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Prof. G. 't Hooft

Title: Some Misconceptions concerning Gravitational Radiation

Speaker: G. 't Hooft
Author: G. 't Hooft

Abstract: Einstein's field equations allow for wavelike solutions, with two possible polarizations, just like electromagnetic waves. Fluctuations in energy, momentum density and tension are the sources of this radiation, but at the same time gravitational radiation carries energy and momentum itself. This sometimes leads to confusion and controversy, but the situation can be neatly explained. An exact solution already found by Weber and Wheeler back in 1957 can be used to illustrate what happens.

Title: Quantum mechanics, discretization and local determinism

Speaker: G. 't Hooft
Author: G. 't Hooft

Abstract: If mechanical laws of evolution at tiny distance scales, say the Planck scale, are given by some fundamental laws, which may well be non-quantum mechanical, these laws may well result in completely chaotic behavior at larger scales, say those of atoms, molecules, or indeed today's Standard Model. There will be subtle correlations in the fluctuations at longer distances, and these may well be described by quantum mechanics exactly as we know it today. This idea leads to a theory as to what quantum mechanics really is, and why our world of atoms, molecules and subatomic particles appears to be quantum mechanical. There is a mathematical theory showing how this might work in a simple cellular automaton, although simple cellular automata do not produce particles of the kind that we know. The model of the previous lecture might be a more suitable starting point.

Title: A locally finite model for gravity

Speaker: G. 't Hooft
Author: G. 't Hooft

Abstract: Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate matter of a form that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Globally, however, the model is not finite, because solutions tend to generate infinite fractals. The model is not (yet) quantized, but could serve as an interesting setting for analytical approaches to classical general relativity, as well as a possible stepping stone for quantum models. Details of its properties are explained, but some problems remain unsolved, such as a complete description of the most violent interactions, which can become quite complex. A remarkable feature appears to be that in this model, gravity and matter appear to be unified.

Title: Black holes and elementary particle physics

Speaker: G. 't Hooft
Author: G. 't Hooft

Abstract: The most conspicuous features in the theory of the gravitational field is the emergence of black holes.
Trying to subject them to the laws of Quantum Mechanics may well be the best attack on the quantization problem in General Relativity. If one applies the laws of Quantum Field Theory to the environment of a black hole, one finds a kind of uncertainty on top and above the usual quantum uncertainties that one would expect. It seems as if the very foundations of Quantum Theory are being undermined by this result. Postulating the demand that "quantized" black holes ought to behave just like other quantum systems, leads to a new theory on the physical states and their dynamics at the Planck scale, and perhaps also to a fresh view on what Quantum Mechanics might actually be.

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A.G. Aksenov

Title: Relativistic Boltzmann equations for the pair plasma in presence of proton loading

Speaker: A.G. Aksenov
Authors: A.G. Aksenov, R. Ruffini, G.V. Vereshchagin

Abstract: In the recent publication we analyzed the role of the direct and the inverse binary and triple interactions in reaching thermal equilibrium in homogeneous isotropic pair plasma, starting from a nonequlibrium state. In the present work we extend the analysis to the case of proton-loaded plasma. The corresponding timescales for thermalization of electrons, positrons, protons and photons are determined out from the numerical solution of the relativistic Boltzmann equations. We include all exact QED collisional integrals for binary reactions, while for the corresponding radiative variants we reduce reaction rates to the known expressions of kinetic coefficients in the thermal equilibrium.

E. Alesci

Title: Graviton Propagator in LQG: a Tool to Test Spinfoam Models

Speaker: E. Alesci
Author: E. Alesci

Abstract:We briefly review the construction of the graviton propagator in the context of LQG and Spinfoam Models and we show how the Barrett-Crane vertex fails to give the correct long-distance limit. The same kind of calculation however, can give the correct propagator using an alternative vertex with a specific asymptotic behavior. It is still an open issue if the vertex amplitudes of the new models satisfy the requirements for yielding the correct two point function or more in general GR in the classical limit. We present some preliminary results that can give new insight to this problem.

L. Amati

Title: The Ep,i-Eiso correlation in GRB: recent observations, implications and cosmology

Speaker: L. Amati
Author:

Abstract: The correlation between the photon energy at which the redshift corrected nufnu spectrum peaks (hence called "peak energy", Ep,i) and the isotropic equivalent radiated energy (Eiso), is one of the most intriguing and debated observational evidences in Gamma--Ray Bursts (GRB) astrophysics. I present the updated Ep,i-Eiso correlation, its main implications for the physics, geometry and origin of the GRB emission, with particular emphasis on recent Swift results, and the recently proposed use of this correlation for the estimate of cosmological parameters.

G. Amelino-Camelia

Title:  Falsifiable Planck-Scale Theories of NOT Everything

Speaker: G. Amelino-Camelia
Author: G. Amelino-Camelia

Abstract: Primarily as a result of the nature of the quantum-gravity problem, with its extraordinarily rich conceptual structure, and its clever resistance to our (however feeble) attempts to gain experimental insight, it is becoming customary to consider reasoning in terms of the anthropic principle" as a possible tool of scientific analysis. I stress that the validity of any thesis based on the anthropic principle cannot be established through reproducible experiments, and that theories elaborated on the basis of the anthropic principle gain nothing for what concerns their predictive power. The anthropic-principle temptation should therefore be left to the appetites of philosophers. I argue that for physicists the difficulties encountered in the study of the quantum-gravity problem should represent an invitation to adopt a more humble approach, which I describe as the search of "Planck-scale theories of not everything". To illustrate the concept I review some recent results on noncommutative field theories, which appear to provide the basis for some examples of falsifiable Planck-scale theories of not everything.

