Monomial Basis Research Articles

The Singular Support of the Ising Model

Abstract We prove a new quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson–Slater $q$-series identity of Rogers–Ramanujan type and to Nahm sums for the matrix $\left (\begin {smallmatrix}8&3\\3&2 \end {smallmatrix}\right ) $. We find, as consequences, an explicit monomial basis for the Ising model and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new $q$-series identities of the Rogers–Ramanujan–Slater type associated with the three irreducible modules of the Virasoro Lie algebra of central charge $1/2$. We give a combinatorial interpretation to the identity associated with the vacuum module.

Quantum queer supergroups via υ-differential operators

By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup Uυ(qn). The underlying space of this representation is a deformed polynomial superalgebra in 2n2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of Uυ(qn). A monomial basis M for Uυ(qn) plays a key role in proving the isomorphism. In this way, we may present the quantum queer supergroup Uυ(qn) by another new basis L together with some explicit multiplication formulas by the generators. As an application, similar presentations are obtained for queer q-Schur superalgebras via the above mentioned homogeneous components.The existence of the bases M and L and the new presentation show that the seminal construction of quantum gln established by Beilinson–Lusztig–MacPherson thirty years ago extends to this “queer” quantum supergroup via a completely different approach.

Hall algebras and quantum symmetric pairs I: Foundations

A quantum symmetric pair consists of a quantum group U $\mathbf {U}$ and its coideal subalgebra U ς ı ${\mathbf {U}}^{\imath }_{\bm{\varsigma }}$ with parameters ς $\bm{\varsigma }$ (called an ı $\imath$ quantum group). We initiate a Hall algebra approach for the categorification of ı $\imath$ quantum groups. A universal ı $\imath$ quantum group U ∼ ı $\widetilde{\mathbf {U}}^{\imath }$ is introduced and U ς ı ${\mathbf {U}}^{\imath }_{\bm{\varsigma }}$ is recovered by a central reduction of U ∼ ı $\widetilde{\mathbf {U}}^{\imath }$ . The semi-derived Ringel–Hall algebras of the first author and Peng, which are closely related to semi-derived Hall algebras of Gorsky and motivated by Bridgeland's work, are extended to the setting of 1-Gorenstein algebras, as shown in the Appendix by the first author. A new class of 1-Gorenstein algebras (called ı $\imath$ quiver algebras) arising from acyclic quivers with involutions is introduced. The semi-derived Ringel–Hall algebras for the Dynkin ı $\imath$ quiver algebras are shown to be isomorphic to the universal quasi-split ı $\imath$ quantum groups of finite type. Monomial bases and PBW bases for these Hall algebras and ı $\imath$ quantum groups are constructed.

Diffeomorphically Learning Stable Koopman Operators

System representations inspired by the infinite-dimensional Koopman operator (generator) are increasingly considered for predictive modeling. Due to the operator’s linearity, a range of nonlinear systems admit linear predictor representations -allowing for simplified prediction, analysis and control. However, finding meaningful finite-dimensional representations for prediction is difficult as it involves determining features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) – a generally unsupervised problem. In this work, we present Koopmanizing Flows -a novel continuous-time framework for supervised learning of linear predictors for a class of nonlinear dynamics. In our model construction a latent diffeomorphically related linear system unfolds into a linear predictor through the composition with a monomial basis. The lifting, its linear dynamics and state reconstruction are learned simultaneously, while an unconstrained parameterization of Hurwitz matrices ensures asymptotic stability regardless of the operator approximation accuracy. The superior efficacy of Koopmanizing Flows is demonstrated in comparison to a state-of-the-art method on the well-known LASA handwriting benchmark.

A note on the $\mathcal{A}$-generators of the polynomial algebra of six variables and applications

Let $ \mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb Z_2[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra of $n$ generators $x_1, x_2, \ldots, x_n$ with the degree of each $x_i$ being 1. We investigate the Peterson hit problem for the polynomial algebra $ \mathcal P_{n},$ regarded as a module over the mod-$2$ Steenrod algebra, $ \mathcal{A}.$ For $n>4,$ this problem remains unsolvable, even with the aid of computers in the case of $n=5.$ In this article, we study the hit problem for the case $n=6$ in degree $d_s=6(2^s -1)+3.2^s,$ with $s$ an arbitrary nonnegative integer. By considering $ \mathbb Z_2$ as a trivial $ \mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $ \mathbb Z_2$-vector space $ \mathbb Z_2 {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ The main goal of the current article is to explicitly determine an admissible monomial basis of the $ \mathbb Z_2$vector space $ \mathbb Z_2{\otimes}_{\mathcal{A}}\mathcal P_n$ for $n=6$ in some degrees. One of the most important applications of the hit problem is to investigate homomorphism introduced by Singer, which is a homomorphism $$ \varphi_n :\text{Tor}^{\mathcal A}_{n, n+d} (\mathbb Z_2,\mathbb Z_2) \longrightarrow (\mathbb{Z}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n})_d^{GL(n; \mathbb Z_2)}$$ from the homology of the Steenrod algebra to the subspace of $(\mathbb{Z}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n})_d$ consisting of all the $GL(n; \mathbb Z_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, $\text{Tor}^{\mathcal A}_{n, n+d}(\mathbb Z_2,\mathbb Z_2).$ The behavior of the sixth Singer algebraic transfer in degree $d_s=6(2^s -1)+3.2^s$ was also discussed at the end of this paper.

