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The need to carry out experiments and physical tests with models is today questioned with the use of the computer; which constitutes an important optimization and a shape search tool. But to what extent? Frei Otto, a German architect born in 1925 in Siegmar, devoted much of his career to the study of shapes and materials from experimental models. Experimentation in architecture was the central axis of his work. Although the principles of these models cannot be applied directly to buildings, they served a pure research purpose and offered a theoretical point of view in the design of structures. He established an unprecedented working methodology. His experimental models allowed him a further understanding of how materials work at their best when they are under tension, the soap film minimal surface models for tensile membranes and the hanging chain catenary form models for compressive arch and shell structures. This allowed him not only to find an optimal lightweight structure, but also to have a more environmentally friendly approach to architecture. This research work interrogates, reads, interprets and reconstructs Frei Otto’s soap film models in order to find a method of working with digital tools that allows us to deepen the study of these forms. After this study, both methods, analog and digital, will be compared to see which one is better to work with nowadays, Award-winning

"The 3rd Japan-Taiwan Joint Conference on Differential Geometry" was held on November 1-3, 2021. This volume records the abstracts and the slides of talks presented in this conference.

"The 3rd Japan-Taiwan Joint Conference on Differential Geometry" was held on November 1-3, 2021. This volume records the abstracts and the slides of talks presented in this conference.

"The 3rd Japan-Taiwan Joint Conference on Differential Geometry" was held on November 1-3, 2021. This volume records the abstracts and the slides of talks presented in this conference.

"The 3rd Japan-Taiwan Joint Conference on Differential Geometry" was held on November 1-3, 2021. This volume records the abstracts and the slides of talks presented in this conference.

Almost every tissue in the human body is curved in a certain way. Examples are the extracellular matrix of different tissues such as trabecular bone, different acini and blood vessels. As such, the ability to create complex curved structures is crucial in the development of biomimetic biomaterials that could be used, for example, for tissue regeneration purposes. Currently, most of these different curved substrates and porous materials are fabricated with 3D-printing techniques. These techniques, however, have limitations. The 3D-printed structures, for example, have limited resolution and the production process is not compatible with planar functionality-inducing processes. A solution for these problems could be the concept of shape-shifting. Shape-shifting is the process through which an object transforms itself into a different shape under the influence of an external stimulus, such as temperature or light. A special interest goes to shape-shifting of initially flat materials (2D) into different complex 3D structures. This method has as main advantage that planar printing, patterning or other 2D processing techniques can be used on the planar (non-shape-shifted) state. In this research, the most important principles of shape-shifting of curved hyperbolic surfaces are explored. With this technique, the benefits of both hyperbolic surfaces and shape-shifting can be combined and exploited. A new way of hyperbolic shape-shifting is introduced. This is done by using a passive rigid frame and an active shape memory polymer (SMP). The passive material determines how and where the structure will fold, while the active SMP generates the force in order to fold and forms a curved (hyperbolic) surface spanned between the frame. A simple square patch design consisting of four rigid beams and an SMP was used as the basis of this research. When activated, this patch forms a saddle shaped (hyperbolic) surface. The design, activation and materials of the patch were changed and ma, Biomedical Engineering

Almost every tissue in the human body is curved in a certain way. Examples are the extracellular matrix of different tissues such as trabecular bone, different acini and blood vessels. As such, the ability to create complex curved structures is crucial in the development of biomimetic biomaterials that could be used, for example, for tissue regeneration purposes. Currently, most of these different curved substrates and porous materials are fabricated with 3D-printing techniques. These techniques, however, have limitations. The 3D-printed structures, for example, have limited resolution and the production process is not compatible with planar functionality-inducing processes. A solution for these problems could be the concept of shape-shifting. Shape-shifting is the process through which an object transforms itself into a different shape under the influence of an external stimulus, such as temperature or light. A special interest goes to shape-shifting of initially flat materials (2D) into different complex 3D structures. This method has as main advantage that planar printing, patterning or other 2D processing techniques can be used on the planar (non-shape-shifted) state. In this research, the most important principles of shape-shifting of curved hyperbolic surfaces are explored. With this technique, the benefits of both hyperbolic surfaces and shape-shifting can be combined and exploited. A new way of hyperbolic shape-shifting is introduced. This is done by using a passive rigid frame and an active shape memory polymer (SMP). The passive material determines how and where the structure will fold, while the active SMP generates the force in order to fold and forms a curved (hyperbolic) surface spanned between the frame. A simple square patch design consisting of four rigid beams and an SMP was used as the basis of this research. When activated, this patch forms a saddle shaped (hyperbolic) surface. The design, activation and materials of the patch were changed and ma, Biomedical Engineering