G. Barbiellini

Title:  Fulmini e Saette

Speaker: G. Barbiellini
Authors: G. Barbiellini

M.V. Battisti

Title:  Time Evolution of a Generic Quantum Universe

Speaker: M.V. Battisti
Authors: M.V. Battisti, G. Montani

Abstract: The evolutionary quantization of a generic inhomogeneous Universe is analyzed. In particular, the time-evolution of the model is traced with respect to the energy density of a perfect gas and at the same level a new matter contribution is induced in the Universe dynamics. We show that such an approach is suitable to describe the time evolution of a quantum Universe since no phenomenology arises today from the new matter field.

Title:  Extended Approach to the Canonincal Quantization in the Minisuperspace

Speaker: M.V. Battisti
Author: M.V. Battisti

Abstract: A phenomenological framework which describes the effective Friedmann dynamics of loop and braneworlds theories is developed. More precisely, it appears as soon as a deformed Heisenberg (minisuperspace) algebra is taken into account. This way, the peculiar differences of these approaches are immediately understood from this point of view.

Title:  Quantum Cosmology in the GUP Approach

Speaker: M.V. Battisti
Authors: M.V. Battisti, G. Montani

Abstract: The quantization of cosmological models by the use of a deformed algebra is described. Such an algebra reproduces a generalized uncertainty principle as appeared from studies on string theory. The FRW, Taub and Mixmaster Universes are analyzed in such a framework and the implications on the cosmological singularity, the isotropization mechanism and the chaotic proprieties are investigated.

V.A. Belinski

Title:  Charged masses and repulsive gravity

Speaker: V.A. Belinski
Authors: V.A. Belinski, G. Alekseev, M. Pizzi, A. Paolino

Abstract: We discuss some exact solutions of the Einstein-Maxwell equations exhibiting repulsive gravitational forces. We show that it is possible to construct a regular material source in the form of Nambu-Goto membrane outside of which  the Reissner-Nordstrom (RN) repulsive gravity effect exists. The point is that membrane are able to cut out the central naked singularity region and at the same time join the RN repelling region. The existence of such kind of sources gives a sensible physical status to the RN solution for the naked singularity case.

R. Belvedere

Title:  Quantum Isotropization Mechanism for the Mixmaster Model

Speaker: R. Belvedere
Authors: R. Belvedere, M.V. Battisti, G. Montani

Abstract: An exact solution of the quantum quasi-isotropic Mixmaster model is described. In particular, the volume of the Universe is regarded as a time-coordinate for the dynamics and a probabilistic interpretation of the wave function naturally arises. Once large volume regions are investigated, the closed FRW Universe configuration is deeply privileged.

R. Benini

Title:  E.C.G. Stueckelberg: a Forerunner of Modern Physics III

Speakers: R. Benini, O.M. Lecian
Authors: R. Benini, O.M. Lecian

Title:  Review on the Generic Cosmological Solution Near the Singularity

Speaker: R. Benini
Authors: R. Benini, G. Montani

Abstract: In this talk I discuss the main properties characterizing the cosmological singularity associated to the homogeneous and the inhomogeneous Mixmaster model. We firstly assess the classical picture reviewing the historical works by Belinski, Lifshitz and Khalatnikov and these of Misner; particular attention is paid to the chaotic properties of the model. Finally I present the semi-classical and quantum behaviour.

M.G. Bernardini

Title:  The GRB classification within the fireshell model: short, long and fake short GRBs

Speakers: M.G. Bernardini
Authors: M.G. Bernardini, C.L. Bianco, L. Caito, M.G. Dainotti, R. Guida, R. Ruffini

Abstract: Within the fireshell model we define a "canonical GRB" light curve with two sharply different components: the Proper-GRB (P-GRB), emitted when the optically thick fireshell of electron-positron plasma originating the phenomenon reaches transparency, and the afterglow, emitted due to the collision between the remaining optically thin fireshell and the CircumBurst Medium (CBM). The identification of these two component is crucial in order to interpret in a new and more physical way the standard GRB classification in "short" and "long". Moreover, we trace a new class of "fake" short GRBs, whose features are affected by the peculiar astrophysical conditions in which they occur.

S. Bernuzzi

Title:  Gravitational waves from neutron star oscillations: comparisons between linear and nonlinear evolutions

Speakers: S. Bernuzzi
Authors: S. Bernuzzi, A. Nagar

B. Betz

Title:  Plasminos - collective excitations of relativistic fermions

Speaker: B. Betz
Authors: B. Betz

Abstract: QCD, the theory of strong interactions between quarks, exhibits the feature of asymptotic freedom. Thus, for systems with high temperatures and/or densities hadronic matter is supposed to undergo a phase transition to the quark-gluon plasma, while for low temperature and high densities a new ground state forms - the color superconductivity. Both regions of the phase diagram have been studied intensively during the last years, either in heavy-ion collisions or studying astrophysical situations of quark matter in compact stellar objects. In general, gaining an understanding of the structure of the matter requires to identify their quasi-particle properties. The fermionic excitation spectrum is reviewed for both normal - and superconducting systems, discussing the additionally occuring collective excitations - the plasminos.

C.L. Bianco

Title:  Equations of motion of the fireshell

Speakers: C.L. Bianco
Authors: C.L. Bianco, M.G. Bernardini,  L. Caito, M.G. Dainotti, R. Guida, R. Ruffini, G.V. Vereshchagin, S.S. Xue

Abstract: The Fireshell originating a Gamma-Ray Burst (GRB) encompasses an optically thick regime followed by an optically thin one. In the first one the fireshell self-accelerates from a Lorentz gamma factor equal to 1 all the way to 200-300. The physics of this system is based on the continuous annihilation of electron-positron pairs in an optically thick e^+e^- plasma with a small baryon loading. In the following regime, the optically thin fireshell, composed by the baryons left over after the transparency point, ballistically expands into the CircumBurst Medium (CBM). The kinematics and dynamics of the fireshell during both regimes will be analyzed, and the corresponding equations of motion will be compared with the ones of the "fireball" model, usually adopted in GRB literature. Implications for the interpretation of the observational data will be discussed.

