Canonical Basis Research Articles

Adapting tree algorithms for partial enumeration of Extreme Pathways sets

Analysis of metabolic networks through characterising flux vectors such as Elementary Flux Modes and Extreme Pathways has shown to be valuable in many application within systems biology. However, their application is still limited to medium-scale metabolic networks due to the combinatorial explosion of the set size. Enumeration of the full set in genome-scale metabolic networks remains impossible, but few techniques exist for partial enumeration. Currently, an adapted version of the Canonical Basis Approach for generation of Extreme Pathways has allowed partial enumeration with good sampling quality through a filter. However, computational efficiency currently limits its application, especially when moving towards larger networks. One approach to increase this efficiency significantly is through candidate narrowing based on bit pattern trees. Here, many of the candidate modes are removed every iteration based on their ability to pass a rank test. However, the depth of the trees should be adapted based on the filter setting and the current amount of candidates to process. In this work, a novel algorithm is presented using candidate narrowing in a stochastic partial enumeration of Extreme Pathways. A case study for the central carbon metabolism of Escherichia coli showcases the adaptive tree depth and the overall performance increase for different settings.

Spiritual-semantic approach as a methodological basis for the analysing religious music (on the example of V. Stepurko’s cycle “Monologues of the Ages”)

Spiritual-semantic approach as a methodological basis for the analysing religious music (on the example of V. Stepurko’s cycle “Monologues of the Ages”)

A basis for the space of weakly holomorphic Drinfeld modular forms of level T

In this article, we explicitly construct a canonical basis for the space of certain weakly holomorphic Drinfeld modular forms for Γ0(T) (resp., for Γ0+(T)) and compute the generating function satisfied by the basis elements. We also give an explicit expression for the action of the Θ-operator, which depends on the divisor of meromorphic Drinfeld modular forms.

Ramanujan type congruences for quotients of Klein forms

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p>19. Canonical bases for modular forms of level 5≤p≤19 are constructed by summing weight one Hecke Eisenstein series of levels 5≤p≤19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ1(p) and Γ(p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ1(p) as a direct sum of subspaces of modular forms for Γ(p) of the form qr/pZ[[q]]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences.

Covert Scattering Control in Metamaterials with Non-Locally Encoded Hidden Symmetry.

Symmetries and tunability are of fundamental importance in wave scattering control, but symmetries are often obvious upon visual inspection, which constitutes a significant vulnerability of metamaterial wave devices to reverse-engineering risks. Here, it is theoretically and experimentally shown that a symmetry in the reduced basis of the "primary meta-atoms" that are directly connected to the outside world is sufficient; meanwhile, a suitable topology of non-local interactions between them, mediated by the internal "secondary" meta-atoms, can hide the symmetry from sight in the canonical basis. Covert symmetry-based scattering control in a cable-network metamaterial featuring a hidden parity ( ) symmetry in combination with hidden- -symmetry-preserving and hidden- -symmetry-breaking tuning mechanisms is experimentally demonstrated. Physical-layer security in wired communications is achieved using the domain-wise hidden -symmetry as a shared secret between the sender and the legitimate receiver. Within the approximation of negligible absorption, the first tuning of a complex scattering metamaterial without mirror symmetry to feature exceptional points (EPs) of -symmetric reflectionless states, as well as quasi-bound states in the continuum, is reported. These results are reproduced in metamaterials involving non-reciprocal interactions between meta-atoms, including the first observation of reflectionless EPs in a non-reciprocalsystem.

Modbed track: Visualization of modified bases in single-molecule sequencing

Recent advances in long-read sequencing technologies have not only dramatically increased sequencing read length but also have improved the accuracy of detecting chemical modifications to the canonical nucleotide bases, thus opening exciting venues to investigate the epigenome. Currently, the ability to visualize modified bases from long-read sequencing data in genome browsers is still limited, preventing users from easily and fully exploring these type of data. To address this limitation, the WashU Epigenome Browser introduces the modbed track type, which provides visualization of modification details in each single read as well as aggregated modifications of individual or multiple molecules across a dynamic range of resolutions. The modbed file can be uploaded for visualization as a local track or viewed with an accessible URL freely on the WashU Epigenome Browser at https://epigenomegateway.wustl.edu/.

Elementary coupling coefficients for the Wigner supermultiplet symmetry

An algorithm for evaluating Wigner coefficients of U(4) ⊃ SUS(2) ⊗ SUT(2) for the coupling [n14,n24,n34,n44]⊗[1,0,0,0]↓[m14,m24,m34,m44] with arbitrary irreducible representation [n14,n24,n34,n44] is provided based on the algebraic expressions with the expansion coefficients of the U(4)⊃SUS(2)⊗SUT(2) states in terms of those in the U(4) canonical basis and the Clebsch-Gordan coefficients of U(4) in the canonical basis. The state expansion coefficients are evaluated as the components of the null space vectors of the spin-isospin projection matrices. Program summaryProgram Title: Wigner-U4-SU2xSU2CPC Library link to program files:https://doi.org/10.17632/4b7gf2phh2.1Licensing provisions: GPLv3Programming language: Wolfram Mathematica version 9.0.1.0 and beyondNature of problem: Numerical evaluation of the elementary Wigner coefficients (EWCs) of U(4)⊃SUS(2)⊗SUT(2).Solution method: Mathematica implementation of the EWCs of U(4)⊃SUS(2)⊗SUT(2) [1] using the expansion coefficients of U(4)⊃SUS(2)⊗SUT(2) states in terms of those of the U(4) canonical basis determined as components of the null space vectors of the spin-isospin projection matrices [2].

