An experimental approach on calcarenite reservoir dilation factor

The time of flight of a plane wavefront generated from an acoustic pulse is expected to decrease when the medium length between the wave emitter and receiver is shortened. This simple idea is extrapolated to the case of reservoir compaction in order to obtain a geophysical parameter R (dilation factor) that relates the rock deformation to the variation of time of flight (also called time-lapse time shift in 4D seismics) or acoustic velocity of a plane wave propagating in the same direction of deformation. Interpretation of a few laboratory compressive tests with simultaneous ultrasonic acquisition, performed on oil-saturated calcarenite samples, are presented and discussed. The samples were subjected to several stress regimes and simultaneous ultrasonic acquisitions. Despite the formerly ultrasonic acquisition rate limitations, it was possible to obtain R values for various lateral-vertical stress ratios for each sample's linear and nonlinear stress-strain trends.

-NONUNIFORM MULTIRESOLUTION ANALYSIS

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal.158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

Frame multiresolution analysis of continuous piecewise linear functions

The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function [Formula: see text] that is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. For [Formula: see text] and [Formula: see text], it is shown that if a function [Formula: see text] is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text], for [Formula: see text], and generates MRA with dilation factor [Formula: see text], then [Formula: see text]. Conversely, for [Formula: see text], it is shown that there exists a [Formula: see text], as satisfying the above conditions, that generates MRA with dilation factor [Formula: see text]. The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for [Formula: see text], we prove that if [Formula: see text] generates FMRA with dilation factor [Formula: see text], then [Formula: see text]. For [Formula: see text], we prove similar results when [Formula: see text]. In addition, for [Formula: see text] we prove that there exists a function [Formula: see text] as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.

Symmetric Decompositions and the Veronese Construction

We study rational generating functions of sequences $\{a_n\}_{n\geq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences $\{a_{rn}\}_{n\geq 0}$. We prove that if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities, then the symmetric decomposition of the numerator for $\{a_{rn}\}_{n\geq 0}$ is real-rooted whenever $r\geq \max \{s,d+1-s\}$. Moreover, if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is symmetric, then we show that the symmetric decomposition for $\{a_{rn}\}_{n\geq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^\ast $-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $r\geq \max \{s,d+1-s\}$. Moreover, if the polytope is Gorenstein, then this decomposition is interlacing.

On Franklin wavelets

The Franklin wavelet (piecewise linear wavelet) is a wavelet function [Formula: see text] with dilation factor 2 which is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for all [Formula: see text]; which is constructed from a multiresolution analysis(MRA) with a scaling function [Formula: see text] which is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. Wavelets with composite dilation factor have the ability to produce long and narrow window functions, with varied orientations, well-suited to applications of image processing; also it is more effective and flexible for the construction of multiscale image representation. This motivates us to consider the following problem to construct some composite dilation Franklin wavelets. Given a natural number [Formula: see text], we ask, is there a continuous [Formula: see text] that is linear on [Formula: see text] and [Formula: see text], for [Formula: see text], [Formula: see text] and for all [Formula: see text], which generates an MRA with dilation factor [Formula: see text] to construct an orthonormal [Formula: see text]- wavelet, i.e. a wavelet with dilation factor [Formula: see text]? For [Formula: see text] and [Formula: see text] we have proved that [Formula: see text] generates an MRA with dilation factor [Formula: see text] only if [Formula: see text]. Next for [Formula: see text] and [Formula: see text], we have proved that, if [Formula: see text] generates an MRA with dilation factor [Formula: see text], then [Formula: see text]. Also, for [Formula: see text], we have proved existence of a continuous [Formula: see text] which generates an MRA with dilation factor [Formula: see text]. We have also constructed the corresponding orthonormal [Formula: see text]-wavelet.

Does the Inertia of a Cell Depend Upon its Information-Content?

ABSTRACTA time dilation factor is derived from information entropy via the holographic principle. By imagining the embryonic cell as a hologram, time dilation during embryogenesis is predicted from the entropy of active genetic information and shown to be consistent with observed lengths of cell duration. The connection between entropy and cell duration as well as the corollary of increase in cellular mass with the reduction in information entropy are discussed. This preprint is the second in a series exploring the value of Developmental Spacetime as a framework for falsifiable predictions about the universe.