Further results on factorization of meromorphic solutions of partial differential equations

This paper is a continuation of Hu-Yang [2]. Here we extend Malmquist type theorem ofalgebraic differential equations of Steinmetz [3] and Tu [4] to higher order partial differential equations. The results also generalize Theorems 4.2 and 4.3 in [2].     

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence C_N . The mixed middle curvature and mixed curvature on C_N are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].     

To the question on the almost everywhere convergence of the Riesz means of double orthogonal series

Sufficient conditions of the classical type ensuring the almost everywhere (a.e.) convergence of the nonnegative-order Riesz means of double orthogonal series are indicated. Analogies of the onedimensional results of Kolmogoroff [7] and Kaczmarz?CZygmund [5, 12] have been obtained for the Cesaro means and those of Zygmund [13] for the Riesz means. These analogies establish the a.e. equiconvergence of the lacunary subsequences of rectangular partial sums and of the entire sequence of Riesz means, generalize the corresponding results of Moricz [9] for the Cesaro a.e. summability by (C, 1, 1), (C, 1, 0), and (C, 0, 1) methods of double orthogonal series, and were announced earlier without proofs in the author??s work [3].     

The General Solution of Abel-Type Functional Equations

The general measurable solution of (A) was found by Stamate [8]. Aczél [3] and Lajkô [6] proved that the general solution of (A) for unknown functions ψ, g, h: ? → ? are (1), (2) and (3), respectively. Filipescu [5] found the general measurable solution of (B). We establish an elementary prof for the general solution of equation (A) (Theorem 1.). Our method is suitable for finding the general solution of (B) (Theorem 2.).     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H ¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L ² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$ Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].     

Exponential polynomials on commutative hypergroups

Polynomials and exponential polynomials play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative groups. Recently several new results have been published in this field [2–4,6]. Spectral analysis and spectral synthesis has been studied on some types of commutative hypergroups, as well. However, a satisfactory definition of exponential monomials on general commutative hypergroups has not been available so far. In [5,7,8] and [9], the authors use a special concept on polynomial and Sturm–Liouville-hypergroups. Here we give a general definition which covers the known special cases.     

Algebraische Unabhängigkeit der Werte gewisser Lückenreihen in nicht-archimedisch bewerteten Körpern

In 1844 Liouville proved the transcendence of α = ∑_h≥1 10^{?h h!} over Q. The number α can be considered as the value of the gap power series ∧(x) =∑_h≥1 at tne point 1/10 Since then, the above result has been generalized in this direction by different authors by applying improved “Liouville-estimates”. For instance, in 1973 Cijsouw and Tijdeman [2] showed that a gap series with algebraic coefficients takes on transcendental values (over Q) at non-zero algebraic points under some conditions on the growth of the coefficients and the gaps. In 1988 Bundschuh [1] resp. Zhu [9] proved the algebraic independence (over Q) of the values of several gap series at different algebraic points. In particular this result includes the algebraic independence of A(α₁),…, α(α_s) for non-zero algebraic numbers α₁,…, α_s of distinct absolute values less than 1. Moreover in [1] a set of continuum-many algebraically independent numbers was constructed. In 1978 Geijsel [4] obtained a result analogous to that of Cijsouw and Tijdeman underlying a non-archimedian valued function field over a finite field, and in 1983 Sieburg [7] was concerned with the algebraic independence of “Liouville-series” in non-archimedian valued fields of characteristic zero. In this paper some of the results of [1] resp. [9] will be transfered to the situation of some non-archimedian valued fields. If the characteristic of the field is prime, we have to require stronger conditions as in the “classical case”. An example shows that in this case the numbers A(c*i),..., A(aa) need not to be algebraically independent. But a set of continuum-many algebraically independent numbers still exists. In characteristic zero, results of the same kind will be obtained like in the “classical case”.     

Algebraic Hecke characters and strictly compatible systems of abelian mod p Galois representations over global fields

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.     

Further remarks on the non-degeneracy condition

The structure of the set of positive solutions of the semilinear elliptic boundary value problem depends on a certain non-degeneracy condition, which was proved by K.J. Brown [2] and T. Ouyang and J. Shi [12], with a shorter proof given later by P. Korman [8]. In this note we present a more general result, communicated to us by L. Nirenberg [13]. We also discuss the extensions in cases when the domain D is in R ², and it is either symmetric or convex.     

The Totally Nonnegative Part of G/P is a CW Complex

The totally nonnegative part of a partial ag variety G/P has been shown in [18], [17] to be a union of semialgebraic cells. Moreover, the closure of a cell was shown in [19] to be a union of smaller cells. In this paper we provide glueing maps for each of the cells to prove that (G/P)_?0 is a CW complex. This generalizes a result of Postnikov, Speyer and the second author [15] for Grassmannians.     

