scholarly journals HAAR BASES FOR $L^2(\mathbb{Q}_2^2)$ GENERATED BY ONE WAVELET FUNCTION

2012 ◽  
Vol 10 (05) ◽  
pp. 1250042 ◽  
Author(s):  
S. ALBEVERIO ◽  
M. SKOPINA
Keyword(s):  
Wavelet Function ◽  
Orthogonal Basis ◽  
Explicit Description ◽  
Two Dimensional ◽  
Wavelet Bases ◽  
The Real

The concept of p-adic quincunx Haar MRA was introduced and studied in Ref. 12. In contrast to the real setting, infinitely many different wavelet bases are generated by a p-adic MRA. We give an explicit description for all wavelet functions corresponding to the quincunx Haar MRA. Each one generates an orthogonal basis, one of them was presented in Ref. 12. A connection between quincunx Haar MRA and two-dimensional separable Haar MRA is also found.

scholarly journals Resonance Vibration of Mistuned Bladed Discs in Stationary Operations

1997 ◽  
Author(s):  
Romuald Rządkowski
Keyword(s):  
Numerical Model ◽  
Compressible Flow ◽  
Two Dimensional ◽  
Rotating System ◽  
Response Level ◽  
Stationary Response ◽  
The Real ◽  
Disc Model ◽  
Real Geometry ◽  
Airfoil Blades

A numerical model for the calculation of resonance stationary response of mistuned bladed disc is presented. The bladed disc model includes all important effects on a rotating system of the real geometry. The excitation forces were calculated by a code on the basis of two-dimensional compressible flow (to M < 0.8) for thin airfoil blades. The calculations presented in this paper show that centrifugal stress, and the values of excitation forces, play an important role in considering the influence of mistuning on the response level.

Two-dimensional bifurcation phenomena in thermal convection in horizontal, concentric annuli containing saturated porous media

1988 ◽  
Vol 187 ◽  
pp. 267-300 ◽  
Author(s):  
K. Himasekhar ◽  
Haim H. Bau
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Axis ◽  
Thermal Convection ◽  
Saturated Porous Medium ◽  
Nonlinear Partial Differential Equations ◽  
Perturbation Expansion ◽  
Two Dimensional ◽  
Partial Differential ◽  
The Real

A saturated porous medium confined between two horizontal cylinders is considered. As a result of a temperature difference between the cylinders, thermal convection is induced in the medium. The flow structure is investigated in a parameter space (R, Ra) where R is the radii ratio and Ra is the Darcy-Rayleigh number. In particular, the cases of R = 2, 2½, 21/4 and 2½ are considered. The fluid motion is described by the two-dimensional Darcy-Oberbeck-Boussinesq's (DOB) equations, which we solve using regular perturbation expansion. Terms up to O(Ra60) are calculated to obtain a series presentation for the Nusselt number Nu in the form \[ Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}. \] This series has a limited range of utility due to singularities of the function Nu(Ra). The singularities lie both on and off the real axis in the complex Ra plane. For R = 2, the nearest singularity lies off the real axis, has no physical significance, and unnecessarily limits the range of utility of the aforementioned series. For R = 2½, 2¼ and 21/8, the singularity nearest to the origin is real and indicates that the function Nu(Ra) is no longer unique beyond the singular point.Depending on the radii ratio, the loss of uniqueness may occur as a result of either (perfect) bifurcations or the appearance of isolated solutions (imperfect bifurcations). The structure of the multiple solutions is investigated by solving the DOB equations numerically. The nonlinear partial differential equations are converted into a truncated set of ordinary differential equations via projection. The steady-state problem is solved using Newton's technique. At each step the determinant of the Jacobian is evaluated. Bifurcation points are identified with singularities of the Jacobian. Linear stability analysis is used to determine the stability of various solution branches. The results we obtained from solving the DOB equations using perturbation expansion are compared with those we obtained from solving the nonlinear partial differential equations numerically and are found to agree well.

