Non-negative Unit Research Articles

Geodesic Basis Function Neural Network.

In the learning of existing radial basis function neural networks-based methods, it is difficult to propagate errors back. This leads to an inconsistency between the learning and recognition task. This article proposes a geodesic basis function neural network with subclass extension learning (GBFNN-ScE). The geodesic basis function (GBF), which is defined here for the first time, uses the geodetic distance in the manifold as a measure to obtain the response of the sample with respect to the local center. To learn network parameters by back-propagating errors for the purpose of correct classification, a specific GBF based on a pruned gamma encoding cosine function is constructed. This function has a concise and explicit expression on the hyperspherical manifold, which is conducive to the realization of error back propagation. In the preprocessing layer, a sample unitization method with no loss of information, nonnegative unit hyperspherical crown (NUHC) mapping, is proposed. The sample can be mapped to the support set of the GBF. To alleviate the problem that one-hot encoding is not effective enough in the differential expression of data labels within a class, a subclass extension (ScE) learning strategy is proposed. The ScE learning strategy can help the learned network be more robust. For the working of GBFNN-ScE, the original sample is projected onto the support set of GBF through the NUHC mapping. Then the mapped samples are sent to the nonlinear computation units composed of GBFs in the hidden layer. Finally, the response obtained in the hidden layer is weighted by the learned weight to obtain the network output. This article theoretically proves that the separability of the data with ScE learning is stronger. The experimental results show that the proposed GBFNN-ScE has a better performance in recognition tasks than existing methods. The ablation experiments show that the ideas of the GBFNN-ScE contribute to the algorithm performance.

A Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸r

An integral quadratic form q : ℤn → ℤ is usually identified with a bilinear form 𝕓∨q:ℤn×ℤn→ℤ satisfying q(x)=𝕓∨q(x,x) for any vector x in ℤn, and such that its Gram matrix with respect to the canonical basis of ℤn is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type 𝔸r (r ≥ 1) and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49–82, 2021] and [Fundamenta Informaticae 185(3):221–246, 2022].

Coxeter Invariants for Non-negative Unit Forms of Dynkin Type 𝔸r

Two integral quadratic unit forms are called strongly Gram congruent if their upper triangular Gram matrices are ℤ-congruent. The paper gives a combinatorial strong Gram invariant for those unit forms that are non-negative of Dynkin type 𝔸r (for r ≥ 1), within the framework introduced in [Fundamenta Informaticae 184(1):49–82, 2021], and uses it to determine all corresponding Coxeter polynomials and (reduced) Coxeter numbers.

A Graph Theoretical Framework for the Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸n

In the context of signed line graphs, this article introduces a modified inflation technique to study strong Gram congruence of non-negative (integral quadratic) unit forms, and uses it to show that weak and strong Gram congruence coincide among positive unit forms of Dynkin type 𝔸n. The concept of inverse of a quiver is also introduced, and is used to obtain and analyze the Coxeter matrix of non-negative unit forms of Dynkin type 𝔸n. With these tools, connected principal unit forms of Dynkin type 𝔸n are also classified up to strong congruence.

Constraint-based Recommender System for Commodity Realization

In this paper, we suggest a novel recommender system where a set of appropriate propositions is formed by measuring how user query features are close to space of all possible propositions. The system is for e-traders selling commodities. A commodity has hierarchical-structure properties which are mapped to the respective numerical scales. The scales are normalized so that a query from a potential customer and any possible proposition from the e-trader is a multidimensional point of a nonnegative unit hypercube put on the coordinate origin. The user can weight levels. The distance between the query and propositions are measured by the respective metric in the Euclidean arithmetic space. The best proposition is defined by the shortest distance. Top N propositions are defined by N shortest distances. The system does not depend on any user experience, nor on the e-trader tendency to impose one’s preferences on the customer.

The eigenvectors to the p-spectral radius of general hypergraphs

Let $$\mathcal {A}$$ be the adjacency tensor of a general hypergraph H. For any real number $$p\ge 1$$ , the p-spectral radius $$\lambda ^{(p)}(H)$$ of H is defined as $$\lambda ^{(p)}(H)=\max \{x^{\mathrm {T}}(\mathcal {A}x)\,|\,x\in {\mathbb {R}}^n, \Vert x\Vert _p=1\}$$ . In this paper we present some bounds on entries of the nonnegative unit eigenvector corresponding to the p-spectral radius of H, which generalize the relevant results of uniform hypergraphs/graphs in the literature.

