Option pricing with discrete dividend payments is still a challenge. This paper proposes a novel model by taking the dividends into consideration, and establishes the option price theorem for obtaining the option price. Theoretical analysis shows that the proposed new model can fully take the impact of dividend payments on option price such as the dividend paying time, amount and number, and hence it can produce an accurate price for option. We also conduct a theoretical comparison of the pricing between the newly-proposed model and classic/benchmark, with which the relation and pricing differences between the new model and these models are deeply detected. The numerical results also show that the proposed model can produce highly accurate prices for options and has strong pricing robustness. Based on this, our model can be an excellent alternative of pricing European options written on the underlying asset paying discrete dividends.

Let $T\in \mathcal{B(H)}$ be a bounded linear operator on a complex Hilbert space $\mathcal{H}$. The properties of the generalized pencil $T=P +\alpha Q+\beta PQ$ of pair $(P, Q)$ of projections at $(\alpha, \beta)\in \mathbb{C}^2$ are investigated. Using Halmos decomposition theory for orthogonal projections we give some equivalent conditions for which $T$ is the generalized pencil and study the spectrum properties of this generalized pencil $T$. We prove that the generalized pencil $T$ is similar to a diagonal operator under some conditions. The spectrum relations among the generalized pencil $T$ and projections $P$, $Q$ are established. Further, we give the necessary and sufficient conditions under which the generalized pencil $T$ is a Fredholm operator, a compact operator or a selfadjoint operator, respectively. Finally, the generalized pencils of pairs of idempotents are studied.

It is conjectured by Professor Zhi-Wei Sun that for each given odd prime $p>100, $ there always exists an solution $(x,y,z)\in[1,p]^3$ to the Pythagoras equation $x^2+y^2=z^2$ such that $x,y,z$ are quadratic residues or non-residues modulo $p$ respectively (eight cases in total). In this paper, we are able to prove the above assertion for all sufficiently large primes $p$, and the method is based on the recent Burgess bound for character sums of forms in many variables due to Lillian B. Pierce and Junyan Xu.

We study perturbation bounds and convergence rates for uniformly ergodic Markov chains on general state spaces in terms of uniform moments of the first hitting times on some set. For reversible and non-negative definite Markov chains, we first investigate the geometrically ergodic convergence rates by the spectral theory. Based on the estimates, together with a first passage formula, we then get the convergence rates and perturbation bounds of uniform ergodicity. If the Markov chain is only reversible, we transfer to study the skeleton chain with transition kernel P². Finally, we investigate perturbation bounds for general Markov chains.

The Chinese mathematician Jiaxi Lu introduced the concepts of LD design and LD^* design in the process of resolving the existence problem of large sets of Steiner triple systems (LSTSs). Lu also presented several recursive constructions and direct constructions for the two types of designs, which played vital roles in producing LSTSs. In order to deal with the remaining six possible exceptions for the existence of LSTSs, Teirlinck still employed LD designs in the recursive constructions using pairwise balanced designs and then he resolved the problem of LSTSs completely. This paper proves that the necessary conditions are all sufficient for the existence of LD designs and there leaves only four possible exceptions for LD^* designs.

This paper considers a supercritical Galton-Watson process with immigration $\{X_n\}_{n\geq0}$. It is well-known that there is a sequence of constants $\{c_n\}_{n\geq0}$ such that $X_n/c_n\to V$ almost surely as $n\rightarrow\infty$. Using Cramér transforms, we obtain lower deviations for the process $\{X_n\}_{n\geq0}$, which refer to the asymptotic properties of $P(X_n=k)$ for sufficiently large $k$ satisfying $k_n\leq k\leq c_n$ and $k_n\rightarrow \infty$.