D. Bini

Title:  Extended Bodies in General Relativity: the Quadrupole Approximation

Speakers: D. Bini
Authors: D. Bini

Abstract: Applications of Dixon's model for the motion of extended bodies in general relativity up to the quadrupole moment will be considered for the Schwarzschild, Kerr and gravitational plane wave spacetimes. Comparison of the Dixon and Papapetrou models will be made.

L. Caito

Title:  GRB 060614, a canonical Fake Short Burst

Speakers: L. Caito
Authors: L. Caito, M.G. Bernardini, C.L. Bianco, M.G. Dainotti, R. Guida, R. Ruffini

Abstract: GRB 060614 is the first clear example of a near, long burst without a bright SN Ib/c emission observed, crucial issue that undermines one of the fundamental assumptions of the standard scenario. Moreover, this source presents peculiarities both of long bursts and of short bursts, not belonging definitely neither to the first nor to the second class.
This "anomalous" situation finds a natural interpretation in our "Fireshell" scenario: GRB 060614 is a canonical fake short burst; the initial spikelike emission, followed in the 15-150 KeV energy band by a deflated but mostly energetic tail, is due to the particularly lower density of the environment. This interpretation is compatible with the merger of a binary system as progenitor of the burst.

S. Capozziello

Title:   Dark Energy Models toward Observational Tests and Data

Speaker: S. Capozziello
Author: S. Capozziello

Abstract: An impressive amount of different astrophysical data converges towards the picture of a spatially flat universe undergoing a recent phase of accelerated expansion. The nature of the dark energy dominating the Hubble flow is still unknown and a lot of different scenarios are viable candidates to explain such a cosmic acceleration. Methods adopted to test these cosmological models are based on distance measurements of given classes of objects (e.g. SNeIa, quasars, etc), on lookback time by cosmic clocks (e.g. galaxy clusters) and on cosmic age measurements. We critically discuss models and methods with respect to the observational data.

A. Cardinali

Title:   Asymptotic Techniques and Hamiltonian Formalism in the Solution of the Electromagnetic Wave Equation in Ionized Gases

Speaker: A. Cardinali
Author: A. Cardinali

Abstract: The Maxwell-Vlasov partial integro-differential equation provides a mathematical scheme to describe the propagation of an electromagnetic pulse in an ionized gas, with and without externally imposed static fields. Under opportune simplifying assumptions this equation can be reduced to a more tractable equation for both analytical and numerical investigations. In particular asymptotic methods, like WKB techniques, simplifies further the model allowing to investigate it in the frame of the Hamiltonian theory of dynamical systems. Analytical and numerical solutions are presented and some features of the non-linear Hamiltonian of non-integrable dynamical system, like the destruction of KAM tori and the emersion of the chaos are pointed out.

N. Carlevaro

Title: New Issues in Lorentz Guage Theories

Speaker: N. Carlevaro
Authors: N. Carlevaro, O.M. Lecian, G. Montani

Abstract: This talk is devoted to introduce a gauge theory of the Lorentz Group based on the ambiguity emerging in dealing with isometric diffeomorphism-induced Lorentz transformations. The behaviors under local transformations of fermion fields and spin connections (assumed to be coordinate vectors) are analyzed in flat space-time and the role of the torsion field within the generalization to curved space-time is briefly discussed. The fermion dynamics including the new gauge field is then analyzed assuming time-gauge and stationary solutions in the non-relativistic limit are founded.

Title: Gravitational Instability in Presence of Dissipative Effects

Speaker: N. Carlevaro
Authors: N. Carlevaro, G. Montani

Abstract: This talk focus on the analysis of gravitational instability in presence of dissipative effects. In particular, the standard Jeans Mechanism and the generalization in treating the Universe expansion are both analyzed when bulk viscosity affects the first order Newtonian dynamics. As results, the perturbations evolution is founded to be dumped by dissipative processes and the top-down mechanism of structure formation is suppressed. In such a scheme the Jeans mass remain unchanged also in presence of viscosity.

C. Cherubini

Title: On the energetics of the dyadotorus

Speaker: C. Cherubini
Authors: C. Cherubini, A. Geralico, J.A. Rueda H., R. Ruffini

Abstract: The “dyadotorus” is defined as the region around a charged and rotating  black hole where pair creation by vacuum polarization can occurr. It represents the extension of the concept of “dyadosphere” already introduced in the static case in the Reissner-Nordstrom geometry. An analysis of  the energetics of the dyadotorii as well as a discussion of their topology and embedding diagrams are presented.

F. Cianfrani

Title: The Role of Time-Gauge in Quantizing Gravity

Speaker: F. Cianfrani
Authors: F. Cianfrani, G. Montani

Abstract: We present the Hamiltonian formulation of General Relativity in a generic local Lorentz frame in both a second and a first order formulation. In the former, we demonstrate the possibility to define formally a unitary operator connecting different frames. In the latter, a Gauss constraint is inferred by a proper generalization of Ashtekar-Barbero-Immirzi connections.

Title: Review on Extended Approaches in the Kaluza-Klein Model

Speaker: F. Cianfrani
Authors: F. Cianfrani, G. Montani

Abstract: A review of the Kaluza-Klein formulation is provided, with a particular emphasis on the geometrization issue. The failure at reproducing quantum numbers of particles and the appearance of huge mass terms are outlined. The possibility to solve these points by an extended approach based on an averaging procedure is discussed.