A novel method for generating a canonical basis for decision implications based on object-induced three-way operators

Three-way concept analysis can express not only the information that the objects of study have in common but also the information that they do not have in common, which is an extension of classical formal concept analysis. The decision implication is a special representation of the implication in the decision formal context, which reveals the dependency between conditional attributes and decision attributes. This paper is devoted to the study of a novel method for generating a canonical basis for decision implications based on object-induced operators (OE operators). First, the decision implication logic based on OE operators is described from the two aspects of semantics and syntax, which is applied to the decision formal context. Then, a method of obtaining the canonical basis for decision implications based on decision the premise of the OE operators is given, and the completeness, the nonredundancy and the optimality of the canonical basis are proven. Finally, a method of generating true premises based on OE operators is proposed by means of an incremental method, and the canonical basis for decision implications is then generated by the true premises based on OE operators.

Partitioning qubits in hypergraph product codes to implement logical gates

The promise of high-rate low-density parity check (LDPC) codes to substantially reduce the overhead of fault-tolerant quantum computation depends on constructing efficient, fault-tolerant implementations of logical gates on such codes. Transversal gates are the simplest type of fault-tolerant gate, but the potential of transversal gates on LDPC codes has hitherto been largely neglected. We investigate the transversal gates that can be implemented in hypergraph product codes, a class of LDPC codes. Our analysis is aided by the construction of a symplectic canonical basis for the logical operators of hypergraph product codes, a result that may be of independent interest. We show that in these codes transversal gates can implement Hadamard (up to logical SWAP gates) and control-Z on all logical qubits. Moreover, we show that sequences of transversal operations, interleaved with error correction, allow implementation of entangling gates between arbitrary pairs of logical qubits in the same code block. We thereby demonstrate that transversal gates can be used as the basis for universal quantum computing on LDPC codes, when supplemented with state injection.

Inquest for the interaction of canonical and non-canonical DNA/RNA bases with ternary based 2D Si2BN and doped Si2BN for biosensing applications

Density functional theory (DFT) is invoked to investigate the interaction between the canonical (CN) and non-canonical (NC) bases with pristine Si2BN (Si2BN) and Phosphorous-doped Si2BN (P-dop-Si2BN) sheets. Inquest for the better sensing substrate is decided through the adsorption energy calculation which reveals that doping of phosphorous atom enhances the adsorption strength of AT (-83.74 kcal/mol) AU (-82.77 kcal/mol) and GC (-96.36 kcal/mol) base pairs. The CN and NC bases have higher adsorption energy than the previous reported values which concludes that the P-dop-Si2BN sheet will be optimal substrate to sense the bases. Meanwhile, the selected CN and NC (except hypoxanthine) bases interact with sheet in parallel manner which infers the π-π interaction with Si2BN and P-dop-Si2BN sheets. The energy gap variation (ΔEg%) of the P-dop-Si2BN complexes has a noticeable change, ranging from −24.75 to −197.28% which thrust the sensitivity of the P-dop-Si2BN sheet over the detection of CN and NC bases. The natural population analysis (NPA) and electron density difference map (EDDM) confirms that charges are transferred from CN and NC bases to Si2BN and P-dop-Si2BN sheet. The optical property of the P-dop-Si2BN complexes reveals that the noticeable red and blue shift in the visible and near-infrared regions (778 nm to 1143 nm) has been observed. Therefore, the above results conclude that the P-dop-Si2BN sheet plays a potential candidate to detect the CN and NC bases which contribute to the development of biosensors and DNA/RNA sequencing devices. Communicated by Ramaswamy H. Sarma

Master integrals for electroweak corrections to gg→γγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$gg\\rightarrow \\gamma \\gamma $$\\end{document}: light quark contributions

We present a calculation of the master integrals (MI’s) required for the calculation of the Electroweak corrections to gg→γγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$gg\\rightarrow \\gamma \\gamma $$\\end{document} production in which the process contains a light quark loop. The integrals can be broken down into five categories based on the flow of the heavy vector bosons throughout the loop. Three of the families are planar, and two are non-planar. We determine a canonical basis for each family which allows an efficient solution of the resulting differential equations via iterated integrals. We calculate the families in relevant physical kinematics and obtain an efficient numerical evaluation based on an implementation of Chen-iterated integrals.

Can methylated purine bases act as photoionization hotspots?

