Basis Of Vector Space Research Articles

Comparability of Metrics and Norms in terms of Basis of Exponential Vector Space

In this work, we study and compare two methods for passing from a dual basis to the basis of a finite-dimensional vector space by recalling the passage from a basis to its dual. For the inverse transition, we clarify the kernel method for linear forms and the matrix inversion method, the former exploiting the properties of linear forms and orthogonality, while the latter relies on the explicit inversion of transition matrices, with remarks for each approach depending on the context of application. The study shows that, although matrix inversion is a more direct method, the kernel method can offer a more elegant and efficient alternative in certain cases, especially when the vector space has particular structures.

Abstract We show that the ring of regular functions of every smooth affine log Calabi–Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb {C}$ -algebra structure with structure coefficients given by the geometric construction of Keel and Yu [ The Frobenius structure theorem for affine log Calabi–Yau varieties containing a torus , Ann. Math. 198 (2023), 419–536]. To prove this result, we first give a canonical compactification of the mirror family associated with a pair $(Y,D)$ constructed by Gross, Hacking and Keel [ Mirror symmetry for log Calabi–Yau surfaces I , Publ. Math. Inst. Hautes Ètudes Sci. 122 (2015), 65168] where $Y$ is a smooth projective rational surface, $D$ is an anti-canonical cycle of rational curves, and $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from Ruddat and Siebert [ Period integrals from wall structures via tropical cycles, canonical oordinates in mirror symmetry and analyticity of toric degenerations , Publ. Math. Inst. Hautes Ètudes Sci. 132 (2020), 1–82] and use this to give a modular interpretation of the compactified mirror family.

In this paper, we present a candidate of a vector space basis for the noncommutative algebra [Formula: see text] of the quantum symplectic sphere for every [Formula: see text]. The algebra [Formula: see text] is defined as a certain subalgebra of the quantum symplectic group [Formula: see text]. A nontrivial application of the Diamond Lemma is used to construct the vector space basis and the conjecture is supported by computer experiments for [Formula: see text].

We consider homogeneous binomial ideals I=(f1,…,fn) in K[x1,…,xn], where fi=aixidi−bimi and ai≠0. When such an ideal is a complete intersection, we show that the monomials which are not divisible by xidi for i=1,…,n form a vector space basis for the corresponding quotient, and we describe the Macaulay dual generator in terms of a directed graph that we associate to I. These two properties can be seen as a natural generalization of well-known properties for monomial complete intersections. Moreover, we give a description of the radical of the resultant of I in terms of the directed graph.

Rota’s basis conjecture holds for random bases of vector spaces

We present two results on multiqubit Werner states, defined to be those states that are invariant under the collective action of any given single-qubit unitary that acts simultaneously on all the qubits. Motivated by the desire to characterize entanglement properties of Werner states, we construct a basis for the real linear vector space of Werner invariant Hermitian operators on the Hilbert space of pure states; it follows that any mixed Werner state can be written as a mixture of these basis operators with unique coefficients. Continuing a study of ‘polygon diagram’ Werner states constructed in earlier work, with a goal to connect diagrams to entanglement properties, we consider a family of multiqubit states that generalize the singlet, and show that their 2-qubit reduced density matrices are separable.

The squaring operation and the hit problem for the polynomial algebra in a type of generic degree

We consider a data-driven method, which combines Koopman operator theory with Extended Dynamic Mode Decomposition. We apply this method to the hypergeometric equation which is the Fuchsian equation with three regular singular points. The space of solutions at any of its singular points is a two-dimensional linear vector space on the field of reals when the independent variable is restricted to take values in the real axis and the unknown function is restricted to be a real-valued function of a real variable. A basis of the linear vector space of solutions is spanned by the hypergeometric function and its products with appropriate powers of the independent variable or the logarithmic function depending on the roots of the indicial equation of the hypergeometric equation. With our work, we obtain a new representation of the fundamental solutions of the hypergeometric equation and relate them to the spectral analysis of the finite approximation of the Koopman operator associated with the hypergeometric equation. We expect that the usefulness of our results will come more to the fore when we extend our study into the complex domain.

In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic set for which any suitable integral solution corresponds to a standard integral table algebra with integral multiplicities, and then try to find the smallest suitable solution. The main results of this paper show the eigenvalues of association schemes of rank 4 and nonsymmetric association schemes of rank 5 will always be cyclotomic. In the rank 5 cases, the results rely on calculations done by computer for Gröbner bases or for bases of rational vector spaces spanned by polynomials. We give several examples of feasible parameter sets for small symmetric association schemes of rank 5 that have noncyclotomic eigenvalues.

The objective of this article is to study about the representation of symbolic 3-plithogenic matrices by linear transformations between symbolic 3-plithogenic vector spaces, where it proves that every symbolic 3- plithogenic matrix can be represented uniquely by a linear transformation between symbolic 3-plithogenic vector spaces. On the other hand, this work introduces an algorithm to compute a basis of any symbolic 3-plithogenic vector space depending on the classical basis of its corresponding classical vector space.

When considered as orthogonal bases in distinct vector spaces, the unit vectors of polarization directions and the Laguerre–Gaussian modes of polarization amplitude are inseparable, constituting a so-called classical entangled light beam. Equating this classical entanglement to quantum entanglement necessary for computing purpose, we show that the parallelism featured in Shor’s factoring algorithm is equivalent to the concurrent light-path propagation of an entangled beam or pulse train. A gedanken experiment is proposed for executing the key algorithmic steps of modular exponentiation and Fourier transform on a target integer N using only classical manipulations on the amplitudes and polarization directions. The multiplicative order associated with the sought-after integer factors is identified through a four-hole diffraction interference from sources obtained from the entangled beam profile. The unique mapping from the fringe patterns to the computed order is demonstrated through simulations for the case N=15.