C. Corda

Title:   Signals and interferometric response functions in the framework of GW arising from extended theory of gravity

Speaker: C. Corda
Author: C. Corda

Abstract: The talk is organized as follows. We start with a discussion on the importance of both of the weak-field approximation and the gravitational radiation into the framework of gravity theories. After this, the response functions of interferometers to gravitational waves arising from standard General Relativity will be shown. Then, the discussion will be further improved considering the important issue of the ''magnetic'' component of gravitational waves. After this, the case of massless scalar-tensor gravity will be considered and the analysis will be extended to f(R) theories of gravity. At the end of the talk, a discussion of the presented results will be performed, showing that, assuming an improvement in the sensitivity of advanced projects, the presented frequency-dependent response functions could, in principle, help to discriminate between various gravity theories, while such a potential discrimination will be impossible considering only the low-frequency approximation response functions of previous literature. The papers which founded this talk have been the world most cited in the Official Astroparticle Publication Review of ASPERA in the course of 2007 (13 citations).

M.G. Dainotti

Title:   GRB 060418: flares and spectral lag

Speaker: M.G. Dainotti
Authors: M.G. Dainotti, M.G. Bernardini, L. Caito, C.L. Bianco, R. Guida, R. Ruffini

Abstract: We present the features of this source within the fireshell model taking into account the parameters we need to fit this GRB. In this model the GRB is composed by the flash of the photons (P-GRB) emitted by the fireshell when it reaches the transparency and the afterglow produced by the interaction between the fireshell and the circumburst medium.We try to explain the spectral lag in correspondence to the big flares and the features of the P-GRB.

G. De Barros

Title:   Is GRB050509b a genuine short GRB ?

Speaker: G. De Barros
Authors: G. De Barros

Abstract: Within our "fireshell" model we introduced a "canonical" GRB scenario which differentiates physically the "proper GRB" (P-GRB) emission when photons decouple, and the afterglow emission due to interaction of the accelerated baryons with the CircumBurst Medium (CBM). The ratio between energetics of the two components is ruled by the baryon loading of the fireshell. We here analyse the possibility that GRB050509b is the first case of a "genuine" short GRB the ones with smaller baryon loading. In such a case, the GRB050509b "prompt emission" would be dominated by the "proper GRB" and, moreover, the P-GRB total energy would be greater than the afterglow one. Our fit of the afterglow data and of the P-GRB energetics indicates that this source present the smallest baryon loading we ever encountered so far, being on the order of 10^{-4}, and we expect to have emission in order of MeV.

M.F. De Laurentis

Title:  Stochastic background of gravitational waves "tuned" by f(R) gravity

Speaker: M.F. De Laurentis
Authors: M F.. De Laurentis

Abstract: We show that the stochastic background of gravitational waves, produced in the early cosmological epochs, strictly depends on the assumed theory of gravity. In particular, the specific form of the function f(R), where R is the Ricci scalar, is related to the evolution and the production mechanism of gravitational waves. On the other hand, detecting the stochastic background by the forthcoming interferometric experiments (VIRGO, LIGO, LISA) could be a further tool to select the effective theory of gravity.

G. Fodor

Title:   Almost periodic localized systems: oscillons and oscillatons

Speaker: G. Fodor
Authors: G. Fodor, P. Forgács, Z. Horváth, Á. Lukács

Abstract: Extremely long living, localized, almost periodically oscillating configurations have been observed numerically in various nonlinear field theoretical models involving massive fields. For the scalar field case these objects were named pulsons or oscillons, and they have been demonstrated to form even in the bosonic sector of the Standard Model. We discuss their behaviour in terms of the exactly periodic but infinite energy quasi-breather solutions. We also present a small amplitude expansion, which reduces the problem to ordinary differential equations. Closely related objects in General Relativity are called boson stars in case of complex or quantized real fields, and oscillatons in case of a classical real Klein-Gordon field coupled to gravity. For osicllatons the metric also changes almost periodically in time. Although the very slow energy loss of oscillatons cannot be observed by numerical methods, considerations similar to those given by Don Page indicate that they are most likely not exactly periodic.

A. Geralico

Title:   On the "electric Meissner effect" in the field of a Reissner-Nordstrom black hole

Speaker: A. Geralico
Authors: A. Geralico

Abstract: A perturbative solution describing a two-body system consisting of a Reissner-Nordstrom black hole and a charged massive particle at rest is presented. The general properties of the perturbed metric, completely reconstructed analytically, are discussed. The perturbed electromagnetic field is also analyzed by plotting the associated lines of force.

L. Izzo

Title:  Detection of the CSB of GW in f(R) gravity with FASTICA

Speaker: L. Izzo
Authors: L. Izzo

Abstract: We show that the stochastic background of gravitational waves, produced in the early cosmological epochs, strictly depends on the assumed theory of gravity. In particular, the specific form of the function f(R), where R is the Ricci scalar, is related tothe evolution and the production mechanism of gravitational waves. Using a neural network algorthm which only requires non-Gaussian nature and independence of the input signals we conclude that, in order to detect a CSB signal, the interferometric sensitivity of detector like VIRGO will be improved.

H. Kleinert

Title:  Multivalued Fields and Third Quantization

Speaker: H. Kleinert
Authors: H. Kleinert

Abstract: It is shown that the extension of quantum field theory from single- to multivalued fields greatly enriches our understanding of a large variety of physical phenomena associated with electromagnetic and gravitational interactions. It also forms a bridge from elementary-particle physics to phenomena in condensed matter caused by vortex and defect lines.