Cílem diplomové práce je vyhodnocení pevnosti v tlaku 3D tištěných těles, jejichž vnitřní výplň je tvořena strukturou minimálních ploch. Vybráno bylo pět struktur trojitě periodických minimálních ploch, TPMP (Triply Periodic Minimal Surface, TPMS), které tvoří výplně pro tři tvary těles. Tato tělesa byla podrobena zkoušce v tlaku a sledovány byly změny pevnosti a modulu pružnosti vůči tělesu pevného bloku stejného tvaru, vůči voštinové výplni a mezi jednotlivými tvary a strukturami navzájem. Jednotlivé struktury minimálních ploch byly navíc tištěny samostatně jako části nekonečné mříže a podrobeny tlakové zkoušce s pozorováním deformace. 3D tisk probíhal na komerční FDM/FFF 3D tiskárně Flashforge Dreamer. Za materiál byl zvolen terpolymer ASA (akrylonitril-styren-akrylát), jako vhodný zástupce konstrukčních termoplastů pro FDM/FFF 3D tisk., The goal of this Diploma thesis is evaluation of the compressive strength of 3D printed objects with infill built by Triply Periodic Minimal Surface structures. Five Minimal Surface structures as an infill for three solids shapes were chosen. Thees objects had been tested in the compressive strength and the changes of the strength and modulus had been compered with the solid shapes with solid infill, with Honeycomb infill and between every Minimal Surface structure and the solid bulk shape. In addition the parts of an infinite lattice of Minimal surface was printed and tested in compressive strength and the deformation was visually determined. 3D print was realized on Flashforge Dreamer FDM/FFF 3D printer. Thermoplastic terpolymer ASA (acrylonitrile-styrene-acrylate) was chosen as an example of the constructional plastic for the FDM/FFF 3D printing., Fakulta chemicko-technologická, Diplomant přednesl výsledky své diplomové práce a zodpověděl dotazy členů komise., Dokončená práce s úspěšnou obhajobou

Two-dimensional (2D) materials such as graphene have been a topic of intense study as these atomically thin materials have shown enhanced characteristics ideal for applications such as renewable energy storage and electrochemical sensing, which has led researchers into a frenzy akin to that of prospectors during the gold rush. While there still is gold to be found, there is also fool’s gold when systematic studies are not performed. In order to develop new processes for fabricating 3D high surface area architectures, we investigated mechanisms at a fundamental level throughout this work. It is known that the surfaces of freestanding 2D graphene sheets exhibits record-breaking electrical and thermal conductivity, remarkable chemical stability, mechanical strength, and naturally high specific surface area, however, current 2D techniques lay the graphene sheets onto substrates, typically decreasing electron transport and always reducing accessible surface area. In order to use sides of a graphene sheet for applications that require high surface area, we took inspiration from origami. A single paper cannot support itself, but when it becomes part of a 3-dimensional (3D) structure, support becomes natural. The first phase of my work involved the use of minimal surface inspired 3D nickel templates to create free-standing graphene structures with inherited morphology, resulting in porous well-defined sponge-like structures of free-standing graphene. The concept of graphene shaped into minimal surfaces architectures is of great interest; theories predict unique mechanical properties, such as a young’s modulus of 1 TPa for single crystal graphene and an exceptionally high stiffness for minimal surface architectures. Few studies have been performed of fabricating graphene architectures with well defined pore sizes, co-continuous pores, and controlled layer thickness needed for enhancing mass transport and electronic conductivity. Studies were also performed involving atomisti

Triply Periodic Minimal Surfaces (TPMS) have earned great popularity in porous meta- biomaterials by cause of unique mass transfer properties, tunable mechanical properties, large pore space and high surface to volume ratio. Numerical and experimental studies suggest a great potential for TPMS-based architectures in the treatment of segmental bone defects, where supporting scaffolds are crucial in bridging the gap unable to heal naturally. However, current studies are limited to investigating the properties of classic TPMS archi- tectures. In this thesis, I present the design, fabrication and characterisation of a novel type of meta-biomaterials inspired by TPMS morphology. I show a practical application of Weierstrass equations to define Schwarz P, Schwarz D and Gyroid surfaces, which pro- vides an opportunity to remove fundamental patches from the original surface and design unique structures. Moreover, one of the developed designs applied basics of graph theory to construct the material which mimics the randomness of the trabecular bone structure, while preserving the benefits of regular TPMS morphology. The results demonstrate the improved performance of the modified TPMS scaffolds compared to classic structures in terms of permeability and compliance properties, crucial for the materials applied in or- thopaedic implants. The porosity and permeability values of the developed metamaterial were found to be in the range of trabecular bone. Additionally, the possibilities to simulate the mass transfer properties of the novel material were examined in COMSOL and valu- able qualitative results, like fluid velocity distribution and flow-induced wall shear stress, retrieved. Overall my thesis introduces an innovative idea for biomaterial design and adds a strong argument in favour of TPMS morphology application in the orthopaedic material research field., Biomedical Engineering

