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Research Lines

Research Lines.

The main fields of activity at IUMPA are Algebra, Dynamical Systems, Fuzzy Mathematics, Geometry, Mathematical Analysis, Mathematical Methods in Synthetic Biology, Mathematical Physics, Numerical Methods, Operations Research, Photonics and Topology

Algebra is an important branch of mathematics concerning with abstract structures that emerge as a generalization of ordinary arithmetic operations and their invariants. Although abstract algebra has being traditionally considered an area of pure mathematics, it presents various applications in other branches of science and technology such as physics, chemistry, computer science or more specifically cryptography and code data transmission.

IUMPA has a group of researchers working on Algebra who are dealing mainly with group theory and its applicactions, supported by a competitive project from the Spanish Ministry (MICINN), coordinated with other groups of the Universitat de València and the Universidad de Zaragoza. Some of these researchers also collaborate and are involved with other external research projects.

Essentially, the main interest of the algebra team focuses on the internal structure of finite or infinite groups (with finite conditions) such as their actions, their representations, the lattices of their subgroups and arithmetical properties. The interest from the applied side is the interaction of groups and semigroups with current problems in the context of automata and formal languages.

IUMPA has a group of researchers working on Algebra who are dealing mainly with group theory and its applicactions, supported by a competitive project from the Spanish Ministry (MICINN), coordinated with other groups of the Universitat de València and the Universidad de Zaragoza. Some of these researchers also collaborate and are involved with other external research projects.

Essentially, the main interest of the algebra team focuses on the internal structure of finite or infinite groups (with finite conditions) such as their actions, their representations, the lattices of their subgroups and arithmetical properties. The interest from the applied side is the interaction of groups and semigroups with current problems in the context of automata and formal languages.

**Groups:**Algebra Research Group (Group Theory)

Dynamical systems is the study of the long-term behaviour of evolving systems. The modern theory of dynamical systems originated with fundamental questions concerning the stability and evolution of the solar system, attempts to answer the asymptotic behaviour of both discrete and continuous systems.Although dynamical systems and linear chaos are still relatively young, there is no doubt that the field is becoming more and more important. Currently, IUMPA has two groups of researchers working in various branches of this field and two research projects of the MEC. These two projects are mainly concerned with hypercyclicity, linear chaos of operators and semigroups, ergodic theory for operators, local singularities and singularities of differential applications.

Fuzzy mathematics is related to fuzzy set theory, fuzzy logic and fuzzy topology. It is connected with many other branches in mathematics and engineering. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work on Fuzzy sets.Fuzzy topology was one of the first subjects studied in this area. One of the most interesting problems that appeared in this area was to obtain a formal and acceptable definition of a fuzzy metric space. Many people have studied these spaces from different points of view. The concept of fussy metric was coined by George and Veeramani. This definition is adequate and deserves special attention because the class of topological metric spaces agrees with the class of George Veeramnai fuzzy metric spaces.
In recent years, the interest in using coloured images has increased significantly in a wide range of applications. Studies on how to apply fuzzy metrics to processing images and construction of new filters for coloured images have been carried out.

Geometry is a wide field of mathematics and provides essential tools for the development of several theories inside and outside mathematics.

We are mainly concerned about the study of the singularities of maps of several complex variables defined over complex analytic varieties, therefore our research topics belong to singularity theory and intersect fields like algebraic geometry, complex analytic geometry and commutative algebra.

We study several kinds of algebraic invariants attached to maps defined over varieties, we are interested in the relation between these invariants and the effective computation of them. We also analyze the meaning of the constancy of such invariants in deformations of maps in terms of several equivalence relations of maps (for instance, topological trivialy, Whitney equisingularity and bi-Lipschitz equivalence).

We are mainly concerned about the study of the singularities of maps of several complex variables defined over complex analytic varieties, therefore our research topics belong to singularity theory and intersect fields like algebraic geometry, complex analytic geometry and commutative algebra.

We study several kinds of algebraic invariants attached to maps defined over varieties, we are interested in the relation between these invariants and the effective computation of them. We also analyze the meaning of the constancy of such invariants in deformations of maps in terms of several equivalence relations of maps (for instance, topological trivialy, Whitney equisingularity and bi-Lipschitz equivalence).

This branch of mathematics deals with functions of real or complex variable.One of the fundamental tools in this area is the concept of limit. Mathematical analysis began its development due to the rigorous formulation of differential and integral calculus proposed by Newton and Leibniz. Its evolution dates back to 17