The direct photoionization of DNA canonical bases under ultraviolet radiation is difficult due to the high ionization potentials. According to previous quantum chemical calculations, methylation can have great influence on the ionization potential. Are methylated nucleobases prone to photoionization and cause DNA damage? As an important epigenetic modification in transcription, expression, and regulation of genes, it is of great biological significance to explore the effect of methylation on base photoionization from the experimental perspective. Herein, we study the photoionization behavior of methylated purines 6 mA and 6mG at 266 nm using a nanosecond transient UV-Vis spectroscopy. The hydrated electron and methylated base radicals are observed, indicating the occurrence of photoionization for both 6mG and 6 mA. We measured one-photon ionization yields to be (5.0 ± 0.2) × 10-3 and (1.4 ± 0.2) × 10-3 for 6mG and 6 mA, respectively. These are higher than those of (dA)20 and (dA20 )·(dT20 ) previously reported, indicating that methylation significantly promotes base photoionization with a stronger effect than base stacking, consistent with calculations in literature. Given that the hydrated electrons and methylated base radicals from photoionization can trigger a cascade of deleterious reactions, the methylated purine bases may act as hotspots of DNA photoionization damage of living organisms.

Groupwise Shape Correspondence Refinement with a Region of Interest Focus

AbstractWhile collections of scan shapes are becoming more prevalent in many real‐world applications, finding accurate and dense correspondences across multiple shapes remains a challenging task. In this work, we introduce a new approach for refining non‐rigid correspondences among a collection of 3D shapes undergoing non‐rigid deformation. Our approach incorporates a Region Of Interest (ROI) into the refinement process, which is specified by the user on one shape within the collection. Based on the functional map framework and more specifically on the notion of cycle‐consistency, our formulation improves the overall matching consistency while prioritizing that of the region of interest. Specifically, the initial pairwise correspondences are refined by first defining the localized harmonics that are confined within the transferred ROI on each shape, and subsequently applying the CCLB (Canonical Consistent Latent Basis) framework both on the global and the localized harmonics. This leads to an enhanced matching accuracy for both the ROIs and the overall shapes across the collection. We evaluate our method on various synthetic and real scan datasets, in comparison with the state‐of‐the‐art techniques.

Rational Cherednik algebras of G(ℓ,p,n) from the Coulomb perspective

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n). In particular, we: (1) show a relationship to the Coulomb branch construction of Braverman, Finkelberg, and Nakajima, and 3-dimensional quantum field theory; (2) show that the spherical Cherednik algebra carries the structure of a principal Galois order; (3) construct a graded lift of category O and the larger category of Dunkl-Opdam modules, whose simple modules have the properties of a dual canonical basis and (4) give the first classification of simple Dunkl-Opdam modules for the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n).

On Structural Rank and Resilience of Sparsity Patterns

A sparsity pattern in $\mathbb{R}^{n \times m}$, for $m\geq n$, is a vector subspace of matrices admitting a basis consisting of canonical basis vectors in $\mathbb{R}^{n \times m}$. We represent a sparsity pattern by a matrix with $0/\star$-entries, where $\star$-entries are arbitrary real numbers and $0$-entries are equal to $0$. We say that a sparsity pattern has full structural rank if the maximal rank of matrices contained in it is $n$. In this paper, we investigate the degree of resilience of patterns with full structural rank: We address questions such as how many $\star$-entries can be removed without decreasing the structural rank and, reciprocally, how many $\star$-entries one needs to add so as to increase the said degree of resilience to reach a target. Our approach goes by translating these questions into max-flow problems on appropriately defined bipartite graphs. Based on these translations, we provide algorithms that solve the problems in polynomial time.

Tame quivers and affine bases I: A Hall algebra approach to the canonical bases

In [17], Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztig's canonical basis and obtains a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang ([14]) and closely related with the work of Beck-Nakajima ([2]). A crucial method in our construction is a generalization of that in [8].

ıSchur duality and Kazhdan-Lusztig basis expanded

Expanding the classical works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an ıSchur duality between an ıquantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wang's quasi-split ıSchur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the ıcanonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the ıSchur duality.

An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions

This paper is concerned with the numerical solution of Volterra integral equations of the third kind with non-smooth solutions based on the recursive approach of the spectral Tau method. To this end, a new set of the fractional version of canonical basis polynomials (called FC-polynomials) is introduced. The approximate polynomial solution (called Tau-solution) is expressed in terms of FC-polynomials. The fractional structure of Tau-solution allows recovering the standard degree of accuracy of spectral methods even in the case of non-smooth solutions. The convergence analysis of the method is carried out. The obtained numerical results show the accuracy and efficiency of the method compared to other existing methods.

Representations of orientifold Khovanov–Lauda–Rouquier algebras and the Enomoto–Kashiwara algebra

We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial.

Two-loop master integrals for a planar and a non-planar topology relevant for single top production

We provide analytic results for two-loop four-point master integrals with one massive propagator and one massive leg relevant to single top production. Canonical bases of master integrals are constructed and the Simplified Differential Equations approach is employed for their analytic solution. The necessary boundary terms are computed in closed form in the dimensional regulator, allowing us to obtain analytic results in terms of multiple polylogarithms of arbitrary transcendental weight. We provide explicit solutions of all two-loop master integrals up to transcendental weight six and discuss their numerical evaluation for Euclidean and physical phase-space points.