Indefinite Function Research Articles

Kreĭn twin support vector machines for imbalanced data classification

Conventional classification assumes a balanced sample distribution among classes. However, such a premise leads to biased performance over the majority class (with the highest number of instances). The Twin Support Vector Machines (TWSVM) obtained great prominence due to their low computational burden compared to the standard SVM. Besides, traditional machine learning seeks methods whose solution depends on a convex problem or semi-positive definite similarity matrices. Yet, this kind of matrix cannot adequately represent many real-world applications. The above defines the need to use non-negative measures as an indefinite function in a Reproducing Kernel Kreĭn Space (RKKS). This paper proposes a novel approach called Kreĭn Twin Support Vector Machines (KTSVM), which appropriately incorporates indefinite kernels within a TWSVM-based gradient optimization. To code pertinent input patterns within an imbalanced data discrimination, our KTSVM employs an implicit mapping to a RKKS. Also, our approach takes advantage of the TWSVM scheme’s benefits by creating two parallel hyperplanes. This promotes the KTSVM optimization in a gradient-descent framework. Results obtained on synthetic and real-world datasets demonstrate that our solution performs better in terms of imbalanced data classification than state-of-the-art techniques.

Fixed-time periodic stabilization of discontinuous reaction–diffusion Cohen–Grossberg neural networks

This paper aims to study the fixed-time stabilization of a class of delayed discontinuous reaction–diffusion Cohen–Grossberg neural networks. Firstly, by providing some relaxed conditions containing indefinite functions and based on inequality techniques, a new fixed-time stability lemma is given, which can improve the traditional ones. Secondly, based on state-dependent switching laws, the periodic wave solution of the formulated networks is transformed into the periodic solution of ordinary differential system. By utilizing differential inclusions theory and coincidence theorem, the existence of periodic solutions is obtained. Thirdly, based on the new fixed-time stability lemma, the periodic solutions are stabilized at zero in a fixed-time, which is a new topic on reaction–diffusion networks. Moreover, the established criteria are all delay-dependent, which are less conservative than the previous delay-independent ones for ensuring the stabilization of delayed reaction–diffusion networks. Finally, two examples give numerical explanations of the proposed results and highlight the influence of delays.

Variational Approach for a Robin Problem Involving Non Standard Growth Conditions

In this manuscript, we study a Robin problem driven by the p(x)-Laplacian with two parameters. −div(|∇w|p(x)−2∇w)=λV(x)|w|q(x)−2w,x∈Q, |∇w|p(x)−2∂w∂n+θ(x)|w|p(x)−2=βV1(x)|w|r(x)−2w.x∈∂Q. Here, Q is a regular bounded domain in RN, λ,β>0, p,q are continuous functions on Q¯, ∂w∂n is the outer unit normal derivative on ∂Q, θ∈L∞(∂Q), such that essinfx∈∂Qθ(x)>0,V is an indefinite function in Ls(x)(Q) and V1 is a non-negative one in Ls1(x)(∂Q). Using variational tools, we show the existence of a non-trivial solution.

On Indefinite Quadratic Optimization over the Intersection of Balls and Linear Constraints

In this paper, we study the minimization of an indefinite quadratic function over the intersection of balls and linear inequality constraints (QOBL). Using the hyperplanes induced by the intersection of each pair of balls, we show that the optimal solution of QOBL can be found by solving several extended trust-region subproblems (e-TRS). To solve e-TRS, we use the alternating direction method of multipliers approach and a branch and bound algorithm. Numerical experiments show the efficiency of the proposed approach compared to the CVX and the extended adaptive ellipsoid-based algorithm.

The u-plane integral, mock modularity and enumerative geometry

We revisit the low-energy effective U(1) action of topologically twisted {mathcal {N}}=2 SYM theory with gauge group of rank one on a generic oriented smooth four-manifold X with nontrivial fundamental group. After including a specific new set of mathcal Q-exact operators to the known action, we express the integrand of the path integral of the low-energy U(1) theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces X=Sigma _g times mathbb {CP}^1 for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface Sigma _g, via an equivalent topological A-model on mathbb {CP}^1, we will be able to express the generating function of genus zero Gromov–Witten invariants of the moduli space of flat rank one connections over Sigma _g in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory.

