Gröbner bases with coefficients in rings

This article gives a short introduction to the theory of Gröbner bases in a class of rings, which includes rings of differential operators and polynomial rings over commutative noetherian rings. A definition of reduced Gröbner bases for these rings is proposed.     

An efficient algorithm for decomposing multivariate polynomials and its applications to cryptography

In this paper, we present an efficient and general algorithm for decomposing multivariate polynomials of the same arbitrary degree. This problem, also known as the Functional Decomposition Problem (FDP), is classical in computer algebra. It is the first general method addressing the decomposition of multivariate polynomials (any degree, any number of polynomials). As a byproduct, our approach can be also used to recover an ideal I from its kth power I^k. The complexity of the algorithm depends on the ratio between the number of variables (n) and the number of polynomials (u). For example, polynomials of degree four can be decomposed in , when this ratio is smaller than . This work was initially motivated by a cryptographic application, namely the cryptanalysis of 2R⁻ schemes. From a cryptographic point of view, the new algorithm is so efficient that the principle of two-round schemes, including 2R⁻ schemes, becomes useless. Besides, we believe that our algorithm is of independent interest.     

Computations in the ring of quaternionic polynomials

In this paper we study basic division properties in the ring of regular quaternionic polynomials. We obtain a Bezout-like theorem and we calculate the module syzygy for any vector of polynomials.     

Computing difference-differential dimension polynomials by relative Gröbner bases in difference-differential modules

In this paper we present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural double filtration. The approach is based on a method of special Gröbner bases with respect to “generalized term orders” on N^m×Zⁿ${\mathbb{N}}^m\times {\mathbb{Z}}^n$ and on difference-differential modules. We define a special type of reduction for two generalized term orders in a free left module over a ring of difference-differential operators. Then the concept of relative Gröbner bases w.r.t. two generalized term orders is defined. An algorithm for constructing these relative Gröbner bases is presented and verified. Using relative Gröbner bases, we are able to compute difference-differential dimension polynomials in two variables.     

Embedding k(n−k) edge-disjoint spanning trees in arrangement graphs

The arrangement graphs are a class of generalized star graphs. In this paper we construct a graph that consists of the maximum number of directed edge-disjoint spanning trees in an arrangement graph. The paths that connect the common root node to any given node through different spanning trees are node-disjoint, and the lengths of these paths differ from the shortest possible lengths by a small additive constant. This graph can be used to derive fault-tolerant algorithms for broadcasting and scattering problems without prior knowledge of the faulty elements of the network.     

(h,k)-Arbiters for h-out-of-k mutual exclusion problem

h-Out-of-k mutual exclusion is a generalization of the 1-mutual exclusion problem, where there are k units of shared resources and each process requests h (1hk) units at the same time. Though k-arbiter has been shown to be a quorum-based solution to this problem, quorums in k-arbiter are much larger than those in the 1-coterie for 1-mutual exclusion. Thus, the algorithm based on k-arbiter needs many messages. This paper introduces the new notion that each request uses different quorums depending on the number of units of its request. Based on the notion, this paper defines two (h,k)-arbiters for h-out-of-k mutual exclusion: a uniform (h,k)-arbiter and a (k+1)-cube (h,k)-arbiter. The quorums in each (h,k)-arbiter are not larger than the ones in the corresponding k-arbiter; consequently, it is more efficient to use (h,k)-arbiters than the k-arbiters. A uniform (h,k)-arbiter is a generalization of the majority coterie for 1-mutual exclusion. A (k+1)-cube (h,k)-arbiter is a generalization of square grid coterie for 1-mutual exclusion.     

