This article introduces the Clairaut conformal Riemannian map. This notion includes the previously studied notions of Clairaut conformal submersion, Clairaut Riemannian submersion, and the Clairaut Riemannian map as particular cases, and is well known in the classical theory of surfaces. Toward this, we find the necessary and sufficient condition for a conformal Riemannian map $\varphi : M \to N$ between Riemannian manifolds to be a Clairaut conformal Riemannian map with girth $s = e^f$. We show that the fibers of $\varphi $ are totally umbilical with mean curvature vector field the negative gradient of the logarithm of the girth function, that is, $-\nabla f$. Using this, we obtain a local splitting of M as a warped product and a usual product, if the horizontal space is integrable (under some appropriate hypothesis). We also provide some examples of the Clairaut conformal Riemannian maps to confirm our main theorem. We observe that the Laplacian of the logarithmic girth, that is, of f, on the total manifold takes the special form. It reduces to the Laplacian on the horizontal distribution, and if it is nonnegative, the universal covering space of M becomes a product manifold, under some hypothesis on f. Analysis of the Laplacian of f also yields the splitting of the universal covering space of M as a warped product under some appropriate conditions. We calculate the sectional curvature and mixed sectional curvature of M when f is a distance function. We also find the relationships between the total manifold and the fibers being symmetrical and, in particular, having constant sectional curvature, and from there, we compare their universal covering spaces, if fibers are also complete, provided f is a distance function. We also find a condition on the curvature tensor of the fibers to be semi-symmetric, provided that the total manifold is semi-symmetric and f is a distance function. In turn, this gives the warped product of symmetric, semi-symmetric spaces into two symmetric, semi-symmetric subspaces (under some hypothesis). Also if the Hessian or the Laplacian of the Riemannian curvature tensor fields is zero, or has a harmonic curvature tensor, then the fibers of $\varphi $ also satisfy the same property, if f is also a distance function. By obtaining Bochner-type formulas for Clairaut conformal Riemannian maps, we establish the relations between the divergences of the Ricci curvature tensor on fibers and horizontal space and the corresponding scalar curvature. We also study the horizontal Killing vector field of constant length and show that they are parallel under appropriate hypotheses. This in turn gives the splitting of the total manifold, if it admits a horizontal parallel Killing vector field and if the horizontal space is integrable. Finally, assuming that $\nabla f$ is a nontrivial gradient Ricci soliton on M, we prove that any vertical vector field is incompressible and hence the volume form of the fiber is invariant under the flow of the vector field.

Morphological changes occurring in naturally weathered micas in seven residual soils from Ontario were studied. Splitting, exfoliation of the (100) planes, surface coats, filling of spaces with soil material as well as cracking and crumbling of (001) surfaces were the main morphological changes found to be taking place during natural weathering of mica particles. The significance of the observed feature in relation to the general theory of mica weathering is discussed.

We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n < 2^{-n}$ for infinitely many ${n \in \mathbb {N}}$. We also call a set $A\subseteq \mathbb {N}$ regainingly approximable if it is c.e. and the strongly left-computable number $2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. K-trivial, we construct such an $\alpha $ such that ${K(\alpha \restriction n)>n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

Theory of stable models is the mathematical basis of answer set programming. Several results in that theory refer to the concept of the positive dependency graph of a logic program. We describe a modification of that concept and show that the new understanding of positive dependency makes it possible to strengthen some of these results.

Motivated by recent results and questions of Raghavan and Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $\kappa =\lambda ^+$ for some $\lambda \geq \omega $ and $\mathfrak {b}(\kappa )=\kappa ^+$ then $\mathfrak {a}_e(\kappa )=\mathfrak {a}_p(\kappa )=\kappa ^+$ . If, additionally, $2^{<\lambda }=\lambda $ then $\mathfrak {a}_g(\kappa )=\kappa ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak {d}(\kappa )$ in terms of $\mathfrak {r}(\kappa )$ , including $\mathfrak {d}(\kappa )\leq \mathfrak {r}_\sigma (\kappa )\leq \operatorname {\mathrm {cf}}([\mathfrak {r}(\kappa )]^\omega )$ , and $\mathfrak {d}(\kappa )\leq \mathfrak {r}(\kappa )$ whenever $\mathfrak {r}(\kappa )<\mathfrak {b}(\kappa )^{+\kappa }$ or $\operatorname {\mathrm {cf}}(\mathfrak {r}(\kappa ))\leq \kappa $ holds.

