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In this paper, the author investigates the role and contribution of pure and applied mathematics to architecture and fine arts in a unified manner. To this end, a thorough and explanatory historical overview of this diachronic and interdisciplinary topic, from the ages of ancient Mediterranean cultures to the so-called western civilization of the late twentieth century is presented. In this framework, the author first examines the fundamental role of traditional mathematics (e.g. Descriptive and Projective Geometry) in architectural design and fine arts, and in the sequel the discussion extends to the outstanding contribution of modern and computational mathematics (NURBS, Fractals, Boolean matrices, Graph Theory, etc.) to these issues. Besides, the important role of computer-aided design (CAD) is mentioned and emphasized. Indeed, CAD is an exceptional scientific and technological achievement, the scientific background of which is essentially a combination of Informatics, Discrete Mathematics and Descriptive Geometry. In addition, various existing problems that sometimes hinder the application of the science of mathematics to architecture and the fine arts are highlighted and demonstrated. Finally, given that the most appropriate mathematical background for the graduate studies in architectural schools along with the schools of fine arts is a very difficult and rather questionable issue, some suggestions are made in order to encourage and strengthen the relationship among applied mathematics, architecture and fine arts.

An upper bound of in-plane shear modulus for a general class of unidirectional fibrous polymeric composites of periodic structure is presented in an explicit form, according to linear elasticity approach. The fibers that reinforce the polymeric matrix are continuous, parallel to each other, and thus, no agglomeration occurs. To formulate the fiber arrangement inside the matrix, a reliable geometric body-centered model was adopted. The theoretical predictions were compared with experimental data found in the literature, together with theoretical results obtained from other workers, and a reasonable accordance was observed.

An experimental technique is presented to evaluate mechanical and thermal load-induced microstructural damage, based on the Electron Probe Micro-Analysis (EPMA) principle, by which certain ana­lyti­cal potentialities of Scanning Electron Micro­scopy (SEM) are used. The aim of the study is to apply this technique in the case of metal matrix composites (MMCs). Based on earlier findings it is shown that by this technique the damage-con­trolled micro­­structural integrity distribution ahead of an edge-notch under tension is deter­mined with sufficient reliability. To demonstrate this fact two MMCs were tested. Dog-bone specimens with an edge notch were subjected to slow tension up to their ultimate stress, the loading process was terminated and the EPMA technique was applied. The procedure was also applied after sudden cooling of the specimens by immersion in liquid hydrogen. It is shown that the MMC with larger differences between the elastic and plastic constants of the inclusion and the matrix exhibit increased prone­ness to mechanical and thermal load-induced damage. The findings obtained are discussed on the basis of the dominating com­bined influence of macro-microscopic mechanical and thermo-elastic stress concentration processes on the matrix-particle interfacial fracture strength.

This computational work aims at studying the three – dimensional turbulent wind flow around two square buildings with pyramid roofs of trivial architectural forms for an arbitrary geographical location. The overall investigation substantially reduces to the fundamental problem of an external three – dimensional turbulent flow field past a mounted obstacle of predefined shape. The novelty of this research is that the independency of the numerical solution for any possible distribution of mesh points was demonstrated in a theoretical manner without the necessity of changing the original grid with simultaneous repetition of the computational process.

In this work, the authors introduce an octahedral body centered model transformed into a nine-layer spherical model, to simulate the periodic microstructure of particulate composites. This model takes into account the vicinity of internal and neighboring particles in the form of their deterministic configurations inside the matrix, along with the concept of interphase on the thermomechanical properties of the overall material. The latter is assumed to be homogeneous and isotropic. Next, by the use of this model, in association with classical elasticity approach, a closed form expression to calculate the thermal expansion coefficient of this category of composites is derived The theoretical predictions were compared with experimental results as well as with theoretical values yielded by formulae derived from other workers and they were found to be in good agreement.

