Generalization Of Bernoulli Research Articles

Applicability of Bernoulli's theorem to upstream and downstream of the bow shock

In this research, the validity of Bernoulli's theorem for the solar wind crossing the Earth's bow shock has been investigated based on space explorations. Using solar wind plasma and interplanetary magnetic field data from the OMNI database, along with magnetosheath plasma and magnetic field data measured by the magnetospheric multiscale satellite in 2016 and 2018, 95 events were selected to study the Bernoulli invariant from the upstream to the downstream of Earth's bow shock (from the unshocked solar wind region to the shocked magnetosheath region). Statistical investigations show that the differences in the characteristic energy of plasmas between the solar wind region and the magnetosheath region are less than 5%, regardless of the upstream solar wind speed and the satellite's location in the magnetosheath. These results quantitatively support the generalization of Bernoulli's equation to the solar wind flowing through the bow shock, including the contribution of magnetic field energy.

Finding and improving bounds of real functions by thermodynamic arguments

The possibility of stating the second law of thermodynamics in terms of the increasing behaviour of a physical property establishes a connection between that branch of physics and the theory of algebraic inequalities. We use this connection to show how some well-known inequalities, such as the standard bounds for the logarithmic function or generalizations of Bernoulli's inequality, can be derived by thermodynamic methods. Additionally, we show that by comparing the global entropy production in processes implemented with decreasing levels of irreversibility but subject to the same change of state of one particular system, we can find progressively better bounds for the real function that represents the entropy variation of the system. As an application, some new families of bounds for the function log (1+x) are obtained by this method.

Jupiter and Saturn Multi-layer Models Rotating Differentially

In a past work, models for Jupiter were constructed in base to a set of concentric distorted spheroids (“spheroidals”) rotating differentially—whose semi-axes are independent of one another—a task that was achieved with a law of rotation deduced from a generalization of Bernoulli’s theorem, and which holds exclusively for axial-symmetric masses. The shape of the mass is that of a spheroid whose surface equation contains an extra term, d/z4, where d is a parameter which measures the degree of distortion. Each layer rotates with its own profile of angular velocity. The rotation law has a simple dependence on the derivative of the gravitational potential. No magnetic fields or equations of state were involved. The multi- structures were demanded, firstly, to reproduce the gravitational moments of the planets, as surveyed by space missions; and, secondly, to be equilibrium figures. For the calculation of the gravitational moments, a minimization procedure was employed. Paying attention on the outermost laye—the relevant one in the present context—of the formerly reported models for Jupiter, we became aware that they all share an angular velocity profile that decreases from the pole towards the equator, an event that, so far, has not been verified observationally. Since figures with profiles of the opposite tendency turned out to be also possible, they should be included as candidates for our purpose, as effectively they are herein as a complement of that work. The same procedure is here entailed to Saturn, for which figures to show one or the other tendencies are as well obtained. The dual behavior of the rotation profiles may be explained by arguments involving the centripetal force. According to this standpoint, the double behavior is a consequence of the algebraic sign assigned to d: if positive, so that the surface is more bloated than that of a spheroid, the decreasing tendency results; whereas if negative, so that the surface is more depressed, the increasing tendency shows up. This, in turn, is because for d negative the radial force increases more rapidly from pole to the equator than for d positive. We point out that the rotation profiles of the current figures are determined from their equilibrium, rather than being imposed ad initio.

A Differential Rotation Law for Stars and Fluid Planets

We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number <i>l</i> of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let <i>l</i> tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed <i>ad initio</i>. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.

Some (p, q)-analogues of Apostol type numbers and polynomials

We consider a new class of generating functions of the generalizations of Bernoulli and Euler polynomials in terms of (p, q)-integers. By making use of these generating functions, we derive (p, q)-generalizations of several old and new identities concerning Apostol–Bernoulli and Apostol–Euler polynomials. Finally, we define the (p, q)-generalization of Stirling polynomials of the second kind of order v, and provide a link between the (p, q)-generalization of Bernoulli polynomials of order v and the (p, q)-generalization of Stirling polynomials of the second kind of order v.

Ward’s Generalized Special Functions

This paper considers generalizations of Bernoulli and Euler numbers to clarify and extend some known relations studied by Morgan Ward. It does this with the Euler-Maclaurin sum formula. It relates the mappings to category theory as a means of applying the ideas further.

Bernoulli–Dunkl and Apostol–Euler–Dunkl polynomials with applications to series involving zeros of Bessel functions

We introduce Bernoulli–Dunkl and Apostol–Euler–Dunkl polynomials as generalizations of Bernoulli and Apostol–Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to find the sum of many different series involving the zeros of Bessel functions.

