Artinian level algebras of codimension 3

In this paper, we continue the study of which $h$-vectors $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$ can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition $\beta_{2,d+2}(I^{\rm lex})=\beta_{1,d+1}(I^{\rm lex})$, where $I^{\rm lex}$ is the lex-segment ideal associated with an ideal $I$. Our approach is to adopt an homological method called {\it Cancellation Principle}: the minimal free resolution of $I$ is obtained from that of $I^{\rm lex}$ by canceling some adjacent terms of the same shift. We prove that when $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, $R/I$ can be an Artinian level $k$-algebra only if either $h_{d-1}<h_d<h_{d+1}$ or $h_{d-1}=h_d=h_{d+1}=d+1$ holds. We also apply our results to show that for $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$, the Hilbert function of an Artinian algebra of codimension 3 with the condition $h_{d-1}=h_d<h_{d+1}$, (a) if $h_d\leq 3d+2$, then $h$-vector \H cannot be level, and (b) if $h_d\geq 3d+3$, then there is a level algebra with Hilbert function \H for some value of $h_{d+1}$.Comment: 15 page

Entanglement-assisted codeword stabilized quantum codes with imperfect ebits

Quantum error correcting codes (QECCs) in quantum communi- cation systems has been known to exhibit improved performance with the use of error-free entanglement bits (ebits). In practical situations, ebits inevitably suffer from errors, and as a result, the error-correcting capability of the code is diminished. Prior studies have proposed two different schemes as a solu- tion. One uses only one QECC to correct errors on the receiver's side (i.e., Bob) and on the sender's side (i.e., Alice). The other uses different QECCs on each side. In this paper, we present a method to correct errors on both sides by using single nonadditive Entanglement-assisted codeword stabilized quantum error correcting code(EACWS QECC). We use the property that the number of effective error patterns decreases as much as the number of ebits. This property results in a greater number of logical codewords using the same number of physical qubits

Generic Initial Ideals And Graded Artinian Level Algebras Not Having The Weak-Lefschetz Property

We find a sufficient condition that \H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function $\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1})$ cannot be level if $h_d\le 2d+3$, and that there exists a level O-sequence of codimension 3 of type \H for $h_d \ge 2d+k$ for $k\ge 4$. Furthermore, we show that \H is not level if $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, and also prove that any codimension 3 Artinian graded algebra $A=R/I$ cannot be level if \beta_{1,d+2}(\Gin(I))=\beta_{2,d+2}(\Gin(I)). In this case, the Hilbert function of $A$ does not have to satisfy the condition $h_{d-1}>h_d=h_{d+1}$. Moreover, we show that every codimension $n$ graded Artinian level algebra having the Weak-Lefschetz Property has the strictly unimodal Hilbert function having a growth condition on $(h_{d-1}-h_{d}) \le (n-1)(h_d-h_{d+1})$ for every $d > \theta$ where $$h_0...>h_{s-1}>h_s.$$ In particular, we find that if $A$ is of codimension 3, then $(h_{d-1}-h_{d}) < 2(h_d-h_{d+1})$ for every $\theta< d <s$ and $h_{s-1}\le 3 h_s$, and prove that if $A$ is a codimension 3 Artinian algebra with an $h$-vector $(1,3,h_2,...,h_s)$ such that h_{d-1}-h_d=2(h_d-h_{d+1})>0 \quad \text{and} \quad \soc(A)_{d-1}=0 for some $r_1(A)<d<s$, then $(I_{\le d+1})$ is $(d+1)$-regular and \dim_k\soc(A)_d=h_d-h_{d+1}.Comment: 25 page

Minimum Weight Perfect Matching via Blossom Belief Propagation

Max-product Belief Propagation (BP) is a popular message-passing algorithm for computing a Maximum-A-Posteriori (MAP) assignment over a distribution represented by a Graphical Model (GM). It has been shown that BP can solve a number of combinatorial optimization problems including minimum weight matching, shortest path, network flow and vertex cover under the following common assumption: the respective Linear Programming (LP) relaxation is tight, i.e., no integrality gap is present. However, when LP shows an integrality gap, no model has been known which can be solved systematically via sequential applications of BP. In this paper, we develop the first such algorithm, coined Blossom-BP, for solving the minimum weight matching problem over arbitrary graphs. Each step of the sequential algorithm requires applying BP over a modified graph constructed by contractions and expansions of blossoms, i.e., odd sets of vertices. Our scheme guarantees termination in O(n^2) of BP runs, where n is the number of vertices in the original graph. In essence, the Blossom-BP offers a distributed version of the celebrated Edmonds' Blossom algorithm by jumping at once over many sub-steps with a single BP. Moreover, our result provides an interpretation of the Edmonds' algorithm as a sequence of LPs

Di(hydroperoxy)alkane Adducts of Phosphine Oxides: Safe, Solid, Stoichiometric and Soluble Oxidizing Agents

