A general result on precise asymptotics for linear processes of positively associated sequences

Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers satisfying ∑j=0^∞｜aj｜〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E｜ε1｜^2＋δ′〈 for some δ′〉0 and μ（n）=O（n^-ρ） for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}.

The incidence of oxygen desaturation during rapid sequence induction and intubation

BACKGROUND: Rapid sequence induction and intubation(RSII) is an emergency airway management technique for patients with a risk of pulmonary aspiration. It involves preoxygenation, administration of predetermined doses of induction and paralytic drugs, avoidance of mask ventilation, and laryngoscopy followed by tracheal intubation and keeping cricoid pressure applied till endotracheal tube cuff be inflated. Oxygen desaturation has been seen during RSII. We assessed the incidence of oxygen desaturation during RSII.METHODS: An institution-based observational study was conducted from March 3 to May 4, 2014 in our hospital. All patients who were operated upon under general anesthesia with RSII during the study period were included. A checklist was prepared for data collection.RESULTS: A total of 153 patients were included in this study with a response rate of 91.6%. Appropriate drugs for RSII, equipments for RSII, equipments for diffi cult intubation, suction machine with a catheter, a monitor and an oxygen backup such as ambu bag were not prepared for 41(26.8%), 50(32.7%), 51(33.3%), 38(24.8%) and 25(16.3%) patients respectively. Cricoid pressure was not applied at all for 17(11.1%) patients and 53(34.6%) patients were ventilated after induction of anesthesia but before intubation and endotracheal cuff inflation. A total of 55(35.9%) patients desaturated during RSII(SPO2<95%). The minimum, maximum and mean oxygen desaturations were 26%, 94% and 70.9% respectively. The oxygen desaturation was in the range of <50%, 50%–64%, 65%–74%, 75%–84%, 85%–89 % and 90%–94% for 6(3.9%), 7(4.6%), 5(3.3%), 10(6.5%), 13(8.5%) and 14(9.2%) patients respectively.CONCLUSION: The incidence of oxygen desaturation during RSII was high in our hospital. Preoperative patient optimization and training about the techniques of RSII should be emphasized.

CONVERGENCE OF ISHIKAWA TYPE ITERATIVE SEQUENCE WITH ERRORS FOR QUASI-CONTRACTIVE MAPPINGS IN CONVEX METRIC SPACES

Some convergence theorems of Ishikawa type iterative sequence with errors for nonlinear general quasi-contractive mapping in convex metric spaces are proved. The results not only extend and improve the corresponding results of L. B. Ciric, Q. H. Liu, H. E. Rhoades and H. K. Xu, et al., but also give an affirmative answer to the open question of Rhoades-Naimpally- Singh in convex metric spaces.

Techniques of rapid sequence induction and intubation at a university teaching hospital

BACKGROUND: Rapid sequence induction and intubation(RSII) is a medical procedure involving a prompt induction of general anesthesia by using cricoid pressure that prevents regurgitation of gastric contents. The factors affecting RSII are prophylaxis for aspiration, preoxygenation, drug and equipment preparation for RSII, ventilation after induction till intubation and patient condition. We sometimes saw diffi culties with the practice of this technique in our hospital operation theatres. The aim of this study was to assess the techniques of rapid sequence induction and intubation.METHODS: Hospital based observational study was conducted with a standardized checklist. All patients who were operated upon under general anesthesia during the study period were included. The techniques of RSII were observed during the induction of anesthesia by trained anesthetists.RESULTS: Altogether 140 patients were included in this study with a response rate of 95.2%. Prophylaxis was not given to 130 patients(92.2%), and appropriate drugs were not used for RSII in 73 patients(52.1%), equipments for diffi cult intubation in 21(15%), suction machines with catheter not connected and turned on in 122(87.1%), ventilation for patients after induction and before intubation in 41(29.3%), cricoid pressure released before cuff inflation in 12(12.1%), and difficult intubation in 8(5.7%), respectively. RSII with cricoid pressure was applied appropriately in 94(67.1%) patients, but cricoid pressure was not used in 46(32.9%) patients.CONCLUSIONS: The techniques of rapid sequence induction and intubation was low. Training should be given for anesthetists about the techniques of RSII.

