Zeros

In this paper, by using standard techniques we shall obtain results with relaxed hypothesis which give zero bounds for the larger class of polynomials. Our results not only generalizes several well-known results but also provide better information about the location of zeros. We also obtain a similar result for analytic functions. In addition to this, we show by examples that our result gives better information on the zero bounds of polynomials than some known results.

In this paper, we give a numerically reliable algorithm to compute the zeros of a periodic descriptor system. The algorithm is a variant of the staircase algorithm applied to the system pencil of an equivalent lifted time-invariant state-space system and extracts a low-order pencil which contains the zeros (both ÿnite and inÿnite) as well as the Kronecker structure of the periodic descriptor system. The proposed algorithm is e cient in terms of complexity by exploiting the structure of the pencil and is exclusively based on orthogonal transformations, which ensures some form of numerical stability.

In this paper, by using standard techniques we shall obtain results with relaxed hypothesis which give zero bounds for the larger class of polynomials. Our results not only generalizes several well-known results but also provide better information about the location of zeros. We also obtain a similar result for analytic functions. In addition to this, we show by examples that our result gives better information on the zero bounds of polynomials than some known results.

A special case of a fundamental theorem of Schneider asserts that if j (τ) is algebraic (where j is the classical modular invariant), then any zero z not in Q.L τ := Q ⊕ Qτ of the Weierstrass function ℘ (τ, •) attached to the lattice L τ = Z ⊕ Zτ is transcendental. In this note we generalize this result to holomorphic Jacobi forms of weight k and index m ∈ N with algebraic Fourier coefficients.

A special case of a fundamental theorem of Schneider asserts that if j (τ) is algebraic (where j is the classical modular invariant), then any zero z not in Q.L τ := Q ⊕ Qτ of the Weierstrass function ℘ (τ, •) attached to the lattice L τ = Z ⊕ Zτ is transcendental. In this note we generalize this result to holomorphic Jacobi forms of weight k and index m ∈ N with algebraic Fourier coefficients.

Let f be a piecewise analytic (but not analytic) function in @[a, b], k > 0, and let p,* be the sequence of polynomials of best uniform approximation to f on [a, b]. It is well known that every point of [a, b] is a limit point of the zeros of the p,*. Let x E [a, b], and suppose that f is analytic at x and f(x) # 0. The main purpose of this paper is to show that there exists a constant y (which depends only on x) such that there is no zero of p,* within the circle of radius (y/n) log n centered at X, for all sufficiently large values of n.

Let f be a piecewise analytic (but not analytic) function in @[a, b], k > 0, and let p,* be the sequence of polynomials of best uniform approximation to f on [a, b]. It is well known that every point of [a, b] is a limit point of the zeros of the p,*. Let x E [a, b], and suppose that f is analytic at x and f(x) # 0. The main purpose of this paper is to show that there exists a constant y (which depends only on x) such that there is no zero of p,* within the circle of radius (y/n) log n centered at X, for all sufficiently large values of n.

An equivalent, but variant form of Riemann's functional equation is explored, and several discoveries are made. Properties of Riemann's zeta function (s), from which a necessary and sufficient condition for the existence of zeros in the critical strip, are deduced. This in turn, by an indirect route, eventually produces a simple, solvable, differential equation for arg((s)) on the critical line s = 1∕2 + i , the consequences of which are explored, and the "LogZeta" function is introduced. A singular linear transform between the real and imaginary components of and ′ on the critical line is derived, and an implicit relationship for locating a zero (= 0) on the critical line is found between the arguments of (1∕2 + i) and � (1∕2 + i). Notably, the Volchkov criterion, a Riemann Hypothesis (RH) equivalent, is analytically evaluated and verified to be half equivalent to RH, but RH is not proven. Numerical results are presented, some of which lead to the identification of anomalous zeros, whose existence in turn suggests that well-established, traditional derivations such as the Volchkov criterion and counting theorems require re-examination. It is proven that the derivative � (1∕2 + i) will never vanish on the perforated critical line (≠ 0). Traditional asymptotic and counting results are obtained in an untraditional manner, yielding insight into the nature of (1∕2 + i) as well as very accurate asymptotic estimates for distribution bounds and the density of zeros on the critical line.

