Parallel addition in non-standard numeration systems

We consider numeration systems where digits are integers and the base is an algebraic number $\beta$ such that $|\beta|>1$ and $\beta$ satisfies a polynomial where one coefficient is dominant in a certain sense. For this class of bases $\beta$, we can find an alphabet of signed-digits on which addition is realizable by a parallel algorithm in constant time. This algorithm is a kind of generalization of the one of Avizienis. We also discuss the question of cardinality of the used alphabet, and we are able to modify our algorithm in order to work with a smaller alphabet. We then prove that $\beta$ satisfies this dominance condition if and only if it has no conjugate of modulus 1. When the base $\beta$ is the Golden Mean, we further refine the construction to obtain a parallel algorithm on the alphabet $\{-1,0,1\}$. This alphabet cannot be reduced any more

kk-block parallel addition versus 11-block parallel addition in non-standard numeration systems

Parallel addition in integer base is used for speeding up multiplication and division algorithms. $k$-block parallel addition has been introduced by Kornerup in 1999: instead of manipulating single digits, one works with blocks of fixed length $k$. The aim of this paper is to investigate how such notion influences the relationship between the base and the cardinality of the alphabet allowing parallel addition. In this paper, we mainly focus on a certain class of real bases --- the so-called Parry numbers. We give lower bounds on the cardinality of alphabets of non-negative integer digits allowing block parallel addition. By considering quadratic Pisot bases, we are able to show that these bounds cannot be improved in general and we give explicit parallel algorithms for addition in these cases. We also consider the $d$-bonacci base, which satisfies the equation $X^d = X^{d-1} + X^{d-2} + \cdots + X + 1$. If in a base being a $d$-bonacci number $1$-block parallel addition is possible on the alphabet $\mathcal{A}$, then $\#\mathcal{A} \geq d+1$; on the other hand, there exists a $k\in\mathbb{N}$ such that $k$-block parallel addition in this base is possible on the alphabet $\{0,1,2\}$, which cannot be reduced. In particular, addition in the Tribonacci base is $14$-block parallel on alphabet $\{0,1,2\}$.Comment: 21 page

From positional representation of numbers to positional representation of vectors

To represent real m-dimensional vectors, a positional vector system given by a non-singular matrix M ∈ ℤm×m and a digit set Ɗ ⊂ ℤm is used. If m = 1, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which are transformable from the case m = 1 to higher dimensions. We focus on an algorithm for parallel addition and on systems allowing an eventually periodic representation of vectors with rational coordinates

Theoretical Informatics and Applications Will be set by the publisher Informatique Théorique et Applications MINIMAL DIGIT SETS FOR PARALLEL ADDITION IN NON-STANDARD NUMERATION SYSTEMS

Abstract. We study parallel algorithms for addition of numbers having finite representation in a positional numeration system defined by a base β in C and a finite digit set A of contiguous integers containing 0. For a fixed base β, we focus on the question of the size of the alphabet allowing to perform addition in constant time independently of the length of representation of the summands. We produce lower bounds on the size of such alphabet A. For several types of well studied bases (negative integer, complex numbers −1 + ı, 2ı, and ı √ 2, quadratic Pisot unit, and the non-integer rational base), we give explicit parallel algorithms performing addition in constant time. Moreover we show that digit sets used by these algorithms are the smallest possible

Distinct genetic control of parasite elimination, dissemination, and disease after Leishmania major infection

Elimination of pathogens is the basis of host resistance to infections; however, relationship between persisting pathogens and disease has not been clarified. Leishmania major infection in mice is an important model of host–pathogen relationship. Infected BALB/c mice exhibit high parasite numbers in lymph nodes and spleens, and a chronic disease with skin lesions, splenomegaly, and hepatomegaly, increased serum IgE levels and cytokine imbalance. Although numerous gene loci affecting these disease symptoms have been reported, genes controlling parasites’ elimination or dissemination have never been mapped. We therefore compared genetics of the clinical and immunologic symptomatology with parasite load in (BALB/c × CcS-11) F2 hybrids and mapped five loci, two of which control parasite elimination or dissemination. Lmr5 influences parasite loads in spleens (and skin lesions, splenomegaly, and serum IgE, IL-4, and IFNγ levels), and Lmr20 determines parasite numbers in draining lymph nodes (and serum levels of IgE and IFNγ), but no skin or visceral pathology. Three additional loci do not affect parasite numbers but influence significantly the disease phenotype—Lmr21: skin lesions and IFNγ levels, Lmr22: IL-4 levels, Lmr23: IFNγ levels, indicating that development of L. major-caused disease includes critical regulations additional to control of parasite spread

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

Genetic Control of Resistance to Trypanosoma brucei brucei Infection in Mice

Trypanosoma brucei are extracellular protozoa transmitted to mammalian host by the tsetse fly. They developed several mechanisms that subvert host's immune defenses. Therefore analysis of genes affecting host's resistance to infection can reveal critical aspects of host-parasite interactions. Trypanosoma brucei brucei infects many animal species including livestock, with particularly severe effects in horses and dogs. Mouse strains differ greatly in susceptibility to T. b. brucei. However, genes controlling susceptibility to this parasite have not been mapped. We analyzed the genetic control of survival after T. b. brucei infection using CcS/Dem recombinant congenic (RC) strains, each of which contains a different random set of 12.5% genes of their donor parental strain STS/A on the BALB/c genetic background. The RC strain CcS-11 is even more susceptible to parasites than BALB/c or STS/A. In F2 hybrids between BALB/c and CcS-11 we detected and mapped four loci, Tbbr1-4 (Trypanosoma brucei brucei response 1–4), that control survival after T. b. brucei infection. Tbbr1 (chromosome 3) and Tbbr2 (chromosome 12) have independent effects, Tbbr3 (chromosome 7) and Tbbr4 (chromosome 19) were detected by their mutual inter-genic interaction. Tbbr2 was precision mapped to a segment of 2.15 Mb that contains 26 genes