Operator Algebras Research Articles

Circuit and graver walks and linear and integer programming

We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.

Differential Operators on Non-compact Harmonic Manifolds

We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function. Furthermore, we show that the algebra of differential operators that commute with taking averages over geodesic spheres is generated by the Laplacian. As an application of this, we show that the heat-semi group is dense in the radial L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} space of a non-compact simply connected harmonic manifold. In the process, we provide a characterisation of the eigenfunctions of the Laplacian and therefore of the eigenfunctions of all differential operators commuting with taking spherical averages.

Leveraging recommendations using a multiplex graph database

PurposeBy applying targeted graph algorithms, the method used by the authors enables effective prediction of user interactions and thus fulfils the complex requirements of modern recommender systems. This study sets a new benchmark for multidimensional recommendation strategies and offers a path towards more advanced and user-centric models.Design/methodology/approachTo improve multidimensional data recommendation systems, multiplex graph structures are useful to capture various types of user interactions. This paper presents a novel framework that uses a graph database to compute and manipulate multiplex graphs. The approach enables flexible dimension management and increases expressive power through a specialised algebra designed for multiplex graph manipulation.FindingsThe authors compare the multiplex graph approach with traditional matrix methods, in particular random walk with restart, and show that the method not only provides deeper insights into user preferences by integrating scores from different layers of the multiplex graph, but also outperforming matrix-based approaches in most configurations. The results highlight the potential of multiplex graphs for developing sophisticated and customised recommender systems that significantly improve both performance and explainability.Originality/valueThe study provides a formal specification of a multiplex graph construction based on interaction and content-based information; and the study also developed an algebra dedicated to multiplex graphs, enabling robust and precise graph manipulations necessary for effective recommendation queries. The authors implement these algebraic operations within the Neo4j graph database system with a thorough analysis and experimentation with three different data sets, benchmarked against traditional matrix-based methods.

Generalized Cesàro operators in the disc algebra and in Hardy spaces

Generalized Cesàro operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document}, for t∈[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in [0,1)$$\\end{document}, are investigated when they act on the disc algebra A(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A({\\mathbb {D}})$$\\end{document} and on the Hardy spaces Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p$$\\end{document}, for 1≤p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p \\le \\infty $$\\end{document}. We study the continuity, compactness, spectrum and point spectrum of Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} as well as their linear dynamics and mean ergodicity on these spaces.

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

Algebraic algorithms for vector bundles over curves

In this paper, we represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and images of global homomorphisms, isomorphisms between vector bundles, cohomology groups, extensions, and splitting into a direct sum of indecomposables. Most algorithms are deterministic except for computing isomorphisms when the base field is infinite. Some algorithms are only polynomial time if we may compute Hermite forms of pseudo-matrices in polynomial time. All algorithms rely exclusively on algebraic operations in function fields. For applications, we give an algorithm enumerating isomorphism classes of vector bundles on an elliptic curve, and to construct algebraic geometry codes over vector bundles. We implement all our algorithms into a SageMath package.

Nonlinear mixed ∗-Jordan n-derivations on ∗-algebras

Let [Formula: see text] be a ∗-algebra with unity [Formula: see text] and a nontrivial projection [Formula: see text]. In this paper, it is shown that if a map [Formula: see text] satisfies [Formula: see text] [Formula: see text] for all [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text], where [Formula: see text], then [Formula: see text] is an additive ∗-derivation. Moreover, we shall call such a map [Formula: see text], a nonlinear mixed ∗-Jordan [Formula: see text]-derivation. As applications, we will characterize nonlinear mixed ∗-Jordan [Formula: see text]-derivation in standard operator algebras and von Neumann algebras.

Inclusions of simple C⁎-algebras arising from compact group actions

Inclusions of operator algebras have long been studied. In particular, inclusions arising from actions of compact groups on factors were studied by Izumi-Longo-Popa and others. The correspondence between intermediate subfactors and subgroups is called the Galois correspondence. Analogues for actions on C⁎-algebras have been studied by Izumi, Cameron-Smith, Peligrad, and others. In this article, we give examples of compact group actions on simple C⁎-algebras for which the Galois correspondence holds.

