Typical Rank Research Articles

Soil Bacteria in Archaeology: What Could Rank Abundance Functions Tell Us About Ancient Human Impacts on Microbial Communities?

Metagenomic analysis of soil bacterial communities based on 16S rRNA reflects a typical community composition containing a low number of high-abundance types and a very high number of low-abundance types. Here, the formation of characteristic rank order functions of bacterial abundance is investigated by modelling the dynamics of soil bacterial communities, assuming a succession of different bacterial populations that grow rapidly and decay more slowly. We found that the characteristic shape of typical rank order functions is well reflected by simulations. In addition, our model allowed us to investigate strong disturbances in the soil, which could be expected in cases of strongly changing local environmental conditions in soil, e.g., after translocation and covering of soil material. Such events could lead to the formation of shoulders in the rank order functions. Abundance rank orders observed in cases of some archaeological soil samples do indeed show such a shoulder and could be well interpreted by simulated rank order functions. As a result, it can be concluded that the investigations herein support our hypothesis that abundance rank orders contain information about the temporal order of developing bacterial types in changing communities and thus store information about local environmental conditions in the past, including ancient humans’ impact on soil. This information can be used for interpretation of archeological findings and for reconstruction of different former human activities, as well as knowledge on the translocation of soil material in the past.

On the typical rank of elliptic curves over $${\mathbb {Q}}(T)$$

As an application of Turán sieve, we give upper bounds for the number of elliptic curves defined over \({\mathbb {Q}}(T)\) in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan’s conjecture (Cowan in Conjecture: 100% of elliptic surfaces over \({\mathbb {Q}}\) have rank zero. Preprint. https://arxiv.org/pdf/2009.08622.pdf, 2020) in the case \(m,n\le 2\).

Fame and Ultrafame: Measuring and comparing daily levels of ‘being talked about’ for United States’ presidents, their rivals, God, countries, and K-pop.


 
 
 When building a global brand of any kind—a political actor, clothing style, or belief system— developing widespread awareness is a primary goal. Short of knowing any of the stories or products of a brand, being talked about in whatever fashion—raw fame—is, as Oscar Wilde would have it, better than not being talked about at all. Here, we measure, examine, and contrast the day-to-day raw fame dynamics on Twitter for US Presidents and major US Presidential candidates from 2008 to 2020: Barack Obama, John McCain, Mitt Romney, Hillary Clinton, Donald Trump, and Joe Biden. We assign “lexical fame” to be the number and (Zipfian) rank of the (lowercased) mentions made for each individual across all languages. We show that all five political figures have at some point reached extraordinary volume levels of what we define to be “lexical ultrafame”: An overall rank of approximately 300 or less which is largely the realm of function words and demarcated by the highly stable rank of ‘god’. By this measure, ‘trump’ has become enduringly ultrafamous, from the 2016 election on. We use typical ranks for country names and function words as standards to improve perception of scale. We quantify relative fame rates and find that in the eight weeks leading up the 2008 and 2012 elections, ‘obama’ held a 1000:757 volume ratio over ‘mccain’ and 1000:892 over ‘romney’, well short of the 1000:544 and 1000:504 volumes favoring ‘trump’ over ‘hillary’ and ‘biden’ in the 8 weeks leading up to the 2016 and 2020 elections. Finally, we track how only one other entity has more sustained ultrafame than ‘trump’ on Twitter: The K-pop (Korean pop) band BTS. We chart the dramatic rise of BTS, finding their Twitter handle ‘@bts twt’ has been able to compete with ‘a’ and ‘the’, reaching a rank of three at the day scale and a rank of one at the quarter-hour scale. Our findings for BTS more generally point to K-pop’s growing economic, social, and political power.
 
 

Strata-based quantification of distributional uncertainty in socio-economic indicators: A comparative study of Indian states

This paper reports a comprehensive study of distributional uncertainty in a few socio-economic indicators across the various states of India over the years 2001–2011. We show that the discrete generalized beta (DGB) distribution, a typical rank order distribution, provide excellent fits to the district-wise empirical data for the population size, literacy rate (LR) and work participation rate (WPR) within every states in India, through its two distributional parameters. Moreover, taking resort to the entropy formulation of the DGB distribution, a proposed uncertainty percentage (UP) unveils the dynamics of the uncertainty of LR and WPR in all states of India. We have also commented on the changes in the estimated parameters and the UP values from the years 2001–2011. Additionally, a gender based analysis of the distribution of these important socio-economic variables within different states of India has also been discussed. Interestingly, it has been observed that, although the distributions of the numbers of literate and working people has a direct (linear) correspondence with that of the population size, the literacy and work-participation rates are distributed independently of the population distributions. This detailed study of distributional uncertainty or inequalities of different socio-economic indicators within and between the various strata along with their relations with the population may help policy makers to frame better policies for a sustainable and equitable development of the country.