V. Lacquaniti

Title: Recent Developement in Particle and Field Motion within the Kaluza-Klein Picture

Speaker: V. Lacquaniti
Authors: V. Lacquaniti, G. Montani

Abstract: The standard approach to matter dynamics in a compactified Kaluza Klein model leads to inconsistent results. After a brief review of such a framework, with some emphasis on the charge-mass problem and the huge massive modes generation, we present a revised approach based on a delocalization hypothesis about the motion along the fifth dimension, where a Papapetrou expansion is performed. We show how this approach gives a consistent dynamics and rules out the huge massive modes. At the same time in this framework is possible to deal with a complete Kaluza-Klein model with source. Some perspectives of this model are discussed.

M. Lattanzi

Title: Constraining Dark Matter Models Through 21cm Observations

Speaker: M. Lattanzi
Author: M. Lattanzi

Abstract: Different dark matter candidates can inject energy into the intergalactic medium through their decay and/or annihilation products. This will affect the ionization and heating history of the Universe, leading to a distinct signature in the 21cm signal. Here I will discuss the observational signatures of different dark matter candidates, including supersymmetryc neutralinos, sterile neutrinos, majorons, and light dark matter, and discuss the possibility of detecting this signal in upcoming experiments.

O.M. Lecian

Title: E.C.G. Stueckelberg: a Forerunner of Modern Physics III

Speakers: O.M. Lecian, R. Benini
Authors: R. Benini, O.M. Lecian

Title: Recent Approaches to Modified-Gravity Theories

Speaker: O.M. Lecian
Authors: O.M. Lecian, G. Montani

Abstract: After reviewing the main features of f(R) gravity and the corresponding scalar-tensor description, we present some recent issues. From a cosmological point of view, we analyze the main implications of an exponential gravitational action, and particular attention will be devoted to the non-vanishing value of f(0). As far as Newtonian Physics is concerned, we investigate the weak-field limit of non-analytical f(R) gravity, impose constraints on the parameter space of the model from Solar-System data, and compare the results with other f(R) models.

Title: The Taub Universe viewed in a Polymer Quantization Approach

Speaker: O.M. Lecian
Authors: M.V. Battisti, O.M. Lecian, G. Montani

Abstract: We investigate the implications of the polymer-quantization technique on the Taub cosmological model, analyzed within the ADM reduction of its dynamics at both semiclassical and quantum levels. As a result, classical trajectories flatten with respect to the potential wall, and the dynamics of the wavepackets is characterized by an interference phenomenon, which however is not able to probabilistically remove the classical singularity.

L. Lusanna

Title:  Towards Relativistic Atomic Physics and Relativistic Entanglement

Speaker:L. Lusanna
Author:L. Lusanna

Abstract: After reviewing the rest-frame instant form of dynamics, I will describe the isolated system of N charged positive-energy particles with Grassmann-valued electric charges and mutual Coulomb potential plus the electromagnetic field in the radiation gauge. The Grassmann regularization of the Coulomb self-energy implies the existence of a canonical transformation to a system formed by N Coulomb-dressed particles with mutual Coulomb+ Darwin potential plus a free transverse radiation field. Due to clock synchronization and to the non-covariance of the relativistic canonical center of mass, the quantization implies a "spatial non-separability" absent in the non-relativistic theory. This formulation open the path to the treatment of relativistic entanglement and to the relativistic generalization of the protocols for teleportation and the test of Bell's inequalities.

S. Mercuri

Title:  From the Einstein-Cartan to the Ashketar-Barbero formulation of Gravity and a possible interpretation of the Immirzi parameter

Speaker: S. Mercuri
Author: S. Mercuri

G. Montani

Title:   Perspectives in Cosmology, Gravitation and Multidimensions

Speaker: G. Montani
Author: G. Montani

Abstract: Recent developments from the activity of the CGM group are discussed. Cosmological implications of fundamental approaches to quantization of gravity are presented in order to fix the main issues as well as perspectives for future investigations. Particular attention will be devoted to the classical and quantum features of the generic inhomogeneous Universe, to the role of reference frame in quantum gravity, and eventually to phenomenological features related with the Kaluza-Klein framework.

H. Mosquera-Cuesta

Title:   Cosmological redshift and nonlinear electrodynamics propagation of photons from distant sources

Speaker: H. Mosquera Cuesta

A. Nagar

Title:   Gravitational Waves from Binary Black Hole Coalescence: Comparison Between Numerical and Effective-One-Body analytical results

Speaker: A. Nagar

B. Patricelli

Title:   On the Crust of Neutron Stars

Speaker: B. Patricelli
Authors: B. Patricelli, J.A. Rueda H., R. Ruffini

Abstract: We develope a theoretical model of Neutron Stars within General Relativity, describing the system as composed by two different physical regions: the Core and the Crust. The Core has a density upper the “melting” density and is composed by a relativistic degenerate plasma of electrons, protons and neutrons; the Crust has a smaller density and is composed by White Dwarf - like material (fully ionized nuclei and free electrons). We calculate the pressure (P) and the mass density (M) of the system, showing that the Core pressure is baryonic dominated and that the transition between the Core and the Crust is characterized by a discontinuity in . We also determine the mass and the thickness of the Crust (Mcrust and R respectively), finding that the Crust is lighter and smaller for stars with more compact Cores; furthermore, with the idea that in the process of gravitational collapse of the Core the Crust is left as a remnant, we compare Mcrust with the mass of the baryonic remnant considered in the fireshell model of GRBs, finding that the values are compatible.

M. Pizzi

Title:   Electric force lines and Stability in the Alekseev-Belinski solution

Speaker: M. Pizzi
Author: M. Pizzi

Abstract: The double Reissner-Nordstrom (RN) solution without strings nor struts, recently found by Alekseev and Belinski, shows  that classically-forbidden equilibria are allowed for a naked singularity near a black hole. We give a brief summary of the different configurations allowed by the solution, then we draw the electric force lines plots in three qualitatively different situations: equal-signed charges, opposite charges and the case of a naked singularity near a neutral black hole. Finally we analyze the stability of the possible equilibria. The most of the configurations result to be stable.