Counting zeros of polynomials over finite fields is one of the most important topics in arithmetic algebraic geometry. In this paper, we consider the problem for complete symmetric polynomials. The homogeneous complete symmetric polynomial of degree $m$ in the $k$-variables $\{x_1,x_2,\ldots,x_k\}$ is defined to be $h_m(x_1,x_2,\ldots$, $x_k):=\sum_{1\leq i_1\leq i_2\leq \cdots \leq i_m\leq k}x_{i_1}x_{i_2}\cdots x_{i_m}.$ A complete symmetric polynomial of degree $m$ over $\mathbb{F}$_q in the $k$-variables $\{x_1,x_2,\ldots,x_k\}$ is defined to be $h(x_1,\ldots$, $x_k):=\sum_{e=0}^m a_eh_e(x_1,x_2,\ldots$, $x_k),$ where $a_e\in$ $\mathbb{F}$_q and $a_m\not=0$. Let $N_q(h):= \#\{(x_1,\ldots, x_k)\in$ $\mathbb{F}$_q |$ h(x_1,\ldots, x_k)=0\}$ denote the number of $\mathbb{F}$_q-rational points on the affine hypersurface defined by $h(x_1,\ldots, x_k)=0.$ In this paper, we improve the bounds given in [J. Zhang and D. Wan, "Rational points on complete symmetric hypersurfaces over finite fields", Discrete Mathematics, 343(11): 112072, 2020] and [D. Wan and J. Zhang, "Complete symmetric polynomials over finite fields have many rational zeros" Scientia Sinica Mathematica, 51(10): 1677-1684, 2021]. Explicitly, we obtain the following new bounds:
(1) Let $h(x_1,\ldots, x_k)$ be a complete symmetric polynomial in $k\geq 3$ variables over $\mathbb{F}$_q of degree $m$ with $1\leq m\leq q-2$. If $q$ is odd, then $N_q(h)\geq\!\frac{\lceil \frac{q-1}{m+1}\rceil}{q-\lceil \frac{q-1}{m+1}\rceil}(q-m-1)q^{k-2}.$
(2) Let $h(x_1,\ldots, x_k)$ be a complete symmetric polynomial in $k\geq 4$ variables over $\mathbb{F}$_q of degree $m$ with $1\leq m\leq q-2$. If $q$ is even, then $N_q(h)\geq\!\frac{\lceil \frac{q-1}{m+1}\rceil}{q-\lceil \frac{q-1}{m+1}\rceil}(q-\frac{m+1}{2})(q-1)q^{k-3}.$\newline Note that our new bounds roughly improve the bounds mentioned in the above two papers by the factor $\frac{q^2}{6m}$ for small degree $m$.

In this paper we study some determinants and permanents. In particular, we investigate the new-type determinants $$\det [(i^2+cij+dj^2)^{p-2}]_{0≤ i,j≤ p-1}{and}det [(i^2+cij+dj^2)^{p-2}]_{1≤ i,j≤ p-1} $$ modulo an odd prime $p$, where $c$ and $d$ are integers. We also pose some conjectures for further research.

In this paper, we study the estimation of the partial linear spatial autoregressive model with the response variable missing at random. Based on the nonparametric B-spline method, the marginal maximum Likelihood estimation and the EM algorithms for the maximum Likelihood estimation and the pseudo-restricted maximum Likelihood estimation are proposed, respectively. Numerical simulations are carried out under different sample sizes, missing rates and spatial weight matrix settings to compare the performances of the three methods. Finally, the effectiveness of the three estimation methods are verified by a real data analysis.

Let $A$ be a free abelian group of rank $n$. It is well known that the automorphism group $\operatorname{Aut}(A)$ of $A$ is $\operatorname{GL}(n,\mathbb{Z})$. Let $f(\lambda)=\lambda^{n}+a_{n-1}\lambda^{n-1}+\cdots+a_{1}\lambda+a_{0}$ be an irreducible polynomial in $\mathbb{Z}[\lambda]$, where $a_{0}=\pm1$. Let $T=\langle\alpha\rangle$ be an infinite cyclic group. Let $\alpha$ act on $A$ via the automorphism of $A$ induced by the Frobenius companion matrix of the monic polynomial $f(\lambda)$. Assume that $G=A\rtimes T$. Let $p$ be a prime. We prove that $G$ is a residually-finite $p$-group if and only if $p$ divides $f(1)$.

In this article, the concept of lightlike growth surface is proposed by evolving a lightlike curve as dictated direction and growth velocity in Minkowski 3-space. The geometric structure of the lightlike growth surfaces are investigated by the aid of the structure function of its generating lightlike curve. Meanwhile, the expression forms of the lightlike growth surfaces initiated by the lightlike helices are explored accompanied with several typical examples to characterize the growth process of such kind of surfaces explicitly.

We mainly consider the existence of random uniform exponential attractors in the weighted space of infinite sequences for second order lattice systems with quasi-periodic forces and multiplicative white noise. We first present some sufficient conditions for the existence of a random uniform exponential attractor for a jointly continuous random dynamical system defined on a product space of weighted space of infinite sequences. Secondly, by using Ornstein-Uhlenbeck process, a reversible variable substitution is constructed to transform the stochastic second-order lattice system (SDE) with white noise into a random system (RDE) without white noise, whose solutions generate a jointly continuous random dynamical system. Then we verify the Lipschitz continuity of the jointly continuous random dynamical system and decompose the difference between the two solutions of system into a sum of the two parts, and estimate the expectations of some random variables. Finally, we obtain the existence of random uniform exponential attractors for the considered system.

