Dilation Matrix Research Articles

Characterization of the convergent multivariate nonnegative cascade algorithm with a dilation matrix

In this paper, we characterize the convergence of the multivariate cascade algorithm with an [Formula: see text] dilation matrix [Formula: see text] and a nonnegative finite mask, which generalizes the result obtained recently. Moreover, we present two examples, which demonstrate the power and applicability of our main result.

Wetting mechanism and alteration of nano-sized shale pores: Insights from contrast variation small angle neutron scattering

Wettability of tight shale is crucial for fluid flow and mass transport process in energy geosciences. However, understanding the interfacial chemistry and wetting mechanisms at sub-nano-pore scales remains a formidable challenge. In this study, the Contrast Variation technique of Small Angle Neutron Scattering (CV-SANS) is employed to investigate shale’s interfacial chemistry using reagents that possess a range of different polarities, including water, n-decane, toluene, and dimethyl methanamide. Through five different experimental strategies, we have demonstrated a successful modification of shale wettability, ranging from enhancement, weakening, to reversal. Delving into the mechanisms, we illustrated the crucial role of pre-existing liquid films in these changes, where the uniquely co-existing polar and non-polar functional groups in dimethyl methanamide acted as a conduit for interfacial chemistry adjustments. Furthermore, a solvent immersion led to matrix dilation as well as liberation of residual oil-occupied pores, resulting in altered pore size distributions, with hydrogen bonding playing a significant role in the polar groups. Interestingly, despite shale exhibiting a stronger affinity for oil over water, hydrophilic solvents induced more substantial dilation than lipophilic ones. Collectively, this work elucidates the dynamic change of interfacial chemistry via the configuration of polarity using chemical reagents, and the CV-SANS technique underscores its invaluable utilities in decoding the interfacial wettability traits in nanopore space of shale.

A simple method to construct multivariate dual framelets with high-order vanishing moments

When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield [Formula: see text] of [Formula: see text] that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters.

Efficient solution of the non-unitary time-dependent Schrodinger equation on a quantum computer with complex absorbing potential

We explore the possibility of adding complex absorbing potential at the boundaries when solving the one-dimensional real-time Schrödinger evolution on a grid using a quantum computer with a fully quantum algorithm described on a n qubit register. Due to the complex potential, the evolution mixes real- and imaginary-time propagation and the wave function can potentially be continuously absorbed during the time propagation. We use the dilation quantum algorithm to treat the imaginary-time evolution in parallel to the real-time propagation. This method has the advantage of using only one reservoir qubit at a time, that is measured with a certain success probability to implement the desired imaginary-time evolution. We propose a specific prescription for the dilation method where the success probability is directly linked to the physical norm of the continuously absorbed state evolving on the mesh. We expect that the proposed prescription will have the advantage of keeping a high probability of success in most physical situations. Applications of the method are made on one-dimensional wave functions evolving on a mesh. Results obtained on a quantum computer identify with those obtained on a classical computer. We finally give a detailed discussion on the complexity of implementing the dilation matrix. Due to the local nature of the potential, for n qubits, the dilation matrix only requires 2n CNOT and 2n unitary rotation for each time step, whereas it would require of the order of 4n+1 C-NOT gates to implement it using the best-known algorithm for general unitary matrices.

Some compactly supported Riesz wavelets associated to any Ed(2)(ℤ) dilation

Let [Formula: see text] be a [Formula: see text] dilation matrix with integer entries and with determinant [Formula: see text]. The existence of Riesz wavelets with compact support associated to [Formula: see text] is proved. Our proof is constructive and the generators of these Riesz wavelets may be taken to be symmetric, with high regularity and many vanishing moments. In our construction, we first study the structure of the quotient group [Formula: see text]. Afterwards and perhaps the main advance in this paper is the fact that we obtain a family of trigonometric polynomials on [Formula: see text] with zeros only in [Formula: see text]. At this point, we are able to get scaling functions with compact support of a multiresolution analysis. In addition, Riesz wavelets with compact support hold by standard multiresolution techniques. Finally, we give some examples of Riesz wavelets where we emphasize on a numerical estimation of their regularity.

