Dynamic adjustment: an electoral method for relaxed double proportionality

We describe an electoral system for distributing seats in a parliament. It gives proportionality for the political parties and close to proportionality for constituencies. The system suggested here is a version of the system used in Sweden and other Nordic countries with permanent seats in each constituency and adjustment seats to give proportionality on the national level. In the national election of 2010 the current Swedish system failed to give proportionality between parties. We examine here one possible cure for this unwanted behavior. The main difference compared to the current Swedish system is that the number of adjustment seats is not fixed, but rather dynamically determined to be as low as possible and still insure proportionality between parties.     

基于身份的动态数字签名方案

数字签名是解决信息安全问题的重要途径,用于鉴别用户身份.随着计算机、网络的发展,安全的用户数字签名显得尤为重要.目前,现代的数字签名技术正向智能化、密码化、多因素、大容量和快速响应方向发展.结合数论中的中国剩余定理及RSA公钥体制,提出了一种基于身份的动态数字签名方案.     

Jacobi's Two-Square Theorem and Related Identities

We show that Jacobi's two-square theorem is an almost immediate consequence of a famous identity of his, and draw combinatorial conclusions from two identities of Ramanujan.     

On order of elements in hypergroups and p-Sylow subhypergroups

In this paper we define a left order, right order and order of an element in a hypergroup, which has a left scalar identity, right scalar identity and scalar identity, respectively. Using left and right orders, we define and investigate some classes of subhypergroups, especially p-Sylow subhypergroups of a hypergroup. Moreover, some examples have been proposed to show that the Sylow theorems do not hold for polygroups.     

Error minimization methods in biproportional apportionment

One of the most active research lines in the area of electoral systems to date deals with the Biproportional Apportionment Problem, which arises in those proportional systems where seats must be allocated to parties within territorial constituencies. A matrix of the vote counts of the parties within the constituencies is given, and one has to convert the vote matrix into an integer matrix of seats ??as proportional as possible?? to it, subject to the constraints that each constituency be granted its pre-specified number of seats, each party be allotted the total number of seats it is entitled to on the basis of its national vote count, and a zero-vote zero-seat condition be satisfied. The matrix of seats must simultaneously meet the integrality and the proportionality requirement, and this not infrequently gives rise to self-contradictory procedures in the electoral laws of some countries. Here we discuss a class of methods for Biproportional Apportionment characterized by an ??error minimization?? approach. If the integrality requirement is relaxed, fractional seat allocations (target shares) can be obtained so as to achieve proportionality at least in theory. In order to restore integrality, one then looks for integral apportionments that are as close as possible to the ideal ones in a suitable metric. This leads to the formulation of constrained optimization problems called ??best approximation problems?? which are solvable in polynomial time through the use of network flow techniques. These error minimization methods can be viewed as an alternative to the classical axiomatic approach introduced by Balinski and Demange (in Math Oper Res 14:700?C719, 1989a; Math Program 45:193?C210, 1989b). We provide an empirical comparison between these two approaches with a real example from the Italian Elections and a theoretical discussion about the axioms that are not necessarily satisfied by the error minimization methods.     

一个公平分配席位的新方案

提出了公平累加分配席位的方法,并证明了该方法在一定条件下满足公平分摊原则和席位单调性原则.并进一步指出该方法具备更多的公平性质.     

Bidimensional allocation of seats via zero-one matrices with given line sums

In some proportional electoral systems with more than one constituency the number of seats allotted to each constituency is pre-specified, as well as, the number of seats that each party has to receive at a national level. “Bidimensional allocation” of seats to parties within constituencies consists of converting the vote matrix V into an integer matrix of seats “as proportional as possible” to V, satisfying constituency and party totals and an additional “zero-vote zero-seat” condition. In the current Italian electoral law this Bidimensional Allocation Problem (or Biproportional Apportionment Problem—BAP) is ruled by an erroneous procedure that may produce an infeasible allocation, actually one that is not able to satisfy all the above conditions simultaneously. In this paper we focus on the feasibility aspect of BAP and, basing on the theory of (0,1)-matrices with given line sums, we formulate it for the first time as a “Matrix Feasibility Problem”. Starting from some previous results provided by Gale and Ryser in the 60’s, we consider the additional constraint that some cells of the output matrix must be equal to zero and extend the results by Gale and Ryser to this case. For specific configurations of zeros in the vote matrix we show that a modified version of the Ryser procedure works well, and we also state necessary and sufficient conditions for the existence of a feasible solution. Since our analysis concerns only special cases, its application to the electoral problem is still limited. In spite of this, in the paper we provide new results in the area of combinatorial matrix theory for (0,1)-matrices with fixed zeros which have also a practical application in some problems related to graphs.     

