Abstract:
Introduction. Today, many medical procedures are based on image analysis with the aim of providing accurate diagnosis and optimal treatment. The goal of this report was to present clinical implications of computer-assisted geometric design of carotid aneurysms. Material and methods. In this study, the three-dimensional reconstruction was based on the approximation power of the parametric spline function, which achieves interpolation and surface fitting of the arterial information obtained by conventional angiography. Two views of conventional angiograms (anterioposterior and lateral views) were used with a personal computer and commercial software. Results. This method of three-dimensional generated images was applied in 45 cases of cerebral aneurysms in carotid bifurcation. 3D reconstructions were made in approximately 20 minutes. They contributed to assessing vascular structures, and it was possible to rotate the three-dimensional image in different angles. Conclusion. Three-dimensional reconstruction of cerebral vessels is very useful for discussing surgical strategies preoperatively. Furthermore, it could also be used in endovascular procedures. .

Abstract:
The theoretical aspects of spin-rotation coupling are presented. The approach is based on the general covariance principle. It is shown that the gyrogravitational ratio of the bare spin-1/2 and the spin-1 particles is equal unity. That is why spin couples with rotation as an ordinary angular momentum. This result is the rigorous substantiation of the cranking model. To observe the phenomenon, the experiment with hydrogen-like ions in a storage ring is suggested. It is found that the splitting of the $1 ^2!S_{1/2}, F=1/2$ hyperfine state of the $^{140}{\rm Pr}^{58+}$ and $^{142}{\rm Pm}^{60+}$ ions circulating in the storage ring ESR in Darmstadt along a helical trajectory is about 4.5 MHz. We argue that such splitting can be experimentally determined by means of the ionic interferometry.

Abstract:
The previously obtained analytical asymptotic expressions for the Gell-Mann - Low function \beta(g) and anomalous dimensions of \phi^4 theory in the limit g\to\infty are based on the parametric representation of the form g = f(t), \beta(g) = f1(t) (where t\sim g_0^{-1/2} is the running parameter related to the bare charge g_0), which is simplified in the complex t plane near a zero of one of the functional integrals. In the present paper, it is shown that the parametric representation has a singularity at t\to 0; for this reason, similar results can be obtained for real values of g_0. The problem of the correct transition to the strong coupling regime is simultaneously solved; in particular, the constancy of the bare or renormalized mass is not a correct condition of this transition. A partial proof is given for the theorem of the renormalizability in the strong coupling region.

Abstract:
With the aim of applications to solving general integral equations, we introduce and study in this paper a special class of bi-Carleman kernels on $\mathbb{R}\times\mathbb{R}$, called $K^\infty$ kernels of Mercer type, whose property of being infinitely smooth is stable under passage to certain left and right multiples of their associated integral operators. An expansion theorem in absolutely and uniformly convergent bilinear series concerning kernels of this class is proved extending to a general non-Hermitian setting both Mercer's and Kadota's Expansion Theorems for positive definite kernels. Another theorem proved in this paper identifies families of those bounded operators on a separable Hilbert space $\mathcal{H}$ that can be simultaneously transformed by the same unitary equivalence transformation into bi-Carleman integral operators on $L^2(\mathbb{R})$, whose kernels are $K^\infty$ kernels of Mercer type; its singleton version implies in particular that any bi-integral operator is unitarily equivalent to an integral operator with such a kernel.

Abstract:
In this paper, we reduce the general linear integral equation of the third kind in $L^2(Y,\mu)$, with largely arbitrary kernel and coefficient, to an equivalent integral equation either of the second kind or of the first kind in $L^2(\mathbb{R})$, with the kernel being the linear pencil of bounded infinitely differentiable bi-Carleman kernels expandable in absolutely and uniformly convergent bilinear series. The reduction is done by using unitary equivalence transformations.

Abstract:
In this paper, we study an infinite system of Fredholm series of polynomials in $\lambda$, formed, in the classical way, for a continuous Hilbert-Schmidt kernel on $\mathbb{R}\times\mathbb{R}$ of the form $\boldsymbol{H}(s,t)-\lambda\boldsymbol{S}(s,t)$, where $\lambda$ is a complex parameter. We prove a convergence of these series in the complex plane with respect to sup-norms of various spaces of continuous functions vanishing at infinity. The convergence results enable us to solve explicitly an integral equation of the second kind in $L^2(\mathbb{R})$, whose kernel is of the above form, by mimicking the classical Fredholm-determinant method.

Abstract:
We prove that, at regular values lying in a strong convergence region, the resolvent kernels for a continuous bi-Carleman kernel vanishing at infinity can be expressed as uniform limits of sequences of resolvent kernels for its approximating subkernels of Hilbert-Schmidt type

Abstract:
In this paper, we give a characterization of all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral operator in $L_2(R)$ with bounded and arbitrarily smooth Carleman kernel on $R^2$. In addition, we give an explicit construction of corresponding unitary operators.

Abstract:
In this paper, we characterize the families of those bounded linear operators on a separable Hilbert space which are simultaneously unitarily equivalent to integral bi-Carleman operators on $L_2(R)$ having arbitrarily smooth kernels of Mercer type. The main result is a qualitative sharpening of an earlier result of [7].