An experimental technique is presented by which mechanical damage in materials can be evaluated. It is based on the Electron Probe Micro-Analysis (EPMA) principle, by which certain analytical possibilities of Scanning Electron Microscopy (SEM) are used. The main scope of the study was to apply this technique for the determination of microcracking damage in edge-cracked metal matrix composites (MMCs). Based on earlier studies this investigation has shown that by this technique the microcracking distribution ahead of an edge-crack under tension can be determined with relatively high reliability. To demonstrate this fact two types of MMCs were investigated. The specimens (shaped as dog-bones with an edge-crack) were subjected to slow tensile loading up to their ultimate stress. At this stress the loading process was terminated and the EPMA technique was applied.

In this study an attempt was made to introduce some novel operational techniques for the characterization of materials’ fracture, assisted by certain approaches of continuum brittle damage mechanics. This was done by using certain variant operational parameters, i.e., initial damage, damage at failure, structural instability number, strain energy dissipation deficiency number, damage process intensity number and failure strain energy efficiency number. The ultimate scope is to characterize the influence of initial damage, on the above parameters. Tensile tests were carried out, taking into account the basic assumptions of continuum damage mechanics allowing analysis of brittle damage to be applied to elastic-brittle solids under low dynamic loading. A polymeric material, commercially known as Plexiglas, was selected, which fulfils satisfactorily, most of the desired theoretical assumptions.

In this paper, the authors introduce a body-centered polyhedral model capable of simulating the periodic structure of particulate composites, in order to estimate the thermal conductivity of a general class of this type of materials. This model takes into account the arrangement of internal and neighboring particles in the form of three distinct deterministic configurations, along with the concept of interphase on the thermal and mechanical properties of the composite, which is assumed to be macroscopically homogeneous and isotropic. Next, by means of this advanced multilayer model, an explicit analytical expression to evaluate the thermal conductivity of this type of composites is derived. The theoretical predictions were compared with the experimental results found in the literature, as well as with the theoretical values yielded by some reliable formulae derived from other workers, and a reasonable agreement was observed.

A lower bound of the longitudinal elastic modulus of polymer composite materials reinforced with unidirectional continuous fibres is obtained by means of a Differential Calculus approach. In the mathematical derivations, the concept of interphase between the fibre and matrix was also taken into consideration. The three phases are considered as isotropic. The results obtained from the proposed formula were compared with those arising from some reliable and accurate theoretical models as well as with experimental data found in the literature, and a reasonable agreement was observed.

In this work, the effect of moisture absorption on the mechanical properties of particulate composite materials is studied. Moisture absorption constitutes a main parameter affecting the thermomechanical behaviour of composites, since it causes plasticization of the polymer matrix with a concurrent swelling. In the present work, the influence of water absorption on the flexural properties of particle-reinforced composites was thoroughly investigated. It was found that during the process of moisture absorption there exists a variation of the flexural properties closely related to the degradation of the mechanical behaviour of the composite, as well as the percentage amount of moisture absorbed. Experiments were carried out with composite made of epoxy resin reinforced with low-content iron particles. The variation of ultimate stress, breaking strain, deflection, elastic modulus and Poisson ratio due to water absorption was examined.

A continuation of our ongoing investigation into Joule - Thomson inversion curves for van der Waals gas, is performed from a mathematical viewpoint. The methodology basis of our analysis is the quadratic polynomial theory. In this context, focusing on the parametric equation of inversion curves in a P - V frame of reference we obtain a qualitative illustration of variables T, V by means of two inequality relations. However, we should elucidate that these inequalities are valid only for the intersection points between the family of Joule - Thomson inversion curves and the isothermal spinodal lines, provided that they are both sketched in a common P - V coordinate system. The mathematical treatment of the parametric equation of these curves has been carried out in a rigorous manner and no further restriction is introduced for the variables T, V. Thus, the proposed inequalities have a wider range of validity when compared with those that had been previously presented by the author and therefore their possible applications to P - V - T surfaces of van der Waals gas, are also wider.