Approximation of functions by wavelet expansions with dilation matrix

In this paper, we obtain the degree of approximation of a function f in Lp(1 ? p ? ?) norm under general conditions of the pointwise and uniform convergence of wavelet expansions associated with the multiresolution analysis with dilation matrix. Our results show that the degree has the exponential decay (faster than any polynomial) for the function f in Lp(R) on a finite interval (a, b).

Two families of compactly supported Parseval framelets in L2(Rd)

For any dilation matrix with integral entries A∈Rd×d, d≥1, we construct two families of Parseval wavelet frames in L2(Rd). Both families have compact support and any desired number of vanishing moments. The first family has |det⁡A| generators. The second family has any desired degree of regularity. For the members of this family, the number of generators depends on the dilation matrix A and the dimension d, but never exceeds |det⁡A|+d. Our construction involves trigonometric polynomials developed by Heller to obtain refinable functions, the Oblique Extension Principle, and a slight generalization of a theorem of Lai and Stöckler.

Two-Dimensional Wavelet with Matrix Dilation M = 2I and its Application in Solving Integral Equations

In this study, using a one-dimensionl MRA we constructed a two-dimensional wavelet as well as four masks which are not related to the MRA. Finally, we provide some examples to prove the applicability of our construction in case of finding numerical solution of two-dimensional first kind Fredholm integral equations.

Multivariate Symmetric Interpolating Dual Multiwavelet Frames

The construction of symmetric multiwavelets in the multivariate case with useful in applications properties is a challenging task, mainly due to the complexity of the matrix extension problem. Nevertheless, for the interpolating case, a general technique can be developed. For an appropriate pair of symmetry group H and matrix dilation M and for a given H-symmetric interpolating refinable matrix mask, a method for the construction of H-symmetric dual refinable matrix masks with a preassigned order of sum rule is suggested. Wavelet matrix masks are constructed using a certain explicit matrix extension algorithm, and their symmetry properties are studied via its polyphase components. The resulting multiwavelet systems form dual multiwavelet frames, where wavelet functions have symmetry properties, preassigned order of vanishing moments and preassigned order of the balancing property. Several examples are presented.

Matrix dilation and Hausdorff operators on modulation spaces

In this paper, we establish the asymptotic estimates for the norms of the matrix dilation operators on modulation spaces. As an application, we study the boundedness on modulation spaces of Hausdorff operators. The definition of Hausdorff operators is also revisited for fitting our study.

Fully coupled thermo-hydro-mechanical approach to model fracture response to injection process in enhanced geothermal systems using displacement discontinuity and finite element method

Assessing, predicting, and controlling injection-induced seismicity is a major challenge for developing enhanced geothermal systems (EGS) due to the complexity of coupled thermo-hydro-mechanical (THM) processes. This study aimed to develop a fully coupled numerical model to assess the complex behavior of a low permeable matrix-fracture during non-isothermal single-phase fluid injection process. A thermo-poroelastic displacement discontinuity (DD) method combining with different forms of finite element method are implemented to encapsulate the fractured medium response and transport processes, respectively. The nonlinear characteristics of normal (changing the joint to hydraulic fracture status) and shear (changing the stick to slip fracture status) fracture deformation are taken into account through fracture constitutive relations. Developed numerical approach was applied to simulate cool water injection into fracture/matrix systems, analyzing the role of coupled processes on spatiotemporal variation of matrix-fracture stresses, temperature and pore pressure and assessing induced seismicity (slippage) and permeability alteration. Numerical simulations demonstrate a fully coupled relation between matrix dilation, shrinkage, and non-linear fracture deformation. The redistribution of dynamic and kinematic parameters along with non-linear fracture deformation showed that although the poroelastic effects are dominant in the early stages, thermoelastic effects dominate in the long-term injection stages. • A fully-coupled Thermo-Hydro-Mechanical approach was developed to model injection process in EGS. • Displacement discontinuity and finite element methods were combined to encapsulate fracture deformation and transport processes, respectively. • Fracture constitutive laws were implemented to model non-linear crack deformation. • Poroelastic effects are dominant in the early stages, and thermoelastic effects dominate in the long-term injection stages.

