited states of this system belong to different universality classes, characterized by infinite-randomness quantum critical behavior. Using strong-disorder renormalization group techniques, we show that the crossover between the zero and finite energy density regimes is universal. We analytically derive a flow equation describing the unitary dynamics of this isolated system at finite energy density from which we obtain universal scaling functions along the crossover.

The density of the strong symmetric genus values of p-groups

ABSTRACTLet G be a finite group. The strong symmetric genus σ0(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Let p a prime, and let Jp be the set of integers g for which there is a p-group of strong symmetric genus g. We show that the set Jp has density zero in the set of positive integers.

Global solution to 3D problem of a compressible viscous micropolar fluid with spherical symmetry and a free boundary

We consider a nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic, in the domain bounded with two concentric spheres in R3. In this paper we establish the existence of a global solution to the free boundary problem defined with the homogeneous boundary conditions for velocity, microrotation, heat flux on the fixed border and homogeneous boundary conditions for strain, microrotation, and heat flux on the free boundary. We suppose that the initial data are spherically symmetric, with positive initial density and temperature, having the zero density on the free boundary. Because of the spherical symmetry, the starting three-dimensional problem is transformed to the one-dimensional problem in Lagrangian coordinates in the domain that is a segment. The solution to our problem is obtained as a limit of the sequence of approximate solutions derived from suitable semi-discrete finite difference approximate systems. By using the derived bounded estimates of the approximate solutions and the results of the weak and strong compactness, we establish the convergence to the generalized solution of our problem globally in time.

Pattern Size in Gaussian Fields from Spinodal Decomposition

We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition in the Cahn-Hilliard model. These are somewhat universal patterns due to an overlay of eigenfunctions of the Laplacian with a similar wave-number. Similar structures appear in other models like reaction-diffusion systems describing animal coats' patterns or vegetation patterns in desertification. Our main result studies random functions given by cosine Fourier series with independent Gaussian coefficients, that dominate the dynamics in the Cahn-Hilliard model. This is not a cosine process, as the sum is taken over domains in Fourier space that not only grow and scale with a parameter of order $1/\varepsilon$, but also move to infinity. Moreover, the model under consideration is neither stationary nor isotropic. To study the pattern size of nodal domains we consider the density of zeros on any straight line through the spatial domain. Using a theorem by Edelman and Kostlan and weighted ergodic theorems that ensure the convergence of the moving sums, we show that the average distance of zeros is asymptotically of order $\varepsilon$ with a precisely given constant.

On the a-points of derivatives of the Riemann zeta function

We prove three results on the a-points of derivatives of the Riemann zeta function. The first result is a formula of the Riemann–von Mangoldt type; we estimate the number of a-points of derivatives of the Riemann zeta function. The second result is on certain exponential sum involving a-points. The third result is an analogue of the zero density theorem. We count the a-points of derivatives of the Riemann zeta function in \(1/2-(\log \log T)^2/\log T<\mathrm{Re}\, s<1/2+(\log \log T)^2/\log T\).

Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

We study the scaling asymptotics of the eigenspace projection kernels $\Pi_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 \Delta + |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $\Pi_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of $\hat{H}_{\hbar}$ have different orders in the Planck constant $\hbar$ in the allowed and forbidden regions: In the allowed region the density is of order $\hbar^{-1}$ while it is $\hbar^{-1/2}$ in the forbidden region. Our main result on nodal sets is that the density of zeros is of order $\hbar^{-\frac{2}{3}}$ in an $\hbar^{\frac{2}{3}}$-tube around the caustic. This tube radius is the `critical radius'. For annuli of larger inner and outer radii $\hbar^{\alpha}$ with $0< \alpha < \frac{2}{3}$ we obtain density results which interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff $(d-2)$-dimensional measure of the intersection of the nodal set with the caustic is of order $\hbar^{- \frac{2}{3}}$.

Inferential and forward projection modeling to evaluate options for controlling invasive mammals on islands.

