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We construct a two discrete parameter family of compact minimal surfaces embedded in the Berger sphere which may be considered as the analogue of the helicoidal Karcher-Scherk surfaces.

This appendix gives a lower bound of the Billingsley-Hausdorff dimension of a level set related to Birkhoff average in the “non-compact” symbolic space $\mathbb{N}^{\mathbb{N}}$, defined by Gibbs measure.

In 1985 Crapo introduced in [3] a new mathematical object that he called geometry of circuits. Four years later, in 1989, Manin and Schechtman defined in [13] the same object and called it discriminantal arrangement, the name by which it is known now a days. Those discriminantal arrangements $\mathcal{B}(n,k,\mathcal{A}^0)$ are builded from an arrangement $\mathcal{A}^0$ of $n$ hyperplanes in general position in a $k$-dimensional space and their combinatorics depends on the arrangement $\mathcal{A}^0$. On this basis, in 1997 Bayer and Brandt (see [2]) distinguished two different type of arrangements $\mathcal{A}^0$ calling very generic the ones for which the intersection lattice of $\mathcal{B}(n,k,\mathcal{A}^0)$ has maximum cardinality and non-very generic the others. Results on the combinatorics of $\mathcal{B}(n,k,\mathcal{A}^0)$ in the very generic case already appear in Crapo [3] and in 1997 in Athanasiadis [1] while the first known result on non-very generic case is due to Libgober and the first author in 2018. In their paper [12] they provided a necessary and sufficient condition on $\mathcal{A}^0$ for which the cardinality of rank 2 intersections in $\mathcal{B}(n,k,\mathcal{A}^0)$ is not maximal anymore. In this paper we further develop their result providing a sufficient condition on $\mathcal{A}^0$ for which the cardinality of rank r, $r \ge 2$, intersections in $\mathcal{B}(n,k,\mathcal{A}^0)$ decreases.

The content of this paper is a generalization of a theorem by Joseph Rabinoff: if P is a finite family of pointed and rational polyhedra in Nℝ such that there exists a fan in Nℝ that contains all the recession cones of the polyhedra of P, if k is a complete non-archimedean field, if S is a connected and regular k-analytic space (in the sense of Berkovich) and Y is a closed k-analytic subset of U P × k S which is relative complete intersection and contained in the relative interior of U P × k S over S, then the quasifiniteness of π : Y → S implies its flatness and finiteness; moreover, all the finite fibers of π have the same length. This namely gives a analytic justification to the concept of stable intersection used in the theory of tropical intersection. [ABSTRACT FROM AUTHOR]

Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite.

The paper summarizes the main core of the last results that we obtained in [8, 4, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann–Du-Bois Reymond equation.Under our assumptions, there exists a minimizing sequence of Lipschitz functions. A first consequence is that we can exclude the presence of the Lavrentiev phenomenon. Moreover, under a further mild growth assumption satisfied by the minimal length functional, fully described in the paper, the above sequence may be taken with the same Lipschitz rank, even when the initial datum and initial value vary on a compact set. The Lipschitz regularity of the value function follows.

In this paper, we show that there are solutions of degree r of the equation of Pell–Abel on some real hyperelliptic curve of genus g if and only if r > g . This result, which is known to the experts, has consequences, which seem to be unknown to the experts. First, we deduce the existence of a primitive kdifferential on an hyperelliptic curve of genus g with a unique zero of order k(2g −2) for every (k, g ) 6= (2, 2). Moreover, we show that there exists a non Weierstrass point of order n modulo a Weierstrass point on a hyperelliptic curve of genus g if and only if n > 2g . [ABSTRACT FROM AUTHOR]

Our aim in this paper is to show that the modulus of smoothness and the $K$-functionals constructed from the Sobolev-type space corresponding to the Dunkl operator are equivalent on the interval $(-1,1)$.

Let $R$ be a commutative noetherian local ring which is singular and has an isolated singularity. Let $\mathsf {D_{sg}}(R)$ be the singularity category of $R$ in the sense of Buchweitz and Orlov. In this paper, we find real numbers $t$ such that the complexity $\delta _t(G,X)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich vanishes for any split generator $G$ of $\mathsf {D_{sg}}(R)$ and any object $X$ of $\mathsf {D_{sg}}(R)$. In particular, the entropy $\mathrm{h}_t(F)$ of an exact endofunctor $F$ of $\mathsf {D_{sg}}(R)$ is not defined for such numbers $t$.

