Suppose Thatf Research Articles

Monodromie unipotente maximale, congruences “à la Lucas” et indépendance algébrique

Let f ( z ) f(z) be in 1 + z Q [ [ z ] ] 1+z\mathbb {Q}[[z]] and S \mathcal {S} be an infinite set of prime numbers such that, for all p ∈ S p\in \mathcal {S} , we can reduce f ( z ) f(z) modulo p p . We let f ( z ) ∣ p f(z)_{\mid p} denote the reduction of f ( z ) f(z) modulo p p . Generally, when f ( z ) f(z) is D-finite, f ∣ p ( z ) f_{\mid p}(z) is algebraic over F p ( z ) \mathbb {F}_p(z) . It turns out that if f ∣ p ( z ) f_{\mid p}(z) is a solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , we can use this type of equations to obtain results of transcendence and algebraic independence over Q ( z ) \mathbb {Q}(z) . In the present paper, we look for conditions on the differential operators annihilating f ( z ) f(z) to guarantee the existence of these particular equations. Suppose that f ( z ) f(z) is solution of a differential operator H ∈ Q ( z ) [ d / d z ] \mathcal {H}\in \mathbb {Q}(z)[d/dz] having a strong Frobenius structure for all p ∈ S p\in \mathcal {S} and we also suppose that f ( z ) f(z) annihilates a Fuchsian differential operator D ∈ Q ( z ) [ d / d z ] \mathcal {D}\in \mathbb {Q}(z)[d/dz] such that zero is a regular singular point of D \mathcal {D} and the exponents of D \mathcal {D} at zero are equal to zero. Our main result states that, for almost every prime p ∈ S p\in \mathcal {S} , f ∣ p ( z ) f_{\mid p}(z) is solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , where A p ( z ) A_p(z) is a rational function with coefficients in F p \mathbb {F}_p of height less than or equal to C p 2 l Cp^{2l} with C C a positive constant that does not depend on p p . We also study the algebraic independence of these power series over Q ( z ) \mathbb {Q}(z) .

Extended Newton-type Method for Nonsmooth Generalized Equation under n , α -point-based Approximation

Let X and Y be Banach spaces and Ω ⊆ X . Let f : Ω ⟶ Y be a single valued function which is nonsmooth. Suppose that F : X⇉ 2 Y is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation 0 ∈ f x + F x and analyze its semilocal and local convergence under the conditions that f + F − 1 is Lipschitz-like and f admits a certain type of approximation which generalizes the concept of point-based approximation so-called n , α -point-based approximation. Applications of n , α -point-based approximation are provided for smooth functions in the cases n = 1 and n = 2 as well as for normal maps. In particular, when 0 < α < 1 and the derivative of f , denoted ∇ f , is ℓ , α -Hölder continuous, we have shown that f admits 1 , α -point-based approximation for n = 1 while f admits 2 , α -point-based approximation for n = 2 , when 0 < α < 1 and the second derivative of f , denoted ∇ 2 f , is K , α -Hölder. Moreover, we have constructed an n , α -point-based approximation for the normal maps f C + F when f has an n , α -point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study.

Generalized derivations vanishing on co-commutator identities in prime rings

Let R be a noncommutative prime ring of char (R)? 2 with Utumi quotient ring U and extended centroid C and I a nonzero two sided ideal of R. Suppose that F(? 0), G and H are three generalized derivations of R and f (x1,...,xn) is a multilinear polynomial over C, which is not central valued on R. If F(G(f(r))f(r)- f(r)H(f(r))) = 0 for all r = (r1,..., rn) ? In, then we obtain information about the structure of R and describe the all possible forms of the maps F, G and H. This result generalizes many known results recently proved by several authors ([1], [4], [5], [8], [9], [13], [15]).