In this paper, a further mathematical analysis of some previous results of the author’s ongoing investigation regarding Prandtl’s equations is accomplished. In particular, the objective of the present work is the performance of an approximate explicit solution to Prandtl’s system of equations for a laminar, isothermal, incompressible and steady boundary layer flow of a Newtonian fluid, over an infinite horizontal flat plate.

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

Dynamic mechanical analysis (DMA) is a versatile technique that complements the information given by the more traditional thermal analysis techniques such as differential scanning calorimetry (DSC), thermogravimetric analysis (TGA), and thermal mechanical analysis (TMA). Dynamic constants such as storage modulus, loss modulus, and loss factor are temperature dependent and provide information about interfacial bonding between reinforced fibre and polymer matrix of composite material. To study the above mentioned properties at the glass transition region, for unidirectional fibrous composites reinforced with continuous fibers a reliable model was applied. In particular, the composite material was considered as composed of three phases with the intermediate phase between matrix and fibres, the interphase, to have variable properties depending on those of main phases and the mode of preparation of the overall material. The glass transition temperature is defined as the point at which the specific volume versus temperature curve changes abruptly slope marking the region between rubbery polymer and glassy polymer nature. Hence, the behaviour of unidirectional fibrous composites was investigated at this region. Examination of the glass transition temperature, which constitutes an upper limit for the structurally important glassy region through the loss factor, was performed by its consideration as a combination of glass transition temperature of matrix and interphase.

In this paper, an upper bound of the longitudinal elastic modulus of unidirectional fibrous composites is proposed according to strength of materials approach, on the premise that the fiber is much stiffer than the matrix. In the mathematical derivations, the concept of boundary interphase between fiber and matrix was also taken into account and the main objective of this work is the attainment of an upper bound for the interphase stiffness with respect to fiber concentration by volume. The novel element here is that the authors have not taken into consideration any specific variation law to approximate the interphase modulus. The theoretical results arising from the proposed formula were compared with those obtained from some reliable theoretical models as well as with experimental data found in the literature, and a satisfactory agreement was observed.

In this paper, an icosahedral non – body centered model to simulate the periodic structure of homogeneous particulate composites, by predicting the particle arrangement, is presented. This model has yielded three different variations which correspond at three different deterministic particle configurations. In addition, the concept of boundary interphase between matrix and inclusions was taken into account. Thus, the influence of particle vicinity was examined in parallel with the interphase concept on the thermomechanical properties of the overall material. Next, by the use of this model the authors derived a closed – form expression to estimate the thermal conductivity of this type of composites. To test the validity of the model, the theoretical values arising from the proposed formula were compared with other theoretical predictions obtained from several accurate formulae found in the literature and an adequate accordance was observed.

This paper examines a consequence of spinodal curve condition on the Joule – Thomson inversion curves, for van der Waals real gas. To this end, we consider the differential form of Joule – Thomson coefficient, and concentrate on the parametrical algebraic equation of the corresponding family of curves in a P – V Cartesian frame of reference. The aim of this work is not the prediction of Joule – Thomson inversion curves for the van der Waals Eos, but to derive inequality relations amongst the variables T, V and the parameters which appear in this constitutive law. Here, we should clarify beforehand that all proposed expressions concern only the intersection points between the set of inversion curves and the isothermal spinodal lines, which are drawn in the same P – V frame of reference. Key words: JT inversion curves, spinodal curve condition, van der Waals Eos, P – V frame of reference.

In this work, the authors present an approximate solution to three point bending equation for a simply supported beam of rectangular section. The deflection of the beam due to a perpendicular point – wise load is estimated for high rates of its curvature. After the deformation of the beam, the shape of its cross sectional area is assumed to remain as is. The proposed solution consists of binomial power series and does not contain elliptic integrals or other special functions. Hence, it may be more appropriate for usual engineering practices. Key words: Simply supported beam, three point bending, large deflection.