A search for a heavy Standard Model Higgs boson decaying via H → Z Z → + − qq, where = e, μ, is presented. The search is performed using a data set of pp collisions at √ s = 7 TeV, corresponding to an integrated luminosity of 1.04 fb −1 collected in 2011 by the ATLAS detector at the CERN LHC collider. No significant excess of events above the estimated background is found. Upper limits at 95% confidence level on the production cross section (relative to that expected from the Standard Model) of a Higgs boson with a mass in the range between 200 and 600 GeV are derived. Within this mass range, there is at present insufficient sensitivity to exclude a Standard Model Higgs boson. For a Higgs boson with a mass of 360 GeV, where the sensitivity is maximal, the observed and expected cross section upper limits are factors of 1.7 and 2.7, respectively, larger than the Standard Model prediction.

It is well known that Sobolev-type orthogonal polynomials with respect to measures supported on the real line satisfy higher-order recurrence relations and these can be expressed as a (2N + 1)-banded symmetric semi-infinite matrix. In this paper, we state the connection between these (2N + 1)-banded matrices and the Jacobi matrices associated with the three-term recurrence relation satisfied by the standard sequence of orthonormal polynomials with respect to the 2-iterated Christoffel transformation of the measure.

The complex zeros of the orthogonal Laguerre polynomials L a n x for a < −n, ultraspherical polynomials P λ n x for λ < −n, Jacobi polynomials P a,β n x for a < −n, β < −n, a β < −2 n 1 , orthonormal Al-Salam-Carlitz II polynomials P a n x; q for a < 0, 0 < q < 1, and q-Laguerre polynomials L a n x; q for a < −n, 0 < q < 1 are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.

Two new limits involving Chebyshev polynomials of the first and second kinds are given. These limits are useful in certain engineering applications. The proofs are based on the Mehler-Heine theorem for Jacobi polynomials. Let Plp34'(x) denote the Jacobi polynomials. It is evident from the representation [ 1, (4.21.2)]

This paper deals with Mehler-Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points. c

Let S n be polynomials orthogonal with respect to the inner product ð f ; gÞ S ¼ Z N 0 fg dm 0 þ l Z N 0 f 0 g 0 dm 1 ; where dm 0 ¼ x a e Àx dx; dm 1 ¼ x aþ1 e Àx xÀx dx þ Md x with a4 À 1; xp0; MX0; and l40: A strong asymptotic on ð0; NÞ; a Mehler-Heine type formula, a Plancherel-Rotach type exterior asymptotic as well as an upper estimate for S n are obtained. As a consequence, we give asymptotic results for the zeros and critical points of S n and the distribution of contracted zeros. Some numerical examples are shown.

A search for the Standard Model Higgs boson in the two photon decay channel is reported, using 1.08 fb −1 of proton-proton collision data at a centre-of-mass energy of 7 TeV recorded by the ATLAS detector. No significant excess is observed in the investigated mass range of 110-150 GeV. Upper limits on the cross-section times branching ratio of between 2.0 and 5.8 times the Standard Model prediction are derived for this mass range. 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2).

This paper illustrates the theory behind the development of a General form of a numerically accurate and stable software for determining the lateral natural frequencies, mode shapes, stability and unbalance response of multi-shaft rotor bearing systems which may be gear coupled and supported on flexible anisotropic bearings, such as hydrodynamic bearings as well as rigid rolling elements. Gyroscopic effects, shear deformation and hysteretic damping are included in the analysis. To achieve the improved accuracy and numerical convergence to all relevant roots of the characteristic equations the Riccati transfer matrix is utilized rather than the traditional transfer matrix approach. This more accurate approach introduces poles into the characteristic equation. After cancellation of all poles, convergence to all relevant roots is guaranteed. In this study two different methods of handling in-span conditions are presented and it is proven that after elimination of poles the Riccati tran...

In this paper certain polynomial inequalities for the polar derivative and the generalized polar derivative with restricted zeros are given, which generalize and refine some well-known polynomial inequalities due to Lax, Tur ́n, AlSaeedi, Rather, Ali, Shafi, and Dar and others.