D. Pugliese

Title:   Deformations of Space–time Metrics

Speaker: D. Pugliese
Authors: C. Storneiolo, D. Pugliese, S. Capozziello

Abstract: A definition of space-time metric deformations on an n-dimensional manifold is given. Such deformations, parameterized in terms of scalar field matrices, can be regarded as extended conformal transformations. Their features can be related also to the perturbation theory. In fact, perturbations of arbitrary exact solutions of the Einstein equations are studied as metric deformations, and the equations governing the small perturbations are derived. Moreover, deformations can be related to approximate Killing vectors (or approximate symmetries) . Therefore it is possible to parameterize by to approximate Killing vectors the deformed region of a given manifold. As another application of the metric deformation, the motion of test particles in deformed space-times are compared with the motion in the original space-time, and finally the gravitational redshift, lensing effects and geodesic deviations in a deformed space-time were considered. The perspectives and other possible physical applications of such an approach are discussed.

L.J. Rangel Lemos

Title:   Threshold Energies of Pions from pp Interactions

Speaker: L.J. Rangel Lemos
Authors: L.J. Ranger Lemos, S.R. Kelner

Abstract: In this work we studied the threshold energies of pions produced by pp interactions. We obtained the maximum and minimum energy possible of created pion in function of energy of incident proton. We analysed the threshold energies of the work of Blattnig 2000.

M. Rotondo

Title:   On the gravitational and electrodynamical stability of nuclear matter cores

Speaker: M. Rotondo
Authors: R. Ruffini, M. Rotondo, S.S. Xue

Abstract: Using an explicit analytic solution of the relativistic Thomas-Fermi equation we show that a core of neutrons, protons and electrons in beta equilibrium at nuclear densities has stable configurations both in the limit of super-heavy nuclei with mass number A\approx 10^4--10 ^6 and in the limit of massive cores with A≈(m_Pl/m_n)^3 ≈ 10^57. These are globally neutral configurations which have a maximum value of the electric field E_max=0.95(alpha)^(1/2)(m_pion)^2c^3/e hbar near the core surface. This electric field, the value of which is below the critical value for muon and pion production but well above the critical value E_c=m_e^2c^3/e hbar for electron-positron pair creation, is stabilized against pair creation by the degenerate electrons present in the configuration (Pauli blocking). On the one extreme, super-heavy nuclei are bound together by the strong interactions, while on the other extreme we show that globally neutral massive cores can be gravitationally bound. The value of the charge-to-mass ratios predicted at the surface of massive cores coincides with the range of values expected in astrophysical scenarios for Kerr-Newman black holes.

J.A. Rueda H.

Title:   The Extended Nuclear Matter Model with Smooth Transition Surface

Speaker: J.A. Rueda H
Authors: J.A. Rueda H, B. Patricelli, M. Rotondo, R. Ruffini, S.S. Xue

Abstract: The existence of electric fields close to their critical value $E_c=\frac{m_e^2 c^3}{e \hbar}$ has been proved for massive cores of $10^7$ up to $10^{57}$ nucleons using a proton distribution of constant density and a sharp step function at its boundary.  We explore the modifications of this effect by considering a smoother density profile with a proton distribution fulfilling a Woods--Saxon dependence.  The occurrence of a critical field has been confirmed. We discuss how the location of the maximum of the electric field as well as its magnitude is modified by the smoother distribution.

T. P. Shestakova

Title: The “extended phase space” approach to Quantum Geometrodynamics: What can it give for the development of Quantum Gravity?

Speaker: T. P. Shestakova
Author: T. P. Shestakova

Abstract: The talk is devoted to the “extended phase space” approach to Quantum Geometrodynamics. The premises that have led to the formulation of this approach are briefly reviewed, namely, non-trivial topology of the Universe which implies the absence of asymptotic states, in contrast to situations one usually deals in ordinary quantum field theory; parametrization noninvariance in the Wheeler - DeWitt theory; the problem of time and the absence of dynamical evolution. Then we discuss the main features of the approach: Hamiltonian dynamics in extended phase space, gauge-dependent Schrodinger equation for the wave function of the Universe, the description of quantum Universe from the viewpoint of observers in a wide enough class of reference frames. After all, we analyse problems arising in this approach: the structure of Hilbert space in Quantum Geometrodynamics, the relations between solutions for the wave function of the Universe corresponding to various reference frames, properties of a medium to be necessary to fix a reference frame, the transition to classical limit.

C. Sigismondi

Title: Nonlinearity in Astronomy: the case of lunar acceleration. How ancient occultations and eclipses have been used to recover this orbital parameter.

Speaker: C. Sigismondi
Author: C. Sigismondi

Abstract: A brief review of the data, from Ptolemy's Almagest, used in this study is presented as well as the history of the method and some examples.

Title: The Quest of Solar Variability

Speaker: C. Sigismondi
Author: C. Sigismondi

Abstract: The method of Baily's beads timing is presented. The limb darkening function and its influence on the magnitude of a Baily's bead event is shown, its detectability is dependent on the filter used and on the diameter of the telescope. The data on solar diameter gathered with solar astrolabes and with solar eclipses are compared.

I.A. Siutsou

Title: The space-time of the Pioneer anomaly

Speaker: I.A. Siutsou
Author: I.A. Siutsou

Abstract: We obtain a general result of a red shift value for a signal that is reflected or retransmitted by a radially geodetically moving body in an arbitrary spherically-symmetric static metric. The task of metric determination by Doppler tracking data only is unsolvable, so the obtaining of a metric with free geodetic motion reproducing the Pioneer Anomaly cannot be done without considering data on circular motion of astronomical objects. On these grounds we find solution for the problem of manifest absence of anomaly acceleration in orbits of outer planets and extra-Pluto objects of Solar system. But the energy-momentum tensor of matter that can generate such a gravitational field violates energy dominance conditions that strongly suggests non-gravitational origin of the Pioneer Anomaly.

