On the 2-adic valuation of the cardinality of elliptic curves over finite extensions of $$\mathbb {F}_{\varvec{q}} $$

In this paper we study the difference between the 2-adic valuations of the cardinalities \( \# E( \mathbb {F}_{q^k} ) \) and \( \# E( \mathbb {F}_q ) \) of an elliptic curve E over \( \mathbb {F}_q \). We also deduce information about the structure of the 2-Sylow subgroup \( E[ 2^\infty ]( \mathbb {F}_{q^k} ) \) from the exponents of \( E[ 2^\infty ]( \mathbb {F}_q ) \).     

Map determined by rank-$$\varvec{}$$ matrice for relatively mall $$\varvec{}$$

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \).     

Ground State Solutions for Asymptotically Periodic Kirchhoff-Type Equations with Asymptotically Cubic or Super-cubic Nonlinearities

This paper is concerned with the following Kirchhoff-type equation

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3}|\nabla {u}|^2\mathrm {d}x\right) \triangle u+V(x)u=f(x, u), \quad x\in \mathbb {R}^{3}, \end{aligned}$$

where \(V\in \mathcal {C}(\mathbb {R}^{3}, (0,\infty ))\), \(f\in \mathcal {C}({\mathbb {R}}^{3}\times \mathbb {R}, \mathbb {R})\), V(x) and f(x, t) are periodic or asymptotically periodic in x. Using weaker assumptions \(\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s)\mathrm {d}s}{|t|^3}=\infty \) uniformly in \(x\in \mathbb {R}^3\) and

$$\begin{aligned}&\left[ \frac{f(x,\tau )}{\tau ^3}-\frac{f(x,t\tau )}{(t\tau )^3} \right] \mathrm {sign}(1-t) +\theta _0V(x)\frac{|1-t^2|}{(t\tau )^2}\ge 0, \quad \\&\quad \forall x\in \mathbb {R}^3,\ t>0, \ \tau \ne 0 \end{aligned}$$

with a constant \(\theta _0\in (0,1)\), instead of the common assumption \(\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s)\mathrm {d}s}{|t|^4}=\infty \) uniformly in \(x\in \mathbb {R}^3\) and the usual Nehari-type monotonic condition on \(f(x,t)/|t|^3\), we establish the existence of Nehari-type ground state solutions of the above problem, which generalizes and improves the recent results of Qin et al. (Comput Math Appl 71:1524–1536, 2016) and Zhang and Zhang (J Math Anal Appl 423:1671–1692, 2015). In particular, our results unify asymptotically cubic and super-cubic nonlinearities.

     

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let \(p\in (1,\infty )\) and \(q\in [1,\infty )\). In this article, the authors characterize the Triebel-Lizorkin space \({F}^{\alpha }_{p,q}(\mathbb {R}^{n})\) with smoothness order α ∈ (0, 2) via the Lusin-area function and the \(g_{\lambda }^{*}\)-function in terms of difference between f(x) and its ball average \(B_{t}f(x):=\frac 1{|B(x,t)|}{\int }_{B(x,t)}f(y)\,dy\) over the ball B(x, t) centered at \(x\in \mathbb {R}^{n}\) with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) via pointwise inequalities, involving ball averages, in spirit close to Haj?asz gradients, here some interesting phenomena naturally appear that, in the end-point case when α = 2, some of these pointwise inequalities characterize the Triebel-Lizorkin spaces \(F^{2}_{p,2}(\mathbb {R}^{n})\), while not \(F^{2}_{p,\infty }(\mathbb {R}^{n})\), and that some of other obtained pointwise characterizations are only known to hold true for \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) with \(p\in (1,\infty )\), α ∈ (0, 2) or α ∈ (n/p, 2). In particular, some new pointwise characterizations of Haj?asz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than 1 on spaces of homogeneous type.     

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)

     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in $$\mathbb {R}^{N}$$

In this article we study the problem

$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$

where \(\Delta ^{2}:=\Delta (\Delta )\) is the biharmonic operator, \(a,b>0\) are constants, \(N\le 7,\) \(p\in (4,2_{*})\) for \(2_{*}\) defined below, and \(V(x)\in C(\mathbb {R}^{N},\mathbb {R})\). Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.