Two-dimensional (2D) materials such as graphene have been a topic of intense study as these atomically thin materials have shown enhanced characteristics ideal for applications such as renewable energy storage and electrochemical sensing, which has led researchers into a frenzy akin to that of prospectors during the gold rush. While there still is gold to be found, there is also fool’s gold when systematic studies are not performed. In order to develop new processes for fabricating 3D high surface area architectures, we investigated mechanisms at a fundamental level throughout this work. It is known that the surfaces of freestanding 2D graphene sheets exhibits record-breaking electrical and thermal conductivity, remarkable chemical stability, mechanical strength, and naturally high specific surface area, however, current 2D techniques lay the graphene sheets onto substrates, typically decreasing electron transport and always reducing accessible surface area. In order to use sides of a graphene sheet for applications that require high surface area, we took inspiration from origami. A single paper cannot support itself, but when it becomes part of a 3-dimensional (3D) structure, support becomes natural. The first phase of my work involved the use of minimal surface inspired 3D nickel templates to create free-standing graphene structures with inherited morphology, resulting in porous well-defined sponge-like structures of free-standing graphene. The concept of graphene shaped into minimal surfaces architectures is of great interest; theories predict unique mechanical properties, such as a young’s modulus of 1 TPa for single crystal graphene and an exceptionally high stiffness for minimal surface architectures. Few studies have been performed of fabricating graphene architectures with well defined pore sizes, co-continuous pores, and controlled layer thickness needed for enhancing mass transport and electronic conductivity. Studies were also performed involving atomisti

Cellular membranes display an incredibly diverse range of shapes, both in the plasma membrane and at membrane bound organelles. These morphologies are intricately related to cellular functions, enabling and regulating fundamental membrane processes. However, the biophysical mechanisms at the origin of these complex geometries are not fully understood from the standpoint of membrane-protein coupling. In this study, we focused on a minimal model of helicoidal ramps representative of specialized endoplasmic reticulum compartments. Given a helicoidal membrane geometry, we asked what is the distribution of spontaneous curvature required to maintain this shape at mechanical equilibrium? Based on the Helfrich energy of elastic membranes with spontaneous curvature, we derived the shape equation for minimal surfaces, and applied it to helicoids. We showed the existence of switches in the sign of the spontaneous curvature associated with geometric variations of the membrane structures. Furthermore, for a prescribed gradient of spontaneous curvature along the exterior boundaries, we identified configurations of the helicoidal ramps that are confined between two infinitely large energy barriers. Overall our results suggest possible mechanisms for geometric control of helicoidal ramps in membrane organelles based on curvature-inducing proteins.

Triply Periodic Minimal Surfaces (TPMS) have earned great popularity in porous meta- biomaterials by cause of unique mass transfer properties, tunable mechanical properties, large pore space and high surface to volume ratio. Numerical and experimental studies suggest a great potential for TPMS-based architectures in the treatment of segmental bone defects, where supporting scaffolds are crucial in bridging the gap unable to heal naturally. However, current studies are limited to investigating the properties of classic TPMS archi- tectures. In this thesis, I present the design, fabrication and characterisation of a novel type of meta-biomaterials inspired by TPMS morphology. I show a practical application of Weierstrass equations to define Schwarz P, Schwarz D and Gyroid surfaces, which pro- vides an opportunity to remove fundamental patches from the original surface and design unique structures. Moreover, one of the developed designs applied basics of graph theory to construct the material which mimics the randomness of the trabecular bone structure, while preserving the benefits of regular TPMS morphology. The results demonstrate the improved performance of the modified TPMS scaffolds compared to classic structures in terms of permeability and compliance properties, crucial for the materials applied in or- thopaedic implants. The porosity and permeability values of the developed metamaterial were found to be in the range of trabecular bone. Additionally, the possibilities to simulate the mass transfer properties of the novel material were examined in COMSOL and valu- able qualitative results, like fluid velocity distribution and flow-induced wall shear stress, retrieved. Overall my thesis introduces an innovative idea for biomaterial design and adds a strong argument in favour of TPMS morphology application in the orthopaedic material research field., Biomedical Engineering

We discuss in this note the stickiness phenomena for nonlocal minimal surfaces. Classical minimal surfaces in convex domains do not stick to the boundary of the domain, hence examples of stickiness can be obtained only by removing the assumption of convexity. On the other hand, in the nonlocal framework, stickiness is "generic". We provide various examples from the literature, and focus on the case of complete stickiness in highly nonlocal regimes.