A search for the Standard Model Higgs boson in the two photon decay channel is reported, using 1.08 fb −1 of proton-proton collision data at a centre-of-mass energy of 7 TeV recorded by the ATLAS detector. No significant excess is observed in the investigated mass range of 110-150 GeV. Upper limits on the cross-section times branching ratio of between 2.0 and 5.8 times the Standard Model prediction are derived for this mass range. 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2).

The number of zeros in (−1, 1) of the Jacobi function of second kind Q (α,β) n (x), α, β > −1, that is the second solution of the differential equation (1 − x 2)y (x) + (β − α − (α + β + 2)x)y (x) + n(n + α + β + 1)y(x) = 0 , is determined for every n ∈ N and for all values of the parameters α > −1 and β > −1. It turns out that this number depends essentially on α and β as well as on the specific normalization of the function Q (α,β) n (x). Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind.

For a wide class of Sobolev orthogonal polynomials, it is proved that their zeros are contained in a compact subset of the complex plane and the asymptotic zero distribution is obtained. With this information, the n-th root asymptotic behavior outside the compact set containing all the zeros is given.

The existing technique of manual calcu The existing technique of manual calcu The existing technique of manual calcu The existing technique of manual calcu studies studies studies studies is laborious and time consuming. is laborious and time consuming. is laborious and time consuming. is laborious and time consuming. simple fast mathematical iteration is simple fast mathematical iteration is simple fast mathematical iteration is simple fast mathematical iteration is were formed and te were formed and te were formed and te were formed and tested for convergence. A pair of complementary formulas with highest rate of sted for convergence. A pair of complementary formulas with highest rate of sted for convergence. A pair of complementary formulas with highest rate of sted for convergence. A pair of complementary formulas with highest rate of convergence was selected. The process of convergence was selected. The process of convergence was selected. The process of convergence was selected. The process of were divided into steps which were were divided into steps which were were divided into steps which were were divided into steps which were g g g geometric properties of the root eometric properties of the root eometric properties of the root eometric properties of the root-de de de described in the literature. Root scribed in the literature. Root scribed in the literature. Root scribed in the literature. Root-loci are drawn to scale instead of rough sketches. loci are drawn to scale instead of rough sketches. loci are drawn to scale instead of rough sketches. loci are drawn to scale instead of rough sketches.

The number of zeros in (−1, 1) of the Jacobi function of second kind Q (α,β) n (x), α, β > −1, that is the second solution of the differential equation (1 − x 2)y (x) + (β − α − (α + β + 2)x)y (x) + n(n + α + β + 1)y(x) = 0 , is determined for every n ∈ N and for all values of the parameters α > −1 and β > −1. It turns out that this number depends essentially on α and β as well as on the specific normalization of the function Q (α,β) n (x). Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind.

Denote by x M,N n,k (α), k = 1,. .. , n, the zeros of the Laguerre-Sobolev-Type polynomials L (α,M,N) n (x) orthogonal with respect to the inner product p, q = 1 Γ(α + 1) ∞ 0 p(x)q(x)x α e −x dx + M p(0)q(0) + N p (0)q (0) where α > −1, M ≥ 0 and N ≥ 0. We prove that x M,N n,k (α) interlace with the zeros of Laguerre orthogonal polynomials L (α) n (x) and establish monotonicity with respect to the parameters M and N of x M,0 n,k (α) and x 0,N n,k (α). Moreover, we find N 0 such that x M,N n,n (α) < 0 for all N > N 0 , where x M,N n,n (α) is the smallest zero of L (α,M,N) n (x). Further, we present monotonicity and asymptotic relations of certain functions involving x M,0 n,k (α) and x 0,N n,k (α).

Design and analysis of control systems often become difficult due to the complexity of the system model and the design techniques involved. This paper presents the development of a Tools Box in Microsoft Excel for control engineer that uses root locus as a time domain technique for system design and analysis. The Tool Box can also serve as a computer-aided graphical analytical tool for trainers. The work was done in two phases: the first phase is the development of a programmable algorithms for root locus using the angle condition and bisection method while the second phase is the implementation of the developed algorithms. The implementation was done using Microsoft Excel (R) Visual Basic Application (VBA). Results of simulations for different systems show the potential of the Tool Box as an alternative for other software package and the ease of using it on the readily available Microsoft Excel environment.

