The non-abelian tensor product of Lie superalgebras and Schur- and Baer-type theorems

We figure out the explicit expression for the trace of the field equations associated to generic higher derivative theories of gravity endowed with Lagrangians depending upon the metric and its Riemann tensor, together with arbitrary order covariant derivatives of the Riemann tensor. Then an equality linking the Lagrangian density with the covariant divergence of a vector field is put forward in terms of the trace of the field equations. As a significant application, we particularly concentrate on a broad range of higher derivative theories of gravity with the Lagrangian density constructed from the contraction of the product for metric tensors with the product of the Riemann tensors and the arbitrary order covariant derivatives of the Riemann tensor. By utilizing the trace for the equations of motion, such a type of Lagrangian density is expressed as the covariant divergence of a vector field.

We prove that the maximal dimension of a subspace [Formula: see text] of the generic tensor product of [Formula: see text] symbol algebras of prime degree [Formula: see text] with [Formula: see text] for all [Formula: see text] is [Formula: see text]. The same upper bound is thus obtained for [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] being 0 for all [Formula: see text]. We make use of the fact that given any subset [Formula: see text] of [Formula: see text] ([Formula: see text] times) with [Formula: see text], for all [Formula: see text] there exist [Formula: see text], [Formula: see text] in [Formula: see text] and [Formula: see text] in [Formula: see text] such that [Formula: see text].

Abstract We investigate a one-point restriction of conformal blocks on $$(\mathbb {P}^1,\infty ,1,0)$$ ( P 1 , ∞ , 1 , 0 ) associated with modules over a vertex operator algebra V . By restricting the module attached to the point $$\infty $$ ∞ to its bottom degree, we obtain a new formula for computing fusion rules in terms of a new left A ( V )-module $$M^1 \odot M^2$$ M 1 ⊙ M 2 over the Zhu algebra A ( V ), constructed from two V -modules $$M^1$$ M 1 and $$M^2$$ M 2 . As a consequence, for strongly rational vertex operator algebras, the construction of $$M^1 \odot M^2$$ M 1 ⊙ M 2 induces the fusion tensor product on the module category $$\textsf{Mod}(A(V))$$ Mod ( A ( V ) ) .

Abstract The incompatibility of measurements is the key feature of quantum theory that distinguishes it from the classical description of nature. Here, we consider groups of d-outcome quantum observables with prime d represented by non-Hermitian unitary operators whose eigenvalues are d'th roots of unity. We additionally assume that these observables mutually commute up to a scalar factor being one of the d'th roots of unity. By representing commutation relations of these observables via a frustration graph, we show that for such a group, there exists a single unitary transforming them into a tensor product of generalized Pauli matrices and some ancillary mutually commuting operators. Building on this result, we derive upper bounds on the sum of the squares of the absolute values and the sum of the expected values of the observables forming a group. Such bounds are of particular importance to deriving uncertainty relations or constructing entanglement witnesses, and are also useful in inflation technique. We finally utilize these bounds to compute the generalized geometric measure of entanglement for qudit stabilizer subspaces.

We introduce a quantum-electrodynamical time-dependent density functional theory with a tensor-product representation (QED-TDDFT-TP) to model molecules strongly coupled to quantized cavity fields. By combining real-space electronic wave functions with truncated Fock-space photon states, the method captures light-matter correlations at a computational cost close to standard DFT. Benchmark calculations show good agreement with QED-FCI and QED-CASCI for ground-state energies and polaritonic spectra. Applications to weakly bound dimers─including (H2)2, Ar2, (H2O)2, and HF─demonstrate that cavity confinement can significantly alter binding energies and geometries in a polarization-dependent manner. The framework provides an accurate and scalable tool for studying cavity-modified molecular structure and interactions.

We evaluated the contributions of ten intrinsic and extrinsic factors readily available from website data to individual home sale prices for three major U.S. cities using a P-spline generalized additive model (GAM). We identified the relative significance of each factor by evaluating the change in the adjusted R2 value resulting from its removal from the model. We combined this with information from correlation matrices to identify the added predictive value of a factor. For these three cities, the tests revealed that living area and location (latitude, longitude) had the strongest impact on explained variance, and each factor independently added predictive value. Relative impacts of the other factors were city-dependent. We utilized this information to develop an improved GAM with superior concurvity values. The improved GAM required the use of linear orthogonalization of factors combined with smoothing functions based on tensor products of correlated factors.