In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three C2-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to “fill the hole” in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

A method is presented for design of curved surface of frame supported membrane structures. Since a uniform stress state is desired, the minimal surface is often chosen as an ideal target curved surface. Another ideal shape is a developable surface that can be generated from a plane sheet. Therefore, in this paper, the minimal surface, the developable surface, and an intermediate surface between them are chosen as the target curved surface. The cutting patterns are generated by reducing the uniform stress from the plane sheet obtained from the curved surface, where correction coefficients are incorporated. In the numerical examples, the properties of selfequilibrium shape and its stress distribution are investigated for surfaces generated from the three kinds of target curved surfaces.

There are many interesting phenomena concerning soap film. One of them is the soap film catenoid. The catenoid is the equilibrium shape of the soap film that is stretched between two circular rings. When the two rings move farther apart, the radius of the neck of the soap film will decrease until it reaches zero and the soap film is split. In our simulation, we show the evolution of the soap film when the rings move apart before the film splits. We use the BMO algorithm for the evolution of a surface accelerated by the mean curvature., Selected Papers from the International Symposium on Computational Science - International Symposium on Computational Science Kanazawa University, Japan

The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of noninterpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments.

Овај рад се бави оптималношћу метода за генерисање почетнегеометрије претходно затегнутих, тензилних конструкција, анализом и поређењем резултата ових метода, као и софтверских пакетау којем су имплементиране..., This work deals with optimality of the existing form.finding methods for generation of initial geometry of tensile structures...

The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a soap film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Adopting a variational approach, we define an energy associated with shape deformations of the system and then derive general equilibrium and (linear) stability conditions by considering the first and second variations of the energy functional. We analyze in detail the transition to instability of flat circular configurations, which are ground states for the system in the absence of surface tension, when the latter is progressively increased. Such a theoretical study is particularly useful here, since the many different perturbations that can lead to instability make it challenging to perform an exhaustive experimental investigation. We generalize previous results, since we allow the filament to possess a curved intrinsic shape and also to display anisotropic flexural properties (as happens when the cross section of the filament is noncircular). This is accomplished by using a rod energy which is familiar from the modeling of DNA filaments. We find that the presence of intrinsic curvature is necessary to obtain a first buckling mode which is not purely tangent to the spanning surface. We also elucidate the role of twisting buckling modes, which become relevant in the presence of flexural anisotropy.

We present a topologically-­based doubly curved building system, based on a single bending thin plate element. The system extends Buckminster Fuller’s plydome research, by proposing an elastic form­finding technique through the introduction of strategic singularities in a periodic grid of originally coplanar plates. The potential of this technique is explored and showcased through the design and manufacture of a large scale prototype.

Nature provides us with a vast source of inspiration. However, given industrial designers' open-mindedness and inquisitiveness, a surprisingly limited set of nature-derived symbols continues to be popular in this creative discipline. Rather than designing products mimicking nature, it is probably more rewarding designing them based on the natural principles leading to its growth and form. However, the constraints related to mass produced products make designing with the often complex forms found in nature a daunting task for a human designer. In this paper, we demonstrate, through the implementation of two generative design systems, how fairly complex everyday objects based on three-dimensional natural-mathematical morphologies can be designed, evaluated and produced using mass production techniques; that digital and analogue methods can be linked to create an aesthetic and functional whole beyond purely decorative mimicry. The output from the generative design system made it possible to produce a fully developed, "ready-for-sale" product, with potential for large-scale production. This is a step towards enabling industrial designers the same level of form articulation as has been available to artists and architects, even though the constraints on the design activity are much different.

Mesoporous silica crystals have a large variety of structures mainly due to the versatility of their structure template. The configuration and the chemical state of the templating micellar surfactants, together with the kinetic process of silica will determine the final outcome of the synthesis. Increasing the understanding of the complex formation processes involved will enable a possibilityto fine tune the material for specific uses, today focused into the fields of photoniccrystals, drug delivery, catalysis and separation technology. In this thesis emphasis is put on (1) increasing the understanding the formation mechanism yielding the different species of mesoporous silica crystals through an in depth study of quasicrystallinity (2) Characterization and description of the structural complexity through various characterization techniquesand also by studying the kinetic structural transformation phenomenon related to the minimal G- and D-surfaces. (3) The structural studies of the versatile surfactant liquid crystals for establishing a thermodynamically stable basis to evaluate the kinetic mesoporous silica growth processes. Furthermorethe thesis both enlightens the possibilities of and contributes to the developmentof electron microscopy characterization techniques. In these studies, electron microscopy is largely employed in the characterization to give a thorough picture of the mesoporous structures. This is combined with the sample preparation techniques cross-section polishing and ionslicing. Low voltage scanning electron microscopy is utilized for studying the surfaces and cross-sections of various materials at the limit of the resolution. Here, a deep understanding of the electron beam-material interaction is used for a better interpretation of the detected signals. Transmission electron microscopyis combined with electron crystallographic reconstruction to yield a three dimensional structural model. For determination of the quasicrystallinity level for a s, At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 7: Manuscript.

