40 research outputs found
Some results on the representation theory of vertex operator algebras and integer partition identities
No full textInteger partition identities such as the Rogers-Ramanujan identities have deep relations with the representation theory of vertex operator algebras, among many other fields of mathematics and physics. Such identities, when written in generating function form typically take the shape ``product side'' = ``sum side.'' In some vertex-operator-algebraic settings, the product sides arise naturally, and the problem is to explain, interpret and prove the sum sides, while some other settings pose an opposite problem. In this thesis, we provide some results on both types of problems. In Part I of this thesis, we interpret the sum sides of the Göllnitz-Gordon identities using Lepowsky-Wilson's -algebraic constructions applied to certain principally twisted level 2 standard modules for . In Part II, we give, following Dong-Lepowsky, explicit constructions for certain higher level twisted intertwining operators for ; these constructions are inspired by a desire to interpret Andrews-Baxter's -series theoretic ``motivated proof'' of the Rogers-Ramanujan identities and more generally, motivated proofs of the Gordon-Andrews and the Andrews-Bressoud identities given by Lepowsky-Zhu and Kanade-Lepowsky-Russell-Sills, respectively. These motived proofs are about explaining the ``sum sides'' starting with the ``product sides.'' In Part III, following an idea of J. Lepowsky, we introduce and analyze a Koszul complex related to the principal subspace of the level 1 vacuum module of ; this construction is expected to yield a ``character formula'' for the principal subspaces, thereby explaining the emergence of ``product sides.''Ph.D.Includes bibliographical referencesby Shashank Kanad
An affine Weyl group interpretation of the "motivated proofs" of the Rogers-Ramanujan and Gordon-Andrews-Bressoud identities
No full textA “motivated proof” of the Rogers-Ramanujan identities was given by G. E. Andrews and R. J. Baxter. This proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu, and later to the even-moduli case of the Andrew-Bressoud identities by S. Kanade, Lepowsky, M. C. Russell and A. Sills. We present a reinterpretation of these proofs, with new motivation coming from the affine Weyl group of .Ph.D.Includes bibliographical referencesby Bud B. Coulso
Some results in the representation theory of strongly graded vertex algebras
No full textIn the first part of this thesis, we study strongly graded vertex algebras and their strongly graded modules, which are conformal vertex algebras and their modules with a second, compatible grading by an abelian group satisfying certain grading restriction conditions. We consider a tensor product of strongly graded vertex algebras and its tensor product strongly graded modules. We prove that a tensor product of strongly graded irreducible modules for a tensor product of strongly graded vertex algebras is irreducible, and that such irreducible modules, up to equivalence, exhaust certain naturally defined strongly graded irreducible modules for a tensor product of strongly graded vertex algebras. We also prove that certain naturally defined strongly graded modules for the tensor product strongly graded vertex algebra are completely reducible if and only if every strongly graded module for each of the tensor product factors is completely reducible. These results generalize the corresponding known results for vertex operator algebras and their modules. In the second part, we derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded conformal vertex algebra under suitable assumptions. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.Ph. D.Includes bibliographical referencesby Jinwei Yan
The image of the lepowsky homomorphism for the group F-20 4
No full textLet Go be a semisimple Lie group, let Ko be a maximal compact subgroup of Go and let k ?g denote the complexification of their Lie algebras. Let G be the adjoint group of g and let K be the connected Lie subgroup of G with Lie algebra ad(k). If U(g) is the universal enveloping algebra of g, then U(g)K will denote the centralizer of K in U(g). Also let P :U(g) →U(k) ⊗ U(a) be the projection map corresponding to the direct sum U(g)=(U(k) ⊗ U(a)) ⊗ U(g)n associated to an Iwasawa decomposition of Go adapted to Ko. In this paper, we give a characterization of the image of U(g)K under the injective antihomorphism P :U(g)K →U(k)M ⊗ U(a), considered by Lepowsky in [12], when Go is isomorphic to the rank 1 real form F -204 of the exceptional Lie group F4. © 2012 The Author(s).Fil: Brega, Alfredo Oscar. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomia y Física. Sección Matemática. Grupo En Teoria de Lie; ArgentinaFil: Cagliero, Leandro Roberto. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomia y Física. Sección Matemática. Grupo En Teoria de Lie; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Tirao, Juan Alfredo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomia y Física. Sección Matemática. Grupo En Teoria de Lie; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin
Absent-Minded and Robotic Inspectors: Nuclear Verification Techniques with Minimal Access to Items, Sites, and Information
No full textNuclear nonproliferation and arms control are predicated on declarations, data exchange, and inspections to verify agreed upon limits on nuclear weapons or materials. While it is difficult to anticipate the objectives of future international agreements, they are likely to require new verification approaches, preserving the thoroughness of onsite inspections — which have traditionally played an essential role in nuclear monitoring and verification — while resolving some concerns about intrusiveness. This would ensure a higher level of privacy for the host, who might otherwise be skeptical about approaches that could potentially reveal sensitive information. Considering the future of verification, this thesis explores verification scenarios with progressively stricter access constraints, ranging from current-day arms control to a conceptual extreme. A novel radiation detection approach is proposed, characterized, and demonstrated for three different settings. The demonstrated correctness of the three proposed approaches underscores the prospect of minimal-access inspection approaches for nuclear verification. Motivated by the verification regime of recent nuclear arms control agreements, minimal access to treaty-accountable items is enabled by confirming their absence, which avoids direct measurements of sensitive items. Since current detection methods primarily rely on neutron emissions to detect plutonium, this work provides a complementary gamma ray-based measurement protocol and prototype device capable of detecting uranium weapon components. Broadening the scope of the imposed limitation, and applicable to both nuclear safeguards and arms control, remote inspections — where the inspector is physically separated from the inspected site — are explored to address minimal inspector access to sensitive sites. To realize this concept, the development and characterization of a robotic neutron detector is detailed, providing single-measurement directional determination, source localization, and template matching capabilities. Progressing further along the continuum of minimal access, the final scenario envisioned advances the concept of robotic inspections by limiting observations to only those that are strictly necessary for performing the verification task. The culmination of this work is an “absent-minded robotic inspector” for minimal access to, and storage or revelation of, sensitive information, defined to include the radiation measurements, any observable features of the search environment, and even the site’s layout
3D Printing for Drug Manufacturing: A Perspective on the Future of Pharmaceuticals
No full textSince a three-dimensional (3D) printed drug was first approved by the Food and Drug Administration in 2015, there has been a growing interest in 3D printing for drug manufacturing. There are multiple 3D printing methods – including selective laser sintering, binder deposition, stereolithography, inkjet printing, extrusion-based printing, and fused deposition modeling – which are compatible with printing drug products, in addition to both polymer filaments and hydrogels as materials for drug carriers. We see the adaptability of 3D printing as a revolutionary force in the pharmaceutical industry. Release characteristics of drugs may be controlled by complex 3D printed geometries and architectures. Precise and unique doses can be engineered and fabricated via 3D printing according to individual prescriptions. On-demand printing of drug products can be implemented for drugs with limited shelf life or for patient-specific medications, offering an alternative to traditional compounding pharmacies. For these reasons, 3D printing for drug manufacturing is the future of pharmaceuticals, making personalized medicine possible while also transforming pharmacies.</jats:p
Emerging Anti-Fouling Methods: Towards Reusability of 3D-Printed Devices for Biomedical Applications