     

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

Dispersion and Limit Theorems for Random Walks Associated with Hypergeometric Functions of Type <Emphasis Type="Italic">BC</Emphasis>

The spherical functions of the non-compact Grassmann manifolds \(G_{p,q}({\mathbb {F}})=G/K\) over the (skew-)fields \({\mathbb {F}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}\) with rank \(q\ge 1\) and dimension parameter \(p>q\) can be described as Heckman–Opdam hypergeometric functions of type BC, where the double coset space G /  / K is identified with the Weyl chamber \( C_q^B\subset {\mathbb {R}}^q\) of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. Rösler and the author in an explicit way such that both formulas can be extended analytically to all real parameters \(p\in [2q-1,\infty [\), and that associated commutative convolution structures \(*_p\) on \(C_q^B\) exist. In this paper, we study the associated moment functions and the dispersion of probability measures on \(C_q^B\) with the aid of this generalized integral representation. This leads to strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on \((C_q^B, *_p)\) where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitly. For integers p, all results have interpretations for G-invariant random walks on the Grassmannians G / K. Besides the BC-cases, we also study the spaces \(GL(q,{\mathbb {F}})/U(q,{\mathbb {F}})\), which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case \(q=1\), the results of this paper are well known in the context of Jacobi-type hypergroups on \([0,\infty [\).     

On the semiclassical functional calculus for <Emphasis Type="Italic">h</Emphasis>-dependent functions

We study the functional calculus for operators of the form \(f_h(P(h))\) within the theory of semiclassical pseudodifferential operators, where \(\{f_h\}_{h\in (0,1]}\subset \mathrm{C^\infty _c}({{\mathbb {R}}})\) denotes a family of h-dependent functions satisfying some regularity conditions, and P(h) is either an appropriate self-adjoint semiclassical pseudodifferential operator in \(\mathrm{L}^2({{\mathbb {R}}}^n)\) or a Schrödinger operator in \(\mathrm{L}^2(M), M\) being a closed Riemannian manifold of dimension n. The main result is an explicit semiclassical trace formula with remainder estimate that is well-suited for studying the spectrum of P(h) in spectral windows of width of order \(h^\delta \), where \(0\le \delta <\frac{1}{2}\).     

Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients

Let \(L=-\mathrm{div}(A\nabla )\) be a second order divergence form elliptic operator and A an accretive \(n\times n\) matrix with bounded measurable complex coefficients in \({\mathbb R}^n\). Let \(\nabla b\in L^n({\mathbb R}^n)\,(n>2)\). In this paper, we prove that the commutator generated by b and the square root of L, which is defined by \([b,\sqrt{L}]f(x)=b(x)\sqrt{L}f(x)-\sqrt{L}(bf)(x)\), is bounded from the homogenous Sobolev space \({\dot{L}}_1^2({\mathbb R}^n)\) to \(L^2({\mathbb R}^n)\).     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .

     

Multiple of Homoclinic Solutions for a Perturbed Dynamical Systems with Combined Nonlinearities

In this paper, we study the existence of multiple and infinite homoclinic solutions for the following perturbed dynamical systems

$$\begin{aligned} \ddot{x}+A\cdot \dot{x}-L(t)\cdot x+\nabla W(t,x)=f(t), \end{aligned}$$

where \(t\in {\mathbb R}, x\in {\mathbb R}^N,\) A is an antisymmetric constant matrix, the matrix L(t) is not necessary positive definite for all \(t\in {\mathbb R}\) nor coercive, the nonlinearity \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) involves a combination of superquadratic and subquadratic terms and is allowed to be sign-changing and \(f\in C({\mathbb R},{\mathbb R}^{N})\cap L^{2}({\mathbb R},{\mathbb R}^{N}).\) Recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results.

     

Riemann Localisation on the Sphere

This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\), \(d\ge 2\), we mean that for a suitable subset X of \(\mathbb {L}_{p}(\mathbb {S}^{d})\), \(1\le p\le \infty \), the \(\mathbb {L}_{p}\)-norm of the Fourier local convolution of \(f\in X\) converges to zero as the degree goes to infinity. The Fourier local convolution of f at \(\mathbf {x}\in \mathbb {S}^{d}\) is the Fourier convolution with a modified version of f obtained by replacing values of f by zero on a neighbourhood of \(\mathbf {x}\). The failure of Riemann localisation for \(d>2\) can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.     