Mesoporous silica crystals have a large variety of structures mainly due to the versatility of their structure template. The configuration and the chemical state of the templating micellar surfactants, together with the kinetic process of silica will determine the final outcome of the synthesis. Increasing the understanding of the complex formation processes involved will enable a possibilityto fine tune the material for specific uses, today focused into the fields of photoniccrystals, drug delivery, catalysis and separation technology. In this thesis emphasis is put on (1) increasing the understanding the formation mechanism yielding the different species of mesoporous silica crystals through an in depth study of quasicrystallinity (2) Characterization and description of the structural complexity through various characterization techniquesand also by studying the kinetic structural transformation phenomenon related to the minimal G- and D-surfaces. (3) The structural studies of the versatile surfactant liquid crystals for establishing a thermodynamically stable basis to evaluate the kinetic mesoporous silica growth processes. Furthermorethe thesis both enlightens the possibilities of and contributes to the developmentof electron microscopy characterization techniques. In these studies, electron microscopy is largely employed in the characterization to give a thorough picture of the mesoporous structures. This is combined with the sample preparation techniques cross-section polishing and ionslicing. Low voltage scanning electron microscopy is utilized for studying the surfaces and cross-sections of various materials at the limit of the resolution. Here, a deep understanding of the electron beam-material interaction is used for a better interpretation of the detected signals. Transmission electron microscopyis combined with electron crystallographic reconstruction to yield a three dimensional structural model. For determination of the quasicrystallinity level for a s, At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 7: Manuscript.

Mesoporous silica crystals have a large variety of structures mainly due to the versatility of their structure template. The configuration and the chemical state of the templating micellar surfactants, together with the kinetic process of silica will determine the final outcome of the synthesis. Increasing the understanding of the complex formation processes involved will enable a possibilityto fine tune the material for specific uses, today focused into the fields of photoniccrystals, drug delivery, catalysis and separation technology. In this thesis emphasis is put on (1) increasing the understanding the formation mechanism yielding the different species of mesoporous silica crystals through an in depth study of quasicrystallinity (2) Characterization and description of the structural complexity through various characterization techniquesand also by studying the kinetic structural transformation phenomenon related to the minimal G- and D-surfaces. (3) The structural studies of the versatile surfactant liquid crystals for establishing a thermodynamically stable basis to evaluate the kinetic mesoporous silica growth processes. Furthermorethe thesis both enlightens the possibilities of and contributes to the developmentof electron microscopy characterization techniques. In these studies, electron microscopy is largely employed in the characterization to give a thorough picture of the mesoporous structures. This is combined with the sample preparation techniques cross-section polishing and ionslicing. Low voltage scanning electron microscopy is utilized for studying the surfaces and cross-sections of various materials at the limit of the resolution. Here, a deep understanding of the electron beam-material interaction is used for a better interpretation of the detected signals. Transmission electron microscopyis combined with electron crystallographic reconstruction to yield a three dimensional structural model. For determination of the quasicrystallinity level for a s, At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 7: Manuscript.

Mesoporous silica crystals have a large variety of structures mainly due to the versatility of their structure template. The configuration and the chemical state of the templating micellar surfactants, together with the kinetic process of silica will determine the final outcome of the synthesis. Increasing the understanding of the complex formation processes involved will enable a possibilityto fine tune the material for specific uses, today focused into the fields of photoniccrystals, drug delivery, catalysis and separation technology. In this thesis emphasis is put on (1) increasing the understanding the formation mechanism yielding the different species of mesoporous silica crystals through an in depth study of quasicrystallinity (2) Characterization and description of the structural complexity through various characterization techniquesand also by studying the kinetic structural transformation phenomenon related to the minimal G- and D-surfaces. (3) The structural studies of the versatile surfactant liquid crystals for establishing a thermodynamically stable basis to evaluate the kinetic mesoporous silica growth processes. Furthermorethe thesis both enlightens the possibilities of and contributes to the developmentof electron microscopy characterization techniques. In these studies, electron microscopy is largely employed in the characterization to give a thorough picture of the mesoporous structures. This is combined with the sample preparation techniques cross-section polishing and ionslicing. Low voltage scanning electron microscopy is utilized for studying the surfaces and cross-sections of various materials at the limit of the resolution. Here, a deep understanding of the electron beam-material interaction is used for a better interpretation of the detected signals. Transmission electron microscopyis combined with electron crystallographic reconstruction to yield a three dimensional structural model. For determination of the quasicrystallinity level for a s, At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 7: Manuscript.