On the quintuple product identity

In this note we present a new proof of the quintuple product identity which is based on our study of order theta functions with characteristics and the identities they satisfy. In this context the quintuple product identity is another example of an identity which when phrased in terms of theta functions, rather than infinite products and sums, has a simpler form and is much less mysterious.

     

HOPF-JACOBSON RADICAL FOR COMODULE ALGEBRAS

51.IntroductionFisher[7]discussedtheHopf-JacobsonradicalJH(A)oftheH--modulealgebraA,whereHisanirreducibleHopfalgebra.In[5],CatgeneralizedFisher'sresult.HeprovedthatJH(A)#HgJ(A#H),whereHisanarbitraryHopfalgebra,AanH--modulealgebra.AversionoftheChevalley-JacobsondensitytheoremforH-modulealgebrawasalsoprovedbyCans].Dually,LiuGuilong[8]definedandstudiedtheHopf-JacobsonradicalofH-comodulealgebraA.Inthispaper,westudyfurthertheHopf-JacobsonradicaloftheH--comodulealgebraA,andwegiveaver…     

The minimal excludant and Schmidt's partition theorem

In this work, we study Schmidt's partition theorem in a combinatorial manner, and find a strong refinement which connects the minimal excludant of ordinary partitions to the length of Schmidt's partitions. As a byproduct, we obtain a bivariate form of an identity recorded in Ramanujan's lost notebook.     

Thresholds of the Divisor Methods

We give an algorithm which permits calculating the maximum and minimum vote shares that allow a party to obtain h seats, that is, the threshold of exclusion and the threshold of representation. These have already been studied for some methods (such as d'Hondt or Sainte-Laguë), and are here generalized to any divisor method, and to any number of seats. The thresholds depend on the size of the constituency, the number of parties running in the constituency, and the divisor method used. Finally, we give some consequences, including a characterization of the d'Hondt method.     

Bounds on Malapportionment

Uniformly sized constituencies give voters similar influence on election outcomes. When constituencies are set up, seats are allocated to the administrative units, such as states or counties, using apportionment methods. According to the impossibility result of Balinski and Young, none of the methods satisfying basic monotonicity properties assign a rounded proportional number of seats (the Hare-quota). We study the malapportionment of constituencies and provide a simple bound as a function of the house size for an important class of divisor methods, a popular, monotonic family of techniques.     

A heuristic algorithm for a dial-a-ride problem with time windows,multiple capacities,and multiple objectives

The paper describes a system for the solution of a static dial-a-ride routing and scheduling problem with time windows (DARPTW). The problem statement and initialization of the development project was made by the Copenhagen Fire-Fighting Service (CFFS). The CFFS needed a new system for scheduling elderly and disabled persons, involving about 50.000 requests per year. The problem is characterized by, among other things, multiple capacities and multiple objectives. The capacities refer to the fact that a vehicle may be equipped with e.g. normal seats, children seats or wheel chair places. The objectives relate to a number of concerns such as e.g. short driving time, high vehicle utilization or low costs. A solution algorithm REBUS based on an insertion heuristics was developed. The algorithm permits in a flexible way weighting of the various goals such that the solution reflects the user's preferences. The algorithm is implemented in a dynamic environment intended for on-line scheduling. Thus, a new request for service is treated in less than 1 second, permitting an interactive user interface.     

