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In a recent paper [JUKIĆ, D.: A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set, J. Comput. Appl. Math. 375 (2020)], a new existence level was introduced and then was used to obtain a necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set. In this paper, we determined that existence level for the residual sum of squares of the power-law regression with an unknown location parameter, and so we obtained a necessary and sufficient condition which guarantee the existence of the least squares estimate.

In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.

In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions e 0 {e_{0}} and e 1 {e_{1}} . It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

[EN] This paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Frobenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Frobenius method, in which the random input coefficients may be expressed via a Karhunen-Loeve expansion., This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewer, which have improved the final version of the paper.

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

It is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

We consider iterated integrals of $\log\zeta(s)$ on certain vertical and horizontal lines. Here, the function $\zeta(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log \zeta(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of an $m$-times iterated integral on the critical line is equivalent to the Riemann Hypothesis., Comment: 15 pages

In 1977, Colville, Davis, and Keimel [Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375] proved that a positive derivation on an Archimedean f-algebra A has its range in the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras.

In this paper we discuss coefficient problems for functions in the class 𝒞 0 ⁢ ( k ) {{\mathcal{C}}_{0}(k)} . This family is a subset of 𝒞 {{\mathcal{C}}} , the class of close-to-convex functions, consisting of functions which are convex in the positive direction of the real axis. Our main aim is to find some bounds of the difference of successive coefficients depending on the fixed second coefficient. Under this assumption we also estimate | a n + 1 | - | a n | {|a_{n+1}|-|a_{n}|} and | a n | {|a_{n}|} . Moreover, it is proved that Re ⁡ { a n } ≥ 0 {\operatorname{Re}\{a_{n}\}\geq 0} for all f ∈ 𝒞 0 ⁢ ( k ) {f\in{\mathcal{C}}_{0}(k)} .

In this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations x n + 1 = 1 + 2 y n − k 3 + y n − k , y n + 1 = 1 + 2 z n − k 3 + z n − k , z n + 1 = 1 + 2 x n − k 3 + x n − k , $$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$ where n, k∈ ℕ0, the initial values x −k , x −k+1, …, x 0, y −k , y −k+1, …, y 0, z −k , z −k+1, …, z 1 and z 0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

In this paper, we consider the long time behavior of solutions for 3D incompressible MHD equations with fractional Laplacian. Firstly, in a periodic bounded domain, we prove the existence of a global attractor. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and magnetic fields. Finally, in the whole space ℝ 3 {\mathbb{R}^{3}} , we establish the sharp algebraic decay rate of solutions to the generalized MHD system provided that the parameters satisfy α , β ∈ ( 0 , 2 ] {\alpha,\beta\in(0,2]} .

In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

In this paper we define and consider some familiar subsets of analytic functions associated with sine functions in the region of unit disk on the complex plane. For these classes our aim is to find the Hankel determinant of order three.

In this paper, we concern with a 2nth-order discrete system. Using the critical point theory, we establish various sets of sufficient conditions for the existence of periodic solutions with prescribed minimal period. To the best of our knowledge, this is the first time to discuss the periodic solutions with prescribed minimal period for a 2nth-order discrete system.

This paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in ℝ N {\mathbb{R}^{N}} ( N ∈ ℕ + {N\in\mathbb{N}^{+}} ), provided that the initial datum is sufficiently small. In addition, the L p {L^{p}} -norm ( 1 ≤ p ≤ ∞ {1\leq p\leq\infty} ) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.

In this paper, a new improved smoothing Newton algorithm for the nonlinear complementarity problem was proposed. This method has two-fold advantages. First, compared with the classical smoothing Newton method, our proposed method needn’t nonsingular of the smoothing approximation function; second, the method also inherits the advantage of the classical smoothing Newton method, it only needs to solve one linear system of equations at each iteration. Without the need of strict complementarity conditions and the assumption of P0 property, we get the global and local quadratic convergence properties of the proposed method. Numerical experiments show that the efficiency of the proposed method.

