There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A fundamental component of the European natural disaster management policy is the detection of potential flood-prone areas, which is directly connected to the European Directive (2007/60). This study presents a framework for mapping potential flooding areas incorporating geographic information systems (GIS), fuzzy logic and clustering techniques, and multi-criteria evaluation methods. Factors are divided in different groups which do not have the same level of trade off. These groups are related to geophysical, morphological, climatological/meteorological and hydrological characteristics of the basin as well as to anthropogenic land use. GIS and numerical simulation are used for geographic data acquisition and processing. The selected factor maps are considered in order to estimate the spatial distribution of the potential flood prone areas. Using these maps, the study area is classified into five categories of flood vulnerable areas. The Multi-Criteria Analysis (MCA) techniques consist of the crisp and fuzzy analytical hierarchy processes (AHP) and are enhanced with different standardization methods. The classification is based on different clustering techniques and it is applied in two approaches. In the first approach, all criteria are normalized before the MCA process and then, the clustering techniques are applied to derive the final flood prone area maps. In the second approach, the criteria are clustered before and after the MCA process for the potential flood prone area mapping. The methodology is demonstrated in Xerias River watershed, Thessaly region, Greece. Xerias River floodplain was repeatedly flooded in the last few years. These floods had major impacts on agricultural areas, transportation networks and infrastructure. Historical flood inundation data has been used for the validation of the methodology. Results show that multiple MCA techniques should be taken into account in initial low-cost detection surveys of flood-prone areas and/or in preliminary analysis of flood hazard mapping.

Multi-Criteria Analysis Framework for Potential Flood Prone Areas Mapping

In the last decades the Moore-Penrose pseudoinverse has found a wide range of applications in many areas of Science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the solution of linear integral equations, etc. The existence of such applications alone should attract the interest of students and researchers in the Moore-Penrose pseudoinverse and in related sub jects, like the singular values decomposition theorem for matrices. In this note we present a tutorial review of the theory of the Moore-Penrose pseudoinverse. We present the first definitions and some motivations and, after obtaining some basic results, we center our discussion on the Spectral Theorem and present an algorithmically simple expression for the computation of the Moore-Penrose pseudoinverse of a given matrix. We do not claim originality of the results. We rather intend to present a complete and self-contained tutorial review, useful for those more devoted to applications, for those more theoretically oriented and for those who already have some working knowledge of the sub ject.

The Moore-Penrose Pseudoinverse. A Tutorial Review of the Theory

In the last decades, the Moore–Penrose pseudoinverse has found a wide range of applications in many areas of science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the solution of linear integral equations, etc. The existence of such applications alone should attract the interest of students and researchers in the Moore–Penrose pseudoinverse and in related subjects, such as the singular value decomposition theorem for matrices. In this note, we present a tutorial review of the theory of the Moore–Penrose pseudoinverse. We present the first definitions and some motivations, and after obtaining some basic results, we center our discussion on the spectral theorem and present an algorithmically simple expression for the computation of the Moore–Penrose pseudoinverse of a given matrix. We do not claim originality of the results. We rather intend to present a complete and self-contained tutorial review, useful for those more devoted to applications, for those more theoretically oriented, and for those who already have some working knowledge of the subject.

The Moore-Penrose Pseudoinverse: A Tutorial Review of the Theory

Der speziellen Relativitatstheorie liegt folgendes Postulat zugrunde, welchem auch durch die Galilei-Newtonsche Mechanik Genuge geleistet wird: Wird ein Koordinatensystem K so gewahlt, das in bezug auf dasselbe die physikalischen Gesetze in ihrer einfachsten Form gelten, so gelten dieselben Gesetze auch in Bezug auf jedes andere Koordinatensystem K′, das relativ zu K in gleichformiger Translationsbewegung begriffen ist. Dieses Postulat nennen wir „spezielles Relativitatsprinzip“. Durch das Wort „speziell“ soll angedeutet werden, das das Prinzip auf den Fall beschrankt ist, das K′ eine gleichformige Translationsbewegung gegen K ausfuhrt, das sich aber die Gleichwertigkeit von K′ und K nicht auf den Fall ungleichformiger Bewegung von K′ gegen K erstreckt.

Die Grundlage der allgemeinen Relativitätstheorie

In this paper, we propose enhancements to Beetle Antennae search &#x0028 BAS &#x0029 algorithm, called BAS-ADAM, to smoothen the convergence behavior and avoid trapping in local-minima for a highly non-convex objective function. We achieve this by adaptively adjusting the step-size in each iteration using the adaptive moment estimation &#x0028 ADAM &#x0029 update rule. The proposed algorithm also increases the convergence rate in a narrow valley. A key feature of the ADAM update rule is the ability to adjust the step-size for each dimension separately instead of using the same step-size. Since ADAM is traditionally used with gradient-based optimization algorithms, therefore we first propose a gradient estimation model without the need to differentiate the objective function. Resultantly, it demonstrates excellent performance and fast convergence rate in searching for the optimum of non-convex functions. The efficiency of the proposed algorithm was tested on three different benchmark problems, including the training of a high-dimensional neural network. The performance is compared with particle swarm optimizer &#x0028 PSO &#x0029 and the original BAS algorithm.

/pdf/bas-adam-an-adam-based-approach-to-improve-the-performance-1mel0kdceb.pdf

BAS-ADAM: an ADAM based approach to improve the performance of beetle antennae search optimizer

An improved method for the computation of the Moore―Penrose inverse matrix

In this article we provide a fast computational method in order to calculate the Moore-Penrose inverse of singular square matrices and of rectangular matrices. The proposed method proves to be much faster and has significantly better accuracy than the already proposed methods, while works for full and sparse matrices.

An improved method for the computation of the Moore-Penrose inverse matrix

Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The results are verified using specific algebraic approach, based on proposed definitions and previously verified properties. The approach used here is new even in the matrix case.

Vasilios N. Katsikis

Papers

BAS-ADAM: an ADAM based approach to improve the performance of beetle antennae search optimizer

An improved method for the computation of the Moore―Penrose inverse matrix

An improved method for the computation of the Moore-Penrose inverse matrix

Core and core-EP inverses of tensors

MATLAB - A Fundamental Tool for Scientific Computing and Engineering Applications - Volume 3