An EF2X Allocation Protocol for Restricted Additive Valuations

We study the problem of fairly allocating a set of $m$ indivisible goods to aset of $n$ agents. Envy-freeness up to any good (EFX) criteria -- whichrequires that no agent prefers the bundle of another agent after removal of anysingle good -- is known to be a remarkable analogous of envy-freeness when theresource is a set of indivisible goods. In this paper, we investigate EFXnotion for the restricted additive valuations, that is, every good has somenon-negative value, and every agent is interested in only some of the goods. We introduce a natural relaxation of EFX called EFkX which requires that noagent envies another agent after removal of any $k$ goods. Our maincontribution is an algorithm that finds a complete (i.e., no good is discarded)EF2X allocation for the restricted additive valuations. In our algorithm wedevise new concepts, namely "configuration" and "envy-elimination" that mightbe of independent interest. We also use our new tools to find an EFX allocation for restricted additivevaluations that discards at most $\lfloor n/2 \rfloor -1$ goods. This improvesthe state of the art for the restricted additive valuations by a factor of $2$.<br

Braided Cyclic Cocycles and Non-Associative Geometry

We use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeld-type cochain twist. These are the so-called quasialgebras and include the octonions as braided-commutative but nonassociative coordinate rings, as well as quasialgebra versions \CC_{q}(G) of the standard q-deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalised automorphism $\sigma$, and obtain the required generalisation of cyclic cohomology. We show that this \emph{braided cyclic cocohomology} is invariant under a cochain twist. We also extend to our generalisation the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra. The paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry, as well as the algebraic noncommutative torus as an associative example.Comment: 36 pages latex, 9 figure

Developing a Finite Element Model to Investigate Second Metatarsal Stress During Running

This is the author accepted manuscript. The final version is available from UKACM via the link in this recordSecond metatarsal (2MT) stress fracture is a common and burdensome injury amongst runners, however understanding of the risk factors leading to injury is limited. Finite Element (FE) modelling represents a viable biorealistic alternative to invasive studies and simple beam theory models. This study shows the design and validation of a simple subject-specific FE model of the 2MT incorporating geometrically accurate soft tissue and loading. Results show a good comparison with both recent models and bone staple strain gauge data.Engineering and Physical Sciences Research Council (EPSRC

The effect of homozygous deletion of the BBOX1 and Fibin genes on carnitine level and acyl carnitine profile.

BACKGROUND: Carnitine is a key molecule in energy metabolism that helps transport activated fatty acids into the mitochondria. Its homeostasis is achieved through oral intake, renal reabsorption and de novo biosynthesis. Unlike dietary intake and renal reabsorption, the importance of de novo biosynthesis pathway in carnitine homeostasis remains unclear, due to lack of animal models and description of a single patient defective in this pathway. CASE PRESENTATION: We identified by array comparative genomic hybridization a 42 months-old girl homozygote for a 221 Kb interstitial deletions at 11p14.2, that overlaps the genes encoding Fibin and butyrobetaine-gamma 2-oxoglutarate dioxygenase 1 (BBOX1), an enzyme essential for the biosynthesis of carnitine de novo. She presented microcephaly, speech delay, growth retardation and minor facial anomalies. The levels of almost all evaluated metabolites were normal. Her serum level of free carnitine was at the lower limit of the reference range, while her acylcarnitine to free carnitine ratio was normal. CONCLUSIONS: We present an individual with a completely defective carnitine de novo biosynthesis. This condition results in mildly decreased free carnitine level, but not in clinical manifestations characteristic of carnitine deficiency disorders, suggesting that dietary carnitine intake and renal reabsorption are sufficient to carnitine homeostasis. Our results also demonstrate that haploinsufficiency of BBOX1 and/or Fibin is not associated with Primrose syndrome as previously suggested

What can(not) be measured with ton-scale dark matter direct detection experiments

Direct searches for dark matter have prompted in recent years a great deal of excitement within the astroparticle physics community, but the compatibility between signal claims and null results of different experiments is far from being a settled issue. In this context, we study here the prospects for constraining the dark matter parameter space with the next generation of ton-scale detectors. Using realistic experimental capabilities for a wide range of targets (including fluorine, sodium, argon, germanium, iodine and xenon), the role of target complementarity is analysed in detail while including the impact of astrophysical uncertainties in a self-consistent manner. We show explicitly that a multi-target signal in future direct detection facilities can determine the sign of the ratio of scalar couplings $f_n/f_p$, but not its scale. This implies that the scalar-proton cross-section is left essentially unconstrained if the assumption $f_p\sim f_n$ is relaxed. Instead, we find that both the axial-proton cross-section and the ratio of axial couplings $a_n/a_p$ can be measured with fair accuracy if multi-ton instruments using sodium and iodine will eventually come online. Moreover, it turns out that future direct detection data can easily discriminate between elastic and inelastic scatterings. Finally, we argue that, with weak assumptions regarding the WIMP couplings and the astrophysics, only the dark matter mass and the inelastic parameter (i.e. mass splitting) may be inferred from the recoil spectra -- specifically, we anticipate an accuracy of tens of GeV (tens of keV) in the measurement of the dark matter mass (inelastic parameter).Comment: 31 pages, 7 figures, 7 table

EFX Allocations: Simplifications and Improvements

The existence of EFX allocations is a fundamental open problem in discretefair division. Given a set of agents and indivisible goods, the goal is todetermine the existence of an allocation where no agent envies anotherfollowing the removal of any single good from the other agent's bundle. Sincethe general problem has been illusive, progress is made on two fronts: $(i)$proving existence when the number of agents is small, $(ii)$ proving existenceof relaxations of EFX. In this paper, we improve results on both fronts (andsimplify in one of the cases). We prove the existence of EFX allocations with three agents, restricting onlyone agent to have an MMS-feasible valuation function (a strict generalizationof nice-cancelable valuation functions introduced by Berger et al. whichsubsumes additive, budget-additive and unit demand valuation functions). Theother agents may have any monotone valuation functions. Our proof technique issignificantly simpler and shorter than the proof by Chaudhury et al. onexistence of EFX allocations when there are three agents with additivevaluation functions and therefore more accessible. Secondly, we consider relaxations of EFX allocations, namely, approximate-EFXallocations and EFX allocations with few unallocated goods (charity). Chaudhuryet al. showed the existence of $(1-\epsilon)$-EFX allocation with$O((n/\epsilon)^{\frac{4}{5}})$ charity by establishing a connection to aproblem in extremal combinatorics. We improve their result and prove theexistence of $(1-\epsilon)$-EFX allocations with $\tilde{O}((n/\epsilon)^{\frac{1}{2}})$ charity. In fact, some of our techniques can be usedto prove improved upper-bounds on a problem in zero-sum combinatoricsintroduced by Alon and Krivelevich.<br