Kinetics of flow stress in ultra-pure tantalum single crystals in stress/temperature regime III

The temperature dependence of the critical resolved shear stress (CRSS), τ, of ultra-pure tantalum single crystals (RRR ≥ 14000) observed below 250 K for a range of shear-strain rates [(g)\dot] = 2×10 ^{- 5} - 6×10 ^{- 3}  \texts ^{- 1} \dot{\gamma } = 2\times 10^{ - 5} - 6\times 10^{ - 3} \,{\text{s}}^{ - 1} was analyzed within the framework of a kink-pair nucleation model of flow stress. The CRSS/strain-rate data follow the model formulation t ^{1/ 2} = C + D  ln[(g)\dot] \tau^{ 1/ 2} = C + D\,{ \ln }\dot{\gamma } , where C and D are positive constants, for each deformation temperature T in the range 78–250 K. Evaluation of the various slip-parameters of flow stress points to (211)[[`1]11] [\bar{1}11] slip system responsible for the yielding of ultra-pure tantalum single crystals in the so-called stress/temperature regime III (T < 250 K). The value of the pre-exponential factor [(g)\dot]_\texto \dot{\gamma }_{\text{o}} in the Arrhenius-type equation for the shear-strain rate [(g)\dot] \dot{\gamma } is found to be of the order of 10⁵ s⁻¹, which is substantially lower than that ( [(g)\dot]_\texto ~ 10⁷  \texts ^{- 1} ) \left( {\dot{\gamma }_{\text{o}} \sim 10^{7} \,{\text{s}}^{ - 1} } \right) determined in the stress/temperature regime II (250–400 K) and contradicts the assumption invariably made in most of the theoretical models of flow stress that [(g)\dot]_\texto \dot{\gamma }_{\text{o}} is a constant over a wide temperature range.     

Physicochemical Behavior of Some Amino Acids/Glycylglycine in Aqueous <Emphasis Type="SmallCaps">D</Emphasis>-Galactose Solutions at Different Temperatures

The apparent molar volumes ([`(V<sub>2</sub>)]){(\overline{V_2})} for glycine (Gly), l-alanine (Ala), phenylalanine (Phe), and glycylglycine (Gly-Gly) in 0.10 m aqueous d-galactose solutions have been determined from density measurements at (298.15, 303.15, 308.15, and 313.15) K. The data for ([`(V₂)]){(\overline{V_2})} were utilized to estimate the partial molar volume at infinite dilution ([`(V<sub>2</sub><sup>0</sup>)]){(\overline{V_2^0})} , and experimental slope (S<sub>v</sub><sup>*</sup>){(S_{\rm v}^\ast)} . The transfer volume, ([`(V₂⁰)]_(tr)){(\overline{V_{2}^0}_{\rm (tr)})} , and hydration number, (n _H) were also evaluated. The viscosity data were used to evaluate A- and B-coefficients of the Jones–Dole equation, the free energy of activation of viscous flow per mole of the solvent (Dm₁^0* ){\left(\Delta \mu_{1}^{0\ast} \right)} and the solute (Dm₂^0* ){\left(\Delta \mu _{2}^{0\ast} \right)} . The molar refractivity (R _D) was calculated from refractive index data. The results were discussed in terms of hydrophilic–ionic, hydrophilic–hydrophobic, and hydrophobic–hydrophobic interactions, and structure-making/-breaking ability of the solute (AAs/peptide) in aqueous d-galactose solutions.     