Understanding the behavior of a black-box model with probabilistic inputs can be based on the decomposition of a parameter of interest (e.g., its variance) into contributions attributed to each coalition of inputs (i.e., subsets of inputs). In this paper, we produce conditions for obtaining unambiguous and interpretable decompositions of very general parameters of interest. This allows recovering known decompositions, holding under weaker assumptions than the literature states.

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek in [26]. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen [24], Gedrich and Gruenberg [17] that were originally proved in the realm of commutative Noetherian rings.

The optimal transport map between the standard Gaussian measure and an $\alpha $-strongly log-concave probability measure is $\alpha ^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp–Lieb and Cramér–Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli’s original result. We also extend Caffarelli’s theorem to the setting in which the Hessians of the log-densities of the measures are bounded by arbitrary positive definite commuting matrices.

This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature.

The last decade has seen an abundance of congruences for $b_\ell (n)$, the number of $\ell $-regular partitions of $n$. Notably absent are congruences modulo $4$ for $b_3(n)$. In this paper, we introduce Ramanujan type congruences modulo $4$ for $b_3(2n)$ involving some primes $p$ congruent to $ 11, 13, 17, 19, 23$ modulo $24$.

A major problem in knot theory is to decide whether the Jones polynomial detects the unknot. In this paper we study a weaker related problem, namely whether the Jones polynomial reduced modulo an integer m detects the unknot. The answer is known to be negative for m = 2r with r ≥ 1 and m = 3. Here we show that if the answer is negative for some m, then it is negative for mr with any r ≥ 1. In particular, for any r ≥ 1, we construct nontrivial knots whose Jones polynomial is trivial modulo 3r. [ABSTRACT FROM AUTHOR]

A conjecture by the second author, proven by Bonnafé–Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra.In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $\mathfrak{m}$ of $R$ satisfies the rank-one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Sklyanin algebras.In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank-one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.

Let $\mathrm{F}$ be a non-archimedean locally compact field and let $\mathrm{G}_{n}$ be the group $\mathrm{PGL}_{n}(\mathrm{F})$. In this paper we construct a tower $(\tilde{\mathrm{X}}_{k})_{k\geqslant 0}$ of graphs fibred over the one-skeleton of the Bruhat–Tits building of $\mathrm{G}_{n}$. We prove that a non-spherical and irreducible generic complex representation of $\mathrm{G}_{n}$ can be realized as a quotient of the compactly supported cohomology of the graph $\tilde{\mathrm{X}}_{k}$ for $k$ large enough. Moreover, when the representation is cuspidal then it has a unique realization in a such model.

In this paper, we study the alternating Euler $T$-sums and $\tilde{S}$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $\tilde{S}$-sums, from which, the parity theorems of Hoffman’s double and triple $t$-values and Kaneko–Tsumura’s double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $\tilde{S}$-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.

In this paper, we present certain results concerning the distribution of zeros of polynomials of a quaternionic variable and with quaternionic coefficients. We obtain ring shaped regions of Eneström–Kakeya type for the zeros of these polynomials and also extend some classical results from the complex to quaternionic setting.

In this paper, we introduce and study a set relative to singularities of plurisubharmonic functions. We prove that this set is countable under the condition $h>0$ on $\mathbb{B}\setminus \lbrace 0\rbrace .$

In this paper, we consider a general single species model in a heterogeneous environment of $n$ patches ($n\ge 2$), where each patch follows a generalized logistic law. First, we prove the global stability of the model. Second, in the case of perfect mixing, i.e. when the migration rate tends to infinity, the total population follows a generalized logistic law with a carrying capacity which in general is different from the sum of the $n$ carrying capacities. Next, we give some properties of the total equilibrium population and we compute its derivative at no dispersal. In some particular cases, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might be greater or smaller than the sum of the $n $ carrying capacities. Finally, we study an example of two-patch model where the first patch follows a logistic law and the second a Richard’s law, we give a complete classification of the model parameter space as to whether dispersal is beneficial or detrimental to the sum of two carrying capacities.

The present paper deals with lines contained in a smooth complex cubic threefold. It is well-known that the set of lines of the second type on a cubic threefold is a curve on its Fano surface. Here we give a description of the singularities of this curve.