Improved Multiple Lyapunov Functions of input-to-output Stability and input-to-state Stability for Switched Systems

This paper investigates the Lyapunov characterizations of input-to-output stability (IOS) and input-to-state stability (ISS) for nonlinear switched systems. The restriction on the negativity of derivatives of Lyapunov functions is relaxed; additionally, subsystems are not restricted to be ISS/IOS throughout the state space. These problems are overcome by a newly proposed method, which extends multiple Lyapunov functions (MLFs) to an indefinite form. Indefinite multiple Lyapunov functions (iMLFs) are proposed on the basis of an inequality to estimate the upper boundary of the system state. Moreover, we apply the iMLFs method to verify the extended notions of IOS in a switched system. More relaxed sufficient conditions for ISS and IOS are proposed, a simulation experiment is carried out on a numerical example of the system, which demonstrates their effectiveness and superiority.

Extension of Interaction Aggregation Operators for the Analysis of Cryptocurrency Market Under q-Rung Orthopair Fuzzy Hypersoft Set

One of the substantial innovations achieved through digitalization is cryptocurrencies, also known as simulated or digital currencies, which have been deliberated in the modern era as a new platform particularly suitable for financiers. Several cryptocurrencies, such as Bitcoin, Ethereum, Binance Coin, and Tether, do not trust a dominant expert. The classification and conduction of insecurity and the confirmation of digital currencies complicate decision-making. q-rung orthopair fuzzy hypersoft sets are an emerging arena of research intended to report the confidential restrictions of q-rung orthopair fuzzy soft sets on multiparameter indefinite functions. Such a function maps a tuple of sub-parameters to a power set of the universe. It emphasizes allocating attributes to their corresponding sub-attribute values in disjoint sets. These structures sort it an innovative systematic tool for addressing the obstacles of hesitancy. The q-rung orthopair fuzzy hypersoft set (q-ROFHSS) expertly compacts with tentative and ambagious facts equated to the existing q- rung orthopair fuzzy soft set and Pythagorean fuzzy hypersoft set (PFHSS). It is the most compelling mode for enlarging imprecise data in decision-making (DM). This investigation’s ultimate impartiality is presenting interactional algebraic operational laws for q-ROFHSS. Furthermore, some interaction aggregation operators (AOs) have been anticipated via our proposed operational laws, such as q-rung orthopair fuzzy hypersoft interactive weighted average (q-ROFHSIWA) and q-rung orthopair fuzzy hypersoft interactive weighted geometric (q-ROFHSIWG) operators with their essential properties. In reality, a mathematical illustration of DM obstacles is pondered to substantiate the proven technique’s dominance. Based on the projected interaction AOs, robust multi-criteria group decision-making (MCGDM) design has been offered, which carries the most practical consequences associated with predominant MCGDM methods. The significance spectacle is that the intentional methodology is more operative and steady in bearing weird facts based on q-ROFHSS.

New Fixed-Time Stability Theorems for Delayed Fractional-Order Systems and Applications

This paper examines the problem of improved fixed-time stability for generalized delayed fractional-order systems (FOSs). As a first step, some stability conditions are presented in two theorems to verify fixed-time stability (F-TS) of FOSs by using Lyapunov stability theory and indefinite Lyapunov functionals, where the fractional-order derivative of the indefinite functions may not exist. Furthermore, the corresponding estimated settling time of FOSs is also provided. Second, the results are extended to study fractional-order neural networks (FONNs) with time-delays in fixed-time synchronization. Using the Dirac delta functions, we propose an explicit saturated impulsive controller to synchronize the master and slave systems. Moreover, by constructing suitable indefinite Lyapunov-Krasovskii functions (LKF), we derive algebraic conditions to guarantee the fixed-time synchronization of the addressed FONNs. The simulation results demonstrate the feasibility and efficacy of the proposed method.

Multiplicity and concentration of solutions for a fractional Schrödinger–Poisson system with sign-changing potential

This paper is concerned with the following fractional Schrödinger–Poisson system: where is a parameter, , , , is allowed to be sign-changing and f is an indefinite function. We require that with having a potential well Ω whose depth is controlled by λ and for all . Under some suitable assumptions on and , we prove that the above system has at least two nontrivial solutions. Moreover, the phenomenon of concentration of the solutions is also explored.

