Search

Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

One of the main results stated in Theorem 4.4 of our article, which appears in Trans. Amer. Math. Soc. 372 (2019), no. 10, 7191-7223, is that the structure algebra K[M(X,r)], over a field K, of a finite bijective left non-degenerate solution (X,r) of the Yang-Baxter equation is a module-finite central extension of a commutative affine subalgebra. This is proven by showing that the structure monoid M(X,r) is central-by-finite. This however is not true, even in case (X,r) is a (left and right) non-degenerate involutive solution. The proof contains a subtle mistake. However, it turns out that the monoid M(X,r) is abelian-by-finite and thus the conclusion that K[M(X,r)] is a module-finite normal extension of a commutative affine subalgebra remains valid. In particular, K[M(X,r)] is Noetherian and satisfies a polynomial identity. The aim of this paper is to give a proof of this result. In doing so, we also strengthen Lemma 5.3 (and its consequences, namely Lemma 5.4 and Proposition 5.5) showing that these results on the prime spectrum of the structure monoid hold even if the assumption that the solution (X,r) is square-free is omitted. [ABSTRACT FROM AUTHOR]

In this paper, I generalize Holzer's theorem for semi-diagonal equation. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the following nonlinear waveequation with variable exponents: u t t − div (| ∇ u | r (⋅) − 2 ∇ u) + | u t | m (⋅) − 2 u t = 0. By using a lemma by Komornik, we prove the decay estimates for the solution under suitable assumptions on the variable exponents m , r and the initial data. We also give two numerical applications to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The blended solutions are more general than the unblended solutions for polynomial-like iterative equation with multivalued functions. In this paper we study the blended solutions of polynomial-like iterative equation with multivalued functions. [ABSTRACT FROM AUTHOR]

A one-sided limit order book is modeled as a noncooperative game for several players. Agents offer various quantities of an asset at different prices, competing to fulfill an incoming order, whose size is not known a priori. Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random incoming order. In a previous paper, the existence of a Nash equilibrium was established by means of a fixed point argument. The main issue discussed in the present paper is whether this equilibrium can be obtained from the unique solution to a two-point boundary value problem, for a suitable system of discontinuous ordinary differential equations. Some additional assumptions are introduced, which yield a positive answer. In particular, this is true when there are exactly two players, or when all players assign the same exponential probability distribution to the incoming order. In both of these cases, we also prove that the Nash equilibrium is unique. A counterexample shows that these assumptions cannot be removed, in general. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q . In this paper, some new classes of permutation polynomials of the form ( x p m − x + δ ) s + x over F p 2 m are obtained by determining the number of solutions of certain equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following differential analog of the Noether normalization lemma: for every d-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map on to the d-dimensional affine space. Equivalently, for every integral differential algebra A over differential field of zero characteristic there exist differentially independent b1, . . . , bd such that A is differentially algebraic over subalgebra B differentially generated by b1, . . . , bd, and whenever p ⊂ B is a prime differential ideal, there exists a prime differential ideal q ⊂ A such that p = B ∩ q. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations. [ABSTRACT FROM AUTHOR]

A ring R is said to be left uniquely generated if R a = R b in R implies that a = u b for some unit u in R . These rings have been of interest since Kaplansky introduced them in 1949 in his classic study of elementary divisors. Writing l ( b ) = { r ∈ R | r b = 0 } , a theorem of Canfell asserts that R is left uniquely generated if and only if, whenever R a + l ( b ) = R where a , b ∈ R , then a − u ∈ l ( b ) for some unit u in R . By analogy with the stable range 1 condition we call a ring with this property left annihilator-stable. In this paper we exploit this perspective on the left UG rings to construct new examples and derive new results. For example, writing J for the Jacobson radical, we show that a semiregular ring R is left annihilator-stable if and only if R / J is unit-regular, an analogue of Bass' theorem that semilocal rings have stable range 1. [ABSTRACT FROM AUTHOR]

