Modular Theory Research Articles

Research on the Modular Design Method and Application of Prefabricated Residential Buildings

As one of the key ways to realize the industrialization and green development of construction, prefabricated construction is conducive to saving resources and energy and improving labor productivity and quality. Aiming to solve the problem of the lack of standardization in the design of prefabricated residential buildings, which leads to the components not being universally used and the industrial characteristics not being fully embodied, while excessive standardization leads to a lack of personalization and flexibility, the modular design theory is applied to the standardized design of prefabricated residential buildings in this study. The application route of modular design theory in the standardized design is constructed, that is, “system decomposition—module design—module combination”. Taking residential buildings within a height of 54 m as an example, each basic functional module is standardized and combined into standard plans. At the same time, the functional space module design based on modular coordination and the module combination design based on the trinity of “modulus, pattern, and mode” are discussed. This research is of great significance for giving full play to the comprehensive benefits of prefabricated concrete structures in quality improvement, cost reduction, and rapid assembly.

Tomita-Takesaki theory and quantum concurrence

Tomita-Takesaki theory and quantum concurrence

Mini-Workshop: Standard Subspaces in Quantum Field Theory and Representation Theory

Real standard subspaces of complex Hilbert spaces are long known to provide the right language for Tomita–Takesaki modular theory of von Neumann algebras. In recent years they have also become an object of prominent interest in mathematical quantum field theory (QFT) and unitary representation theory of Lie groups. This workshop brought together mathematicians and physicists working with standard subspaces, particularly in QFT (construction of QFT models, characterization of entropy, information-theoretic aspects), nets of standard subspaces on causal homogeneous spaces and aspects of reflection positivity and euclidean models related to standard subspaces and modular theory.

Evolving brain network dynamics in early childhood: Insights from modular graph metrics

Modular dynamic graph theory metrics effectively capture the patterns of dynamic information interaction during human brain development. While existing research has employed modular algorithms to examine the overall impact of dynamic changes in community structure throughout development, there is a notable gap in understanding the cross-community dynamic changes within different functional networks during early childhood and their potential contributions to the efficiency of brain information transmission. This study seeks to address this gap by tracing the trajectories of cross-community structural changes within early childhood functional networks and modeling their contributions to information transmission efficiency. We analyzed 194 functional imaging scans from 83 children aged 2 to 8 years, who participated in passive viewing functional magnetic resonance imaging sessions. Utilizing sliding windows and modular algorithms, we evaluated three spatiotemporal metrics—temporal flexibility, spatiotemporal diversity, and within-community spatiotemporal diversity—and four centrality metrics: within-community degree centrality, eigenvector centrality, between-community degree centrality, and between-community eigenvector centrality. Mixed-effects linear models revealed significant age-related increases in the temporal flexibility of the default mode network (DMN), executive control network (ECN), and salience network (SN), indicating frequent adjustments in community structure within these networks during early childhood. Additionally, the spatiotemporal diversity of the SN also displayed significant age-related increases, highlighting its broad pattern of cross-community dynamic interactions. Conversely, within-community spatiotemporal diversity in the language network exhibited significant age-related decreases, reflecting the network's gradual functional specialization. Furthermore, our findings indicated significant age-related increases in between-community degree centrality across the DMN, ECN, SN, language network, and dorsal attention network, while between-community eigenvector centrality also increased significantly for the DMN, ECN, and SN. However, within-community eigenvector centrality remained stable across all functional networks during early childhood. These results suggest that while centrality of cross-community interactions in early childhood functional networks increases, centrality within communities remains stable. Finally, mediation analysis was conducted to explore the relationships between age, brain dynamic graph metrics, and both global and local efficiency based on community structure. The results indicated that the dynamic graph metrics of the SN primarily mediated the relationship between age and the decrease in global efficiency, while those of the DMN, language network, ECN, dorsal attention network, and SN primarily mediated the relationship between age and the increase in local efficiency. This pattern suggests a developmental trajectory in early childhood from global information integration to local information segregation, with the SN playing a pivotal role in this transformation. This study provides novel insights into the mechanisms by which early childhood brain functional development impacts information transmission efficiency through cross-community adjustments in functional networks.