A multi-criteria approach for assessing the potential of renewable energy sources for electricity generation: Case Serbia

When assessing the potential of renewable energy alternatives for electricity generation, it is necessary to implement a multi-perspective approach that includes economic, technical, environmental, and socio-political criteria. For the evaluation of criteria and alternatives, the Fuzzy Analytical Hierarchy Process (FAHP) method is applied. The obtained weights are formed according to the values of energy indicators and expert judgments. The use of a fuzzy numerical value scale for the assessment of expert judgments and energy indicators provides a more sensitive scoring system for differences between values than the typical rank of absolute energy indicator values. This approach is implemented for assessment of the potential of renewable energy sources that could be utilized in the Serbian electricity sector. The results of the FAHP reveal that hydropower and biomass have the highest potential among available renewable energy sources.

Refinements of the algorithms needed to retrieve atmospheric vertical thermodynamic profiles from GNSS radio occultations

Abstract In this work we consider some refinements to the Boundary Profile eValuation (BPV) method, developed by Vespe et al. (2004), Vespe and Persia (2006), Vespe (2016), Andrisani and Vespe (2020) in order to retrieve atmospheric thermodynamic profiles from Radio Occultation (RO) observations. Former BPV models solved typical rank deficiencies in treating RO data by applying parametric dry refractivity models for the atmosphere, such as the CIRA86aQ or Hopfield (1969). The involved parameters were selected by fitting those dry models with the refractivity observed in the stratosphere, where water vapor concentrations are negligible, with the usual Least Square Error (LSE) method. Dry refractivity is then extrapolated down to the Earth surface. Nevertheless, unphysical occurrence of negative values of humidity concentrations, typical for the so called “simple models” (Kursinski and Kursinski, 2013) can occur in the lower troposphere. This problem is solved with a non-negative residuals constrained fitting for the total refractivity, obtained with different mathematical methodologies. Other refinements include the departure from the ideal state equation for a more realistic state equation concerning compressibility factors for the dry and wet components of the atmosphere. An application to 848 RO observations from COSMIC_1 space mission, covering tropical and middle latitudes during the months of January, March and May 2009, together with a statistical comparison with other atmospheric models and experimental data, was made in order to estimate the accuracy of the BPV method. The results show a description about the thermodynamic state of the atmosphere in line with other recognized atmospheric models, in particular for what concerns the tropics and the southern middle latitudes.

Typical ranks in symmetric matrix completion

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called typical ranks . We give a combinatorial description of the patterns of specified entries of n × n symmetric matrices that have n as a typical rank. Moreover, we describe exactly when such a generic partial matrix is minimally completable to rank n . We also characterize the typical ranks for patterns of entries with low maximal typical rank.

Humidity Profiles Retrieved From GNSS Radio Occultations by a Non-negative Residual Constrained Least Square Error Method

We performed some refinements to the Boundary Profile eValuation (BPV) method, developed by Vespe and collaborators, for the retrieval of vertical atmospheric humidity profiles from Radio Occultation (RO) observations. Previous BPV models solve typical rank deficiencies in treating RO data by recurring to parametric dry atmospheric refractivity models, such as the Hopfield or the CIRA86aQ. The involved parameters were selected by fitting observed RO bending angles (BA) through the stratosphere where the humidity is negligible. Total refractivity was then obtained from observed bending angles via a variational method. Such approach furnishes a valid alternative to the usual Abel inversion. Humidity profiles were finally achieved by subtracting the dry refractivity contributions from the total refractivity profiles. Nevertheless, unphysical behaviors of “negative” values of humidity can occur when the dry refractivity profiles are extrapolated toward the lower part of the troposphere. In order to avoid such unphysical behavior, in this work we recur to a second fit for the total refractivity data, through a least square error method having a non-negative residual constrain. This is mathematically achieved with a modification of the usual least square error functional to minimize, by means of an additional exponential fast increasing term with respect to negative residuals. A Levenberg-Marquardt method relative to this new functional has been developed too, in order to numerically estimates the relative minimizers. With this approach new dry refractivity profiles, more suited to be extrapolated in the troposphere, are recovered. We tested this new method with a series of almost 450 of GPS-RO observations from the FORMOSAT-3 COSMIC-1 Space Mission in 2009. The results show that unphysical negative values for partial water vapor pressures are removed, without loosing, but in most cases even improving, the accuracy of the retrieved values, as compared with data from ECMWF and NCEP meteorological analysis, COSMIC data analysis as well, and RAOB balloons excursions suggests

