Multidimensional Setting Research Articles

Solitons in nonlinear lattices

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

On evolving deformation microstructures in non-convex partially damaged solids

The paper outlines a relaxation method based on a particular isotropic microstructure evolution and applies it to the model problem of rate independent, partially damaged solids. The method uses an incremental variational formulation for standard dissipative materials. In an incremental setting at finite time steps, the formulation defines a quasi-hyperelastic stress potential. The existence of this potential allows a typical incremental boundary value problem of damage mechanics to be expressed in terms of a principle of minimum incremental work. Mathematical existence theorems of minimizers then induce a definition of the material stability in terms of the sequential weak lower semicontinuity of the incremental functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of weak convexity notions of the stress potential. Furthermore, the variational setting opens up the possibility to analyze the development of deformation microstructures in the post-critical range of unstable inelastic materials based on energy relaxation methods. In partially damaged solids, accumulated damage may yield non-convex stress potentials which indicate instability and formation of fine-scale microstructures. These microstructures can be resolved by use of relaxation techniques associated with the construction of convex hulls. We propose a particular relaxation method for partially damaged solids and investigate it in one- and multi-dimensional settings. To this end, we introduce a new isotropic microstructure which provides a simple approximation of the multi-dimensional rank-one convex hull. The development of those isotropic microstructures is investigated for homogeneous and inhomogeneous numerical simulations.

Some gap power series in multidimensional setting

We study extensions of classical theorems on gap power series of a complex variable to the multidimensional case.

On the Spectral Gap of Brownian Motion with Jump Boundary

In this paper we consider the Brownian motion with jump boundary and present a new proof of a recent result of Li, Leung and Rakesh concerning the exact convergence rate in the one-dimensional case. Our methods are dierent and mainly probabilistic relying on coupling methods adapted to the special situation under investigation. Moreover we answer a question raised by Ben-Ari and Pinsky concerning the dependence of the spectral gap from the jump distribution in a multi-dimensional setting.

On an inequality for convex functions with some applications on fractional derivatives and fractional integrals

The main goal of the paper is to state and prove the new general inequality for convex and increasing functions. We introduce some new inequalities by involving some fractional integrals and fractional derivatives of Riemman-Liouville, Canavati, Hadamard and Erdelyi-Kober type and apply our result to multidimensional setting to obtain new results involving mixed Riemman-Liouville fractional integrals. 1.

Voter turnout in a multidimensional policy space

Many factors influence the likelihood of citizens turning out to vote. In this paper we focus our attention on issue voting, that is, on the likelihood that different policies offered by politicians affect the probability of voting. If voters consider both the benefits and the costs of voting, rational voters will only vote when politicians offer differentiated policies. In a multidimensional policy space this implies that citizens only vote when they perceive enough difference on the issues they care about the most. We investigate the role of voter abstention due to indifference in a unidimensional and a multidimensional policy setting using data from the US National Election Studies for 1972–2000 and find support for our predictions: voters perceiving a small difference between the platforms of the Democratic and Republican parties are less likely to vote; and voters who perceive the two parties as more different on a larger number of issues are significantly more likely to vote.

Identification problem for damped sine-Gordon equation with point sources

We establish the existence and uniqueness of solutions for sine-Gordon equations in a multidimensional setting. The equations contain a point-like source. Furthermore, the continuity and the Gâteaux differentiability of the solution map is established. An identification problem for parameters governing the equations is set, and is shown to have a solution. The objective function is proved to be Fréchet differentiable with respect to the parameters. An expression for the Fréchet derivative in terms of the solutions of the direct and the adjoint systems is presented. A criterion for optimal parameters is formulated as a bang-bang control principle. An application of these results to the one-dimensional sine-Gordon equation is considered.

A Characterization of Multidimensional S-Automatic Sequences

An infinite word is S-automatic if, for all n≥0, its (n+1)st letter is the output of a deterministic automaton fed with the representation of n in the considered numeration system S. In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d≥2, we state that a multidimensional infinite word x:ℕ d →Σ over a finite alphabet Σ is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word.