In this manuscript, new algebraic and analytic aspects of the orthogonal polynomials satisfying $R_{II}$ type recurrence relation given by \begin{align*} \mathcal{P}_{n+1}(x) = (x-c_n)\mathcal{P}_n(x)-λ_n (x-a_n)(x-b_n)\mathcal{P}_{n-1}(x), \quad n \geq 0, \end{align*} where $λ_n$ is a positive chain sequence and $a_n$, $b_n$, $c_n$ are sequences of real or complex numbers with $\mathcal{P}_{-1}(x) = 0$ and $\mathcal{P}_0(x) = 1$ are investigated when the recurrence coefficients are perturbed. Specifically, representation of new perturbed polynomials (co-polynomials of $R_{II}$ type) in terms of original ones with the interlacing and monotonicity properties of zeros are given. For finite perturbations, a transfer matrix approach is used to obtain new structural relations. Effect of co-dilation in the corresponding chain sequences and their consequences onto the unit circle are analysed. A particular perturbation in the corresponding chain sequence called complementary chain sequence...

We propose an approach to modeling and estimating discrete choice demand that allows for a large number of zero sale observations, rich unobserved heterogeneity, and endogenous prices. We do so by modeling small market sizes through Poisson arrivals. Each of these arriving consumers then solves a standard discrete choice problem. We present a Bayesian IV estimation approach that addresses sampling error in product shares and scales well to rich data environments. The data requirements are traditional market-level data and measures of consumer search intensity. After presenting simulation studies, we consider an empirical application of air travel demand where product-level sales are sparse. We find considerable variation in demand over time. Periods of peak demand feature both larger market sizes and consumers with higher willingness to pay. This amplifies cyclicality. However, observed frequent price and capacity adjustments offset some of this compounding effect. Ali Hortaçsu Kenn...

We prove that if ω is uniformly distributed on [0, 1], then as T → ∞, t → ζ(iωT +it+1/2) converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between probabilistic number theory and the theory of multiplicative chaos-the latter is known to be connected to various branches of modern probability theory and mathematical physics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let δ T approach zero slowly enough as T → ∞, then t → ζ(1/2 + iδ T t + iωT) is asymptotically a product of a divergent scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.

We consider the orthogonal polynomials on [−1, 1] with respect to the weight w c (x) = h (x) (1 − x) α (1 + x) β Ξ c (x) , α, β > −1, where h is real analytic and strictly positive on [−1, 1], and Ξ c is a step-like function: Ξ c (x) = 1 for x ∈ [−1, 0) and Ξ c (x) = c 2 , c > 0, for x ∈ [0, 1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in C, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n → ∞. In particular, we prove for w c a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior. For the asymptotic analysis we use the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at x = 0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.

Let C λ n (x), n = 0, 1,. .. , λ > −1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal in (−1, 1) with respect to the weight function (1 − x 2) λ−1/2. Denote by x nk (λ), k = 1,. .. , n, the zeros of C λ n (x) enumerated in decreasing order. In this short note we prove that, for any n ∈ IN , the product (λ + 1) 3/2 x n1 (λ) is a convex function of λ if λ ≥ 0. The result is applied to obtain some inequalities for the largest zeros of C λ n (x). If x nk (α), k = 1,. .. , n, are the zeros of Laguerre polynomial L α n (x), also enumerated in decreasing order, we prove that x n1 (λ)/(α + 1) is a convex function of α for α > −1.

The number of zeros in (−1, 1) of the Jacobi function of second kind Q (α,β) n (x), α, β > −1, that is the second solution of the differential equation (1 − x 2)y (x) + (β − α − (α + β + 2)x)y (x) + n(n + α + β + 1)y(x) = 0 , is determined for every n ∈ N and for all values of the parameters α > −1 and β > −1. It turns out that this number depends essentially on α and β as well as on the specific normalization of the function Q (α,β) n (x). Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind.