XEM: Tensor accelerator for AB21 supercomputing artificial intelligence processor

AbstractAs computing systems become increasingly larger, high‐performance computing (HPC) is gaining importance. In particular, as hyperscale artificial intelligence (AI) applications, such as large language models emerge, HPC has become important even in the field of AI. Important operations in hyperscale AI and HPC are mainly linear algebraic operations based on tensors. An AB21 supercomputing AI processor has been proposed to accelerate such applications. This study proposes a XEM accelerator to accelerate linear algebraic operations in an AB21 processor effectively. The XEM accelerator has outer product‐based parallel floating‐point units that can efficiently process tensor operations. We provide hardware details of the XEM architecture and introduce new instructions for controlling the XEM accelerator. Additionally, hardware characteristic analyses based on chip fabrication and simulator‐based functional verification are conducted. In the future, the performance and functionalities of the XEM accelerator will be verified using an AB21 processor.

Understanding of linear operators through Wigner analysis

In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators belong to (weighted) modulation spaces, particularly in Sjöstrand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes.Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schrödinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations S∈Sp(d,R). The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator T on Rd into a pseudodifferential operator K on R2d. This transformation involves a symbol σ well-localized around the manifold defined by z=Sw.

Relative monadicity

We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense ▪-functor j:A→E, a ▪-functor r:D→E is j-monadic if and only if r admits a left j-relative adjoint and creates j-absolute colimits. This provides a refinement of the classical monadicity theorem – characterising those categories whose objects are given by those of E equipped with algebraic structure – in which the arities of the algebraic operations are valued in A. In particular, when j=1, we recover a formal monadicity theorem. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the (j-relative) monadicity of the composite of a ▪-functor with a (j-relatively) monadic ▪-functor.

Similarity joins and clustering for SPARQL

The SPARQL standard provides operators to retrieve exact matches on data, such as graph patterns, filters and grouping. This work proposes and evaluates two new algebraic operators for SPARQL 1.1 that return similarity-based results instead of exact results. First, a similarity join operator is presented, which brings together similar mappings from two sets of solution mappings. Second, a clustering solution modifier is introduced, which instead of grouping solution mappings according to exact values, brings them together by using similarity criteria. For both cases, a variety of algorithms are proposed and analysed, and use-case queries that showcase the relevance and usefulness of the novel operators are presented. For similarity joins, experimental results are provided by comparing different physical operators over a set of real world queries, as well as comparing our implementation to the closest work found in the literature, DBSimJoin, a PostgreSQL extension that supports similarity joins. For clustering, synthetic queries are designed in order to measure the performance of the different algorithms implemented.

A geometric classification of the holomorphic vertex operator algebras of central charge 24

A geometric classification of the holomorphic vertex operator algebras of central charge 24

Verifiable attribute-based multi-keyword search scheme with sensitive information hiding for cloud-assisted e-healthcare sharing systems

Cloud-assisted e-healthcare sharing systems (EHSSs) play an increasingly pivotal role in the contemporary healthcare field. By outsourcing electronic medical records (EMRs) to the cloud, hospitals can alleviate local storage and management burdens while facilitating data sharing. Due to the highly sensitive nature of EMRs, encryption is necessary before storing them on the cloud. Attribute-based keyword search (ABKS) enables the privacy protection of EMRs with efficient search services. However, there remain some limitations in practical application. Firstly, most ABKS schemes only support single keyword queries, resulting in inaccurate results and wastage of computing and bandwidth resources. Secondly, since sensitive information within EMRs is encrypted as a whole, different data users (including internal doctors and external researchers) should have varying access rights to prevent leakage of this sensitive information. Thirdly, incorrect search results could lead to misdiagnosis or endanger patients' lives and affect researchers' decision-making processes. To effectively tackle these challenges, this paper proposes a verifiable attribute-based multi-keyword search scheme with sensitive information hiding (VABMKS-SIH) for cloud-assisted EHSSs, where we present a secure model for multi-keyword search with two-level access structure by incorporating an improved blindness filtering technique into ciphertext-policy attribute-based encryption (CP-ABE) within existing keyword search framework. Our scheme employs a super-increasing sequence to aggregate multiple filtered data blocks into one unified ciphertext, thereby greatly reducing communication overhead during the transmission phases of ciphertext. To check the correctness of returned results, we introduce a lightweight algebraic signature algorithm based on fundamental algebraic operations. A security analysis demonstrates that VABMKS-SIH is provably secure under the random oracle mode. Additionally, we also evaluate the proposed scheme's performance to demonstrate its utility in cloud-assisted EHSSs.