Some elementary properties of Laurent phenomenon algebras

<abstract><p>Let $ \Sigma $ be a Laurent phenomenon (LP) seed of rank $ n $, $ \mathcal{A}(\Sigma) $, $ \mathcal{U}(\Sigma) $, and $ \mathcal{L}(\Sigma) $ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $ \mathcal{A}(\Sigma) $ is uniquely defined by its cluster and any two seeds of $ \mathcal{A}(\Sigma) $ with $ n-1 $ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $ \mathcal{U}(\Sigma) $ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $ \Sigma $. Besides, we obtain the standard monomial bases of $ \mathcal{L}(\Sigma) $. We also prove that $ \mathcal{U}(\Sigma) $ coincides with $ \mathcal{L}(\Sigma) $ under certain conditions.</p></abstract>

Set superpartitions and superspace duality modules

Abstract The superspace ring $\Omega _n$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $\Omega _n$ , the authors previously defined a family of doubly graded quotients ${\mathbb {W}}_{n,k}$ of $\Omega _n$ , which carry an action of the symmetric group ${\mathfrak {S}}_n$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules ${\mathbb {W}}_{n,k}$ in greater detail. We describe a monomial basis of ${\mathbb {W}}_{n,k}$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions. These are ordered set partitions $(B_1 \mid \cdots \mid B_k)$ of $\{1,\dots ,n\}$ in which the nonminimal elements of any block $B_i$ may be barred or unbarred.

Solving parametric systems of polynomial equations over the reals through Hermite matrices

We design a new algorithm for solving parametric systems of equations having finitely many complex solutions for generic values of the parameters. More precisely, let f=(f1,…,fm)⊂Q[y][x] with y=(y1,…,yt) and x=(x1,…,xn), V⊂Ct×Cn be the algebraic set defined by the simultaneous vanishing of the fi's and π be the projection (y,x)→y. Under the assumptions that f admits finitely many complex solutions when specializing y to generic values and that the ideal generated by f is radical, we solve the following algorithmic problem. On input f, we compute semi-algebraic formulas defining open semi-algebraic sets S1,…,Sℓ in the parameters' space Rt such that ∪i=1ℓSi is dense in Rt and, for 1≤i≤ℓ, the number of real points in V∩π−1(η) is invariant when η ranges over Si.This algorithm exploits special properties of some well chosen monomial bases in the quotient algebra Q(y)[x]/I where I⊂Q(y)[x] is the ideal generated by f in Q(y)[x] as well as the specialization property of the so-called Hermite matrices which represent Hermite's quadratic forms. This allows us to obtain “compact” representations of the semi-algebraic sets Si by means of semi-algebraic formulas encoding the signature of a given symmetric matrix.When f satisfies extra genericity assumptions (such as regularity), we use the theory of Gröbner bases to derive complexity bounds both on the number of arithmetic operations in Q and the degree of the output polynomials. More precisely, letting d be the maximal degrees of the fi's and D=n(d−1)dn, we prove that, on a generic input f=(f1,…,fn), one can compute those semi-algebraic formulas using O˜((t+Dt)23tn2t+1d3nt+2(n+t)+1) arithmetic operations in Q and that the polynomials involved in these formulas have degree bounded by D.We report on practical experiments which illustrate the efficiency of this algorithm, both on generic parametric systems and parametric systems coming from applications since it allows us to solve systems which were out of reach on the current state-of-the-art.

Smooth and robust multi-mode shaped commands

Smooth robust shaped commands with general polynomial and waveform profiles are proposed for eliminating residual vibrations in multi-mode damped systems. The command time length of the proposed commands is selectable which helps in designing an optimum command that results of a desired reduction in the induced transient vibration, utilizes the maximum input capability of the actuator, and enhances the command robustness to the uncertainties in the system parameters. The commands are generated using the system natural frequencies and damping ratios. This is attractive for complicated multi-mode damped systems where mathematical models are difficult to derive. The continuous-time profiles of the proposed commands that are based on the monomial and Fourier bases render the possibility of enhancing the robustness and smoothness in order to eliminate jerks, i.e., inrush current, and inaccurate timing and delay of the actuator. The efficacy of their performance in eliminating residual vibrations is validated using numerical simulations and experiment. Their sensitivity to system modeling uncertainties is further reduced using three different approaches; relaxing the command length, flattening the sensitivity curve at and adjacent to the modeling frequencies, and introducing virtual frequencies adjacent to the design frequencies to enhance the bandwidth of the reduced sensitivity region.