Algebraic Boundaries Among Typical Ranks for Real Binary Forms of Arbitrary Degree

We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci.

Labels of real projective varieties

Let X be a complex projective variety defined over $${\mathbb {R}}$$. Recently, Bernardi and the first author introduced the notion of admissible rank with respect to X. This rank takes into account only decompositions that are stable under complex conjugation. Such a decomposition carries a label, i.e., a pair of integers recording the cardinality of its totally real part. We study basic properties of admissible ranks for varieties, along with special examples of curves; for instance, for rational normal curves admissible and complex ranks coincide. Along the way, we introduce the scheme theoretic version of admissible rank. Finally, analogously to the situation of real ranks, we analyze typical labels, i.e., those arising as labels of a full-dimensional Euclidean open set. We highlight similarities and differences with typical ranks.

Effects of calcium on the evolution of nitrogen during pyrolysis of a typical low rank coal

This study aims to investigate the effects of calcium on the migration of nitrogen in coal (coal-N) to N-containing gas species, particularly, NH3 and HCN (volatile-N) in volatiles, as well as the chemical transformation of the N in char during coal pyrolysis under different temperatures. The pyrolysis experiments of Shengli brown coal and its derived coal samples loaded with different contents of calcium were conducted under 600–800 °C in a novel fluidized bed reactor. The experimental results showed that during coal pyrolysis, the generation of NH3 is mainly derived from secondary reactions among volatiles, tar and char with the catalytic effect of mineral matter, especially calcium in coal. Increasing pyrolysis temperature from 600 to 800 °C could enhance the release of N in coal to volatiles. Meanwhile, the increased pyrolysis temperature could also inhibit the generation of NH3 while facilitating the formation of HCN. The release of HCN is more sensitive to pyrolysis temperatures. Specifically, under higher pyrolysis temperatures, more N-containing structures in coal would become thermally unstable and crack into HCN; On the other hand, higher pyrolysis temperature could also enhance the decomposition of N in coal to N-containing species in tar or N2, thus reducing the release of HCN and NH3. Nitrogen in tar could either undergo secondary decomposition reactions, generating NH3, HCN, N2 and other N-containing species in gas phase, or experience condensation polymerization by forming macromolecular structure and be retained in char at high pyrolysis temperatures. Calcium could significantly restrain the release of N from coal, thus reducing the yields of NH3 and HCN. During coal pyrolysis, calcium catalytically enhances the fracture and combination of chemical bonds, generating abundant free radicals. These free radicals could continuously attack N-containing structures and consequently release the N-containing gaseous products, such as NH3, HCN, N2 etc., resulting in the decrease of N in char. Calcium also plays important roles in nitrogen transformation in char during coal pyrolysis by catalytically intensifying the transformation of N in char from pyridinic nitrogen (N-6) and pyrrolic nitrogen (N-5) to quaternary type nitrogen (N-Q) during coal pyrolysis.

Typical ranks of semi-tall real 3-tensors

ABSTRACT Let m, n and p be integers with and . We showed in previous papers that if , then typical ranks of -tensors over the real number field are p and p+1 if and only if there exists a non-singular bilinear map . We also showed that the ‘if’ part is also valid in the case where . In this paper, we consider the case where and show that the typical ranks of -tensors over the real number field are p and p+1 in several cases including the case where there is no non-singular bilinear map . In particular, we show that the ‘only if’ part of the above mentioned fact is not valid for the case .