Morita invariance of unbounded bivariant K-theory

We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov’s bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a refinement of the usual interior Kasparov product. We illustrate our results by proving C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-versions of well-known C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebraic Morita equivalences in the context of hereditary subalgebras, conformal equivalences and crossed products by discrete groups.

Analysis and comparison of high-performance computing solvers for minimisation problems in signal processing

Several physics and engineering applications involve the solution of a minimisation problem to compute an approximation of the input signal. Modern hardware and software use high-performance computing to solve problems and considerably reduce execution time. In this paper, different optimisation methods are compared and analysed for the solution of two classes of non-linear minimisation problems for signal approximation and denoising with different constraints and involving computationally expensive operations, i.e., (i) the global optimisers divide rectangle-local and the improved stochastic ranking evolution strategy, and (ii) the local optimisers principal axis, the Limited-memory Broyden, Fletcher, Goldfarb, Shanno, and the constrained optimisation by linear approximations. The proposed approximation and denoising minimisation problems are attractive due to their numerical and analytical properties, and their analysis is general enough to be extended to most signal-processing problems. As the main contribution and novelty, our analysis combines an efficient implementation of signal approximation and denoising on arbitrary domains, a comparison of the main optimisation methods and their high-performance computing implementations, and a scalability analysis of the main algebraic operations involved in the solution of the problem, such as the solution of linear systems and singular value decomposition. Our analysis is also general regarding the signal processing problem, variables, constraints (e.g., bounded, non-linear), domains (e.g., structured and unstructured grids, dimensionality), high-performance computing hardware (e.g., cloud computing, homogeneous vs. heterogeneous). Experimental tests are performed on the CINECA Marconi100 cluster at the 26th position in the “top500” list and consider several parameters, such as functional computation, convergence, execution time, and scalability. Our experimental tests are discussed on real-case applications, such as the reconstruction of the solution of the fluid flow field equation on an unstructured grid and the denoising of a satellite image affected by speckle noise. The experimental results show that principal axis is the best optimiser in terms of minima computation: the efficiency of the approximation is 38% with 256 processes, while the denoising has 46% with 32 processes.

A graphics processing unit accelerated sparse direct solver and preconditioner with block low rank compression

We present the GPU implementation efforts and challenges of the sparse solver package STRUMPACK. The code is made publicly available on github with a permissive BSD license. STRUMPACK implements an approximate multifrontal solver, a sparse LU factorization which makes use of compression methods to accelerate time to solution and reduce memory usage. Multiple compression schemes based on rank-structured and hierarchical matrix approximations are supported, including hierarchically semi-separable, hierarchically off-diagonal butterfly, and block low rank. In this paper, we present the GPU implementation of the block low rank (BLR) compression method within a multifrontal solver. Our GPU implementation relies on highly optimized vendor libraries such as cuBLAS and cuSOLVER for NVIDIA GPUs, rocBLAS and rocSOLVER for AMD GPUs and the Intel oneAPI Math Kernel Library (oneMKL) for Intel GPUs. Additionally, we rely on external open source libraries such as SLATE (Software for Linear Algebra Targeting Exascale), MAGMA (Matrix Algebra on GPU and Multi-core Architectures), and KBLAS (KAUST BLAS). SLATE is used as a GPU-capable ScaLAPACK replacement. From MAGMA we use variable sized batched dense linear algebra operations such as GEMM, TRSM and LU with partial pivoting. KBLAS provides efficient (batched) low rank matrix compression for NVIDIA GPUs using an adaptive randomized sampling scheme. The resulting sparse solver and preconditioner runs on NVIDIA, AMD and Intel GPUs. Interfaces are available from PETSc, Trilinos and MFEM, or the solver can be used directly in user code. We report results for a range of benchmark applications, using the Perlmutter system from NERSC, Frontier from ORNL, and Aurora from ALCF. For a high frequency wave equation on a regular mesh, using 32 Perlmutter compute nodes, the factorization phase of the exact GPU solver is about 6.5× faster compared to the CPU-only solver. The BLR-enabled GPU solver is about 13.8× faster than the CPU exact solver. For a collection of SuiteSparse matrices, the STRUMPACK exact factorization on a single GPU is on average 1.9× faster than NVIDIA’s cuDSS solver.