The aim of the paper is to study problem of image segmentation and missing boundaries completion introduced in [Mikula, K.—Sarti, A.––Sgallarri, A.: Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation, Comput. Vis. Sci. 9 (2006), 23–31], [Mikula, K.—Sarti, A.—Sgallari, F.: Co-volume level set method in subjective surface based medical image segmentation, in: Handbook of Medical Image Analysis: Segmentation and Registration Models (J. Suri et al., eds.), Springer, New York, 583–626, 2005], [Mikula, K.—Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing, Numer. Math. 89 (2001), 561–590] and [Tibenský, M.: VyužitieMetód Založených na Level Set Rovnici v Spracovaní Obrazu, Faculty of mathematics, physics and informatics, Comenius University, Bratislava, 2016]. We generalize approach presented in [Eymard, R.—Handlovičová, A.—Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA J. Numer. Anal. 31 (2011), 813–846] and apply it in the field of image segmentation. The so called regularised Riemannian mean curvature flow equation is presented and the construction of the numerical scheme based on the finite volume method approach is explained. The principle of the level set, for the first time given in [Osher, S.—Sethian, J. A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), 12–49] is used. Based on the ideas from [Eymard, R.—Handlovičová, A.– –Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA J. Numer. Anal. 31 (2011), 813–846] we prove the stability estimates on the numerical solution and the uniqueness of the numerical solution. In the last section, there is a proof of the convergence of the numerical scheme to the weak solution of the regularised Riemannian mean curvature flow equation and the proof of the convergence of the approximation of the numerical gradient is mentioned as well.

In this paper, we present an effective algorithm for globally solving quadratic programs with quadratic constraints, which has wide application in engineering design, engineering optimization, route optimization, etc. By utilizing new parametric linearization technique, we can derive the parametric linear programming relaxation problem of the quadratic programs with quadratic constraints. To improve the computational speed of the proposed algorithm, some interval reduction operations are used to compress the investigated interval. By subsequently partitioning the initial box and solving a sequence of parametric linear programming relaxation problems the proposed algorithm is convergent to the global optimal solution of the initial problem. Finally, compared with some known algorithms, numerical experimental results demonstrate that the proposed algorithm has higher computational efficiency.

In this paper, the nonlinear Thomas-Fermi equation for neutral atoms by using the fractional order of rational Chebyshev functions of the second kind (FRC2), FU n α ( t , L ) ${\rm{FU}}_{\rm{n}}^\alpha \left( {{\rm{t}},{\rm{L}}} \right)$ (t, L), on an unbounded domain is solved, where L is an arbitrary parameter. Boyd (Chebyshev and Fourier Spectral Methods, 2ed, 2000) has presented a method for calculating the optimal approximate amount of L and we have used the same method for calculating the amount of L. With the aid of quasilinearization and FRC2 collocation methods, the equation is converted to a sequence of linear algebraic equations. An excellent approximation solution of y(t), y′ (t), and y ′ (0) is obtained.

Let H be a Hilbert space and B(H) the set of all bounded linear operators on H. In the paper we consider the generalized concept of order relations on B(H) which was proposed by Šemrl and which covers the star partial order, the left-star partial order, the right-star partial order, and the minus partial order. We also connect this concept with the sharp partial order.

In this paper we are going to apply the Henstock-Kurzweil integrals defined on an unbounded intervals to differential and integral equations defined on such intervals. To deal with linear differential equations we examine convolution involving functions integrable in Henstock-Kurzweil sense. In the case of nonlinear Hammerstein integral equation as well as Volterra integral equation we look for solutions in the space of functions of bounded variation in the sense of Jordan.

A general linear model can be given in certain multiple partitioned forms, and there exist submodels associated with the given full model. In this situation, we can make statistical inferences from the full model and submodels, respectively. It has been realized that there do exist links between inference results obtained from the full model and its submodels, and thus it would be of interest to establish certain links among estimators of parameter spaces under these models. In this approach the methodology of additive matrix decompositions plays an important role to obtain satisfactory conclusions. In this paper, we consider the problem of establishing additive decompositions of estimators in the context of a general linear model with partial parameter restrictions. We will demonstrate how to decompose best linear unbiased estimators (BLUEs) under the constrained general linear model (CGLM) as the sums of estimators under submodels with parameter restrictions by using a variety of effective tools in matrix analysis. The derivation of our main results is based on heavy algebraic operations of the given matrices and their generalized inverses in the CGLM, while the whole contributions illustrate various skillful uses of state-of-the-art matrix analysis techniques in the statistical inference of linear regression models.

