The biased random walk on supercritical Galton-Watson trees is known to exhibit a multiscale phenomenon in the slow regime: the maximal displacement of the walk in the first $n$ steps is of order $(\log n)^3$, whereas the typical displacement of the walk at the $n$-th step is of order $(\log n)^2$. Our main result reveals another multiscale property of biased walks: the maximal potential energy of the biased walks is of order $(\log n)^2$ in contrast with its typical size, which is of order $\log n$. The proof relies on analyzing the intricate multiscale structure of the potential energy.

Given a forbidden graph $H$ and a function $f(n)$, the Ramsey-Turán number $\textbf{RT}\left( {n,H,f\left( n \right)} \right)$ is the maximum number of edges of an $H$-free graph on $n$ vertices with independence number less than ${f\left( n \right)}$. For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any red/blue edge coloring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. Denote $G+H$ by the join graph obtained from disjoint $G$ and $H$ by adding all edges between them completely. We first show that for any fixed graph $H$, if there are two constants $p:=p(H)>0$ and $q:=q(H)>1$ such that $R(H,K_n)\le \frac{pn^q}{(\log n)^{q-1}}$, then $\textbf{RT}(n,K_2+H,o(n^{\frac{1}{q}}(\log n)^{1-\frac{1}{q}}))=o(n^2),$ which extends several previous results. Moreover, we show that for any fixed forest $F$ of order $k\ge3$, and for any $0<\delta<1$ and sufficiently large $n$, \begin{align*} \textbf{RT}( {n,F+F,n^\delta} )\le n^{2-(1-\delta)/\lceil\frac{(k-1)(2-\delta)}{1-\delta}\rceil}. \end{align*} As a corollary, we have an upper bound for ${\bf{RT}}( {n,K_{2,2,2},n^{\delta}})$ for any $0<\delta<1$.

Given a simple graph $G=(V, E)$ and its (proper) total coloring $\phi$ with elements of the set $\{1, 2,\cdots, k\}$, let $w_{\phi}(v)$ denote the sum of the color of $v$ and the colors of all edges incident with $v$. If for each edge $uv\in E$, $w_{\phi}(u)\neq w_{\phi}(v)$, we call $\phi$ a neighbor sum distinguishing total coloring of $G$. Let $L=\{L_x\, |\, x\in V\cup E\}$ be a set of lists of real numbers, each of size $k$. The neighbor sum distinguishing total choosability of $G$ is the smallest $k$ for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from $L_x$ for each $x\in V\cup E$, and we denote it by ${\rm ch}''_{\sum}(G)$. The known results of neighbor sum distinguishing total choosability are mainly about planar graphs. In this paper, we focus on $1$-planar graphs. A graph is $1$-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. We prove that ${\rm ch}''_{\sum}(G)\leq \Delta+4$ for any $1$-planar graph $G$ with $\Delta\geq 15$, where $\Delta$ is the maximum degree of $G$.

In this paper, by using the concentration-compactness principle and a version of symmetry mountain pass theorem, we establish the existence and multiplicity of solutions to the following $p$-biharmonic problem with critical nonlinearity: $$\Bigg\{\begin{array}{ll} \Delta_p^2u=f(x,u)+\mu|u|^{p^*-2}u ~&\text{in}~\Omega, \\ u=\dfrac{\partial u}{\partial \nu}=0 ~&\text{on}~\partial \Omega, \end{array}$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ $(N\geq 3)$ with smooth boundary, $\Delta_p^2u=\Delta(|\Delta u|^{p-2}\Delta u),$ $1 < p< \frac{N}{2}$, $p^*=\frac{Np}{N-2p},$ $\frac{\partial u}{\partial \nu}$ is the outer normal derivative, $\mu$ is a positive parameter and $f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ is a Carathéodory function.

For an $r$-uniform hypergraph $F$, the anti-Ramsey number ${\rm ar}(n,r,F)$ is the minimum number $c$ of colors such that an $n$-vertex $r$-uniform complete hypergraph equipped any edge-coloring with at least $c$ colors unavoidably contains a rainbow copy of $F$. In this paper, we determine the anti-Ramsey number for cycles of length three in $r$-uniform hypergraphs for $r\geq 3$, including linear cycles, loose cycles and Berge cycles.