Extensions of access structures and their cryptographic applications

In secret sharing schemes a secret is distributed among a set of users ${\mathcal{P}}In secret sharing schemes a secret is distributed among a set of users P{\mathcal{P}} in such a way that only some sets, the authorized sets, can recover it. The family Γ of authorized sets is called the access structure. To design new cryptographic protocols, we introduce in this work the concept of extension of an access structure: given a monotone family G ì 2^P{{\it \Gamma} \subset 2^\mathcal{P}} and a larger set P^￠ = P è[(P)\tilde]{\mathcal{P}^{\prime} = \mathcal{P} \cup \tilde{\mathcal{P}}}, a monotone access structure G^￠ ì 2^P￠{{\it \Gamma}^{\prime}\subset 2^{\mathcal{P}^{\prime}}} is an extension of Γ if the following two conditions are satisfied: (1) The set P{\mathcal{P}} is a minimal subset of Γ′, i.e. P ? G^￠{\mathcal{P} \in {\it \Gamma}^{\prime}} and P - {R_i} ? G^￠{\mathcal{P} - \{R_i\}\notin {\it \Gamma}^{\prime}} for every R_i ? P{R_i \in \mathcal{P}}, (2) A subset A ì P{A \subset \mathcal{P}} is in Γ if and only if the subset A è[(P)\tilde]{A \cup \tilde{\mathcal{P}}} is in Γ′. As our first contribution, we give an explicit construction of an extension Γ′ of a vector space access structure Γ, and we prove that Γ′ is also a vector space access structure. Although the definition may seem a bit artificial at first, it is well motivated from a cryptographic point of view. Indeed, our second contribution is to show that the concept of extension of an access structure can be used to design encryption schemes with access structures that are chosen ad-hoc at the time of encryption. Specifically, we design and analyze a dynamic distributed encryption scheme and a ciphertext-policy attribute-based encryption scheme. In some cases, the new schemes enjoy better properties than existing ones.     

The structure of the unidirectionally solidified Al-Al2Au eutectic

The structure of a number of unidirectionally solidified Al-Al₂Au alloys of eutectic and off-eutectic compositions has been investigated over a wide range of growth rates (1.6×10^–4 to 1.66×10^–2cm sec^–1) using a thermal gradient of approximately 80 to 100 lamellar interface || (001)_{Al 2 Au} || (01 1) _Al [ 1 1 0 ]_{Al 2 Au} || [ 1 0 0 ] _Al growth direction of lamellae and rods || [ 1 1 0 ]_{Al2 Au} || [ 1 0 0 ]_Al \begin{gathered} lamellar interface \left\| {(001)_{Al_{ 2} Au} } \right.\left\| {(01 1)} \right._{Al} \hfill \\ \left[ {1 1 0} \right]_{Al_{ 2} Au} \left\| {\left[ {1 0 0} \right]} \right._{Al} \hfill \\ growth direction of \hfill \\ lamellae and rods \left\| {\left[ {1 1 0} \right]_{Al_2 Au} \left\| {\left[ {1 0 0} \right]_{Al} } \right.} \right. \hfill \\ \end{gathered}     

Influence of F-ligands on Half-metallicity of Cubic Spinel Materials

O-ligands of Fe₃O₄ were substituted partly by F-ions, and then the magnetic and electronic properties of spinel compounds containing F-ligands were studied systematically based on the first principle calculation. Their electronic structures, and then the mechanisms of magnetic and electric properties were analyzed. It is shown that spinel materials have no half-metallicity if the O-ligands of Fe-ions on A-sites are substituted by F-ions. On the other hand, the concentration of F-ligands of Fe-ions on B-sites can cause a very important influence on the half-metallicity of spinel materials. There are no half-metallicity if x≤0.5. If x>0.5, materials have evidently half-metallicity. The Fermi level moves to higher energy, and then the half-metallicity are more stable with increasing concentration of F-ions. The main mechanism is that there is stronger crystal field with increasing concentration of F-ligands, which causes greater splitting of up-spin sub-bands. The electronic structures of a Fe-ion are Fe t_2g³_-t¹_2gˉe_g²_-\textrm{t}_{\mathrm{2g}}^{3}{}_{\uparrow}\textrm{t}^{1}_{\mathrm{2g\downarrow}}\textrm{e}_{\mathrm{g}}^{2}{}_{\uparrow} .     

Elastic,piezoelectric, and dielectric properties of lanthanum-gallium tantalate crystals at temperatures of 20–600°C

Measurements of the temperature variations in the relative dielectric constants e₃₃^T/e₀ \varepsilon_{33}^T/{\varepsilon_0} , e₁₁^T/e₀ \varepsilon_{11}^T/{\varepsilon_0} ,and e₂₂^T/e₀ \varepsilon_{22}^T/\varepsilon {}_0 , conductivities g₁₁, g₂₂, and g₃₃, elastic constants C₁₁^D C_{11}^D , C₆₆^D C_{66}^D , and S₂₂^E S_{22}^E , and electromechanical coupling constants k_t, k₂₆, and k₁₂ of lanthanum-gallium tantalate over temperatures of 20–600°C are reported. Data are presented on the effect of lifetime (breakdown) testing on these characteristics of the piezoelectric crystals after retention at a temperature of 625°C for 250 h.     

