Abstract:
A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some $\rho > 0$, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility $\chi_{1,1}$ has been evaluated exactly. For $\rho =1$ we find that $\chi_{1,1}$ is finite at the bulk critical temperature $T_c$, in contrast with the recently derived value $\gamma_{1,1}=1$ in the case of just one global spherical constraint. The result $\gamma_{1,1}=1$ is recovered only if $\rho =\rho_c= 2-(12 K_c)^{-1}$, where $K_c$ is the dimensionless critical coupling. When $\rho > \rho_c$, $\chi_{1,1}$ diverges exponentially as $T\to T_c^{+}$. An effective hamiltonian which leads to an exactly solvable model with $\gamma_{1,1}=2$, the value for the $n\to \infty $ limit of the corresponding O(n) model, is proposed too.

Abstract:
A d-dimensional finite quantum model system confined to a general hypercubical geometry with linear spatial size L and ``temporal size'' 1/T (T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. Because of its close relation with the system of quantum rotors it represents an effective model for studying the low-temperature behaviour of quantum Heisenberg antiferromagnets. Close to the zero-temperature quantum critical point the ideas of finite-size scaling are used for studying the critical behaviour of the model. For a film geometry in different space dimensions $\half\sigma

Abstract:
The scaling properties of the free energy and some of universal amplitudes of a group of models belonging to the universality class of the quantum nonlinear sigma model and the O(n) quantum $\phi^4$ model in the limit $n\to \infty$ as well as the quantum spherical model, with nearest-neighbor and long-range interactions (decreasing at long distances $r$ as $1/r^{d+\sigma}$) is presented.

Abstract:
A $d$-dimensional quantum model system confined to a general hypercubical geometry with linear spatial size $L$ and ``temporal size'' $1/T $ ($T$ - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. For a film geometry in different space dimensions $\frac 12\sigma

Abstract:
A $d$--dimensional quantum model in the spherical approximation confined to a general geometry of the form $L^{d-d^{\prime}} \times\infty^{d^{\prime}}\times L_{\tau}^{z}$ ($L$--linear space size and $L_{\tau}$--temporal size) and subjected to periodic boundary conditions is considered. Because of its close relation with the quantum rotors model it can be regarded as an effective model for studying the low-temperature behavior of the quantum Heisenberg antiferromagnets. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1

Abstract:
We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the $d$-dimensional spherical model with a power law long-range interaction decaying at large distances $r$ as $r^{-d-\sigma}$, where $\sigmaT_c$ and $TT_c$ it decays as $L^{-d-\sigma}$, where $L$ is the thickness of the film. We consider both the case of a finite system that has no phase transition of its own, when $d-1<\sigma$, as well as the case with $d-1>\sigma$, when one observes a dimensional crossover from $d$ to a $d-1$ dimensional critical behavior. The behavior of the force along the phase coexistence line for a magnetic field H=0 and $T1$ and a decreasing one for $\sigma<1$. For any $d$ and $\sigma$ the minimum of the force at $T=T_c$ is always achieved at some $H\ne 0$.

Abstract:
The three dimensional mean spherical model on a hypercubic lattice with a film geometry $L\times \infty ^2$ under periodic boundary conditions is considered in the presence of an external magnetic field $H$. The universal Casimir amplitude $\Delta $ and the Binder's cumulant ratio $B$ are calculated exactly and found to be $\Delta =-2\zeta (3)/(5\pi)\approx -0.153051$ and $B=2\pi /(\sqrt{5}\ln ^3[(1+\sqrt{5})/2]).$ A discussion on the relations between the finite temperature $C$-function, usually defined for quantum systems, and the excess free energy (due to the finite-size contributions to the free energy of the system) scaling function is presented. It is demonstrated that the $C$-function of the model equals 4/5 at the bulk critical temperature $T_c$. It is analytically shown that the excess free energy is a monotonically increasing function of the temperature $T$ and of the magnetic field $|H|$ in the vicinity of $T_c.$ This property is supposed to hold for any classical $d$-dimensional $O(n),n>2,$ model with a film geometry under periodic boundary conditions when $d\leq 3$. An analytical evidence is also presented to confirm that the Casimir force in the system is negative both below and in the vicinity of the bulk critical temperature $T_c.$

Abstract:
The behavior of the finite-temperature C-function, defined by Neto and Fradkin [Nucl. Phys. B {\bf 400}, 525 (1993)], is analyzed within a d -dimensional exactly solvable lattice model, recently proposed by Vojta [Phys. Rev. B {\bf 53}, 710 (1996)], which is of the same universality class as the quantum nonlinear O(n) sigma model in the limit $n\to \infty$. The scaling functions of C for the cases d=1 (absence of long-range order), d=2 (existence of a quantum critical point), d=4 (existence of a line of finite temperature critical points that ends up with a quantum critical point) are derived and analyzed. The locations of regions where C is monotonically increasing (which depend significantly on d) are exactly determined. The results are interpreted within the finite-size scaling theory that has to be modified for d=4. PACS number(s): 05.20.-y, 05.50.+q, 75.10.Hk, 75.10.Jm, 63.70.+h, 05.30-d, 02.30

Abstract:
let r be a commutative unitary ring of prime characteristic p which is a direct product of indecomposable subrings and let g be a multiplicative abelian group such that g0/gp is nite. we characterize the isomorphism class of the unit group u(rg) of the group algebra rg. this strengthens recent results due to mollov-nachev (commun. algebra, 2006) and danchev (studia babes bolyai - mat., 2011).

Abstract:
. An extension in the terms of $\sigma$-summable abelian $p$-groups of the classical Dieudonn\'e criterion (Portugaliae Mathematica, 1952) for direct sums of \hbox{$p$-cyclics} (= cyclic $p$-groups) is given. Specifically, it is proved that $G$ is a $\sigma$-summable abelian $p$-group if $A$ is its balanced $\sigma$-summable abelian $p$-subgroup so that $G/A$ is a \hbox{$\sigma$-summable} abelian $p$-group.