Relational bulk reconstruction from modular flow

The entanglement wedge reconstruction paradigm in AdS/CFT states that for a bulk qudit within the entanglement wedge of a boundary subregion A¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{A} $$\\end{document}, operators acting on the bulk qudit can be reconstructed as CFT operators on A¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{A} $$\\end{document}. This naturally fits within the framework of quantum error correction, with the CFT states containing the bulk qudit forming a code protected against the erasure of the boundary subregion A. In this paper, we set up and study a framework for relational bulk reconstruction in holography: given two code subspaces both protected against erasure of the boundary region A, the goal is to relate the operator reconstructions between the two spaces. To accomplish this, we assume that the two code subspaces are smoothly connected by a one-parameter family of codes all protected against the erasure of A, and that the maximally-entangled states on these codes are all full-rank. We argue that such code subspaces can naturally be constructed in holography in a “measurement-based” setting. In this setting, we derive a flow equation for the operator reconstruction of a fixed code subspace operator using modular theory which can, in principle, be integrated to relate the reconstructed operators all along the flow. We observe a striking resemblance between our formulas for relational bulk reconstruction and the infinite-time limit of Connes cocycle flow, and take some steps towards making this connection more rigorous. We also provide alternative derivations of our reconstruction formulas in terms of a canonical reconstruction map we call the modular reflection operator.

The logical structure of modular semantic theories of software systems

AbstractThis paper studies the structure of semantic theories over modular computational systems and applies the algebraic Theory of Institutions to provide a logical representation of such theories. A modular semantic theory is here defined by a cluster of semantic theories, each for a single program's module, and by a set of relations connecting models of different semantic theories. A semantic theory of a single module is provided in terms of the set of Σ‐models mapped from the category Th of Σ‐theories and generating a hierarchy of structures from an abstract model to a concrete model of data. The collection of abstract models representing different modules of a program is formalised as the category of institutions INS, where theory morphisms express refinements, integrations, and compositions between couples of modules. Finally, it is required that a morphism in INS at any level occurs iff the same morphism occurs at the lower level alongside the Th hierarchy.

Unruh-De Witt detectors, Bell-CHSH inequality and Tomita-Takesaki theory

The interaction between Unruh-De Witt spin 1/2 detectors and a real scalar field is scrutinized by making use of the Tomita-Takesaki modular theory as applied to the Von Neumann algebra of the Weyl operators. The use of the modular theory enables to evaluate in an exact way the trace over the quantum field degrees of freedom. The resulting density matrix is employed to the study of the Bell-CHSH correlator. It turns out that, as a consequence of the interaction with the quantum field, the violation of the Bell-CHSH inequality exhibits a decreasing as compared to the case in which the scalar field is absent.

Modular Representation Theory and Commutative Banach Algebras

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to C \mathbb {C} . One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.

Universal predictions of Siegel modular invariant theories near the fixed points

We analyze a general class of locally supersymmetric, CP and modular invariant models of lepton masses depending on two complex moduli taking values in the vicinity of a fixed point, where the theory enjoys a residual symmetry under a finite group. Like in models that depend on a single modulus, we find that all physical quantities exhibit a universal scaling with the distance from the fixed point. There is no dependence on the level of the construction, the weights of matter multiplets and their representations, with the only restriction that electroweak lepton doublets transform as irreducible triplets of the finite modular group. Also the form of the kinetic terms, which here are assumed to be neither minimal nor flavor blind, is irrelevant to the outcome. The result is remarkably simple and the whole class of examined theories gives rise to five independent patterns of neutrino mass matrices. Only in one of them, the predicted scaling agrees with the observed neutrino mass ratios and lepton mixing angles, exactly as in single modulus theories living close to τ = i.

Quasi-free Isomorphisms of Second Quantization Algebras and Modular Theory

Using Araki–Yamagami’s characterization of quasi-equivalence for quasi-free representations of the CCRs, we provide an abstract criterion for the existence of isomorphisms of second quantization local von Neumann algebras induced by Bogolubov transformations in terms of the respective one particle modular operators. We discuss possible applications to the problem of local normality of vacua of Klein-Gordon fields with different masses.

Tensor-Triangular Geometry and Interactions

The workshop brought together experts in a rapidly growing field of tensor triangular geometry highlighting applications to and techniques coming from homotopy theory, algebraic geometry, modular representation theory, motivic homotopy theory and noncommutative algebra.

