Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, and Cp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. The p-adic norm is normalized so that |p|p 1/p. Let N be the set of natural numbers and Z N ∪ {0}. Let UD Zp be the space of uniformly differentiable functions on Zp. For f ∈ ...

1. A review of the classical setting 2. Elliptic units for real quadratic fields 2.1. p-adic measures 2.2. Double integrals 2.3. Splitting a two-cocycle 2.4. The main conjecture 2.5. Modular symbols and Dedekind sums 2.6. Measures and the Bruhat-Tits tree 2.7. Indefinite integrals 2.8. The action of complex conjugation and of Up 3. Special values of zeta functions 3.1. The zeta function 3.2. Va...

and Applied Analysis 3 By 2.2 , we get μf,−q ( a pZp ) 2 q 2 qp −qa ∫ Zp f ( a px ) dμ−qpn x . 2.3 Therefore, by 2.3 , we obtain the following theorem. Theorem 2.1. For f, g ∈ C Zp , one has μαf βg,−q αμf,−q βμg,−q, 2.4 where α, β are constants. From 2.2 and 2.4 , we note that ∣μf,−q ( a pZp )∣∣ ≤ M‖f‖∞, 2.5 where ‖f‖∞ supx∈Zp |f x | and M is some positive constant. Now, we recall the definitio...

Let $p$ be an analytic function defined on the open unit disc $mathbb{D}$ with $p(0)=1.$ The conditions on $alpha$ and $beta$ are derived for $p(z)$ to be subordinate to $1+4z/3+2z^{2}/3=:varphi_{C}(z)$ when $(1-alpha)p(z)+alpha p^{2}(z)+beta zp'(z)/p(z)$ is subordinate to $e^{z}$. Similar problems were investigated for $p(z)$ to lie in a region bounded by lemniscate of Bernoulli $|w^{2}-1|=1$ ...

BACKGROUND The mammalian oocyte extracellular matrix known as the zona pellucida (ZP) acts as a barrier to accomplish sperm fusion with the female gamete. Although penetration of the ZP is a limiting event to achieve fertilization, this is one of the least comprehended stages of gamete interaction. Even though previous studies suggest that proteases of sperm origin contribute to facilitate the ...

Background: The slow-freezing method is one of the most popular techniques to freeze human oocytes and it is also used in routine in vitro-fertilization (IVF) programs, particularly in those countries where embryo freezing is not allowed. Slow-freezing damages some of the structures of the egg, among which the meiotic spindle (MS) and the zona pellucida (ZP). Polarized Light Microscopy (PLM) pe...

Zona pellucida (ZP) hardening (resistance to proteolysis) has been classically identified as a post-fertilization event that contributes to the block to polyspermy. Di-(N-succinimidyl)-3,3'-dithiodipropionate (DSP), a permeable amine-reactive cross-linker, was recently shown to induce pre-fertilization ZP hardening and to improve porcine IVF productivity. The objectives of this study were to in...

BACKGROUND The aim of this study was to determine the effect of culture medium osmolality, in the range known to occur in the male and female reproductive tracts, on human sperm tyrosine phosphorylation and sperm-zona pellucida (ZP) interaction in vitro. METHODS Motile sperm (2x10(6)), selected by swim-up from semen of normozoospermic men with normal sperm-ZP binding, were incubated with or w...

Zona pellucida (ZP) is an extracellular matrix surrounding and protecting mammalian and fish oocytes, which is responsible for sperm binding. Mammalian ZP consists of three to four glycoproteins, called ZP1, ZP2, ZP3, ZP4. These proteins polymerize into long interconnected filaments, through a common structural unit, known as the ZP domain, which consists of two domains, ZP-N and ZP-C. ZP is re...

We show that a closed orientable 3-manifold M admits an action of Zp with fixed point set S 1 iff M can be obtained as the result of surgery on a p-periodic framed link L and Zp acts freely on the components of L. We prove a similar theorem for free Zp-actions. As an interesting application, we prove the following, rather unexpected result: for any M as above and for any odd prime p, H1(M,Zp) 6...