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The family of exponential pseudo-splines is the non-stationary counterpart of the pseudo-splines and includes the exponential B-spline functions as special members. Among the family of the exponential pseudo-splines, there also exists the subclass consisting of interpolatory cardinal functions, which can be obtained as the limits of the exponentials reproducing subdivision. In this paper, we mainly focus on this subclass of exponential pseudo-splines and propose their dual refinable functions with explicit form of symbols. Based on this result, we obtain the corresponding biorthogonal wavelets using the non-stationary Multiresolution Analysis (MRA). We verify the stability of the refinable and wavelet functions and show that both of them have exponential vanishing moments, a generalization of the usual vanishing moments. Thus, these refinable and wavelet functions can form a non-stationary generalization of the Coifman biorthogonal wavelet systems constructed using the masks of the D–D interpolatory subdivision.

This paper presents a variant scheme of the cubic exponential B-spline scheme, which, with two parameters, can generate curves with different shapes. This variant scheme is obtained based on the iteration from the generation of exponentials and a suitably chosen function. For such a scheme, we show its C2-convergence and analyze the effect of the parameters on the shape of the generated curves and also discuss its convexity preservation. In addition, a non-uniform version of this variant scheme is derived in order to locally control the shape of the generated curves. Numerical examples are given to illustrate the performance of the new schemes in this paper.

In this paper, we show that the univariate interpolatory subdivision schemes with the optimal approximation order, which are the D-D interpolatory ones, can be obtained from the cubic B-spline scheme. Such schemes are derived in this paper by suitably using the push-back operation, a connection between the approximating and interpolatory subdivision. Then, the process to obtain the D-D schemes is generalized to the bivariate case and bivariate interpolatory schemes with the optimal approximation order are generated based on the triangular meshes. Compared with the existing bivariate interpolatory schemes with the optimal approximation order, the newly constructed ones own some advantages, such as a better performance in terms of the number of nonzero coefficients in the mask. Several examples are given and explicitly computed.

In this paper, a family of non-stationary combined ternary $5$-point subdivision schemes with multiple variable parameters is proposed. The construction of the scheme is based on the generalized ternary subdivision scheme of order $4$, which is built upon refinement of a family of generalized B-splines, using the variable displacements. For such a non-stationary scheme, we study its smoothness and get that it can generate $C^{2}$ interpolating limit curves and $C^{4}$ approximating limit curves. Besides, we investigate the exponential polynomial generation/reproduction property and approximation order. It can generate/reproduce certain exponential polynomials with suitable choices of the variable parameters, and reach approximation order $5$.

In this paper, a non-stationary combined subdivision scheme is presented, which can unify several existing non-stationary approximating and interpolatory subdivision schemes. This scheme is obtained by generalizing the connection between the approximating and interpolatory schemes in the stationary case, first formalized by Maillot & Stam using a push-back operator, to the non-stationary case. For such a combined scheme, we investigate its C l convergence and exponential polynomial generation/reproduction property and get that it can reach C 4 degree of smoothness and generate/reproduce certain exponential polynomials with suitable choices of the parameters. Besides, we give a more generalized combined scheme for the purpose of generating and reproducing more general exponential polynomials. The performance of our new schemes is illustrated by several numerical examples.

In this paper, by suitably using the so-called push-back operation, a connection between the approximating and interpolatory subdivision, a new family of nonstationary subdivision schemes is presented. Each scheme of this family is a quasi-interpolatory scheme and reproduces a certain space of exponential polynomials. This new family of schemes unifies and extends quite a number of the existing interpolatory schemes reproducing exponential polynomials and noninterpolatory schemes like the cubic exponential B-spline scheme. For these new schemes, we investigate their convergence, smoothness, and accuracy and show that they can reach higher smoothness orders than the interpolatory schemes with the same reproduction property and better accuracy than the exponential B-spline schemes. Several examples are given to illustrate the performance of these new schemes.

In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters μ and ν. By changing the values of μ and ν, the sensitivity of the shape of the subdivision curve to the initial control value v0 can be changed and different kinds of curves can then be obtained by adjusting the value of v0. For this new scheme, we show that, with any admissible choice of μ and ν, it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.

