Canonical Basis Research Articles

Resummation methods for Master Integrals

We present in detail two resummation methods emerging from the application of the Simplified Differential Equations approach to a canonical basis of master integrals. The first one is a method which allows for an easy determination of the boundary conditions, since it finds relations between the boundaries of the basis elements and the second one indicates how using the x → 1 limit to the solutions of a canonical basis, one can obtain the solutions to a canonical basis for the same problem with one mass less. Both methods utilise the residue matrices for the letters {0, 1} of the canonical differential equation. As proof of concept, we apply these methods to a canonical basis for the three-loop ladder-box with one external mass off-shell, obtaining subsequently a canonical basis for the massless three-loop ladder-box as well as its solution.

Quantization of canonical bases and the quantum symplectic double

We describe a natural q-deformation of Fock and Goncharov’s canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type A. We then describe an extension of this construction involving a cluster variety called the symplectic double.

Coherent IC-sheaves on type 𝐴_{𝑛} affine Grassmannians and dual canonical basis of affine type 𝐴₁

The convolution ring K G L n ( O ) ⋊ C × ( G r G L n ) K^{GL_n(\mathcal {O})\rtimes \mathbb {C}^\times }(\mathrm {Gr}_{GL_n}) was identified with a quantum unipotent cell of the loop group L S L 2 LSL_2 in Cautis and Williams [J. Amer. Math. Soc. 32 (2019), pp. 709–778]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.

APPLICATION OF MULTIDIMENSIONAL TIME MODEL TO DYNAMICAL RELATION OF POISSON SPIKE TRAINS IN NEURAL ION CURRENT MODELS AND FORMATION OF NON-CANONICAL BASES, ISLANDS, AND G-QUADRUPLEXES IN DNA, MRNA, AND RNA AT OR NEAR THE TRANSCRIPTION

Ground breaking application of mathematics and biochemistry to explain formation of non-canonical bases, islands, G-quadruplex structures, and analog bases in DNA and mRNA at or near the transcription with connection to neural networks is implemented using statistical and stochastic methods apparatus with the addition of quantum principles. As a result the usual transience of Poisson spike trains (PST) becomes very instrumental tool for finding periodical type of solutions to Fokker-Plank (FP) stochastic differential equation (SDE). The present study develops new multidimensional methods of finding solutions to SDE. This is based on more rigorous approach to mathematical apparatus through Kolmogorov-Chentsov continuity theorem (KCCT) that allows the stochastic processes with jumps under certain conditions to have γ-Holder continuous modification, which is used as basis for finding analogous parallels in dynamics of formation of CpG and non-CpG islands (CpGI or non CpGI), repeats of G-quadruplexes, and non canonical bases during DNA (de)- methylation and neural networks.

Convergence of polarizations, toric degenerations, and Newton–Okounkov bodies

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\{J_s\}$ of the complex structure on $X$ and bases $\mathcal{B}_s$ of $H^0(X,L, J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s \to \infty$, the basis elements approach dirac-delta distributions centered at Bohr-Sommerfeld fibers of a moment map associated to $X$ and its toric degeneration. The theory of Newton-Okounkov bodies and its associated toric degenerations shows that the technical hypotheses mentioned above hold in some generality. Our results significantly generalize previous results in geometric quantization which prove independence of polarization between Kahler quantizations and real polarizations. As an example, in the case of general flag varieties $X=G/B$ and for certain choices of $\lambda$, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of $V_{\lambda}^*$ to a collection of dirac delta functions supported at the Bohr-Sommerfeld fibres corresponding exactly to the lattice points of a Littelmann-Berenstein-Zelevinsky string polytope $\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$.

Model theory of R-trees

We show the theory of pointed $\R$-trees with radius at most $r$ is axiomatizable in a suitable continuous signature. We identify the model companion $\rbRT_r$ of this theory and study its properties. In particular, the model companion is complete and has quantifier elimination; it is stable but not superstable. We identify its independence relation and find built-in canonical bases for non-algebraic types. Among the models of $\rbRT_r$ are $\R$-trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of $\rbRT_r$; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of $\rbRT_r$. Among other things, we show that the space of $2$-types over the empty set is nonseparable. Also, we characterize the principal types of finite tuples (over the empty set) and use this information to conclude that $\rbRT_r$ has no atomic model.