Three and Four Centre Oscillators and Their Application in Nuclear Physics

1974 ◽  
Vol 29 (7) ◽  
pp. 1003-1010 ◽  
Author(s):  
Peter Bergmann ◽  
Hans-Joachim Scheefer
Keyword(s):  
Analytical Solution ◽  
Schrödinger Equation ◽  
Nuclear Physics ◽  
Schrodinger Equation ◽  
Numerical Procedure ◽  
Orthogonal Basis ◽  
Two Dimensional ◽  
Schmidt's Method ◽  
Symmetry Properties

The extension of the nuclear two-centre-oscillator to three and four centres is investigated. Some special symmetry-properties are required. In two cases an analytical solution of the Schrödinger equation is possible. A numerical procedure is developed which enables the diagonalization of the Hamiltonian in a non-orthogonal basis without applying Schmidt's method of orthonormalization. This is important for calculations of arbitrary two-dimensional arrangements of the centres.

The forcing partial order on the three times punctured disk

1997 ◽  
Vol 17 (3) ◽  
pp. 593-610 ◽  
Author(s):  
MICHAEL HANDEL
Keyword(s):  
Partial Order ◽  
Mapping Class Group ◽  
Class Group ◽  
Conjugacy Classes ◽  
Explicit Description ◽  
Mapping Class ◽  
Two Dimensional

The two-dimensional analogue of the Sharkovski order on periods for maps of the interval restricts to a partial order on essential pseudo-Anosov conjugacy classes in the mapping class group of the $n$-times punctured disk. In this paper we give an explicit description of this restricted partial order in the case when $n=3$.

scholarly journals XIV. Some applications of conformal transformation to problems in hydrodynamics

1915 ◽  
Vol 215 (523-537) ◽  
pp. 439-487 ◽  
Keyword(s):  
Real Axis ◽  
Half Plane ◽  
Conformal Transformation ◽  
Difficult Problem ◽  
Two Dimensional ◽  
The Real ◽  
Dimensional Flow ◽  
Conformal Representation ◽  
Two Dimensional Flow ◽  
Interpretation Of Results

The problem of the conformal representation of the part of the plane of a variable z , which is bounded by a rectilineal polygon, upon the half-plane of a variable w bounded by the real axis, is solved (save for an integration) by the well-known transformation of Schwarz dz = CII ( w — a r ) - ar /π dw , where C , a 1 a 2 , &c., are real constants, and π— α 1 , π— α 2 , &c., are the internal angles of the rectilineal polygon. A more difficult problem is that of the conformal representation upon the half-plane of w of a region in the z plane whose boundary is partly curved; it is with this problem that the present paper is concerned, always however with a view to interpretation of results in terms of the two-dimensional flow of liquid in regions having particular types of boundary.

scholarly journals Band functions in the presence of magnetic steps

2015 ◽  
Vol 26 (01) ◽  
pp. 161-184 ◽  
Author(s):  
P. D. Hislop ◽  
N. Popoff ◽  
N. Raymond ◽  
M. P. Sundqvist
Keyword(s):  
Magnetic Field ◽  
Asymptotic Expansion ◽  
Magnetic Fields ◽  
Essential Spectrum ◽  
Effective Potential ◽  
Explicit Description ◽  
Natural Order ◽  
Two Dimensional ◽  
Piecewise Constant ◽  
The Magnetic Field

We complete the analysis of the band functions for two-dimensional magnetic Schrödinger operators with piecewise constant magnetic fields. The discontinuity of the magnetic field can create edge currents that flow along the discontinuity, which have been described by physicists. Properties of these edge currents are directly related to the behavior of the band functions. The effective potential of the fiber operator is an asymmetric double well (eventually degenerated) and the analysis of the splitting of the bands incorporates the asymmetry. If the magnetic field vanishes, the reduced operator has essential spectrum and we provide an explicit description of the band functions located below the essential spectrum. For non-degenerate magnetic steps, we provide an asymptotic expansion of the band functions at infinity. We prove that when the ratio of the two magnetic fields is rational, a splitting of the band functions occurs and has a natural order, predicted by numerical computations.

scholarly journals TRIANGULAR A−STATISTICAL RELATIVE UNIFORM CONVERGENCE FOR DOUBLE SEQUENCES OF POSITIVE LINEAR OPERATORS

2021 ◽  
pp. 065
Author(s):  
Selin Çınar
Keyword(s):  
Uniform Convergence ◽  
Approximation Theorem ◽  
Dimensional Space ◽  
Linear Operators ◽  
Two Dimensional ◽  
Double Sequences ◽  
The Real ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Relative Uniform Convergence

In this paper, we introduce the concept of triangular A-statistical relative convergence for double sequences of functions defined on a compactsubset of the real two-dimensional space. Based upon this new convergencemethod, we prove Korovkin-type approximation theorem. Finally, we give some further developments.

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