Finite Generation of Algebras Associated to Powers of Ideals

We study generalized symbolic powers and form ideals of powers of ideals and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.Comment: 17 pages. The only change in this version is to add new reference

On monoids, 2-firs, and semifirs

Several authors have studied the question of when the monoid ring DM of a monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir, or a 2-fir (definitions recalled in section 1). It is known that for M nontrivial, a necessary condition for any of these properties to hold is that D be a division ring. Under that assumption, necessary and sufficient conditions on M are known for DM to be a right or left fir, and various conditions on M have been proved necessary or sufficient for DM to be a 2-fir or semifir. A sufficient condition for DM to be a semifir is that M be a direct limit of monoids which are free products of free monoids and free groups. W.Dicks has conjectured that this is also necessary. However F.Ced\'o has given an example of a monoid M which is not such a direct limit, but satisfies the known necessary conditions for DM to be a semifir. It is an open question whether for this M, the rings DM are semifirs. We note some reformulations of the known necessary conditions for DM to be a 2-fir or a semifir, motivate Ced\'o's construction and a variant, and recover Ced\'o's results for both constructions. Any homomorphism from a monoid M into \Z induces a \Z-grading on DM, and we show that for the two monoids in question, the rings DM are "homogeneous semifirs" with respect to all such nontrivial \Z-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free. If M is a monoid such that DM is an n-fir, and N a "well-behaved" submonoid of M, we obtain results on DN. Using these, we show that for M a monoid such that DM is a 2-fir, mutual commutativity is an equivalence relation on nonidentity elements of M, and each equivalence class, together with the identity element, is a directed union of infinite cyclic groups or infinite cyclic monoids. Several open questions are noted.Comment: 28 pages. To appear, Semigroup Forum. Some clarifications and corrections from previous versio

On the existence of FF-thresholds and related limits

The $F$-thresholds are important numerical invariants in prime characteristic, whose existence had been established only under certain assumptions. We show the existence of $F$-thresholds in full generality. We study properties of standard graded algebras over a field for which $F$-pure threshold and $F$-threshold at the irrelevant maximal ideal agree. We also exhibit explicit bounds for the $a$-invariants and Castelnuovo-Mumford regularity of Frobenius powers of ideals in terms of $F$-thresholds and $F$-pure thresholds, obtaining existence of related limits in certain cases.Comment: 21 page

Bounding symbolic powers via asymptotic multiplier ideals

We revisit a bound on symbolic powers found by Ein-Lazarsfeld-Smith and subsequently improved by Takagi-Yoshida. We show that the original argument of Ein-Lazarsfeld-Smith actually gives the same improvement. On the other hand, we show by examples that any further improvement based on the same technique appears unlikely. This is primarily an exposition; only some examples and remarks might be new.Comment: 10 pages. Primarily exposition. Originally written as appendix to lecture notes by Brian Harbourne. v2: Minor changes. v3: Final version, appeared in Ann. Univ. Pedagog. Crac. Stud. Mat

Observation of Transparency of Erbium-doped Silicon nitride in photonic crystal nanobeam cavities

One-dimensional nanobeam photonic crystal cavities are fabricated in an Er-doped amorphous silicon nitride layer. Photoluminescence from the cavities around 1.54 um is studied at cryogenic and room temperatures at different optical pump powers. The resonators demonstrate Purcell enhanced absorption and emission rates, also confirmed by time-resolved measurements. Resonances exhibit linewidth narrowing with pump power, signifying absorption bleaching and the onset of stimulated emission in the material at both 5.5 K and room temperature. We estimate from the cavity linewidths that Er has been pumped to transparency at the cavity resonance wavelength.Comment: 10 pages, 7 figure

Nonperturbative Effects on T_c of Interacting Bose Gases in Power-Law Traps

The critical temperature T_c of an interacting Bose gas trapped in a general power-law potential V(x)=\sum_i U_i|x_i|^{p_i} is calculated with the help of variational perturbation theory. It is shown that the interaction-induced shift in T_c fulfills the relation (T_c-T_c^0)/T_c^0= D_1(eta)a + D'(eta)a^{2 eta}+ O(a^2) with T_c^0 the critical temperature of the trapped ideal gas, a the s-wave scattering length divided by the thermal wavelength at T_c, and eta=1/2+\sum_i 1/p_i the potential-shape parameter. The terms D_1(eta)a and D'(eta) a^{2 eta} describe the leading-order perturbative and nonperturbative contributions to the critical temperature, respectively. This result quantitatively shows how an increasingly inhomogeneous potential suppresses the influence of critical fluctuations. The appearance of the a^{2 eta} contribution is qualitatively explained in terms of the Ginzburg criterion.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper (including all PS fonts) at http://www.physik.fu-berlin.de/~kleinert/35

Effective dynamics of scalar perturbations in a flat Friedmann-Robertson-Walker spacetime in Loop Quantum Cosmology

We study the evolution of a homogeneous and isotropic spacetime whose spatial sections have three-torus topology, coupled to a massless scalar field with small scalar perturbations within loop quantum cosmology. We consider a proposal for the effective dynamics based on a previous hybrid quantization completed by us. Consequently, we introduce a convenient gauge fixing and adopt reduced canonical variables adapted to that hybrid quantum description. Besides, we keep backreaction contributions on the background coming from terms quadratic in the perturbations in the action of the system. We carry out a numerical analysis assuming that the inhomogeneities were in a massless vacuum state at distant past (where the initial data are set). At distant future, we observe a statistical amplification of the modes amplitude in the infrared region, as well as a phase synchronization arising from quantum gravity phenomena. A description of the perturbations in terms of the Mukhanov-Sasaki gauge invariants provides the same qualitative results. Finally, we analyze some consequences of the backreaction in our effective description.Comment: 15 pages, 9 figures. Accepted for publication in Phys. Rev.