On sets of singular rotations for translation invariant differentiation bases formed by intervals

We study the dependence of differential properties of an indefinite integral on a rotation of the coordinate system. Namely, the following problem is studied: For a summable function f, what kind of a set may be the set of rotations θ for which {\int f} is not differentiable with respect to the θ-rotation of a given basis B? For translation invariant bases B formed by two-dimensional intervals, some classes of sets of singular rotations are found. In particular, for such bases with symmetric structure, a characterization of at most countable sets of singular rotations is found.

Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

Transformation groups on real plane and their differential invariants

Complete sets of bases of differential invariants, operators of invariant differentiation, and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of finite-dimensional Lie algebras on the real plane are revisited.

Avian Extended Junctional SR: 3-D Geometry Rendered in Stereo

Extended junctional sarcoplasmic reticulum (EJSR) is an invariant differentiation of the sarcoplasmic reticulum (SR) in bird cardiac myocytes (CM) and central to excitation-contraction coupling (ECC). EJSR occurs as both continuous and discontinuous extensions of junctional sarcoplasmic reticulum (JSR), and surrounds and pervades the Z/I band as the “ EJSR Z-rete” whose geometry has mechanistic implications for the function of “couplings” in ECC, in general. “Peripheral coupling(s)” (PC) in birds, and the additional “interior coupling(s)” (IC) at transverse tubules (TT) in mammals, are formed by tight apposition to plasmalemma of JSR, a specialized calcium (Ca) store of the SR. Free SR (FSR; i.e. free of JSR/EJSR specializations) is the rest of the smooth, tubular SR network, which connects intercalated patches of EJSR forming the EJSR Z-retes and, elsewhere, displays both longitudinal and transverse geometries in surrounding the contractile material for the purpose of sequestering Ca after each muscle contraction. Except for EJSR having no plasmalemmal contact, morphologically, EJSR and JSR are homologues:1 both have similar sizes; are studded (approx. 32 nm center-to-center) with junctional processes (JP; ryanodine receptor (RyR)/-Ca-release channels);