Additiven-commuting maps on semiprime rings

AbstractLetRbe a semiprime ring with the extended centroidCandQthe maximal right ring of quotients ofR. Set [y,x]1= [y,x] =yx−xyforx,y∈Qand inductively [y,x]k= [[y,x]k−1,x] fork> 1. Suppose thatf:R→Qis an additive map satisfying [f(x),x]n= 0 for allx∈R, wherenis a fixed positive integer. Then it can be shown that there exist λ ∈Cand an additive mapμ:R→Csuch thatf(x) = λx+μ(x) for allx∈R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

A weighted maximal inequality for differentially subordinate martingales

The paper contains the proof of a weighted Fefferman-Stein inequality in a probabilistic setting. Suppose that f = ( f n ) n ≥ 0 f=(f_n)_{n\geq 0} , g = ( g n ) n ≥ 0 g=(g_n)_{n\geq 0} are martingales such that g g is differentially subordinate to f f , and let w = ( w n ) n ≥ 0 w=(w_n)_{n\geq 0} be a weight, i.e., a nonnegative, uniformly integrable martingale. Denoting by M f = sup n ≥ 0 | f n | Mf=\sup _{n\geq 0}|f_n| , M w = sup n ≥ 0 w n Mw=\sup _{n\geq 0}w_n the maximal functions of f f and w w , we prove the weighted inequality | | g | | L 1 ( w ) ≤ C | | M f | | L 1 ( M w ) , \begin{equation*} ||g||_{L^1(w)}\leq C||Mf||_{L^1(Mw)}, \end{equation*} where C = 3 + 2 + 4 ln ⁡ 2 = 7.186802 … C=3+\sqrt {2}+4\ln 2=7.186802\ldots . The proof rests on the existence of a special function enjoying appropriate majorization and concavity.

IDENTITIES WITH ADDITIVE MAPPINGS IN SEMIPRIME RINGS

Abstract. The aim of this paper is to prove the next result. Let n > 1be an integer and let R be a n!-torsion free semiprime ring. Suppose thatf : R →R is an additive mapping satisfying the relation [f(x),x n ] = 0for all x ∈R. Then f is commuting on R. 1. Introduction and the maintheoremThroughout, R will represent an associative ring with a center Z(R). Letn > 1 be an integer. A ring R is n-torsion free if nx = 0, x ∈ R, impliesx = 0. The Lie product (or a commutator) of elements x,y ∈ R will be denotedby [x,y] (i.e., [x,y] = xy − yx). Recall that a ring R is prime if aRb = {0},a,b ∈ R, implies that either a = 0 or b = 0. Furthermore, a ring R is calledsemiprime if aRa = {0}, a ∈ R, implies a = 0. We will denote by C and Q theextended centroid and the maximal right ring of quotients of a semiprime ringR, respectively. For the explanation of the extended centroid as well as themaximal right ring of quotients of a semiprime ring we refer the reader to [4].As usual, the socle of a ring R will be denoted by soc(R).An additive mapping D : R → R is called a derivation on R if D(xy) =D(x)y + xD(y) holds for all pairs x,y ∈ R. An additive mapping f : R →R is called centralizing on R if [f(x),x] ∈ Z(R) holds for all x ∈ R. In aspecial case, when [f(x),x] = 0 for all x ∈ R, the mapping f is said to becommuting on R. A classical result of Posner [21] (Posner’s second theorem)states that the existence of a nonzero centralizing derivation on a prime ringforces the ring to be commutative. Posner’s second theorem in general cannotbe proved for semiprime rings as shows the following example. Let R

Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems

Abstract Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let α > 0 , and let A be an α-inverse strongly-monotone mapping of C into H. Let T be a generalized hybrid mapping of C into H. Let B and W be maximal monotone operators on H such that the domains of B and W are included in C. Let 0 < k < 1 , and let g be a k-contraction of H into itself. Let V be a γ ¯ -strongly monotone and L-Lipschitzian continuous operator with γ ¯ > 0 and L > 0 . Take μ , γ ∈ R as follows: 0 < μ < 2 γ ¯ L 2 , 0 < γ < γ ¯ − L 2 μ 2 k . Suppose that F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 ≠ ∅ , where F ( T ) and ( A + B ) − 1 0 , W − 1 0 are the set of fixed points of T and the sets of zero points of A + B and W, respectively. In this paper, we prove a strong convergence theorem for finding a point z 0 of F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 , where z 0 is a unique fixed point of P F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 ( I − V + γ g ) . This point z 0 ∈ F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 is also a unique solution of the variational inequality 〈 ( V − γ g ) z 0 , q − z 0 〉 ≥ 0 , ∀ q ∈ F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 . Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi (Nonlinear Anal. 73:1562-1568, 2010). MSC:47H05, 47H10, 58E35.