In a recent paper, Vishik proved the global well-posedness of the two-dimensional Euler equation in the critical Besov space B2,12. In the present paper we prove that Navier–Stokes system is globally well-posed in B2,12, with uniform estimates on the viscosity. We prove also a global result of inviscid limit. The convergence rate in L2 is of order ν. To cite this article: T. Hmidi, S. Keraani, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

In this paper we define a continuous version of multiple zeta functions. They can be analytically continued to meromorphic functions on $\mathbb{C}^r$ with only simple poles at some special hyperplanes. The evaluations of these functions at positive integers (continuous multiple zeta values) satisfy the shuffle product. We give a detailed analysis about the depth structure of continuous multiple zeta values. There are also sum formulas for continuous multiple zeta values. Lastly we calculate some special continuous multiple zeta values in terms of special values of multiple polylogarithms.

This paper develops Rio’s method [11] to prove the weak law of large numbers for maximal partial sums of pairwise independent random variables. The method allows us to avoid using the Kolmogorov maximal inequality. As an application, a weak law of large numbers for maximal partial sums of pairwise independent random variables under a uniform integrability condition is also established. The sharpness of the result is illustrated by an example.

In this paper, we analyse the long-time behavior of solutions to a coupled system describing the motion of a rigid disk in a 2D viscous incompressible fluid. Following previous approaches in [4, 15, 17] we look at the problem in the system of coordinates associated with the center of mass of the disk. Doing so, we introduce a further nonlinearity to the classical Navier Stokes equations. In comparison with the classical nonlinearities, this new term lacks time and space integrability, thus complicating strongly the analysis of the long-time behavior of solutions.We provide herein two refined tools: a refined analysis of the Gagliardo–Nirenberg inequalities and a thorough description of fractional powers of the so-called fluid-structure operator [2]. On the basis of these two tools we extend decay estimates obtained in [4] to arbitrary initial data and show local stability of the Lamb-Oseen vortex in the spirit of [7, 8].

The Grassmann convexity conjecture, formulated in [8], gives a conjectural formula for the maximal total number of real zeroes of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. The conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group. The formula has already been shown to be a correct lower bound and to give a correct upper bound in several small dimensional cases. In this paper we obtain a general explicit upper bound.

We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schrödinger operator. They improve, in this particular situation, the results obtained in the companion paper [38].The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [40], together with estimates on the spectral gap [33, 1]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest.

The generalized Frobenius number is the largest integer represented in at most $p$ ways by a linear combination of nonnegative integers of given positive integers $a_1,a_2,\,\dots ,\,a_k$. When $p=0$, it reduces to the classical Frobenius number. In this paper, we give the generalized Frobenius number when $a_j=(b^{n+j-1}-1)/ (b-1)$ ($b\ge 2$) as a generalization of the result of $p=0$ in [16].

Let $\Psi $ be a non-constant complex-valued analytic function defined on a connected, open set containing the $L^p$-spectrum of the Laplacian $\mathcal{L}$ on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup $T(t)=e^{t\Psi (\mathcal{L})}$ to be chaotic on $L^{p}$-spaces. We also study the chaotic dynamics of the semigroup $T(t)=e^{t(a\mathcal{L}+b)}$ separately and obtain a sharp range of $b$ for which $T(t)$ is chaotic on $L^{p}$-spaces. It includes some of the important semigroups such as the heat semigroup and the Schrödinger semigroup.

This paper concerns the energy conservation for the weak solutions to the Navier–Stokes–Maxwell system. Although the Maxwell equation with hyperbolic nature, we still establish a $L^qL^p$ type condition guarantee validity of the energy equality for the weak solutions. We mention that there no regularity assumption on the electric field $E$.

In this paper, we consider the asymptotic behavior of solutions of Monge–Ampère equations with general boundary value conditions in half spaces, which reveals the accurate effect of boundary value condition on asymptotic behavior and improves the result in [13].

