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1.
Let f,gi,i=1,…,l,hj,j=1,…,m, be polynomials on Rn and S?{xRngi(x)=0,i=1,…,l,hj(x)≥0,j=1,…,m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.  相似文献   

2.
In this paper, we study orthogonal polynomials with respect to the inner product (f, g)S(N) =〈u, fg〉+∑m=1N λmu, f(m)g(m) 〉, where λm≥0 form=1,…,N, anduis a semiclassical, positive definite linear functional. For these non-standard orthogonal polynomials, algebraic and differential properties are obtained, as well as their representation in terms of the standard orthogonal polynomials associated withu.  相似文献   

3.
For any additive character ψ and multiplicative character χ on a finite field Fq, and rational functions f,g in Fq(x), we show that the elementary Stepanov-Schmidt method can be used to obtain the corresponding Weil bound for the sum ∑xFq?Sχ(g(x))ψ(f(x)) where S is the set of the poles of f and g. We also determine precisely the number of characteristic values ωi of modulus q1/2 and the number of modulus 1.  相似文献   

4.
5.
The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u|Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.  相似文献   

6.
The Abel equation of the second kind
[g0(x)+g1(x)u]u=f0(x)+f1(x)u+f2(x)u2  相似文献   

7.
Let ? be a binary relation on A×X, and suppose that there are real valued functions f on A and g on X such that, for all ax, byA×X, ax ? by if and only if f (a)+g(x) ? f(b)+g(y). This paper establishes uniqueness properties for f and g when A is a finite set, X is a real interval with g increasing on X, and for any a, b and x there is a y for which f(a)+g(x)=f(b)+g(y). The resultant uniqueness properties occupy an intermediate position among uniqueness properties for other structural cases of two-factor additive measurement.It is shown that f is unique up to a positive affine transformation (αf1 with α > 0), but that g is unique up to a similar positive affine transformation (αg2) if and only if the ratio [f(a)?f(b)]/[f(a)?f(c)] is irrational for some a, b, cA. When the f ratios are rational for all cases where they are defined, there will be a half-open interval (x0, x1) in X such that the restriction of g on (x0, x1) can be any increasing function for which sup {g(x)?g(x0): x0 ? x < x1} does not exceed a specified bound, and, when g is thus defines on (x0, x1), it will be uniquely determined on the rest of X. In general, g must be continuous only in the ‘irrational’ case.  相似文献   

8.
An investigation is made of the polynomials fk(n) = S(n + k, n) and gk(n) = (?1)ks(n, n ? k), where S and s denote the Stirling numbers of the second and first kind, respectively. The main result gives a combinatorial interpretation of the coefficients of the polynomial (1 ? x)2k+1Σn=0fk(n)xn analogous to the well-known combinatorial interpretation of the Eulerian numbers in terms of descents of permutations.  相似文献   

9.
This work is a continuation and extension of our earlier articles on irreducible polynomials. We investigate the irreducibility of polynomials of the form g(f(x)) over an arbitrary but fixed totally real algebraic number field L, where g(x) and f(x) are monic polynomials with integer coefficients in L, g is irreducible over L and its splitting field is a totally imaginary quadratic extension of a totally real number field. A consequence of our main result is as follows. If g is fixed then, apart from certain exceptions f of bounded degree, g(f(x)) is irreducible over L for all f having distinct roots in a given totally real number field.  相似文献   

10.
We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x 2)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S n ?1 (f) = S n ?1 (f, x) is the projection onto the subspace of algebraic polynomials p n = p n (x) of degree at most n, i.e., S n (p n ) = p n ; in addition, S n ?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S n ?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ n (x) of the partial sums S n ?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .  相似文献   

11.
In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =(a,b),a function G ∈ S(w):= { f:∫I | f(x)| w(x)d x < ∞} satisfying the conditions G 2j(x) ≥ 0,x ∈(a,b),j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S(w) with G ≥ 0 satisfying sup n ∑λ0knG(xkn) k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.  相似文献   

12.
Let k be a field of characteristic zero and f(t),g(t) be polynomials in k[t]. For a plane curve parameterized by x=f(t),y=g(t), Abhyankar developed the notion of Taylor resultant (Mathematical Surveys and Monographs, Vol. 35, American Mathematical Society, Providence, RI, 1990) which enables one to find its singularities without knowing its defining polynomial. This concept was generalized as D-resultant by Yu and Van den Essen (Proc. Amer. Math. Soc. 125(3) (1997) 689), which works over an arbitrary field. In this paper, we extend this to a curve in affine n-space parameterized by x1=f1(t),…,xn=fn(t) over an arbitrary ground field k, where f1,…,fnk[t]. This approach compares to the usual approach of computing the ideal of the curve first. It provides an efficient algorithm of computing the singularities of such parametric curves using Gröbner bases. Computational examples worked out by symbolic computation packages are included.  相似文献   