一种O(2.983n)时间复杂度的最优联盟结构生成算法

首先,在有限整数集上建立有效拆分关系,在联盟集上建立有效二部分解关系,并设计了一种EOCS(effective optimal coalition structure)算法.该算法采用自底向上方式,只对具有有效二部分解关系的联盟进行二部分解来求联盟的优值,从而降低了二部分解的数量.随后,利用函数的克林闭包特性证明了EOCS算法的正确性,利用积分极限定理证明了EOCS算法时间复杂度的下界是O(2.818n),用时间序列分析方法求出了EOCS算法的上界是O(2.983n).最后,将EOCS算法与其他算法作了对比,指出无论联盟值满足何种概率分布,EOCS算法都能在O(2.983n)时间内找出最优联盟结构.Rothkopf提出的DP(dynamic programming)算法和Rahwan提出的IDP(improved dynamic programming)算法能够在O(3n)时间内求出最优联盟结构.所作的EOCS算法设计、正确性证明、时间复杂度的上下界分析都是对Rothkopf及Rahwan等人相关工作的改进和提高.     

一类可分离SAT问题的O(1.890n)精确算法

布尔可满足性问题（SAT）是指对于给定的布尔公式,是否存在一个可满足的真值指派.这是第1个被证明的NP完全问题,一般认为不存在多项式时间算法,除非P=NP.学者们大都研究了子句长度不超过k的SAT问题（k-SAT）,从全局搜索到局部搜索,给出了大量的相对有效算法,包括随机算法和确定算法.目前,最好算法的时间复杂度不超过O（（2-2/k）ⁿ）,当k=3时,最好算法时间复杂度为O（1.308ⁿ）.而对于更一般的与子句长度k无关的SAT问题,很少有文献涉及.引入了一类可分离SAT问题,即3-正则可分离可满足性问题（3-RSSAT）,证明了3-RSSAT是NP完全问题,给出了一般SAT问题3-正则可分离性的O（1.890ⁿ）判定算法.然后,利用矩阵相乘算法的研究成果,给出了3-RSSAT问题的O（1.890ⁿ）精确算法,该算法与子句长度无关.     

The Inf-Sup Condition for the Mapped Q k ?P k?1 disc Element in Arbitrary Space Dimensions

 One of the most popular pairs of finite elements for solving mixed formulations of the Stokes and Navier–Stokes problem is the Q _k −P _k−1 ^disc element. Two possible versions of the discontinuous pressure space can be considered: one can either use an unmapped version of the P _k−1 ^disc space consisting of piecewise polynomial functions of degree at most k−1 on each cell or define a mapped version where the pressure space is defined as the image of a polynomial space on a reference cell. Since the reference transformation is in general not affine but multilinear, the two variants are not equal on arbitrary meshes. It is well-known, that the inf-sup condition is satisfied for the first variant. In the present paper we show that the latter approach satisfies the inf-sup condition as well for k≥2 in any space dimension. Received January 31, 2001; revised May 2, 2002 Published online: July 26, 2002     

Exact 3-satisfiability is decidable in time O(20.16254n)

Let F = C ₁ C _m be a Boolean formula in conjunctive normal form over a set V of n propositional variables, s.t. each clause C _i contains at most three literals l over V. Solving the problem exact 3-satisfiability (X3SAT) for F means to decide whether there is a truth assignment setting exactly one literal in each clause of F to true (1). As is well known X3SAT is NP-complete [6]. By exploiting a perfect matching reduction we prove that X3SAT is deterministically decidable in time O(2^0.18674n ). Thereby we improve a result in [2,3] stating X3SAT O(2^0.2072n ) and a bound of O(2^0.200002n ) for the corresponding enumeration problem #X3SAT stated in a preprint [1]. After that by a more involved deterministic case analysis we are able to show that X3SAT O(2^0.16254n ).An extended abstract of this paper was presented at the Fifth International Symposium on the Theory and Applications of Satisfiability Testing (SAT 2002).     

MAX(1)和MARG(1)中公式改名的复杂性

改名是一个将变元映射到变元本身或它的补的函数,变元改名是公式变元集合上的一个置换,文字改名是一个改名和一个变元改名的组合.研究CNF公式的改名有助于改进DPLL算法.考虑判定问题"对于给定的CNF公式H和F是否存在一个变元(或文字)改名ψ使得ψ(H)=F?"的计算复杂性.MAX(1)和MARG(1)是极小不可满足公式的两个子类,这两个子类中的公式可以用树表示.树同构的判定问题在线性时间内是可解的.证明了对于MAX(1)和MARG(1)中的公式,文字改名问题在线性时间内可解,变元改名问题在平方次时间内可解.     

