Self-contracted curves in Riemannian manifolds

It is established that every self-contracted curve in a Riemannian manifold has finite length, provided its image is contained in a compact set.     

Pathological solutions to the Euler–Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems

In this paper, we prove that if $L(x,u,v)\in C^3(R^3\to R)$, $L_{vv}>0$ and $L\ge \alpha |v|+\beta$, $\alpha >0$, then all problems (1), (2) admit solutions in the class $W^{1,1}[a,b]$, which are in fact $C^3$-regular provided there are no pathological solutions to the Euler equation (5). Here $u\in C^3[c,d[$ is called a pathological solution to equation (5) if the equation holds in $[c,d[$, $|\dot{u}(x)|\to \infty$ as $x\to d$, and ${6u6}_{C[c,d]}<\infty$. We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.     

One-dimensional cadmium(II)/bicinate(−1) complexes : The role of the alkali metal ion used in the reaction medium

The initial employment of N,N-bis(2-hydroxyethyl)glycine (bicine; bicH₃) in CdCl₂ chemistry is reported, and the syntheses, IR spectra and crystal structures of the 1D coordination polymers [CdCl(bicH₂)]_n·nH₂O (1·H₂O) and [CdNaCl₂(bicH₂)(MeOH)]_n (2) are described. The identity of the products depends on the solvent, the reaction temperature and the alkali metal ion of the base used. The structure of 1·H₂O consists of zig-zag chains. The 7-coordinate Cd^II atoms are bridged by η¹:η¹:μ₂ carboxylate groups of the 2.21111 (Harris notation) bicH₂⁻ ligand. The coordination geometry of the metal center can be either described as a very distorted pentagonal bipyramidal or as a distorted capped octahedral. In the structure of 2 the Cd^II atoms form an almost linear chain with neighboring Na^I atoms on opposite sites of the chain. Every pair of Cd^II atoms is linked by two chloro ligands and the two oxygen atoms of the bicinate carboxylate group. The Cd^II and Na^I atoms are bridged by one μ₂ carboxylate bicinate oxygen and one μ₃ chloro ligand. The 3.2_1,21₁1₂1₂1₂ coordination mode of bicH₂⁻ is unprecedented. The CdCl₄(O_carboxylate)₂ and Na(O_hydroxyl)₂(O_carboxylate)(O_MeOH)NCl coordination spheres are octahedral and trigonal prismatic, respectively. IR data of the complexes are discussed in terms of the coordination modes of bicH₂⁻ and the known structures.     

Interaction of Fe(III) with herbicide-carboxylato ligands – Di-, tri- and tetra-nuclear compounds: Structure and magnetic behavior

The iron complexes with the phenoxyalkanoic acids 3,4-D, 2,3-D and 2,4,5-T in the presence or not of a nitrogen donor heterocyclic ligand, phen, were prepared and characterized. Interaction of Fe(III) with phenoxyalkanoic acids and phen leads to dinuclear neutral complexes while the absence of phen favours trinuclear cationic or tetranuclear neutral forms. The crystal structure of hexakis(2,3-dichlorophenoxyacetato)tris(methanol)oxotri-iron(III) chloride–methanol(1/3), [Fe₃O(2,3-D)₆(MeOH)₃]Cl · 3MeOH (2), and tetrakis(dimethyl-sulfoxide)octakis(2,4,5-trichlorophenoxyacetato)dioxotetra-iron(III) methanol(1/2)–water(1/1)–dimethylsulfoxide(1/0.8), {[Fe₄O₂(2,4,5-T)₈(dmso)₄] · 2MeOH · H₂O · 0.8dmso} (3), have been determined and refined by least-squares methods using three-dimensional Mo Kα data.     

Forecasting of compound Erlang demand

Intermittent demand items dominate service and repair inventories in many industries and they are known to be the source of dramatic inefficiencies in the defence sector. However, research in forecasting such items has been limited. Previous work in this area has been developed upon the assumption of a Bernoulli or a Poisson demand arrival process. Nevertheless, intermittent demand patterns may often deviate from the memory-less assumption. In this work we extend analytically previous important results to model intermittent demand based on a compound Erlang process, and we provide a comprehensive categorisation scheme to be used for forecasting purposes. In a numerical investigation we assess the benefit of departing from the memory-less assumption and we provide insights into how the degree of determinism inherent in the process affects forecast accuracy. Operationalised suggestions are offered to managers and software manufacturers dealing with intermittent demand items.     

Intermittent demand: Linking forecasting to inventory obsolescence

The standard method to forecast intermittent demand is that by Croston. This method is available in ERP-type solutions such as SAP and specialised forecasting software packages (e.g. Forecast Pro), and often applied in practice. It uses exponential smoothing to separately update the estimated demand size and demand interval whenever a positive demand occurs, and their ratio provides the forecast of demand per period. The Croston method has two important disadvantages. First and foremost, not updating after (many) periods with zero demand renders the method unsuitable for dealing with obsolescence issues. Second, the method is positively biased and this is true for all points in time (i.e. considering the forecasts made at an arbitrary time period) and issue points only (i.e. considering the forecasts following a positive demand occurrence only). The second issue has been addressed in the literature by the proposal of an estimator (Syntetos-Boylan Approximation, SBA) that is approximately unbiased. In this paper, we propose a new method that overcomes both these shortcomings while not adding complexity. Different from the Croston method, the new method is unbiased (for all points in time) and it updates the demand probability instead of the demand interval, doing so in every period. The comparative merits of the new estimator are assessed by means of an extensive simulation experiment. The results indicate its superior performance and enable insights to be gained into the linkage between demand forecasting and obsolescence.