The paper is devoted to a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential
The paper is devoted to a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential
On the basis of a prescribed quadratic Lagrangian, an algorithm of synthesis for an electric circuit is suggested here. That is, the circuit evolution equations are equivalent to the relevant Euler–Lagrange equations. The proposed synthesis is a systematic approach that allows one to realize any finite-dimensional physical system described by a quadratic Lagrangian in a lossless electric circuit so that their evolution equations are equivalent. The synthesized circuit is composed of (i) capacitors and inductors of positive or negative values for the respective capacitances and inductances, and (ii) gyrators. The circuit topological design is based on the set of
A family
The paper is devoted to the study of deformations of Artinian algebras and zero-dimensional germs of varieties. In particular, an approach is developed to solving the open problem about the nonexistence of rigid Artinian algebras; it is based essentially on the use of the canonical duality in the cotangent complex. Thus, it is shown that there are no rigid Gorenstein Artinian algebras and rigid almost complete intersections. The proof of the latter statement is based on the properties of the torsion functors. More precisely, the tensor product of the conormal and canonical modules of the corresponding Artinian algebra is calculated. In this case, the homology and cohomology groups of higher degrees are also found. Among other things, some estimates are obtained for the dimension of the spaces of the first lower and upper cotangent functors of Artinian algebras, and the relationship between them is described. In conclusion, several examples of nonsmoothable Artinian noncomplete intersections are examined, and some unusual properties of such algebras are discussed.
A version of the extra-zero conjecture, formulated by the first named author, is proved for
Atsuyama’s result on the geometry of symmetric spaces of type EIII is generalized to the case of arbitrary fields of characteristic not 2 or 3. As an application, a variant of the “chain lemma” for microweight tori in groups of type
A hierarchy of Lax pairs with
A necessary and sufficient Blaschke-type condition is obtained for the zeros of derivatoves of Nevanlinna class functions.
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