Number Of Sign Changes Research Articles

A successive approximation method to precisely measure leakage current of the rechargeable Lithium coin battery

When the rechargeable Lithium coin battery is employed as the storage component for indoor energy harvesting, the leakage current of the battery cannot be ignored, especially in ultra-low-power applications. The leakage current of the Lithium coin battery is commonly believed in the low μA range. However the exact value is unknown. An experimental method to measure leakage current by applying a known charge current in μAs to a stabilized post-charge battery to observe the sign of the battery terminal voltage change is proposed. When the applied charge current is larger than the leakage current, a positive sign (terminal voltage increase) can be observed. Otherwise a negative sign appears. By gradually changing the charge current using the successive approximation search algorithm, the leakage current will finally converge to the applied charge current. The convergence happens after the sign changes in two successive search procedures, demonstrating that charge currents are approaching the leakage current from both directions. The accuracy of the approximation can be improved by adjusting the number of sign changes in the successive approximation search algorithm. The proposed method permits the full capacity of a rechargeable Lithium coin battery to be utilised in ultra-low-power applications.

On the q-Exponents of Generalized Modular Functions

We consider two problems concerning oscillations of the q-exponents of generalized modular functions. First, we prove a quantitative result for the number of sign changes for the sequence of q-exponents at primes in short intervals. Second, we investigate the growth of the q-exponents.

Oscillations of coefficients of Dirichlet series attached to automorphic forms

For $m\ge 2$, let $\pi$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_\pi(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $\pi$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_\pi(n)$ as $x \rightarrow \infty$. For $m\ge 3$ and $\pi$ not a symmetric power of a $GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $\pi$, their bound is better than all previous bounds. In this paper, we further improve the bound of Golfeld and Sengupta. We also prove a quantitative result for the number of sign changes of the coefficients of certain automorphic $L$-functions, provided the coefficients are real numbers.

SIGN OF FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF‐INTEGRAL WEIGHT

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

A Zero-Crossing Rate Property of Power Complementary Analysis Filterbank Outputs

We establish zero-crossing rate (ZCR) relations between the input and the subbands of a maximally decimated $M$ -channel power complementary analysis filterbank when the input is a stationary Gaussian process. The ZCR at lag $\ell $ is defined as the number of sign changes between the samples of a sequence and its $\ell $ -sample shifted version, normalized by the sequence length. We derive the relationship between the ZCR of the Gaussian process at lags that are integer multiples of $M$ and the subband ZCRs. Based on this result, we propose a robust iterative autocorrelation estimator for a signal consisting of a sum of sinusoids of fixed amplitudes and uniformly distributed random phases. Simulation results show that the performance of the proposed estimator is better than the sample autocorrelation over the SNR range of $ - 6$ to 15 dB. Validation on a segment of a trumpet signal showed similar performance gains.

Sign changes of Hecke eigenvalues

Let f be a holomorphic or Maass Hecke cusp form for the full modular group and write $${\lambda_f(n)}$$ for the corresponding Hecke eigenvalues. We are interested in the signs of those eigenvalues. In the holomorphic case, we show that for some positive constant $${\delta}$$ and every large enough x, the sequence $${(\lambda_f(n))_{n \leq x}}$$ has at least $${\delta x}$$ sign changes. Furthermore we show that half of non-zero $${\lambda_f(n)}$$ are positive and half are negative. In the Maass case, it is not yet known that the coefficients are non-lacunary, but our method is robust enough to show that on the relative set of non-zero coefficients there is a positive proportion of sign changes. In both cases previous lower bounds for the number of sign changes were of the form x δ for some δ < 1.

Optimizing dynamical changes of structural balance in signed network based on memetic algorithm

In dynamical evolution of structural balance, unbalanced signed networks evolve to structurally balanced ones. In this paper, we compute the least number of sign changes in the evolution of structural balance. It is suggested that there be a certain bias towards flipping positive or flipping negative signs. The number of flipped signs is quantified by an objective function. Moreover, a memetic algorithm is proposed to optimize the objective function. Experiments show that our algorithm is efficient and effective to optimize dynamical evolution of structural balance.

