7,614 research outputs found
Artinian level algebras of codimension 3
In this paper, we continue the study of which -vectors can be the Hilbert function of a level algebra by
investigating Artinian level algebras of codimension 3 with the condition
, where is
the lex-segment ideal associated with an ideal . Our approach is to adopt an
homological method called {\it Cancellation Principle}: the minimal free
resolution of is obtained from that of by canceling some
adjacent terms of the same shift.
We prove that when ,
can be an Artinian level -algebra only if either
or holds. We also apply our results to show that for
, the Hilbert function of an Artinian
algebra of codimension 3 with the condition ,
(a) if , then -vector \H cannot be level, and
(b) if , then there is a level algebra with Hilbert function
\H for some value of .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 cannot be level
if , and that there exists a level O-sequence of codimension 3 of
type \H for for . Furthermore, we show that \H is
not level if , and also
prove that any codimension 3 Artinian graded algebra cannot be level if
\beta_{1,d+2}(\Gin(I))=\beta_{2,d+2}(\Gin(I)). In this case, the Hilbert
function of does not have to satisfy the condition .
Moreover, we show that every codimension graded Artinian level algebra
having the Weak-Lefschetz Property has the strictly unimodal Hilbert function
having a growth condition on for every
where
In particular, we find that if is of codimension 3, then for every and , and prove that
if is a codimension 3 Artinian algebra with an -vector
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 , then is
-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
Recursive Chain-of-Feedback Prevents Performance Degradation from Redundant Prompting
Large Language Models (LLMs) frequently struggle with complex reasoning
tasks, failing to construct logically sound steps towards the solution. In
response to this behavior, users often try prompting the LLMs repeatedly in
hopes of reaching a better response. This paper studies such repetitive
behavior and its effect by defining a novel setting, Chain-of-Feedback (CoF).
The setting takes questions that require multi-step reasoning as an input. Upon
response, we repetitively prompt meaningless feedback (e.g. 'make another
attempt') requesting additional trials. Surprisingly, our preliminary results
show that repeated meaningless feedback gradually decreases the quality of the
responses, eventually leading to a larger deviation from the intended outcome.
To alleviate these troubles, we propose a novel method, Recursive
Chain-of-Feedback (R-CoF). Following the logic of recursion in computer
science, R-CoF recursively revises the initially incorrect response by breaking
down each incorrect reasoning step into smaller individual problems. Our
preliminary results show that majority of questions that LLMs fail to respond
correctly can be answered using R-CoF without any sample data outlining the
logical process.Comment: Still Ongoing Work; 8 Pages; 2 Figure
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