Teichmüller sequences on trajectories invariant under a Kleinian group

We extend the results of [T2] to the situation where there is a compatibility with the action of a Kleinian group. A classical Techmuller sequence is a sequence of quasiconformal mapsf i with complex dilatations of the form\(k_i \bar \varphi /\left| \varphi \right|\), where ϕ is a quadratic differential and 0<-k i<1 are numbers such thatk i→1 asi→∞. We proved in [T2] that if τ is a vertical trajectory associated to ϕ, then there is often, for instance if the sequence is normalized so thatf i fix 3 points, a subsequence such thatf i tend either toward a constant or an injective map of τ. If there is compatibility with the action of a non-elementary finitely generated Kleinian groupG, we can given a precise characterization which of these cases occurs. Suppose thatf i induce isomorphisms ϕi ofG onto another Kleinian group and that ϕi have algebraic limit ϕ. If the quadratic differential is defined on a component of the ordinary set ofG, if there are no parabolic elements, and if τ is extended maximally so that all branches coming together at a singular point are included, then we can state the main result as follows. The limit is a constantc if the stabilizerG τ of τ is elementary; and, if it is non-elementary, then the limit is injective. In the first case, ϕ(g) is parabolic with fixpointc wheneverg∈G τ is of infinite order; and in the latter case, the limitf is an embedding of τ in a natural topology of τ, andf embeds τ into a component of the limit set of ϕG whose stabilizer is ϕG τ. Various extensions and generalizations are presented.

Poset fiber theorems

Suppose thatf:P→Qf:P \to Qis a poset map whose fibersf−1(Q≤q)f^{-1}(Q_{\le q})are sufficiently well connected. Our main result is a formula expressing the homotopy type ofPPin terms ofQQand the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, Cohen-Macaulay, and equivariant versions are given, and some applications are discussed.

Minimal systems of generators for central division algebra and semi-simple algebra

Suppose thatF is a field of characteristic zero and (i)D is a finite-dimensional central division algabra overF; (ii)A is a finite-dimensional semi-simple algabra overF. It is proved that asF-algebras,D andA can be generated by two elements respectively.

The multiple-point schemes of a finite curvilinear map of codimension one

LetX andY be smooth varieties of dimensionsn−1 andn over an arbitrary algebraically closed field,f: X→Y a finite map that is birational onto its image. Suppose thatf is curvilinear; that is, for allxeX, the Jacobian ϱf(x) has rank at leastn−2. Forr≥1, consider the subschemeN r ofY defined by the (r−1)th Fitting ideal of the $$\mathcal{O}_Y $$ -module $$f_ * \mathcal{O}_X $$ , and setM r ∶=f −1 N r . In this setting—in fact, in a more general setting—we prove the following statements, which show thatM r andN r behave like reasonable schemes of source and targetr-fold points off. If each component ofM r , or equivalently ofN r , has the minimal possible dimensionn−r, thenM r andN r are Cohen-Macaulay, and their fundamental cycles satisfy the relation,f *[M r ]=r[N r ]. Now, suppose that each component ofM s , or ofN s , has dimensionn−s fors=1,...,r+1. Then the blowup Bl(N r ,N r+1 ) is equal to the Hilbert scheme Hilb and the blowup Bl(M r ,M r+1 ) is equal to the universal subscheme Univ of Hilb × Y X; moreover, Hilb and Univ are Gorenstein. In addition, the structure maph:Hilb →Y is finite and birational onto its image; and its conductor is equal to the ideal $$\mathcal{J}_r $$ ofN r+1 inN r , and is locally self-linked. Reciprocally, $$h_ * \mathcal{O}_{Hilb_f^r } $$ is equal to $$\mathcal{H}om(\mathcal{J}_r ,\mathcal{O}_{N_r } )$$ . Moreover,h * [h −1 N r+1 ]=(r+1)[N r+1 ]. Similar assertions hold for the structure maph 1: Univ →X ifr≥2.

