Field of Science

Some geometrical aspects of 3- and 4-spaces

A couple of abstracts about the geomtery of space:
Historically, there have been many attempts to produce the appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's quaternions, Gibbs' vector system using the dot and cross products. We illustrate however, that Clifford's geometric algebra (GA) provides the most elegant description of physical space. Supporting this conclusion, we firstly show how geometric algebra subsumes the key elements of the competing formalisms and secondly we show how it provides an intuitive representation and manipulation of the basic concepts of points, lines, areas and volumes. We also provide two examples where GA has been found to provide an improved description of two key physical phenomena, electromagnetism and quantum theory, without using tensors or complex vector spaces. This paper also provides pedagogical tutorial-style coverage of the various basic applications of geometric algebra in physics.
James M. Chappell, Azhar Iqbal & Derek Abbott (2011). Geometric Algebra: A natural representation of three-space, arXiv:
We indicate that Heron's formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms, and other elementary 4-dimensional solids.
J. Scott Carter & David A. Mullens (2015). Some Elementary Aspects of 4-dimensional Geometry, arXiv:
There's also a minimalistic introduction to euclidean planes.

Water on Mars

"Our quest on Mars has been to 'follow the water', in our search for life in the universe, and now we have convincing science that validates what we've long suspected. This is a significant development, as it appears to confirm that water -- albeit briny -- is flowing today on the surface of Mars."
John Grunsfeld from the press release. The abstract of the paper follows:
Determining whether liquid water exists on the Martian surface is central to understanding the hydrologic cycle and potential for extant life on Mars. Recurring slope lineae, narrow streaks of low reflectance compared to the surrounding terrain, appear and grow incrementally in the downslope direction during warm seasons when temperatures reach about 250–300 K, a pattern consistent with the transient flow of a volatile species. Brine flows (or seeps) have been proposed to explain the formation of recurring slope lineae, yet no direct evidence for either liquid water or hydrated salts has been found4. Here we analyse spectral data from the Compact Reconnaissance Imaging Spectrometer for Mars instrument onboard the Mars Reconnaissance Orbiter from four different locations where recurring slope lineae are present. We find evidence for hydrated salts at all four locations in the seasons when recurring slope lineae are most extensive, which suggests that the source of hydration is recurring slope lineae activity. The hydrated salts most consistent with the spectral absorption features we detect are magnesium perchlorate, magnesium chlorate and sodium perchlorate. Our findings strongly support the hypothesis that recurring slope lineae form as a result of contemporary water activity on Mars.

Ojha, L., Wilhelm, M., Murchie, S., McEwen, A., Wray, J., Hanley, J., Massé, M., & Chojnacki, M. (2015). Spectral evidence for hydrated salts in recurring slope lineae on Mars Nature Geoscience DOI: 10.1038/ngeo2546

Freedom and truth in mathematics

The very essence of #mathematics is its freedom. (Georg #Cantor)
The way we deal with today's numbers in schools is essentially the same manner used by our ancestors Pythagoreans, who saw the numbers as concrete objects, of course, but in a way that prevented them from conceiving the infinity. The only ancient mathematician who approached the infinity was Archimedes, but in the history of mathematics can be considered a fairly unique case of lack of development mainly due to the isolation of the mathematicians at that time and of noticeable difference in quality between the Sicilian and colleagues. In order to return to touch the wall of infinity and use them in a profitable way the Earth had to wait the arrival of Georg Cantor.
The German mathematician actually faced numbers, revolutionizing mathematics, using essentially sets and logic, two tools that enabled him not only to approach, but even manipulate the infinite thanks to the transfinite numbers. Leading his steps was probably the following conviction:
The very essence of mathematics is its freedom.
According to Daniel Bonevac, this veritable mantra, written in 1883, is emblematic of the Cantor's libertarian approach to mathematics. With this milestone, Bonevac try to write a theory of mathematical truth, in order to explain some facts more or less established:
1) that the mathematical statements are either necessarily true or necessarily false;
2) that mathematical truth is derived primarily from logical truth;
3) that the existence in mathematics involves a kind of modality, which requires only the consistency or the constructability.

Hyperbolic Pascal triangles and other stories

A new set of mathematical abstracs. We start with the hyperbolic Pascal trianlges:

Fibonacci and Pell sequences in the hyperbolic Pascal triangle
In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearence of some binary recurrences in a fixed hyperbolic triangle are investigated.
Hacene Belbachir, László Németh & László Szalay (2015). Hyperbolic Pascal triangles, arXiv:

Alexander Gerst's timelapse

Watch Earth roll by through the perspective of ESA astronaut Alexander Gerst in this six-minute timelapse video from space. Combining 12 500 images taken by Alexander during his six-month Blue Dot mission on the International Space Station this Ultra High Definition video shows the best our beautiful planet has to offer.
Marvel at the auroras, sunrises, clouds, stars, oceans, the Milky Way, the International Space Station, lightning, cities at night, spacecraft and the thin band of atmosphere that protects us from space.
Often while conducting scientific experiments or docking spacecraft Alexander would set cameras to automatically take pictures at regular intervals. Combining these images gives the timelapse effect seen in this video.
Watch the video in 4K resolution for the best effect and find out more about Alexander Gerst's Blue Dot mission.