This paper introduces the new class of periodic multivariate GARCH models in their periodic BEKK specification. Semi-polynomial Markov chains combined with algebraic geometry are used to obtain some properties like irreducibility. We impose weak conditions to obtain the strict periodic stationarity and the geometric ergodicity of the process, via the theory of positive linear operators on a cone : it is supposed that zero belongs to the support of the driving noise density which is absolutely continuous with respect to the Lebesgue measure and the spectral radius of a matrix built from the periodic coefficients of the model is smaller than one.

Let $\Gamma$ denote a bipartite Q-polynomial distance-regular graph with vertex set $X$, valency $k\geq 3$ and diameter $D\geq 3$. Let $A$ be the adjacency matrix of $\Gamma$ and let $A^*:=A^*(x)$ be the dual adjacency matrix of $\Gamma$ with respect to a fixed vertex $x \in X$. Let $T:=T(x)$ denote the Terwilliger algebra of $\Gamma$ generated by $A$ and $A^*$. In this paper, we first describe the relations between $A$ and $A^*$. Then we determine the dimensions of both $T$ and the center of $T$, and moreover we give a basis of $T$.

In this paper, we consider the Cauchy problem of the $d$-dimensional damping incompressible magnetohydrodynamics system without dissipation. Precisely, this system includes a velocity damped term and a magnetic damped term. We establish the existence and uniqueness of global solutions to this damped system in the critical Besov spaces by means of the Fourier frequency localization and Bony paraproduct decomposition.

In this paper, we propose a new method for spread option pricing under the multivariate irreducible diffusions without jumps and with different types of jumps by the expansion of the transition density function. By the quasi-Lamperti transform, which unitizes the diffusion matrix at the initial time, and applying the small-time It$\mathrm{\hat{o}}$-Taylor expansion method, we derive explicit recursive formulas for the expansion coefficients of transition densities and spread option prices for multivariate diffusions with jumps in return. It is worth mentioning that we also give the closed-form formula of spread option price whose underlying asset price processes contain a Merton jump and a double exponential jump, which is innovative compared with current literature. The theoretical proof of convergence is presented in detail.

In this paper, $\bar{\partial}$-dressing method based on a local $3\times 3$ matrix $\bar{\partial}$-problem with non-normalization boundary conditions is used to investigate coupled two-component Kundu-Eckhaus equations. Firstly, we propose a new compatible system with singular dispersion relation, that is time spectral problem and spatial spectral problem of coupled two-component Kundu-Eckhaus equations via constraint equations. Then, we derive a hierarchy of nonlinear evolution equations by introducing a recursive operator. At last, by solving constraint matrixes, a spectral transform matrix is given which is sufficiently important for finding soliton solutions of potential function, and we obtain $N$-soliton solutions of coupled two-component Kundu-Eckhaus equations.

This paper is devoted to the following fractional relativistic Schrödinger equation: \begin{equation*} (-\Delta+m^{2})^su+V(x)u=f(x,u), \qquad x\in \mathbb{R}^N, \end{equation*} where $(-\Delta+m^{2})^s$ is the fractional relativistic Schrödinger operator, $s\in (0, 1), m>0,$ $V : \mathbb{R}^N \to \mathbb{R}$ is a continuous potential and $f: \mathbb{R}^N\times\mathbb{R} \to \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth. We consider the case where the potential $V$ is indefinite so that the relativistic Schrödinger operator $(-\Delta+m^{2})^s+V$ possesses a finite-dimensional negative space. With the help of extension method and Morse theory, the existence of a nontrivial solution for the above problem is obtained.

