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.     

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.     

Modeling social services performance: a four-stage DEA approach to evaluate fundraising efficiency,capacity building,service quality,and effectiveness in the nonprofit sector

The Swedish electoral system exhibits significant levels of proportionality compared with the systems used in other countries. However, it has several deficiencies that could be corrected. Therefore, this paper (a) evaluates the current Swedish electoral system by identifying the imbalances in the representation of political parties and the sizes of the electoral constituencies that can occur and (b) presents two proposals for improvement that seek to correct the previously identified deficiencies. The first proposal consists of a slight modification of the current system that applies when parties get more seats than they proportionally deserve according to their global number of votes, as occurred in the 2010 election. In this case, the proposal includes a criterion so that the overrepresented parties return their excess seats. The second proposal relies on the implementation of biproportional allotments and on the replacement of the electoral thresholds of 4 % of the total votes nationwide and 12 % of the votes in a given constituency by a new threshold based on a reduction in the number of votes of the parties. The application of any of these proposals to the Swedish election held in 2010 reveals that the deficiencies in the representation of political parties are eliminated. Furthermore, the second proposal also corrects the deficiencies in the sizes of the electoral constituencies for Sweden.     

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225?C234, 2005; Appl. Math. E-Notes 7:214?C221, 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian??s accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.     

Foulkes characters, Eulerian idempotents, and an amazing matrix

John Holte (Am. Math. Mon. 104:138?C149, 1997) introduced a family of ??amazing matrices?? which give the transition probabilities of ??carries?? when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra (Brenti and Welker, Adv. Appl. Math. 42:545?C556, 2009; Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009) and in the analysis of riffle shuffling (Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009). We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday (Cyclic Homology, 1992) in work on Hochschild homology. The connections give new closed formulae for Foulkes characters and allow explicit computation of natural correlation functions in the original carries problem.     

Updating the regularization parameter in the adaptive cubic regularization algorithm

The adaptive cubic regularization method (Cartis et al. in Math. Program. Ser. A 127(2):245?C295, 2011; Math. Program. Ser. A. 130(2):295?C319, 2011) has been recently proposed for solving unconstrained minimization problems. At each iteration of this method, the objective function is replaced by a cubic approximation which comprises an adaptive regularization parameter whose role is related to the local Lipschitz constant of the objective??s Hessian. We present new updating strategies for this parameter based on interpolation techniques, which improve the overall numerical performance of the algorithm. Numerical experiments on large nonlinear least-squares problems are provided.     

On the Existence of Solitary Traveling Waves for Generalized Hertzian Chains

We consider the question of existence of ??bell-shaped?? (i.e., nonincreasing for x>0 and nondecreasing for x<0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p=3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego (Proc. Am. Math. Soc. 133:1763, 2005) formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke?CWattis result (Commun. Math. Phys. 161:391, 1994) by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model.     

Some Consequences of a Three-Party Split in a UK General Election

The British electoral system is unique in Europe in being of the first-past-the-post variety. The apparent emergence of a strong third party renders any prediction exercise a good deal more difficult, although some political commentators appear oblivious to that fact. It would appear that a transition matrix approach is capable of providing the deeper insights needed to explore the consequences of alternative voting-behaviour patterns. Unfortunately data of this form are not currently collected, but it is possible to postulate that transition matrices of a particular form could be of interest to the three parties. By associating such transition probabilities with the 1983 results for each of the 633 mainland constituencies, one can derive interesting relationships between the number of seats secured by each party.A range of computer analyses was performed, and this article sets out some of the more interesting results, some of which came as something of a surprise.     

General two-function topological minimax theorems

Using the level set method of Joó (Acta Math Hung 54(1–2):163–172, 1989), a general two-function topological minimax theorem are proved. The theorem improves and generalizes the known results shown by Cheng and Lin (Acta Math Hung 73(1–2):65–69, 1996), Lin and Cheng (Acta Math Hung 100(3):177–186, 2003), and Frenk and Kassay (Math Program Ser A 105(1):145–155, 2006).     

Further mathematical results concerning Burgers fluids and their generalizations

In this paper, we extend the earlier work by Quintanilla and Rajagopal (Math Methods Appl Sci 29: 2133?C2147, 2006) and establish qualitative new results for a proper generalization of Burgers?? original work that stems form a general thermodynamic framework. Such fluids have been used to describe the behavior of several geological materials such as asphalt and the earth??s mantle as well as polymeric fluids. We study questions concerning stability, uniqueness and continuous dependence on initial data for the solutions of the flows of these fluids. We show that if certain conditions are not satisfied by the material moduli, the solutions could be unstable. The spatial behavior of the solutions is also analyzed.     

