On varieties in multiple-projective spaces

AbstractIn this paper, mP will denote a projective space of dimension m, and (m,n)P will denote a doubly-projective space of dimension m+n, namely the space of all pairs of points (x¦y), where x varies in mP and y in nP. Just so, (m,n,s)P will denote a triply-projective space of dimension m+n+s, and so on.A variety V of dimension d in mP has just one degree g, namely the number of points of intersection of V with d generic linear hyperplanes (ux)=0, where (ux) means ∑uixi. On the other hand, a variety V' of dimension d in (m,n)P has several degree ga,b (a+b=d), defined as follows: ga,b is the number of points of intersection of V' with a hyperplanes (ux)=0 and b hyperplanes (vy)=0.Let xo, …, xm, yo, …, yn be the homogeneous coordinates of a point in m+n+1P. It sometimes happens that the equations of a variety V in m+n+1P are not only homogeneous in all variables x and y together, but even homogeneous in the x's and in the y's. In this case the same set of equations also defines a variety V' in the doubly-projective space (m+n)P. If d is the dimension of V', the dimension of V is d+1, for to every point (x¦y) of V' corresponds a whole straight line of points (xα, yβ) in V.In some cases it is easier to determine the degrees ga,b of V' than to determine the degree g of V. For this reason, it is desirable to have a rule that enables us to calculate g from the ga,b's. Such a rule will be proved here. It says:The degree g is the sum of all ga,b with a+b=d.In the case of multiply-projective spaces (m,n,…)P the same rule holds: g is the sum of the ga,b,… with a+b+…=d.Examples of applications of this rule will be given at the end

Pamela, a parking analysis model for predicting effects in local areas

To improve existing parking models and to meet several additional requirements of practitioners, a parking analysis model at the scale of local areas is developed. The model called Pamela which stands for Parking Analysis Model for predicting Effects in Local Areas. Pamela simulates at the local level different travel and parking decisions from the moment an individual has decided to leave home for weekly or non-weekly shopping until the moment the individual has completed her/his activity, leaves the chosen parking facility and goes home. Three different choice models form the heart of Pamela: (i) a parking choice set composition model to generate the car drivers parking choice set, (ii) a combined travel choice model combining the choice of shopping destination, travel mode and parking/bicycle stall, and (iii) an adaptive parking choice model that describes car drivers’ reactions when facing a fully occupied parking facility. The models include a variety of characteristics related to shopping destination, travel mode, and parking and storage facility. In addition, the adaptive parking choice model also includes characteristics that describe the situation of the parking facility at the moment a car drivers enters a fully occupied parking facility. All included models are estimated using stated choice data collected in the town of Veldhoven and the city of Eindhoven, the Netherlands. For each part of Pamela a stated choice experiment is set up and presented to residents of Veldhoven and Eindhoven in a home sent questionnaire. The data of 1024 residents are used for the analyses. The data are analyzed using mixed logit models that include both mean (consisting of means and standard deviations) and context (only means) effects where context effects represent the difference between weekly and non-weekly shopping. Most estimation results are satisfactory indicating that the estimated models give a good representation of the respondents’ stated choice behavior. The percentage correctly predicted choices varies from almost 36 (in the case of the combined travel choice model) to more than 70 (in the case of the parking consideration set model) percent. In all cases the mixed multinomial logit model performs better than the traditional multinomial logit model. Most effects of the included model attributes are as expected. Regarding the composition of parking choice sets it appears that the characteristics parking costs and maximum parking duration influences the probability of a parking facility to be included in the car drivers’ choice set mostly. At some distance these characteristics are followed by the chance of a free space and walking distance between parking facility and nearest supermarket/department store. The effects found for the characteristics differ significantly for weekly and non-weekly shopping visits. Looking to the combined travel choice behavior, it appears that most influential characteristics are in order of influence: travel time of bicycle, parking costs, travel time bus, maximum parking duration, and supply of shops. Also in this case differences in influence are found between weekly and non-weekly shopping visits. Car drivers’ adaptive parking choice is mostly influenced by the expected waiting time, the number of parking facilities visited before entering the fully occupied parking, and the chance of getting a parking fine. Differences between weekly and non-weekly shopping visits only exist for number of parking facilities visited before and number of cars waiting for a free space. The validity of the estimated models is tested by applying the models to the town of Veghel, a comparable town to Veldhoven. Because of the available observations, only the parking choice set composition and the combined travel choice models for weekly shopping trips could be validated. Regarding the performance of the models, it appears that the consideration set model is well able to predict the composition of parking consideration sets that are observed in Veghel. On average the model predicts in approximately 67 percent the presence or non-presence correctly. The performance of combined travel choice model is low, especially at the individual level. At the aggregate level the model is able to explain 84 percent of the distribution across the choice alternatives. However, at the individual level only 9 percent of the choices were correctly predicted which is somewhat better than the null model (4 percent correctly predicted). The model mainly predicts choice combinations that include the car as travel mode. To illustrate the working of Pamela a micro-simulation is worked out using the multiagent system NetLogo. A hypothetical setting is created consisting of three shopping centers, nine parking facilities, and three bicycle stalls. The simulation includes the whole process from the generation of a traveler until the traveler’s move from the shopping center to her/his home location. Besides the estimated model parameters the simulation is complemented with additional data retrieved from empirical data (type of shopping) and the data collection (shopping duration). The simulation is used to evaluate the following three different transport policies: leveling out the parking costs for all parking facilities, setting all storage costs of bicycle stalls to ‘no charge’, and equalizing walking distance between parking facilities and nearest supermarket or department store to 150 meters. The travel decisions of 500 residents are simulated for a base situation and the three transport policies. To level out random effects, the simulation is carried out ten times and all results are averaged over these ten simulation runs. The simulation shows the changes in destination, travel mode and parking/storage choice at an overall level (daytime period from 8:00 – 20:00 hours) and at the level of time slices (every minute during the day time period). It also shows for each travel mode the changes in average and total distance traveled of all included residents during the daytime period

Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems

B{\'e}zout 's theorem states that dense generic systems of n multivariate quadratic equations in n variables have 2 n solutions over algebraically closed fields. When only a small subset M of monomials appear in the equations (fewnomial systems), the number of solutions may decrease dramatically. We focus in this work on subsets of quadratic monomials M such that generic systems with support M do not admit any solution at all. For these systems, Hilbert's Nullstellensatz ensures the existence of algebraic certificates of inconsistency. However, up to our knowledge all known bounds on the sizes of such certificates -including those which take into account the Newton polytopes of the polynomials- are exponential in n. Our main results show that if the inequality 2|M| -- 2n $\le$ \sqrt 1 + 8{\nu} -- 1 holds for a quadratic fewnomial system -- where {\nu} is the matching number of a graph associated with M, and |M| is the cardinality of M -- then there exists generically a certificate of inconsistency of linear size (measured as the number of coefficients in the ground field K). Moreover this certificate can be computed within a polynomial number of arithmetic operations. Next, we evaluate how often this inequality holds, and we give evidence that the probability that the inequality is satisfied depends strongly on the number of squares. More precisely, we show that if M is picked uniformly at random among the subsets of n + k + 1 quadratic monomials containing at least $\Omega$(n 1/2+$\epsilon$) squares, then the probability that the inequality holds tends to 1 as n grows. Interestingly, this phenomenon is related with the matching number of random graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results showing that certificates in inconsistency can be computed for systems with more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201

Car drivers' evaluation of parking garages

Because of growing competition between parking facilities in inner city areas, managers of parking garages are looking more carefully to the requirements of (segments of) their customers (e.g., Visser, 2000). According to Visser, the increase of operation costs requires an optimal operation of parking facilities. Parking operators have to act more professionally and focus more on the requirements and evaluation of their costumers, which concern various characteristics of parking garages