We give the definition of geominimal L_p (p ≥ 1) integral curvature. We mainly prove the existence and uniqueness of the L_p entropy Petty body for the convex body containing the origin in its interiors. Moreover, we also study the continuity of the geominimal L_p integral curvature and the L_p entropy Petty body.

Constructing examples of groups is an important aspect in the theory of groups. We will study the residual finiteness of two concrete matrix groups. Let $p$ be a prime, let $C=\langle c\rangle$ be an infinite cyclic group, let $R=\mathbb{Z}C$ be the integral group ring over $C$, and let $U(n,R)$ be the upper unitriangular group over $R$ of order $n$, where $n\geq 2$, which is a nilpotent group of infinite rank of class $n-1$. Firstly, we prove that $U(n,R)$ is a residually finite $p$-group. Secondly, let $ G=\langle\alpha\rangle\ltimes U(3,R)$, where $\alpha={\rm diag}(c,1,c)$ is a diagonal matrix of order 3. We will study the structure of $G$ and prove that $G$ is a residually finite $p$-group, $G$ is a 3-generated soluble group of derived length 3. Moreover, we will construct two quotient groups of $G$, neither of which is residually finite. These two quotient groups seem to be more elementary and concrete than the classical examples discovered by Hall.

The curvature integral inequalities play an important role in geometric inequalities. In this paper, we first obtain an integral inequality about periodic functions by using the Fourier analysis method. Furthermore, we obtain the strengthened form of the famous Ros inequality on the plane. On the other hand, by applying the obtained lemma, we combine Green-Osher inequality with Steiner polynomial, then the curvature integral inequalities of higher power of planar convex curve are obtained. These inequalities are generalizations and improvements of known Green-Osher inequalities on the Euclidean plane.

In this paper, using complex analysis methods, we will show that an $h^p$ harmonic function $u$ can be decomposed into the sum of one singular function and one absolutely continuously function on unit ball $B$ of $\mathbb{R}^n$ for $p\geq 1.$ Then we will obtain the corresponding results of functions in $h^p$ space of the upper half space of $\mathbb{R}^n$ by the Kelvin transform.

In this paper, we explicitly compute the Kodaira-Spencer map over a quaternionic Shimura curve over the field of rational numbers, and also compute its effect on the metrics of the Hodge bundle. The former is based on moduli interpretation and deformation theory, and the latter is based on the theory of complex abelian varieties.

Projective codes over finite fields have important applications in combinatorial designs and strongly regular graphs. In this paper, we first construct a family of linear codes and then study their parameters and weight distributions in four cases. It turns out that the proposed linear codes are projective and are optimal in two cases. The duals of these codes are either optimal or almost optimal according to the sphere-packing bound. As applications, these codes are used to construct $t$-designs and strongly regular graphs.

Quantum detection investigates the injectivity of quantum measurements on quantum states. Since every measurement of quantum system can be characterized by a positive operator-valued measure (POVM), and every Parseval frame corresponds to a rank-one POVM. In this paper, we mainly consider the quantum injectivity problem of Gabor frames, and give a sufficient condition for the quantum injectivity of a Gabor frame $\left\{\pi (m,n) \varphi \right\}_{(m,n) \in \Lambda}$, namely it is a full Gabor frame and satisfies $\langle \pi (m,n) \varphi,\varphi \rangle \ne 0$ for $m=0,\ 1 \le n \le \frac{N}{2}$, $1 \le m \le \frac{N-1}{2},\ 0 \le n \le N-1$ and $\frac{N-1}{2} < m \le \frac{N}{2},\ 0 \le n \le \frac{N}{2}$. We also give its stability with a quantitative error estimate and its applications for low dimensional cases.

We prove that there is a positive proportion of positive integers which can be uniquely represented as the sum of a Fibonacci number and a prime. We also study the integers of the form $p+a_k$, where $p$ is a prime and $\{ a_k\}$ is an exponential type sequence of integers.

