Three problems on the lengths of increasing runs

Let U₁, U₂,… be a sequence of independent, uniform (0, 1) r.v.'s and let R₁, R₂,… be the lengths of increasing runs of {U_i}, i.e., X₁=R₁=inf{i:U_i+1<U_i},…, X_n=R₁+R₂+?+R_n=inf{i:i>X_n?1,U_i+1<U_i}. The first theorem states that the sequence $\text{(}\text{3}\text{2}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(X}{}_{\mathrm{n}}\text{?2n)}$ can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R₁+R₂+?+R_τ(n)?n, $\text{R}{}^1{}_{\mathrm{n}}\text{=n?(R}{}_1\text{+R}{}_2\text{+?+R}{}_{\mathrm{\tau (n)}}\text{)}$ and $\text{M}{}_{\mathrm{n}}\text{=}\text{max}\text{{R}{}_1\text{,R}{}_2\text{,…,R}{}_{\mathrm{\tau (n)}}\text{,R}{}^1{}_{\mathrm{n}}\text{}}$. Here M_n is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of M_n.The limit distribution of the lengths of increasing runs is our third problem.     

Finiteness of Ricci flat supersymmetric non-linear σ-models

Combining the constraints of Kähler differential geometry with the universality of the normal coordinate expansion in the background field method, we study the ultraviolet behavior of 2-dimensional supersymmetric non-linear -models with target space an arbitrary riemannian manifoldM. We show that the constraint ofN=2 supersymmetry requires that all counterterms to the metric beyond one-loop order be cohomologically trivial. It follows that such supersymmetric non-linear -models defined on locally symmetric spaces are super-renormalizable and thatN=4 models are on-shell ultraviolet finite to all orders of perturbation theory.     

Corrigendum to “(p,1)-total labelling of graphs” [Discrete Math. 308 (2008) 496–513]

We give a corrected proof of Proposition 51 of the paper cited in the title.     

Twin Paradox and the Logical Foundation of Relativity Theory

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that~the Twin Paradox becomes provable in AccRel, but it is not provable without IND.     

Towards relativistic ECP / DFT description of chemical bonding in E112 compounds: spin-orbit and correlation effects in E112X versus HgX (X=H, Au)

The relativistic effective core potential (RECP) approach combined with the spin-orbit DFT electron correlation treatment was applied to the study of the bonding of eka-mercury (E112) and mercury with hydrogen and gold atoms. Highly accurate small-core shape-consistent RECPs derived from Hartree-Fock-Dirac-Breit atomic calculations with Fermi nuclear model were employed. The accuracy of the DFT correlation treatment was checked by comparing the results in the scalar-relativistic (spin-orbit-free) limit with those of high level scalar-relativistic correlation calculations within the same RECP model. E112H was predicted to be slightly more stable than its lighter homologue (HgH). The E112-Au bond energy is expected to be ca. 25–30 % weaker than that of Hg-Au. The role of correlations and magnetic (spin-dependent) interactions in E112-X and Hg-X (X=H, Au) bonding is discussed. The present computational procedure can be readily applied to much larger systems and seems to be a promising tool for simulating E112 adsorption on metal surfaces.     

Genetic analysis of the GRIK2 modifier effect in Huntington's disease

Background  

In Huntington's disease (HD), age at neurological onset is inversely correlated with the length of the CAG trinucleotide repeat mutation, but can be modified by genetic factors beyond the HD gene. Association of a relatively infrequent 16 TAA allele of a trinucleotide repeat polymorphism in the GRIK2 3'UTR with earlier than expected age at neurological onset has been suggested to reflect linkage disequilibrium with a functional polymorphism in GRIK2 or an adjacent gene.     

Stable isotope geochemistry of calcrete nodules and septarian concretions in a Quaternary 'red clay' paleovertisol from Hungary

Calcrete nodules and concretions in unusually large amounts are embedded in the Quaternary clay-rich (Vertisol-type) 'red clay' soil-sedimentary complex at the pediment of the Mátra Mountains (Hungary). Stable isotope signatures were studied in nodules and septarian concretions, uncommon due to their several millimeter sized calcite crystals filling voids and fractures, to reveal their origin. The isotope composition of calcrete covers a wide range: delta18O=-5.9 to-10.4 per thousand and delta13C=-8.9 to-12.3 per thousand (vs. V-PDB). Isotope compositions support pedogenic (sensu stricto) and/or shallow groundwater origin for the calcrete nodules and concretions, the role of 'evolved' (isotopically modified) groundwaters in the formation of secondary carbonate was possibly subordinate. Late-stage, large, Mn-rich euhedral calcite crystals in concretions have the lowest delta13C values, which are interpreted as a result of larger contribution of isotopically light organic carbon due to decomposition of organic matter under reducing conditions. Precipitation of late calcite crystals in concretions occurred in early diagenetic environment after shallow burial of the 'red clay' paleovertisol.