On closed graph and implicit function theorems for multifunctions

AbstractWe give several general implicit function and closed graph theorems for ser-valued functions. Let Z be a normed space,X, Ymetric spaces withXcomplete. Letf: X ⇉ Z,F: X × Y⇉ Z be multifunctions withz0∈ f(x0) ∩ F(x0, y0) such thatfis open at (x0, y0) andf‘approximates’Fin an appropriate sense. Suppose thatf−1(z) is closed,F(x, y)is compact for eachx, yandzand suppose thatF(x0, ·) is lower semi-continuous aty0. ThenF(·,y) is of closed graph ‘locally’, is open atx0, and there exists a functionx(·) withx(y)→x0fory→y0such thatz0∈F(x(y), (y))for allyneary0. A more general form dealing with the non-linear rate situation is also established.

The Incongruence of Consecutive Values of Polynomials

Suppose thatf∈ Z|x|. PutDf(n) = min{k> 0|f(1), ... ,f(n) are pairwise incongruent modulok}. Special cases of this function were previously considered, using methods from elementary number theory. Results from the theory of finite fields are used to prove a theorem that for allfin a large subset of Z|x| provides a characterization ofDf(n) for allnsufficiently large. This theorem partially encompasses results due to Bremser, Schumer and Washington and to Moree and Mullen, who characterizedDf(n) for cyclic, respectively, Dickson polynomials.

On almost everywhere convergence

Suppose thatf is an element ofL2(Rn) whose orbit under the action ofSO(n) spans a finite-dimensional subspace. Then the spherical partial sums of the inverse Fourier transform off converge almost everywhere.

A boundary Morera theorem

LetD ⊂CN,N ≥ 2 be a bounded open set withC2 boundary and letL be an open connected set of affine complex hyperplanes inCN containing a hyperplane that misses\(\bar D\). LetE = ∪Λ∈LΛ, Γ =E ∩bD. Suppose thatf ∈C(Γ) and assume that $$\int_{\Lambda \cap bD} {f\alpha = 0} $$ for every Λ ∈L that meetsbD transversely and for every (N −1,N −2)-form with constant coefficients. Thenf satisfies the weak tangential Cauchy-Riemann equations on Γ. It is shown by an example that the condition thatL contains a hyperplane that misses\(\bar D\) cannot be dropped.

On spaces of Baire I functions over K-analytic spaces

Suppose thatℱ is a relatively countably compact subset of B1(X), the space of Baire I functions over a K-analytic space X equipped with the pointwise convergence topology. It is proved that (1) the closure ofℱ is a strongly countably compact Frechet-Urysohn space; (2) ifℱ is ℵ1 -compact,ℱ is a bicompactum; (3) if X is a paracompact space, the closure ofℱ is a bicompactum.

$$A$$ and the vanishing topology of discriminants

Suppose thatf: ℂn, 0→ℂp, 0 is finitely\(A\)-determined withn≧p. We define a “Milnor fiber” for the discriminant off; it is the discriminant of a “stabilization” off. We prove that this “discriminant Milnor fiber” has the homotopy type of a wedge of spheres of dimensionp−1, whose number we denote byµΔ(f). One of the main theorems of the paper is a “μ=τ” type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then\(\mu _\Delta (f) \geqq A_e \)-codium,with equality iff is weighted homogeneous. Outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization. These results are further extended to nonlinear sections of free divisors.

Coconvex approximation of continuous functions

Suppose thatf(x)∈C[−1, 1],f(x) is convex in [0,1] and concave in [−1,0] with the new concept of regular convexity-turning points,we obtain the following estimates of Jackson type for coconvex approximation off(x) by algebraic polynomials of degree ⩽n: $$\tilde E_n (f) \leqslant C \omega \left( {f, \frac{1}{n}} \right)$$ .

The extrapolation of band-limited signals in L2 with sampling errors

In this paper we will provide a theory for an extrapolation algorithm of band-limited signals with sampling errors or low noises. The main result is as follows: Suppose thatf(t) is in L2 and is a continuous Ω band-limited signal. Then for any positive numbers T, A(>T) and e, there exists a δ>0 such that when the sampling errors (or noises) off(t) on [−T, T] is less than δ we can extrapolate the values off(t) on the interval [−A, A] such that the difference between the extrapolated value and the exact value off(t) does not exceed e.

Periodic point free homeomorphism of 𝑇²

Suppose that f : T 2 → T 2 f:{T^2} \to {T^2} is an orientation preserving homeomorphism of the torus that is homotopic to the identity and that has no periodic points. We show that there is a direction θ \theta and a number ρ \rho such that every orbit of f f has rotation number ρ \rho in the direction θ \theta .