In this paper, we first introduce the concept of primitive Lie superalgebras and investigation three types of primitive Lie superalgebra and some related structural properties. Then, the concept of chief factors of Lie superalgebra is introduced. According to the properties of the third type of primitive Lie superalgebra, the L connection relation between the chief factors of Lie superalgebra is given. Finally, we introduce the CAP-subalgebras of Lie superalgebra L, and prove that L is solvable if all maximal subalgebras of Lie superalgebra are CAP-subalgebras.

We consider the uniqueness of the meromorphic solution $f(z)$ of the second order linear difference equation $ p_2(z)y(z+2)+p_1(z)y(z+1)+p_0(z)y(z)=0, $ where $p_2(z), p_1(z), p_0(z)$ are nonzero polynomials with $p_2(z)+p_1(z)+p_0(z)\not\equiv0$. We give the forms of $f(z)$ if $f(z)$ shares $0, 1, \infty$ CM with any meromorphic function $g(z)$. Furthermore, if $g(z)$ is also a solution of the above equation, we obtain the exact forms of this equation. As a corollary, we see that if a meromorphic function $g(z)$ shares $0, 1, \infty$ CM with the gamma function $\Gamma(z)$, then $g(z)\equiv \Gamma (z)$.

The main purpose of this paper is to give the expressions of meromorphic solutions of the following non-linear differential equation \begin{equation*}f^{n}(z)+P_{d}(z,f)=p_{1}{\rm e}^{\alpha_{1}z}+p_{2}{e}^{\alpha_{2}z}+p_{3}{\rm e}^{\alpha_{3}z}\end{equation*} under certain conditions, where n ≥ 3 is an integer, $P_{d}(z,f)\not\equiv0$ is a differential polynomial in f of degree d ≤ n - 1 with small function coefficients, p_j (j = 1, 2, 3) are non-zero constants, α_j (j = 1, 2, 3) are three distinct non-zero constants. Moreover, some examples are given to illustrate our results.

The main purpose of this paper is using the elementary methods and the properties of the solutions of some congruence equations to study the calculating problem of a certain fourth power means of the generalized exponential sums, and give three interesting identities for them.

In this note, we confirm a conjecture on the existence of test functions for trilinear zeta integrals with regular support, for representations with maximal exponent strictly less than 1/22.

We mainly use the nonnegative continuous function $\{\zeta_n(r)\}_{n\ge 0}$ to redefine the Bohr radius for the class of analytic functions satisfying ${\rm Re} f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of the alternating series $A_f(r)$ with analytic functions $f$ of the form $f(z)=\sum_{n=0}^{\infty}a_{pn+m}z^{pn+m}$ in $|z|<1$. In the latter case, one can also get information about Bohr radius for even and odd analytic functions. Moreover, the relationships between the majorant series $M_f(r)$ and the odd and the even bits of $f(z)$ are also established. We will prove that most of results are sharp.

We introduce a new Tseng’s extragradient algorithm for solving pseudomonotone variational inequality problems in Hilbert spaces. We prove that the sequence generated by our proposed algorithm converges strongly to an element of solution set for variational inequality problems. The results obtained in this paper extend and improve many recent ones in the literature.

Composite quantile regression has good properties in robustness and estimation efficiency. For the longitudinal data single-index models, we propose profile composite quantile regression based estimating equations and smooth-threshold variable selection methods. The new methods can incorporate the intra-subject correlation by using copula functions, and inherit the advantages of composite quantile regression. Under some mild conditions, the asymptotical properties are established. Simulation studies and real data analysis are included to illustrate the finite sample performance.

We investigate the asymptotic approximation of solutions of initial and boundary value problems for set-valued differential equations with maxima. By introducing the concepts of Hausdorff measure and semideviation measure, the approximate relationship of the solutions between the original equations and the average equations is discussed via the averaging method when the average limit of the right-hand function exists or does not exist.

In this paper, by establishing new monotone formulas along the shortening flows for plane curves, we present new proofs for three geometric inequalities. In particular, a new proof of the Ros theorem on $\mathbb R^2$ is given by the classical curve shortening flow. And new proofs of the Ros theorem on $\mathbb R^2$ and its refined form and an entropy inequality for plane curve are given by an area-preserving curve shortening flow.

Let $k\in \{5, 6\}$ and $\eta$ be any given real number. Suppose that $\lambda_1, \lambda_2, \ldots, \lambda_7$ are nonzero real numbers, not all of the same sign and $\lambda_1/\lambda_2$ is irrational. It is proved that the inequality $|\lambda_1x_1^2+\lambda_2x_2^3+\lambda_3x_3^3+\lambda_4x_4^3+\lambda_5x_5^3+\lambda_6x_6^4+\lambda_7x_7^k+\eta|<(\max_{1\leq j\leq 7} x_j)^{-\sigma}$ has infinitely many solutions in positive integers $x_1, x_2, \ldots, x_7 $ for $0<\sigma<\frac{1}{12(k-3)}$. This result constitutes an improvement upon that of Li and Gong.