Non-linear viscoelasticity of vapor grown carbon nanofiber/polystyrene composites

The strain (γ) dependences of viscoelasticity and electrical resistance (R) of vapor grown carbon nanofiber (VGCF)/polystyrene (PS) composites have been studied using simultaneous measurement technique. The composites containing at least 4 vol.% VGCF present two regions of strain softening at γ < 10% and γ > 30%, respectively. Using strain amplification factor introduced by hydrodynamic effects as vertical and horizontal shifting factors, the curves of dynamic storage modulus (G′) and loss modulus (G″) as a function of γ for the composites can be superposed, respectively, on those for the pure matrix at γ ≥ 30%. Significant deflection from the master curves can be observed at γ < 30%. R tested as a function of γ provides direct evidences for breakdown of filler–filler interactions even by a small strain perturbation. It is suggested that breakdown of filler–filler interactions plays a vital role in strain softening at small strains, whereas the matrix provides the main contribution to strain softening at large strains. Dynamic moduli G_{\textf,(0.1% ,j)}^￠ G_{{{\text{f,}}(0.1\% ,\varphi )}}^{\prime } and G_{\textf,(0.1% ,j)}^￠￠ G_{{{\text{f,}}(0.1\% ,\varphi )}}^{\prime \prime } of the filler phase at volume fraction φ at 0.1% strain are used to account for the viscoelastic contribution of the initial filler structure. Ratios of dynamic moduli of the filler phase to the composite at 0.1% strain, G_\textf,0.1% ^￠ /G_\textc,0.1% ^￠ G_{{{\text{f,}}0.1\% }}^{\prime } /G_{{{\text{c,}}0.1\% }}^{\prime } and G_\textf,0.1% ^￠￠ /G_\textc,0.1% ^￠￠ G_{{{\text{f,}}0.1\% }}^{\prime \prime } /G_{{{\text{c,}}0.1\% }}^{\prime \prime } , exhibit percolation-like transition as a function of φ, which is in consistence with the electric percolation transition.     

Characterization of Pb-doped Sr-Ferrites at Room Temperature

We report the magnetic characteristics of Pb-doped Sr-ferrites at room temperature. The polycrystalline samples of the series Sr_0.5Pb_0.5^{2 +} Fe_{12 - x}Pb_x^{3 +} O₁₉\mathrm{Sr}_{0.5}\mathrm{Pb}_{0.5}^{2 +} \mathrm{Fe}_{12 - x}\mathrm{Pb}_{x}^{3 +} \mathrm{O}_{19} (x=0, 0.2, 0.4, 0.6, 0.8, 1.0) have been prepared by the standard ceramic technique with the aim to study the magnetic properties including coercivity, remanence, and energy at room temperature. The measurements show decreasing trends in coercivity and remanence from 4682 Oe to 1783 Oe and from 1833 G to 1511 G for the sample with x-content of 0.0 to 1.0, respectively. A minute addition of Pb affects the behavior of the materials by decreasing its energy product.     

The lower bound on the second-order nonlinearity of a class of Boolean functions with high nonlinearity

The r-th order nonlinearity of Boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers. It is also very useful in coding theory, since it is related to the covering radii of Reed-Muller codes. By investigating the lower bound of the nonlinearity of the derivative of the function f, this paper tightens the lower bound of the second-order nonlinearity of a class of Boolean functions over F2n{F_{2^n}} with high nonlinearity in the form f(x) = tr(λ x d ), where l ? F2r*, d=22r+2r+1{\lambda\in F_{2^r}^*, d=2^{2r}+2^{r}+1} and n = 4r.     

Shear-thickening behaviour of concentrated polymer dispersions under steady and oscillatory shear