Tensor product factorization-based numerical algorithms for the inverses of generalized banded Toeplitz matrices

To address the challenges of high miss rates in subcentimeter nodules, false positives caused by vascular adhesion, and insufficient multi-scale feature fusion in lung CT analysis, a multi-stage detection model named MLND-IU, which incorporates an improved U-Net++ architecture, is proposed. The three-stage framework begins with an enhanced RetinaNet optimized by a dynamic focal loss to generate candidate regions with high sensitivity while mitigating class imbalance. The second stage introduces AG-UNet++ with a novel Dense Attention Bridging Module (DABM), which employs a tensor product fusion of channel and deformable spatial attention across densely connected skip pathways to amplify feature representation for 3-5 mm nodules. The final stage employs a 3D Contextual Pyramid Module (3D-CPM) to integrate multi-slice morphological and contextual features, thereby reducing vascular false positives. Ablation studies indicated that the second stage improved the Dice coefficient by 21.1% compared with the first stage (paired t-test, p < 0.01, independent validation on LIDC-IDRI). The third stage further reduced the false positives per scan (FP/Scan) to 1.4, corresponding to an 87.3% reduction compared to the baseline. Multicenter validation on the LIDC-IDRI (n = 1,018) and DSB2017 (n = 1,595) datasets resulted in a segmentation Dice coefficient of 92.7%, a sensitivity of 93.4% for nodules smaller than 6 mm (compared to radiologists' sensitivity of 68.5%, p = 0.003), and an AUC of 0.84 for malignancy classification, representing a 19.2% improvement over conventional methods. With a processing time of 2.3 seconds per case, the proposed framework presents a clinically viable solution for early lung cancer screening by simultaneously improving small nodule detection and suppressing false positives.

Abstract We prove that any tensor product factorization with a commutative tensor factor of a modular group algebra over a prime field comes from a direct product decomposition of the group basis. This extends previous work by Carlson and Kovács for the commutative case and answers one of their questions in certain cases.

Transition metal complexes present significant challenges for electronic structure theory due to strong electron correlation arising from partially filled d-orbitals. We compare our recently developed Tensor Product Selected Configuration Interaction (TPSCI) with Density Matrix Renormalization Group (DMRG) for computing exchange coupling constants in six transition metal systems, including dinuclear Cr, Fe, and Mn complexes and a tetranuclear Ni-cubane. TPSCI uses a locally correlated tensor product state basis to capture electronic structure efficiently while maintaining interpretability. From calculations on active spaces ranging from (22e,29o) to (42e,49o), we find that TPSCI consistently yields higher variational energies than DMRG due to truncation of local cluster states, but provides magnetic exchange coupling constants (J) generally within 10-30 cm-1 of DMRG results. Key advantages include natural multistate capability enabling direct J extrapolation with smaller statistical errors, and computational efficiency for challenging systems. However, cluster state truncation represents a fundamental limitation requiring careful convergence testing, particularly for large local cluster dimensions. We identify specific failure cases where current truncation schemes break down, highlighting the need for improved cluster state selection methods and distributed memory implementations to realize TPSCI's full potential for strongly correlated systems.

We study multiplicative Chern insulators (MCIs) as canonical examples of multiplicative topological phases of matter. Constructing the MCI Bloch Hamiltonian as a symmetry-protected tensor product of two topologically nontrivial parent Chern insulators (CIs), we study two-dimensional (2D) MCIs and introduce three-dimensional mixed MCIs, constructed by requiring the two 2D parent Hamiltonians share only one momentum component. We study the 2D MCI response to time-reversal symmetric flux insertion, observing a 4 π Aharonov-Bohm effect, relating these topological states to fractional quantum Hall states via the microscopic field theories of the quantum skyrmion Hall effect. As part of this response, we observe evidence of quantization of a proposed topological invariant for compactified many-body states to a rational number, suggesting higher-dimensional topology may also be relevant. Finally, we study effects of bulk perturbations breaking the symmetry-protected tensor-product structure of the child Hamiltonian, finding the MCI evolves adiabatically into a topological skyrmion phase.

Abstract We study the integration problem over the s -dimensional unit cube on four types of Banach spaces of integrands. First we consider Haar wavelet spaces, consisting of functions whose Haar wavelet coefficients exhibit a certain decay behavior measured by a parameter $$\alpha &gt;0$$ α &gt; 0 . We study the worst case error of integration over the norm unit ball and provide upper error bounds for quasi-Monte Carlo (QMC) cubature rules based on arbitrary ( t , m , s )-nets as well as matching lower error bounds for arbitrary cubature rules. These results show that using arbitrary ( t , m , s )-nets as sample points yields the best possible rate of convergence. Afterwards we study spaces of integrands of fractional smoothness $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and state a sharp Koksma-Hlawka-type inequality. More precisely, we show that on those spaces the worst case error of integration is equal to the corresponding fractional discrepancy. Those spaces can be continuously embedded into tensor product Bessel potential spaces, also known as Sobolev spaces of dominated mixed smoothness, with the same set of parameters. The latter spaces can be embedded into suitable Besov spaces of dominating mixed smoothness $$\alpha $$ α , which in turn can be embedded into the Haar wavelet spaces with the same set of parameters. Therefore our upper error bounds on Haar wavelet spaces for QMC cubatures based on ( t , m , s )-nets transfer (with possibly different constants) to the corresponding spaces of integrands of fractional smoothness and to Sobolev and Besov spaces of dominating mixed smoothness. Moreover, known lower error bounds for periodic Sobolev and Besov spaces of dominating mixed smoothness show that QMC integration based on arbitrary ( t , m , s )-nets yields the best possible convergence rate on periodic as well as on non-periodic Sobolev and Besov spaces of dominating smoothness.

