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The Internet generates large volumes of data at a high rate, in particular, posts on social networks. Although social network data has numerous semantic adulterations, and is not intended to be a source of geo-spatial information, in the text of posts we find pieces of important information about how people relate to their environment, which can be used to identify interesting aspects of how human beings interact with portions of land based on their activities. This research proposes a methodology for the identification of land uses using Natural Language Processing (NLP) from the contents of the popular social network Twitter. It will be approached by identifying keywords with linguistic patterns from the text, and the geographical coordinates associated with the publication. Context-specific innovations are introduced to deal with data across South America and, in particular, in the city of Arequipa, Peru. The objective is to identify the five main land uses: residential, commercial, institutional-governmental, industrial-offices and unbuilt land. Within the framework of urban planning and sustainable urban management, the methodology contributes to the optimization of the identification techniques applied for the updating of land use cadastres, since the results achieved an accuracy of about 90%, which motivates its application in the real context. In addition, it would allow the identification of land use categories at a more detailed level, in situations such as a complex/mixed distribution building based on the amount of data collected. Finally, the methodology makes land use information available in a more up-to-date fashion and, above all, avoids the high economic cost of the non-automatic production of land use maps for cities, mostly in developing countries.

The Jacobi polynomials $\hat{P}_n^{(\alpha,\beta)}(x)$ conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function $(1-x)^\alpha (1+x)^\beta, \alpha,\beta>-1,$ on the interval $[-1,+1]$. The spreading of its associated probability density (i.e., the Rakhmanov density) over the orthogonality support has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic $\mathfrak{L}_{q}$-norms (Shannon and R\'enyi entropies) and the monotonic complexity-like measures of Cram\'er-Rao, Fisher-Shannon and LMC (L\'opez-Ruiz, Mancini and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non-handy way; specially when the degree or the parameters $(\alpha,\beta)$ are large. In this work, we determine in a simple, compact form the entropic and complexity-like properties of the Jacobi polynomials in the two extremal situations: ($n\rightarrow \infty$; fixed $\alpha,\beta$) and ($\alpha\rightarrow \infty$; fixed $n,\beta$). These two asymptotics are relevant \textit{per se} and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of numerous supersymmetric quantum-mechanical systems., Comment: 11 pages, 1 figure

The main monotonic statistical complexity-like measures of the Rakhmanov's probability density associated to the hypergeometric orthogonal polynomials (HOPs) in a real continuous variable, each of them quantifying two configurational facets of spreading, are examined in this work beyond the Cram\'er-Rao one. The Fisher-Shannon and LMC (L\'opez-Ruiz-Mancini-Calvet) complexity measures, which have two entropic components, are analytically expressed in terms of the degree and the orthogonality weight's parameter(s) of the polynomials. The degree and parameter asymptotics of these two-fold spreading measures are shown for the parameter-dependent families of HOPs of Laguerre and Gegenbauer types. This is done by using the asymptotics of the R\'enyi and Shannon entropies, which are closely connected to the $\mathfrak{L}_{q}$-norms of these polynomials, when the weight's parameter tends towards infinity. The degree and parameter asymptotics of these Laguerre and Gegenbauer algebraic norms control the radial and angular charge and momentum distributions of numerous relevant multidimensional physical systems with a spherically-symmetric quantum-mechanical potential in the high-energy (Rydberg) and high-dimensional (quasi-classical) states, respectively. This is because the corresponding states' wavefunctions are expressed by means of the Laguerre and Gegenbauer polynomials in both position and momentum spaces., Comment: 23 pages

Biclustering is a powerful approach to search for patterns in data, as it can be driven by a function that measures the quality of diverse types of patterns of interest. However, due to its computational complexity, the exploration of the search space is usually guided by an algorithmic strategy, sometimes introducing random factors that simplify the computational cost (e.g. greedy search or evolutionary computation). Shifting patterns are specially interesting as they account constant fluctuations in data, i.e. they capture situations in which all the values in the pattern move up or down for one dimension maintaining the range amplitude for all the dimensions. This behaviour is very common in nature, e.g. in the analysis of gene expression data, where a subset of genes might go up or down for a subset of patients or experimental conditions, identifying functionally coherent categories. Boolean reasoning was recently revealed as an appropriate methodology to address the search for constant biclusters. In this work, this direction is extended to search for more general biclusters that include shifting patterns. The mathematical foundations are described in order to associate Boolean concepts with shifting patterns, and the methodology is presented to show that the induction of shifting patterns by means of Boolean reasoning is due to the ability of finding all inclusion--maximal {\delta}-shifting patterns. Experiments with a real dataset show the potential of our approach at finding biclusters with {\delta}-shifting patterns, which have been evaluated with the mean squared residue (MSR), providing an excellent performance at finding results very close to zero., Comment: 29 pages, 8 figures

The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydrogenic systems are obtained from first principles; that is, they are given in terms of the states' principal and magnetic hyperquantum numbers $(n,\mu_1,\mu_2,\ldots,\mu_{D-1})$, the system's dimensionality $D$ and the nuclear charge $Z$ in an analytical, compact form. Explicit expressions for the total Shannon entropies are given for the quasi-spherical states, which conform a relevant class of specific states of the $D$-dimensional hydrogenic system characterized by the hyperquantum numbers $\mu_1=\mu_2\ldots=\mu_{D-1}=n-1$, including the ground state.

Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric Fisher-R\'enyi complexity, named $(p,\beta,\lambda)$-Fisher--R\'enyi complexity. This complexity measure quantifies the combined balance of the spreading and the gradient contents of $\rho$, and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the $(p,\beta,\lambda)$-Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential.

In this work the one-parameter Fisher-R\'enyi measure of complexity for general $d$-dimensional probability distributions is introduced and its main analytic properties are discussed. Then, this quantity is determined for the hydrogenic systems in terms of the quantum numbers of the quantum states and the nuclear charge.

We introduce and discuss the notion of monotonicity for the complexity measures of general probability distributions, patterned after the resource theory of quantum entanglement. Then, we explore whether this property is satisfied by the three main intrinsic measures of complexity (Cramer-Rao, Fisher-Shannon, LMC) and some of their generalizations., Comment: Version accepted in Physics Letters A

The R\'enyi and Shannon entropies are information-theoretic measures which have enabled to formulate the position-momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density-functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment $\langle r^2\rangle$ for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the R\'enyi maximization procedure with a covariance constraint due to Costa, Hero and Vignat \cite{CosHer03}. The contributions to these bounds coming from the radial and angular parts of the physical wavefunctions are explicitly given., Comment: 8 pages

The momentum entropic moments and R\'enyi entropies of a one-dimensional particle in an infinite well potential are found by means of explicit calculations of some Dirichlet-like trigonometric integrals. The associated spreading lengths and quantum uncertainty-like sums are also provided., Comment: 14 pages

The analytic information theory of discrete distributions was initiated in 1998 by C. Knessl, P. Jacquet and S. Szpankowski who addressed the precise evaluation of the Renyi and Shannon entropies of the Poisson, Pascal (or negative binomial) and binomial distributions. They were able to derive various asymptotic approximations and, at times, lower and upper bounds for these quantities. Here we extend these investigations in a twofold way. First, we consider a much larger class of distributions, the Rakhmanov distributions $\rho_n(x)=\omega(x)y_n^2(x)$, where $\{y_n(x)\}$ denote the sequences of discrete hypergeometric-type polynomials which are orthogonal with respect to the weight function $\omega(x)$ of Poisson, Pascal, binomial and hypergeometric types; that is the polynomials of Charlier, Meixner, Kravchuk and Hahn. Second, we obtain the explicit expressions for the relative Fisher information of these four families of Rakhmanov distributions with respect to their respective weight functions., Comment: 18 pages, 10 figures

The Cram\'er-Rao, Fisher-Shannon and LMC shape complexity measures have been recently shown to play a relevant role to study the internal disorder of finite many-body systems (e.g., atoms, molecules, nuclei). They highlight amongst the bunch of complexities appeared in the literature by their mathematical simplicity and transparent physical interpretation. They are composed by two spreading measures (variance, Fisher information and Shannon entropies) of the single-particle probability density of the systems. Here we discuss the physico-mathematical knowledge of these two-component complexities for hydrogenic systems with standard ($D = 3$) and non-standard dimensionalities. These systems include not only the hydrogen atom and its isoelectronic series but also a large diversity of quantum systems such as, e.g. exotic atoms, antimatter atoms, some qubits systems and Rydberg atoms, ions and molecules., Comment: 13 pages, 1 figure

The macroscopic properties of a quantum system strongly depend on the spreading of the physical eigenfunctions (wavefunctions) of its Hamiltonian operador over its confined domain. The wavefunctions are often controlled by classical or hypergeometric-type orthogonal polynomials (Hermite, Laguerre and Jacobi). Here we discuss the spreading of these polynomials over its orthogonality interval by means of various information-theoretic quantities which grasp some facets of the polynomial distribution not yet analyzed. We consider the information-theoretic lengths closely related to the Fisher information and R\'enyi and Shannon entropies, which quantify the polynomial spreading far beyond the celebrated standard deviation., Comment: 17 pages

The asymptotics of the weighted $L_{p}$-norms of Hermite polynomials, which describes the R\'enyi entropy of order $p$ of the associated quantum oscillator probability density, is determined for $n\to\infty$ and $p>0$. Then, it is applied to the calculation of the R\'enyi entropy of the quantum-mechanical probability density of the highly-excited (Rydberg) states of the isotropic oscillator., Comment: 11 pages

The uncertainty relation based on the Shannon entropies of the probability densities in position and momentum spaces is improved for quantum systems in arbitrary $D$-dimensional spherically symmetric potentials. To find it, we have used the $L^p$ -- $L^q$ norm inequality of L. De Carli and the logarithmic uncertainty relation for the Hankel transform of S. Omri. Applications to some relevant three-dimensional central potentials are shown., Comment: 13 pages, 3 figures

The position-momentum uncertainty-like inequality based on moments of arbitrary order for d-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the Renyi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in d-dimensional physics: the hydrogenic and oscillator-like systems., Comment: 31 pages, 9 figures

The internal disorder of a D-dimensional hydrogenic system, which is strongly associated to the non-uniformity of the quantum-mechanical density of its physical states, is investigated by means of the shape complexity in the two reciprocal spaces. This quantity, which is the product of the disequilibrium and the Shannon entropic power, is mathematically expressed for both ground and excited stationary states in terms of certain entropic functionals of Laguerre and Gegenbauer (or ultraspherical) polynomials. Emphasis is made in the ground and circular states., Comment: 12 pages, 0 figures. Published in Mathematical Physics and Field Theory