Game as a psychological program of the subject’s life activity

Modern features of human life activity are characterized by the accumulation of risks of critical uncertainty, lack of predictability and rational basis for productive decision-making and social interaction. It actualizes the large-scale problems of various genesis. In particular, it is talked about global problems of the military-political, medical-social and cultural-economic context. The hybrid geopolitical confrontation, the COVID-19 sanitary crisis, the precariat-movements Gilets jaunes and BLM and others are the examples of this. Therefore, a modern person is faced with the critical urgency to find and use reliable means of life activity in such conditions. Scientific psychology has the task of creating and providing practical effective mental tools for a person’s life in the complex social conditions of everyday life. The development of such tools should be carried out on a multidisciplinary methodological basis. It is argued that the most important tool of human life under these circumstances is the phenomenon of the game which is in the subject field of philosophical, psychological, mathematical, sociological, cultural and political discourses. Comparative scientific analysis confirms that the game is a universal onto-phenomenal form and a productive program of subject’s rational life activity; a way of the person’s and a society’s being in difficult conditions. In the field of scientific methodology, the game becomes a canonical multidisciplinary basis for research of this range of problems. This opportunity is given to the game by its ability to ensure the processes of systemgenesis of human and social group, because it is the game as a metaprogram of the subject’s life activity that optimally ensures the implementation of the fundamental principles of systemgenesis: a) ordered integrity; b) self-stabilization; c) self-organization, d) hierarchization and e) interaction of system components. Among these principles, the leading role is played by the principle of mutual cooperation, which is provided by the mechanisms of releasing the components of the system from excessive degrees of freedom, creating a model of useful results for it. These principles are axiomatically and isomorphically implemented both at the level of personality and at the level of social communication. This is due to the game programming of the subject, which exactly corresponds to the situation relevant to him. A universal system-forming factor in the genesis of a personal game program here is usefulness. The game combines time-space indicators, indicators of the result quality, and the mechanisms of its verification. The game introduces a program-genetic context of influence on the formation of human behavior from the mode of imitation to the mode of full-fledged action in moments of uncertainty. The full-fledged application of the game as a metaprogram of activity in difficult conditions spreads in society the priorities of intersubjective communicative rationality. The canonicity of the game here is manifested in the isomorphism of its content and participation with the mechanisms of Pareto-optimality (mathematical game theory), communicative rationality (philosophy), and taxonomy of the types of leading human activity (psychology of activity).

An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras

Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the t-analogs of q-characters of simple modules of quantum affine algebras.

Diagram automorphisms and canonical bases for quantum affine algebras

Let Uq− be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra g, and U_q− the algebra corresponding to the orbit algebra of g obtained from a diagram automorphism σ on g. Let Bσ be the set of σ-fixed elements in the canonical basis of Uq−, and B_ the canonical basis of U_q−. Lusztig proved that there exists a canonical bijection Bσ≃B_ based on his geometric construction of canonical bases. In this paper, we prove (the signed bases version of) this fact, in the case where g is finite or affine type, in an elementary way, in the sense that we don't appeal to the geometric theory of canonical bases nor Kashiwara's theory of crystal bases. We also discuss the correspondence for PBW-bases, by using a new type of PBW-bases of Uq− obtained by Muthiah-Tingley, which is a generalization of PBW-bases constructed by Beck-Nakajima.

On the numerical computation of bivariate Lagrange polynomials

In this paper we propose a simple procedure for numerically computing the Lagrange interpolation polynomial on a unisolvent set of points in the plane. We suggest the use of the canonical polynomial basis centered at the barycenter of the set of points and the PA=LU decomposition for solving the associated Vandermonde system to compute the coefficients of the Taylor polynomial. We show that the 1-norm condition number of the Vandermonde matrix is an upper bound for the Lebesgue constant of the interpolation node set in the unit disk. Therefore, the analysis of the condition number can be useful to select the unisolvent set of nodes in a set of scattered nodes. Numerical experiments show the efficiency and accuracy of the proposed method.

P-Blocks Relative to a Character of a Normal Subgroup II

Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let $$\theta $$ be a G-invariant irreducible character of N. In Rizo (J Algebra 514:254–272, 2018), we introduced a canonical partition of the set $$\mathrm{Irr}(G|\theta )$$ of irreducible constituents of the induced character $$\theta ^G$$ , relative to the prime p. We call the elements of this partition the $$\theta $$ -blocks. In this paper, we construct a canonical basis of the complex space of class functions defined on $$\{ x \in G \, |\, x_p \in N\}$$ , which supersedes previous non-canonical constructions. This allows us to define $$\theta $$ -decomposition numbers in a natural way. We also prove that the elements of the partition of $${\text {Irr}}(G|\theta )$$ established by these $$\theta $$ -decomposition numbers are the $$\theta $$ -blocks.

Master integrals for two-loop QCD corrections to quark quasi PDFs

We compute the master integrals for the two-loop QCD corrections to quark quasi parton distribution functions (PDFs) in the large momentum effective theory (LaMET). With a proper canonical basis, we derive the analytical results for the three families of master integrals using the method of differential equations. The final expressions for the master integrals are given in terms of Goncharov polylogarithms. These results allow us to extract the two-loop short-distant matching coefficients between quark quasi and lightcone PDFs in LaMET, and are valuable to improve the determination of the nucleon PDFs from first principles in future.