In this paper, a geometric body-centered model to simulate the periodic structure of unidirectional fibrous composites is presented. To this end, three prescribed configurations are introduced to predict in a deterministic manner the arrangement of internal and neighboring fibers inside the matrix. Thus, three different representative volume elements (RVEs) are established. Furthermore, the concept of the interphase has been taken into account, stating that each individual fiber is encircled by a thin layer of variable thermomechanical properties. Next, these three unit cells are transformed in a unified manner to a coaxial multilayer cylinder model. This advanced model includes the influence of fiber contiguity in parallel with the interphase concept on the thermomechanical properties of the overall material. Then, by the use of this model, the authors propose explicit expressions to evaluate the longitudinal and transverse thermal conductivity of this type of composite. The theoretical predictions were compared with experimental results, as well as with theoretical values yielded by some reliable formulae derived from other workers, and a reasonable agreement was found.

In this paper, the thermal conductivity coefficients of unidirectional composites have been estimated. In the present analysis, a model which takes into account the influence of the boundary interphase has been proposed. The thermal conductivity coefficients of the composite were calculated by using this concept to determine the role of the interphase which mainly depends on the quality of adhesion between fibers and matrix. The interphase matter is the part of the polymer matrix lying on the close vicinity of the fiber bounding surface. It has been primarily assumed that the interphase is inhomogeneous with thermophysical properties varying from the fiber surface to the matrix. Five laws of variation have been taken into consideration in order to derive closed form expressions for the thermal conductivity coefficients in which the role of the interphase layer occurs, possessing properties different from those of the constituent phases. A thermal analysis method known in the literature has been adopted in...

In this paper, the author investigates a generic type of the two dimensional incompressible viscous boundary layer flows near the inflection point of a smooth curve. In particular, an approximate evaluation of velocity distribution in an "adjacent" region of the inflection point of this curve is derived. Here, it is a priory assumed that this point is unique. Besides, the flow field is supposed to be steady throughout and Prandtl's simplified assumptions for two dimensional boundary layer flows are also taken into consideration. Besides, if the cross - section of a structure can be simulated by such a curve one should denote that its inflection point is from technical aspect a very suitable place for emplacing a wind generator compared with the top of a curve, which is a stationary point, because if a wind generator was located on the top of a curve, it could be exhibited in unexpected strong blasts.

In this paper, we will attempt to derive an analytical representation of velocity field for the concept of an internal air flow inside a vortex tube machine of Hilsch-Ranque type. The basic assumption here is that the flux field is assumed to be steady and incompressible.Τhe vector which denotes instantaneous velocity, for the description of the flux field viaLagrangian stand point, will be expressed in a cylindrical coordinate system with orthonormal basis in the center line of the cross section of this vortex tube. We also assume here, that the aforementioned flux field varies only in radial direction, hence velocity vector is actually an one-dimensional damped circular helix trajectory with variable pitch. Key words: Vortex tube, homographic function, vortex lines, helix trajectory.

In this paper, an icosahedral non-body-centered model is presented to simulate the periodic structure of a general class of homogeneous particulate composites, by predicting the particle arrangement. This model yielded three different variations, which correspond to three different deterministic particle configurations. In addition, the concept of a boundary interphase between matrix and inclusions was taken into account. In this framework, the influence of particle vicinity on the thermomechanical properties of the overall material was examined in parallel with the concept of boundary interphase. The simultaneous consideration of these two basic influential factors constitutes the novelty of this work. Next, by the use of this advanced model, the authors derived a closed-form expression to estimate the thermal conductivity of this type of composite. To test the validity of the model, the theoretical predictions arising from the proposed formula were compared with experimental data found in the literature, together with theoretical results obtained from several accurate formulae derived from other workers, and an adequate accordance was observed.

