For the stability of degenerate parabolic equations, the usual boundary value condition may be overdetermined. How to find an optimal partial boundary value condition has been a long-standing ...problem. In this paper, we develop the weak characteristic function method to establish a reasonable partial boundary value condition, no matter whether the spatial variable domain Ω is bounded or unbounded. By means of the Kruzkov bi-variables method, the stability of entropy solutions dependent on this partial boundary value condition is proved.
We say that two meromorphic functions f and g sub-weighted share a value a∈C‾ with level k if they have the same set of a-points counted with multiplicity for the value a, where all a-points with ...multiplicity exceeding k are omitted. In this paper, we investigate the relation between the characteristic functions of two meromorphic functions sub-weighted sharing three values. We show that if two non-constants meromorphic functions f and g share sub-weighted three distinct values a1,a2,a3 with level k1,k2,k3 respectively, then(1−ϵ−δϵ)T(r,f)≤(2+ϵ+δϵ)T(r,g)+S(r,g) for every positive number ϵ, where δϵ=(24ϵ2+16ϵ4)(1k1+1+1k2+1+1k3+1). Our result is the extension of the previous result of P. Li and C. C. Yan 7 and others.
This note finds a new characterization of complete Nevanlinna-Pick kernels on the Euclidean unit ball.
The classical theory of Sz.-Nagy and Foias about the characteristic function is extended in this ...note to a commuting tuple T of bounded operators satisfying the natural positivity condition of 1/k-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characteristic function is a multiplier from Hk⊗E to Hk⊗F, factoring a certain positive operator, for suitable Hilbert spaces E and F depending on T. There is a converse, which roughly says that if a kernel k admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction (1/k-contraction where k is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain.
Rényi entropy based on characteristic function has been used as an information measure contained in wide-sense and real stationary vector autoregressive and moving average (VARMA) processes. These ...classes of processes have been extended by fractionally integrated VARMA (VARFIMA) ones, composed of a VARMA process, a vector of fractional differencing parameters, and independent and identically distributed multivariate normal random errors. Such processes have often been used to explicitly account for persistence to incorporate long-term correlations into multivariate data. The purpose of this paper is to extend Rényi entropy from VARMA to VARFIMA processes, addressing long-memory behavior of time series by adding a fractional differencing parameter. The characteristic function of the process can be derived directly from the asymptotic form of the impulse response function using the Wold representation. Then, assuming multivariate Gaussian white noise with known fractional differencing, autoregressive and moving average matrix parameters, the differential and Rényi entropies and Kullback–Leibler and Rényi divergences were obtained by evaluating the variance-covariance matrix identified with VARFIMA process distribution. The influences of the fractional differencing parameters on the Rényi entropy increment were analyzed, as were comparisons between VARFIMA processes using the Kullback–Leibler and Rényi divergences. Finally, numerical examples and an application to U.S. daily temperature time series are presented.
•Rényi entropy has been used as a measure of information contained in VARFIMA processes.•VARFIMA processes account for persistence to incorporate the long-term correlations in the multivariate data.•We extend the Rényi entropy from VARMA to VARFIMA processes by adding a fractional differencing parameter.•Characteristic function is derived directly from the asymptotic form of the impulse response function.•Numerical examples and an application to U.S. daily temperature time series are presented.
Inverse spectral problem for a self-adjoint differential operator, which is the sum of the operator of the third derivative on a finite interval and of the operator of multiplication by a real ...function (potential), is solved. Closed system of integral linear equations is obtained. Via solution to this system, the potential is calculated. It is shown that the main parameters of the obtained system of equations are expressed via spectral data of the initial operator. It is established that the potential is unambiguously defined by the four spectra.
In this letter, we derive new results for the statistics of the ratio of two complex Gaussian random variables (RVs), where the numerator and denominator may have arbitrary means and are possibly ...correlated. Exact expressions are derived for the joint probability density function (pdf) of the real and imaginary parts, for the joint pdf of the amplitude and phase, and also for the joint characteristic function (cf) of the real and imaginary parts, which generalize the existing results. Then, we show an example application of the derived pdf to the symbol error probability (SEP) analysis for a single antenna communication system with imperfect channel state information (ICSI).
Based on a quantum mechanical approach, we investigate moment- (or M-) indeterminate probability densities by way of the characteristic function and self-adjoint operators. The approach leads to new ...methods to construct classes of M-indeterminate probability densities.
Subdivide n-dimensional space En into disjoint n-dimensional unit cubes, and let the set of these cubes be Λn. The vertices of each cube are called nodes. Let LPT be the set of allcontinuous curve ...passing through node P(s1,s2,...,sn)and nodeT(t1,t2,...,tn) . ∀Γ ∊ LpT, α∊Λn define the characteristic function of Γ to oc, χΓ(α)={ 0,whenμ(α∩Γ)=0,1,whenμ(α∩Γ)>0, Where μ(α ∩ Γ) is Lebesgue measure of point set α ∩ Γ. Name all its non-empty element as αΓ(1),αΓ(2),...,αΓ(r(Γ)), and let λk(αΓ(k)) be the weighting coefficients of αΓ(k)(k=1,2...,r(Γ)) on Γ . This paper investigated the value of Γ∈LPTinf{Σk=1r(Γ)λk(αΓ(k))χ(αΓ(k))}, and by using discrete mathematics theory and construction method, proved that if all weighting coefficients are equal to 1, then max {m1,m2,⋯mn}≤Γ∈LPTinf{Σk=1r(Γ)λk(αΓ(k))χ(αΓ(k))}≤Σk=1n(−1)k−1Σ1≤i1<i2<⋯<ik≤ngcd(mi1,mi2,⋯mik) Where mi=|si−ti|, i = 1,2, ..., n. The paper also generalizes the result to a wedge-shaped case.
PDF