We show there exists a continuous compactly supported function ' de ned on the reals satsifying '(x) = c0'(2x) + c1'(2x 1) + c2'(2x 2) if and only if c1 = 1, c0 + c2 = 1, and jc0j; jc2j < 1. The functions ' are examples of scaling functions from the theory of wavelets. x

In this paper, we obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary t-norms f(2x+ y) + f(2x− y) = 4[f(x + y) + f(x− y)] + 2[f(2x) − 4f(x)]− 6f(y).

In this paper, we obtain the general solution and the generalized Hyers-Ulam Rassias stability of the functional equation f(2x+ y) + f(2x− y) = 4(f(x+ y) + f(x− y))− 3 7 (f(2y)− 2f(y)) + 2f(2x) − 8f(x).

BACKGROUND AND AIMS Dioscorea alata is a polyploid species with a ploidy level ranging from diploid (2n = 2x = 40) to tetraploid (2n = 4x = 80). Ploidy increase is correlated with better agronomic performance. The lack of knowledge about the origin of D. alata spontaneous polyploids (triploids and tetraploids) limits the efficiency of polyploid breeding. The objective of the present study was t...

The present paper deals with meiotic studies in 15 species belonging to 6 genera of the tribe Cichorieae from various localities of Western Himalayas. The chromosome number has been reported for the first time in Hieracium crocatum (2n = 10) and Lactuca lessertiana (2n = 2x = 16). Further, intraspecific variability has been reported for the first time in H. umbellatum (2n = 2x = 10 and 2n = 6x ...

Penicillin-binding protein 2x (PBP 2x) of Streptococcus pneumoniae is one of the high-molecular-weight PBPs involved in the development of intrinsic beta-lactam resistance. Point mutations in the PBP 2x genes (pbpX) have now been characterized in five independent spontaneous laboratory mutants in order to identify protein regions which are important for interaction with beta-lactam antibiotics....

Recent advances in plant genetic engineering offer substantial benefits to farmers throughout the world. Genetic research has identified many exogenous genes that could considerably decrease production costs through transgene-mediated resistance to insect, viral, fungal and bacterial pathogens. Potato can be produced from true potato seed (TPS) through a sexual polyploidization step, known as 4...

We give a systematic overview of techniques to compute efficient arithmetic modulo 2p±1. This is useful for computations in the supersingular isogeny Diffie-Hellman (SIDH) keyexchange protocol which is one of the more recent contenders in the post-quantum public-key arena. One of the main computational bottlenecks in this key-exchange protocol is computing modular arithmetic in a finite field d...