A polynomial recognition of unit forms using graph-based strategies

The units forms are algebraic expressions that have important role in representation theory of algebras. We identified that existing algorithms have exponential time complexity for weakly nonnegative and weakly positive types. In this paper we introduce a polynomial algorithm for the recognition of weakly nonnegative unit forms. The related algorithm identifies hypercritical restrictions in a given unit form, testing every subgraph of 9 vertices of the unit form associated graph. By adding Depth First Search approach, a similar strategy could be used in the recognition of weakly positive unit forms. We also present the most popular methods to decide whether or not a unit form is weakly nonnegative or weakly positive, we analyze their time complexity and we compare the results with our algorithms.

A Polynomial Recognition of Unit Forms

In this paper we introduce a polynomial algorithm for the recognition of weakly nonnegative unit forms. The algorithm identify hypercritical restrictions testing every 9-point subset of the quadratic form associated graph. With Depth First Search strategy, we use a similar approach for the weakly positive recognition.

$h$-Polynomials via Reduced Forms

The flow polytope $F_{\widetilde{G}}$ is the set of nonnegative unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope $F_{\widetilde{G}}$ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial $\prod_{(i,j)\in E(G)}x_{ij}$. We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but also regular flag triangulations of flow polytopes. We prove that reduced forms in the subdivision algebra are generalizations of $h$-polynomials of the triangulations of flow polytopes. We deduce several corollaries of the above results, most notably proving certain cases of a conjecture of Kirillov about the nonnegativity of reduced forms in the noncommutative quasi-classical Yang-Baxter algebra.

The extended affine Lie algebra associated with a connected non-negative unit form

Given a connected non-negative unit form we construct an extended affine Lie algebra by giving a Chevalley basis for it. We also obtain this algebra as a quotient of an algebra defined by means of generalized Serre relations by M. Barot, D. Kussin and H. Lenzing. This is done in an analogous way to the construction of the simply-laced affine Kac–Moody algebras. Thus, we obtain a family of extended affine Lie algebras of simply-laced Dynkin type and arbitrary nullity. Furthermore, there is a one-to-one correspondence between these Lie algebras and the equivalence classes of connected non-negative unit forms.

Hölder type matrix inequalities of Pate, Blakley, and Roy extended to the inner product of Frobenius

Suppose m, n, and k are positive integers, and let 〈·,·〉 be the standard inner product on the spaces Rp, p>0. Recently Pate has shown that if D is an m×n non-negative real matrix, and u and v are non-negative unit vectors in Rn and Rm, respectively, then〈(DDt)kDu,v〉⩾〈Du,v〉2k+1,with equality if and only if 〈(DDt)kDu,v〉=0, or there exists α>0 such that Du=αv and Dtv=αu. This extends to non-symmetric non-square matrices a 1965 result of Blakley and Roy, and resolves a special case of a graph theoretic inequality conjectured by Sidorenko. We generalize the above, obtaining pure matrix inequalities involving the Frobenius inner product, 〈·,·〉f. In particular, we show that if k is a positive integer, and D, X, and Y are non-negative matrices that are m×n,n×p, and m×p, respectively, then∑i=1p‖xi‖‖yi‖2k〈D(DtD)kX,Y〉f⩾(〈DX,Y〉f)2k+1,where X has columns x1,x2,…,xp, Y has columns y1,y2,…,yp, and ‖·‖ is the 2-norm. Necessary and sufficient conditions for equality are also given.

Single item lot-sizing with non-decreasing capacities

We consider the single item lot-sizing problem with capacities that are non-decreasing over time. When the cost function is (i) non-speculative or Wagner–Whitin (for instance, constant unit production costs and non-negative unit holding costs) and (ii) the production set-up costs are non-increasing over time, it is known that the minimum cost lot-sizing problem is polynomially solvable using dynamic programming. When the capacities are non-decreasing, we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot-sizing problem to optimality when the objective function satisfies (i) and (ii). The formulation is based on mixing set relaxations and reduces to the (known) convex hull of solutions when the capacities are constant over time. We illustrate the use and potential effectiveness of this improved LP formulation on a few test instances, including instances with and without Wagner–Whitin costs, and with both non-decreasing and arbitrary capacities over time.

Single Item Lot-Sizing with Non-Decreasing Capacities

We consider the single item lot-sizing problem with capacities that are non-decreasing over time. When the cost function is i) non-speculative or Wagner-Whitin (for instance, constant unit production costs and non-negative unit holding costs), and ii) the production set-up costs are non-increasing over time, it is known that the minimum cost lotsizing problem is polynomially solvable using dynamic programming. When the capacities are non-decreasing, we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot-sizing problem to optimality when the objective function satisfies i) and ii). The formulation is based on mixing set relaxations and reduces to the (known) convex hull of solutions when the capacities are constant over time. We illustrate the use and effectiveness of this improved LP formulation on a few test instances, including instances with and without Wagner-Whitin costs, and with both non-decreasing and arbitrary capacities over time.