Framelet scaling set with matrix dilation in L2(R2)

Framelet scaling set with matrix dilation in L2(R2)

Self-affine $2$-attractors and tiles

We study two-digit attractors (2-attractors) in $\mathbb{R}^d$, which are self-affine compact sets defined by two affine contractions with the same linear part. They have widely been studied in the literature under various names (integer self-affine 2-tiles, twindragons, two-digit tiles, 2-reptiles and so on) due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in discrete geometry and in number theory. We obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$ and show that all of them are pairwise homeomorphic but not diffeomorphic. In the general, nonisotropic, case we prove that a 2-attractor is uniquely defined by the spectrum of the dilation matrix up to affine similarity. We estimate the number of different 2-attractors in $\mathbb{R}^d$ by analysing integer unitary expanding polynomials with free coefficient $\pm2$. The total number of such polynomials is estimated using the Mahler measure. We present several infinite series of such polynomials. For some 2-attractors their Hölder exponents are found. Some of our results are extended to attractors with an arbitrary number of digits. Bibliography: 63 titles.

Coorbit spaces associated to integrably admissible dilation groups

This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue spaces with respect to the quasi-regular representation of the semi-direct product of Euclidean space and a suitable matrix dilation group. The class of dilation groups that we allow, the so-called integrably admissible dilation groups, contains the matrix groups yielding an irreducible, square-integrable quasi-regular representation as a proper subclass. The obtained scale of coorbit spaces extends therefore the well-studied wavelet coorbit spaces associated to discrete series representations. We show that for any integrably admissible dilation group there exists a convienent space of smooth, admissible analyzing vectors that can be used to define a consistent coorbit space possessing all the essential properties that are known to hold in the setting of discrete series representations. In particular, the obtained coorbit spaces can be realized as Besov-type decomposition spaces by means of a Littlewood—Paley-type characterization. The classes of anisotropic Besov spaces associated to expansive matrices are shown to coincide precisely with the coorbit spaces induced by the integrably admissible one-parameter groups.

Dilations of unitary tuples

We study the space of all $d$-tuples of unitaries $u=(u_1,\ldots, u_d)$ using dilation theory and matrix ranges. Given two $d$-tuples $u$ and $v$ generating C*-algebras $\mathcal A$ and $\mathcal B$, we seek the minimal dilation constant $c=c(u,v)$ such that $u\prec cv$, by which we mean that $u$ is a compression of some $*$-isomorphic copy of $cv$. This gives rise to a metric \[ d_D(u,v)=\log\max\{c(u,v),c(v,u)\} \] on the set of equivalence classes of $*$-isomorphic tuples of unitaries. We also consider the metric \[ d_{HR}(u,v)=\inf\left\{\|u'-v'\|:u',v'\in B(H)^d, u'\sim u\textrm{ and } v'\sim v\right\}, \] and we show the inequality \[ d_{HR}(u,v)\leq K d_D(u,v)^{1/2}. \] Let $u_\Theta$ be the universal unitary tuple $(u_1,\ldots,u_d)$ satisfying $u_\ell u_k=e^{i\theta_{k,\ell}} u_k u_\ell$, where $\Theta=(\theta_{k,\ell})$ is a real antisymmetric matrix. We find that $c(u_\Theta, u_{\Theta'})\leq e^{\frac{1}{4}\|\Theta-\Theta'\|}$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $\Theta\mapsto U(\Theta)\in B(H)^d$ such that $U(\Theta)\sim u_\Theta$ and \[ \|U(\Theta)-U({\Theta'})\|\leq K\|\Theta-\Theta'\|^{1/2}. \] Of special interest are: the universal $d$-tuple of noncommuting unitaries ${\mathrm u}$, the $d$-tuple of free Haar unitaries $u_f$, and the universal $d$-tuple of commuting unitaries $u_0$. We obtain the bounds \[ 2\sqrt{1-\frac{1}{d}}\leq c(u_f,u_0)\leq 2\sqrt{1-\frac{1}{2d}}. \] From this, we recover Passer's upper bound for the universal unitaries $c({\mathrm u},u_0)\leq\sqrt{2d}$. In the case $d=3$ we obtain the new lower bound $c({\mathrm u},u_0)\geq 1.858$ improving on the previously known lower bound $c({\mathrm u},u_0)\geq\sqrt{3}$.