From Control Theory to Gravitational Waves

When D:ξ→η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1:η→ξ such that Dξ = η implies D1η = 0. When D is involutive, the procedure provides successive first-order involutive operators D1,...,Dn when the ground manifold has dimension n. Conversely, when D1 is given, a much more difficult “inverse problem” is to look for an operator D:ξ→η having the generating CC D1η = 0. If this is possible, that is when the differential module defined by D1 is “torsion-free”, that is when there does not exist any observable quantity which is a sum of derivatives of η that could be a solution of an autonomous OD or PD equation for itself, one shall say that the operator D1 is parametrized by D. The parametrization is said to be “minimum” if the differential module defined by D does not contain a free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test with five steps using double differential duality. We prove and illustrate through many explicit examples the fact that a control system is controllable if and only if it can be parametrized. Accordingly, the controllability of any OD or PD control system is a “built in” property not depending on the choice of the input and output variables among the system variables. In the OD case and when D1 is formally surjective, controllability just amounts to the formal injectivity of ad(D1), even in the variable coefficients case, a result still not acknowledged by the control community. Among other applications, the parametrization of the Cauchy stress operator in arbitrary dimension n has attracted many famous scientists (G. B. Airy in 1863 for n = 2, J. C. Maxwell in 1870, E. Beltrami in 1892 for n = 3, and A. Einstein in 1915 for n = 4). We prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized and the comparison needs no comment. As a byproduct, they are all based on a confusion between the so-called div operator D2 induced from the Bianchi operator and the Cauchy operator, adjoint of the Killing operator D which is parametrizing the Riemann operator D1 for an arbitrary n. This purely mathematical result deeply questions the origin and existence of gravitational waves, both with the mathematical foundations of general relativity. As a matter of fact, this new framework provides a totally open domain of applications for computer algebra as the quoted test can be studied by means of Pommaret bases and related recent packages.

A GENERAL CONSTITUTIVE RELATION FOR FATIGUE CRACK GROWTH ANALYSIS OF METAL STRUCTURES

Crack growth rate curves are the fundamental material property for metal structures under fatigue loading. Although there are many crack growth rate curves available in the literature, few of them showed the capability to explain various special phenomena observed in tests. A modified constitutive relation recently proposed by McEvily and his co-workers showed very promising capability. This modified constitutive relation is further generalized by (1) introducing an unstable fracture condition; (2) defining a virtual strength to replace the yield stress; and (3) defining an overload and underload parameter. The performances of this general constitutive relation for fatigue crack growth is extensively studied and it is found that this general constitutive relation is able to explain various phenomena observed with particular strong capability on load sequence effect.

The Mathematical Foundations of General Relativity Revisited

The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 2) JANET VERSUS SPENCER: The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of electromagnetism and gravitation must be revisited within this formal framework, though striking it may look like. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.

Generating Compatibility Conditions and General Relativity

The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingly, this paper can be considered as a natural continuation of a previous paper recently published in JMP under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with the degree of generality introduced by A. Einstein in his 1930 letters to E. Cartan. One of the motivating examples that we provide is so striking that it is even difficult to imagine that such an example could exist. We hope this paper could be used as a source of testing examples for future applications of computer algebra in general relativity and, more generally, in mathematical physics.

General Term of Sequence

In the study of number sequences,we learn to uncover how the sequence grows.Then we make a generalisation about the number sequence by stating the general term in algebraic form. Below is an activity using matches(or toothpicks,etc)to build and expand a number pattern. Given the pattern in the figures as follows:

ON DOMINATING SEQUENCE METHOD IN THE POINT ESTIMATE AND SMALE THEOREM

It can be optimized for the work of the point estimate on the Newton iteration, reported by Smale at the 20th Congress of Mathematicians in 1986. It has been proved that iffα(z,f) , for every analytic map f:E→F, z ∈ is an approximate zero of f,whereE and F are real or complex Banach spaces. In addition, if α,(z,f) , z is anapproximate zero of the second kind of f.

GENERALIZATION OF SEQUENCING PROBLEMS OF THE OPTIMUM SERVICE FOR MODEL P-J

With regard to sequencing problems of optimum service for Model P-J, the author has already introduced a method of optimal sequencing. This paper will develop a further generalization. For the sake of convenience, this paper will adopt the same symbols as those in (1, 2)Suppose that there are K-service groups.

Detection of QTL(quantitative trait loci) associated with wood density by evaluating genetic structure and linkage disequilibrium of teak

To find the quantitative trait loci associated with wood density in teak(Tectona grandis L.f.), 21 co-dominant markers including 13 site specific recombinase and 8 EST-based co-dominant markers designed from lignin biosynthesis genes were applied to 174 teak plus tree clones at the National Germplasm Bank, Chandrapur,India. The germplasm bank exhibited 10.6% coefficient of variation for wood densities with 84.5 ± 31.3 genetic polymorphism(%). The highly panmictic set of genotypes(FST= 0.035 ± 0.004) harbored 96.47 ± 0.40 genetic variability(%). The average allelic frequency of the 21 codominant markers was 0.65 ± 0.11 with 12.9% pairs of loci in significant LD(p\0.05, R^2 values [ 0.1), confirming their suitability for a strong marker-trait association study. The marker CCoAMT-1 was significantly(p\0.01) associated with wood density showing stability by both GLM and MLM models and explained 4.3% of the phenotypic effect. The marker from the EST representing CCoAMT can be further developed for gene-assisted selection of elite genotypes of teak with greater wood density. Therefore, we believe that the report will help accelerate the genetic improvement and advance the breeding program of the species.