Non-Negativity of a Quadratic form with Applications to Panel Data Estimation, Forecasting and Optimization

For a symmetric matrix B, we determine the class of Q such that Q t BQ is non-negative definite and apply it to panel data estimation and forecasting: the Hausman test for testing the endogeneity of the random effects in panel data models. We show that the test can be performed if the estimated error variances in the fixed and random effects models satisfy a specific inequality. If it fails, we discuss the restrictions under which the test can be performed. We show that estimators satisfying the inequality exist. Furthermore, we discuss an application to a constrained quadratic minimization problem with an indefinite objective function.

The Interrogative-Indefinite Puzzle in the Context of Biblical Hebrew

The biblical corpus features a number of verses in which interrogative pronouns appear in non-interrogative contexts. The same phenomenon is observed in many other languages and gives rise to the question known in the linguistic literature as “the interrogative-indefinite puzzle,” namely, what is the natural connection between the interrogative and indefinite functions. This paper seeks to explore how this question should be examined in the context of the Biblical Hebrew data. It will be argued that a consideration of typological observations can yield important insights into this question. Subsequently, it proposes a formal semantic analysis of the indefinite pronouns in question and shows how the proposed approach can help explain their distribution.

Temperature Factor Effect on the Disturbance Propagation in Hypersonic Flow past a Yawed Flat Plate

The flow in the three-dimensional laminar boundary layer on a yawed flat plate of finite length is studied in the regime of strong viscous-inviscid interaction with a hypersonic flow. In the vicinity of the leading edge the flow functions are expanded in series under the assumption that the base pressure dependent on the transverse coordinate is given at the trailing edge of the plate. It is established that the expansions obtained include an indefinite function and its derivative with respect to the transverse coordinate. The corresponding boundary value problems are formulated and numerically solved, the eigenvalues are found, and it is shown that the exponent in the third term of the expansion differs from that in the second term only by one. The plate surface temperature effect on the flow parameters and the initiation of three-dimensional disturbances is investigated.

On the stressed state of the joint upon bending beam with an elastic coating

Joint deformation of two elastic layers is considered, one of them being a coating applied to the surface of another thicker layer. It is assumed that slippage is absent in the plane of layer contact. External load is normal to the coating surface and unchanging in width. In the absence of lateral loading, the junction can be considered a two-layer beam and described by the equations of the plane elasticity using the asymptotic method. The required functions, normal and tangential stresses, and also the components of the displacement of an arbitrary point of the beam and coating are expanded in power series of the small parameter (accepted as a half thickness of the corresponding layer). In contrast to the known asymptotic expansions, an alternative asymptotic method is proposed in which all the required functions are asymptotically equivalent, i.e., expanded in asymptotic series of the same structure. When all positive powers of the small parameter are present in the series, the asymptotic algorithm leads to the appearance of two independent recurrent systems of linear equations, which greatly simplifies their solution. In each approach for one elastic layer the algorithm generates five indefinite functions of the longitudinal coordinate. Due to a rapid convergence of the asymptotic series, the first asymptotic approximation is used to construct the mathematical model of the deformation of a beam-to-coating coupling. Ten indefinite functions of the coordinate x provide fulfilling of eight conditions of continuity of the stress-strain state in the linkage depth, two functions remaining indefinite. Equilibrium equations (a system of two linear differential equations of the sixth order) are derived for them proceeding from the principle of minimum potential energy of deformation and corresponding boundary conditions (that can be realized both in static and kinematic variants). Numerical results are obtained in static approximation for two types of surface loading — steady and sinusoidally varying — along the length of the coating. An emphasis is made on the necessity of allowing for tangential stresses when studying the stress state of the coating as they can be rather significant upon bending.

Generation of Three-Dimensional Disturbances in a Boundary Layer on Strong Interaction with a Hypersonic Flow

Laminar boundary layer flow over an infinite-span, finite-length flat plate is investigated in the regime of strong interaction with a hypersonic gas flow. Under the assumption that an additional condition dependent on the transverse coordinate can be imposed on the trailing edge of the plate the flow functions are expanded in power series in the vicinity of the leading edge. It is shown that these expansions include an indefinite function dependent on the transverse coordinate. The corresponding boundary value problems are formulated and solved and the eigenvalues are determined. It is established that in this case the two-dimensional boundary layer can rearrange itself into a three-dimensional boundary layer.