Given a distribution of pebbles on the vertices of a connected graph $G$, a pebbling move on $G$ consists of taking two pebbles off one vertex and placing one on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices $u$ and $w$ that are adjacent to a vertex $v$, and an extra pebble is added at vertex $v$. The rubbling number of $G$, denoted by $\rho(G)$, is the smallest number $m$ such that for every distribution of $m$ pebbles on $G$ and every vertex $v$, at least one pebble can be moved to $v$ by a sequence of rubbling moves. The optimal rubbling number of $G$, denoted by $\rho_{opt}(G)$, is the smallest number $k$ such that for some distribution of $k$ pebbles on $G$, one pebble can be moved to any vertex of $G$. In this paper, we determine $\rho(G)$ for a non-complete bipartite graph $G\in B(s,t)$ with $\delta(G)\ge \lceil \frac{2s+1}{3}\rceil$, give an upper bound of $\rho(G)$ for $G\in B(s,t)$ with $\delta(G)\ge \lceil \frac{s+1}{2}\rceil$, and also obtain $\rho_{opt}(G)$ for a non-complete bipartite graph $G\in B(s,t)$ with $\delta(G)\ge \lceil \frac{s+1}{2}\rceil$, where $B(s,t)$ is the set of all connected bipartite graphs with partite sets of size $s$ and $t$ ($s\ge t$) and $\delta(G)$ is the minimum degree of $G$.

This paper concerned with the orbital stability of solitary waves for the mKdV-Schrödinger system with cubic-quintic nonlinear terms through detailed spectral analysis and abstract stability theorem. First, we derived the explicit solitary wave solutions by assuming the solution expression. Then, through using the orbital stability theory developed by Grillakis et al., we established a general criteria for assessing the orbital stability for solitary waves of this system.

In this paper we consider a kind of predator-prey model named Holling-Tanner model. Firstly, we prove all solutions of this model to be bounded from above. Secondly, we find a positive invariant set of the model, and prove the existence of stable limit cycle in this invariant set by Poincaré-Bendixson theorem for the unstable equilibrium. Thirdly, we get the region of parameters in which the corresponding stable equilibrium are also globally asymptotically stable. Lastly, we give a bifurcation diagram and illustration with two limit cycles for special parameters through numerical simulation. By our knowledge, the invariant set constructed in this paper is better than that in the book written by Murray.

In this paper, the existence, uniqueness and Strichartz type estimates to solutions of multipoınt problem for abstract linear and nonlinear wave equations are obtained. The equation includes a linear operator $A$ defined in a Hilbert space $H$. We obtain the existence, uniqueness regularity properties, and Strichartz type estimates to solutions of a wide class of wave equations which appear in the fields of elastic rod, hydro-dynamical process, plasma, materials science and physics, by choosing the space $H$ and the operator $A$.

In this article, we study the empirical likelihood (EL) method for autoregressive models with spatial errors. The EL ratio statistics are constructed for the parameters of the models. It is shown that the limiting distributions of the EL ratio statistics are chi-square distributions, which are used to construct confidence intervals for the parameters of the models. A simulation study is conducted to compare the performances of the EL based and the normal approximation (NA) based confidence intervals. Simulation results show that the confidence intervals based on EL are superior to the NA based confidence intervals.

In this paper, we consider a Dirichlet-Laplacian with a drift problem. We prove existence of weak solutions of it on the Nehari manifold. Then, we show that the associated energy functional has a minimizer on a rearrangement class generated by a determined function. Finally, we prove a duality theorem for this boundary value problem.

The link of (2+1)-dimensional Harry Dym equation with the modified Kadomtsev-Petviashvili equation by the reciprocal transformation is shown. With the help of Darboux transformation, exact solutions of the (2+1)-dimensional Harry Dym equation are constructed and represented in terms of Wronskians.

In this paper, we propose an efficient and accurate method for pricing Guaranteed Minimum Death Benefit (GMDB) under time-changed Lévy processes. Suppose that the GMDB payoff depends on a dollar cost averaging (DCA) style periodic investment, and the activity rate process in stochastic time change is modeled by a square-root process. We develop a recursive method to derive the closed form valuation formula by using the frame duality projection method. Numerical examples are reported for demonstrating the effectiveness of our approach and illustrating the interplay between contract parameters and the valuation.