A class of vertex operator algebras generated by Virasoro vectors

A class of vertex operator algebras generated by Virasoro vectors

On intermediate exceptional series

The Freudenthal–Tits magic square m(A1,A2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {m}(\\mathbb {A}_1,\\mathbb {A}_2)$$\\end{document} for A=R,C,H,O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {A}=\\mathbb {R},\\mathbb {C},\\mathbb {H},\\mathbb {O}$$\\end{document} of semi-simple Lie algebras can be extended by including the sextonions S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}$$\\end{document}. A series of non-reductive Lie algebras naturally appear in the new row associated with the sextonions, which we will call the intermediate exceptional series, with the largest one as the intermediate Lie algebra E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document} constructed by Landsberg–Manivel. We study various aspects of the intermediate vertex operator (super)algebras associated with the intermediate exceptional series, including rationality, coset constructions, irreducible modules, (super)characters and modular linear differential equations. For all gI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}_I$$\\end{document} belonging to the intermediate exceptional series, the intermediate VOA L1(gI)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1(\\mathfrak {g}_I)$$\\end{document} has characters of irreducible modules coinciding with those of the simple rational C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}-cofinite W-algebra W-h∨/6(g,fθ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{-h^\\vee /6}(\\mathfrak {g},f_\ heta )$$\\end{document} studied by Kawasetsu, with g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g} $$\\end{document} belonging to the Cvitanović–Deligne exceptional series. We propose some new intermediate VOA Lk(gI)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_k(\\mathfrak {g}_I)$$\\end{document} with integer level k and investigate their properties. For example, for the intermediate Lie algebra D6+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{6+1/2}$$\\end{document} between D6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_6$$\\end{document} and E7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_7$$\\end{document} in the subexceptional series and also in Vogel’s projective plane, we find that the intermediate VOA L2(D6+1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2(D_{6+1/2})$$\\end{document} has a simple current extension to a SVOA with four irreducible Neveu–Schwarz modules, and the supercharacters can be realized by a fermionic Hecke operator on the N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=1$$\\end{document} Virasoro minimal model L(c16,2,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L(c_{16,2},0)$$\\end{document}. We also provide some (super) coset constructions such as L2(E7)/L2(D6+1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2(E_7)/L_2(D_{6+1/2})$$\\end{document} and L1(D6+1/2)⊗2/L2(D6+1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1(D_{6+1/2})^{\\otimes 2}\\!/L_2(D_{6+1/2})$$\\end{document}. In the end, we find that the theta blocks associated with the intermediate exceptional series produce some new holomorphic Jacobi forms of critical weight and lattice index.

ANALYSIS OF STUDENTS' ABILITY TO SOLVE MATHEMATICAL PROBLEMS ON CONTEXTUAL PROBLEMS INVOLVING ALGEBRAIC FORMS

This research is motivated by the fact that problem solving is a basic ability that every student must have. However, observations in the field show that students' mathematical problem-solving skills, especially in the material of algebraic form operations, are still relatively low. This study aims to describe students' mathematical problem-solving skills based on Polya's theory, which consists of four stages: (1) understanding the problem, (2) planning problem solving, (3) implementing the plan, and (4) re-examining the results. This study involved 26 ninth grade students as subjects. The method used was descriptive qualitative with data collection techniques including tests, interviews, and documentation, data analysis conducted based on four levels of problem-solving ability: excellent, good, sufficient, and deficient. The results showed that students' mathematical problem-solving ability on algebraic form operation material obtained as a whole was still relatively low with a scale of 42%. The implications of these findings indicate the need for the alication of more varied and contextualized teaching strategies to improve students' problem-solving skills. This research can be the basis for teachers in providing further assistance to students who are experiencing difficulties, focusing on improving skills in understanding and solving mathematical problems as a whole.