DYNAMICAL MCDUFF-TYPE PROPERTIES FOR GROUP ACTIONS ON VON NEUMANN ALGEBRAS

Abstract We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II $_1$ factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C $^*$ -dynamics. Given a countable discrete group G and an amenable action $G\curvearrowright M$ on any separably acting semifinite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing G-action is suitably absorbed at the level of each fibre in the direct integral decomposition of M, then it is tensorially absorbed by the action on M. As a direct application of Ocneanu’s theorem, we deduce that if M has the McDuff property, then every amenable G-action on M has the equivariant McDuff property, regardless whether M is assumed to be injective or not. By employing Tomita–Takesaki theory, we can extend the latter result to the general case, where M is not assumed to be semifinite.

A short proof of Tomita's theorem

Tomita-Takesaki theory associates a positive operator called the “modular operator” with a von Neumann algebra and a cyclic-separating vector. Tomita's theorem says that the unitary flow generated by the modular operator leaves the algebra invariant. I give a new, short proof of this theorem which only uses the analytic structure of unitary flows, and which avoids operator-valued Fourier transforms (as in van Daele's proof) and operator-valued Mellin transforms (as in Zsidó's and Woronowicz's proofs). The proof is similar to one given by Bratteli and Robinson in the special case that the modular operator is bounded.

On the normalisation of the modular forms in modular invariant theories of flavour

The problem of normalisation of the modular forms in modular invariant lepton and quark flavour models is discussed. Modular invariant normalisations of the modular forms are proposed.

Cores of Partitions in Rectangles

For a positive integer $t \geq 2$, the $t$-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of $t$-cores of partitions contained in an {$r\times s$} rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle that are themselves $t$-cores and compute its asymptotics for large $r,s$. We then prove that the number of partitions inside the rectangle whose $t$-cores are a fixed partition $\rho$ is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the $t$-core of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer $n$ (Ann. Appl. Prob. 2023). In particular, we show that in the limit as $r,s \to \infty$ maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter $\alpha = (t-1)/2$ and rate parameter $\beta$ that depends on the aspect ratio.

A note on two-times measurement entropy production and modular theory

A note on two-times measurement entropy production and modular theory

Ergodic states on type III1 factors and ergodic actions

Abstract Since the early days of Tomita–Takesaki theory, it is known that a von Neumann algebra 𝑀 that admits a state 𝜑 with trivial centralizer M φ M_{\varphi} must be a type III1 factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense G δ G_{\delta} set among all faithful normal states on any III1 factor with separable predual. Through Connes’ Radon–Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper.

Entropy-Area Law and Temperature of de Sitter Horizons from Modular Theory

Abstract We derive an entropy-area law for the future horizon of an observer in diamonds inside a static patch of de Sitter space-time, taking into account the back reaction of quantum matter fields. We prove the positivity and convexity of the relative entropy for coherent states using Tomita–Takesaki modular theory, from which the quantum null energy condition for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature that is measured by an observer at rest exhibits subleading quantum corrections with respect to the well known cosmological horizon temperature H/(2π).

Concrete constructions or messy mangroves? How modelling contextual effects on constructional alternations reflect theoretical assumptions of language structure

Abstract Depending on the theory of language employed, the paradigmatic and lexical variation associated with a given composite form-meaning pair is treated in different ways. First, variation can be treated as independent of the constructional semantics, an approach typical of modular theories. Second, paradigmatic variation can be considered indicative of constructional semantics; its variation constituting networks of closely related families of constructions. This is a common approach in construction grammar. Third, there exists a trend in cognitive linguistics and construction grammar to treat grammatical constructions as non-discrete emergent clusters of many-to-many form-meaning mappings. This study explores the possibility of extending current methods for quantitatively modelling construction grammar to an approach that does not assume discrete grammatical constructions. The speaker choice examined consists of the English future constructions will and BE going to and their use in contemporary informal British English. The constructions are examined with the behavioural profile approach. Three different regression modelling methods are applied to the grammatical alternations, each operationalizing one of the theoretical assumptions. While the results show that all three approaches are feasible and comparable in predictive accuracy, model interpretation becomes increasingly difficult with added complexity.

Smith theory and cyclic base change functoriality

Abstract Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbf {Z}/p\mathbf {Z}$ -extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For $\mathbf {Z}/p\mathbf {Z}$ -extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along $\mathbf {Z}/p\mathbf {Z}$ -extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from modular representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod p spherical Hecke algebras, in a joint appendix with Gus Lonergan.