On M?bius Spaces Admitting Automorphism Groups of Type II.2

In 1987 the first author extended C. Hering??s classification for M?bius planes (cf. Hering in Math Z 87:252?C262, 1965) to higher dimensional M?bius spaces (cf. Kroll in Result Math 12:357?C365, 1987). Actually the classification was done for groups of automorphisms of a M?bius space. In this paper we are concerned with M?bius spaces admitting a group of type II.2. Our main result is a generalization of a result of Yaqub (Math Z 142:281?C292, 1975, Theorem 1).     

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.     

On second variation of Perelman’s Ricci shrinker entropy

In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman??s ??-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of (Cao in Adv Lect Math 11:1?C38, 2010). Moreover, we obtain a necessary condition for linearly stable shrinkers in terms of the least eigenvalue and its multiplicity of certain Lichnerowicz type operator associated to the second variation.     

On the Arf invariant in historical perspective, part?2

This is a continuation of our previous article (Lorenz and Roquette in Math. Semesterber. 57(1):73?C102, 2010) in this journal, where we discussed the paper of Cahit Arf (J. Reine Angew. Math. 183:148?C167, 1941). There he introduced what today is called the ??Arf invariant?? of a quadratic form over a field of characteristic?2. After our article had appeared we obtained some new information about the present state of the theory. When we say ??new?? then this means that this was new to us, i.e., we had not been aware before of the literature on the subject. (See our list of references.) It seems worthwhile to us to report here about it, as far as it is relevant for the assessment of Arf??s paper in its historical perspective.     

On Ulm’s method for Fréchet differentiable operators

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Izv. Akad. Nauk Est. SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) do not. We also show that under the same hypotheses and computational cost as (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) finer error sequences can be obtained. Numerical examples are also provided further validating the results.     

Axiomatizing core extensions

We give an axiomatization of the aspiration core on the domain of all TU-games using a relaxed feasibility condition, non-emptiness, individual rationality, and generalized versions of the reduced game property (consistency) and superadditivity. Our axioms also characterize the C-core (Guesnerie and Oddou, Econ Lett 3(4):301?C306, 1979; Sun et?al. J Math Econ 44(7?C8):853?C860, 2008) and the core on appropriate subdomains. The main result of the paper generalizes Peleg??s (J Math Econ 14(2):203?C214, 1985) core axiomatization to the entire family of TU-games.     

Projection pencils of quadrics and Ivory��s theorem

Using selfadjoint regular endomorphisms, the authors of Stachel and Wallner (Sib Math J 45(4):785?C794, 2004) defined, for an indefinite inner product, a variant of the notion of confocality for the Euclidean space. Our aim is to give a definition that is a common generalization of the usual confocality, and the variant in Stachel and Wallner (Sib Math J 45(4):785?C794, 2004). We use this definition to prove a more general form of Ivory??s theorem.     

Ghost seats in parliaments

In several European Countries citizens’ movements have formed parties whose programme is to reform electoral legislation so that blank votes are transformed into vacant seats in parliament. The first objective of this paper is to offer some tools in order to quantify the potential effect of a reform that would allow citizens to cast a dissatisfaction vote. We show that such a reform would reduce the ease to pass a proposal and modify the distribution of power among parties. The second objective of this paper is to show that this reform could have an impact in parliaments. This is illustrated by the case of the Basque Parliament from 1980 to 2016. At the theoretical level this paper provides some relations between measures of success.     

Inversion of Catalan matrix plus one

Motivated by a statistical problem, in (Aggarwala and Lamoureux in Am. Math. Mon., 109:371–377, 2002) it is shown how to invert a linear combination of the Pascal matrix with the identity matrix. Continuing this idea, we invert various linear combinations of the Catalan matrix, introduced in (Stanimirovi? et al. in Appl. Math. Comput. 215:796–805, 2009), with the identity matrix. In (Aggarwala and Lamoureux in Am. Math. Mon., 109:371–377, 2002) the occurrence of the polylogarithm function is observed. Inverses of linear combinations of the Catalan and the identity matrix are expressed in terms of Catalan numbers, the pochhammer function and the generalized hypergeometric function.     

Correction: Nearly Quartic Mappings in β-Homogeneous F-Spaces [Results Math. 63 (2013) 529–541]

In Chang et al. (Results Math. 63:529–541, 2013), Eshaghi Gordji et al. proved the Hyers-Ulam stability of a quartic functional equation in β-homogeneous F-spaces. In the main step of the proof of Chang et al. (Results Math. 63:529–541, 2013, Theorem 2.2), there is a fatal error. We correct the statement of Chang et al. (Results Math. 63:529–541, 2013, Theorem 2.2).