The universal Kolyvagin recursion implies the Kolyvagin recursion

let U_z be the universal norm distribution and M a fixed power of prime p, by using the double complex method employed by Anderson, we study the universal Kolyvagin recursion occurred in the canonical basis in the zero-th cohomology group of U_z/M U_z. We furthermore show that the universal Kolyvagin recursion implies the Kolyvagin recursion in the theory of Euler systems(cf. Theorem 4.5.4 of K. Rubin's book Euler systems). One certainly hopes this could lead a new way to find new Euler systems.Comment: 14 page

On the universal norm distribution

We introduce and study the universal norm distribution in this paper, which generalizes the concepts of universal ordinary distribution and the universal Euler system. We study the Anderson type resolution of the universal norm distribution and then use this resolution to study the group cohomology of the universal norm distribution.Comment: 18 pages; correct typos, minor changes in Section 2.1, Proposition 4.5 adde

Spectral sequences of universal distribution and Sinnott's index formula

We prove an abstract index formula about Sinnott's symbol between two different lattices. We also develop the theory of the universal distribution and predistribution in a double complex point of view. The theory of spectral sequence is used to interpret the index formula and to analyze the cohomology of the universal distribution. Combing these results, we successfully prove Sinnott's index formula about the Stickelberger ideal. In addition, the {1, -1}-cohomology groups of the universal distribution and the universal predistribution are obtained.Comment: 23 page

On a conjecture of Wan about limiting Newton polygons

We show that for a monic polynomial $f(x)$ over a number field $K$ containing a global permutation polynomial of degree $>1$ as its composition factor, the Newton Polygon of $f\mod\mathfrak p$ does not converge for $\mathfrak p$ passing through all finite places of $K$. In the rational number field case, our result is the "only if" part of a conjecture of Wan about limiting Newton polygons

A Common Information-Based Multiple Access Protocol Achieving Full Throughput and Linear Delay

We consider a multiple access communication system where multiple users share a common collision channel. Each user observes its local traffic and the feedback from the channel. At each time instant the feedback from the channel is one of three messages: no transmission, successful transmission, collision. The objective is to design a transmission protocol that coordinates the users' transmissions and achieves high throughput and low delay. We present a decentralized Common Information-Based Multiple Access (CIMA) protocol that has the following features: (i) it achieves the full throughput region of the collision channel; (ii) it results in a delay that is linear in the number of users, and is significantly lower than that of CSMA protocols; (iii) it avoids collisions without channel sensing

Newton polygons of LL-functions of polynomials xd+axd−1x^d+ax^{d-1} with p≡−1 mod dp\equiv-1\bmod d

For prime $p\equiv-1\bmod d$ and $q$ a power of $p$, we obtain the slopes of the $q$-adic Newton polygons of $L$-functions of $x^d+ax^{d-1}\in \mathbb{F}_q[x]$ with respect to finite characters $\chi$ when $p$ is larger than an explicit bound depending only on $d$ and $\log_p q$. The main tools are Dwork's trace formula and Zhu's rigid transform theorem.Comment: 8 page

Counting the solutions of λ1x1k1+⋯+λtxtkt≡c mod n\lambda_1 x_1^{k_1}+\cdots +\lambda_t x_t^{k_t}\equiv c\bmod{n}

Given a polynomial $Q(x_1,\cdots, x_t)=\lambda_1 x_1^{k_1}+\cdots +\lambda_t x_t^{k_t}$, for every $c\in \mathbb{Z}$ and $n\geq 2$, we study the number of solutions $N_J(Q;c,n)$ of the congruence equation $Q(x_1,\cdots, x_t)\equiv c\bmod{n}$ in $(\mathbb{Z}/n\mathbb{Z})^t$ such that $x_i\in (\mathbb{Z}/n\mathbb{Z})^\times$ for $i\in J\subseteq I= \{1,\cdots, t\}$. We deduce formulas and an algorithm to study $N_J(Q; c,p^a)$ for $p$ any prime number and $a\geq 1$ any integer. As consequences of our main results, we completely solve: the counting problem of $Q(x_i)=\sum\limits_{i\in I}\lambda_i x_i$ for any prime $p$ and any subset $J$ of $I$; the counting problem of $Q(x_i)=\sum\limits_{i\in I}\lambda_i x^2_i$ in the case $t=2$ for any $p$ and $J$, and the case $t$ general for any $p$ and $J$ satisfying $\min\{v_p(\lambda_i)\mid i\in I\}=\min\{v_p(\lambda_i)\mid i\in J\}$; the counting problem of $Q(x_i)=\sum\limits_{i\in I}\lambda_i x^k_i$ in the case $t=2$ for any $p\nmid k$ and any $J$, and in the case $t$ general for any $p\nmid k$ and $J$ satisfying $\min\{v_p(\lambda_i)\mid i\in I\}=\min\{v_p(\lambda_i)\mid i\in J\}$.Comment: 22 page

Linear complexity of generalized cyclotomic sequences of period 2pm2p^{m}

In this paper, we construct two generalized cyclotomic binary sequences of period $2p^{m}$ based on the generalized cyclotomy and compute their linear complexity, showing that they are of high linear complexity when $m\geq 2$

Signaling for Decentralized Routing in a Queueing Network

A discrete-time decentralized routing problem in a service system consisting of two service stations and two controllers is investigated. Each controller is affiliated with one station. Each station has an infinite size buffer. Exogenous customer arrivals at each station occur with rate $\lambda$. Service times at each station have rate $\mu$. At any time, a controller can route one of the customers waiting in its own station to the other station. Each controller knows perfectly the queue length in its own station and observes the exogenous arrivals to its own station as well as the arrivals of customers sent from the other station. At the beginning, each controller has a probability mass function (PMF) on the number of customers in the other station. These PMFs are common knowledge between the two controllers. At each time a holding cost is incurred at each station due to the customers waiting at that station. The objective is to determine routing policies for the two controllers that minimize either the total expected holding cost over a finite horizon or the average cost per unit time over an infinite horizon. In this problem there is implicit communication between the two controllers; whenever a controller decides to send or not to send a customer from its own station to the other station it communicates information about its queue length to the other station. This implicit communication through control actions is referred to as signaling in decentralized control. Signaling results in complex communication and decision problems. In spite of the complexity of signaling involved, it is shown that an optimal signaling strategy is described by a threshold policy which depends on the common information between the two controllers; this threshold policy is explicitly determined

A note on cyclotomic Euler systems and the double complex method

In this note we give the Kolyvagin recursion in cyclotomic Euler systens a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of the chosen field satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution a la Kubert. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion.Comment: 17 page