An adaptive Dendrite-HDMR metamodeling technique for high dimensional problems

High dimensional model representation (HDMR), decomposing the high-dimensional problem into summands of different order component terms, has been widely researched to work out the dilemma of “curse-of-dimensionality” when using surrogate techniques to approximate high-dimensional problems in engineering design. However, the available one-metamodel-based HDMRs usually encounter the predicament of prediction uncertainty, while current multi-metamodels-based HDMRs cannot provide simple explicit expressions for black-box problems, and have high computational complexity in terms of constructing the model by the explored points and predicting the responses of unobserved locations. Therefore, aimed at such problems, a new stand-alone HDMR metamodeling technique, termed as Dendrite-HDMR, is proposed in this study based on the hierarchical Cut-HDMR and the white-box machine learning algorithm, Dendrite Net. The proposed Dendrite-HDMR not only provides succinct and explicit expressions in the form of Taylor expansion, but also has relatively higher accuracy and stronger stability for most mathematical functions than other classical HDMRs with the assistance of the proposed adaptive sampling strategy, named KKMC, in which k-means clustering algorithm, k-Nearest Neighbor classification algorithm and the maximum curvature information of the provided expression are utilized to sample new points to refine the model. Finally, the Dendrite-HDMR technique is applied to solve the design optimization problem of the solid launch vehicle propulsion system with the purpose of improving the impulse-weight ratio, which represents the design level of the propulsion system.

On Analysis of Banhatti Indices for Hyaluronic Acid Curcumin and Hydroxychloroquine

Topological indices are numerical numbers assigned to the graph/structure and are useful to predict certain physical/chemical properties. In this paper, we give explicit expressions of novel Banhatti indices, namely, first K Banhatti index B 1 G , second K Banhatti index B 2 G , first K hyper-Banhatti index HB 1 G , second K hyper-Banhatti index HB 2 G , and K Banhatti harmonic index H b G for hyaluronic acid curcumin and hydroxychloroquine. The multiplicative version of these indices is also computed for these structures.

Construction of Type 2 Poly-Changhee Polynomials and Its Applications

In this paper, we introduce type 2 poly-Changhee polynomials by using the polyexponential function. We derive some explicit expressions and identities for these polynomials, and we also prove some relationships between poly-Changhee polynomials and Stirling numbers of the first and second kind. Also, we introduce the unipoly-Changhee polynomials by employing unipoly function and give multifarious properties. Furthermore, we provide a correlation between the unipoly-Changhee polynomials and the classical Changhee polynomials.

Multiparametric Rational Solutions of Order N to the KPI Equation and the Explicit Case of Order 3

We present multiparametric rational solutions to the Kadomtsev-Petviashvili equation (KPI). These solutions of order N depend on 2N − 2 real parameters. Explicit expressions of the solutions at order 3 are given. They can be expressed as a quotient of a polynomial of degree 2N(N +1)−2 in x, y and t by a polynomial of degree 2N(N +1) in x, y and t, depending on 2N − 2 real parameters. We study the patterns of their modulus in the (x,y) plane for different values of time t and parameters.

Classification of Metaplectic Fusion Categories

In this paper, we study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for so(2p+1)2. These categories describe non-abelian anyons dubbed ‘metaplectic anyons’. We obtain explicit expressions for all the F- and R-symbols. Based on these, we conjecture a classification for their monoidal equivalence classes from an analysis of their gauge invariants and define a function which gives us the number of classes.

Bifurcations of Nonlinear Waves in the Generalized KdV–mKdV-Like Equation

Using bifurcation analytic method of dynamical systems, we investigate the nonlinear waves and their bifurcations of the generalized KdV–mKdV-like equation. We obtain the following results : (i) Three types of new explicit expressions of nonlinear waves are obtained. They are trigonometric expressions, exp-function expressions, and hyperbolic expressions. (ii) Under different parameteric conditions, these expressions represent different waves, such as solitary waves, kink waves, 1-blow-up waves, 2-blow-up waves, smooth periodic waves and periodic blow-up waves. (iii) Two kinds of new interesting bifurcation phenomena are revealed. The first phenomenon is that the single-sided periodic blow-up waves can bifurcate from double-sided periodic blow-up waves. The second phenomenon is that the double-sided 1-blow-up waves can bifurcate from 2-blow-up waves. Furthermore, we show that the new expressions encompass many existing results.

Effect of quantum fluctuations on soliton regimes in microlasers

We present a theoretical investigation of effect of quantum fluctuations on laser solitons. Derivation of the stochastic equation, linearized with respect to quantum perturbations is carried out and the solutions are found. Explicit expressions are obtained for the time dependence of the soliton coordinates and momentum dispersion (variance) for perturbations averaged over the reservoir. It is shown that the dispersion of the soliton momentum becomes constant. It is shown that the dispersion of quantum perturbations tends to infinity near the Andronov-Hopf bifurcation threshold. The magnitude of quantum perturbations near the threshold of the appearance of hysteresis is estimated. It is shown that quantum perturbations do not significantly noise the soliton profile even with a very low intensity tending to zero. The number of photons in such solitons without supporting radiation, can reach unity.

Optimal Portfolio Choice with Estimation Risk: No Risk-Free Asset Case

We propose an optimal combining strategy to mitigate estimation risk for the popular mean-variance portfolio choice problem in the case without a risk-free asset. We find that our strategy performs well in general, and it can be applied to known estimated rules and the resulting new rules outperform the original ones. We further obtain the exact distribution of the out-of-sample returns and explicit expressions of the expected out-of-sample utilities of the combining strategy, providing not only a fast and accurate way of evaluating the performance, but also analytical insights into the portfolio construction. This paper was accepted by Tyler Shumway, finance.

The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks

The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let L n 8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of L n 8,4 , we get the corresponding Möbius graph M Q n 8,4 . In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of M Q n 8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒ A and ℒ S of order 4 n . Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of M Q n 8,4 .