The rheological behaviour of a 58 vol.% dispersion of styrene/acrylate particles in ethylene glycol was investigated using a plate-on-plate rheometer. Experimental results showed that the concentrated polymer dispersion exhibited a strong shear-thickening transition under both steady shear and dynamic oscillatory conditions. The low-frequency dynamic oscillatory behaviour could be reasonably interpreted in terms of the steady shear behaviour. Accordingly, the critical dynamic shear rate [(g)\dot]_{\textc_d} , \dot{\gamma }_{{{\text{c\_d}}}} , agreed well with the critical shear rate obtained in steady flow [(g)\dot]_{\textc_s} , \dot{\gamma }_{{{\text{c\_s}}}} , where [(g)\dot]_{\textc_d} \dot{\gamma }_{{{\text{c\_d}}}} was calculated as the maximum shear rate by the critical dynamic shear strain γ _c and the frequency ω, i.e. [(g)\dot]_{\textc_d} = wg_\textc . \dot{\gamma }_{{{\text{c\_d}}}} = \omega \gamma_{\text{c}} . However, during high-frequency dynamic oscillation, it was observed that the shear thickening occurred only when an apparent critical shear strain was reached, which could not be fully explained by the wall-slipping effect. Based on freeze fracture microscopic observations, the effect of the micro-sized flocculation of particles on the rheology of concentrated dispersions was also discussed.     

Reduction of a NiO thin film deposited by PLD on a single crystal YSZ (111) substrate

The NiO/YSZ interface prepared by depositing NiO on a single crystal YSZ (111) substrate has been investigated by transmission electron microscopy. As deposited, a very thin nickel layer ascribing to the nonstoichiometry at the very beginning growth of NiO and an amorphous silica phase resulting from silicon segregation were present at the interface. The orientational relationship of NiO (1[`1] 1) (1\overline{1} 1) //Ni (1[`1] 1) (1\overline{1} 1) //YSZ (1[`1] 1) (1\overline{1} 1) with NiO [110]//Ni [110]//YSZ [110] was observed. The microstructural and chemical changes at the NiO/YSZ interface after being heated in vacuum and hydrogen indicated different reduction mechanisms. In vacuum, the reaction \textNiO ? \textNi + 1/ 2 \text O ₂ ( \textg ) {\text{NiO}} \to {\text{Ni}} + 1/ 2 {\text{ O}}_{ 2} \left( {\text{g}} \right) was prevailing at the interface between NiO and pre-existing Ni, which led to the thickening of nickel layer. In hydrogen, the reduction initiated on the NiO surface was dominant, following the chemical equation H₂ + O_O (NiO) → H₂O (g) + V_O ^.. (NiO) + 2e (Ni).     

Phase and morphology evolution of calcium carbonate precipitated by carbonation of hydrated lime

Phase and morphology evolution of CaCO₃ precipitated during carbonation of lime pastes via the reaction Ca(OH)₂ + CO₂ → CaCO₃ + H₂O has been investigated under different conditions (pCO₂ ≈ 10^−3.5 atm at 60 % RH and 93 % RH; pCO₂ = 1 atm at 93 % RH) using XRD, FTIR, TGA, and SEM. Simulations of the pore solution chemistry for different stages and conditions of carbonation were performed using the PHREEQC code to investigate the evolution of the chemistry of the system. Results indicate initial precipitation of amorphous calcium carbonate (ACC) which in turn transforms into scalenohedral calcite under excess Ca²⁺ ions. Because of their polar character, { 21[`3]4 } \left\{ {21\bar{3}4} \right\} scalenohedral faces (type S) interact more strongly with excess Ca<sup>2+</sup> than non-polar { 10[`1]4 } \left\{ {10\bar{1}4} \right\} rhombohedral faces (type F), an effect that ultimately favors the stabilization of { 21[`3]4 } \left\{ {21\bar{3}4} \right\} faces. Following the full consumption of Ca<sup>2+</sup> ions and further dissolution of CO<sub>2</sub> leading to a pH drop of the pore solution, { 21[`3]4 } \left\{ {21\bar{3}4} \right\} scalenohedra are subjected to dissolution. This eventually results in re-precipitation of { 10[`1]4 } \left\{ {10\bar{1}4} \right\} rhombohedra at close-to-neutral pH. This crystallization sequence progresses through the carbonated depth with a strong dependence on the degree of exposure to CO₂, which is controlled by the carbonated pore structure governing the diffusion of CO₂. Both the carbonation process and the scalenohedral-to-rhombohedral transformation are kinetically favored under high RH and high pCO₂. Supersaturation plays a critical role on the nucleation density and size of CaCO₃ crystals. These results have important implications in understanding the behavior of ancient and modern lime mortars for applications in architectural heritage conservation.     