In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function $-T_{\nu ,\alpha ,\beta }(s)$ is completely monotonic in $s$ and absolutely monotonic in $\nu $ if and only if $\beta \ge 1$, where $T_{\nu ,\alpha ,\beta }(s)=K_{\nu }^2(s)-\beta K_{\nu -\alpha }(s)K_{\nu +\alpha }(s)$ defined on $s>0$ and $K_{\nu }(s)$ is the modified Bessel function of the second kind of order $\nu $. Finally, we determine the necessary and sufficient conditions for the functions $s \mapsto T_{\mu ,\alpha ,1}(s)/T_{\nu ,\alpha ,1}(s)$, $s \mapsto (T_{\mu ,\alpha ,1}(s) + T_{\nu ,\alpha ,1}(s))/(2T_{(\mu +\nu )/2,\alpha ,1}(s))$, and $s \mapsto \frac{\mathrm{d}^{n_1}}{\mathrm{d} \nu ^{n_1}} T_{\nu ,\alpha ,1}(s)/\frac{\mathrm{d}^{n_2}}{\mathrm{d} \nu ^{n_2}} T_{\nu ,\alpha ,1}(s)$ to be monotonic in $s\in (0,\infty )$ by employing the monotonicity rules.

This paper is concerned with the blow-up of solutions to the following hyperbolic-elliptic chemotaxis system: \begin{equation*} {\left\lbrace \begin{array}{ll} u_{t} =-\nabla \cdot (\chi u \nabla v)+g(u), \qquad x\in \Omega , \ t>0,\\ \;\;\; 0 =\Delta v-v+u, \hspace{58.33328pt}x\in \Omega , \ t>0, \end{array}\right.} \end{equation*} under homogeneous Neumann boundary conditions in a bounded domain $ \Omega \subset \mathbb{R}^{n}, n\ge 1,$ with smooth boundary and the function $g$ is assumed to generalize the logistic source: \begin{equation*} g(s)\le a s - b s^{\gamma }\ \text{ for} \ s>0 \end{equation*} with $1

We present in this paper a consistent nonparametric test for heteroscedasticity when data are of functional kind. The latter is constructed by evaluating the difference between the conditional and unconditional variances. We show the asymptotic normality of the statistical test under the null hypothesis. In addition, we prove that this test is consistent against all deviations from homoscedasticity condition. [ABSTRACT FROM AUTHOR]

Résumé Dans cet article, nous établissons une identité pour des polynômes d'Appell généralisant des formules explicites pour les nombres et polynômes de Bernoulli généralisés. In this paper, we establish an identity for some Appell polynomials generalizing explicit formulas for generalized Bernoulli numbers and polynomials. [ABSTRACT FROM AUTHOR]

Statistical physics and stochastic modelling in economic sciences share the same mathematical bases given by the Gibbs distribution, but system characteristics are different. For instance, an economic system can be described by a Bose–Einstein statistics with few non-degenerate states and an infinitesimal “temperature”; under such conditions, the approximation of the most probable configuration is invalid. Therefore, the calculus of the exact solution needs using a Metropolis algorithm, which estimates a Gibbs distribution. This paper presents a much more efficient algorithm. For small systems, the exact distribution on the canonical set can be computed, and then this distribution is compared to the solutions of the old and new algorithms. [ABSTRACT FROM AUTHOR]

In the framework of semiclassical resonances, we make more precise the link between the polynomial estimates of the extension of the resolvent and the propagation of the singularities through the trapped set. This approach makes it possible to eliminate infinity and to concentrate the study near the trapped set. It has allowed us in previous papers to obtain the asymptotic of resonances in various geometric situations. [ABSTRACT FROM AUTHOR]

In this paper, we disprove a recent monotonicity conjecture of C. Procesi on the generating function for monotone walks on the symmetric group, an object which is equivalent to the Weingarten function of the unitary group.

In this paper, we address the problem of estimating a covariance matrix of a multivariate Gaussian distribution, from a decision theoretic point of view, relative to a Stein type loss function. We investigate the case where the covariance matrix is invertible and the case when it is non–invertible in a unified approach.

In this paper, we prove the theorem announced in the title.

In this paper, we derive the $\Gamma $-limit of some optimal material distribution problems as the exponent goes to infinity.

In this paper we study the motion of a rigid body driven by Newton’s law immersed in a stationary incompressible Stokes flow occupying a bounded simply connected domain. The aim is that of trajectory tracking of the solid by the means of a control in the form of Dirichlet boundary data on the outside boundary of the fluid domain. We show that it is possible to exactly achieve any smooth trajectory for the solid that stays away from the external boundary, by the means of such a remote control. The proof relies on some density methods for the Stokes system, as well as a reformulation of the solid equations into an ODE.