The research closely related to the digitalization of signals is analysis of time-frequency, Fourier analysis, wavelets and their applications to processing signals and image filters.^{th}century and is connected with problems in physics. Although mathematical analysis is an area of theoretical research, it has recently undergone a profound transformation, and nowadays it is a useful tool for digital image process and resizing or scaling process. Large and well-recognized leading experts work at IUMPA on different areas of mathematical analysis. They are in charge of six research projects of the Ministry of Education and Science (MEC). The research interests of the group span topics such as real and complex analysis, harmonic analysis and especially functional analysis and operator theory. In the more theoretical aspect many abstract spaces are studied, for instance: locally convex, vector measure, spaces of continuous, differentiable or analytic functions and Fréchet spaces. Also, the geometry of Banach spaces, partial linear differential equations, hypercyclicity, linear chaos of operators and semigroups, descriptive set topology and infinite-dimensional holomorphy are studied.The mathematical modelling group works at IUMPA on developing mathematical models that describe heat transfer processes. As it is known, the basic equation that rules these processes is the equation of heat, which is a differential equation with second partial derivatives of parabolic type. There are different reformulations and modifications of the same equation. For example, the addition of convective terms allows us to study the mass transfer phenomena; and the addition of a second derivative with respect to time turns the parabolic equation into a hyperbolic one which describes heat waves.
There are two examples of models that are developed in this area and related to the analysis of geothermal heating and industrial grinding parts. The first one refers to the usual air-conditioning systems in buildings, which use refrigeration/heating units situated in roofs or terraces that operates on transferring or removing heat in the air environment. Opposite to this, there exist the geothermal systems which extract or transfer heat to the soil surrounding of the building through a buried water circuit. This technology allows huge energy savings compared to conventional air conditioning. Grinding metal parts through grinding wheel has great industrial interest. This process generates large quantities of heat by friction between the grinding wheel and the piece.
Mathematical modelling area at IUMPA deals with such problems by means of differential equations. This area analyses mathematical aspects such as the existence and stability of the solutions, the spectrum, and properties of the associated Cauchy problem which, from a technological point of view, leads to the optimization of air conditioning and thermal grinding techniques.

**Research Groups:**Interdisciplinary Modelling Group- InterTech

The numerical method group develops algorithms and computer codes to simulate and study the transfer of heat and mass problems; problems of electromagnetic propagation in photonic systems or to optimise process in biological systems.The numerical tools that are developed can be applied to the simulation of complex physical problems (by means of Monte Carlo techniques) and to the resolution of ordinary differential equations and linear and nonlinear partial deferential equations using methods of discretization (finite differences and finite elements) and modal techniques.
Very often, the computational needs of these codes require supercomputing strategies in order to be executed (parallel codes / grid environments). This fact has increased the close collaboration with the Institute of Information Technology Applications at the UPV, and the participation of IUMPA in the HPC-Europe.

**Research Groups:**Interdisciplinary Modelling Group- InterTechThis line of research is currently working on two areas: the combinatorial optimization and general systems theory.The first area focuses on solving and modelling some real optimization vehicle routing problems, considering factors such as the capacity of vehicles, forbidden or penalized turns, time-dependent costs, etc., whith the help of the Graph Theory. In the last years, the operational research group has maintained a close cooperation with several departments related to the area of computer science, by applying its ideas about routes and graph to the fast-developed wireless network communication technology, with the purpose to obtain better network protocols of communication. General theory of systems for the study of demography is adapting von Foerster-McKendrick?s models considering factors such as fertility, death and migration to relate them with quality of life variables, which are considered as real engines of a demographic change.
In collaboration with researchers from other departments, the operational research group has begun to study the dynamics of the human brain. The observation of the change in the time variables and the level of activation allows to construct models of coupled differential equations. These equations describe the brain dynamics and they can be used also to find methods of intervention to solve pathologies of personality.

**Research Groups:**Vehicle Routing Optimization and General Systems Theory Photonics deals with mathematical modelling and numerical simulation of phenomena associated with the propagation of light waves and matter. Especially, it focuses on nonlinear phenomena in micro and nano scales. Its research can be divided into two large blocks: photonics and nonlinear nanophotonics; and cold atoms and matter waves. Both areas have a great technological potential and are seen as basic oriented research.

Problems of nonlinear wave propagation of light in micro and nanostructured spaces (on either temporary or spatial domain) have a great technological interest. The replacement of the present electronic technology with a new technology based on photonic circuits is one of the great challenges of nanotechnology. This area develops both mathematical models and innovative numerical algorithms, in order to deal efficiently

The field of cold atoms and matter waves includes the study of the behaviour of matter under temperatures near absolute zero. This framework upraises new quantum phenomena, analysed by nonlinear equations (similar to optics nonlinear equations). The present experimental control of these systems at low temperatures allows foreseeing new technological applications arising from quantum nature of matter at the atomic scale. Concepts such as nanotechnology or quantum computing are involved in these applications. In this way, our experience in nonlinear optical modelling is used to propose new ideas to this promising field.

Problems of nonlinear wave propagation of light in micro and nanostructured spaces (on either temporary or spatial domain) have a great technological interest. The replacement of the present electronic technology with a new technology based on photonic circuits is one of the great challenges of nanotechnology. This area develops both mathematical models and innovative numerical algorithms, in order to deal efficiently

**and creatively with the analysis of these systems.**The field of cold atoms and matter waves includes the study of the behaviour of matter under temperatures near absolute zero. This framework upraises new quantum phenomena, analysed by nonlinear equations (similar to optics nonlinear equations). The present experimental control of these systems at low temperatures allows foreseeing new technological applications arising from quantum nature of matter at the atomic scale. Concepts such as nanotechnology or quantum computing are involved in these applications. In this way, our experience in nonlinear optical modelling is used to propose new ideas to this promising field.