Despite its importance and wide use as oxidizing agent, aqueous H₂O₂ has disadvantages. It easily decomposes, and when the substrates are not water-soluble, biphasic reaction mixtures are required. Thus, oxidizing agents that are anhydrous and soluble in organic solvents are desired. To this purpose, several H₂O₂ adducts of phosphine oxides, for example [tBu₃PO• H₂O₂]2₂and [Ph₃PO• H₂O₂]₂ H₂O₂, have been synthesized and characterized. These adducts represent an extension to the adducts previously reported by the Bluemel group, and display comparable physical properties. Furthermore, di(hydroperoxy)alkane adducts, R₃PO•(HOO)₂CR'R" (R, R', R" = alkyl, aryl), were synthesized and fully characterized. These adducts can be constructed using a wide variety of alkanes and phosphine oxides. All di(hydroperoxy)alkane adducts are structurally well defined as proven by single crystal X-ray analysis, and they contain two active oxygen atoms per assembly. These adducts of the type R₃PO•(HOO)₂CR'R" are highly soluble in organic solvents, allowing for oxidation reactions to occur in one phase. Moreover, there are many beneficial features to be harvested from their well-defined molecular structure and relatively anhydrous character. For example, selective and fast oxidation of dialkylsulfides to corresponding sulfoxides can be accomplished, without overoxidation to sulfones, because the solid oxidizing agents can easily be administered stoichiometrically. The adducts can also successfully oxidize substrates sensitive to hydrolysis, such as Ph₂P-PPh₂, without cleaving the P-P bond. The R₃PO•(HOO)₂CR'R" adducts are robust and practically no decomposition is found after storing the solids for 100 days at 4 °C. At room temperature, the adducts slowly decompose over time, via the release of oxygen gas. When exposed to higher temperatures or mechanical stress such as hammering or grinding, no sudden release of energy and/or oxygen was observed, attesting to the stability of the adducts. In the presence of catalytic amounts of acid, adducts with di(hydroperoxy)cycloalkane moieties decompose by undergoing a Baeyer-Villiger oxidation, and the di(hydroperoxy)cycloalkanes are transformed into the corresponding lactones. The R₃PO•(HOO)₂CR'R" adducts are stable, solid, stoichiometric and soluble materials, and can serve as an excellent complement to aqueous H₂O₂ as oxidizing agents

Di(hydroperoxy)alkane Adducts of Phosphine Oxides: Safe, Solid, Stoichiometric and Soluble Oxidizing Agents

Despite its importance and wide use as oxidizing agent, aqueous H₂O₂ has disadvantages. It easily decomposes, and when the substrates are not water-soluble, biphasic reaction mixtures are required. Thus, oxidizing agents that are anhydrous and soluble in organic solvents are desired. To this purpose, several H₂O₂ adducts of phosphine oxides, for example [tBu₃PO• H₂O₂]2₂and [Ph₃PO• H₂O₂]₂ H₂O₂, have been synthesized and characterized. These adducts represent an extension to the adducts previously reported by the Bluemel group, and display comparable physical properties. Furthermore, di(hydroperoxy)alkane adducts, R₃PO•(HOO)₂CR'R" (R, R', R" = alkyl, aryl), were synthesized and fully characterized. These adducts can be constructed using a wide variety of alkanes and phosphine oxides. All di(hydroperoxy)alkane adducts are structurally well defined as proven by single crystal X-ray analysis, and they contain two active oxygen atoms per assembly. These adducts of the type R₃PO•(HOO)₂CR'R" are highly soluble in organic solvents, allowing for oxidation reactions to occur in one phase. Moreover, there are many beneficial features to be harvested from their well-defined molecular structure and relatively anhydrous character. For example, selective and fast oxidation of dialkylsulfides to corresponding sulfoxides can be accomplished, without overoxidation to sulfones, because the solid oxidizing agents can easily be administered stoichiometrically. The adducts can also successfully oxidize substrates sensitive to hydrolysis, such as Ph₂P-PPh₂, without cleaving the P-P bond. The R₃PO•(HOO)₂CR'R" adducts are robust and practically no decomposition is found after storing the solids for 100 days at 4 °C. At room temperature, the adducts slowly decompose over time, via the release of oxygen gas. When exposed to higher temperatures or mechanical stress such as hammering or grinding, no sudden release of energy and/or oxygen was observed, attesting to the stability of the adducts. In the presence of catalytic amounts of acid, adducts with di(hydroperoxy)cycloalkane moieties decompose by undergoing a Baeyer-Villiger oxidation, and the di(hydroperoxy)cycloalkanes are transformed into the corresponding lactones. The R₃PO•(HOO)₂CR'R" adducts are stable, solid, stoichiometric and soluble materials, and can serve as an excellent complement to aqueous H₂O₂ as oxidizing agents