A homogeneous polynomial of degree $d$ in $n+1$ variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of $d$ and $n$ for which a general polynomial of degree $d$ in $n+1$ variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces., 25 pages. Proof of Proposition 17 and Lemma 18 fixed, some typos corrected

The theory of multivariate neural network operators in a Kantorovich type version is here introduced and studied. The main results concerns the approximation of multivariate data, with respect to the uniform and Lp norms, for continuous and Lp functions, respectively. The above family of operators, are based upon kernels generated by sigmoidal functions. Multivariate approximation by constructive neural network algorithms are useful for applications to neurocomputing processes involving high dimensional data. At the end of the paper, several examples of sigmoidal functions for which the above theory holds have been presented.

The purpose of this paper is to study a nondifferentiable multiobjective fractional programming problem (MFP) in which each component of objective functions contains the support function of a compact convex set. For a differentiable function, we introduce the class of second-order ( C , α , ρ , d ) - V {(C,\alpha,\rho,d)-V} -type-I convex functions. Further, Mond–Weir- and Wolfe-type duals are formulated for this problem and appropriate duality results are proved under the aforesaid assumptions.

In the present paper we prove a separation theorem for t-Wright convex and t-Wright concave functionals. We also consider the problem of supporting the t-Wright concave functions by t-Wright affine ones. As a consequence of these results we prove some decomposition theorems for t-Wright convex functions.

The purpose of this paper is to establish the rate of convergence in terms of the weighted modulus of continuity and Lipschitz type maximal function for the q-Szász-beta operators. We also study the rate of A-statistical convergence. Lastly, we modify these operators using King type approach to obtain better approximation.

In this paper, we first establish some refinements of the weighted generalized trapezoid inequality for functions of bounded variation in terms of a cumulative variation function, and then obtain some perturbed versions of the weighted generalized trapezoid inequality. Several applications for selfadjoint operators on complex Hilbert spaces are also given.

In this paper we prove a stability result for the functional equation 1 L ⁢ ∑ λ ∈ K F ⁢ ( x + λ ⁢ y ) = α ⁢ ( y ) ⁢ G ⁢ ( x ) + H ⁢ ( y ) , x , y ∈ S , \frac{1}{L}\sum_{\lambda\in K}F(x+\lambda y)=\alpha(y)G(x)+H(y),\quad x,y\in S, for the functions F , G , H : S → X {F,G,H\colon S\to X} , α : S → 𝕂 {\alpha\colon S\to\mathbb{K}} , where ( S , + ) {(S,+)} is an abelian semigroup, K is a finite subgroup of the automorphism group of S, L := | K | {L:=|K|} and ( X , ∥ ⋅ ∥ ) {(X,\|\cdot\|)} is the Banach space over a field 𝕂 ∈ { ℝ , ℂ } {\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}} .

Let π be a unitary cuspidal automorphic representation for GL m ⁢ ( 𝔸 ℚ ) {\mathrm{GL}_{m}(\mathbb{A}_{\mathbb{Q}})} , and let L ⁢ ( s , π ) {L(s,\pi)} be the automorphic L-function attached to π, which has a Dirichlet series expression in the half-plane ℜ ⁡ s > 1 {\Re s>1} , i.e. L ⁢ ( s , π ) = ∑ n = 1 ∞ λ π ⁢ ( n ) n s . L(s,\pi)=\sum_{n=1}^{\infty}\frac{\lambda_{\pi}(n)}{n^{s}}. In this paper we are interested in the upper bound of the fourth power moment of λ π ⁢ ( n ) {\lambda_{\pi}(n)} , i.e. ∑ n ≤ x λ π ⁢ ( n ) 4 {\sum_{n\leq x}\lambda_{\pi}(n)^{4}} . If m = 2 {m=2} , we are able to consider the sixteenth power moment of λ π ⁢ ( n ) {\lambda_{\pi}(n)} . As an application, we consider the lower bound of ∑ n ≤ x | λ π ⁢ ( n ) | {\sum_{n\leq x}\lvert\lambda_{\pi}(n)\rvert} , which improves previous results.

