791 research outputs found
Bostonia. Volume 4
Founded in 1900, Bostonia magazine is Boston University's main alumni publication, which covers alumni and student life, as well as university activities, events, and programs
Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy
The algebraic matrix hierarchy approach based on affine Lie algebras
leads to a variety of 1+1 soliton equations. By varying the rank of the
underlying algebra as well as its gradation in the affine setting, one
encompasses the set of the soliton equations of the constrained KP hierarchy.
The soliton solutions are then obtained as elements of the orbits of the
dressing transformations constructed in terms of representations of the vertex
operators of the affine algebras realized in the unconventional
gradations. Such soliton solutions exhibit non-trivial dependence on the KdV
(odd) time flows and KP (odd and even) time flows which distinguishes them from
the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of
the constrained KP hierarchy.Comment: LaTeX, 13pg
Combinatorial Hopf algebras and Towers of Algebras
Bergeron and Li have introduced a set of axioms which guarantee that the
Grothendieck groups of a tower of algebras can be
endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap,
and independently Lam and Shimozono constructed dual graded graphs from
primitive elements in Hopf algebras. In this paper we apply the composition of
these constructions to towers of algebras. We show that if a tower
gives rise to graded dual Hopf algebras then we must
have where .Comment: 7 page
Nonstandard Drinfeld-Sokolov reduction
Subject to some conditions, the input data for the Drinfeld-Sokolov
construction of KdV type hierarchies is a quadruplet (\A,\Lambda, d_1, d_0),
where the are -gradations of a loop algebra \A and \Lambda\in \A
is a semisimple element of nonzero -grade. A new sufficient condition on
the quadruplet under which the construction works is proposed and examples are
presented. The proposal relies on splitting the -grade zero part of \A
into a vector space direct sum of two subalgebras. This permits one to
interpret certain Gelfand-Dickey type systems associated with a nonstandard
splitting of the algebra of pseudo-differential operators in the
Drinfeld-Sokolov framework.Comment: 19 pages, LaTeX fil
Regular Conjugacy Classes in the Weyl Group and Integrable Hierarchies
Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade
one regular semisimple elements from non-equivalent Heisenberg subalgebras of a
loop algebra \G\otimes{\bf C}[\lambda,\lambda^{-1}] are studied. The graded
Heisenberg subalgebras containing such elements are labelled by the regular
conjugacy classes in the Weyl group {\bf W}(\G) of the simple Lie algebra
\G. A representative w\in {\bf W}(\G) of a regular conjugacy class can be
lifted to an inner automorphism of \G given by , where is the defining vector of an subalgebra
of \G.The grading is then defined by the operator and any grade one regular element from the
Heisenberg subalgebra associated to takes the form , where and is included in an
subalgebra containing . The largest eigenvalue of is
except for some cases in , . We explain how these Lie
algebraic results follow from known results and apply them to construct
integrable systems.If the largest eigenvalue is , then
using any grade one regular element from the Heisenberg subalgebra associated
to we can construct a KdV system possessing the standard \W-algebra
defined by as its second Poisson bracket algebra. For \G a classical
Lie algebra, we derive pseudo-differential Lax operators for those
non-principal KdV systems that can be obtained as discrete reductions of KdV
systems related to . Non-abelian Toda systems are also considered.Comment: 44 pages, ENSLAPP-L-493/94, substantial revision, SWAT-95-77. (use
OLATEX (preferred) or LATEX
Generalized Drinfeld-Sokolov Reductions and KdV Type Hierarchies
Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local
reductions of Hamiltonian flows generated by monodromy invariants on the dual
of a loop algebra. Following earlier work of De Groot et al, reductions based
upon graded regular elements of arbitrary Heisenberg subalgebras are
considered. We show that, in the case of the nontwisted loop algebra
, graded regular elements exist only in those Heisenberg
subalgebras which correspond either to the partitions of into the sum of
equal numbers or to equal numbers plus one . We prove that the
reduction belonging to the grade regular elements in the case yields
the matrix version of the Gelfand-Dickey -KdV hierarchy,
generalizing the scalar case considered by DS. The methods of DS are
utilized throughout the analysis, but formulating the reduction entirely within
the Hamiltonian framework provided by the classical r-matrix approach leads to
some simplifications even for .Comment: 43 page
PD-L1/PD-1 Pattern of Expression Within the Bone Marrow Immune Microenvironment in Smoldering Myeloma and Active Multiple Myeloma Patients
Background: The PD-1/PD-L1 axis has recently emerged as an immune checkpoint that controls antitumor immune responses also in hematological malignancies. However, the use of anti-PD-L1/PD-1 antibodies in multiple myeloma (MM) patients still remains debated, at least in part because of discordant literature data on PD-L1/PD-1 expression by MM cells and bone marrow (BM) microenvironment cells. The unmet need to identify patients which could benefit from this therapeutic approach prompts us to evaluate the BM expression profile of PD-L1/PD-1 axis across the different stages of the monoclonal gammopathies. Methods: The PD-L1/PD-1 axis was evaluated by flow cytometry in the BM samples of a total cohort of 141 patients with monoclonal gammopathies including 24 patients with Monoclonal Gammopathy of Undetermined Significance (MGUS), 38 patients with smoldering MM (SMM), and 79 patients with active MM, including either newly diagnosed or relapsed-refractory patients. Then, data were correlated with the main immunological and clinical features of the patients. Results: First, we did not find any significant difference between MM and SMM patients in terms of PD-L1/PD-1 expression, on both BM myeloid (CD14+) and lymphoid subsets. On the other hand, PD-L1 expression by CD138+ MM cells was higher in both SMM and MM as compared to MGUS patients. Second, the analysis on the total cohort of MM and SMM patients revealed that PD-L1 is expressed at higher level in CD14+CD16+ non-classical monocytes compared with classical CD14+CD16â cells, independently from the stage of disease. Moreover, PD-L1 expression on CD14+ cells was inversely correlated with BM serum levels of the anti-tumoral cytokine, IL-27. Interestingly, relapsed MM patients showed an inverted CD4+/CD8+ ratio along with high levels of pro-tumoral IL-6 and a positive correlation between Ă14+PD-L1+ and Ă8+PD-1+ cells as compared to both SMM and newly diagnosed MM patients suggesting a highly compromised immune-compartment with low amount of CD4+ effector cells. Conclusions: Our data indicate that SMM and active MM patients share a similar PD-L1/PD-1 BM immune profile, suggesting that SMM patients could be an interesting target for PD-L1/PD-1 inhibition therapy, in light of their less compromised and more responsive immune-compartment
Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction
The matrix version of the -KdV hierarchy has been recently
treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian
symmetry reduction applied to a Poisson submanifold in the dual of the Lie
algebra . Here a
series of extensions of this matrix Gelfand-Dickey system is derived by means
of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra
using the natural
embedding for any positive integer. The
hierarchies obtained admit a description in terms of a matrix
pseudo-differential operator comprising an -KdV type positive part and a
non-trivial negative part. This system has been investigated previously in the
case as a constrained KP system. In this paper the previous results are
considerably extended and a systematic study is presented on the basis of the
Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson
brackets and makes clear the conformal (-algebra) structures related to
the KdV type hierarchies. Discrete reductions and modified versions of the
extended -KdV hierarchies are also discussed.Comment: 60 pages, plain TE
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