**Research Groups:**Interdisciplinary Modelling Group- InterTech

Many of the actual challenges faced by our modern world have such scope and complexity that they require mathematicians collaborating with experts from other domains to make significant progress. In this context, a new inherently multidisciplinary area of great potential and high strategic value appears: Synthetic Biology. The prior objective of synthetic biology is designing and implementing new applications of biological systems by means of mathematical modelling methods and simulation as main elements of the design and implementation of new ideas enabling the systematic introduction of quantification and prediction in the system.In fact, synthetic biology combines methods of mathematical modelling and biology to form a research area which aims to introduce new biological circuits (built of proteins and nucleic acids) in cells; using a standardization process similar to the one used in electronics. Thus such cells can be transformed into small programmed biological robots to perform predetermined tasks.
Researchers of IUMPA maintain an intense activity in this area since many resources have been dedicated to this new line of research due to its participation in two important European projects.

**Research Groups:**Interdisciplinary Modelling Group- InterTech

General topology is the branch of mathematics which formalizes and explores the ideas of approximity and limit. For this reason, it is a basic tool for the development of many other branches of mathematics.

Its origin lies in Cantor papers published between 1879 and 1884 concerning with problems of uniqueness for trigonometric series. At the beginning of the 20th century, Fréchet and Riesz introduced independently, the concept of abstract space endowed with a topological structure. However the first satisfactory definition about topological spaces using the concept of neighbourhoods was given by Hausdorfff in 1914.

Until the middle of the 20th century, metric compact spaces and uniform spaces attracted the attention of the researchers in this area. In 1962, Pervin proved that every topological space is quasiuniformable. This important result increased significantly the interest in asymmetric topological structures (quasiuniformities, quasimetrics, quasinorms, etc.). In recent years, these structures have had interesting applications in algorithmic complexity, computational programming, computational biology, etc.

IUMPA has a group of researchers dedicated to topology and its applications with two existing competitive projects (MEC, coordinated with a group of the “Universidad del País Vasco” and other group funded by “Generalitat Valenciana”). The main research topics are: asymmetric structures, the hyperspaces, metrics and fuzzy quasimetrics, as well as connections to algebraic structures and applications to various fields of the computer science (complexity of algorithms and programs, semantics, databases, etc.).

Its origin lies in Cantor papers published between 1879 and 1884 concerning with problems of uniqueness for trigonometric series. At the beginning of the 20th century, Fréchet and Riesz introduced independently, the concept of abstract space endowed with a topological structure. However the first satisfactory definition about topological spaces using the concept of neighbourhoods was given by Hausdorfff in 1914.

Until the middle of the 20th century, metric compact spaces and uniform spaces attracted the attention of the researchers in this area. In 1962, Pervin proved that every topological space is quasiuniformable. This important result increased significantly the interest in asymmetric topological structures (quasiuniformities, quasimetrics, quasinorms, etc.). In recent years, these structures have had interesting applications in algorithmic complexity, computational programming, computational biology, etc.

IUMPA has a group of researchers dedicated to topology and its applications with two existing competitive projects (MEC, coordinated with a group of the “Universidad del País Vasco” and other group funded by “Generalitat Valenciana”). The main research topics are: asymmetric structures, the hyperspaces, metrics and fuzzy quasimetrics, as well as connections to algebraic structures and applications to various fields of the computer science (complexity of algorithms and programs, semantics, databases, etc.).

**Research Groups:**Topology and its Applications

Wave Propagation in Periodic Media and Acoustics

Phononic crystals are periodic distributions of two elastic materials; however, in the case when one of these materials is a fluid, the system is called a sonic crystal (SC).Over the last 20 years, the exploitation of the particular dispersion relation of these structures has revealed very interesting physical properties. The existence of ranges of frequencies, called bandgaps (BGs), in which non-propagating modes can be excited in the system, has been observed in a wide range of frequencies due to the scalability of the systems. The explanation of these bandgaps implies the resolution of a complex eigenvalues problem for the Helmholtz equation.
The complex part of the eigenvalues has physical existence: they correspond to evanescent waves. The evanescent properties in periodic composites have shown several interesting possibilities as imaging with super-resolution. Other properties of the dispersion relation have been used to control the wave propagation through these periodic structures as the control the spatial dispersion of the waves inside the periodic structures in order to obtain both the self-collimating effect, negative refraction,…
On the other hand our group have worked intensively on the optimization of this systems using heuristic techniques as multiobjective genetic algorithms, allowing the design of tailored filters. Now our work is focused on another type of distributions of elastic materials, as the ones based on fractal geometry o quasi-uniform structures. Also, a big effort has been done in the funcional approximation of signals coming from acoustics experiments and the mathematical analysis of engineering devices based on Sonic Crystal, more specifically Sonic Crystal Acoustic Barriers (SCAB) that have been acoustically standardized as traffic noise reducing device.

**Research Group:**FUNAPHY