Double optimal stopping times and dynamic pricing problem: description of the mathematical model

In many industries, managers face the problem of selling a given stock of items by a deadline. We investigate the problem of dynamically pricing such inventories when demand is price sensitive and stochastic and the firm’s objective is to maximize expected revenues. Examples that fit this framework include retailers selling fashion and seasonal goods and the travel and leisure industry, which markets space such as seats on airline flights, cabins on vacation cruises, hotels renting rooms before midnight and theaters selling seats before curtain time that become worthless if not sold by a specific time. Given a fixed number of seats, rooms, or coats, the objective for these industries is to maximize revenues in excess of salvage value. When demand is price sensitive and stochastic, pricing is an effective tool to maximize revenues. In this paper, we address the problem of deciding the optimal timing of a double price changes from a given initial price to given lower or higher prices. Under mild conditions, it is shown that it is optimal to decrease the initial price as soon as the time-to-go falls below a time threshold and increase the price if time-to-go is longer than adequate time threshold. These thresholds depend on the number of yet unsold items.      

用再生核表示小波变换

本文研究了调制高斯函数的小波变换．利用再生核函数的特殊技巧，得到了该小波变换的等距恒等式和像空间的结构，同时给出了该小波变换的采样定理．使得小波变换能用再生核函数表示．这为一般的小波变换的像空间的研究提供了理论基础．     

Cubic Identities of Theta Functions

Many remarkable cubic theorems involving theta functions can be found in Ramanujan's Lost Notebook. Using addition formulas, the Jacobi triple product identity and the quintuple product identity, we establish several theorems to prove Ramanujan's cubic identities.     

Government formation in a two dimensional policy space

Given any allocation of parliament seats among parties, we characterize all the stable government configurations (supported by at least a majority of the parliament) in terms of winning coalitions and policy outcomes. We consider a two dimensional policy space and we assume that there are four parties that care mainly about holding office, and only instrumentally about policy. We find that for any distribution of seats in the parliament only two scenarios are possible: either there is a party that is a member of almost all equilibrium coalitions (dominant party scenario) or there is a party that is never a member of an equilibrium coalition (dominated party scenario). We characterize the key party for each possible scenario and we show that it is sufficient that the key party has intense preferences over one the issues to guarantee the formation of a stable government coalition.     

The give-up problem for blocked regional lists with multi-winners

The current electoral law for the Italian Parliament prescribes blocked, linearly ordered lists of candidates for each party within each constituency. The peculiarity of the Italian electoral system is that a party can present the same candidate in different constituencies. There are several seats at stake in each constituency; these seats are allocated to the parties proportionally to the total number of votes they get. If the blocked list mechanism-which assigns the seats obtained by a party in a constituency to the first candidates of the corresponding ordered list-causes some candidates to win in more than one constituency, they may retain only one of the seats, giving up all the remaining ones. Thus, the problem arises for a party to find a suitable “schedule of give-ups” that produces the final set of winners for that party. In order to do this, we assume that such decision is centralized and based on some models of global (inter-regional) preferences over the set of candidates. In this paper, we introduce two classes of models to formulate the “give-up problem”, i.e., utility and ordinal models, and we show that for both of them some natural formulations of the problem can be efficiently solved by network flows techniques.     

某情形洛必达法则的相关证明

分析部分教材对极限式分母趋于无穷大情形下的洛必达法则的证明,指出相关文献给出的证明过程存在局限甚至失误,随后通过引入一个恒等式,给出关于此情形下洛必达法则的一个完整证明,此外,还运用上、下极限给出此情形下洛必达法则的一个新证明．     

Determinant preserving maps: an infinite dimensional version of a theorem of Frobenius

In this paper we investigate the structure of maps on classes of Hilbert space operators leaving the determinant of linear combinations invariant. Our main result is an infinite dimensional version of the famous theorem of Frobenius about determinant preserving linear maps on matrix algebras. In this theorem of ours, we use the notion of (Fredholm) determinant of bounded Hilbert space operators which differ from the identity by an element of the trace class. The other result of the paper describes the structure of those transformations on sets of positive semidefinite matrices which preserve the determinant of linear combinations with fixed coefficients.