No full textMicrofluidic devices are used in a myriad of biomedical applications such as cancer screening, drug testing, and point-of-care diagnostics. Three-dimensional (3D) printing offers a low-cost, rapid prototyping, efficient fabrication method, as compared to the costly—in terms of time, labor, and resources—traditional fabrication method of soft lithography of poly(dimethylsiloxane) (PDMS). Various 3D printing methods are applicable, including fused deposition modeling, stereolithography, and photopolymer inkjet printing. Additionally, several materials are available that have low-viscosity in their raw form and, after printing and curing, exhibit high material strength, optical transparency, and biocompatibility. These features make 3D-printed microfluidic chips ideal for biomedical applications. However, for developing devices capable of long-term use, fouling—by nonspecific protein absorption and bacterial adhesion due to the intrinsic hydrophobicity of most 3D-printed materials—presents a barrier to reusability. For this reason, there is a growing interest in anti-fouling methods and materials. Traditional and emerging approaches to anti-fouling are presented in regard to their applicability to microfluidic chips, with a particular interest in approaches compatible with 3D-printed chips
Partition identities arising from the standard A(2)2-modules of level 4
No full textIn this dissertation, we propose a set of new partition identities, arising from a twisted vertex operator construction of the level 4 standard modules for the affine Kac-Moody algebra of type A(2)2 . These identities have an interesting new feature, absent from previously known examples of this type. This work is a continuation of a long line of research of constructing standard modules for affine Kac-Moody algebras via vertex operators, and the associated combinatorial identities. The interplay between representation theory and combinatorial identities was exemplified by the vertex-algebraic proof of the famous Rogers-Ramanujan-type identities using standard A(1)1-modules by J. Lepowsky and R. Wilson. In his Ph.D. thesis, S. Capparelli proposed new combinatorial identities using a twisted vertex operator construction of the standard A(2)2-modules of level 3, which were later proved independently by G. Andrews, S. Capparelli, and M. Tamba-C. Xie. We begin with an obvious spanning set for each of the level 4 standard modules for A(2)2 , and reduce this spanning set using various relations. Most of these relations come from certain product generating function identities which are valid for all the level 4 modules. There are also other ad-hoc relations specific to a particular module of level 4. In this way, we reduce our spanning sets to match with the graded dimensions of the said modules as closely as possible. We conjecture and present strong evidence for three partition identities based on the spanning sets for the three standard A(2)2-modules of level 4. One surprising result of our work is the discovery of relations of arbitrary length. Consequently, the partitions corresponding to these spanning sets cannot be described by difference conditions of finite length. The spanning set result proves one inequality of the proposed identities. There is strong evidence for the validity of the conjecture (i.e., the opposite inequality), since it has been verified to hold for partitions of n ≤ 170, and n = 180, 190 and 200.Ph.D.Includes bibliographical referencesby Debajyoti Nand
Extensions of the Jacobi identity for generalized vertex algebras
No full textAbstractIn this paper the (generalized) Jacobi identity for generalized vertex algebras is extended to multi-operator identities. This work is based on the book of Dong and Lepowsky on generalized vertex algebras and relative vertex operators (Dong and Lepowsky, 1993), on the work of the author on the Jacobi identity for vertex operator algebras in Husu (1993), Section 1, and on the work in Husu (1993) and Husu (1995) on the A1(1)-case of the Jacobi identity for relative vertex operators. The multi-operator extensions of the (generalized) Jacobi identity (in the present paper) generalize the corresponding results on vertex operator algebras in Husu (1993), Section 1, and provide a general framework both for the vertex operator construction of generating function identities for standard A1(1)-modules in Husu (1995) and for further applications of the vertex operator construction to other affine Lie algebras
Non-Negative Integral Level Affine Lie Algebra Tensor Categories and Their Associativity Isomorphisms
No full textFor a finite-dimensional simple Lie algebra , we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra at a fixed level with a certain tensor category of finite-dimensional -modules. More precisely, the category of level a"" standard -modules is the module category for the simple vertex operator algebra , and as is well known, this category is equivalent as an abelian category to , the category of finite-dimensional modules for the Zhu's algebra , which is a quotient of . Our main result is a direct construction using Knizhnik-Zamolodchikov equations of the associativity isomorphisms in induced from the associativity isomorphisms constructed by Huang and Lepowsky in . This construction shows that is closely related to the Drinfeld category of [[h]]-modules used by Kazhdan and Lusztig to identify categories of -modules at irrational and most negative rational levels with categories of quantum group modules.SCI(E)[email protected]