A palaeomagnetic pole position, derived from a precisely dated primary remanence, with minimal uncertainties due to secular variation and structural correction, has been obtained for China’s largest dyke swarm, which trends for about 1000 km in a NNW direction across the North China craton. Positive palaeomagnetic contact tests on two dykes signify that the remanent magnetization is primary and formed during initial cooling of the intrusions. The age of one of these dykes, based on U–Pb dating of primary zircon, is 1769.1 ± 2.5 Ma. The mean palaeomagnetic direction for 19 dykes, after structural correction, is D = 36°, I = − 5°, k = 63, α95 = 4°, yielding a palaeomagnetic pole at Plat=36°N, Plong=247°E, dp = 2°, dm = 4° and a palaeolatitude of 2.6°S. Comparison of this pole position with others of similar age from the Canadian Shield allows a continental reconstruction that is compatible with a more or less unchanged configuration of Laurentia, Siberia and the North China craton since about 1800 Ma

[1] We tested single-domain (SD) and multidomain (MD) theories of the acquisition and thermal demagnetization of thermoremanent magnetization (TRM) and partial TRM (pTRM) in Thellier paleointensity determination, using as samples plagioclase and mafic minerals extracted from Matachewan diabase dikes. Susceptibility, hysteresis, and coercivity data indicate that the plagioclase contains mainly SD grains of magnetite while the mafic minerals (clinopyroxene, biotite, and actinolite) contain mainly MD magnetite. The extracted minerals were used in simulated Thellier experiments with laboratory TRM as a proxy for natural remanent magnetization (NRM). The plagioclase behaved ideally, as predicted by Neel SD theory, giving a linear NRM-pTRM (Arai) plot with positive pTRM checks. The mafic minerals behaved nonideally, with strongly curved Arai plots sagging below the ideal SD line, in accord with observations for MD magnetites and with predictions that in heating to moderate temperatures T, more TRM is lost than can be regained as pTRM, causing convex-down Arai plots. In experiments with pTRM parallel to NRM, all pTRM tail checks were zero, but pTRM checks increasingly deviated from the Arai curve toward the SD line as T increased. However, in repeat Thellier experiments, pTRMs at all T agreed with the original pTRMs. The negative pTRM checks were therefore not due to physicochemical alteration but rather to departure from initial states with different magnetic histories than those of the original pTRMs. The inability of pTRM checks to reliably distinguish between MD behavior and irreversible alteration could lead to overestimates of paleointensity from straight-line fits to the low-T “unaltered” part of the Arai curve or underestimates from fits to high-T data. Alternating field cleaning to 15 mT before each Thellier step made the Arai plot only marginally less curved. Low-temperature demagnetization (LTD, zero-field cycling through the 120 K Verwey transition) hardly changed the curvature of the Arai plot for pTRM parallel to NRM. For pTRM perpendicular to NRM, LTD led to an inflected Arai curve straddling the ideal SD line. Data of either form would not give reliable paleointensity values. No presently available pretreatment renders the paleointensity data of MD grains usable. SD and nearly SD grains which behave ideally in the Thellier experiment remain the only useful recorders of absolute paleointensity.

The ca. 2 Ga Kapuskasing zone, a 500-km-long belt of uplifted crust, cuts across the regional structural trend of the Archean Superior Province in Canada and disrupts the continuity of 2.45 Ga Matachewan dikes that form a major swarm in its vicinity. As expressed by positive gravity and magnetic anomalies, it appears to end about 100 km east of Lake Superior. Until now, the nature of this termination has remained obscure. Regional variations in paleomagnetic polarity and in the intensity of feldspar clouding caused by magnetite exsolution are two physical properties of the dike swarm that depend upon postemplacement erosion level. They show that the Kapuskasing zone, as a narrow, fault-bounded crustal uplift, continues, after sinistral offset, for a further 60 km to the southwest before terminating in a major cross fault. Dual magnetic polarity and variable levels of feldspar clouding are features found in many other Proterozoic dike swarms worldwide, and therefore, in principle, may be used to document other examples of craton response to tectonism.