Greek Astronomical Calendars and their Relation to the Athenian Civil Calendar

Several investigations have been devoted to the Athenian calendar and to the cycles of Meton and Kallippos. However, most authors have not clearly distinguished between true and mean lunar months, nor between astronomical calendars and the Athenian calendar. In investigating the Athenian calendar, many authors have made use of the regular successions of full and hollow months described by Geminos in his Isagoge, without first making sure that these months were in actual use at Athens. Discussion as to whether ‘the month' began with the astronomical New Moon or with the visibility of the crescent might have been avoided if the authors had realised that the word ‘month' has several meanings and that in every particular case the meaning has to be inferred from the context. Peasants or soldiers, far away from civilisation, would start their month with the visible crescent, astronomers would make it begin at the day of true or mean New Moon, and cities would adapt their festival calendar to the needs of the moment, intercalating or omitting days in such a way that the festivals can be held at the days prescribed by law or tradition. Of course, it may happen any time that a civil month coincides with the astronomical or with the observed lunar month, but in absence of definite evidence we never have the right to identify a civil month with an astronomical mont

Efficient numerical diagonalization of hermitian 3x3 matrices

A very common problem in science is the numerical diagonalization of symmetric or hermitian 3x3 matrices. Since standard "black box" packages may be too inefficient if the number of matrices is large, we study several alternatives. We consider optimized implementations of the Jacobi, QL, and Cuppen algorithms and compare them with an analytical method relying on Cardano's formula for the eigenvalues and on vector cross products for the eigenvectors. Jacobi is the most accurate, but also the slowest method, while QL and Cuppen are good general purpose algorithms. The analytical algorithm outperforms the others by more than a factor of 2, but becomes inaccurate or may even fail completely if the matrix entries differ greatly in magnitude. This can mostly be circumvented by using a hybrid method, which falls back to QL if conditions are such that the analytical calculation might become too inaccurate. For all algorithms, we give an overview of the underlying mathematical ideas, and present detailed benchmark results. C and Fortran implementations of our code are available for download from http://www.mpi-hd.mpg.de/~globes/3x3/ .Comment: 13 pages, no figures, new hybrid algorithm added, matches published version, typo in Eq. (39) corrected; software library available at http://www.mpi-hd.mpg.de/~globes/3x3

Hybrid Dirac Fields

Hybrid Dirac fields are fields that are general superpositions of the annihilation and creation parts of four Dirac spin 1/2 fields, $\psi^{(\pm)}(x;\pm m)$, whose annihilation and creation parts obey the Dirac equation with mass $m$ and mass $-m$. We discuss a specific case of such fields, which has been called ``homeotic.'' We show for this case, as is true in general for hybrid Dirac fields (except the ordinary fields whose annihilation and creation parts both obey one or the other Dirac equation), that (1) any interacting theory violates both Lorentz covariance and causality, (2) the discrete transformations $\mathcal{C}$, and $\mathcal{CPT}$ map the pair $\psi_h(x)$ and $\bar{\psi}_h(x)$ into fields that are not linear combinations of this pair, and (3) the chiral projections of $\psi_h(x)$ are sums of the usual Dirac fields with masses $m$ and $-m$; on these chiral projections $\mathcal{C}$, and $\mathcal{CPT}$ are defined in the usual way, their interactions do not violate $\mathcal{CPT}$, and interactions of chiral projections are Lorentz covariant and causal. In short, the main claims concerning ``homeotic'' fields are incorrect.Comment: 10 pages, acknowledgement added, to appear in Phys. Lett.

Towards Mixed Gr{\"o}bner Basis Algorithms: the Multihomogeneous and Sparse Case

One of the biggest open problems in computational algebra is the design of efficient algorithms for Gr{\"o}bner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of unmixed polynomial systems, that is systems with polynomials having the same support, using the approach of Faug{\`e}re, Spaenlehauer, and Svartz [ISSAC'14]. We present two algorithms for sparse Gr{\"o}bner bases computations for mixed systems. The first one computes with mixed sparse systems and exploits the supports of the polynomials. Under regularity assumptions, it performs no reductions to zero. For mixed, square, and 0-dimensional multihomogeneous polynomial systems, we present a dedicated, and potentially more efficient, algorithm that exploits different algebraic properties that performs no reduction to zero. We give an explicit bound for the maximal degree appearing in the computations