Composite B-spline regularized delta functions for the immersed boundary method: Divergence-free interpolation and gradient-preserving force spreading.

Higher-form symmetry in a tensor product Hilbert space is always emergent: The symmetry generators become genuinely topological only when the Gauss law is energetically enforced at low energies. In this Letter, we present a general method for defining the 't Hooft anomaly of higher-form symmetries in lattice models built on a tensor product Hilbert space. In (2+1)D, for given Gauss-law operators realized by finite-depth circuits that generate a finite 1-form G symmetry, we construct an index representing a cohomology class in H^{4}(B^{2}G,U(1)), which characterizes the corresponding 't Hooft anomaly. This construction generalizes the Else-Nayak characterization of 0-form symmetry anomalies. More broadly, under the assumption of a specified formulation of the p-form G symmetry action and Hilbert space structure in arbitrary d spatial dimensions, we show how to characterize the 't Hooft anomaly of the symmetry action by an index valued in H^{d+2}(B^{p+1}G,U(1)).

In this paper, a tensor product functional link neural network (TP-FLNN) was applied on the Mackey-Glass chaotic time series in the long-term forecasting. The forecasting performance of TP-FLNN was compared with simple forecasting methods namely naive, drift and average methods. The obtained results show the possibility to consider TP-FLNN in long-term forecasting.

The norms for symmetric and antisymmetric tensor products of the weighted shift operators

UDC 519.17 The forgotten topological index denoted by $F(G)$ of a graph $G=(V,E)$ is defined as follows: $F(G)=\displaystyle\sum\nolimits_{i=1}^{n}\!d_v^3,$ where $d_v$ denotes the degree of the vertex $v.$ We extend the notion of forgotten topological index to signed graphs and introduce the $MS$-index of a signed graph. Moreover, we determine the forgotten topological index for the tensor product, Cartesian product, lexicographic product, strong product, symmetric difference, and the joint of the graphs $G_1$ and $G_2$ in terms of the forgotten topological index, the first Zagreb index, and the $M\kern-1ptS$-index of signed graphs $\Sigma_1=(G_1,\sigma_1)$ and $\Sigma_2=(G_2,\sigma_2),$ along with their sizes and orders.

This paper develops a structural and functional framework for operator algebras acting on nonseparable Banach spaces (NSBS). While classical operator algebras—such as 𝐶∗ - and 𝑊∗ -algebras—are traditionally constructed on separable Hilbert spaces, many physical and mathematical contexts require non-separable or even transfinite structures: quantum field theories with infinitely many degrees of freedom, infinite tensor product systems, and algebras associated with non-measurable state spaces. We extend the classical operator-algebraic formalism to NSBS by introducing approximate operator algebras, defined through directed nets of weakly compact projections and local separable subspaces. This approach restores the analytic machinery of functional calculus, spectra, and representations, while preserving topological and dual properties within locally separable components. The paper establishes several new results concerning approximate ideals, bicommutants, spectral continuity, and weak operator topologies in NSBS. Furthermore, we analyse the correspondence between approximate representations of 𝐶∗ -algebras on NSBS and physical observables in quantum mechanics and field theory. From a physical perspective, the proposed framework provides a rigorous mathematical description of systems with non-countable degrees of freedom, extending von Neumann’s theory of operator algebras beyond separability. Applications include the representation of infinite spin systems, algebras of observables in non-separable Hilbert–Banach settings, and generalised state spaces in quantum statistical mechanics.

A bstract To facilitate a simultaneous treatment of an arbitrary number of colors in representation theory-based descriptions of QCD color structure, we derive an N -independent reduction of SU( N ) tensor products. To this end, we label each irreducible representation by a pair of Young diagrams, with parts acting on quarks and antiquarks. By combining this with a column-wise multiplication of Young diagrams, we generalize the Littlewood-Richardson rule for the product of two Young diagrams to the product of two Young diagram pairs, achieving a general- N decomposition.