let $r$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $r$ and $f(x)=a_0+a_1x+cdots+a_nx^n$ be a nonzero skew polynomial in $r[x;alpha]$. it is proved that if there exists a nonzero skew polynomial $g(x)=b_0+b_1x+cdots+b_mx^m$ in $r[x;alpha]$ such that $g(x)f(x)=c$ is a constant in $r$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $r$ such tha...

in this paper, we introduce the concept of the   generalized aip-rings as a generalization of the generalized quasi-   baer rings   and generalized p.p.-rings. we show that the class of the generalized aip-rings is closed under direct products and morita invariance. we also characterize the 2-by-2 formal upper triangular matrix rings of this new class of rings. finally, we provide sever...

We introduce graded $${\mathbb {E}}_{\infty }$$ -rings and modules over them, study their properties. construct projective schemes associated to connective {N}}$$ -graded in spectral algebraic geometry. Under some finiteness conditions, we show that the $$\infty $$ -category of almost perfect quasi-coherent sheaves a scheme $$\text { {Proj}}\,(A)$$ -ring A can be described terms $${{\mathbb {Z}...

Digital Subtractive Angiography (DSA) has been performed to image human coronary arteries using wiggler radiation from electron storage rings. The significant medical promise of this procedure motivates the development of smaller and less costly x-ray sources. Several exotic sources are candidates for consideration, using effects such as Cherenkov, channeling, coherent bremsstrahlung, laser bac...

In this paper we construct a pushforward-pullback adjoint pair for categories of quasi-coherent sheaves, along a morphism of algebraic stacks, which is represented in algebraic stacks over the site C = Affflat. The construction uses the characterization of algebraic stacks of [H3] and is based on the descent description of the category of quasi-coherent sheaves given in [H2]. We show that an es...

A laser-based method to generate ultrashort pulses of synchrotron radiation in electron storage rings is coherent harmonic generation (CHG) using two undulators to produce coherent radiation at harmonics of the initial laser wavelength by microbunching. The bunching factor and thus the pulse intensity, however, decreases exponentially with increasing harmonic order. Echo-enabled harmonic genera...

We present detailed calculations of low-energy spin dynamics in the " ferric wheel " systems Na:Fe6 and Cs:Fe8 in a magnetic field. We compute by exact diagonalisation the low-energy spectra and matrix elements for total-spin and Néel-vector components, and thus the time-dependent correlation functions of these operators. Comparison of our results with the semiclassical theory of coherent quant...

In storage rings with very low frequency of synchrotron oscillations coherent beam-beam instabilities can be suppressed due to the “phase averaging” effect, which recently was described by S. Krishnagopal and R. Siemann for the incoherent beam-beam instability. In this paper using the Vlasov’s equation we calculate the form factors, which renormalize beam-beam parameters of coherent beam-beam o...

Application of aromaticity indices to geometries of rings of eleven derivatives of fulvene, two salts of cyclopentadienyl, and nine derivatives of heptafulvene gives a coherent view in line with expectation due to the (4n + 2) rule: electron-donating substituents attached to exocyclic carbon atom in fulvenes and electron-accepting substituents attached to heptafulvenes increase significantly th...