Sign changes of Fourier coefficients of Hilbert modular forms

Sign changes of Fourier coefficients of various modular forms have been studied. In this paper, we analyze some sign change properties of Fourier coefficients of Hilbert modular forms, under the assumption that all the coefficients are real. The quantitative results on the number of sign changes in short intervals are also discussed.

Sign changes of Fourier coefficients of half-integral weight cusp forms

We prove a quantitative result for the number of sign changes of the Fourier coefficients of half-integral weight cusp forms in the Kohnen plus space, provided the Fourier coefficients are real numbers.

A short note on sign changes

In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\ge 1}, j=2,3,4$ of the Fourier coefficients of normalized Hecke eigencusp forms for the full modular group SL 2(ℤ). We also prove a similar kind of quantitative result for the number of sign changes of the q-exponents $c(p)~(p \mbox{~vary ~over ~primes})$ of certain generalized modular functions for the congruence subgroup Γ0(N), where N is square-free.

Sign patterns with minimum rank 2 and upper bounds on minimum ranks

A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0}. The minimum rank (resp., rational minimum rank) of a sign pattern matrix 𝒜 is the minimum of the ranks of the real (resp., rational) matrices whose entries have signs equal to the corresponding entries of 𝒜. The notion of a condensed sign pattern is introduced. A new, insightful proof of the rational realizability of the minimum rank of a sign pattern with minimum rank 2 is obtained. Several characterizations of sign patterns with minimum rank 2 are established, along with linear upper bounds for the absolute values of an integer matrix achieving the minimum rank 2. A known upper bound for the minimum rank of a (+, −) sign pattern in terms of the maximum number of sign changes in the rows of the sign pattern is substantially extended to obtain upper bounds for the rational minimum ranks of general sign pattern matrices. The new concept of the number of polynomial sign changes of a sign vector is crucial for this extension. Another known upper bound for the minimum rank of a (+, −) sign pattern in terms of the smallest number of sign changes in the rows of the sign pattern is also extended to all sign patterns using the notion of the number of strict sign changes. Some examples and open problems are also presented.

Nonexistence of solutions of the p-adic strings

We discuss mathematical aspects of the nonexistence of continuous (nontrivial) solutions of boundary value problems for equations of p-adic closed and open strings in the one-dimensional case. We find that the number of sign changes of the solution ψ(t) is not equal to the order of zeros of the function ψn(t) and that nonnegative (nonpositive) solutions do not exist. In the case of even n, if a solution ψ exists, then the orders of zeros of the function ψn and the order of its tangency to positive maximums (minimums) are not divisible by four and therefore have the form 2(2r + 1), r ≥ 0.

On the existence of sign changing bound state solutions of a quasilinear equation

In this paper we establish the existence of bound state solutions having a prescribed number of sign changes for(P)Δmu+f(u)=0,x∈RN,N⩾m>1, where Δmu=∇⋅(|∇u|m−2∇u). Our result is new even for the case of the Laplacian (m=2).

Sign-changing solutions of a quasilinear heat equation with a source term

The Cauchy problem of a heat equation with a source term $$ \psi_t=\Delta \left(|\psi|^{m-1}\psi\right)+|\psi|^{\gamma-1}\psi\ \ \mbox{in}\ \ (0, \infty)\times R^n $$ is considered, where $\gamma>m>1$. We are interested in global solutions with H$\ddot{o}$lder continuity which satisfy the equation in the distribution sense, and with a fixed number of sign changes at any given time $ t > 0$. Through detailed analysis of the self-similarity problem, we prove the existence of two type of such solutions, one with compact support and the other decays to zero as $ | x| \rightarrow \infty$ with an algebraic rate determined uniquely by $ n, m$ and $\gamma$. Our results extend previous study on positive self-similar solutions. Moreover, they demonstrate vital difference from the well-studied semi-linear case of $m = 1$.

Sign preservation properties of some nonlinear transformations

It is shown that the number of sign changes in certain transformations does not increase with each iteration in time. These transformations are composed of linear components defined in terms of totally positive matrices and semi linear components similar to u → ku3. In particular the analysis shows that for certain semi linear parabolic equations discretized using finite difference methods, the number of sign changes does not increase. Mathematics subject classification (2010): Primary 47J05; Secondary 47J25, 15A90, 65N06.