On cyclic self-dual codes

We investigate cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} . We give a decomposition of a repeated-root cyclic codes over \mathbbF_p^r{\mathbb{F}_{p^{r}}} . The decomposition is used to analyze cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} . We obtain a necessary and sufficient condition for the existence of nontrivial cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} , and prove that all cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} are Type I. Finally we classify cyclic self-dual codes of some lengths over \mathbbF₄{\mathbb{F}_{4}} , \mathbbF₈{\mathbb{F}_{8}} , and \mathbbF₁₆{\mathbb{F}_{16}} .     

Structural and optical properties of Cd x Zn(1 − x)Te thin films

Seven Cd _x Zn_{(1 ? x} Te solid solutions with x = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 were synthesized by fusing stoichiometric amounts of CdTe and ZnTe constituents in silica tubes. Each composition was used in the preparation of a group of thin films of different thicknesses. Structural investigation of the obtained films indicates they have a polycrystalline structure with predominant diffraction lines corresponding to (111) (220) and (311) reflecting planes, which can be attributed to the characteristics of growth with the (111) plane. The optical constants (the refractive index n, the absorption index k, and the absorption coefficient α) of Cd _x Zn_{(1 \s -x)} Te thin films were determined in the spectral range 500–2000 nm. At certain wavelengths it was found that the refractive index, n, increases with increasing molar fraction, x. It was also found that plots of α² (hv) and α^1/2 (hv) yield straight lines, corresponding to direct and indirect allowed transitions respectively obeying the following two equations: $$\begin{gathered} E_g^d = 1.583 + 0.277x + 0.197x^2 \hfill \\ E_g^{ind} = 1.281 + 0.111x + 0.302x^2 \hfill \\ \end{gathered}$$      

Equilibration kinetics of (La,Sr)MnO3

The present work reports isothermal changes of oxygen non-stoichiometry for the perovskite-type electrode material (La_0.8Sr_0.2)MnO₃ in the temperature range 945–1255 K. A thermogravimetric method was used to monitor the rate of the gas/solid equilibration. For equilibration degrees larger than 0.5, the equilibration kinetic data can be described by a diffusion equation. The determined chemical diffusion coefficient depends essentially on the oxygen partial pressure. Its temperature dependence can be expressed by the following expressions at low and high p(O₂), respectively:

Critical and Gaussian Conductivity Fluctuations in Granular Ho1?xPrxBa2Cu3O7?δ Superconductor

Fluctuations in the electrical conductivity of polycrystalline Ho_1−x Pr _x Ba₂Cu₃O_7−δ superconductors were investigated with Ho contents from x=0.01 up to 0.10. Samples were prepared by the standard solid-state reaction technique. The method of analysis is based on the determination of the quantity c_s^-1=-\fracddTlnDs\chi_{\sigma}^{-1}=-\frac{d}{dT}\ln\Delta\sigma, where Δσ=σ−σ _R is the fluctuation conductivity. The results show that the resistive transition proceeds in two stages as seen by the temperature derivative of the resistivity near T _C . In the normal phase, Gaussian and critical fluctuation conductivity regimes were identified. The pairing transition splitting, associated with Pr doping and related to the occurrence of a phase separation, as observed in studies with other rare earth elements, was not clearly observed. On approaching the zero-resistance state, our results show a power-law behavior that corresponds to a phase transition from a paracoherent to a coherent state of the granular array. This behavior was not affected by Pr doping.     

Thermal characteristics and crystallization kinetics of tellurite glass with small amount of additive As<Subscript>2</Subscript>O<Subscript>3</Subscript>

The crystallization kinetics of the TeO₂/TiO₂/As₂O₃ glassy system was studied under nonisothermal conditions. The method was applied to the experimental data obtained by differential thermal analysis (DTA), using continuous-heating techniques. In addition, two approaches were used to analyze the dependence of glass transition temperature (T _g) on the heating rate (β): One is the empirical linear relationship between (T _g) and (β); The other approach is the use of straight line from the plot of ln( T_\textg² /b ) \textvs . 1/T_\textg \ln \left( {T_{\text{g}}^{2} /\beta } \right)\,{\text{vs}} .\,1/T_{\text{g}} for evaluation of the activation energy for glass transition. The crystallization results are analyzed, and both the activation energy of crystallization process and the crystallization mechanism are characterized.     