The ubiquitin ligase NEDD4-2/NEDD4L regulates both sodium homeostasis and fibrotic signaling to prevent end-stage renal disease

Kidney disease progression can be affected by Na+ abundance. A key regulator of Na+ homeostasis is the ubiquitin ligase NEDD4-2 and its deficiency leads to increased Na+ transport activity and salt-sensitive progressive kidney damage. However, the mechanisms responsible for high Na+ induced damage remain poorly understood. Here we show that a high Na+ diet compromised kidney function in Nedd4-2-deficient mice, indicative of progression toward end-stage renal disease. Injury was characterized by enhanced tubule dilation and extracellular matrix accumulation, together with sustained activation of both Wnt/β-catenin and TGF-β signaling. Nedd4-2 knockout in cortical collecting duct cells also activated these pathways and led to epithelial–mesenchymal transition. Furthermore, low dietary Na+ rescued kidney disease in Nedd4-2-deficient mice and silenced Wnt/β-catenin and TGF-β signaling. Our study reveals the important role of NEDD4-2-dependent ubiquitination in Na+ homeostasis and protecting against aberrant Wnt/β-catenin/TGF-β signaling in progressive kidney disease.

Structured singular value of implicit systems

Implicit systems provide a general framework in which many important properties of dynamic systems can be studied. Implicit systems are especially relevant to behavioural systems theory, the analysis and synthesis of complex interconnected systems, systems identification and robust control. By incorporating algebraic constraints, implicit models provide additional versatility relative to the standard input–output framework. Problems of robust stability in implicit systems lead in a natural way to non‐standard structured singular value (μ) formulations. In this note, it is shown that for a class of uncertainty structures involving repeated scalar parameters, these problems reduce to a standard μ problem which is well studied and for the solution of which several numerical algorithms are available. Our results are based on a matrix dilation technique and the redefinition of the uncertainty structure of the transformed problem. The main results of the paper are illustrated with a numerical example.

Sufficient Conditions for a Multidimensional System of Periodic Wavelets to be a Frame

We study multidimensional periodic wavelet systems with matrix dilations. We obtain conditions sufficient for such a system to be Bessel. The conditions are given in terms of Fourier coefficients. We propose a method for constructing a wavelet Riesz basis that starts with a suitable sequence of trigonometric polynomials.

Approximation by periodic multivariate quasi-projection operators

Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions φj and a sequence of distributions/functions φ˜j. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions for φj and the compatibility conditions for φj and φ˜j. These estimates are given in terms of the Fourier coefficients of approximated functions and provide analogs of some known non-periodic results. Under some additional assumptions error estimates are given in other terms in particular using the best approximation. A number of examples are provided.

Multiscaling frame multiresolution analysis and associated wavelet frames

Frame multiresolution analysis (FMRA) in [Formula: see text] is an important topic in frame theory and its applications. In this paper, we consider the so-called multiscaling FMRA in [Formula: see text], which has matrix dilations and a finite number of scaling functions. This framework is a generalization of the theories both on monoscaling FMRA and on the classical MRA of multiplicity [Formula: see text]. We characterize wavelet frames and Parseval wavelet frames for [Formula: see text] under the circumstances that they can be associated with a multiscaling FMRA. We give two necessary and sufficient conditions for given functions [Formula: see text] in [Formula: see text] to be multiframe generators of [Formula: see text]. Especially, the second condition depends on the multiscaling FMRA and [Formula: see text] only, does not require the existence of other functions, and is relatively easier to verify. Moreover, for any finitely-generated frame of integer translates, we give explicitly the Fourier transforms of the generators of its canonical dual frame. We illustrate the implementation and an application of the theory with an example.