Successful pest-mammal eradications from remote islands have resulted in important biodiversity benefits. Near-shore islands can also serve as refuges for native biota but require ongoing effort to maintain low-pest or pest-free status. Three management options are available in the presence of reinvasion risk: (1) control-to-zero density, in which immigration may occur but reinvaders are removed; (2) sustained population suppression (to relatively low numbers); or (3) no action. Biodiversity benefits can result from options one and two. The management challenge is to make evidence-based decisions on the selection of an appropriate objective and to identify a financially feasible control strategy that has a high probability of success. This requires understanding the pest species population dynamics and how it will respond to a range of potential management strategies, each with an associated financial cost. We developed a two-stage modeling approach that consisted of (1) Bayesian inferential modeling to estimate parameters for a model of pest population dynamics and control, and (2)a forward projection model to simulate a range of plausible management scenarios and quantify the probability of obtaining zero density within four years. We applied the model to an ongoing, six-year trapping program to control stoats (Mustela erminea) on Resolution Island, New Zealand. Zero density has not yet been achieved. Results demonstrate that management objectives were impeded by a combination of a highly fecund population, insufficient trap attractiveness, and a substantial proportion of the population that did not enter traps. Immigration is known to occur because the founding population arrived on the island by swimming from the mainland. However, immigration rate during this study was indistinguishable from zero. The forward projection modeling showed that control-to-zero density was feasible but required greater than a two-fold budget increase to intensify the trapping rate relative to population growth. The two-stage modeling provides the foundation for a management program in which broad-scale trials of additional trapping effort or improved trap lures would test model predictions and increase our understanding of system dynamics.

Multiplicative atom decomposition of sets of exceptional units in residue class rings

Given the multiplicative group Zn⁎ of units in the ring Zn:=Z/nZ, let Zn⁎⁎ denote the set of exceptional units in Zn, i.e. units u∈Zn⁎ satisfying 1−u∈Zn⁎. A subset of a finite group G containing all generators of any (cyclic) subgroup of G is called an atom of G. Let An⁎ denote the set of all atoms of Zn⁎. By means of An⁎⁎:={A∈An⁎:A⊂Zn⁎⁎}, the set Zn⁎⁎ trivially decomposes into atoms, i.e. Zn⁎⁎=⋃A∈An⁎⁎ as a disjoint union. An explicit construction of that atom decomposition is easily obtained if n is a prime power.We characterise so-called tame integers, i.e. odd n>1 with prime factorisation n=∏piki, say, for which the atom decomposition of Zn⁎⁎ is obtained by multiplicative composition of the atom decompositions of the Zpiki⁎⁎. Moreover, it is shown that the set of tame integers has density zero.

Symplectic n-level densities with restricted support

In this paper, we demonstrate that the alternative form, derived by us in an earlier paper, of the [Formula: see text]-level densities for eigenvalues of matrices from the classical compact group [Formula: see text] is far better suited for comparison with derivations of the [Formula: see text]-level densities of zeros in the family of Dirichlet [Formula: see text]-functions associated with real quadratic characters than the traditional determinantal random matrix formula. Previous authors have found ingenious proofs that the leading order term of the [Formula: see text]-level density of the zeros agrees with the determinantal random matrix result under certain conditions, but here we show that comparison is more straightforward if the more suitable form of the random matrix result is used. For the support of the test function in [Formula: see text] and in [Formula: see text] we compare with existing number theoretical results. For support in [Formula: see text] no rigorous number theoretical result is known for the [Formula: see text]-level densities, but we derive the densities here using random matrix theory in the hope that this may make the path to a rigorous number theoretical result clearer.

Experimental realization of acoustic double zero index metamaterials

Acoustic double zero index metamaterial with simultaneous zero density and infinite bulk modulus induced by Dirac cone at the Brillouin zone center provide a practical solution for applications. The resulted finite impedance of this metamaterial can be designed to match with surrounding materials. However, such metamaterial consists of scatterers with lower sound speed than the matrix, which is fundamentally challenging for air acoustics because the sound speed in air is among the lowest in nature. Here, we experimentally realize double zero index metamaterial by periodically varying waveguide thickness, which modulates the speed sound of high order waveguide mode. With proper designed dimensions of the scatterers, a Dirac cone at 18.7 kHz formed by the degeneracy of one monopolar and two dipolar modes at the Brillouin zone center is obtain. The monopolar mode modulates the bulk modulus and the dipolar modes modulate the density, resulting in simultaneous zero density and infinite bulk modulus. In our exp...

Sufficient Conditions for the Existence of the υ $$ \upsilon $$ -Density of Zeros for an Entire Function of Order Zero

We select subclasses of zero-order entire functions f for which we present sufficient conditions for the existence of the $$ \upsilon $$ -density of zeros of f in terms of the asymptotic behavior of the logarithmic derivative F and regular growth of the Fourier coefficients of F.