Mesoporous silica crystals have a large variety of structures mainly due to the versatility of their structure template. The configuration and the chemical state of the templating micellar surfactants, together with the kinetic process of silica will determine the final outcome of the synthesis. Increasing the understanding of the complex formation processes involved will enable a possibilityto fine tune the material for specific uses, today focused into the fields of photoniccrystals, drug delivery, catalysis and separation technology. In this thesis emphasis is put on (1) increasing the understanding the formation mechanism yielding the different species of mesoporous silica crystals through an in depth study of quasicrystallinity (2) Characterization and description of the structural complexity through various characterization techniquesand also by studying the kinetic structural transformation phenomenon related to the minimal G- and D-surfaces. (3) The structural studies of the versatile surfactant liquid crystals for establishing a thermodynamically stable basis to evaluate the kinetic mesoporous silica growth processes. Furthermorethe thesis both enlightens the possibilities of and contributes to the developmentof electron microscopy characterization techniques. In these studies, electron microscopy is largely employed in the characterization to give a thorough picture of the mesoporous structures. This is combined with the sample preparation techniques cross-section polishing and ionslicing. Low voltage scanning electron microscopy is utilized for studying the surfaces and cross-sections of various materials at the limit of the resolution. Here, a deep understanding of the electron beam-material interaction is used for a better interpretation of the detected signals. Transmission electron microscopyis combined with electron crystallographic reconstruction to yield a three dimensional structural model. For determination of the quasicrystallinity level for a s, At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 7: Manuscript.

We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d <= 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is ( generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras., QC 20120214

An indefinite Riemannian manifold is called neutral it its index is equal to one half of its dimension and an indefinite Kaehler manifold is called neutral Kaehler if its complex index is equal to the half of its complex dimension. In the first part of this article, we extend the notion of slant surfaces in Lorentzian Kaehler surfaces to slant submanifolds in neutral Kaehler manifolds; moreover, we characterize slant submanifolds with parallel canonical structures. By applying the results obtained in the first part we completely classify slant surfaces with parallel mean curvature vector and minimal slant surfaces in the Lorentzian complex plane in the second part of this article.

An indefinite Riemannian manifold is called neutral it its index is equal to one half of its dimension and an indefinite Kaehler manifold is called neutral Kaehler if its complex index is equal to the half of its complex dimension. In the first part of this article, we extend the notion of slant surfaces in Lorentzian Kaehler surfaces to slant submanifolds in neutral Kaehler manifolds; moreover, we characterize slant submanifolds with parallel canonical structures. By applying the results obtained in the first part we completely classify slant surfaces with parallel mean curvature vector and minimal slant surfaces in the Lorentzian complex plane in the second part of this article.

An indefinite Riemannian manifold is called neutral it its index is equal to one half of its dimension and an indefinite Kaehler manifold is called neutral Kaehler if its complex index is equal to the half of its complex dimension. In the first part of this article, we extend the notion of slant surfaces in Lorentzian Kaehler surfaces to slant submanifolds in neutral Kaehler manifolds; moreover, we characterize slant submanifolds with parallel canonical structures. By applying the results obtained in the first part we completely classify slant surfaces with parallel mean curvature vector and minimal slant surfaces in the Lorentzian complex plane in the second part of this article.

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The author defines a tube to be an immersed submanifold u:Mp→Rn+1 and the interval of existence τ(Mp) to be the interval of those t for which the intersection Σt of u(Mp) with the hyperplane xn+1=t in Rn+1 is nonempty and compact. The length of τ(Mp) is called the time of existence of the tube. The tube is minimal if u is a minimal immersion. Denote by vT an orthogonal projection of v into the tangent space of M, ν=eTn+1/∥eTn+1∥, and introduce a vector J, called a vector-flow with coordinates, Jk=∫Σt((ek)T,ν),1≤k≤n+1. The angle between J and en+1 is denoted by α. The main result of the article under review is the following estimate: |τ(M)|≤G(M)∥J∥cos(α)16α2, where G(M) denotes the absolute integral Gauss curvature of M.

The author defines a tube to be an immersed submanifold u:Mp→Rn+1 and the interval of existence τ(Mp) to be the interval of those t for which the intersection Σt of u(Mp) with the hyperplane xn+1=t in Rn+1 is nonempty and compact. The length of τ(Mp) is called the time of existence of the tube. The tube is minimal if u is a minimal immersion. Denote by vT an orthogonal projection of v into the tangent space of M, ν=eTn+1/∥eTn+1∥, and introduce a vector J, called a vector-flow with coordinates, Jk=∫Σt((ek)T,ν),1≤k≤n+1. The angle between J and en+1 is denoted by α. The main result of the article under review is the following estimate: |τ(M)|≤G(M)∥J∥cos(α)16α2, where G(M) denotes the absolute integral Gauss curvature of M.