Sublinearity of the travel-time variance for dependent first-passage percolation

Let $E$ be the set of edges of the $d$-dimensional cubic lattice $\mathbb{Z}^d$, with $d\geq2$, and let $t(e),e\in E$, be nonnegative values. The passage time from a vertex $v$ to a vertex $w$ is defined as $\inf_{\pi:v\rightarrow w}\sum_{e\in\pi}t(e)$, where the infimum is over all paths $\pi$ from $v$ to $w$, and the sum is over all edges $e$ of $\pi$. Benjamini, Kalai and Schramm [2] proved that if the $t(e)$'s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex $v$ is sublinear in the distance from 0 to $v$. This result was extended to a large class of independent, continuously distributed $t$-variables by Bena\"{\i}m and Rossignol [1]. We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the $t(e)$'s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical "Ising landscape."

A refinement of the variation diminishing property of Bézier curves

For a given polynomial F ( t ) = ∑ i = 0 n p i B i n ( t ) , expressed in the Bernstein basis over an interval [ a , b ] , we prove that the number of real roots of F ( t ) in [ a , b ] , counting multiplicities, does not exceed the sum of the number of real roots in [ a , b ] of the polynomial G ( t ) = ∑ i = k l p i B i − k l − k ( t ) (counting multiplicities) with the number of sign changes in the two sequences ( p 0 , … , p k ) and ( p l , … , p n ) for any value k , l with 0 ⩽ k ⩽ l ⩽ n . As a by product of this result, we give new refinements of the classical variation diminishing property of Bézier curves.

Sturm-Sequence Properties of Recurrence Dispersion Functions of Periodic Waveguide Arrays: Theory and Applications

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Sturm-sequence properties are reported for recurrence dispersion functions of periodic waveguide arrays. According to these properties, the number of sign changes in the recurrence sequence at modal cutoff equals the number of zeros of the dispersion function of the array. A generalized differential form of this equality is developed, which applies along any path in the parameter space. It allows deriving explicit analytical design rules for these arrays. The derived rules reveal the possibility of supporting a specific number of TE or TM modes, which is independent of coupling conditions. Even under strong coupling, it is shown that the zeros of the consecutive recurrence dispersion functions are interlaced. A recurrence zero-search algorithm employs this interlacing in resolving closely-spaced zeros of the dispersion functions of both symmetric and asymmetric arrays. The algorithm is applied with the derived rules in maximizing phase and group birefringence of single-mode silicon-on-insulator (SOI) waveguides. Two strip-loaded SOI waveguides are designed with phase and group birefringence of 1.03 and 1.64 at a free-space wavelength of 1.55 <formula formulatype="inline"><tex Notation="TeX">$\mu{\hbox {m}}$</tex></formula>. </para>

On the expectations of maxima of sets of independent random variables

Let X 1 , … , X k and Y 1 , … , Y m be jointly independent copies of random variables X and Y , respectively. For a fixed total number n of random variables, we aim at maximising M ( k , m ) ≔ E max { X 1 , … , X k , Y 1 , … , Y m } in k = n − m ≥ 0 , which corresponds to maximising the expected lifetime of an n -component parallel system whose components can be chosen from two different types. We show that the lattice { M ( k , m ) : k , m ≥ 0 } is concave, give sufficient conditions on X and Y for M ( n , 0 ) to be always or ultimately maximal and derive a bound on the number of sign changes in the sequence M ( n , 0 ) − M ( 0 , n ) , n ≥ 1 . The results are applied to a mixed population of Bienayme–Galton–Watson processes, with the objective to derive the optimal initial composition to maximise the expected time to extinction.

Formula omitted]-stability of periodic stationary solutions of scalar convection–diffusion equations

The aim of this paper is to study the L 1 -stability of periodic stationary solutions of scalar convection–diffusion equations. We obtain dispersion in L 2 for all space dimensions using Kružkov type entropy. And when the space dimension is one, we estimate the number of sign changes of a solution to obtain L 1 -stability.