In this paper, we study the problem of irreversible investment under endowment constraints. We first establish the existence and uniqueness of the result and then demonstrate the necessity and sufficient conditions for optimality. Based on this condition, we provide a characterization for optimal investment plans, which can be obtained by the so-called base capacity solving a backward equation. We may obtain explicit solutions for certain typical cases.

In this paper, we develop a robust variable selection procedure based on the exponential squared loss (ESL) function for the varying coefficient partially nonlinear model. Under certain conditions, some asymptotic properties of the proposed penalized ESL estimator are established. Meanwhile, the proposed procedure can automatically eliminate the irrelevant covariates, and simultaneously estimate the nonzero regression coefficients. Furthermore, we apply the local quadratic approximation (LQA) and minorization-maximization (MM) algorithm to calculate the estimates of non-parametric and parametric parts, and introduce a data-driven method to select the tuning parameters. Simulation studies illustrate that the proposed method is more robust than the classical least squares technique when there are outliers in the dataset. Finally, we apply the proposed procedure to analyze the Boston housing price data. The results reveal that the proposed method has a better prediction ability.

The purpose of this work is to investigate the boundedness of the pullback attractors for the micropolar fluid flows in two-dimensional unbounded domains. Exactly, the $H^1$-boundedness and $H^2$-boundedness of the pullback attractors are established when the external force $F(t,x)$ has different regularity with respect to time variable, respectively.

The stable switching phenomenon of a diffusion mussel-algae system with two delays and half saturation constant is studied. The stability of positive equilibrium and the existence of Hopf bifurcation on two-delay plane are investigated by calculating the stability switching curves. The normal form on the central manifold near Hopf bifurcation point is also derived. It can find that two different delays can induce the stable switches which cannot occur with the same delays. Through numerical simulations, the region of complete stability of system increases with the increase of half-saturation constant, indicating that the half-saturation constant contributes to the stability of system. In addition, double Hopf bifurcations may occur due to the intersection of bifurcation curves. These results show that two different delay and half-saturation constant have important effects on the system. The numerical simulation results validate the correctness of the theoretical analyses.

In the paper, by exploring Stampacchia truncation method, some comparison techniques and variational approaches, we study the existence and regularity of positive solutions for a boundary value problem involving the fractional p-Laplacian, where the nonlinear term satisfies some growth conditions but without the Ambrosetti and Rabinowitz condition.

In this paper, we propose a pricing model of airbag options with discrete monitoring, time-varying barriers, early exercise opportunities, and other popular features simultaneously. We show that the option value is a viscosity solution of a PDE system. In particular, a closed-form solution is obtained in the classic Black-Scholes economy with no early exercise opportunities. For the general case, we develop a numerical algorithm and conduct an extensive numerical analysis after calibrating the model to the CSI 500 index in China. Greek letters, dynamic hedging, and assessment of investing in airbag options are also studied.

This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates. The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element, and the control variable is approximated by piecewise constant functions. The time derivative is discretized by the backward Euler method. Firstly, we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis. Secondly, we derive a priori error estimates for all variables. Thirdly, we present a two-grid scheme and analyze its convergence. In the two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. At last, a numerical example is presented to verify the theoretical results.

In this paper, we consider the time-harmonic electromagnetic scattering by a perfect conductor in a homogeneous chiral environment. For three-dimensional cylindrical structures, it can be simplified as a two-dimensional model problem, which can be modeled by two scalar Helmholtz equations via coupled boundary conditions. The boundary integral equation method is used to prove the unique existence of the weak solution to this problem. Then we apply the linear sampling method to recover the scatterer from one of the far field pattern of wave fields. Some numerical examples are shown to verify the correctness and effectiveness of the proposed method.

In this article, the maximum likelihood estimators (MLEs) of the scale and shape parameters β and λ from the Exponential-Poisson distribution will be considered in moving extremes ranked set sampling (MERSS). These MLEs will be compared in terms of asymptotic efficiencies. The numerical results show that the MLEs obtained via MERSS can serve as effective alternatives to those derived from simple random sampling.