Donaldson-Witten theory and indefinite theta functions

We consider partition functions with insertions of surface operators of topologically twisted mathcal{N}=2 , SU(2) supersymmetric Yang-Mills theory, or Donaldson-Witten theory for short, on a four-manifold. If the metric of the compact four-manifold has positive scalar curvature, Moore and Witten have shown that the partition function is completely determined by the integral over the Coulomb branch parameter a, while more generally the Coulomb branch integral captures the wall-crossing behavior of both Donaldson polynomials and Seiberg-Witten invariants. We show that after addition of a overline{mathcal{Q}} -exact surface operator to the Moore-Witten integrand, the integrand can be written as a total derivative to the anti-holomorphic coordinate ā using Zwegers’ indefinite theta functions. In this way, we reproduce Göttsche’s expressions for Donaldson invariants of rational surfaces in terms of indefinite theta functions for any choice of metric.

Minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint

In this paper, we consider the problem of minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint. A key difficulty with this problem is its nonconvexity. Using Lagrange duality, we show that under a mild assumption, this problem can be solved by solving a linearly constrained convex univariate minimization problem. Finally, the superior efficiency of the new approach compared to the known semidefinite relaxation and a known approach from the literature is demonstrated by solving several randomly generated test problems.

On the existence results for a class of singular elliptic system involving indefinite weight functions and asymptotically linear growth forcing term

In this work, we study the existence of positive solutions to the singular system$$\left\{\begin{array}{ll}-\Delta_{p}u = \lambda a(x)f(v)-u^{-\alpha} & \textrm{ in }\Omega,\\-\Delta_{p}v = \lambda b(x)g(u)-v^{-\alpha} & \textrm{ in }\Omega,\\u = v= 0 & \textrm{ on }\partial \Omega,\end{array}\right.$$where $\lambda $ is positive parameter, $\Delta_{p}u=\textrm{div}(|\nabla u|^{p-2} \nabla u)$, $p>1$, $ \Omega \subset R^{n} $ some for $ n >1 $, is a bounded domain with smooth boundary $\partial \Omega $ , $ 0<\alpha< 1 $, and $f,g: [0,\infty] \to\R$ are continuous, nondecreasing functions which are asymptotically $ p $-linear at $\infty$. We prove the existence of a positive solution for a certain range of $\lambda$ using the method of sub-supersolutions.

PARTIAL INDEFINITE THETA IDENTITIES

Using a result of Warnaar, we prove a number of single- and multi-sum identities in the spirit of Ramanujan’s partial theta identities, but with partial indefinite binary theta functions in the role of partial theta functions. We also calculate the corresponding residual identities and use a result of Ji and Zhao to recast our identities in terms of indefinite ternary theta functions.

On the complex fractional quadratic optimization with a quadratic constraint

In this paper, we study the problem of minimizing the ratio of two complex indefinite quadratic functions subject to a strictly convex quadratic constraint. First, using the known method due to Dinkelbach, we reformulate the fractional problem as a univariate equation. To find the root of the univariate equation, the generalized Newton method is utilized that requires solving a nonconvex quadratic optimization problem at each iteration. To solve these nonconvex quadratic problems, we present an efficient algorithm by a diagonalization scheme that requires solving a univariate minimization problem at each iteration. Moreover, for the homogeneous case, it requires solving a simple linear optimization problem. Our preliminary numerical experiments on several randomly generated test problems show that, the new approach is much faster in finding the global optimal solution than the known semidefinite relaxation approach, especially when solving large scale problems.

A Proposed Technique for Solving Quasi-Concave Quadratic Programming Problems with Bounded Variables

In this paper, a new method is proposed for finding an optimal solution to a Quasi-Concave Quadratic Programming Problem with Bounded Variables in which the objective function involves the product of two indefinite factorized linear functions and constraints functions are in the form of linear inequalities. The proposed method is mainly based upon the primal dual simplex method. The Linear Programming with Bounded Variables (LPBV) algorithm is extended to solve quasi-concave Quadratic Programming with Bounded Variables (QPBV). For developing this method, we use programming language MATHEMATICA. We also illustrate numerical examples to demonstrate our method.Dhaka Univ. J. Sci. 63(2):111-117, 2015 (July)