The Lie algebra associated to a unit form

To any unit form q ( x ) = ∑ i = 1 n x i 2 + ∑ i < j q i j x i x j , q i j ∈ Z , we associate a Lie algebra G ˜ ( q ) —an intersection matrix Lie algebra in the terminology of Slodowy—by means of generalized Serre relations. For a nonnegative unit form the isomorphism type of G ˜ ( q ) is determined by the equivalence class of q. Moreover for q nonnegative and connected with radical of rank zero or one respectively, the algebras G ˜ ( q ) turn out to be exactly the simply-laced Lie algebras which are finite-dimensional simple or affine Kac–Moody, respectively. In case q is connected, nonnegative of corank two and not of Dynkin type A n , the algebra G ( q ) is elliptic.

Root-induced integral quadratic forms

Given an integral quadratic unit form q : Z n → Z and a finite tuple of q-roots r = ( r j ) j∈ J the induced q-root form q r is considered as in [P. Gabriel, A.V. Roiter, Representations of finite dimensional algebras, in: A.I. Kostrikin, I.V. Shafarevich (Eds.), Algebra VIII, Encyclopaedia of the Mathematical Sciences, vol. 73, 1992, Springer (Chapter 6)]. We show that two non-negative unit forms are of the same Dynkin type precisely when they are root-induced one from the other. Moreover, there are only finitely many non-negative unit forms without double edges of a given Dynkin type. Root-induction yields an interesting partial order on the Dynkin types, which is studied in the paper.

Coordinates of maximal roots of weakly non-negative unit forms

Coordinates of maximal roots of weakly non-negative unit forms

On Weakly Non-Negative Unit forms and Tame Algebras

In this paper we consider integral quadratic forms q: IP —>Z of the form q(x) = 2i x] + ^i<jqijX,Xj and call them unit forms. Such a unit form q is said to be weakly non-negative if q(x)^Q whenever Jt2*0, that is, whenever x^O for i = 1,..., n. Our main purpose is to formulate a criterion for weak non-negativity of unit forms. This question arises in various areas of representation theory in connection with the characterization of tame representation type: in the representation theory of quivers [17, 6], partially ordered sets [18, 19], certain classes of bimodules [8, 23, 1.3] and certain classes of algebras [20, 14, 24, 16]. For each of these structures Q, there is a unit form qn, naturally attached to Q, such that the weak non-negativity of qn is an equivalent (or at least a necessary) condition for Q to be of tame representation type. A class of algebras for which tame representation type is equivalent to the weak non-negativity of the associated unit form is described in the last section of this paper. A general criterion for weak non-negativity using determinants was given by Zeldich [28] (see also [21]), another criterion was obtained by de la Pena [21]. Our approach is similar to that in [11], where we gave a characterization of weakly positive unit forms based on the classification of the critical forms. Let us recall some notions. Given a unit form q: IP-*TL and a non-empty subset / = {ix < ... < im) of {1,..., n), we denote by qt the quadratic form qd, where d, Z —*• IT maps the /cth natural basis vector ek onto eik. Such a unit form q, is called the restriction of q to /. We call a unit form q on IP a hypercritical form if q, is weakly non-negative for every proper subset /<={1,..., n), though q itself is not. In these terms, a unit form q is weakly non-negative if and only if q admits no hypercritical restriction. So we have to classify the hypercritical forms. In § 1 we show that they can be obtained by certain elementary transformations from the unit forms naturally attached to the hyperbolic graphs of Fig. 1 on p. 50. In particular, there are only finitely many hypercritical forms. However, their number is large. By a simple process we reduce the problem to the classification of the stretched hypercritical forms. In § 2 we show that, except for a few of them, these are of the shape q(x) = q'(x') + xn -xn-iXn where q' = q{\,...,n-D is critical. (In a similar way, the unit forms attached to the hyperbolic graphs Am, Dm, Em correspond to those

Fluid limits and diffusion approximations for networks of multi-server queues in heavy traffic

A Functional Strong-Law-of-Large-Numbers (known as fluid limit or fluid approximation) and a Functional Central Limit Theorem (known as diffusion approximation) are proved for both open and closed networks of multi-server queues in heavy traffic. The fluid limit is a reflected piecewise linear deterministic process, and the diffusion limit is a reflected Brownian motion; both limiting processes are on the nonnegative orthant for open networks and on the nonnegative unit simplex for closed networks. The results generalize the existing ones for networks of single-server queues.

Some Necessary Conditions for a Non-Negative Unit Impulse Response and for a Positive Real Immittance Function

The fact that the unit impulse response of a network is non-negative imposes certain conditions on the real part of the transfer or immittance function. Such conditions are derived in this paper. In addition, when the real part of the immittance function is non-negative, necessary conditions are imposed upon the impulse response. Such conditions are also presented here. These conditions are in a graphical form, which can be utilized without the necessity of lengthy calculations.