In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

In this paper, we discuss mean values of some Witten L-functions with respect to the “character aspect”. In particular, we focus on the SU ⁢ ( 2 ) ${\mathrm{SU}(2)}$ case. We prove that such mean values turn out to be 0, 1 or to diverge.

In this paper, we give estimates of the L p ⁢ ( ℝ n ) ${L^{p}(\mathbb{R}^{n})}$ ( 1 < p < ∞ ${1 ) and weak type ( 1 , 1 ) ${(1,1)}$ norm of singular integral operator T Ω ${T_{\Omega}}$ with homogeneous kernel defined by T Ω ⁢ f ⁢ ( x ) = p . v . ∫ ℝ n Ω ⁢ ( x - y ) | x - y | n ⁢ f ⁢ ( y ) ⁢ 𝑑 y , $T_{\Omega}f(x)=\mathrm{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n}}% f(y)\,dy,$ where Ω satisfies L 1 ${L^{1}}$ -Dini condition with mean value zero on 𝕊 n - 1 ${\mathbb{S}^{n-1}}$ . More precisely, we show that the following two estimates hold: ∥ T Ω ⁢ f ∥ p ≲ ( ( log ⁡ n ) ⁢ ∥ Ω ∥ 1 + ∫ 0 1 n ω ~ 1 ⁢ ( δ ) δ ⁢ 𝑑 δ ) ⁢ ∥ f ∥ p , $\|T_{\Omega}f\|_{p}\lesssim\Biggl{(}(\log n)\|\Omega\|_{1}+\int_{0}^{\frac{1}{% n}}\frac{\tilde{\omega}_{1}(\delta)}{\delta}\,d\delta\Biggr{)}\|f\|_{p},$ and for any λ > 0 ${\lambda>0}$ , λ ⁢ m ⁢ ( { x ∈ ℝ n : | T Ω ⁢ f ⁢ ( x ) | > λ } ) ≲ ( ( log ⁡ n ) ⁢ ∥ Ω ∥ 1 + ∫ 0 1 n ω ~ 1 ⁢ ( δ ) δ ⁢ 𝑑 δ ) ⁢ ∥ f ∥ 1 , $\lambda m(\{x\in\mathbb{R}^{n}:|T_{\Omega}f(x)|>\lambda\})\lesssim{\Bigg{(}(% \log n)\|\Omega\|_{1}+\int_{0}^{\frac{1}{n}}\frac{\tilde{\omega}_{1}(\delta)}{% \delta}\,d\delta\Biggr{)}}\|f\|_{1},$ where ω ~ 1 ${\tilde{\omega}_{1}}$ denotes the L 1 ${L^{1}}$ integral continuous modulus of Ω defined by translation in ℝ n ${\mathbb{R}^{n}}$ . Moreover, some similar estimates are also established for the Marcinkiewicz integral μ Ω ${\mu_{\Omega}}$ , which was introduced by Stein in [13].

In this paper we obtain the classification of p-nilpotent restricted Lie algebras of dimension at most 4 over a perfect field of positive characteristic p.

The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [12] and [11]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field ℚ {\mathbb{Q}} , since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [8] and Flach completed their results at 2 in [13] and [14].

Let E $E$ be an elliptic curve over ${\mathbb {Q}}$ without complex multiplication. For each prime p $p$ of good reduction, let | E ( p ) | $|E({\mathbb {F}}_p)|$ be the order of the group of points of the reduced curve over p ${\mathbb {F}}_p$ . According to a conjecture of Koblitz, there should be infinitely many such primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ is prime, unless there are some local obstructions predicted by the conjecture. Suppose that E $E$ is a curve without local obstructions (which is the case for most elliptic curves over ${\mathbb {Q}}$ ). We prove in this paper that, under the GRH, there are at least 2 . 778 C E twin x / ( log x ) 2 $2.778 C_E^{\rm twin} x / (\log x)^2$ primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant C E twin $C_E^{\rm twin}$ appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] appearing in the upper bound for the number of primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.