Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms

In this review paper, we treat the topic of fine gradings of Lie algebras. This concept is important not only for investigating the structural properties of the algebras, but, on top of that, the fine gradings are often used as the starting point for studying graded contractions or deformations of the algebras. One basic question tackled in the work is the relation between the terms 'grading' and 'group grading'. Although these terms have originally been claimed to coincide for simple Lie algebras, it was revealed later that the proof of this assertion was incorrect. Therefore, the crucial statements about one-to-one correspondence between fine gradings and MAD-groups had to be revised and re-formulated for fine group gradings instead. However, there is still a hypothesis that the terms 'grading' and 'group grading' coincide for simple complex Lie algebras. We use the MAD-groups as the main tool for finding fine group gradings of the complex Lie algebras $A_3 \cong D_3$, $B_2 \cong C_2$, and $D_2$. Besides, we develop also other methods for finding the fine (group) gradings. They are useful especially for the real forms of the complex algebras, on which they deliver richer results than the MAD-groups. Systematic use is made of the faithful representations of the three Lie algebras by $4\times 4$ matrices: $A_3 = sl(4,\mathbb C)$, $C_2 = sp(4,\mathbb C)$, $D_2 = o(4,\mathbb C)$. The inclusions $sl(4,\mathbb C)\supset sp(4,\mathbb C)$ and $sl(4,\mathbb C) \supset o(4,\mathbb C)$ are important in our presentation, since they allow to employ one of the methods which considerably simplifies the calculations when finding the fine group gradings of the subalgebras $sp(4,\mathbb C)$ and $o(4,\mathbb C)$.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Distinct Transmission Cycles of Leishmania tropica in 2 Adjacent Foci, Northern Israel

TOC summary for table of contents: Infection with Leishmania tropica is emerging because of encroachment of rock hyraxes and transmission by multiple vector species

Genetics of Host Response to Leishmania tropica in Mice – Different Control of Skin Pathology, Chemokine Reaction, and Invasion into Spleen and Liver

Several hundred million people are exposed to the risk of leishmaniasis, a disease caused by intracellular protozoan parasites of several Leishmania species and transmitted by phlebotomine sand flies. In humans, L. tropica causes cutaneous form of leishmaniasis with painful and long-persisting lesions in the site of the insect bite, but the parasites can also penetrate to internal organs. The relationship between the host genes and development of the disease was demonstrated for numerous infectious diseases. However, the search for susceptibility genes in the human population could be a difficult task. In such cases, animal models may help to discover the role of different genes in interactions between the parasite and the host. Unfortunately, the literature contains only a few publications about the use of animals for L. tropica studies. Here, we report an animal model suitable for genetic, pathological and drug studies in L. tropica infection. We show how the host genotype influences different disease symptoms: skin lesions, parasite dissemination to the lymph nodes, spleen and liver, and increase of levels of chemokines CCL2, CCL3 and CCL5 in serum

On-line algorithms for multiplication and division in real and complex numeration systems

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $\beta$ such that $|\beta|>1$, and the digit set $A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit. In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $\beta$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(\beta, A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $\beta$ and a digit set $A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\# A > |\beta|$. For a complex base $\beta$ and symmetric digit set $A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\# A > \beta\overline{\beta} + |\beta + \overline{\beta}|$. Provided that addition and subtraction are realizable in parallel in the system $(\beta, A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $\beta=\frac{3+\sqrt{5}}{2}$ with digits $A=\{-1,0,1\}$; base $\beta=2i$ with digits $A = \{-2,-1, 0,1,2\}$; and base $\beta = -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + \omega$, where $\omega = \exp{\frac{2i\pi}{3}}$, with digits $A = \{0, \pm 1, \pm \omega, \pm \omega^2 \}$

On-line algorithms for multiplication and division in real and complex numeration systems

A positional numeration system is given by a base and by a set of digits. Thebase is a real or complex number $\beta$ such that $|\beta|>1$, and the digitset $A$ is a finite set of digits including $0$. Thus a number can be seen as afinite or infinite string of digits. An on-line algorithm processes the inputpiece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivediand Ercegovac, is a mode of computation where operands and results flow througharithmetic units in a digit serial manner, starting with the most significantdigit. In this paper, we first formulate a generalized version of the on-linealgorithms for multiplication and division of Trivedi and Ercegovac for thecases that $\beta$ is any real or complex number, and digits are real orcomplex. We then define the so-called OL Property, and show that if $(\beta,A)$ has the OL Property, then on-line multiplication and division are feasibleby the Trivedi-Ercegovac algorithms. For a real base $\beta$ and a digit set$A$ of contiguous integers, the system $(\beta, A)$ has the OL Property if $\#A > |\beta|$. For a complex base $\beta$ and symmetric digit set $A$ ofcontiguous integers, the system $(\beta, A)$ has the OL Property if $\# A >\beta\overline{\beta} + |\beta + \overline{\beta}|$. Provided that addition andsubtraction are realizable in parallel in the system $(\beta, A)$ and thatpreprocessing of the denominator is possible, our on-line algorithms formultiplication and division have linear time complexity. Three examples arepresented in detail: base $\beta=\frac{3+\sqrt{5}}{2}$ with digits$A=\{-1,0,1\}$; base $\beta=2i$ with digits $A = \{-2,-1, 0,1,2\}$; and base$\beta = -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + \omega$, where $\omega =\exp{\frac{2i\pi}{3}}$, with digits $A = \{0, \pm 1, \pm \omega, \pm \omega^2\}$.Comment: Extended version of contribution on 23rd IEEE Symposium on Computer Arithmetic ARITH2

Leishmania tropica in Rock Hyraxes (Procavia capensis) in a Focus of Human Cutaneous Leishmaniasis

Cutaneous leishmaniasis, caused by Leishmania tropica, has recently emerged in urban and rural foci of central and northern Israel, and constitutes a major public health concern. Rock hyraxes (Procavia capensis), the suspected natural reservoir, were trapped in the cutaneous leishmaniasis urban focus of Maale Adumim in central Israel and evaluated for L. tropica infection by real-time kinetoplast DNA (kDNA) polymerase chain reaction (PCR) and serology. Real-time PCR on blood and computerized western blot serology analysis was positive for L. tropica in 58% and 80%, respectively, of the hyraxes tested. Phylogenetic analysis of the ribosomal internal transcribed spacer 1 region indicated that similar genotypes were present in humans and hyraxes from the same habitat. The high rates of infection and exposure to L. tropica among hyraxes supports their involvement in the transmission cycle of this parasite, and their potential role as a reservoir for human disease