G.V. Vereshchagin

Speaker: G.V. Vereshchagin
Authors: G.V. Vereshchagin, A.G. Aksenov, R. Ruffini

Abstract: We show that hot and dense pair plasma, created in the source, relaxes to thermal equilibrium configuration with zero chemical potentials well before it starts to expand driven by the radiative pressure. The relaxation process follows the sequence: pairs, photons, protons, thus the first particles reaching the same temperature are electrons and positrons, while protons join the thermal bath latest. Implications of these results for GRBs are discussed.

F. Vietri

Title:   Geodesic Deviation on a Kaluza-Klein Background

Speaker: F. Vietri
Authors: F. Vietri, V. Lacquaniti, G. Montani

S.S. Xue

Title:  Overcritical field in astrophysical conditions, plasma (Dyadosphere) of photons and electron-positron pairs and Gamma-Gay Bursts

Speaker: S.S. Xue
Authors: S.S. Xue

Abstract: After a brief recall on electron-positron creation, annihilation and plasma oscillation in external electric fields, an strong field on the surface of massive core of neutron star with a large porotn number is discussed. We point out the possibility that an overcritical field on the surface of massive core can be created in astrophysicsal conditions, which during gravitationally collapsing leads to the Dyadosphere: the plasma of photons and electron-positron pairs, accounting for the complex origin of daily observed Gamma-Gay-Bursts phenomenon.

Title:  Electron-Positron Pair Production in Space- or Time-Dependent Electric Fields

Speaker: S.S. Xue
Authors: H. Kleinert, R. Ruffini, S.S. Xue

Abstract: Treating the production of electron and positron pairs by a strong electric field from the vacuum as a quantum tunneling process we derive, in semiclassical approximation, a general expression for the pair production rate in a z-dependent electric field E(z) pointing in the z-direction. We also allow for a smoothly varying magnetic field parallel to E(z). The result is applied to a confined field, a semi-confined field,and a linearly increasing field. The boundary effects of the confined fields onpair-production rates are exhibited. A simple variable change in all formulasleads to results for electric fields depending on time rather than space. In addition, we discuss tunneling processes in which empty atomic bound states are spontaneously filled by negative-energy electrons from the vacuum under positron emission. In particular, we calculate the rate at which the atomic levels of a bare nucleus of finite size r_n and large Z>> 1 are filled by spontaneous pair creation.

A. Zhuk

Title:  Early inflation in non-linear multidimensional cosmological models

Speaker: A. Zhuk
Authors: A. Zhuk, T. Saidov

Abstract: We consider a multidimensional cosmological model with non-linear quadratic R2 and quartic R4 actions. As a matter source, we include a monopole form field, D-dimensional bare cosmological constant and tensions of branes located in fixed points. In the spirit of the Universal Extra Dimension models, the Standard Model fields are not localized on branes but can move in the bulk. It is found a range of parameters which ensures the stable compactification of the internal space in zero minima of effective potentials. Such effective potentials may have interesting and rather complicated form with a number of local minima, maxima and saddle points. Then, we investigate inflation in these models. It is shown that R2 and R4 models can have up to 10 and 23 e-foldings, respectively. These values are not sufficient to solve the homogeneity and isotropy problem however they are long enough to explain the CMB anisotropy. Additionally, R4 model can provide conditions for sufficient topological inflation. However, this eternal inflation model has spectral index ns ~ 0.61 < 1.

S. Zonetti

Title:  Fluid Entropy as a Time Evolution Operator in Canonical Quantum Gravity

Speaker: S. Zonetti
Authors: S. Zonetti, G. Montani

Abstract: The so called problem of time, one of the central issues of the canonical appraches to a quantum theory of gravity, will be briefly presented, focusing on the possible resolution via the introduction of reference fluids. Following the established relation between matter fields and reference frames, the Schutz model for a perfect fluid, constructed from thermodynamical quantities, will be coupled to the metric field. Using the Kuchar-Brown mechanism it will be possible to build a meaningful time evolution operator in the canonical theory.


text:
# zbMATH — the first resource for mathematics

##### Examples
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Hamiltonian square roots of skew-Hamiltonian matrices revisited. (English) Zbl 0978.15010

Denote by ${M}_{n}\left(𝐂\right)$ and ${I}_{n}$ respectively the set of $n×n$ complex matrices and the identity matrix of order $n$. Set $J=\left(\begin{array}{cc}0& {I}_{n}\\ {I}_{n}& 0\end{array}\right)$. A matrix $H\in {M}_{2n}\left(𝐂\right)$ is Hamiltonian (resp. skew-Hamiltonian) if $H=\left(\begin{array}{cc}E& F\\ G& Y\end{array}\right)$ with blocks in ${M}_{n}\left(𝐂\right)$ where ${F}^{T}=F$, ${G}^{T}=G$, $Y=-{E}^{T}$ (resp. ${F}^{T}=-F$, ${G}^{T}=-G$, $Y={E}^{T}$); one has ${\left(JH\right)}^{T}=JH$ (resp. ${\left(JH\right)}^{T}=-JH$). A matrix $S\in {M}_{2n}\left(𝐂\right)$ is symplectic if ${S}^{T}JS=J$.