Quality of in vitro-produced embryos is influenced by changes in gene expression in response to adverse conditions. Gene markers for predicting ‘good embryos’ do not exist at present. We propose that the expression of pluripotency markers OCT4–SOX2–NANOG in D9 (day 9) bovine demi-embryos correlated with development at D13 (day 13). Day 8 in vitro-produced blastocysts were split in two cloned halves, one half (D9) was subjected to analysis of pluripotency markers and the other was kept in culture until D13 of development. Embryo development was scored and correlated with its own status at D9 and assigned to one of two categories: G1, arrested/dead; or G2, development up to D13. SOX2 and NANOG expression levels were significantly higher in embryos from G1 and there was also negative correlation between SOX2 and embryo survival to D13 (G3; r = −0.37; P = 0.03). We observed a significant reduction in the expression of the three studied genes from D9 to D13. Furthermore, there was a correlation between the expression of pluripotency markers at D9 and embryo diameter and the expression of trophoblastic markers at D13 (TP1–EOMES–FGF4–CDX2–TKDP1). Finally, the quotient between the relative expression of SOX2 and OCT4 in the D9 blastocysts from G1 and G2 showed that embryos that were considered as competent (G2) had a quotient close to one, while the other group had a quotient of 2.3 due to a higher expression of SOX2. These results might indicate that overexpression of SOX2 at the blastocyst stage had a negative effect on the control of embryonic developmental potential.

We prove the consistency of

$$~~\text{add}\left( \mathcal{N} \right)<\operatorname{cov}\left( \mathcal{N} \right)<\mathfrak{p}\text{=}\mathfrak{s}\text{=}\mathfrak{g}< \text{add}\left( \mathcal{M} \right)=\text{cof}\left( \mathcal{M} \right)<\mathfrak{a}=\mathfrak{r}=\text{non}\left( N \right)=\mathfrak{c}$$

$\text{ZFC}$

A novel generalised successive overrelaxation (GSOR) method for solving generalised saddle point problems is proposed, based on splitting the coefficient matrix. The proposed method is shown to converge under suitable restrictions on the iteration parameters, and we present some illustrative numerical results.

Embryo splitting might be used to increase offspring yield and for molecular analysis of embryo competence. How splitting affects developmental potential of embryos is unknown. This research aimed to study the effect of bovine blastocyst splitting on morphological and gene expression homogeneity of demi-embryos and on embryo competence during elongation. Grade I bovine blastocyst produced in vitro were split into halves and distributed in nine groups (3 × 3 setting according to age and stage before splitting; age: days 7–9; stage: early, expanded and hatched blastocysts). Homogeneity and survival rate in vitro after splitting (12 h, days 10 and 13) and the effect of splitting on embryo development at elongation after embryo transfer (day 17) were assessed morphologically and by RT-qPCR. The genes analysed were OCT4, SOX2, NANOG, CDX2, TP1, TKDP1, EOMES, and BAX. Approximately 90% of split embryos had a well conserved defined inner cell mass (ICM), 70% of the halves had similar size with no differences in gene expression 12 h after splitting. Split embryos cultured further conserved normal and comparable morphology at day 10 of development; this situation changes at day 13 when embryo morphology and gene expression differed markedly among demi-embryos. Split and non-split blastocysts were transferred to recipient cows and were recovered at day 17. Fifty per cent of non-split embryos were larger than 100 mm (33% for split embryos). OCT4, SOX2, TP1 and EOMES levels were down-regulated in elongated embryos derived from split blastocysts. In conclusion, splitting day-8 blastocysts yields homogenous demi-embryos in terms of developmental capability and gene expression, but the initiation of the filamentous stage seems to be affected by the splitting.

The symmetric energy-conserved splitting FDTD scheme developed in is a very new and efficient scheme for computing the Maxwell’s equations. It is based on splitting the whole Maxwell’s equations and matching the x-direction and y-direction electric fields associated to the magnetic field symmetrically. In this paper, we make further study on the scheme for the 2D Maxwell’s equations with the PEC boundary condition. Two new energy-conserved identities of the symmetric EC-S-FDTD scheme in the discrete H1-norm are derived. It is then proved that the scheme is uncondi-tionally stable in the discrete H1-norm. By the new energy-conserved identities, the super-convergence of the symmetric EC-S-FDTD scheme is further proved that it is of second order convergence in both time and space steps in the discrete H1-norm. Numerical experiments are carried out and confirm our theoretical results.

In this paper, we studythe linear Schrödinger equation over the d-dimensional torus,with small values of the perturbing potential.We consider numerical approximations of the associated solutions obtainedby a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable).In this fully discrete setting, we prove that the regularity of the initialdatum is preserved over long times, i.e. times that are exponentially longwith the time discretization parameter. We here refer to Gevrey regularity, and our estimatesturn out to be uniform in the space discretization parameter.This paper extends [G. Dujardin and E. Faou, Numer. Math.97 (2004) 493–535], where a similar result has been obtained inthe semi-discrete situation, i.e. when the mere time variable is discretized and spaceis kept a continuous variable.

Among the comets that were observed to break in two or more fragments, only a few of them are periodic. So far the dynamic study of the relative motion of a secondary nucleus with respect to the primary has supposed that a cometary fragment is subject to a small and continuous radial nongravitational force after separation at rest. This force acts against the solar attraction and varies according to an inverse square law. A small impulse at break up may also be invoked in some case. A different approach is followed in this paper when dealing with a fragment of a periodic comet: after separation the motion of a secondary nucleus is characterized by nongravitational forces which vary according to the same g(r) law currently used for the primary.

Results of the study of comets P/Biela and P/du Toit-Hartley show that the motion of their fragments after separation is characterized by nongravitational parameters which are larger than those of the parent bodies. Both fragments lasted for about 2 full revolutions and three returns.