PCA2GO: a new multivariate statistics based method to identify highly expressed GO-Terms

BackgroundSeveral tools have been developed to explore and search Gene Ontology (GO) databases allowing efficient GO enrichment analysis and GO tree visualization. Nevertheless, identification of highly specific GO-terms in complex data sets is relatively complicated and the display of GO term assignments and GO enrichment analysis by simple tables or pie charts is not optimal. Valuable information such as the hierarchical position of a single GO term within the GO tree (topological ordering), or enrichment within a complex set of biological experiments is not displayed. Pie charts based on GO tree levels are, themselves, one-dimensional graphs, which cannot properly or efficiently represent the hierarchical specificity for the biological system being studied.ResultsHere we present a new method, which we name PCA2GO, capable of GO analysis using complex multidimensional experimental settings. We employed principal component analysis (PCA) and developed a new score, which takes into account the relative frequency of certain GO terms and their specificity (hierarchical position) within the GO graph. We evaluated the correlation between our representation score R and a standard measure of enrichment, namely p-values to convey the versatility of our approach to other methods and point out differences between our method and commonly used enrichment analyses. Although p values and the R score formally measure different quantities they should be correlated, because relative frequencies of GO terms occurrences within a dataset are an indirect measure of protein numbers related to this term. Therefore they are also related to enrichment. We showed that our score enables us to identify more specific GO-terms i.e. those positioned further down the GO-graph than other common tools used for this purpose. PCA2GO allows visualization and detection of multidimensional dependencies both within the acyclic graph (GO tree) and the experimental settings. Our method is intended for the analysis of several experimental sets, not for one set, like standard enrichment tools. To demonstrate the usefulness of our approach we performed a PCA2GO analysis of a fractionated cardiomyocyte protein dataset, which was identified by enhanced liquid chromatography-mass spectrometry (GeLC-MS). The analysis enabled us to detect distinct groups of proteins, which accurately reflect properties of biochemical cell fractions.ConclusionsWe conclude that PCA2GO is an alternative efficient GO analysis tool with unique features for detection and visualization of multidimensional dependencies within the dataset under study. PCA2GO reveals strongly correlated GO terms within the experimental setting (in this case different fractions) by PCA group formation and improves detection of more specific GO terms within experiment dependent GO term groups than standard p value calculations.

Sex Discrimination in Education: Interaction of Ethical and Contextual Challenges in Implementing Equal Opportunities in Hong Kong

Ethical decisions are contextualized in the dialectic of a multidimensional system, including situation, setting, culture, and generation. There may be further gaps between the ethical considerations of professionals and folk values. The experience of promoting equal opportunities in Hong Kong illustrates some of these challenges. Whereas the rule of law under a Western legal system advocates human rights, the traditional emphasis on harmony and preference for balancing in conflict resolution underlie the gaps in the interpretation of these ideals. The case of the Hong Kong secondary school places allocation system highlights the conflicting perspectives of law, gender stereotypes, psychological knowledge, as well as ethical principles of justice and fairness in promoting equal opportunities in educational assessment and placement. The cultural perspective highlighted in this case illustrates the complex contexts of ethical decisions.

The Mega Distributed Lag Model

This paper attempts to describe the graphical behavior of the distributed lag model in an infinite coordinate space. The “mega distributed lag model” (MDL) is a mathematical framework that can examine the simultaneous interrelationships between all involved variables. The multidimensional graphical setting simultaneously reveals all non-linear exposure — response dependencies and delayed effects between lagged and dependent variables — which two-dimensional figures overwhelmingly fail to capture. Under the Omnia Mobilis assumption, each distribution lag function is indexed with respect to time and space. The Mega distributed lag model observes multiple trends in full motion, the final output (determinant) of which is called “the JIM-coefficient”. Hence, this paper tries to analyze different approaches of lag distribution models that can help in the construction of our new model. The mega distributed lag model (MDL) is moving from the uses of the classic 2-dimensional and 3-dimensional graphical modeling to a multidimensional graphical modeling in Econometrics. Finally, this model is an extension of those explored earlier in the field of Econographicology.

Of mice and men: the evolution of animal welfare guidelines for cancer research

Of mice and men: the evolution of animal welfare guidelines for cancer research

Multidimensional Measurement of Richness: Theory and an Application to Germany

Closely following recent innovations in the literature on the multidimensional measurement of poverty, this paper provides similar measures for the top of the distribution using a dual cutoff method to identify individuals, who can be considered as rich in a multidimensional setting. We use this framework to analyze the role of wealth, health and education, in addition to income, as dimensions of multidimensional well-being in Germany. Our analysis shows that more than half of the German population is affluent in at least one dimension and less than 1% is affluent in all four dimensions. The likelihood of being rich in all dimensions is highest for prime-aged males from the West who live in couple households without children. Mobility between different affluence counts between 2002 and 2007 is rather low and existing changes are mostly driven by health and to a lesser extent by wealth.