This paper deals with the solvability of interval matrix equations in max-plus algebra. Max-plus algebra is the algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗, where a ⊕ b = max ⁡ { a , b } and a ⊗ b = a + b . The notation A ⊗ X ⊗ C = B represents an interval max-plus matrix equation, where A , B , and C are given interval matrices. We define four types of solvability of interval max-plus matrix equations, i.e., the tolerance, weak tolerance, left-weak tolerance, and right-weak tolerance solvability. We derive the necessary and sufficient conditions for checking each of them, whereby all can be verified in polynomial time. [ABSTRACT FROM AUTHOR]

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors { x ( i ) } with x ( i ) ∈ R n 1 × ⋯ × n d in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram–Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α i in an arbitrary tensor v = ∑ i α i x ( i ) . The orthonormal Q factor in the QR decomposition X ≡ [ x ( 1 ) , ⋯ , x ( p ) ] = Q R cannot be computed but expressed as X R − 1 when required. The resulting algorithm has an O ( p 2 d n ) computational complexity, with n = max ⁡ n i . Some illustrative examples in the numerical solution of tensor linear equations are presented. [ABSTRACT FROM AUTHOR]

In the very recent work Wang and Wang (2014) proved that the initial value problem for the three dimension incompressible MHD equations has a global solutions ( u , B ) ∈ C ( [ 0 , ∞ ) ; χ − 1 ) , provided that the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients. In this paper, we prove that global solutions u , B that was obtained in Wang and Wang (2014) is asymptotic to zero in sense of the norm of the space χ − 1 as time goes to infinity. Moreover, stability of global solutions is also discussed. [ABSTRACT FROM AUTHOR]

A higher order Kirchhoff type equation ∫ R 2 m ( ∣ ∇ m u ∣ 2 + ∑ γ = 0 m − 1 a γ ( x ) ∣ ∇ γ u ∣ 2 ) d x ( ( − Δ ) m u + ∑ γ = 0 m − 1 ( − 1 ) γ ∇ γ ⋅ ( a γ ( x ) ∇ γ u ) ) = f ( x , u ) ∣ x ∣ β + ϵ h ( x ) in R 2 m is considered in this paper. We assume that the nonlinearity of the equation has exponential critical growth and prove that, for a positive ϵ which is small enough, there are two distinct nontrivial solutions to the equation. When ϵ = 0 , we also prove that the equation has a nontrivial mountain-pass type solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior of solutions for nonlocal dispersal equation with non-homogeneous boundary conditions. We find that the long-time behavior is determined by the boundary value. [ABSTRACT FROM AUTHOR]

R. Coleman and K. McMurdy calculated the stable reduction of X0(p3) (p ≥ 13) on the basis of the rigid geometry in [8]. In the present paper, we determine the stable reduction of X0(p4) for primes p ≥ 13 on the basis of their idea. As irreducible components in the stable reduction of X0(p4), a certain number of copies of the Hermitian curve, with an affine equation ap - a = tp+1, appear. [ABSTRACT FROM AUTHOR]

Finding the solution of the absolute value equation (AVE) A x − | x | = b has attracted much attention in recent years. In this paper, we propose a relaxed nonlinear PHSS-like iterative method, which is more efficient than the Picard-HSS iterative method for the AVE, and is a generalization of the nonlinear HSS-like iteration method for the AVE. By using the theory of nonsmooth analysis, we prove the convergence of the relaxed nonlinear PHSS-like iterative method for the AVE. Numerical experiments are given to demonstrate the feasibility, robustness and effectiveness of the relaxed nonlinear HSS-like method. [ABSTRACT FROM AUTHOR]

This paper describes how to analyse translinear topologies whose response is modified by the body effect by means of the general translinear principle. It proposes two different strategies for obtaining the translinear loop equations from each closed translinear set. Both strategies are tested analysing three topologies, one of them including some transistors in triode. From the results of such analysis, a fast analysis method for topologies with all transistor bodies connected to the same potential is also proposed. Simulated results are given showing good agreement with theoretical expressions. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we consider the slightly L2-supercritical gKdV equations , with the nonlinearity and . In the previous work of the author, we know that there exists a stable self-similar blow-up dynamics for slightly L2-supercritical gKdV equations. Such solutions can be viewed as solutions with a single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time. [ABSTRACT FROM AUTHOR]

We show how the derivation of the Derrida–Lebowitz–Speer–Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy–Widom GOE distribution. [ABSTRACT FROM AUTHOR]