Resolving embarrassing medical conditions with online health information

PurposeReliance on online health information is proliferating and the Internet has the potential to revolutionize the provision of public health information. The anonymity of online health information may be particularly appealing to people seeking advice on ‘embarrassing' health problems. The purpose of this study was to investigate (1) whether data generated by the embarrassingproblems.com health information site showed any temporal patterns in problem resolution, and (2) whether successful resolution of a medical problem using online information varied with the type of medical problem. MethodsWe analyzed the responses of visitors to the embarrassingproblems.com website on the resolution of their problems. The dataset comprised 100,561 responses to information provided on 77 different embarrassing problems grouped into 9 classes of medical problem over an 82-month period. Data were analyzed with a Bernoulli Generalized Linear Model using Bayesian inference. ResultsWe detected a statistically important interaction between embarrassing problem type and the time period in which data were collected, with an improvement in problem resolution over time for all of the classes of medical problem on the website but with a lower rate of increase in resolution for urinary health problems and medical problems associated with the mouth and face. As far as we are aware, this is the first analysis of data of this nature. ConclusionsFindings support the growing recognition that online health information can contribute to the resolution of embarrassing medical problems, but demonstrate that outcomes may vary with medical problem type. The results indicate that building data collection into online information provision can help to refine and focus health information for online users.

Binomial convolution and transformations of Appell polynomials

We show that the set A of Appell sequences is an abelian group under the binomial convolution. This is essentially equivalent to other approaches considered in the literature, in particular, the determinantal approach. We also define a scale transformation Tα:A→A, which is an isomorphism, and a transformation RY:A→A based on expectations with respect to a random variable Y. Using these tools, we give explicit formulas for the determinantal form of Appell sequences, as well as for the formula of representation of powers and the Srivastava–Pintér addition theorem. Various illustrative examples, mainly referring to various generalizations of Bernoulli and Euler polynomials, are discussed in detail.

A generalization of Bernoulli's inequality

We prove a generalization of Bernoulli's inequality and we apply this generalization to sharpen certain Weierstrass product inequalities

Bayesian inference supports the host selection hypothesis in explaining adaptive host specificity by European bitterling

Generalist parasites have the capacity to infect multiple hosts. The temporal pattern of host specificity by generalist parasites is rarely studied, but is critical to understanding what variables underpin infection and thereby the impact of parasites on host species and the way they impose selection on hosts. Here, the temporal dynamics of infection of four species of freshwater mussel by European bitterling fish (Rhodeus amarus) was investigated over three spawning seasons. Bitterling lay their eggs in the gills of freshwater mussels, which suffer reduced growth, oxygen stress, gill damage and elevated mortality as a result of parasitism. The temporal pattern of infection of mussels by European bitterling in multiple populations was examined. Using a Bernoulli Generalized Additive Mixed Model with Bayesian inference it was demonstrated that one mussel species, Unio pictorum, was exploited over the entire bitterling spawning season. As the season progressed, bitterling showed a preference for other mussel species, which were inferior hosts. Temporal changes in host use reflected elevated density-dependent mortality in preferred hosts that were already infected. Plasticity in host specificity by bitterling conformed with the predictions of the host selection hypothesis. The relationship between bitterling and their host mussels differs qualitatively from that of avian brood parasites.

Ecological preference of the diving bell spider Argyroneta aquatica in a resurgence of the Po plain (Northern Italy) (Araneae: Cybaeidae)

The diving bell spider &lt;em&gt;Argyroneta aquatica&lt;/em&gt; is the only known spider to conduct a wholly aquatic life. For this reason, it has been the object of an array of studies concerning different aspects of its peculiar biology such as reproductive behavior and sexual dimorphism, physiology, genetic and silk. On the other hand, besides some empirical observations, the autoecology of this spider is widely understudied. We conducted an ecological study in a resurgence located in the Po Plain (Northern Italy, Province of Vercelli) hosting a relatively rich population of &lt;em&gt;Argyroneta&lt;/em&gt; &lt;em&gt;aquatica&lt;/em&gt;, aiming at identifying the ecological factors driving its presence at the micro-habitat level. By means of a specific sampling methodology, we acquired distributional data of the spiders in the study area and monitored physical-chemical and habitat structure parameters at each plot. We analyzed the data through Bernoulli Generalized Linear Models (GLM). Results pointed out a significant positive effect of the presence of aquatic vegetation in the plot. In addition, the presence of &lt;em&gt;A. aquatica&lt;/em&gt; was significantly associated with areas of the resurgence characterized at the same time by high prey availability and low density of predators. Considering the ecological importance and rarity of this species, we update and revise the data on the distribution of &lt;em&gt;A. aquatica&lt;/em&gt; in Italy.