Computational method for solving weakly singular Fredholm integral equations of the second kind using an advanced barycentric Lagrange interpolation formula

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

Determinantal Formulas for SEM Expansions of Schubert Polynomials

We show that for any permutation w that avoids a certain set of 13 patterns of length 5 and 6, the Schubert polynomial \({\mathfrak {S}}_w\) can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi–Trudi identity. For such w, this determinantal formula is equivalent to a (signed) subtraction-free expansion of \(\mathfrak S_w\) in the basis of standard elementary monomials.

Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

We define generalized standard triples $\boldsymbol{X}$, $\boldsymbol{Y}$, and $L(z) = z\boldsymbol{C}_{1} - \boldsymbol{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\boldsymbol{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\boldsymbol{X}(z \boldsymbol{C}_{1}~-~\boldsymbol{C}_{0})^{-1}\boldsymbol{Y}~=~\boldsymbol{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\boldsymbol{P}$. This representation can be used in constructing so-called algebraic linearizations for matrix polynomials of the form $\boldsymbol{H}(z) = z \boldsymbol{A}(z)\boldsymbol{B}(z) + \boldsymbol{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$. This can be done even if $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\boldsymbol{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k \phi_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases.

The element-free Galerkin method for the dynamic Signorini contact problems with friction in elastic materials

The element-free Galerkin method is proposed for the dynamic Signorini contact problems with friction in elastic materials. The Dirichlet boundary conditions and the constrained conditions are imposed by the penalty method. The error estimates of the element-free Galerkin method indicate that the convergence order depends on the spatial step, the time step, the largest degree of a complete Pascal's monomial basis in the moving least-squares approximation and the penalty factor. Numerical examples verify our theoretical results.

Numerical differentiation on scattered data through multivariate polynomial interpolation

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

On the dimension of [formula omitted] as a module over Steenrod ring

We write P for the polynomial algebra in one variable over the finite field Z2 and P⊗t=Z2[x1,…,xt] for its t-fold tensor product with itself. We grade P⊗t by assigning degree 1 to each generator. We are interested in determining a minimal set of generators for the ring of invariants (P⊗t)Gt as a module over Steenrod ring, A2. Here Gt is a subgroup of the general linear group GL(t,Z2). An equivalent problem is to find a monomial basis of the space of “unhit” elements, Z2⊗A2(P⊗t)Gt in each t and degree n≥0. The structure of this tensor product is proved surprisingly difficult and has been not yet known for t≥5, even for the trivial subgroup Gt={e}. In the present paper, we consider the subgroup Gt={e} for t∈{5,6}, and obtain some new results on A2-generators of (P⊗t)Gt in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [24,26].

A Leray Model for the Orlik–Solomon Algebra

Abstract We construct a combinatorial generalization of the Leray models for hyperplane arrangement complements. Given a matroid and some combinatorial blow-up data, we give a presentation for a bigraded (commutative) differential graded algebra. If the matroid is realizable over $\mathbb {C}$, this is the familiar Morgan model for a hyperplane arrangement complement, embedded in a blowup of projective space. In general, we obtain a cdga that interpolates between the Chow ring of a matroid and the Orlik–Solomon algebra. Our construction can also be expressed in terms of sheaves on combinatorial blowups of geometric lattices. As a key technical device, we construct a monomial basis via a Gröbner basis for the ideal of relations. Combining these ingredients, we show that our algebra is quasi-isomorphic to the classical Orlik–Solomon algebra of the matroid.

Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.

Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation

In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.

Monomial bases for the primitive complex Shephard groups of rank three

For the group algebra of each primitive non-Coxeter Shephard group of rank three, we construct a monomial basis and its explicit multiplication table. First, we find a Gröbner–Shirshov basis for the Shephard group of type L 2. Then, since each of the groups of types L 3 and M 3 has a parabolic subgroup isomorphic to the group of type L 2, by using the sets of minimal right coset representatives of L 2 in L 3 and M 3, respectively, we apply the Gröbner–Shirshov basis technique to find the monomial bases for the Shephard groups of rank three L 3 and M 3. From this, we obtain the operation tables between the elements in each of the groups. Also we explicitly show that the group of type M 3 has a subgroup of index 2 isomorphic to the group of type L 3.

Quantum Supertwistors

In this paper, we give an explicit expression for a star product on the super-Minkowski space written in the supertwistor formalism. The big cell of the super-Grassmannian Gr(2|0,4|1) is identified with the chiral, super-Minkowski space. The super-Grassmannian is a homogeneous space under the action of the complexification SL(4|1) of SU(2,2|1), the superconformal group in dimension 4, signature (1,3), and supersymmetry N=1. The quantization is done by substituting the groups and homogeneous spaces by their quantum deformed counterparts. The calculations are done in Manin’s formalism. When we restrict to the big cell, we can explicitly compute an expression for the super-star product in the Minkowski superspace associated to this deformation and the choice of a certain basis of monomials.