Minimum Parametrization of the Cauchy Stress Operator

When D: ξ→η is a linear differential operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: η→ξ such that Dξ=η implies D1η=0. When D is involutive, the procedure provides successive first order involutive operators D1, ..., Dn, when the ground manifold has dimension n, a result first found by M. Janet as early as in 1920, in a footnote. However, the link between this “Janet sequence” and the “Spencer sequence” first found by the author of this paper in 1978 is still not acknowledged. Conversely, when D1 is given, a more difficult “inverse problem” is to look for an operator D: ξ→η having the generating CC D1η=0. If this is possible, that is when the differential module defined by D1 is torsion-free, one shall say that the operator D1 is parametrized by D and there is no relation in general between D and D2. The parametrization is said to be “minimum” if the differential module defined by D has a vanishing differential rank and is thus a torsion module. The solution of this problem, first found by the author of this paper in 1995, is still not acknowledged. As for the applications of the “differential double duality” theory to standard equations of physics (Cauchy and Maxwell equations can be parametrized while Einstein equations cannot), we do not know other references. When erator in arbitrary dimension=1 as in control theory, the fact that controllability is a “built in” property of a control system, amounting to the existence of a parametrization and thus not depending on the choice of inputs and outputs, even with variable coefficients, is still not acknowledged by engineers. The parametrization of the Cauchy stress operator in arbitrary dimension n has nevertheless attracted, “separately” and without any general “guiding line”, many famous scientists (G.B. Airy in 1863 for n = 2, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for n = 3 , A. Einstein in 1915 for n = 4 ). The aim of this paper is to solve the minimum parametrization problem in arbitrary dimension and to apply it through effective methods that could even be achieved by using computer algebra. Meanwhile, we prove that all these works are using the Einstein operator which is self-adjoint and not the Ricci operator, a fact showing that the Einstein operator, which cannot be parametrized, has already been exhibited by Beltrami more than 20 years before Einstein. As a byproduct, they are all based on the same confusion between the so-called div operator induced from the Bianchi operator D2 and the Cauchy operator which is the formal adjoint of the Killing operator D parametrizing the Riemann operator D1 for an arbitrary n. We prove that this purely mathematical result deeply questions the origin and existence of gravitational waves. We also present the similar motivating situation met in the study of contact structures when n = 3. Like the Michelson and Morley experiment, it is thus an open historical problem to know whether Einstein was aware of these previous works or not, but the comparison needs no comment.

Minkowski, Schwarzschild and Kerr Metrics Revisited

In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. Unhappily, these physicists are neither familiar with the formal theory of systems of partial differential equations and differential modules, nor with the formal theory of Lie pseudogroups. Hence, even if they discovered a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. The purpose of this difficult computational paper is to provide differential and homological methods in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools, which are now available as computer algebra packages, question the mathematical foundations of GR and the origin of gravitational waves.

Certain Results for the 2-Variable Peters Mixed Type and Related Polynomials

In this article, the 2-variable general polynomials are taken as base with Peters polynomials to introduce a family of 2-variable Peters mixed type polynomials.These polynomials are framed within the context of monomiality principle and their properties are established. Certain summation formulae for these polynomials are also derived. Examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored.

Why Gravitational Waves Cannot Exist

The purpose of this short but difficult paper is to revisit the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT) in the light of a modern approach to nonlinear systems of ordinary or partial differential equations, using new methods from Differential Geometry (D.C. Spencer, 1970), Differential Algebra (J.F. Ritt, 1950 and E. Kolchin, 1973) and Algebraic Analysis (M. Kashiwara, 1970). The main idea is to identify the differential indeterminates of Ritt and Kolchin with the jet coordinates of Spencer, in order to study Differential Duality by using only linear differential operators with coefficients in a differential field K. In particular, the linearized second order Einstein operator and the formal adjoint of the Ricci operator are both parametrizing the 4 first order Cauchy stress equations but cannot themselves be parametrized. In the framework of Homological Algebra, this result is not coherent with the vanishing of a certain second extension module and leads to question the proper origin and existence of gravitational waves. As a byproduct, we also prove that gravitation and electromagnetism only depend on the second order jets (called elations by E. Cartan in 1922) of the system of conformal Killing equations because any 1-form with value in the bundle of elations can be decomposed uniquely into the direct sum (R, F) where R is a section of the Ricci bundle of symmetric covariant 2-tensors and the EM field F is a section of the vector bundle of skew-symmetric 2-tensors. No one of these purely mathematical results could have been obtained by any classical approach. Up to the knowledge of the author, it is also the first time that differential algebra in a modern setting is applied to study the specific algebraic feature of most equations to be found in mathematical physics, particularly in GR.

Novel approaches to the multiscalar display of vector data

There are many challenges to achieving quality multiscalar displays of vector data in geographical information system(GIS).Acknowledging and making use of the differences between traditional cartographic generalization and multiscalar display in GIS,this paper focuses on novel approaches to the definition of small objects and the establishment of scale sequence.The ease of implementation and efficacy of the solutions proposed are exemplified and analyzed.Possibilities for further research into the multiscalar display of vector data in GIS are thereby suggested.