Let R R be a polynomial ring over field and M equals circled-plus Underscript n Endscripts upper Subscript n"> M</mm...

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Graded-index optical fiber probes suitable for focusing or collimating the output of an at a wavelength 1.3 µm have become enabling technology in coherence tomography imaging applications vivo bioimaging. Such tips however remain uncommon other photonics applications. This paper provides first numerical study graded-index covering broad range wavelengths spanning from UV to short-infrared. The ...

Let G be a finite group. A collection $$\Pi =\{H_1,\dots ,{H_r}\}$$ of subgroups G, where $$r>1$$ , is said non-trivial partition if every non-identity element belongs to one and only $$H_i$$ for some $$1\leqslant i\leqslant r$$ . We call group that does not admit any partition-free In this paper, we study whose all proper non-cyclic partitions.

We consider the lower central filtration of the free associative algebra An with n generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebraWn of polynomial vector fields on C. We compute the space [An, An]/[An, [An, An...

A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional subscheme of projective 3-space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines. The second and third authors have shown that these curves have very nice combinatorial properties, and they have made a careful study of the even liaison classes of these curves. We build on this ...