Idempotent and <Emphasis Type="Italic">p</Emphasis>-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform \(\widehat{Q}\) of a quadratic function \(Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p\) satisfies \(|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}\) for all \(b\in {\mathbb F}_{p^n}\), where \(0\le s\le n-1\) is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class \(\mathcal {C}_1\) is defined for arbitrary n as \(\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}\), and the larger class \(\mathcal {C}_2\) is defined for even n as \(\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}\). For an odd prime p, the subclass \(\mathcal {D}\) of all p-ary quadratic functions is defined as \(\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}\). We determine the generating function for the distribution of the parameter s for \(\mathcal {C}_1, \mathcal {C}_2\) and \(\mathcal {D}\). As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case \(p > 2\), the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to \(\mathcal {C}_1\) and \(\mathcal {C}_2\) in terms of a generating function.     

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups

We consider the spherical complementary series of rank one Lie groups \(H_n={ SO }_0(n, 1; {\mathbb {F}})\) for \({\mathbb {F}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}\). We prove that there exist finitely many discrete components in its restriction under the subgroup \(H_{n-1}={ SO }_0(n-1, 1; {\mathbb {F}})\). This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of \(G_n=SU(n, 1)\), \(SU(n, 1)\times SU(n, 1)\) and SU(2n, 2) and by the branching of holomorphic representations under the corresponding subgroup \(G_{n-1}\).     

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let \((M,\Omega )\) be a connected symplectic 4-manifold and let \(F=(J,H) :M\rightarrow \mathbb {R}^2\) be a completely integrable system on M with only non-degenerate singularities. Assume that F does not have singularities with hyperbolic blocks and that \(p_1,\ldots ,p_n\) are the focus–focus singularities of F. For each subset \(S=\{i_1,\ldots ,i_j\}\), we will show how to modify F locally around any \(p_i, i \in S\), in order to create a new integrable system \(\widetilde{F}=(J, \widetilde{H}) :M \rightarrow \mathbb {R}^2\) such that its classical spectrum \(\widetilde{F}(M)\) contains j smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of \(\widetilde{F}\). Moreover the focus–focus singularities of \(\widetilde{F}\) are precisely \(p_i\), \(i \in \{1,\ldots ,n\} \setminus S\). The proof is based on Eliasson’s linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.     

Complete mappings and Carlitz rank

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any \(d\ge 2\) and any prime \(p>(d^2-3d+4)^2\) there is no complete mapping polynomial in \(\mathbb {F}_p[x]\) of degree d. For arbitrary finite fields \(\mathbb {F}_q\), we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if \(n<\lfloor q/2\rfloor \), then there is no complete mapping in \(\mathbb {F}_q[x]\) of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank \(n<\lfloor q/2\rfloor \) are from being complete, by studying value sets of \(f+x.\) We provide examples of complete mappings if \(n=\lfloor q/2\rfloor \), which shows that the above bound cannot be improved in general.     

Representation of real Riesz maps on a strong <Emphasis Type="Italic">f</Emphasis>-ring by prime elements of a frame

In classical topology, it is proved that for a topological space X, every bounded Riesz map \(\varphi :C (X) \rightarrow {\mathbb {R}}\) is of the from \({\hat{x}}\) for a point \(x\in X\). In this paper, our main purpose is to prove a version of this result by lattice-valued maps. A ring representation of the from \(A\rightarrow {\mathbb {R}}\) is constructed. This representation is denoted by \(\widetilde{p_c}\) that is an onto f-ring homomorphism for every \(p\in \Sigma L\), where its index c, denotes a cozero lattice-valued map. Also, it is shown that for every Riesz map \(\phi :A\rightarrow {\mathbb {R}} \) and \(c\in F(A, L)\) with specific properties, there exists \(p\in \Sigma L\) such that \(\phi =\phi (1)\widetilde{p_c}\).