Maclaurin's Inequality and a Generalized Bernoulli Inequality

SummaryMaclaurin's inequality is a natural, but nontrivial, generalization of the arithmetic-geometric mean inequality. We present a new proof that is based on an analogous generalization of Bernoulli's inequality. Applications of Maclaurin's inequality to iterative sequences and probability are discussed, along with graph-theoretic versions of the Maclaurin and Bernoulli inequalities.

A new representation of rotational flow fields satisfying Euler's equation of an ideal compressible fluid

A new representation of the solution to Euler's equation of motion is presented by using a system of expressions for compressible rotational flows of an ideal fluid. This is regarded as a generalization of Bernoulli's theorem to compressible rotational flows. The present expressions are derived from the variational principle. The action functional for the principle consists of the main terms of the total kinetic, potential and internal energies, together with three additional terms yielding the equations of continuity, entropy and a third term that provides the rotational component of velocity field. The last term has the form of scalar product satisfying gauge symmetry with respect to both translation and rotation. This is a generalization of the Clebsch transformation from a physical point of view. It is verified that the system of new expressions, in fact, satisfies Euler's equation of motion.

An elementary proof of a generalization of Bernoulli’s formula

An elementary proof of a generalization of Bernoulli’s formula

Maximum Likelihood Detection and Estimation of Bernoulli Generalized Gaussian Processes with Non-Gaussian Colored Noise

Maximum Likelihood Detection and Estimation of Bernoulli Generalized Gaussian Processes with Non-Gaussian Colored Noise

On the Uniform Quantization of a Class of Sparse Sources

We consider the uniform scalar quantization of a class of mixed distributed memoryless sources, namely sources having a Bernoulli Generalized Gaussian (BGG) distribution. Both for low and high resolutions, asymptotic expressions of the distortion for a pth-order moment error measure, and close approximations of the entropy are provided for these sources. Operational rate-distortion functions at high bit rate and their slope factors at low bit rate are derived. The dependence of these results on p and the distribution parameters as well as the relation to the Shannon optimal rate-distortion bound are then discussed. The application of these results to transform coding in two simple cases is finally highlighted.

A thermodynamically general theory for convective vortices

Convective vortices are common features of atmospheres that absorb lower-entropy-energy at higher temperatures than they reject higher-entropy-energy to space. These vortices range from small to large-scale and play an important role in the vertical transport of heat, momentum, and tracer species. Thus, the development of theoretical models for convective vortices is important to our understanding of some of the basic features of planetary atmospheres. The heat engine framework is a useful tool for studying convective vortices. However, current theories assume that convective vortices are reversible heat engines. Since there are questions about how reversible real atmospheric heat engines are, their usefulness for studying real atmospheric vortices is somewhat controversial. In order to reduce this problem, a theory for convective vortices that includes irreversible processes is proposed. The paper's main result is that the proposed theory provides an expression for the pressure drop along streamlines that includes the effects of irreversible processes. It is shown that a simplified version of this expression is a generalization of Bernoulli's equation to convective circulations. It is speculated that the proposed theory not only explains the intensity, but also sheds light on other basic features of convective vortices such as their physical appearance.

Generalizations of Bernoulli's inequality with applications

By using methods on the theory of majorization, some new generalizations of Bernoulli’s inequality are established and some applications of the generalizations are given.

A thermodynamically general theory for convective vortices

Convective vortices are common features of atmospheres that absorb lower-entropy-energy at higher temperatures than they reject higher-entropy-energy to space. These vortices range from small to large-scale and play an important role in the vertical transport of heat, momentum, and tracer species. Thus, the development of theoretical models for convective vortices is important to our understanding of some of the basic features of planetary atmospheres.The heat engine framework is a useful tool for studying convective vortices. However, current theories assume that convective vortices are reversible heat engines. Since there are questions about how reversible real atmospheric heat engines are, their usefulness for studying real atmospheric vortices is somewhat controversial. In order to reduce this problem, a theory for convective vortices that includes irreversible processes is proposed.The paper’s main result is that the proposed theory provides an expression for the pressure drop along streamlines that includes the effects of irreversible processes. It is shown that a simplified version of this expression is a generalization of Bernoulli’s equation to convective circulations. It is speculated that the proposed theory not only explains the intensity, but also sheds light on other basic features of convective vortices such as their physical appearance.