The author defines a tube to be an immersed submanifold u:Mp→Rn+1 and the interval of existence τ(Mp) to be the interval of those t for which the intersection Σt of u(Mp) with the hyperplane xn+1=t in Rn+1 is nonempty and compact. The length of τ(Mp) is called the time of existence of the tube. The tube is minimal if u is a minimal immersion. Denote by vT an orthogonal projection of v into the tangent space of M, ν=eTn+1/∥eTn+1∥, and introduce a vector J, called a vector-flow with coordinates, Jk=∫Σt((ek)T,ν),1≤k≤n+1. The angle between J and en+1 is denoted by α. The main result of the article under review is the following estimate: |τ(M)|≤G(M)∥J∥cos(α)16α2, where G(M) denotes the absolute integral Gauss curvature of M.

The author defines a tube to be an immersed submanifold u:Mp→Rn+1 and the interval of existence τ(Mp) to be the interval of those t for which the intersection Σt of u(Mp) with the hyperplane xn+1=t in Rn+1 is nonempty and compact. The length of τ(Mp) is called the time of existence of the tube. The tube is minimal if u is a minimal immersion. Denote by vT an orthogonal projection of v into the tangent space of M, ν=eTn+1/∥eTn+1∥, and introduce a vector J, called a vector-flow with coordinates, Jk=∫Σt((ek)T,ν),1≤k≤n+1. The angle between J and en+1 is denoted by α. The main result of the article under review is the following estimate: |τ(M)|≤G(M)∥J∥cos(α)16α2, where G(M) denotes the absolute integral Gauss curvature of M.

A least upper bound for the inner radius of an opening in a complete minimal hypersurface contained in a parallel layer is given.

A least upper bound for the inner radius of an opening in a complete minimal hypersurface contained in a parallel layer is given.

A least upper bound for the inner radius of an opening in a complete minimal hypersurface contained in a parallel layer is given.

A least upper bound for the inner radius of an opening in a complete minimal hypersurface contained in a parallel layer is given.

A least upper bound for the inner radius of an opening in a complete minimal hypersurface contained in a parallel layer is given.

Cieľom mojej bakalárskej práce je vytvoriť prehľad špeciálnych plôch a popísať ich vlastnosti. Za najdôležitejšie kategórie plôch som považoval plochy rotačné, minimálne, s predpísanou Gaussovou krivosťou a hlavne Clairotove plochy. Ku každej kategórií doložím z môjho pohľadu najdôležitejšie príklady plôch, s ich parametrizáciou a vlastnosťami a popíšem ich. Ku plochám uvediem množstvo obrázkov vytvorených pomocou MATLABu. V poslednej časti sa budem venovať výlučne Clairotovým plochám, hladaniu geodetík na týchto plochách a ich vykresleniu. Ponúknem originálne obrázky geodetík na čo najväčšom počte plôch., The aim of this thesis is to create an overview of special surfaces and to define their characteristics. Categories of surfaces that I found the most important are surfaces of revolution, minimal, with constant Gaussian curvature, and finally Clairaut surfaces. For every category I'll introduce, in my opinion, the most important examples of surfaces along with their parametrizations and I'll describe them. Surfaces will be accompanied by images, created in MATLAB. In the last part I'm going to focus on Clairot patches, on finding geodesics on these surfaces and their description. I'll show numerous original images of geodesics on diverse surfaces.

В статье рассматривается понятие кусочно-квадратичного приближенного решения уравнения минимальной поверхности, заданного над триангулированной областью. Изучается вопрос об аппроксимации функционала площади в данном классе поверх- ностей и сходимость таких решений при стремлении к нулю мелкости треугольной сетки. В частности, доказана равномерная сходимость кусочно-квадратичных реше- ний и указана оценка скорости этой сходимости через шаг сетки h., The concept of a piecewise-quadratic approximate solution of the minimal surface equation, defined over a triangulated domain, is considered. We study the problem of approximating the area functional in a given class of surfaces and the convergence of such solutions, as the fineness of the triangular grid tends to zero. In particular, the uniform convergence of piecewise-quadratic solutions is proved and the estimate of the rate of that convergence in terms of the step of the h grid is indicated.

