How long does it take to consensus in the Hegselmann-Krause model?

Hegselmann and Krause introduced a discrete-time model of opinion dynamics with agents having limit confidence. It is well known that the dynamics reaches a stable state in a polynomial number of time steps. However, the gap between the known lower and upper bounds for the worst case is still immense. In this paper exact values for the maximum time, needed to reach consensus or to discover that consensus is impossible, are determined using an integer linear programming approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

How many bits of information does an independent variable yield in a multiple regression?

A simple measure from statistical information theory is proposed to measure the contribution of each independent variable in a multiple regression.     

How to Arrange it?

Problem: If you arrange 24 people in 6 rows and each one must have 5 people,how can you range them?Answer: If you range 24 people in 6 rows, you must have 5 people per row. Isit possible to range so? For there are 6 rows and each one should have 5 people,i.e. 5×6= 30 people !     

How many dentists does Sri Lanka need? Modelling to inform policy decisions

This paper describes a model that has been used to inform government policy in Sri Lanka. The model has two components, representing supply and demand for dental-care services. The supply-side component (the dental workforce model) uses system dynamics to represent the career progression of dentists from recruitment and training at the University Dental School, through different career paths through to retirement. The demand-side model calculates a range of future demand scenarios for dental care, based on different assumptions about Sri Lanka’s potential future economic development. The combined model was used by the Sri Lankan government to plan better provision of state-funded dental care and the future university intake of dental students. Data collection for both models was challenging, and required some innovative research. The model itself, and its successful application, raise a number of wider issues concerning model implementation and acceptability by clients and model users.     

How to find many counterfeit coins?

We propose an algorithm for findingm defective coins, that uses at most + 15m weighings on a balance scale, wheren is the number of all coins.     

How many lottery tickets to buy?

We investigate the optimal number of tickets an expected-utility-maximizing individual who participate in a lottery will buy. We show that the expected utility is not always unimodal in the number of tickets. We also show that a risk-averse individual will not buy more than one ticket. (S)he will be willing to own a share in several tickets only if that share is sufficiently small.     

How many Fourier samples are needed for real function reconstruction?

In this paper we present some new results on the reconstruction of structured functions by a small number of equidistantly distributed Fourier samples. In particular, we show that real spline functions of order m with non-uniform knots containing N terms can be uniquely reconstructed by only m+N Fourier samples. Further, linear combinations of N non-equispaced shifts of a known low-pass function Φ can be reconstructed by N+1 Fourier samples. In the bivariate case, we consider the problem of function recovering by a small amount of Fourier samples on different lines through the origin. Our methods are based on the Prony method. The proofs given in this paper are constructive. Some numerical examples show the applicability of the proposed approach.     

How many data clusters are in the Galaxy data set?

Advances in Data Analysis and Classification - In model-based clustering, the Galaxy data set is often used as a benchmark data set to study the performance of different modeling approaches. Aitkin...     

How much is it worth to know the future in online conversion problems?

We answer this question using the competitive ratio as an indicator for the quality of information about the future. Analytical results show that the better the information the better the worst-case competitive ratios. However, experimental analysis gives a slightly different view. We calculate the empirical-case competitive ratios of different variants of a threat-based online algorithm. The results are based on historical data of the German Dax-30 index. We compare our experimental empirical-case results to the analytical worst-case results given in the literature. We show that better information does not always lead to a better performance in real life applications. The empirical-case competitive ratio is not always better with better information, and some a-priori information is more valuable than other for practical settings.     

How to discretize the Total Variation of an image?

Because they are based on finite differences, usual discretizations of the Total Variation lead to aliased images. We propose a new discretization called spectral total variation that agrees with Shannon sampling principles and produces images that can be exactly interpolated. The quality improvement is illustrated experimentally in the case of image deblurring. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

What does it mean to explain the rising of the tippe top?

A fast rotating tippe top (TT) defies our intuition because, when it is launched on its bottom, it flips over to spin on its handle. The existing understanding of the flipping motion of TT is based on analysis of stability of asymptotic solutions for different values of TT parameters: the eccentricity of the center of mass 0 ⩽ α ⩽ 1 and the quotient of main moments of inertia γ = I ₁/I ₃. These results provide conditions for flipping of TT but they say little about dynamics of inversion. I propose here a new approach to study the equations of TT and introduce a Main Equation for the tippe top. This equation enables analysis of dynamics of TT and explains how the axis of symmetry of TT moves on the unit sphere S ². This approach also makes possible to study the relationship between behavior of TT and the law of friction.      

How many diffusions exist on the Vicsek snowflake?

We consider Kusuoka's construction of Dirichlet forms on the Vicsek snowflake. a nested fractal. His method is generalized and all irreducible, local Dirichlet forms which can be constructed in this way are characterized. We end up with a one-parameter family of different possible forms. This proves that Brownian motion on fractals is not unique if the isometry group of the fractal is too small.     

How does the core sit inside the mantle?

The k‐core, defined as the maximal subgraph of minimum degree at least k, of the random graph has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold d_k for the appearance of an extensive k‐core. The aim of the present paper is to describe how the k‐core is “embedded” into the random graph in the following sense. Let and fix . Colour each vertex that belongs to the k‐core of in black and all remaining vertices in white. Here we derive a multi‐type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k‐core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2‐core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k‐core. Based on this the study of the k‐core reduces to the analysis of Warning Propagation on a suitable Galton‐Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 459–482, 2017     

Reducing the debt: is it optimal to outsource an investment?

This paper deals with the problem of outsourcing the debt for a big investment, according two types of contract: either the firm outsources both the investment (and the associated debt) and the exploitation to another firm (for example a private consortium), or the firm supports the debt and the investment but outsources the exploitation. We prove the existence of Stackelberg and Nash equilibria between the firms, for both types of contract. We compare the benefits of these contracts, theoretically and numerically. We conclude with a study of what happens in case of incomplete information, in the sense that the risk aversion coefficient of each partner may be unknown by the other partner.     

How does the method of implicit differentiation use?

The functions are described by expressing one variable explicitly in terms of another variable.For example y=2x~3+x or y= sinx,or,in general,y=f(x).In other words,an explicit relation between x and     

How many points have the same Engel and Sylvester expansions?

In this paper, we prove that the set of points in (0,1] with the same Engel and Sylvester expansions is of Hausdorff dimension .     

How to use the Limit Laws?

We used tables and graphs by using graphing-calculator to guess values of limits,however,we found methods we used don’t always lead to the correct answer.Theproperties of limits have been found.It iseasy to believe they are true.For example,iff(x)is close to Land g(x)is close to M,itis reasonable to get the result that f(x)+g