Universal City-size distributions through rank ordering

We consider a two-parameter discrete generalized beta (DGB) distribution and propose its universal applications to study the size-distribution of the urban agglomerations across various countries in the world, where the urban agglomerations include the small and mid-sized cities along with the heavily populated cities. Our proposition is validated by an exhaustive study with the 3 decades’ census data for India and China and census data of USA for a time window of 8 years. Moreover, we have studied the city size distributions for many different countries, like Brazil, Italy, Sweden, Australia, Uganda etc., from all the continents around the world according to the availability of the data. The detailed analyses exhibit a unique global pattern for the city size distributions, from low to high size, across the world with various geographic and economic conditions. Further analyses based on the entropy of the distribution provide insights on the underlying randomness and spreads of the city sizes within a country. The DGB distribution, a typical rank order (RO) distribution, through its two parameters, not only fits the data on wider range of city sizes better than the well-known power law for all the countries considered, it also helps us to characterize, discriminate and study their evolution over time.

On the algebraic boundaries among typical ranks for real binary forms

We describe the algebraic boundaries of the regions of real binary forms with fixed typical rank and of degree at most eight, showing that they are dual varieties of suitable coincident root loci.

Real rank boundaries and loci of forms

ABSTRACTIn this article, we study forbidden loci and typical ranks of forms with respect to the embeddings of given by the line bundles (2, 2d). We introduce the Ranestad–Schreyer locus corresponding to supports of non-reduced apolar schemes. We show that, in those cases, this is contained in the forbidden locus. Furthermore, for these embeddings, we give a component of the real rank boundary, the hypersurface dividing the minimal typical rank from higher ones. These results generalize to a class of embeddings of . Finally, in connection with real rank boundaries, we give a new interpretation of the hyperdeterminant.

On real typical ranks

We study typical ranks with respect to a real variety X. Examples of such are tensor rank (X is the Segre variety) and symmetric tensor rank (X is the Veronese variety). We show that any rank between the minimal typical rank and the maximal typical rank is also typical. We investigate typical ranks of n-variate symmetric tensors of order d, or equivalently homogeneous polynomials of degree d in n variables, for small values of n and d. We show that 4 is the unique typical rank of real ternary cubics, and quaternary cubics have typical ranks 5 and 6 only. For ternary quartics we show that 6 and 7 are typical ranks and that all typical ranks are between 6 and 8. For ternary quintics we show that the typical ranks are between 7 and 13.

Typical ranks of certain 3-tensors and absolutely full column rank tensors

In this paper, we study typical ranks of 3-tensors and show that there are plural typical ranks for tensors over in the following cases: (1) and , where is the Hurwitz–Radon function, (2) , and , (3) , , and , (4) , , and . (5) , and .

Effective Criteria for Specific Identifiability of Tensors and Forms

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for some third, fourth and sixth order symmetric tensors of small dimension as well as for binary forms of degree at least three. We extended this result to $4 \times 4 \times 4 \times 4$ symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank $8$, which is optimal.

Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals

Let m,n≥3, (m−1)(n−1)+2≤p≤mn, and u=mn−p. The set Ru×n×m of all real tensors with size u×n×m is one to one corresponding to the set of bilinear maps Rm×Rn→Ru. We show that Rm×n×p has plural typical ranks p and p+1 if and only if there exists a nonsingular bilinear map Rm×Rn→Ru. We show that there is a dense open subset O of Ru×n×m such that for any Y∈O, the ideal of maximal minors of a matrix defined by Y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal. Further, we show that there is a dense open subset T of Rn×p×m and continuous surjective open maps ν:O→Ru×p and σ:T→Ru×p, where Ru×p is the set of u×p matrices with entries in R, such that if ν(Y)=σ(T), then rankT=p if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal.

Semialgebraic Geometry of Nonnegative Tensor Rank

We study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. We show that nonnegative, real, and complex ranks are all equal for a general nonnegative tensor of nonnegative rank strictly less than the complex generic rank. In addition, such nonnegative tensors always have unique nonnegative rank-$r$ decompositions if the real tensor space is $r$-identifiable. We determine conditions under which a best nonnegative rank-$r$ approximation has a unique nonnegative rank-$r$ decomposition: For $r \le 3$, this is always the case; for general $r$, this is the case when the best nonnegative rank-$r$ approximation does not lie on the boundary of $D_r$. Many of our general identifiability results also apply to real tensors and real symmetric tensors.