The Reference Prior for Complex Covariance Matrices With Efficient Implementation Strategies

The paper derives the reference prior for complex covariance matrices. The reference prior is a noninformative prior that circumvents some of the weaknesses of common alternatives in multidimensional settings. As a consequence, inference based on this prior renders well-behaving solutions that in many cases outperform traditionally used approaches. The main obstacle is that inference based on this prior require integration over high-dimensional spaces which have no closed form solutions. A focus of the paper is therefore to discuss efficient implementation strategies based on Markov chain Monte Carlo methods. It is identified that certain structures can be treated analytically both for the case where the parameter of interest is the covariance matrix itself but also for cases in which the covariance matrix is a nuisance parameter that characterizes noise color. Evaluation in both these settings also verify the superior performance obtained by using the proposed prior as compared to traditional techniques to treat unknown covariance matrices.

Multidimensional Measurement of Richness: Theory and an Application to Germany

Closely following recent innovations in the literature on the multidimensional measurement of poverty, this paper provides similar measures for the top of the distribution using a dual cutoff method to identify individuals, who can be considered as rich in a multidimensional setting. We use this framework to analyze the role of wealth, health and education, in addition to income, as dimensions of multidimensional well-being in Germany. Our analysis shows that more than half of the German population is affluent in at least one dimension and less than 1% is affluent in all four dimensions. The likelihood of being rich in all dimensions is highest for prime-aged males from the West who live in couple households without children. Mobility between different affluence counts between 2002 and 2007 is rather low and existing changes are mostly driven by health and to a lesser extent by wealth.

On an Inequality of H. G. Hardy

We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally, we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals.

Multipliers of multidimensional Fourier algebras

Let G be a locally compact σ -compact group. Motivated by an earlier notion for discrete groups due to Effros and Ruan, we introduce the multidimensional Fourier algebra An(G) of G . We characterise the completely bounded multidimensional multipliers associated with An(G) in several equivalent ways. In particular, we establish a completely isometric embedding of the space of all n -dimensional completely bounded multipliers into the space of all Schur multipliers on Gn+1 with respect to the (left) Haar measure. We show that in the case G is amenable the space of completely bounded multidimensional multipliers coincides with the multidimensional Fourier-Stieltjes algebra of G introduced by Ylinen. We extend some well-known results for abelian groups to the multidimensional setting. Mathematics subject classification (2010): Primary 47B49; Secondary 47L60, 43A22.

On the Shiftability of Dual-Tree Complex Wavelet Transforms

The dual-tree complex wavelet transform (DT-\BBCWT) is known to exhibit better shift-invariance than the conventional discrete wavelet transform. We propose an amplitude-phase representation of the DT-IBBC WT which, among other things, offers a direct explanation for the improvement in the shift-invariance. The representation is based on the shifting action of the group of fractional Hilbert transform (fHT) operators, which extends the notion of arbitrary phase-shifts from sinusoids to finite-energy signals (wavelets in particular). In particular, we characterize the shift ability of the DT-\BBCWT in terms of the shifting property of the fHTs. At the heart of the representation are certain fundamental invariances of the fHT group, namely that of translation, dilation, and norm, which play a decisive role in establishing the key properties of the transform. It turns out that these fundamental invariances are exclusive to this group. Next, by introducing a generalization of the Bedrosian theorem for the fHT operator, we derive an explicitly understanding of the shifting action of the fHT for the particular family of wavelets obtained through the modulation of lowpass functions (e.g., the Shannon and Gabor wavelet). This, in effect, links the corresponding dual-tree transform with the framework of windowed-Fourier analysis. Finally, we extend these ideas to the multidimensional setting by introducing a directional extension of the fHT, the fractional directional Hilbert transform. In particular, we derive a signal representation involving the superposition of direction-selective wavelets with appropriate phase-shifts, which helps explain the improved shift-invariance of the transform along certain preferential directions.

On a generalization of Kantorovich operators on simplices and hypercubes

In this paper we introduce and study two new sequences of positive linear operators acting on the space of all Lebesgue integrable functions defined, respectively, on the N-dimensional hypercube and on the N-dimensional simplex (N ≥ 1). These operators represent a natural generalization to the multidimensional setting of the ones introduced in [Altomare and Leonessa, Mediterr. J. Math. 3: 363–382, 2006] and, in a particular case, they turn into the multidimensional Kantorovich operators on these frameworks. We study the approximation properties of such operators with respect both to the sup-norm and to the Lp-norm and we give some estimates of their rate of convergence by means of certain moduli of smoothness.

Multidimensional generalized automatic sequences and shape-symmetric morphic words

An infinite word is S -automatic if, for all n ≥ 0 , its ( n + 1 ) th letter is the output of a deterministic automaton fed with the representation of n in the numeration system S . In this paper, we consider an analogous definition in a multidimensional setting and study its relation to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d ≥ 1 , we show that a multidimensional infinite word x : N d → Σ over a finite alphabet Σ is S -automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word.