Cieľom mojej bakalárskej práce je vytvoriť prehľad špeciálnych plôch a popísať ich vlastnosti. Za najdôležitejšie kategórie plôch som považoval plochy rotačné, minimálne, s predpísanou Gaussovou krivosťou a hlavne Clairotove plochy. Ku každej kategórií doložím z môjho pohľadu najdôležitejšie príklady plôch, s ich parametrizáciou a vlastnosťami a popíšem ich. Ku plochám uvediem množstvo obrázkov vytvorených pomocou MATLABu. V poslednej časti sa budem venovať výlučne Clairotovým plochám, hladaniu geodetík na týchto plochách a ich vykresleniu. Ponúknem originálne obrázky geodetík na čo najväčšom počte plôch., The aim of this thesis is to create an overview of special surfaces and to define their characteristics. Categories of surfaces that I found the most important are surfaces of revolution, minimal, with constant Gaussian curvature, and finally Clairaut surfaces. For every category I'll introduce, in my opinion, the most important examples of surfaces along with their parametrizations and I'll describe them. Surfaces will be accompanied by images, created in MATLAB. In the last part I'm going to focus on Clairot patches, on finding geodesics on these surfaces and their description. I'll show numerous original images of geodesics on diverse surfaces.

Cieľom mojej bakalárskej práce je vytvoriť prehľad špeciálnych plôch a popísať ich vlastnosti. Za najdôležitejšie kategórie plôch som považoval plochy rotačné, minimálne, s predpísanou Gaussovou krivosťou a hlavne Clairotove plochy. Ku každej kategórií doložím z môjho pohľadu najdôležitejšie príklady plôch, s ich parametrizáciou a vlastnosťami a popíšem ich. Ku plochám uvediem množstvo obrázkov vytvorených pomocou MATLABu. V poslednej časti sa budem venovať výlučne Clairotovým plochám, hladaniu geodetík na týchto plochách a ich vykresleniu. Ponúknem originálne obrázky geodetík na čo najväčšom počte plôch., The aim of this thesis is to create an overview of special surfaces and to define their characteristics. Categories of surfaces that I found the most important are surfaces of revolution, minimal, with constant Gaussian curvature, and finally Clairaut surfaces. For every category I'll introduce, in my opinion, the most important examples of surfaces along with their parametrizations and I'll describe them. Surfaces will be accompanied by images, created in MATLAB. In the last part I'm going to focus on Clairot patches, on finding geodesics on these surfaces and their description. I'll show numerous original images of geodesics on diverse surfaces.

Cieľom mojej bakalárskej práce je vytvoriť prehľad špeciálnych plôch a popísať ich vlastnosti. Za najdôležitejšie kategórie plôch som považoval plochy rotačné, minimálne, s predpísanou Gaussovou krivosťou a hlavne Clairotove plochy. Ku každej kategórií doložím z môjho pohľadu najdôležitejšie príklady plôch, s ich parametrizáciou a vlastnosťami a popíšem ich. Ku plochám uvediem množstvo obrázkov vytvorených pomocou MATLABu. V poslednej časti sa budem venovať výlučne Clairotovým plochám, hladaniu geodetík na týchto plochách a ich vykresleniu. Ponúknem originálne obrázky geodetík na čo najväčšom počte plôch., The aim of this thesis is to create an overview of special surfaces and to define their characteristics. Categories of surfaces that I found the most important are surfaces of revolution, minimal, with constant Gaussian curvature, and finally Clairaut surfaces. For every category I'll introduce, in my opinion, the most important examples of surfaces along with their parametrizations and I'll describe them. Surfaces will be accompanied by images, created in MATLAB. In the last part I'm going to focus on Clairot patches, on finding geodesics on these surfaces and their description. I'll show numerous original images of geodesics on diverse surfaces.

Cieľom mojej bakalárskej práce je vytvoriť prehľad špeciálnych plôch a popísať ich vlastnosti. Za najdôležitejšie kategórie plôch som považoval plochy rotačné, minimálne, s predpísanou Gaussovou krivosťou a hlavne Clairotove plochy. Ku každej kategórií doložím z môjho pohľadu najdôležitejšie príklady plôch, s ich parametrizáciou a vlastnosťami a popíšem ich. Ku plochám uvediem množstvo obrázkov vytvorených pomocou MATLABu. V poslednej časti sa budem venovať výlučne Clairotovým plochám, hladaniu geodetík na týchto plochách a ich vykresleniu. Ponúknem originálne obrázky geodetík na čo najväčšom počte plôch., The aim of this thesis is to create an overview of special surfaces and to define their characteristics. Categories of surfaces that I found the most important are surfaces of revolution, minimal, with constant Gaussian curvature, and finally Clairaut surfaces. For every category I'll introduce, in my opinion, the most important examples of surfaces along with their parametrizations and I'll describe them. Surfaces will be accompanied by images, created in MATLAB. In the last part I'm going to focus on Clairot patches, on finding geodesics on these surfaces and their description. I'll show numerous original images of geodesics on diverse surfaces.