Ideals of general forms and the ubiquity of the Weak Lefschetz property

Let $d_1,...,d_r$ be positive integers and let $I = (F_1,...,F_r)$ be an ideal generated by general forms of degrees $d_1,...,d_r$, respectively, in a polynomial ring $R$ with $n$ variables. When all the degrees are the same we give a result that says, roughly, that they have as few first syzygies as possible. In the general case, the Hilbert function of $R/I$ has been conjectured by Fr\"oberg. In a previous work the authors showed that in many situations the minimal free resolution of $R/I$ must have redundant terms which are not forced by Koszul (first or higher) syzygies among the $F_i$ (and hence could not be predicted from the Hilbert function), but the only examples came when $r=n+1$. Our second main set of results in this paper show that further examples can be obtained when $n+1 \leq r \leq 2n-2$. We also show that if Fr\"oberg's conjecture on the Hilbert function is true then any such redundant terms in the minimal free resolution must occur in the top two possible degrees of the free module. Related to the Fr\"oberg conjecture is the notion of Weak Lefschetz property. We continue the description of the ubiquity of this property. We show that any ideal of general forms in $k[x_1,x_2,x_3,x_4]$ has it. Then we show that for certain choices of degrees, any complete intersection has it and any almost complete intersection has it. Finally, we show that most of the time Artinian ``hypersurface sections'' of zeroschemes have it.Comment: 24 page

How alternative food networks work in a metropolitan area? An analysis of Solidarity Purchase Groups in Northern Italy

Our paper focuses on Solidarity Purchase Group (SPG) participants located in a highly urbanized area, with the aim to investigate the main motivations underlining their participation in a SPG and provide a characterization of them. To this end, we carried out a survey of 795 participants involved in 125 SPGs in the metropolitan area of Milan (Italy). Taking advantage of a questionnaire with 39 questions, we run a factor analysis and a two-step cluster analysis to identify different profiles of SPG participants. Our results show that the system of values animating metropolitan SPG practitioners does not fully conform to that traditionally attributed to an alternative food network (AFN). In fact, considerations linked to food safety and healthiness prevail on altruistic motives such as environmental sustainability and solidarity toward small producers. Furthermore, metropolitan SPGs do not consider particularly desirable periurban and local food products. Observing the SPGs from this perspective, it emerges as such initiatives can flourish also in those places where the lack of connection with the surrounding territory is counterbalanced by the high motivation to buy products from trusted suppliers who are able to guarantee genuine and safe products, not necessarily located nearby

GHZGHZ state generation of three Josephson qubits in presence of bosonic baths

We analyze an entangling protocol to generate tripartite Greenberger-Horne-Zeilinger states in a system consisting of three superconducting qubits with pairwise coupling. The dynamics of the open quantum system is investigated by taking into account the interaction of each qubit with an independent bosonic bath with an ohmic spectral structure. To this end a microscopic master equation is constructed and exactly solved. We find that the protocol here discussed is stable against decoherence and dissipation due to the presence of the external baths.Comment: 16 pages and 4 figure

Learning Mechanism for Column Formation in the Olfactory Bulb

Sensory discrimination requires distributed arrays of processing units. In the olfactory bulb, the processing units for odor discrimination are believed to involve dendrodendritic synaptic interactions between mitral and granule cells. There is increasing anatomical evidence that these cells are organized in columns, and that the columns processing a given odor are arranged in widely distributed arrays. Experimental evidence is lacking on the underlying learning mechanisms for how these columns and arrays are formed. To gain insight into these mechanisms, we have used a simplified realistic circuit model to test the hypothesis that distributed connectivity can self-organize through an activity-dependent dendrodendritic synaptic mechanism. The results point to action potentials propagating in the mitral cell lateral dendrites as playing a critical role in this mechanism. The model predicts that columns emerge from the interaction between the local temporal dynamics of the action potentials and the synapses that they activate during dendritic propagation. The results suggest a novel and robust learning mechanism for the development of distributed processing units in a cortical structure

TRAIL, DR5 and Caspase 3-dependent Apoptosis in Vessels of Diseased Human Temporomandibular Joint Disc. An Immunohistochemical Study

To evaluate the apoptosis involvement in the angiogenesis as a self-limiting process in patients with temporomandibular joint (TMJ) degenerated disc vessels, we assessed, by immunohistochemistry, the detection of TRAIL, its death receptor DR5 and caspase 3. TRAIL, its death receptor DR5 and caspase 3 expression were studied by immunohistochemistry in 15 TMJ discs displaced without reduction and in 4 unaffected discs. These apoptosis molecules were detected in the intima and media layers of newly formed vessels affected discs. In conclusion, vessels apoptosis activation in TMJ disc with ID could be regarded as a self-limiting process that try to leads to vessel regression; in this way an inhibition of angiogenic vessels may prove a key strategy in limiting pathological angiogenesis, by cutting off blood supply to tumors, or by reducing harmful inflammation

Non-Markovian dissipative dynamics of two coupled qubits in independent reservoirs: a comparison between exact solutions and master equation approaches

The reduced dynamics of two interacting qubits coupled to two independent bosonic baths is investigated. The one-excitation dynamics is derived and compared with that based on the resolution of appropriate non-Markovian master equations. The Nakajima-Zwanzig and the time-convolutionless projection operator techniques are exploited to provide a description of the non-Markovian features of the dynamics of the two-qubits system. The validity of such approximate methods and their range of validity in correspondence to different choices of the parameters describing the system are brought to light.Comment: 6 pages, 3 figures. Submitted to PR

On the shape of a pure O-sequence

An order ideal is a finite poset X of (monic) monomials such that, whenever M is in X and N divides M, then N is in X. If all, say t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree. Equivalently, in the language of commutative algebra, pure O-sequences are the h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of Richard Stanley's early works in this area, and have since played a significant role in at least three disciplines: the study of simplicial complexes and their f-vectors, level algebras, and matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences. Our work, making an extensive use of algebraic and combinatorial techniques, includes: (i) A characterization of the first half of a pure O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel; (ii) A study of (the failing of) the unimodality property; (iii) The problem of enumerating pure O-sequences, including a proof that almost all O-sequences are pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) characterization; (v) A pithy connection of the ICP with Stanley's matroid h-vector conjecture; (vi) A specific study of pure O-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 in characteristic zero. As a corollary, pure O-sequences of codimension 3 and type 2 are unimodal (over any field); (vii) An analysis of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii) Some observations about pure f-vectors, an important special case of pure O-sequences.Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly minor revisions with respect to last year's versio