In this paper, for a second order sublinear dynamic equation with a damping term we will study the lower bounds of the distance between zeros of a solution and/or its derivatives and then establish some new criteria for disconjugacy and disfocality. Our results present a slight improvement to some results proved in the litrature. As a special case when T = R, for a second order linear differential equation, we get some results proved by Brown and Harris as a consequence of our results. The results will be proved by employing the time scales Hölder inequality, the time scales chain rule and some new dynamic Opial-type inequalities designed and proved for this purpose.

We show that a large class of Cantor-like sets of R d , d ≥ 1, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when d ≥ 2. An analogous result is also proved for subsets of R d arising in the study of geodesic flows corresponding to (d+1)-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable numbers in R. Furthermore, we describe a condition on sets, which is fulfilled by a large class, ensuring a large intersection with these Cantor-like sets.

Adding right-handed neutrino singlets and/or fermion triplets to the particle content of the Standard Model allows for the implementation of the seesaw mechanism to give mass to neutrinos and, simultaneously, for the construction of anomaly-free gauge group extensions of the theory. We consider Abelian extensions based on an extra U (1)X gauge symmetry, where X is an arbitrary linear combination of the baryon number B and the individual lepton numbers Le,µ,τ. By requiring cancellation of gauge anomalies, we perform a detailed analysis in order to identify the charge assignments under the new gauge symmetry that lead to neutrino phenomenology compatible with current experiments. In particular, we study how the new symmetry can constrain the flavor structure of the Majorana neutrino mass matrix, leading to two-zero textures with a minimal extra fermion and scalar content. The possibility of distinguishing different gauge symmetries and seesaw realizations at colliders is also briefly discussed.

The method of dimensional-directional analys& is established by a way which consists of the representation of the physical quantities as quaternions. Two dimensionaldirectional bases are obtained for the particular case of Newtonian mechanics. The dimensionaldirectional, the dimensional, the directional and the reduced-dimensional-directional equations are defined. Furthermore, the Principle of Dimensional-directional Homoyeneity is established and the Buckinyham 7z theorem is reformulated. Some examples show the powerfulness of the dimensional-directional analysis. Finally, the conversion matrices for a chanye of dimensionaldirectional basis are 9iven.

Motivated by cognitive experiments providing evidence that large crossing-angles do not impair the readability of a graph drawing, RAC (Right Angle Crossing) drawings were introduced to address the problem of producing readable representations of non-planar graphs by supporting the optimal case in which all crossings form 90 • angles. In this work, we make progress on the problem of finding RAC drawings of graphs of low degree. In this context, a long-standing open question asks whether all degree-3 graphs admit straight-line RAC drawings. This question has been positively answered for the Hamiltonian degree-3 graphs. We improve on this result by extending to the class of 3-edge-colorable degree-3 graphs. When each edge is allowed to have one bend, we prove that degree-4 graphs admit such RAC drawings, a result which was previously known only for degree-3 graphs. Finally, we show that 7-edge-colorable degree-7 graphs admit RAC drawings with two bends per edge. This improves over the previous result on degree-6 graphs. 2012 ACM Subject Classification Theory of computation → Computational geometry; Mathematics of computing → Graph algorithms Keywords and phrases Graph Drawing, RAC graphs, Straight-line and bent drawings Digital Object Identifier 10.4230/LIPIcs.MFCS.2022.

I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

Green marketing has become a fast-developing research field in recent years, even though the term "green product" is still new for consumers in developing and sociallyconstructed economies like Turkey. The aim of this study is to examine the effect of collective self-esteem and social influence on the green product purchase intention of Turkish consumers while delineating other factors that may play a role within this relationship. According to the results of the study which were gathered from 370 consumers, the effects of collective self-esteem, social influence and attitudes towards green products significantly influence green product purchase intention. It can be seen from the results that collective self-esteem and social influence positively affect consumers' green product purchase intention and the relationship between collective self-esteem and social influence on green product purchase intention is partially mediated by attitude toward green products. In addition, generational cohort differences play a significant role. While the impact of social influence on green product purchase intention is stronger for Gen X'ers, there was no generational difference on influence of collective self-esteem on purchase intention. Findings and implications are discussed.