The decentralized and highly scalable nature of structured peer-to-peer networks, based on distributed hash tables (DHTs), makes them a great fit for facilitating the interaction and exchange of information between dynamic and geographically dispersed autonomous entities. The recent emergence of multimedia-based services and applications in the Internet of Things (IoT) has led to a noticeable shift in the type of data traffic generated by sensing devices from structured textual and numerical content to unstructured and bulky multimedia content. The wide semantic spectrum of human recognizable concepts that can be stemmed from multimedia data, e.g., video and audio, introduces a very large semantic content space. The scale of the content space poses a semantic boundary between data consumers and producers in large-scale peer-to-peer publish/subscribe systems. The exact-match query model of DHTs falls short when participants use different terms to describe the same semantic concepts. In this work, we present OpenPubSub, a peer-to-peer content-based approximate semantic publish/subscribe system. We propose a hybrid event routing model that combines rendezvous routing and gossiping over a structured peer-to-peer network. The network is built on the basis of a high-dimensional semantic vector space as opposed to conventional logical key spaces. We propose methods to partition the space, construct a semantic DHT via bootstrapping, perform approximate semantic lookup operations, and cluster nodes based on their shared interests. Results show that for an approximate event matching upper bound recall of 56.7%, rendezvous-based routing achieves up to 54% recall while decreasing the messaging overhead by 44%, whereas, the hybrid routing approach achieves up to 43.8% recall while decreasing the messaging overhead by 59%.

Linear functional analysis historically founded by Fourier and Legendre played a significant role to provide a unified vision of mathematical transformations between vector spaces. The possibility of extending this approach is explored when basis of vector spaces is built Tailored to the Problem Specificity (TPS) and not from the convenience or effectiveness of mathematical calculations. Standardized mathematical transformations, such as Fourier or polynomial transforms, could be extended toward TPS methods, on a basis, which properly encodes specific knowledge about a problem. Transition between methods is illustrated by comparing what happens in conventional Fourier transform with what happened during the development of Jewett Transform, reported in previous articles. The proper use of computational intelligence tools to perform Jewett Transform allowed complexity algorithm optimization, which encourages the search for a general TPS methodology.

This paper deals with an exterior algebra of a vector space whose base field is of positive characteristic. In this work, a minimal set of generators forming the annihilator of even neat elements of such an exterior algebra is exhibited. The annihilator of some special type of even neat elements is determined to prove the conjecture established in [3]. Moreover, a vector space basis for the annihilators under consideration is calculated.

Graded identities for Kac–Moody and Heisenberg algebras with the Cartan grading

The purpose of this research paper is to determinate the level of competence on the concept of basis of a vector space on college level students who finished the Linear Algebra course, providing complementary information to the various studies from a different perspective through the SOLO taxonomy (Structure of the Observed Learning Outcome). An instrument was designed to address the concept of basis from different perspectives and difficulty degrees; it was applied through individual video-recorded interviews. Their answers were analyzed, and it was found that they average level 3 (multistructural) of the taxonomy: the students know the basis definition from an algorithmic or methodological perspective, and they can reproduce some procedures, but they are unable to understand the basis concept.

A new way to formulate the notions of minimal basis and minimal indices is developed in this paper, based on the concept of a filtration of a vector space. The goal is to provide useful new tools for working with these important concepts, as well as to gain deeper insight into their fundamental nature. This approach also readily reveals a strong minimality property of minimal indices, from which follows a characterization of the vector polynomial bases in rational vector spaces. The effectiveness of this new formulation is further illustrated by proving several fundamental properties: the invariance of the minimal indices of a matrix polynomial under field extension, the direct sum property of minimal indices, the polynomial linear combination property, and the predictable degree property.

Abstract The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we observe that knowledge of the equivariant graded Betti numbers (in the sense of commutative algebra) of any irreducible representation in category ${\mathscr{O}}$ is equivalent to knowledge of the Kazhdan–Lusztig character of the irreducible object (we use this observation in joint work with Fishel–Manosalva). We then explore the extent to which Cherednik algebra techniques may be applied to ideals of linear subspace arrangements: we determine when the radical of the polynomial representation of the Cherednik algebra is a radical ideal and, for the cyclotomic rational Cherednik algebra, determine the socle of the polynomial representation and characterize when it is a radical ideal. The subspace arrangements that arise include various generalizations of the $k$-equals arrangement. In the case of the radical, we apply our results with Juteau together with an idea of Etingof–Gorsky–Losev to observe that the quotient is Cohen–Macaulay for positive choices of parameters. In the case of the socle (in cyclotomic type), we give an explicit vector space basis in terms of certain specializations of nonsymmetric Jack polynomials, which in particular determines its minimal generators and Hilbert series and answers a question posed by Feigin and Shramov.

Product identities in two variables x, q expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi’s triple product identity, Watson’s quintuple identity, and Hirschhorn’s septuple identity. We view these series expansions as representations in canonical bases of certain vector spaces of quasiperiodic meromorphic functions (related to sections of line and vector bundles), and find new identities for two nonuple products, an undecuple product, and several two-variable Rogers–Ramanujan type sums. Our main theorem explains a correspondence between the septuple product identity and the two original Rogers–Ramanujan identities; this amounts to an unexpected proportionality of canonical basis vectors, two of which can be viewed as two-variable analogues of fifth-order mock theta functions. We also prove a similar correspondence between an octuple product identity of Ewell and two simpler variations of the Rogers–Ramanujan identities, related to third-order mock theta functions, and conjecture other occurrences of this phenomenon. As applications, we specialize our results to obtain identities for quotients of generalized eta functions and mock theta functions.