In this paper, we formulate and prove the so-called $p$-adic non-commutative analytic subgroup theorem. This result is seen as the $p$-adic analogue of a recent theorem given by Yafaev in [11].

An abstract group $G$ is called totally $2$-closed if $H=H^{(2),\Omega }$ for any set $\Omega $ with $G\cong H\le \mathrm{Sym}(\Omega )$, where $H^{(2),\Omega }$ is the largest subgroup of $\mathrm{Sym}(\Omega )$ whose orbits on $\Omega \times \Omega $ are the same orbits of $H$. In this paper, we classify the finite soluble totally $2$-closed groups. We also prove that the Fitting subgroup of a totally $2$-closed group is a totally $2$-closed group. Finally, we prove that a finite insoluble totally $2$-closed group $G$ of minimal order with non-trivial Fitting subgroup has shape $Z\cdot X$, with $Z=Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian.

In this paper, the class of (complex) quasi-Herglotz functions is introduced as the complex vector space generated by the convex cone of ordinary Herglotz functions. We prove characterization theorems, in particular, an analytic characterization. The subclasses of quasi-Herglotz functions that are identically zero in one half-plane as well as rational quasi-Herglotz functions are investigated in detail. Moreover, we relate to other areas such as weighted Hardy spaces, definitizable functions, the Cauchy transform on the unit circle, sum-rule identities and matrix-valued Herglotz functions.

The main purpose of this paper is to study meromorphic solutions of the first order differential equations with delays \begin{equation*} w(z+1)-w(z-1)+a(z)\left(\frac{w^{\prime }(z)}{w(z)}\right)^k=R(z,w(z)) \end{equation*} and \begin{equation*} w(z+1)+a(z)\left(\frac{w^{\prime }(z)}{w(z)}\right)^k=R(z,w(z)), \end{equation*} where $k$ is a positive integer, $a(z)$ is a rational function, $R(z,w)$ is rational in $w$ with rational coefficients. Some necessary conditions on the degree of $R(z, w)$ are obtained for the equation to admit a transcendental meromorphic solution of minimal hypertype. These are extensions of some previous results due to Halburd, Korhonen, Liu and others. Some examples are given to support our conclusions.

A finite group all of whose complex character values are rational is called a rational group. In this paper, we classify all rational groups whose character degree graphs are disconnected.

We consider an adaptive multiresolution-based lattice Boltzmann scheme, which we have recently introduced and studied from the perspective of the error control and the theory of the equivalent equations. This numerical strategy leads to high compression rates, error control and its high accuracy has been explained on uniform and dynamically adaptive grids. However, one key issue with non-uniform meshes within the framework of lattice Boltzmann schemes is to properly handle acoustic waves passing through a level jump of the grid. It usually yields spurious effects, in particular reflected waves. In this paper, we propose a simple mono-dimensional test-case for the linear wave equation with a fixed adapted mesh characterized by a potentially large level jump. We investigate this configuration with our original strategy and prove that we can handle and control the amplitude of the reflected wave, which is of fourth order in the space step of the finest mesh. Numerical illustrations show that the proposed strategy outperforms the existing methods in the literature and allow to assess the ability of the method to handle the mesh jump properly.

In this paper, we prove a new point-sphere incidence bound in vector spaces over finite fields. More precisely, let $P$ be a set of points and $S$ be a set of spheres in $\mathbb{F}_q^d$. Suppose that $|P|, |S|\le N$, we prove that the number of incidences between $P$ and $S$ satisfies \[ I(P, S)\le N^2q^{-1}+q^{\frac{d-1}{2}}N, \] under some conditions on $d, q$, and radii. This improves the known upper bound $N^2q^{-1}+q^{\frac{d}{2}}N$ in the literature. As an application, we show that for $A\subset \mathbb{F}_q$ with $q^{1/2}\ll |A|\ll q^{\frac{d^2+1}{2d^2}}$, one has \[ \max \left\lbrace |A+A|,~|dA^2|\right\rbrace \gg \frac{|A|^d}{q^{\frac{d-1}{2}}}. \] This improves earlier results on this sum-product type problem over arbitrary finite fields.