The initial boundary value problem of a class of coupled hyperbolic systems with logarithmic source terms is considered. In this article, we classify the initial data for the global existence, finite time blow-up and long time decay of the solution. By using potential well method combined with Sobolev embedding theorem, the sufficient initial conditions of global existence, asymptotic behavior, the upper and lower bounds of blow-up time are derived at low energy level $E(0) < d$. These results are extended in parallel to the critical case $E(0) = d$. Besides, with additional assumptions on initial data, the finite time blow up result is given with arbitrary positive initial energy $E(0) > 0$.

This paper is concerned with the non-isentropic compressible Navier-Stokes/Allen-Cahn equations with the diffusion interface, which is an important mathematical model in the numerical simulation of compressible immiscible two-phase flow. When the space-asymptotic states $(v_\pm, u_\pm, \theta_\pm)$ lie in the rarefaction curve of the Riemann problem of the compressible Euler equations, we prove that the time-asymptotic state of solutions to the 1-D Cauchy problem is the rarefaction wave, that is, the stability of the rarefaction wave, where the strength of the rarefaction wave is not required to be small. Moreover, we consider the general gases including ideal polytropic gas and allow the different space-asymptotic states $\chi_\pm$ for the concentration difference of the mixture fluids. The proof is mainly based on a basic energy method. By product, we give the proof of the uniqueness of the global solutions to the 1-D Cauchy problem.

In this paper we prove vanishing viscosity limits of a coupled chemotaxis-fluid model in a bounded domain $\Omega\subset\mathbb{R}^3$. The proof is based on the Banach's fixed point theorem and the $L^p$-energy method. In addition, the $L^\infty$-estimates and gradient estimates of the heat equations also play a crucial role.

A non-increasing sequence $\pi=(d_1,\cdots,d_n)$ of nonnegative integers is said to be a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. In this case, $G$ is referred to as a realization of $\pi$. In terms of graphic sequences, the Loebl-Komlós-Sós conjecture states that for any integers $k$ and $n$, if $\pi=(d_1,\cdots,d_n)$ is a graphic sequence with $d_{\lceil\frac{n}{2}\rceil}\ge k$, then every realization of $\pi$ contains all trees with $k$ edges as subgraphs. This problem can be viewed as a forcible degree sequence problem. In this paper, we consider a potential degree sequence problem of the Loebl-Komlós-Sós conjecture, that is, we prove that for any integers $k$ and $n$, if $\pi=(d_1,\cdots,d_n)$ is a graphic sequence with $d_{\lceil\frac{n}{2}\rceil}\ge k$, then there is a realization of $\pi$ containing all trees with $k$ edges as subgraphs.

Recurrent event data with a terminal event are commonly encountered in longitudinal follow-up studies. In this paper, we investigate regression analysis of the weighted composite endpoint of recurrent and terminal events with a semiparametric mixed model. Particularly, the weighted composite endpoint is constructed by the severity of all events while leaving the dependence structure among the recurrent and terminal events unspecified. The semiparametric mixed model is flexible since it allows the covariate effects on the rate function of the weighted composite endpoint to be proportional or convergent. For inference on the model parameters, the estimating equation approach and the inverse probability weighting technique are developed. The asymptotic properties of the resulting estimators are established and the finite sample performance of the proposed procedure is evaluated through Monte Carlo simulation studies. We apply the proposed method to a real data set on a medical cost study of chronic heart failure patients for illustration.

A general model of insider trading on a dynamic asset in a finite time interval is proposed, in which an insider possesses the whole information on the dynamic values, noise traders without any information submit orders randomly as a martingale with volatility following a stochastic process, and market makers observe partial information when setting price in a semi-strong efficient way. With the help of filtering theory, BSDE method and dynamic programming principle, we establish a market equilibrium consisting of linear insider trading strategy and linear pricing rule, with the later characterized by price pressure on market orders and price pressure on asset observations. It shows that in the equilibrium, all the information on the risky asset is incorporated into the market price at the end of the transaction, and price pressure on market orders is a submartingale while market depth process is a martingale. Furthermore, as market makers’ information precision on the asset tends to zero, the equilibrium with partial observation of market makers on the risky asset converges to the one without partial observation of market makers, while when market makers observe almost all of information on the asset, the expected profit earned by the insider makes almost zero, which is in accord with our economic intuition. Our results cover some classical results about continuous-time insider trading on a static asset.

