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- Bernoulli number (3)
- Bernoulli polynomial (3)
- Generating function (3)
- Power series (3)
- Catalan number (2)
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- Delta operator (2)
- Euler number (2)
- Euler polynomial (2)
- Eulerian fraction (2)
- Eulerian polynomial (2)
- Riordan arrays (2)
- Sheffer-type polynomial sequence (2)
- Stirling numbers (2)
- Symbolic summation operator (2)
- $k$-Gamma functions (1)
- 39A70 (1)
- 41A15 (1)
- 41A30 (1)
- 42C40 (1)
- 65D07 (1)
- 65D18 (1)
- 65T60 (1)
- AA-sequence (1)
- Abel-Gontscharoff interpolation series. (1)
- Abel-Gontscharoff polynomial (1)
- Abel-Gontscharoff-Gould polynomial (1)
- Abel-Gontscharoff’s Expansion Formula (1)
- And Euler-Frobenius fractions. (1)
- And Laguerre polynomial (1)
- And hitting-time subgroup. (1)
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Articles 1 - 30 of 38
Full-Text Articles in Applied Mathematics
A Quantitative Analysis On Bitmex Perpetual Inverse Futures Xbtusd Contract, Yue Wu
A Quantitative Analysis On Bitmex Perpetual Inverse Futures Xbtusd Contract, Yue Wu
Undergraduate Economic Review
The perpetual inverse futures contract is a recent and most popularly traded cryptocurrency derivative over crypto derivatives exchanges. Exchanges implement a liquidation mechanism that terminates positions which no longer satisfy maintenance requirements. In this study, we use regression, stochastic calculus, and simulation methods to provide a quantitative description of the wealth/return process for holding an XBTUSD contract on BitMEX, examine the funding rate and index price properties, and relate liquidation to leverage as a stopping time problem. The results will help investors understand liquidation to optimize their trading strategy and researchers in studying the design of crypto derivatives.
Pricing Asian Options: Volatility Forecasting As A Source Of Downside Risk, Adam T. Diehl
Pricing Asian Options: Volatility Forecasting As A Source Of Downside Risk, Adam T. Diehl
Undergraduate Economic Review
Asian options are a class of derivative securities whose payoffs average movements in the underlying asset as a means of hedging exposure to unexpected market behavior. We find that despite their volatility smoothing properties, the price of an Asian option is sensitive to the choice of volatility model employed to price them from market data. We estimate the errors induced by two common schemes of forecasting volatility and their potential impact upon trading.
Construction Of Nonlinear Expression For Recursive Number Sequences, Tian-Xiao He
Construction Of Nonlinear Expression For Recursive Number Sequences, Tian-Xiao He
Scholarship
A type of nonlinear expressions of Lucas sequences are established inspired by Hsu [9]. Using the relationships between the Lucas sequence and other linear recurring sequences satisfying the same recurrence relation of order 2, we may transfer the identities of Lucas sequences to the latter.
The Pascal Matrix Function And Its Applications To Bernoulli Numbers And Bernoulli Polynomials And Euler Numbers And Euler Polynomials, Tian-Xiao He, Jeff Liao, Peter Shiue
The Pascal Matrix Function And Its Applications To Bernoulli Numbers And Bernoulli Polynomials And Euler Numbers And Euler Polynomials, Tian-Xiao He, Jeff Liao, Peter Shiue
Scholarship
A Pascal matrix function is introduced by Call and Velleman in [3]. In this paper, we will use the function to give a unied approach in the study of Bernoulli numbers and Bernoulli polynomials. Many well-known and new properties of the Bernoulli numbers and polynomials can be established by using the Pascal matrix function. The approach is also applied to the study of Euler numbers and Euler polynomials.
Elliptic Curves And Cryptography, Linh Nguyen, Andrew Shallue, Faculty Advisor
Elliptic Curves And Cryptography, Linh Nguyen, Andrew Shallue, Faculty Advisor
John Wesley Powell Student Research Conference
No abstract provided.
In Pursuit Of The Ringel-Kotzig Conjecture: Uniform K-Distant Trees Are Graceful, Kimberly Wenger, Daniel Roberts, Faculty Advisor
In Pursuit Of The Ringel-Kotzig Conjecture: Uniform K-Distant Trees Are Graceful, Kimberly Wenger, Daniel Roberts, Faculty Advisor
John Wesley Powell Student Research Conference
Graph labeling has been an active area of research since 1967, when Rosa introduced the concept. Arguably, the biggest open conjecture in the field is referred to as the Ringel-Kotzig conjecture, which states that all trees admit a graceful labeling. In this talk, we will give a bit of background on the problem, as well as present our own results. Namely, that a certain infinite class of trees (called uniform k-distant trees) admits a graceful labeling.
Decomposing Complete Graphs Into A Graph Pair Of Order 6, Yizhe Gao, Daniel Roberts, Faculty Advisor
Decomposing Complete Graphs Into A Graph Pair Of Order 6, Yizhe Gao, Daniel Roberts, Faculty Advisor
John Wesley Powell Student Research Conference
Firstly, a graph G consists of a vertex set V (G), and an edge set E (G) of endpoints which relate two vertices with each edge. Also, a decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list. In the field of graph theory, graph decomposition is an active field of research. A graph pair is a pair of graphs on the same vertex set whose union is the complete graph. Abueida and Daven studied decompositions of complete graphs into graph-pairs of order four and five. We are extending …
Students Ahead Of The Curve In Regional Mathematics Competition, Tia Patsavas
Students Ahead Of The Curve In Regional Mathematics Competition, Tia Patsavas
News and Events
No abstract provided.
Four Named To Endowed Professorships, Kim Hill
Parametric Catalan Numbers And Catalan Triangles, Tian-Xiao He
Parametric Catalan Numbers And Catalan Triangles, Tian-Xiao He
Scholarship
Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schroder numbers and small Schroder numbers, respectively. Using the characterization sequences of Bell-type Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers …
Mathematics Professor’S Publications Add Up, Kim Hill
Mathematics Professor’S Publications Add Up, Kim Hill
News and Events
No abstract provided.
A Unified Approach To Generalized Stirling Functions, Tian-Xiao He
A Unified Approach To Generalized Stirling Functions, Tian-Xiao He
Scholarship
Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in \cite{He11}. Previous well-known Stirling functions introduced by Butzer and Hauss \cite{BH93}, Butzer, Kilbas, and Trujilloet \cite{BKT03} and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed, which extend the corresponding results about the Stirling numbers shown in \cite{HS98} to the defined Stirling functions.
Eulerian Polynomials And B-Splines, Tian-Xiao He
Eulerian Polynomials And B-Splines, Tian-Xiao He
Scholarship
Here presented is the interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, B-splines, respectively. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann-zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed.
Characterizations Of Orthogonal Generalized Gegenbauer-Humbert Polynomials And Orthogonal Sheffer-Type Polynomials, Tian-Xiao He
Characterizations Of Orthogonal Generalized Gegenbauer-Humbert Polynomials And Orthogonal Sheffer-Type Polynomials, Tian-Xiao He
Scholarship
We present characterizations of the orthogonal generalized Gegen-bauer-Humbert polynomial sequences and the orthogonal Sheffer-type polynomial sequences. Using a new polynomial sequence transformation technique presented in [12], we give a method to evaluate the measures and their supports of some orthogonal generalized Gegenbauer-Humbert polynomial sequences.
Riordan Arrays Associated With Laurent Series And Generalized Sheffer-Type Groups, Tian-Xiao He
Riordan Arrays Associated With Laurent Series And Generalized Sheffer-Type Groups, Tian-Xiao He
Scholarship
A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type …
Boundary Type Quadrature Formulas Over Axially Symmetric Regions, Tian-Xiao He
Boundary Type Quadrature Formulas Over Axially Symmetric Regions, Tian-Xiao He
Scholarship
A boundary type quadrature formula (BTQF) is an approximate integration formula with all its of evaluation points lying on the Boundary of the integration domain. This type formulas are particularly useful for the cases when the values of the integrand functions and their derivatives inside the domain are not given or are not easily determined. In this paper, we will establish the BTQFs over sonic axially symmetric regions. We will discuss time following three questions in the construction of BTQFs: (i) What is the highest possible degree of algebraic precision of the BTQF if it exists? (ii) What is the …
A Pair Of Operator Summation Formulas And Their Applications, Tian-Xiao He, Leetsch Hsu, Dongsheng Yin
A Pair Of Operator Summation Formulas And Their Applications, Tian-Xiao He, Leetsch Hsu, Dongsheng Yin
Scholarship
Two types of symbolic summation formulas are reformulated using an extension of Mullin–Rota’s substitution rule in [R. Mullin, G.-C. Rota, On the foundations of combinatorial theory: III. Theory of binomial enumeration, in: B. Harris (Ed.), Graph Theory and its Applications, Academic Press, New York, London, 1970, pp. 167–213], and several applications involving various special formulas and identities are presented as illustrative examples.
On Sequences Of Numbers And Polynomials Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter Shiue
On Sequences Of Numbers And Polynomials Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter Shiue
Scholarship
Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a generalmethod to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.
Sequence Characterization Of Riordan Arrays, Tian-Xiao He, Renzo Sprugnoli
Sequence Characterization Of Riordan Arrays, Tian-Xiao He, Renzo Sprugnoli
Scholarship
In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of …
Characterization Of Compactly Supported Renable Splines With Integer Matrix, Tian-Xiao He, Yujing Guana
Characterization Of Compactly Supported Renable Splines With Integer Matrix, Tian-Xiao He, Yujing Guana
Scholarship
Let M be an integer matrix with absolute values of all its eigenvalues being greater than 1. We give a characterization of compactly supported M-refinable splines f and the conditions that the shifts of f form a Riesz basis.
Padé Spline Functions, Tian-Xiao He
Padé Spline Functions, Tian-Xiao He
Scholarship
We present here the definition of Pad´e spline functions, their expressions, and the estimate of the remainders of pad´e spline expansions. Some algorithms are also given.
A Symbolic Operator Approach To Several Summation Formulas For Power Series Ii, Tian-Xiao He, Peter Shiue, L. C. Hsu
A Symbolic Operator Approach To Several Summation Formulas For Power Series Ii, Tian-Xiao He, Peter Shiue, L. C. Hsu
Scholarship
Here expounded is a kind of symbolic operator method that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.
Symbolization Of Generating Functions; An Application Of The Mullin–Rota Theory Of Binomial Enumeration, Tian-Xiao He, Peter S, Leetsch Hsu
Symbolization Of Generating Functions; An Application Of The Mullin–Rota Theory Of Binomial Enumeration, Tian-Xiao He, Peter S, Leetsch Hsu
Scholarship
We have found that there are more than a dozen classical generating functions that could be suitably symbolized to yield various symbolic sum formulas by employing the Mullin–Rota theory of binomial enumeration. Various special formulas and identities involving well-known number sequences or polynomial sequences are presented as illustrative examples. The convergence of the symbolic summations is discussed.
Fourier Transform Of Bernstein–Bézier Polynomials, Tian-Xiao He, Charles Chui, Qingtang Jiang
Fourier Transform Of Bernstein–Bézier Polynomials, Tian-Xiao He, Charles Chui, Qingtang Jiang
Scholarship
Explicit formulae, in terms of Bernstein–Bézier coefficients, of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on an interval are derived in this paper. Examples are given and discussed to illustrate the general theory. Finally, this consideration is related to the study of refinement masks of spline function vectors.
The Sheffer Group And The Riordan Group, Tian-Xiao He, Peter Shiue, Leetsch Hsu
The Sheffer Group And The Riordan Group, Tian-Xiao He, Peter Shiue, Leetsch Hsu
Scholarship
We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs.
Construction Of Biorthogonal B-Spline Type Wavelet Sequences With Certain Regularities, Tian-Xiao He
Construction Of Biorthogonal B-Spline Type Wavelet Sequences With Certain Regularities, Tian-Xiao He
Scholarship
No abstract provided.
Multivariate Expansion Associated With Sheffer-Type Polynomials And Operators, Tian-Xiao He, Leetsch Hsu, Peter Shiue
Multivariate Expansion Associated With Sheffer-Type Polynomials And Operators, Tian-Xiao He, Leetsch Hsu, Peter Shiue
Scholarship
With the aid of multivariate Sheffer-type polynomials and differential operators, this paper provides two kinds of general expansion formulas, called respectively the first expansion formula and the second expansion formula, that yield a constructive solution to the problem of the expansion of A(ˆt)f([g(t)) (a composition of any given formal power series) and the expansion of the multivariate entire functions in terms of multivariate Sheffer-type polynomials, which may be considered an application of the first expansion formula and the Sheffer-type operators. The results are applicable to combinatorics and special function theory.
An Euler-Type Formula For Ζ(2k +1), Tian-Xiao He, Michael Dancs
An Euler-Type Formula For Ζ(2k +1), Tian-Xiao He, Michael Dancs
Scholarship
In this short paper, we give several new formulas for ζ(n) when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler’s classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.
Numerical Approximation To Ζ(2n+1), Tian-Xiao He, Michael Dancs
Numerical Approximation To Ζ(2n+1), Tian-Xiao He, Michael Dancs
Scholarship
In this short paper, we establish a family of rapidly converging series expansions ζ(2n +1) by discretizing an integral representation given by Cvijovic and Klinowski [3] in Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math. 142 (2002) 435–439. The proofs are elementary, using basic properties of the Bernoulli polynomials.
On The Convergence Of The Summation Formulas Constructed By Using A Symbolic Operator Approach, Tian-Xiao He, Leetsch Hsu, Peter Shiue
On The Convergence Of The Summation Formulas Constructed By Using A Symbolic Operator Approach, Tian-Xiao He, Leetsch Hsu, Peter Shiue
Scholarship
This paper deals with the convergence of the summation of power series of the form Σa ≤ k ≤ bf(k)xk, where 0 ≤ a ≤ b < ∞, and {f(k)} is a given sequence of numbers with k ∈ [a, b) or f(t) a differentiable function defined on [a, b). Here, the summation is found by using the symbolic operator approach shown in [1]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Several examples such as the generalized Euler's transformation series will also be given. In addition, we will compare the convergence of the given series transforms.