We further develop the theory of $W$-graph ideals in a Coxeter system $(W,S)$. We mainly study the structural coefficients of the corresponding modules, the direct and iterative algorithms for the canonical basis elements. Compared with standard recursive algorithms, this algorithm has the advantage of fast computation and memory saving when computing specific canonical basis elements. Due to the generality of the concept of $W$-graph ideal, our results are also the generalizations of those in some classical cases.

This paper studies a novel dynamic single index varying coefficient quantile regression model, which reflects the dynamic interaction between explanatory variables and the response variable, and covers many important models as special cases. In order to improve the interpretability and estimation accuracy, this paper further discusses the semi-varying structure of the model. Firstly, we use the B-spline method to obtain the estimators of the varying coefficient function and the index function. Secondly, the semi-varying model is identified based on the penalty function method. We also propose an estimation procedure for this semi-parametric model. In addition, We establish the consistency and asymptotic normality of each estimator, and both parametric and non-parametric estimators can achieve the optimal convergence rate. Numerical simulations show that the proposed models and estimation methods enjoy excellent properties. Finally, we analyze a NO$_2$ data set to demonstrate the performance of the proposed method in practical applications.

We systematically study the recurrent set and non recurrent set of the continuous time Markov process in the general state space, and focus on the determination methods of the recurrent set and non recurrent set of the Markov process, which provides a strong support for the study of the recurrence of the continuous time Markov process in the general state space.

We study some properties of mean linear independence for the bundle map induced by a diffeomorphisim of smooth Riemannian manifold. We first define a $\theta$ function by using the distances between vectors in frames and then give an equivalent definition of mean linear independence. At the same time, a variational principle on $\theta$ function is also investigated. Second, we prove that any tangent vector belongs to a mean linear independent $k$-frame, where $k$ is the style number of the system at the base point of the vector.

Quantum toroidal algebras or double affine quantum algebras are special cases ($N=2$) of the recently defined quantum $N$-toroidal algebras, which generalize the toroidal Lie algebras and $N$-toroidal Lie algebras. In this paper, we will construct a level one representation of the quantum $N$-toroidal algebra for the exceptional type $G_2$, which can be regarded as a generalization of the basic representation of the quantum affine algebra in type $G_2$.

In this paper, aggregate zero density bounds for a family of automorphic $\mathrm{L}$-functions are deduced from bounds for a sum of integral power moments of such $\mathrm{L}$-functions. More precisely, let $I$ be a set of certain automorphic representations $\pi$, and let $c(\pi)$ be a non-negative coefficient for each $\pi\in I$ such that $\sum_{\pi\in I}c(\pi)$ converges. Assume that \begin{equation*} \sum_{\pi\in I} c(\pi) \int_T^{T+T^\alpha} \bigg| \mathrm{L}\bigg(\frac12+{\rm i}t,\pi\bigg) \bigg|^{2\ell} dt \ll_\varepsilon T^{\theta+\varepsilon} \sum_{\pi\in I} c(\pi) \end{equation*} for certain $\ell\geq1$, $0<\alpha\leq1$ and $\theta\geq\alpha$. Upper bounds for the following aggregate zero density \begin{equation*} \sum_{\pi\in I} c(\pi) N_\pi(\sigma,T,T+T^\alpha) \end{equation*} will be proved, where $N_\pi(\sigma,T_1,T_2)$ is the number of zeros $\rho=\beta+{\rm i}\gamma$ of $\mathrm{L}(s,\pi)$ in $\sigma<\beta<1$ and $T_1\leq\gamma\leq T_2$.

We study the variable selection problem of the spatial autoregressive quantile model with fixed effects. By penalizing the relevant parameters, we can identify the spatial effects, estimate the unknown parameters and select the explanatory variables simultaneously. In addition, we give an algorithm of variable selection and prove the large sample property of penalty estimator. Numerical simulation and real data analysis show the excellent performance of the proposed method.

In this paper, we prove that the unique nonnegative equilibrium of a rational difference equation with higher order is globally attractive. As application, our results not only improve many known ones, but also solve several open problems and conjectures.