13.
The purpose of this paper is to solve the problem of determining the limits of multivariate rational functions.It is essential to decide whether or not limxˉ→0f g=0 for two non-zero polynomials f,g∈R[x1,...,xn]with f(0,...,0)=g(0,...,0)=0.For two such polynomials f and g,we establish two necessary and sufcient conditions for the rational functionf g to have its limit 0 at the origin.Based on these theoretic results,we present an algorithm for deciding whether or not lim(x1,...,xn)→(0,...,0)f g=0,where f,g∈R[x1,...,xn]are two non-zero polynomials.The design of our algorithm involves two existing algorithms:one for computing the rational univariate representations of a complete chain of polynomials,another for catching strictly critical points in a real algebraic variety.The two algorithms are based on the well-known Wu’s method.With the aid of the computer algebraic system Maple,our algorithm has been made into a general program.In the final section of this paper,several examples are given to illustrate the efectiveness of our algorithm.  相似文献   

14.
In this paper, we prove the following result: Let f(z) and g(z) be two nonconstant meromorphic(entire) functions, n ≥ 11(n ≥ 6) a positive integer. If fn(z)f′(z) and gn(z)g′(z) have the same fixed-points, then either f(z) = c1ecz2g(z) = c2e− cz2, where c1c2, and c are three constants satisfying 4(c1c2)n + 1c2 = −1, or f(z) ≡ tg(z) for a constant t such that tn + 1 = 1.  相似文献   

15.
Let T g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T g has a Cantor-type attractor F and a unique invariant measure ??0 supported on F. The corresponding unitary operator (U g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? n,r , n ?? 1, 0 ?? r ?? 2 n?1 ? 1 with eigenfunctions e r (n) (x). Suppose that f ?? C 1([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L p ([?1, 1], d??0), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T g we prove that S n (f) ?? 2?n q n , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.  相似文献   

16.
The gamma class Γ α (g) consists of positive and measurable functions that satisfy f(x+yg(x))/f(x)→exp(αy). In most cases, the auxiliary function g is Beurling varying, i.e. g(x)/x→0 and g∈Γ0(g). Taking h=logf, we find that hEΓ α (g,1), where EΓ α (g,a) is the class of ultimately positive and measurable functions that satisfy (f(x+yg(x))?f(x))/a(x)→αy. In this paper, we discuss local uniform convergence for functions in the classes Γ α (g) and EΓ α (g,a). From this we obtain several representation theorems. We also prove some higher order relations for functions in the classes Γ α (g) and EΓ α (g,a). Some applications conclude the paper.  相似文献   

17.
For a given integer k ∈ ? we determine the possible forms of operators T: C k (?) → C(?) satisfying a generalized Leibniz rule operator equation T(f · g)(x) = Tf(x) · g(x)+f(x) · Tg(x)+S(f, g)(x), f,gC k (?), x ∈ ? for two different types of perturbations S(f, g). In the first case, S is given by a function B in localized form $$S(f,g)(x) = B(x,({f^{(j)}}(x))_{j = 0}^{k - 1},({g^{(j)}}(x))_{j = 0}^{k - 1})$$ involving only derivatives of lower order. We show that, if in addition T annihilates the polynomials of degree ≤ k ? 1, T is a multiple of the k-th derivative. For k = 2 and functions on ? n , we give a characterization of the Laplacian by a similar equation, orthogonal invariance and annihilation of affine functions. In the second setting, we assume S to have the form S(f, g)(x) = Af(x) · Ag(x) where A: C k (?) → C(?) is a general operator. Thus here, S has a product form, but the factor Af(x) is not assumed to depend only on the jet of f at x. We describe the possible forms of T and A satisfying the generalized Leibniz rule; T and A turn out to be closely related. Here, T and A need not to be localized, i.e., Tf(x) and Af(x) may depend on values f(y) for yx.  相似文献   

18.
Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x2+xm. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x0,x0+1,…,x0+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).  相似文献   

19.
We give an exact characterization of permutation polynomials modulo n=2w, w≥2: a polynomial P(x)=a0+a1x +···+adxd with integral coefficients is a permutation polynomial modulo n if and only if a1 is odd, (a2+a4+a6+···) is even, and (a3+a5+a7+···) is even. We also characterize polynomials defining latin squares modulo n=2w, but prove that polynomial multipermutations (that is, a pair of polynomials defining a pair of orthogonal latin squares) modulo n=2wdo not exist.  相似文献   

20.
Every symmetric polynomial p = p(x) = p(x 1,..., x g ) (with real coefficients) in g noncommuting variables x 1,..., x g can be written as a sum and difference of squares of noncommutative polynomials:
$ (SDS) p(x) = \sum\limits_{j = 1}^{\sigma _ + } {f_j^ + (x)^T f_j^ + (x)} - \sum\limits_{\ell = 1}^{\sigma _ - } {f_\ell ^ - (x)^T f_\ell ^ - (x)} , $ (SDS) p(x) = \sum\limits_{j = 1}^{\sigma _ + } {f_j^ + (x)^T f_j^ + (x)} - \sum\limits_{\ell = 1}^{\sigma _ - } {f_\ell ^ - (x)^T f_\ell ^ - (x)} ,   相似文献   

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