Build shape-shifting tries for fast IP lookup in O(n) time

Most data structures for packet forwarding are optimized for IPv4, and less capable of handling IPv6 routing tables. Shape-Shifting Trie (SST) is specifically designed to be scalable to IPv6. To reduce the worst-case lookup time that is proportional to the height of SST, it is desirable to construct a minimum-height SST. Breadth-First Pruning (BFP) algorithm takes O(n²) time to construct a minimum-height SST, where n is the number of nodes in the binary trie corresponding to the routing table. In this paper, we propose a Post-Order Minimum-Height Pruning (POMHP) algorithm that takes only O(n) time to construct a minimum-height SST. We further propose nodes merging algorithm to cut down SST size without affecting SST height.     

A FORTRAN program for plotting mineral stabilities in the Fe---Cu---S---O system in terms of log ( ) or log fO2 vs pH or T

A computer program for the generation of mineral-stability diagrams in terms of log ( ) vs pH or T is presented. Simple modifications of the program to produce log fO₂ vs pH or T diagrams also are documented. Such diagrams are useful particularly in the geochemical interpretation of hydrothermal sulfide ore deposits. Plotting of other geochemical parameters such as log fS₂, mole fractions of aqueous sulfur species, sulfur isotopic compositions, and metal complex solubilities also is possible utilizing the mineral-stability diagram as a base. Both line-printer and digital-plotter methods are explained.     

A semi-empirical model for predicting hourly ground-level fine particulate matter (PM2.5) concentration in southern Ontario from satellite remote sensing and ground-based meteorological measurements

A semi-empirical model is developed to predict the hourly concentration of ground-level fine particulate matter (PM_2.5) coincident to satellite overpass, at a regional scale. The model corrects the aerosol optical depth (AOD) data from the Moderate Resolution Imaging Spectroradiometer (MODIS) by the assimilated parameters characterizing the boundary layer and further adjusts the corrected value according to meteorological conditions near the ground. The model was built and validated using the data collected for southern Ontario, Canada for 2004. Overall, the model is able to explain 65% of the variability in ground-level PM_2.5 concentration. The model-predicted values of PM_2.5 mass concentration are highly correlated with the actual observations. The root-mean-square error of the model is 6.1 µg/m³. The incorporation of ground-level temperature and relative humidity is found to be significant in improving the model predictability. The coarse resolution of the assimilated meteorological fields limits their value in the AOD correction. Although MODIS AOD data is acquired on a daily basis and the valid data coverage can sometimes be very limited due to unfavourable weather conditions, the model provides a cost-effective approach for obtaining supplemental PM_2.5 concentration information in addition to the ground-based monitoring station measurement.     

In2O3 + xBaO (x = 0.5-5 at.%) - A novel material for trace level detection of NOx in the ambient

Indium oxide (In₂O₃) doped with 0.5-5 at.% of Ba was examined for their response towards trace levels of NO_x in the ambient. Crystallographic phase studies, electrical conductivity and sensor studies for NO_x with cross interference for hydrogen, petroleum gas (PG) and ammonia were carried out. Bulk compositions with x ≤ 1 at.% of Ba exhibited high response towards NO_x with extremely low cross interference for hydrogen, PG and ammonia, offering high selectivity. Thin films of 0.5 at.% Ba doped In₂O₃ were deposited using pulsed laser deposition technique using an excimer laser (KrF) operating at a wavelength of (λ) 248 nm with a fluence of ∼3 J/cm² and pulsed at 10 Hz. Thin film sensors exhibited better response towards 3 ppm NO_x quite reliably and reproducibly and offer the potential to develop NO_x sensors (Threshold limit value of NO₂ and NO is 3 and 25 ppm, respectively).     

Poincaré–Birkhoff–Witt theorem for Leibniz -algebras

We construct the universal enveloping algebra of a Leibniz n-algebra and we prove that the category of modules over this algebra is equivalent to the category of representations.We also give a proof of the Poincaré–Birkhoff–Witt theorem for universal enveloping algebras of finite-dimensional Leibniz n-algebras using Gröbner bases in a free associative algebra.