Vanadium participation in cellular events entails in-depth comprehension of its soluble and bioavailable forms bearing physiological ligands in aqueous distributions of binary and ternary systems. Poised to understand the ternary V(V)-H(2)O(2)-amino acid interactions relevant to that metal ion's biological role, we have launched synthetic efforts involving the physiological ligands glycine and H(2)O(2). In a pH-specific fashion, V(2)O(5), glycine, and H(2)O(2) reacted and afforded the unusual complexes (H(3)O)(2)[V(2)(O)(2)(mu(2):eta(2):eta(1)-O(2))(2)(eta(2)-O(2))(2)(C(2)H(5)NO(2))] x 5/4 H(2)O (1) and K(2)[V(2)(O)(2)(mu(2):eta(2):eta(1)-O(2))(2)(eta(2)-O(2))(2)(C(2)H(5)NO(2))] x H(2)O (2). 1 crystallizes in the triclinic space group P1, with a = 7.805(4) A, b = 8.134(5) A, c = 12.010(7) A, alpha = 72.298(9) degrees, beta = 72.991(9) degrees, gamma = 64.111(9) degrees, V = 641.9(6) A(3), and Z = 2. 2 crystallizes in the triclinic space group P1, with a = 7.6766(9) A, b = 7.9534(9) A, c = 11.7494(13) A, alpha = 71.768(2) degrees, beta = 73.233(2) degrees, gamma = 65.660(2) degrees, V = 610.15(12) A(3), and Z = 2. Both complexes 1 and 2 were characterized by UV/visible, LC-MS, FT-IR, Raman, NMR spectroscopy, cyclic voltammetry, and X-ray crystallography. The structures of 1 and 2 reveal the presence of unusual ternary dinuclear vanadium-tetraperoxo-glycine complexes containing [(V(V)=O)(O(2))(2)](-) units interacting through long V-O bonds and an effective glycinate bridge. The latter ligand is present in the dianionic assembly as a bidentate moiety spanning both V(V) centers in a zwitterionic form. The collective physicochemical properties of the two ternary species 1 and 2 project the chemical role of the low molecular mass biosubstrate glycine in binding V(V)-diperoxo units, thereby stabilizing a dinuclear V(V)-tetraperoxo dianion. Structural comparisons of the anions in 1 and 2 with other known dinuclear V(V)-tetraperoxo binary anionic species provide insight into the chemical reactivity of V(V)-diperoxo species in key cellular events such as insulin mimesis and antitumorigenicity, potentially modulated by the presence of glycinate and hydrogen peroxide.

Vanadium involvement in cellular processes requires deep understanding of the nature and properties of its soluble and bioavailable forms arising in aqueous speciations of binary and ternary systems. In an effort to understand the ternary vanadium-H(2)O(2)-ligand interactions relevant to that metal ion's biological role, synthetic efforts were launched involving the physiological ligands betaine (Me(3)N(+)CH(2)CO(2)(-)) and H(2)O(2). In a pH-specific fashion, V(2)O(5), betaine, and H(2)O(2) reacted and afforded three new, unusual, and unique compounds, consistent with the molecular formulation K(2)[V(2)O(2)(O(2))(4){(CH(3))(3)NCH(2)CO(2))}]·H(2)O (1), (NH(4))(2)[V(2)O(2)(O(2))(4){(CH(3))(3)NCH(2)CO(2))}]·0.75H(2)O (2), and {Na(2)[V(2)O(2)(O(2))(4){(CH(3))(3)NCH(2)CO(2))}(2)]}(n)·4nH(2)O (3). All complexes 1-3 were characterized by elemental analysis; UV/visible, FT-IR, Raman, and NMR spectroscopy in solution and the solid state; cyclic voltammetry; TGA-DTG; and X-ray crystallography. The structures of 1 and 2 reveal the presence of unusual ternary dinuclear vanadium-tetraperoxido-betaine complexes containing [(V(V)═O)(O(2))(2)] units interacting through long V-O bonds. The two V(V) ions are bridged through the oxygen terminal of one of the peroxide groups bound to the vanadium centers. The betaine ligand binds only one of the two V(V) ions. In the case of the third complex 3, the two vanadium centers are not immediate neighbors, with Na(+) ions (a) acting as efficient oxygen anchors and through Na-O bonds holding the two vanadium ions in place and (b) providing for oxygen-containing ligand binding leading to a polymeric lattice. In 1 and 3, interesting 2D (honeycomb) and 1D (zigzag chains) topologies of potassium nine-coordinate polyhedra (1) and sodium octahedra (3), respectively, form. The collective physicochemical properties of the three ternary species 1-3 project the chemical role of the low molecular mass biosubstrate betaine in binding V(V)-diperoxido units, thereby stabilizing a dinuclear V(V)-tetraperoxido dianion. Structural comparisons of the anions in 1-3 with other known dinuclear V(V)-tetraperoxido binary anionic species provide insight into the chemical reactivity of V(V)-diperoxido systems and their potential link to cellular events such as insulin mimesis and anitumorigenicity modulated by the presence of betaine.