Toric degenerations of cluster varieties and cluster duality

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

Speech Compressive Sampling Using Approximate Message Passing and a Markov Chain Prior.

By means of compressive sampling (CS), a sparse signal can be efficiently recovered from its far fewer samples than that required by the Nyquist–Shannon sampling theorem. However, recovering a speech signal from its CS samples is a challenging problem, as it is not sparse enough on any existing canonical basis. To solve this problem, we propose a method which combines the approximate message passing (AMP) and Markov chain that exploits the dependence between the modified discrete cosine transform (MDCT) coefficients of a speech signal. To reconstruct the speech signal from CS samples, a turbo framework, which alternately iterates AMP and belief propagation along the Markov chain, is utilized. In addtion, a constrain is set to the turbo iteration to prevent the new method from divergence. Extensive experiments show that, compared to other traditional CS methods, the new method achieves a higher signal-to-noise ratio, and a higher perceptual evaluation of speech quality (PESQ) score. At the same time, it maintaines a better similarity of the energy distribution to the original speech spectrogram. The new method also achieves a comparable speech enhancement effect to the state-of-the-art method.

Deformed relativistic Hartree-Bogoliubov theory in continuum with a point-coupling functional: Examples of even-even Nd isotopes

The aim of this work is to develop the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) theory based on the point-coupling density functionals and extend it to provide a unified description for all even-even nuclei in the nuclear chart by overcoming all possible challenges. The nuclear superfluidity is considered via Bogoliubov transformation. Densities and potentials are expanded in terms of Legendre polynomials to include the axial deformation degrees of freedom. Sophisticated relativistic Hartree-Bogoliubov equations in coordinate space are solved in the DiracWoods-Saxon basis to consider the continuum effects. Numerical checks are performed from light nuclei to heavy nuclei. The techniques to construct the DRHBc mass table for even-even nuclei are explored. The DRHBc theory is extended to study heavier nuclei beyond magnesium isotopes. Taking Nd isotopes as examples, the experimental binding energies, two-neutron separation energies, quadrupole deformations, and charge radii are reproduced rather well. The deformation and continuum play essential roles in the description of nuclear masses and prediction of drip-line nuclei. By examining the single-particle levels in the canonical basis and their contributions to the total density, the thickness of the neutron skin, the particles number in continuum, and the Coulomb barrier, the exotic structures including the neutron skin and the proton radioactivity are predicted.

Isometries of combinatorial Banach spaces

We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.

Analytic P-ideals and Banach spaces

We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals from submeasures, we point out numerous symmetries between the two theories. Also, we investigate a special case, the interactions between combinatorics of families of finite sets, topological properties of the “Schreier type” Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these spaces. Among other results, we present some new examples of Banach spaces and analytic P-ideals, a new characterization of precompact families and its applications to enhance Pták's Lemma and Mazur's Lemma.

Classification of 5-dimentional subalgebras for 6-dimentional nilpotent Lie algebras

In this paper, we consider the classical problem of the classification of subalgebras of small dimensional Lie algebras. We found all 5-dimentional subalgebras of 6-dimentional nilpotent Lie algebras under the field with the zero characteristic. As is known, up to isomorphism all 6-dimensional nilpotent Lie algebras (their number is 32) were received by V. V. Morosov. However, the standard method based on the Campbell – Hausdorf formula is not effective for the determination of subalgebras of Lie 5- or higher dimensional algebras. In our research, we use a new approach to the solution of the problem of the determination of 5-dimensional subalgeras of indicated 6-dimensional nilpotent Lie algerbas, namely, the application of canonical bases for subspaces of vector spaces.

Quark mass effects in two-loop Higgs amplitudes

We provide two two-loop amplitudes relevant for precision Higgs physics. The first is the two-loop amplitude for Higgs boson production through gluon fusion with exact dependence on the top quark mass up to squared order in the dimensional regulator ε. The second result we provide is the two-loop amplitude for the decay of a Higgs boson into a pair of massive bottom quarks through the Higgs-to-gluon coupling in the infinite top mass limit. Both amplitudes are computed by finding canonical bases of master integrals, which we evaluate explicitly in terms of harmonic polylogarithms. We obtain the bare, renormalized and IR-subtracted amplitude and provide the results in terms of building blocks suitable for changing renormalization schemes.

MONOMIAL BASES AND BRANCHING RULES

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra. As an application, we produce monomial bases for representations of the general linear and symplectic Lie algebras associated with natural chains of subalgebras. We also show that our basis in type A is related to both the Gelfand-Tsetlin basis and the Littelmann basis via triangular transition matrices which implies that the triangularity property extends to the matrix connecting the Gelfand-Tsetlin and canonical bases. A similar relationship holds between our basis in type C and a suitably modified version of the basis constructed earlier by the first author.