Complex-valued Wiener measure: An approach via random walk in the complex plane

1999 ◽  
Vol 42 (1) ◽  
pp. 61-67 ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Random Walk ◽  
Complex Plane ◽  
Wiener Measure ◽  
Complex Valued

Complex Approximation and Simultaneous Interpolation on Closed Sets

1977 ◽  
Vol 29 (4) ◽  
pp. 701-706 ◽  
Author(s):  
P. M. Gauthier ◽  
W. Hengartner
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Closed Subset ◽  
Finite Complex ◽  
Complex Approximation ◽  
Closed Sets ◽  
Limit Points ◽  
Simultaneous Interpolation ◽  
Complex Valued

Let ƒ be a complex-valued function denned on a closed subset F of the finite complex plane C, and let {Zn} be a sequence on F without limit points. We wish to find an analytic function g which simultaneously approximates ƒ uniformly on F and interpolates ƒ at the points {Zn}.

scholarly journals An Exploration of the Triplet Periodicity in Nucleotide Sequences with a Mature Self-Adaptive Spectral Rotation Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Bo Chen ◽  
Ping Ji
Keyword(s):  
Random Walk ◽  
Complex Plane ◽  
Query Sequence ◽  
Nucleotide Sequences ◽  
Sequence Pattern ◽  
Coding Regions ◽  
Persistent Pattern ◽  
Triplet Periodicity ◽  
Self Adaptive

Previously, for predicting coding regions in nucleotide sequences, a self-adaptive spectral rotation (SASR) method has been developed, based on a universal statistical feature of the coding regions, named triplet periodicity (TP). It outputs a random walk, that is, TP walk, in the complex plane for the query sequence. Each step in the walk is corresponding to a position in the sequence and generated from a long-term statistic of the TP in the sequence. The coding regions (TP intensive) are then visually discriminated from the noncoding ones (without TP), in the TP walk. In this paper, the behaviors of the walks for random nucleotide sequences are further investigated qualitatively. A slightly leftward trend (a negative noise) in such walks is observed, which is not reported in the previous SASR literatures. An improved SASR, named the mature SASR, is proposed, in order to eliminate the noise and correct the TP walks. Furthermore, a potential sequence pattern opposite to the TP persistent pattern, that is, the TP antipersistent pattern, is explored. The applications of the algorithms on simulated datasets show their capabilities in detecting such a potential sequence pattern.

scholarly journals The problem of missing terms in term by term integration involving divergent integrals

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160567 ◽  
Author(s):  
Eric A. Galapon
Keyword(s):  
Analytic Continuation ◽  
Complex Plane ◽  
Finite Part ◽  
Stieltjes Transform ◽  
Complex Valued

Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity conditions, the problem of missing terms for the incomplete Stieltjes transform, ∫ 0 a f ( x ) ( ω + x ) − 1   d x , and the Stieltjes transform itself, ∫ 0 ∞ f ( x ) ( ω + x ) − 1   d x , is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex-valued function f ( z )( ω + z ) −1 , with the divergent integrals arising from term by term integration interpreted as finite part integrals.

scholarly journals On the spectra of Schrödinger and Jacobi operators with complex-valued quasi-periodic algebro-geometric coefficients

2005 ◽  
Author(s):  
◽  
Vladimir Batchenko
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Complex Plane ◽  
Mean Value ◽  
Qualitative Behavior ◽  
Conditional Stability ◽  
One Dimensional ◽  
Jacobi Operators ◽  
The Mean ◽  
Complex Valued

In this thesis we characterize the spectrum of one-dimensional Schrödinger operators. H = -d2/dx2+V in L2(R; dx) with quasi-periodic complex-valued algebro geometric, potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These results extend to the Lp(R; dx)-setting for p 2 [1,1). In addition, we apply these techniques to the discrete case and characterize the spectrum of one-dimensional Jacobi operators H = aS+ + a-S- b in 2(Z) assuming a, b are complex-valued quasi-periodic algebro-geometric coefficients. In analogy to the case of Schrödinger operators, we prove that the spectrum of H coincides with the conditional stability set of H and can also explicitly be described in terms of the mean value of the Green's function of H. The qualitative behavior of the spectrum of H in the complex plane is similar to the Schrödinger case: the spectrum consists of finitely many bounded simple analytic arcs in the complex plane which may exhibit crossings as well as confluences.

Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules

1986 ◽  
Vol 23 (03) ◽  
pp. 790-802 ◽  
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Random Walk ◽  
State Space ◽  
Complex Plane ◽  
General Class ◽  
Stochastic Behavior ◽  
Vacation Models ◽  
Server Vacations ◽  
Fixed Probability ◽  
Oscillating Random Walk

Processors handling multi-class traffic typically alternate between serving a particular class of traffic and performing other tasks, e.g., secondary service tasks or routine maintenance. The stochastic behavior of such systems is modeled by a newly introduced class of Bernoulli GI/G/1 vacation models. For this model, when a vacation is completed and customers are present, a customer is served. When a customer has just been served and other customers are present, the server accepts a customer with fixed probability p or commences a vacation of prespecified random duration with probability 1 – p. Whenever no customers are present, a vacation is taken. When p = 0 or p = 1 this schedule reduces to the previously introduced single service schedule and the exhaustive service schedule, respectively. An analysis of all three schedules on a state space incorporating server vacations is presented using simple methods in the complex plane. It is shown that the recent decomposition results for exhaustive service extend to the more general class of Bernoulli schedules.

Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules

1986 ◽  
Vol 23 (3) ◽  
pp. 790-802 ◽  
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Random Walk ◽  
State Space ◽  
Complex Plane ◽  
General Class ◽  
Stochastic Behavior ◽  
Vacation Models ◽  
Server Vacations ◽  
Fixed Probability ◽  
Oscillating Random Walk

Processors handling multi-class traffic typically alternate between serving a particular class of traffic and performing other tasks, e.g., secondary service tasks or routine maintenance. The stochastic behavior of such systems is modeled by a newly introduced class of Bernoulli GI/G/1 vacation models. For this model, when a vacation is completed and customers are present, a customer is served. When a customer has just been served and other customers are present, the server accepts a customer with fixed probability p or commences a vacation of prespecified random duration with probability 1 – p. Whenever no customers are present, a vacation is taken. When p = 0 or p = 1 this schedule reduces to the previously introduced single service schedule and the exhaustive service schedule, respectively. An analysis of all three schedules on a state space incorporating server vacations is presented using simple methods in the complex plane. It is shown that the recent decomposition results for exhaustive service extend to the more general class of Bernoulli schedules.

scholarly journals A Comparison of Two Equivalent Real Formulations for Complex-Valued Linear Systems Part 2: Results

2003 ◽  
Vol 1 (4) ◽  
Author(s):  
Abnita Munankarmy ◽  
Michael Heroux
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Complex Plane ◽  
Spectral Properties ◽  
Complex Problem ◽  
Linear Solver ◽  
The Real ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods ◽  
Complex Valued

Many iterative linear solver packages focus on real-valued systems and do not deal well with complex-valued systems, even though preconditioned iterative methods typically apply to both real and complex-valued linear systems. Instead, commonly available packages such as PETSc and Aztec tend to focus on the real-valued systems, while complex-valued systems are seen as a late addition. At the same time, by changing the complex problem into an equivalent real formulation (ERF), a real valued solver can be used. In this paper we consider two ERF’s that can be used to solve complex-valued linear systems. We investigate the spectral properties of each and show how each can be preconditioned to move eigenvalues in a cloud around the point (1,0) in the complex plane. Finally, we consider an interleaved formulation, combining each of the previously mentioned approaches, and show that the interleaved form achieves a better outcome than either separate ERF. [This article is the second part of a sequence of reports. See the December 2002 issue for Part 1—Editor.]

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