In this article we study the arcwise connected component in the Pontryagin dual of an abelian topological group. It is clear that the set of continuous characters that can be lifted over the reals is contained in the arcwise connected component of the dual group. We show that the converse is true if all arcs in the character group are equicontinuous sets. This property is present in Pontryagin duals of pseudocompact groups, of reflexive groups and of groups which are k-spaces as topological spaces. We study the meaning of such a property and its presence in groups and vector spaces endowed with weak topologies. We also characterize the image of the exponential mapping of a dual group as formed by those characters which can be lifted over the reals endowed with the Bohr topology.

A search for the Standard Model Higgs boson in the two photon decay channel is reported, using 1.08 fb −1 of proton-proton collision data at a centre-of-mass energy of 7 TeV recorded by the ATLAS detector. No significant excess is observed in the investigated mass range of 110-150 GeV. Upper limits on the cross-section times branching ratio of between 2.0 and 5.8 times the Standard Model prediction are derived for this mass range. 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2).

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions. We generalize some results appeared in the literature very recently.

The paper considers a problem of approximation of functions by means of their finite number of Fourier coefficients. Convergence acceleration of approximations by the truncated Fourier series is achieved by application of polynomial and rational correction functions. Rational corrections include unknown parameters whose determination is a crucial problem. We consider an approach connected with the roots of the Laguerre polynomials and study the rates of convergence of such approximations.

Design and analysis of control systems often become difficult due to the complexity of the system model and the design techniques involved. This paper presents the development of a Tools Box in Microsoft Excel for control engineer that uses root locus as a time domain technique for system design and analysis. The Tool Box can also serve as a computer-aided graphical analytical tool for trainers. The work was done in two phases: the first phase is the development of a programmable algorithms for root locus using the angle condition and bisection method while the second phase is the implementation of the developed algorithms. The implementation was done using Microsoft Excel (R) Visual Basic Application (VBA). Results of simulations for different systems show the potential of the Tool Box as an alternative for other software package and the ease of using it on the readily available Microsoft Excel environment.

A search for the Standard Model Higgs boson in the two photon decay channel is reported, using 1.08 fb -1 of proton-proton collision data at a centre-of-mass energy of 7 TeV recorded by the ATLAS detector. No significant excess is observed in the investigated mass range of 110-150 GeV. Upper limits on the cross-section times branching ratio of between 2.0 and 5.8 times the Standard Model prediction are derived for this mass range.

The correspondences between dynamics of q-Toda and q-Volterra equations for the coefficients of the Jacobi operator and its resolvent function are established. The main ingredient are orthogonal polynomials which satisfy an Appell condition, with respect to the q-difference operator D q , and a Lax type theorem for the point spectrum of the Jacobi operator associated with these equations. Examples related with the big q-Legendre, discrete q-Hermite I, and little q-Laguerre orthogonal polynomials and q-Toda and q-Volterra equations are given.

A search for the Standard Model Higgs boson in the two photon decay channel is reported, using 1.08 fb −1 of proton-proton collision data at a centre-of-mass energy of 7 TeV recorded by the ATLAS detector. No significant excess is observed in the investigated mass range of 110-150 GeV. Upper limits on the cross-section times branching ratio of between 2.0 and 5.8 times the Standard Model prediction are derived for this mass range. 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2).

A search is made for massive highly ionising particles with lifetimes in excess of 100 ns, with the ATLAS experiment at the Large Hadron Collider, using 3.1 pb −1 of pp collision data taken at √ s = 7 TeV. The signature of energy loss in the ATLAS inner detector and electromagnetic calorimeter is used. No such particles are found and limits on the production cross section for electric charges 6e ≤ |q| ≤ 17e and masses 200 GeV ≤ m ≤ 1000 GeV are set in the range 1 − 12 pb for different hypotheses on the production mechanism.

Abstract: We show a proof of the so-called Riemann Hypothesis (RH) stating that “All the non-trivial zero of the Zeta Function are on the Critical Line”. We prove the RH using the theory of “inner product spaces ” I and l2 Hilbert spaces, where is defined the “functional ” (a,b), named scalar [or inner] product of the vectors a and b. The proof is so simple that we suspect that there could be an error that we are unable to find. [Galetto F. Riemann Hypothesis proved. Academ Arena 2014;6(12):19-22]. (ISSN 1553-992X).