Hints of physics behind standard model?

The LHCb collaboration is studying the decay of mesons $B$ in order to find some violations in standard model rules. In particular LHCb has measured a particular ratio, named $R (D^*)$, between two decay modes of $\overline{B}^0$ and they find a violation from the standard model prediction that is compatible with other similar measures:
In the SM all charged leptons, such as taus ($\tau$) or muons ($\mu$), interact in an identical fashion (or, in physicists' language, have the same "couplings"). This property is called "lepton universality". However, differences in mass between the leptons must be accounted for, and affect decays involving these particles. The $\tau$ lepton is much heavier than the $\mu$ lepton and therefore the SM prediction for the ratio $R(D^*)$ is substantially smaller than 1. This ratio is considered to be precisely calculable thanks to the cancellation of uncertainties associated with the $B$ to $D^*$ meson transition.
But there is another hint of new physics. At the end of July Nature Physics published a new paper from the LHCb collaboration about the possible existence of a new particle:
The LHCb collaboration published in Nature Physics a paper based on run 1 data which reports the determination of the parameter $|V_{ub}|$ describing the transition of a $b$ quark to a $u$ quark. This measurement was made by studying a particular decay of the $\Lambda_b^0$ baryon. Other measurements of $|V_{ub}|$ by previous experiments had returned two sets of inconsistent results, depending on which method was used to determine the parameter. Theorists had suggested that this discrepancy could be explained by the presence a new particle contributing to the decay process, which affected the result differently, depending on the measurement method. Today's result from LHCb removes the need for this new particle, while the puzzle of why the original sets of measurements do not agree persists.
where $|V_{ub}|$ is connected to the Cabibbo-Kobayashi-Maskawa matrix.

Learning abstracts: from mathematics, to the neural trasnformation machines

A coulpe of abstracts about e-learning:
This paper presents a new framework for adding semantics into e-learning system. The proposed approach relies on two principles. The first principle is the automatic addition of semantic information when creating the mathematical contents. The second principle is the collaborative tagging and annotation of the e-learning contents and the use of an ontology to categorize the e-learning contents. The proposed system encodes the mathematical contents using presentation MathML with RDFa annotations. The system allows students to highlight and annotate specific parts of the e-learning contents. The objective is to add meaning into the e-learning contents, to add relationships between contents, and to create a framework to facilitate searching the contents. This semantic information can be used to answer semantic queries (e.g., SPARQL) to retrieve information request of a user. This work is implemented as an embedded code into Moodle e-learning system.
Iyad Abu Doush, Faisal Alkhateeb, Eslam Al Maghayreh, Izzat Alsmadi & Samer Samarah (2012). Annotations, Collaborative Tagging, and Searching Mathematics in E-Learning, arXiv:
As far as Learning Management System is concerned, it offers an integrated platform for educational materials, distribution and management of learning as well as accessibility by a range of users including teachers, learners and content makers especially for distance learning. Usability evaluation is considered as one approach to assess the efficiency of e-Learning systems. It is used to evaluate how well technology and tools are working for users. There are some factors contributing as major reason why LMS is not used effectively and regularly. Learning Management Systems, as major part of e-Learning systems, can benefit from usability research to evaluate the LMS ease of use and satisfaction among its handlers. Many academic institutions worldwide prefer using their own customized Learning Management Systems; this is the case with Moodle, an open source Learning Management Systems platform designed and operated by most of the universities in Sri Lanka. This paper gives an overview of Learning Management Systems used in Sri Lankan universities, and evaluates its usability using some pre-defined usability standards. In addition it measures the effectiveness of Learning Management System by testing the Learning Management Systems. The findings and result of this study as well as the testing are discussed and presented.
Selvarajah Thuseethan, Sivapalan Achchuthan & Sinnathamby Kuhanesan (2015). Usability Evaluation of Learning Management Systems in Sri Lankan Universities, Global Journal of Computer Science and Technology, 15 (1) arXiv:
And the last abstract about the learning processo in neural network:
We propose Neural Transformation Machine (NTram), a novel architecture for sequence-to-sequence learning, which performs the task through a series of nonlinear transformations from the representation of the input sequence (e.g., a Chinese sentence) to the final output sequence (e.g., translation to English). Inspired by the recent Neural Turing Machines [8], we store the intermediate representations in stacked layers of memories, and use read-write operations on the memories to realize the nonlinear transformations of those representations. Those transformations are designed in advance but the parameters are learned from data. Through layer-by-layer transformations, NTram can model complicated relations necessary for applications such as machine translation between distant languages. The architecture can be trained with normal back-propagation on parallel texts, and the learning can be easily scaled up to a large corpus. NTram is broad enough to subsume the state-of-the-art neural translation model in [2] as its special case, while significantly improves upon the model with its deeper architecture. Remarkably, NTram, being purely neural network-based, can achieve performance comparable to the traditional phrase-based machine translation system (Moses) with a small vocabulary and a modest parameter size.
Fandong Meng, Zhengdong Lu, Zhaopeng Tu, Hang Li & Qun Liu (2015). Neural Transformation Machine: A New Architecture for Sequence-to-Sequence Learning, arXiv: