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This paper looks at the method of Medvedev and Scaillet at pricing short-term American options and provides an improved representation of their series solution in which explicit formulae for the coefficients are provided. This then avoids the need for solving complicated, recursive systems and efficiently provides fast solutions. We also compare the method with well-known existing methods and find that it outperforms them in giving more accurate option prices as well as critical stock prices.

Circumcenters play an important role in the design and analysis of accelerating various iterative methods in optimization. In this work, we propose Bregman (pseudo-)circumcenters associated with finite sets. We show the existence and give explicit formulae for the unique backward and forward Bregman pseudo-circumcenters of finite sets. Moreover, we use duality to establish connections between backward and forward Bregman (pseudo-)circumcenters. Various examples are presented to illustrate the backward and forward Bregman (pseudo-)circumcenters of finite sets. Our general framework for circumcenters paves the way for the development of accelerating iterative methods by Bregman circumcenters., Comment: 24 pages and 4 figures

Extending the main result of Lorscheid and Weist (2015), in the first part of this paper we show that every quiver Grassmannian of an indecomposable representation of a quiver of type $\tilde D_{n}$ D ~ n has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. F-polynomials). Along these lines, we obtain explicit formulae for all cluster variables of cluster algebras coming from quivers of type $\tilde D_{n}$ D ~ n .

In many practical applications models exhibiting chance constraints play a role. Since, in practice one is also typically interesting in numerically solving the underlying optimization problems, an interest naturally arises in understanding analytical properties, such as differentiability, of probability functions. However in order to build nonlinear programming methods, not only knowledge of differentiability, but also explicit formulae for gradients are important. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by applications from energy management, wherein we face a variety of non-linear transforms of underlying Elliptical distributions, we investigate probability functions acting on decision dependent union of polyhedra. Union of polyhedra naturally occur as soon as one approaches the components of “difference-of-convex” (DC) functions with their respective cutting plane models. In this work, we will establish that the probability functions are locally Lipschitzian and exhibit explicit formulae for “the” Clarke sub-gradients, under very mild conditions. We also highlight, on a numerical example, that the formulae can be put to use “in practice”.

We develop a functional model for operators arising in the study of boundary-value problems of materials science and mathematical physics. We then provide explicit formulae for the resolvents of the associated extensions of symmetric operators in terms of the associated generalised Dirichlet-to-Neumann maps, which can be utilised in the analysis of the properties of parameter-dependent problems as well as in the study of their spectra., 25 pages, 1 figure

We study the number of ordered k-partitions of a multiset with equal sums, having elements $$\alpha _1,\ldots ,\alpha _n$$ α 1 , … , α n and multiplicities $$m_1,\ldots ,m_n$$ m 1 , … , m n . Denoting this number by $$S_k(\alpha _1, \ldots , \alpha _n; m_1, \ldots , m_n)$$ S k ( α 1 , … , α n ; m 1 , … , m n ) , we find the generating function, derive an integral formula, and illustrate the results by numerical examples. The special case involving the set $$\{1,\dots ,n\}$$ { 1 , ⋯ , n } presents particular interest and leads to the new integer sequences $$S_k(n)$$ S k ( n ) , $$Q_k(n)$$ Q k ( n ) , and $$R_k(n)$$ R k ( n ) , for which we provide explicit formulae and combinatorial interpretations. Conjectures in connection to some superelliptic Diophantine equations and an asymptotic formula are also discussed. The results extend previous work concerning 2- and 3-partitions of multisets.

Herein we shall consider Lorentz boosts and Wigner rotations from a (complexified) quaternionic point of view. We shall demonstrate that for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. We subsequently relate the Wigner rotation to the generic non-associativity of the composition of three 4-velocities, and develop a necessary and sufficient condition for associativity to hold. Finally, we relate the composition of 4-velocities to a specific implementation of the Baker-Campbell-Hausdorff theorem. As compared to ordinary 4x4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae., Comment: 21 pages; no figures

For any given multivariate distribution, explicit formulae for the asymptotic covariances of cumulant vectors of the third and the fourth order are provided here. General expressions for cumulants of elliptically symmetric multivariate distributions are also provided. Utilizing these formulae one can extend several results currently available in the literature, as well as obtain practically useful expressions in terms of population cumulants, and computational formulae in terms of commutator matrices. Results are provided for both symmetric and asymmetric distributions, when the required moments exist. New measures of skewness and kurtosis based on distinct elements are discussed, and other applications to independent component analysis and testing are considered.

In this paper we derive the discretization of the nonlinear Monge-Ampere equation by means of the explicit formulae of the meshless Generalized Finite Difference Method (GFDM) in both two and three dimensional settings (2D and 3D). To do so we implement the Cascadic iterative algorithm. We provide a rigorous proof of the consistency of the GFDM for this elliptic equation and present several examples in 2D and 3D, where the method shows its potential and accuracy. We provide a discussion on the number of total nodes in the domain and the number of local supporting nodes. Also, we give an example with physical meaning where we find a surface whose Gaussian curvature is given.

This paper proposes a multiple-delayed predator–prey system with habitat complexity and harvesting effort, and investigates the dynamical behavior including stability properties and Hopf bifurcation. Firstly, stability of equilibrium points and the existence of Hopf bifurcation are investigated and some critical conditions which guarantee the corresponding results are obtained based on mathematical view. Secondly, the explicit formulae for determining the direction, stability and period of the bifurcating periodic solutions are derived by using the center manifold theory and the normal form theory. Finally, in order to verify the theoretical results, some numerical simulations are done to illustrate the results. It is observed that the level of abundance of prey and predator populations depends on the gestation delay if the gestation delay exceeds some critical values.

Signed graphs have their edges labeled either as positive or negative. Here we introduce two types of signed distance matrix for signed graphs. We characterize balance in signed graphs using these matrices and we obtain explicit formulae for the distance spectrum of some unbalanced signed graphs. We also introduce the notion of distance-compatible signed graphs and partially characterize it.

In this paper we consider pentadiagonal ( n + 1 ) × ( n + 1 ) matrices with two sub-diagonals above and below the main diagonal at distances k and l from the main diagonal where 1 ≤ k l ≤ n . We give explicit formulae for the eigenpairs of imperfect Toeplitz matrices in the case when ( n + 1 ) / 2 ≤ k . Imperfectness means that the first and last k elements of the main diagonal differ from the elements in the middle.

In this paper, a stationary one-dimensional Poisson-Nernst-Planck model with permanent charge is studied under the assumption that n - 1 positively charged ion species have the same valence and the permanent charge is small. By expanding the singular solutions of Poisson-Nernst-Planck model with respect to small permanent charge, the explicit formulae for the zeroth order approximation and the first order approximation of individual flux can be obtained. Based on these explicit formulae, the effects of small permanent charges on individual flux are investigated.

Consider a random simplex $[X_1,\ldots,X_n]$ defined as the convex hull of independent identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^{n-1}$ with the following beta density: $$ f_{n-1,\beta} (x) \propto (1-\|x\|^2)^{\beta} 1_{\{\|x\| < 1\}}, \qquad x\in\mathbb{R}^{n-1}, \quad \beta>-1. $$ Let $J_{n,k}(\beta)$ be the expected internal angle of the simplex $[X_1,\ldots,X_n]$ at its face $[X_1,\ldots,X_k]$. Define $\tilde J_{n,k}(\beta)$ analogously for i.i.d. random points distributed according to the beta' density $$ \tilde f_{n-1,\beta} (x) \propto (1+\|x\|^2)^{-\beta}, \qquad x\in\mathbb{R}^{n-1}, \quad \beta > \frac{n-1}{2}. $$ We derive formulae for $J_{n,k}(\beta)$ and $\tilde J_{n,k}(\beta)$ which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of $\beta$. For $J_{n,1}(\pm 1/2)$ we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry. (i) We compute the expected $f$-vectors of the typical Poisson-Voronoi cells in dimensions up to $10$. (ii) Consider the random polytope $K_{n,d} := [U_1,\ldots,U_n]$ where $U_1,\ldots,U_n$ are i.i.d. random points sampled uniformly inside some $d$-dimensional convex body $K$ with smooth boundary and unit volume. M. Reitzner proved the existence of the limit of the normalized expected $f$-vector of $K_{n,d}$: $$ \lim_{n\to\infty} n^{-{\frac{d-1}{d+1}}}\mathbb E \mathbf f(K_{n,d}) = \mathbf c_d \cdot \Omega(K), $$ where $\Omega(K)$ is the affine surface area of $K$, and $\mathbf c_d$ is an unknown vector not depending on $K$. We compute $\mathbf c_d$ explicitly in dimensions up to $d=10$ and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere., Comment: 30 pages. Minor changes. References updated

For each positive characteristic multiple zeta value (defined by Thakur in Function field arithmetic, World Scientific Publishing, River Edge, 2004), the first and third authors in Chang and Mishiba (Invent Math, 2020, https://doi.org/10.1007/s00222-020-00988-1 ) constructed a t-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson–Thakur series and t-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.

In the present paper, we obtain explicit formulae for geodesics in some left–invariant sub–Finsler problems on Heisenberg groups $\mathbb {H}_{2n+1}$ . Our main assumption is the following: the compact convex set of unit velocities at identity admits a generalization of spherical coordinates. This includes convex hulls and sums of coordinate 2–dimensional sets, all left–invariant sub–Riemannian structures on $\mathbb {H}_{2n+1}$ , and unit balls in Lp–metric for $1\le p\le \infty $ . In the last case, extremals are obtained in terms of incomplete Euler integral of the first kind.

This paper proposes the topology optimization for steady-state heat conduction structures by incorporating the meshless-based generalized finite difference method (GFDM) and the solid isotropic microstructures with penalization interpolation model. In the meshless GFDM numerical scheme, the explicit formulae of the partial differential equation are expressed by the Taylor series expansions and the moving-least squares approximations to address the required partial derivatives of unknown nodal variables. With the relative density of meshless GFDM node as the design variable, the implementation of the topology optimization is formulated involving the minimization of heat potential capacity as the objective function under node number constraint. Moreover, sensitivity of the objective function is derived based on the adjoint method, and sensitivity filtering subsequently suppresses the checkerboard pattern. Next, the update of design variables at each iteration is solved by the optimality criteria method. At last, several numerical examples are illustrated to demonstrate the validity and feasibility of the proposed method.

This study examines several generic properties of weighted consistent projections and indicators of inconsistency in an arbitrary finite dimensional inner product space of square matrices. In the case of weighted Frobenius inner products we present explicit formulae for them in terms of the matrix entries and weights. It extends the recent results, due to Koczkodaj et al. [Fund. Inform. 172 (2020) 379–397], to positive matrices in pairwise comparisons. Moreover, we discuss the possibilities of a proper choice of the inner product weights for numerical applications based on the orthogonal consistent projections.

Explicit formulae for maximum likelihood estimates of the parameters of square root processes and Bessel processes and first and second order approximate sufficient statistics are supplied. Applications of the estimation formulae to simulated interest rate and index time series are supplied, demonstrating the accuracy of the approximations and the extreme speed-up in estimation time. This significantly improved run time for parameter estimation has many applications where ex-ante forecasts are required frequently and immediately, such as in hedging interest rate, index and volatility derivatives based on such models, as well as modelling credit risk, mortality rates, population size and voting behaviour.

The resistance between two nodes in some electronic networks has been studied extensively. Let $$G_n$$ be a generalized phenylene with n 6-cycles and n 4-cycles. Using series and parallel rules and the $$\Delta - Y$$ transformations we obtain explicit formulae for the resistance distance between any two points of $$G_n$$ . To the best of our knowledge $$\{G_n\}_{n=1}^{\infty }$$ is a nontrivial family with diameter going to $$\infty $$ for which all resistance distances have been explicitly calculated. We also determine the maximal resistance distance and the minimal resistance distance in $$G_n$$ . The monotonicity and some asymptotic properties of resistance distances in $$G_n$$ are given. At last some numerical results are discussed, in which we calculate the Kirchhoff indices of a set of benzenoid hydrocarbons; We compare their Kirchhoff indices with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the Kirchhoff index is better than or as good as the models corresponding to the other distance-based indices.

Exponential dispersion models are well used and studied in quantitative risk management and actuarial science. One of the main interests is the risk measurement analysis of such models when facing extreme loss events. In this paper, we propose two multivariate risk measures based on conditional expectation and derive the explicit formulae for exponential dispersion models. In particular, our multivariate risk measures could facilitate a systemic risk measure with explicit expressions for exponential dispersion models subject to any pre-specified “systemic event.” We provide two numerical examples based on practical data to show the advantages of our approach in the context of exponential dispersion models.

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are then obtained, and applications to enumeration of ribbon graphs with even valencies and to the special cubic Hodge integrals are considered., Comment: It was added a remark after Theorem 1; added Appendix A; added references; two statements moved to Introduction; corrected typos. 26 pages

In this work we deduce explicit formulae for the elements of the matrices representing the action of integro-differential operators over the coefficients of generalized Fourier series. Our formulae are obtained by performing operations on the bases of orthogonal polynomials and result directly from the three-term recurrence relation satisfied by the polynomials. Moreover we give exact formulae for the coefficients for some families of orthogonal polynomials. Some tests are given to demonstrate the robustness of the formulas presented.

In this paper, based on the original friction-induced oscillation, we propose a new friction-induced mathematical model associated with moving belt of particle supply device. Two delay feedback control methods are provided to make the new model be stable. Linear stability analysis is carried out to obtain the stability of equilibrium and the stability boundary, corresponding to the critical value of Hopf bifurcation, dividing stable and unstable regimes of equilibrium. Next, we derive explicit formulae for determining the stability of Hopf bifurcating periodic solutions and the direction of Hopf bifurcation. Finally, detailed numerical simulations using MATLAB software are presented to demonstrate the application of the theoretical results, and we also compare the efficacy of the two control forces in new friction-induced models.

We study the size of certain acyclic domains that arise from geometric and combinatorial constructions. These acyclic domains consist of all permutations visited by commuting equivalence classes of maximal reduced decompositions if we consider the symmetric group and, more generally, of all c-singletons of a Cambrian lattice associated to the weak order of a finite Coxeter group. For this reason, we call these sets Cambrian acyclic domains. Extending a closed formula of Galambos--Reiner for a particular acyclic domain called Fishburn's alternating scheme, we provide explicit formulae for the size of any Cambrian acyclic domain and characterize the Cambrian acyclic domains of minimum or maximum size., Comment: 24 pages, 11 figures

Probability functions appearing in chance constraints are an ingredient of many practical applications. Understanding differentiability, and providing explicit formulae for gradients, allow us to build nonlinear programming methods for solving these optimization problems from practice. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by gas network applications, we investigate differentiability of probability functions acting on non-convex quadratic forms. We establish continuous differentiability for the broad class of elliptical random vectors under mild conditions.

In this paper, we mainly investigate a delayed predator–prey model with stage structure for prey and Ivlev-type functional response. Four assumptions about this model are made as follows: (1) there are a single predator and a single prey population in the model; (2) the prey is divided by the age into two stage structures, immature and mature; (3) the trophic transfer from prey to predator is incomplete; (4) there are two time delays due to the time of maturation of prey and the time of gestation of predator. Some properties of equilibria for the model without delays are provided. Further, the stability of equilibria and existence of Hopf bifurcation are studied by discussing the different cases of time delays for the model with delays. We observe that delays can cause a stable equilibrium to become unstable one, even occur Hopf bifurcation when delays pass through their corresponding critical values. Meanwhile, we derive explicit formulae to determine the properties of Hopf bifurcation such as the direction of Hopf bifurcation and the stability of periodic solutions. Numerical simulations of all theoretical analyses are given for verifying our theoretical results. In the present work, we demonstrate that the validity and universality of oscillations induced by the time delays both theoretically and numerically. These results of this paper may be helpful for us to further understand the role of the critical values of time delays in stabilizing the predator–prey model.

We derive a recursive formula for certain relative Gromov–Witten invariants with a maximal tangency condition via the Witten–Dijkgraaf–Verlinde–Verlinde equation. For certain relative pairs, we get explicit formulae of invariants using the recursive formula.

To improve the sensitivity of a Shewhart control chart, it is common among practitioners to use supplementary runs rules. The performance of such runs rules charts is studied in the presence of positive autocorrelation caused by a first-order discrete autoregressive process. This type of data-generating process allows to compute the chart’s run length properties exactly and efficiently, by utilizing the finite Markov chain embedding technique. Explicit formulae are derived for common types of runs rules. Afterwards, a detailed performance study about runs rules charts under serial dependence is presented.

We obtain explicit formulae for the expected values $$E \{ \prod \nolimits _{i=1}^{3}g_i ( X_i ) \}$$ of standard tri-variate Gaussian random vector $${\underline{X}}= \left( X_1, X_2 , X_3 \right) $$ over the set $$g_i (x) \in \left\{ \delta (x), \mathrm {sgn}(x), |x|, x \right\} $$ of nonlinear and linear functions. Based on the results, we also suggest corrections to long-known formulae for two incomplete moments. Applications of the formulae in practical examples are also illustrated concisely. For easy reference for the readers, the explicit formulae derived in this paper and related well-known results are summarized in tables.

In an (hyper)elliptic curve cryptosystem, the most important operation or the most time-consuming operation is the divisor scalar multiplication which consists of a sequence of doubling (of divisor) and addition (of two divisors). Point halving algorithms for elliptic curve cryptosystem and divisor halving algorithms for genus-2 hyperelliptic curve cryptosystem had been successively put forward to take the place of doubling algorithms for speeding up (hyper)elliptic curve cryptosystem. We present an outline for an algorithm for divisor halving on genus-3 hyperelliptic curves over the binary field and give some explicit formulae for a class of genus-3 curves. Our algorithm improves previously known best doubling algorithms in most cases. A halve-and-add binary method for divisor scalar multiplications is presented.

The sparse ( 0 , 1 ) matrix that was introduced by Redheffer has the property that its nth leading principal minor equals ∑ i ≤ n μ ( i ) , where μ is the Mobius function. Recently, a substantially sparser ( 0 , 1 ) matrix that possesses the same property has been described. Let R n denote the nth leading principal submatrix of this sparser matrix. Our main result establishes asymptotically tight estimates for the two dominant eigenvalues of R n . We also state explicit formulae for the eigenvectors of R n that are associated with the eigenvalues that are different from 1. Finally, we state several conjectures about the eigenstructure of a related class of matrix.

The resistance distance between any two vertices of a connected graph $G$ is defined as the net effective resistance between them in the electrical network constructed from $G$ by replacing each edge with a unit resistor. The Kirchhoff index of $G$ is defined as the sum of resistance distances between all pairs of vertices. In this paper, two unary graph operations on $G$ are taken into consideration, with the resulted graphs being denoted by $RT(G)$ and $H(G)$ . Using electrical network approach and combinatorial approach, we derive explicit formulae for resistance distances and Kirchhoff indices of $RT(G)$ and $H(G)$ . It turns out that resistance distances and Kirchhoff indices of $RT(G)$ and $H(G)$ could be expressed in terms of resistance distances and graph invariants of $G$ . Our result generalizes the previously known result on the Kirchhoff index of $RT(G)$ for a regular graph $G$ to the Kirchhoff index of $RT(G)$ for an arbitrary graph $G$ .

The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros. The duality between primes and Riemann zeta function zeros suggests some new goals and aspects to be studied: {\em adelic duality} and the {\em POSet of prime numbers}. The article presents computational evidence of the structure of the imaginary parts $t$ of the non-trivial zeros of the Riemann zeta function $\rho=1/2+it$, called in this article the {\em Riemann Spectrum}, using the study of their distribution. The novelty represents in considering the associated characters $p^{it}$, towards an algebraic point of view, than rather in the sense of Analytic Number Theory. This structure is tentatively interpreted in terms of adelic characters, and the duality of the rationals. Second, the POSet structure of prime numbers studied, is tentatively mirrored via duality in the Riemann spectrum. A direct study of the convergence of their Fourier series, along Pratt trees, is proposed. Further considerations, relating the Riemann Spectrum, adelic characters and distributions, in terms of Hecke idelic characters, local zeta integrals (Mellin transform) and $\omega$-eigen-distributions, are explored following., Comment: 15 pages

In this chapter we study some important functionals of the telegraph processes. We derive the explicit formulae for the densities of the telegraph processes with reflecting and absorbing barriers, as well as first passage times and occupation time distributions. Our presentation corrects some inaccuracies and errors in this field.