Studying the inertias of LCM matrices and revisiting the Bourque-Ligh conjecture

Let $S=\{x_1,x_2,\ldots,x_n\}$ be a finite set of distinct positive integers. Throughout this article we assume that the set $S$ is GCD closed. The LCM matrix $[S]$ of the set $S$ is defined to be the $n\times n$ matrix with $\mathrm{lcm}(x_i,x_j)$ as its $ij$ element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of a GCD closed set $S$ is always invertible, but currently it is a well-known fact that any nontrivial LCM matrix is indefinite and under the right circumstances it can be even singular (even if the set $S$ is assumed to be GCD closed). However, not much more is known about the inertia of LCM matrices in general. The ultimate goal of this article is to improve this situation. Assuming that $S$ is a meet closed set we define an entirely new lattice-theoretic concept by saying that an element $x_i\in S$ generates a double-chain set in $S$ if the set $\mathrm{meetcl}(C_S(x_i))\setminus C_S(x_i)$ can be expressed as a union of two disjoint chains (here the set $C_S(x_i)$ consists of all the elements of the set $S$ that are covered by $x_i$ and $\mathrm{meetcl}(C_S(x_i))$ is the smallest meet closed subset of $S$ that contains the set $C_S(x_i)$). We then proceed by studying the values of the M\"obius function on sets in which every element generates a double-chain set and use the properties of the M\"obius function to explain why the Bourque-Ligh conjecture holds in so many cases and fails in certain very specific instances. After that we turn our attention to the inertia and see that in some cases it is possible to determine the inertia of an LCM matrix simply by looking at the lattice-theoretic structure of $(S,|)$ alone. Finally, we are going to show how to construct LCM matrices in which the majority of the eigenvalues is either negative or positive

Model-independent and model-based local lensing properties of CL0024+1654 from multiply-imaged galaxies

We investigate to which precision local magnification ratios, $\mathcal{J}$, ratios of convergences, $f$, and reduced shears, $g = (g_{1}, g_{2})$, can be determined model-independently for the five resolved multiple images of the source at $z_\mathrm{s}=1.675$ in CL0024. We also determine if a comparison to the respective results obtained by the parametric modelling program Lenstool and by the non-parametric modelling program Grale can detect biases in the lens models. For these model-based approaches we additionally analyse the influence of the number and location of the constraints from multiple images on the local lens properties determined at the positions of the five multiple images of the source at $z_\mathrm{s}=1.675$. All approaches show high agreement on the local values of $\mathcal{J}$, $f$, and $g$. We find that Lenstool obtains the tightest confidence bounds even for convergences around one using constraints from six multiple image systems, while the best Grale model is generated only using constraints from all multiple images with resolved brightness features and adding limited small-scale mass corrections. Yet, confidence bounds as large as the values themselves can occur for convergences close to one in all approaches. Our results are in agreement with previous findings, supporting the light-traces-mass assumption and the merger hypothesis for CL0024. Comparing the three different approaches allows to detect modelling biases. Given that the lens properties remain approximately constant over the extension of the image areas covered by the resolvable brightness features, the model-independent approach determines the local lens properties to a comparable precision but within less than a second. (shortened)Comment: 22 pages, published in A&A 612 A17, comments welcom

Tax evasion, information reporting, and the regressive bias hypothesis

A robust prediction from the tax evasion literature is that optimal auditing induces a regressive bias in e¤ective tax rates compared to statutory rates. If correct, this will have important distributional consequences. Nevertheless, the regressive bias hypothesis has never been tested empirically. Using a unique data set, we provide evidence in favor of the regressive bias prediction but only when controlling for the tax agency�s use of third-party information in predicting true incomes. In aggregate data, the regressive bias vanishes because of the systematic use of third-party information. These results are obtained both in simple reduced-form regressions and in a data-calibrated state-of-the-art model

Deciding the sale of a life policy in the viatical market: Implications on individual welfare

In this paper, we present an economic model that allows a terminally ill policy-holder to decide whether or not to sell (part of) the policy in the viatical settlement market. The viatical settlement market emerged in the late 1980s in response to the AIDS epidemic. Nowadays it is part of the large US market in life settlements. The policies traded in the viatical market are those of terminally ill policyholders expected to die within the next two years. The model is discrete and considers only the next two periods (years), since this is the max- imum remaining lifetime of the policyholder. The decisor has an initial wealth and has to share it between his own consumption and the bequests left to his heirs. We rst introduce the expected utility function of our decisor and then use dynamic programming to deduce the strategy that gives higher utility (not selling/selling (part of) the policy at time zero/selling (part of) the policy at time one). The optima depends on the value of the viaticated policy and on some personal parameters of the individual. We nd an analitical expression for the optimal strategy and perform a sensitivity analysis.expected utility, viatical settlement, dynamic programming