On Core-Walras (Non-) Equivalence for Economies with a Large Commodity Space

Addressing a question raised by Tourky and Yannelis (1998), we show that given any non-separable Banach space as commodity space and giben any atomless measure space of agents, there is an economy fulfilling the usual standard assumptions but having a core allocation not supportable as a Walrasion equilibrium, and in fact, having no Walrasian equilibria at all. We shall also consider the framework of economies with weakly compact consuption sets as developed by Khan and Yannelis (1991). We prove that in this setting the core of an economy with a measure space of traders is non-empty, regardless of wheter or not the commodity space is separable. On the other hand, we show that when the commodity space contains weakly compact subsets that are non-separable, than, again, there are atomless economies for which core-Walras equivalence fails. Thus, in particular, for very large commodity spaces the notion of the core seems to be more robust than that of a Walrasian equilibrium.

Time-resolved prompt-gamma activation analysis at spallation neutron sources and applications to cultural heritage, security, and radiation protection

Abstract The present and future developments of time-resolved prompt-gamma activation analysis (T-PGAA) at pulsed neutron sources is discussed in the framework of the successful history of neutron-activation techniques. A brief description of the state of the art and the most important user facilities using standard prompt-gamma activation analysis (PGAA) is provided. Then, we discuss the challenges and the opportunities for T-PGAA at pulsed neutron sources, and the potential impact for applications to cultural heritage, radiation protection, and security. We notice some inversions of trend needed for the further development of T-PGAA with epithermal and fast neutrons, such as the possibility to use fast and high-efficiency γ-ray scintillators with lower energy resolution (compared to usual high-purity germanium detectors) when the signal from neutron capture resonance is selected. We also suggest how detection systems often used in other fields, such as medical physics, can be of interest and inspiration also in the case of neutron-based investigations. Finally, we present new data of T-PGAA measurements on VESUVIO using neutron energies up to the keV using the scintillators available on the instrument, for samples of gold (of interest in cultural heritage), cadmium (for environmental safety), and tantalum (a material used in biomedical implants)

Note on the Core-Walras Equivalence Problem when the Commodity Space is a Banach Lattice

The core-Walras equivalence problem for an atomless economy is considered in the commodity space setting of Banach lattices. In particular, necessary and sufficient conditions on the commodity space in order for core-Walras equivalence to hold are established. In general, these conditions can be regarded as implying that an economy with a continuum of agents has indeed "many more agents than commodities". However, it turns out that there are special commoditiy spaces in which core-Walras equivalence holds for every atomless economy satisfying certain standard assumptions, but in which an atomless economy does not have the meaning of there being "many more agents than commodities."

Partitioning Edge-Colored Hypergraphs into Few Monochromatic Tight Cycles

Confirming a conjecture of Gy´arf´as, we prove that, for all natural numbers k and r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmikl´ossy [Discrete Math., 340 (2017), pp. 2053–2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number