In a pH-specific fashion, V(2)O(5) and citric acid in the absence and presence of H(2)O(2) reacted and afforded, in the presence of NaOH and (CH(6)N(3))(2)CO(3), two new dinuclear V(V) binary non-peroxo (CH(6)N(3))(6)[V(2)O(4)(C(6)H(4)O(7))(2)].2H(2)O (1) and ternary peroxo (CH(6)N(3))(4)[V(2)O(2)(Omicron(2))(2)(C(6)H(5)O(7))(2)].6Eta(2)Omicron (2) species, respectively. Complexes 1 and 2 were further characterized by elemental analysis, UV/Vis, FT-IR, NMR (solution and solid state Cross Polarization-Magic Angle Spinning (CP-MAS)) and Raman spectroscopies, cyclic voltammetry, and X-ray crystallography. Both 1 and 2 are members of the family of dinuclear V(V)-citrate species bearing citrate with a distinct coordination mode and degree of deprotonation, with 2 being the missing link in the family of pH-structural variants of the ternary V(V)-peroxo-citrate system. Given that 1 and 2 possess distinct structural features, relevant binary V(III), V(IV) and V(V), and ternary V(V) species bearing O- and N-containing ligands were tested in in vitro cell cultures to assess their cellular toxicity and insulin mimetic capacity. The results project a clear profile for all species tested, earmarking the importance of vanadium oxidation state and its ligand environment in influencing further binary and ternary interactions of vanadium arising with variable mass cellular targets, ultimately leading to a specific (non)toxic phenotype and glucose uptake ability.

The intention of this paper is to study the mechanical behavior of asymmetric elastic beams in off – axis four point bending. The overall phenomenon is investigated for high rates of the curvature. An approximate analytical method to estimate the deflection is proposed, rendering it expressible in terms of a finite combination of elementary single – valued functions. The novel element here is the examination of the elastic behavior of the continuum beam without its separation in distinguished regions. Hence, the derived analytical representation of the deflection concerns both its positive and negative rates. Consequently, the proposed explicit solution enables us to draw firm qualitative information about a large category of rectangular continuum beams.

A theoretical model for the evaluation of the thermal expansion coefficient of particulate composites is presented in this paper. The proposed model takes into account the influence of neighbouring spherical inclusions on the thermal expansion of the composite consisting of the matrix and filler. A microstructural cubic model reproducing its the basic cell at a microscopic scale was transformed to a four–phase representative volume element (RVE) in order to apply the theory of elasticity. Theoretical values derived from this model were compared with other theoretical ones obtained from literature as well as with experimental results with aluminium particulate–filled epoxy resin composites carried out by the authors and also by other scientists. The goal is to study the influence of the particle contiguity (or the particle interaction) in order to show that the volume fraction is not the only parameter determining the thermomechanical constants.