The Cauchy problem of the fifth-order nonlinear Schrödinger (foNLS) equation is investigated with nonzero boundary conditions in detailed. Firstly, the spectral analysis of the scattering problem is carried out. A Riemann surface and affine parameters are introduced to transform the original spectral parameter to a new spectral parameter in order to avoid the multi-valued problem. Based on Lax pair of the foNLS equation, the Jost functions are obtained, and their analytical, asymptotic, symmetric properties, as well as the corresponding properties of the scattering matrix are established systematically. For the inverse scattering problem, we discuss the cases that the scattering coefficients have simple zeros and double zeros, respectively, and we further derive their corresponding exact solutions via solving a suitable Riemann-Hilbert problem. Moreover, some interesting phenomena are found when we choose some appropriate parameters for these exact solutions, which are helpful to study the propagation behavior of these solutions.

In this paper, we studied a stochastic predator-prey model of two predators with stage structure. By constructing a suitable stochastic Lyapunov function, the condition of stationary distribution is verified, and we get the sufficient condition for the model to have ergodic stationary distribution. Then, by using the Itô’s formula for the model, the sufficient conditions for the extinction of the predator population are given. Finally, some examples and numerical simulations are illustrated to verify the theoretical results.

Let $w(z)$ be non-rational meromorphic solutions with hyper-order less than $1$ to a family of higher order nonlinear delay differential equations \begin{align*} w(z+1)w(z-1)+a(z)\frac{w^{(k)}(z)}{w(z)}=R(z,w(z)), \qquad k\in\mathbb{N^{+}}, \end{align*} where $a(z)$ is rational, $R(z,w(z))=\frac{P(z,w(z))}{Q(z,w(z))}$ is an irreducible rational function in $w$ with rational coefficients in $z$. This paper mainly show the relationships of the degree of $P(z,w(z))$ and $Q(z,w(z))$ when the above equations exist such solutions $w(z)$. There are also some examples to show that our results are sharp.

This paper studies a model of $n$ insiders with perfect information trading on a risky asset with value normally distributed and disclosed at a random deadline. We propose a concept of information protocol equilibrium under semi-strong efficient pricing, consisting of an $n-$profile of insider trading strategies with terminal residual information protocols, and find that if a common protocol on terminal residual information before trading is obeyed by all insiders, then, in the market with more than two insiders there exists a uique equilibrium only when it requires to release common partial information eventually, or it does not exist if it requires to release all or not any; but in the market with a single insider, the insider may release all private information eventually to make a maximal profit. Thereby, the existence and uniqueness of information protocol equilibrium among $n$ insiders are deduced. Finally, numerical results illustrate some market characteristics of equilibria with different information protocols required before trading.

We consider a system of $k$-Hessian equations: $$ \left\{ \begin{aligned} &S_{k}(\lambda (D^{2}u))=(-u)^{\alpha_{1}}+(-v)^{\beta_{1}},&\quad &{\rm in}\ B,\\ &S_{k}(\lambda (D^{2}v))=(-u)^{\alpha_{2}}, &\quad &{\rm in}\ B,\\ &u=v=0, &\quad \ &{\rm on}\ \partial B,\\ \end{aligned} \right. $$ where $1\leq k\leq n\ (n\geq2),\ \alpha_{1},\alpha_{2}$ and $\beta_{1}$ are positive constants, $B=\{x\in \mathbb{R}^{n}:|x|<1\}$. By giving the complete classification for the constants $\alpha_1$, $\alpha_2$ and $\beta_1$ according to the value of $k$, some sharp conditions are obtained for the existence, uniqueness and nonexistence results of $k$-convex solutions to the above problem.

In this paper, we explore the existence of analytic solutions for an iterative functional equation of the form $$g(g(x))+x=f(g(x))$$ that originates from Painlevé equations. By an invertible transformation, we study the analytic solutions of an auxiliary equation under three different cases, and obtain the invertible analytic solutions for the original equation.