• Title, Summary, Keyword: explicit formula

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Modeling strength of high-performance concrete using genetic operation trees with pruning techniques

  • Peng, Chien-Hua;Yeh, I-Cheng;Lien, Li-Chuan
    • Computers and Concrete
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    • v.6 no.3
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    • pp.203-223
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    • 2009
  • Regression analysis (RA) can establish an explicit formula to predict the strength of High-Performance Concrete (HPC); however, the accuracy of the formula is poor. Back-Propagation Networks (BPNs) can establish a highly accurate model to predict the strength of HPC, but cannot generate an explicit formula. Genetic Operation Trees (GOTs) can establish an explicit formula to predict the strength of HPC that achieves a level of accuracy in between the two aforementioned approaches. Although GOT can produce an explicit formula but the formula is often too complicated so that unable to explain the substantial meaning of the formula. This study developed a Backward Pruning Technique (BPT) to simplify the complexity of GOT formula by replacing each variable of the tip node of operation tree with the median of the variable in the training dataset belonging to the node, and then pruning the node with the most accurate test dataset. Such pruning reduces formula complexity while maintaining the accuracy. 404 experimental datasets were used to compare accuracy and complexity of three model building techniques, RA, BPN and GOT. Results show that the pruned GOT can generate simple and accurate formula for predicting the strength of HPC.

EXPLICIT MINIMUM POLYNOMIAL, EIGENVECTOR AND INVERSE FORMULA OF DOUBLY LESLIE MATRIX

  • WANICHARPICHAT, WIWAT
    • Journal of applied mathematics & informatics
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    • v.33 no.3_4
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    • pp.247-260
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    • 2015
  • The special form of Schur complement is extended to have a Schur's formula to obtains the explicit formula of determinant, inverse, and eigenvector formula of the doubly Leslie matrix which is the generalized forms of the Leslie matrix. It is also a generalized form of the doubly companion matrix, and the companion matrix, respectively. The doubly Leslie matrix is a nonderogatory matrix.

A DEFINITE INTEGRAL FORMULA

  • Choi, Junesang
    • East Asian mathematical journal
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    • v.29 no.5
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    • pp.545-550
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    • 2013
  • A remarkably large number of integral formulas have been investigated and developed. Certain large number of integral formulas are expressed as derivatives of some known functions. Here we choose to recall such a formula to present explicit expressions in terms of Gamma function, Psi function and Polygamma functions. Some simple interesting special cases of our main formulas are also considered. It is also pointed out that the same argument can establish explicit integral formulas for other those expressed in terms of derivatives of some known functions.

A q-ANALOGUE OF QI FORMULA FOR r-DOWLING NUMBERS

  • Cillar, Joy Antonette D.;Corcino, Roberto B.
    • Communications of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.21-41
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    • 2020
  • In this paper, we establish an explicit formula for r-Dowling numbers in terms of r-Whitney Lah and r-Whitney numbers of the second kind. This is a generalization of the Qi formula for Bell numbers in terms of Lah and Stirling numbers of the second kind. Moreover, we define the q, r-Dowling numbers, q, r-Whitney Lah numbers and q, r-Whitney numbers of the first kind and obtain several fundamental properties of these numbers such as orthogonality and inverse relations, recurrence relations, and generating functions. Hence, we derive an analogous Qi formula for q, r-Dowling numbers expressed in terms of q, r-Whitney Lah numbers and q, r-Whitney numbers of the second kind.

EXPLICIT FORMULA FOR COEFFICIENTS OF TODD SERIES OF LATTICE CONES

  • Chae, Hi-Joon;Jun, Byungheup;Lee, Jungyun
    • Communications of the Korean Mathematical Society
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    • v.30 no.2
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    • pp.73-79
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    • 2015
  • Todd series are associated to maximal non-degenerate lattice cones. The coefficients of Todd series of a particular class of lattice cones are closely related to generalized Dedekind sums of higher dimension. We generalize this construction and obtain an explicit formula for coefficients of the Todd series. It turns out that every maximal non-degenerate lattice cone, hence the associated Todd series can be obtained in this way.

SHIODA-TATE FORMULA FOR AN ABELIAN FIBERED VARIETY AND APPLICATIONS

  • Oguiso, Keiji
    • Journal of the Korean Mathematical Society
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    • v.46 no.2
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    • pp.237-248
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    • 2009
  • We give an explicit formula for the Mordell-Weil rank of an abelian fibered variety and some of its applications for an abelian fibered $hyperk{\ddot{a}}hler$ manifold. As a byproduct, we also give an explicit example of an abelian fibered variety in which the Picard number of the generic fiber in the sense of scheme is different from the Picard number of generic closed fibers.

OME PROPERTIES OF THE BERNOULLI NUMBERS OF THE SECOND KIND AND THEIR GENERATING FUNCTION

  • Qi, Feng;Zhao, Jiao-Lian
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1909-1920
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    • 2018
  • In the paper, the authors find a common solution to three series of differential equations related to the generating function of the Bernoulli numbers of the second kind and present a recurrence relation, an explicit formula in terms of the Stirling numbers of the first kind, and a determinantal expression for the Bernoulli numbers of the second kind.

ON p, q-DIFFERENCE OPERATOR

  • Corcino, Roberto B.;Montero, Charles B.
    • Journal of the Korean Mathematical Society
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    • v.49 no.3
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    • pp.537-547
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    • 2012
  • In this paper, we define a $p$, $q$-difference operator and obtain an explicit formula which is used to express the $p$, $q$-analogue of the unified generalization of Stirling numbers and its exponential generating function in terms of the $p$, $q$-difference operator. Explicit formulas for the non-central $q$-Stirling numbers of the second kind and non-central $q$-Lah numbers are derived using the new $q$-analogue of Newton's interpolation formula. Moreover, a $p$, $q$-analogue of Newton's interpolation formula is established.