Common extension of bilateral series for Andrews’ q-Bailey and q-Gauss sums

By means of the modified Abel lemma on summation by parts, we establish a common bilateral series extension of the q-Bailey and q-Gauss sums discovered by Andrews (Duke Math. J. 40 (1973), 525–528).     

An extension of Matkowski’s and Wardowski’s fixed point theorems with applications to functional equations

Aequationes mathematicae - In this paper, we prove a fixed point theorem for a system of maps on the finite product of metric spaces. Our result generalizes the result of Matkowski (Bull Acad Pol...     

Cartan’s Second Main Theorem and Mason’s Theorem for Jackson Difference Operator

Let f : C → Pⁿ be a holomorphic curve of order zero. The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves. In addition, a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial. Furthermore, they extend the Mason’s theorem for m + 1 polynomials. Some examples are constructed to show that their results are accurate.     

On some general Hilbert’s inequalities with non-conjugate parameters

A generalized discrete Hilbert’s and Hardy-Hilbert’s inequality with non-conjugate parameters can be established by means of Euler-Maclaurin summation formula. We derive some general results for homogeneous functions and compare our results with some previously known from the literature. We also obtain the improvements on some earlier results.     

Complex Euler’s groups and values of Euler’s function at complex integer Gauss points

The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z, relatively prime to z. This group is calculated for z=(3+0i) ⁿ : it is isomorphic to the product of three cyclic group or orders (8, 3 ⁿ⁻¹ and 3 ⁿ⁻¹).     

An extension of Barta’s Theorem and geometric applications

We prove an extension of a theorem of Barta and we give some geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We show that the spectrum of the Nadirashvili bounded minimal surfaces in have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations. Bessa and Montenegro were partially supported by CNPq Grant.     

Vanishing coefficients and identities concerning Ramanujan’s parameters

The Ramanujan Journal - In this paper we investigate several infinite products with vanishing Taylor coefficients in arithmetic progressions. These infinite products are closely related to...     

Discretized Newman-Shapiro Operators and Jackson’s Inequality on the Sphere

Applying a quadrature rule with positive weights to some integral operators introduced by Newman and Shapiro we obtain generalized hyperinterpolation operators on the sphere, whose approximation error can be estimated — inspite of the discretisation — by means of the modulus of continuity. The main reason is that the weight distribution satisfies necessarily some regularity condition, which has been used before in hyperinterpolation by Sloan and Womersley, and which turned out to hold always by its own.     

Möbius’s functional equation and Schur’s lemma with applications to the complex unit disk

Möbius addition is defined on the complex open unit disk by

$$\begin{aligned} a\oplus _M b = \dfrac{a+b}{1+\bar{a}b} \end{aligned}$$

and Möbius’s exponential equation takes the form \(L(a\oplus _M b) = L(a)L(b)\), where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Möbius’s exponential equation is connected to Cauchy’s exponential equation. Möbius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Möbius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting.

     

Synchronization of integer order and fractional order Chua’s systems using robust observer

In this paper we present a fractional order Chua’s circuit that behaves chaotically based on the use of a fractional order low pass filter. Next, an integer order robust observer will be designed to synchronize the fractional order Chua’s circuit as well as integer order Chua’s circuit with unknown nonlinearity. This method consists in designing a Luenberger like observer appended with an estimator of the unknown nonlinear function. The estimator assumes that the nonlinear function is slowly varying and that the observer converges quickly and uses the backward difference formula to approximate the state derivative. The efficiency of the proposed method is confirmed using numerical simulations.     

Generalization of B.M. Levitan and M.G. Gasymov’s solvability theorems to the case of indecomposable boundary conditions

Doklady Mathematics -     

Reformulation of Hensel’s Lemma and extension of a theorem of Ore

In this paper, we extend the theorem of Ore regarding factorization of polynomials over p-adic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of Khanduja and Kumar (J Pure Appl Algebra 216:2648–2656, 2012) and Cohen et al. (Mathematika 47:173–196, 2000). As an application, we derive the analogue of Dedekind’s Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in Khanduja and Kumar (Int J Number Theory 4:1019–1025, 2008). The generalized version of Ore’s Theorem leads to an extension of a result of Weintraub dealing with a generalization of Eisenstein Irreducibility Criterion (cf. Weintraub in Proc Am Math Soc 141:1159–1160, 2013). We also give a reformulation of Hensel’s Lemma for polynomials with coefficients in henselian valued fields which is used in the proof of the extended Ore’s Theorem and was proved in Khanduja and Kumar (J Algebra Appl 12:1250125, 2013) in the particular case of complete rank one valued fields.     

On Jensen’s inequality and Hölder’s inequality for g-expectation

In this paper, we shall prove that for n > 1, the n-dimensional Jensen inequality holds for the g-expectation if and only if g is independent of y and linear with respect to z, in other words, the corresponding g-expectation must be linear. A Similar result also holds for the general nonlinear expectation defined in Coquet et al. (Prob. Theory Relat. Fields 123 (2002), 1–27 or Peng (Stochastic Methods in Finance Lectures, LNM 1856, 143–217, Springer-Verlag, Berlin, 2004). As an application of a special n-dimensional Jensen inequality for g-expectation, we give a sufficient condition for g under which the Hölder’s inequality and Minkowski’s inequality for the corresponding g-expectation hold true.     

The Piltz’s problem over a ring of matrices of second order with integer elements

The asymptotic formula for a summatory function of the number of representations of matrices from $ {M_2}\left( \mathbb{Z} \right) $ in the form C = A ₁ A ₂ A ₃ under the condition det C ≤ x is constructed. The second moment of the error term of this asymptotic formula is estimated.     

Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.