Evolution of the microstructure and mechanical properties of eutectic Fe<Subscript>30</Subscript>Ni<Subscript>20</Subscript>Mn<Subscript>35</Subscript>Al<Subscript>15</Subscript>

The microstructure of the eutectic alloy Fe₃₀Ni₂₀Mn₃₅Al₁₅ (in at.%) was modified by cooling at different rates from 1623 K, i.e., above the eutectic temperature. The lamellar spacing decreased with increasing cooling rate, and in water-quenched specimens lamellae widths of ~100 nm were obtained. The orientation relationship between the fcc and B2 lamellae was found to be sensitive to the cooling rate. In a drop-cast alloy the Kurdjumov–Sachs orientation relationship dominated, whereas the orientation relationship in an arc-melted alloy with a faster cooling rate was \textfcc( [`1]12 )//\textB2( 0 1 1 );  \textfcc[ 1[`1]1 ]//\textB2 [ 1[`1]1 ]  \textand \textfcc( 0[`1]1 )//\textB2( 00 1 );\text fcc[ 0 1 1 ]//\textB2[ [`1][`1]0 ] {\text{fcc}}\left( {\bar{1}12} \right)//{\text{B2}}\left( {0 1 1} \right);\;{\text{fcc}}\left[ {1\bar{1}1} \right]//{\text{B2 }}\left[ {1\bar{1}1} \right] \,{\text{and}}\,{\text{fcc}}\left( {0\bar{1}1} \right)//{\text{B2}}\left( {00 1} \right);{\text{ fcc}}\left[ {0 1 1} \right]//{\text{B2}}\left[ {\bar{1}\bar{1}0} \right] . The hardness increased with microstructural refinement, obeying a Hall–Petch-type relationship. The strength of the alloy decreased significantly above 600 K due to softening of the B2 phase.     

Effect of sintering temperature on dielectric properties, vibrational modes and crystal structures of Ba[(Ni0.7Zn0.3)1/3Nb2/3]O3 ceramics

The Ba[(Ni_0.7Zn_0.3)_1/3Nb_2/3]O₃ solid solutions were sintered at 1450, 1500, and 1550 °C for 3 h, respectively, by conventional solid-state sintering method, so as to clarify the effect of the sintering temperatures on vibrational modes, crystal structures, and dielectric properties. Ceramics were characterized by X-ray diffraction (XRD), Raman spectroscopy and Fourier transform far-infrared reflection (FTIR) spectroscopy. Space group and crystal symmetry of Pm[`3]m Pm\bar{3}m were determined by XRD. Lattice vibrational spectra, obtained by Raman and FTIR spectroscopy, show the correlation among polar phonon modes, crystal structures, and dielectric properties of Ba[(Ni_0.7Zn_0.3)_1/3Nb_2/3]O₃ ceramics as a function of the sintering temperatures. The results demonstrate that the dielectric constant e_r \varepsilon_{r} reaches a maximum value of 35.713 at 1500 °C and is related to Raman shifts of the A_1g(O) modes and the FWHM values of the E_g(O) modes, the temperature coefficient of the capacitance t_c \tau_{c} , which gradually increases from −6 × 10⁻⁶/°C to 0, is closely related to the Raman shifts of E_g(O) modes, and the dielectric loss values are closely connected with the full width at half-maximum of E_g(O) active modes and the Raman shifts of of the A_1g(O) modes with the increasing temperatures. FTIR shows that the dielectric properties are closely related to the far-infrared phonon modes. Raman and FTIR active modes were indicated according to the group theory.     

Derivation of a new analytical solution for a general two-dimensional finite-part integral applicable in fracture mechanics

An exact expression is derived for the general finite-part integral over an inclined ellipticaldomain Ω. r denotes the distance of a point in Ω to the singular point $\left({x,y} \right).f = x_{^0 }^i y_0^j \sqrt {Z\left({x_{0,} y_0 }\right)}$ is a general function of the Cartesian co-ordinates x₀,y₀. The boundary of the region Ω represents the equation Z(x₀, y₀)=O. These integrals appear during the numerical solution of plane crack problems in three-dimensional elasticity where they are the dominant part of a hypersingular integral equation. The availability of exact expressions for the integrals with arbitrary integers i and j will increase the accuracy of the numerical results and, simultaneously, lead to quicker numerical results. The considered finite-part integral can be expressed in closed form as function of complete elliptical integrals or Gauss hypergeometric functions, respectively. Formuias for special cases and some i, j values and their numerical verification are given in Appendices II and III.