Quadratic-stretch elasticity

A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed.

On the Dimension of Non-Abelian Tensor Squares of $n$-Lie Algebras

Let $L$ be an $n$-Lie algebra over a field $\F$. In this paper, we introduce the notion of non-abelian tensor square $L\otimes L$ of $L$ and define the central ideal $L\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\square L\cong L^{ab}\square L^{ab}$. Also, we establish the short exact sequence\[0\lra\M(L)\lra\frac{L\otimes L}{L\square L}\lra L^2\lra0\]and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.

On the non-abelian tensor square of all groups of order dividing p5

In this paper, we consider all groups of order dividing [Formula: see text]. We obtain the explicit structure of the non-abelian tensor square, non-abelian exterior square, tensor center, exterior center, the third homotopy group of suspension of an Eilenberg–MacLane space [Formula: see text] and [Formula: see text] of such groups.

On the quantum improved Schwarzschild black hole

Deriving the gravitational effective action directly from exact renormalization group is very complicated, if not impossible. Hence, to study the effects of running gravitational coupling which tends to a non-Gaussian UV fixed point (as it is supposed by the asymptotic safety conjecture), two steps are usually adopted. Cutoff identification and improvement of the gravitational coupling to the running one. As suggested in Ref. 1, a function of all independent curvature invariants seems to be the best choice for cutoff identification of gravitational quantum fluctuations in curved space–time and makes the action improvement, which saves the general covariance of theory, possible. Here, we choose Ricci tensor square for this purpose and then the equation of motion of improved gravitational action and its spherically symmetric vacuum solution are obtained. Indeed, its effect on the massive particles’ trajectory and the black hole thermodynamics is studied.