Computing Final Polynomials and Final Syzygies Using Buchberger’s Gröbner Bases Method

Final polynomials and final syzygies provide an explicit representation of polynomial identities promised by Hilbert’s Nullstellensatz. Such representations have been studied independently by Bokowski [2,3,4] and Whiteley [23,24] to derive invariant algebraic proofs for statements in geometry. In the present paper we relate these methods to some recent developments in computational algebraic geometry. As the main new result we give an algorithm based on B. Buchberger’s Gröbner bases method for computing final polynomials and final syzygies over the complex numbers. Degree upper bound for final polynomials are derived from theorems of Lazard and Brownawell, and a topological criterion is proved for the existence of final syzygies. The second part of this paper is expository and discusses applications of our algorithm to real projective geometry, invariant theory and matrix theory.     

Ideal Divisibility Monoids

Inspired by the monograph of Larsen/McCarthy, [26], in [10] and [11] the author started a series of articles concerning abstract multiplicative ideal theory along the problem lines of [26]. In this paper we turn to multiplicative lattices having the left Priifer property, that is to m-lattices satisfying the implication a¹ + … + a_n ? B ? a₁ +… + a_n ¦? B or even the multiplication property A ? B ? A ¦B, respectively. Clearly, studying such structures includes studying substructures of d-semigroups.     

Generators for cubic surfaces with two skew lines over finite fields

Let S be a smooth cubic surface defined over a field K. As observed by Segre [5] and Manin [3, 4], there is a secant and tangent process on S that generates new K-rational points from old ones. It is natural to ask for the size of a minimal generating set for S(K). In a recent paper, for fields K with at least 13 elements, Siksek [7] showed that if S contains a skew pair of K-lines, then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for # K = 2 or 3.     

Improved sparse fourier approximation results: faster implementations and stronger guarantees

We study the problem of quickly estimating the best k-term Fourier representation for a given periodic function f: [0, 2π] → ?. Solving this problem requires the identification of k of the largest magnitude Fourier series coefficients of f in worst case k ² · log ^O(1) N time. Randomized sublinear-time Monte Carlo algorithms, which have a small probability of failing to output accurate answers for each input signal, have been developed for solving this problem (Gilbert et al. 2002, 2005). These methods were implemented as the Ann Arbor Fast Fourier Transform (AAFFT) and empirically evaluated in Iwen et al. (Commun Math Sci 5(4):981–998, 2007). In this paper we present a new implementation, called the Gopher Fast Fourier Transform (GFFT), of more recently developed sparse Fourier transform techniques (Iwen, Found Comput Math 10(3):303–338, 2010, Appl Comput Harmon Anal, 2012). Experiments indicate that GFFT is faster than AAFFT. In addition to our empirical evaluation, we also consider the existence of sublinear-time Fourier approximation methods with deterministic approximation guarantees for functions whose sequences of Fourier series coefficents are compressible. In particular, we prove the existence of sublinear-time Las Vegas Fourier Transforms which improve on the recent deterministic Fourier approximation results of Iwen (Found Comput Math 10(3):303–338, 2010, Appl Comput Harmon Anal, 2012) for Fourier compressible functions by guaranteeing accurate answers while using an asymptotically near-optimal number of function evaluations.     

The Mellin Transform and Spectral Properties of Toric Varieties

In this paper we apply the results of [W] on the twisted Mellin transform to problems in toric geometry. In particular, we use these results to describe the asymptotics of probability densities associated with the monomial eigenstates, z ^k , $ k \in \mathbb{Z}^{d} $ , in Bargmann space and prove an “upstairs” version of the spectral density theorem of [BGU]. We also obtain for the z ^k ’s, “upstairs” versions of the results of [STZ] on distribution laws for eigenstates on toric varieties.     

C∞ functions in infinite dimension and linear partial differential difference equations with constant coefficients

This work is closed to [2] where a dense linear subspace \(\mathbb{E}\) (E) of the space ?(E) of the Silva C ^∞ functions on E is defined; the dual of \(\mathbb{E}\) (E) is described via the Fourier transform by a Paley-Wiener-Schwartz theorem which is formulated exactly in the same way as in the finite dimensional case. Here we prove existence and approximation result for solutions of linear partial differential difference equations in \(\mathbb{E}\) (E) with constant coefficients. We also obtain a Hahn-Banach type extension theorem for some C^∞ functions defined on a closed subspace of a DFN space, which is analogous to a Boland’s result in the holomorphic case [1].     

Classroom-based interventions in mathematics education: relevance, significance, and applicability

This special issue discusses various pedagogical innovations and myriad of significant findings. This commentary is not a synthesis of these contributions, but a summary of my own reflections on selected aspects of the nine papers comprising the special issue. Four themes subsume these reflections: (1) Gestural Communication (Alibali, Nathan, Church, Wolfgram, Kim and Knuth 2013); (2) Development of Ways of Thinking (Jahnke and Wambach 2013; Lehrer, Kobiela and Weinberg 2013; Mariotti 2013; Roberts and A. Stylianides 2013; Shilling-Traina and G. Stylianides 2013; Tabach, Hershkowitz and Dreyfus 2013); (3) Learning Mathematics through Representation (Saxe, Diakow and Gearhart 2013); and (4) Challenges in Dialogic Teaching (Ruthven and Hofmann 2013).     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.