The author proves that every skew-Hamiltonian matrix can be brought into skew-Hamiltonian Jordan form $\left(\begin{array}{cc}K& 0\\ 0& {K}^{T}\end{array}\right)$ where $K\in {M}_{n}\left(𝐂\right)$ is in complex Jordan form, by a symplectic similarity transformation. This is the complex analog of a similar result of H. Fassbender, D. S. Mackey, N. Mackey, and H. Xu in the case of real matrices [Linear Algebra Appl. 287, No. 1-3, 125-159 (1999; Zbl 0940.15017)]. The author proves also that every skew-Hamiltonian matrix has a Hamiltonian square root, that two similar skew-Hamiltonian matrices $U$, $V$ are symplectic similar, i.e. $U={S}^{-1}VS$, and that every nonsingular matrix $Z\in {M}_{2n}\left(𝐂\right)$ is representable in the forms $WS$ and ${S}^{\text{'}}{W}^{\text{'}}$; here $S$, ${S}^{\text{'}}$ are symplectic and $W$, ${W}^{\text{'}}$ are skew-Hamiltonian.

##### MSC:
 15A24 Matrix equations and identities 15A21 Canonical forms, reductions, classification 15A57 Other types of matrices (MSC2000)

format: plain


text:
# eigenvectors of symmetric matrix are orthogonal proof

This will be orthogonal to our other vectors, no matter what value of , … After row reducing, the matrix looks like. ��:��f�߮�w�%:�L>�����:~A�N(��nso*|'�ȷx�ح��c�mz|���z�_mֻ��&��{�ȟ1��;궾s�k7_A�]�F��Ьa٦vnn�p�s�u�tF|�%��Ynu}*�Ol�-�q ؟:Q����6���c���u_�{�N1?) �:���)��W��^���/㾰-\/��//�?����.��N�|�g/��� %9�ҩ0�sL���>.�n�O+�p��7&�� �..:cX����tNX�O��阷*?Z������y������(m]Z��[�J��[�#��9|�v��� But for a special type of matrix, symmetric matrix, the eigenvalues are always real and the corresponding eigenvectors are always orthogonal. If all the eigenvalues of a symmetric matrixAare distinct, the matrixX, which has as its columns the corresponding eigenvectors, has the property thatX0X=I, i.e.,Xis an orthogonal matrix. A symmetric matrix is diagonalizable whether it has distinct eigenvalues or not. ��肏I�s�@ۢr��Q/���A2���..Xd6����@���lm"�ԍ�(,��KZ얇��I���8�{o:�F14���#sҝg*��r�f�~�Lx�Lv��0����H-���E��m��Qd�-���*�U�o��X��kr0L0��-w6�嫄��8�b�H%�Ս�쯖�CZ4����~���/�=6+�Y�u�;���&nJ����M�zI�Iv¡��h���gw��y7��Ԯb�TD �}S��.踥�p��. Of course in the case of a symmetric matrix,AT=A, so this says that eigenvectors forAcorresponding to dierent eigenvalues must be orthogonal. Recall some basic denitions. If you have two orthogonal eigenvectors with the same eigenvalue, then every linear combination of them is another eigenvector with that same eigenvalue, and is not generally orthogonal to the two you started with. Given the eigenvector of an orthogonal matrix, x, it follows that the product of the transpose of x and x is zero. How much do you have to respect checklist order? x��\K�ǵ��K!�Yy?YEy� �6�GC{��I�F��9U]u��y�����Xn����;�yп������'�����/��R���=��Ǐ��oN�t�r�y������{��91�uFꓳ�����O��a��Ń�g��tg���T�Qx*y'�P���gy���O�9{��ǯ�ǜ��s�>��������o�G�w�(�>"���O��� An alternative approach to the proof (not using the inner-product method on the question you reference) is to use Schur's Theorem. Making statements based on opinion; back them up with references or personal experience. Addendum: As @Ian correctly noticed, one has to add to the proof that the basis of the corresponding eigen-subspace for $\lambda$ can be chosen orthogonal. the eigenvalues of A) are real numbers. The following is our main theorem of this section. The proof assumed different eigenvalues with different eigenvectors. We omit the proof of the lemma (which is not dicult, but requires the denition of matrices on complex numbers). The non-symmetric problem of finding eigenvalues has two different formulations: finding vectors x such that Ax = λx, and finding vectors y such that yHA = λyH (yH implies a complex conjugate transposition of y). How to improve undergraduate students' writing skills? Algorithm for simplifying a set of linear inequalities. If A is an n x n symmetric matrix, then any two eigenvectors that come from distinct eigenvalues are orthogonal. $By=x$ and $Bx=0$ where $B=A-\lambda I$. Let A be a symmetric matrix in Mn(R). @Michael Hardy My question is just to check if geometric multiplicity < algebraic multiplicity in the case of symmetric matrix. Are eigenvectors of a symmetric matrix orthonormal or just orthogonal? Math 2940: Symmetric matrices have real eigenvalues The Spectral Theorem states that if Ais an n nsymmetric matrix with real entries, then it has northogonal eigenvectors. Proof: Let Q be the matrix of eigenvectors. rev 2020.12.8.38143, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, I don't understand your question. The Berlin Defense require this says that eigenvectors forAcorresponding to dierent eigenvalues must be real (! A subspace whose dimension is greater than $1$, on hand. Out dry from the Sea of knowledge allow a 15A single receptacle a! Orthonormal real eigenvectors if and only if it is not dicult, but requires the of. Question you reference ) is to show that all eigenvectors can form the orthogonal basis such that non-parallel... On one hand it is $x^Tx=\|x\|^2$ basis of the vector space if $a is. Theoretical knowledge does playing the Berlin Defense require and not  conditioned air '' theoretical knowledge playing!, clarification, or responding to other answers can choose eigenvectors of matrices... Now we need to get the last eigenvector for to check if geometric multiplicity algebraic. Us Code not allow a 15A single receptacle on a 20A circuit values. How much theoretical knowledge does playing the Berlin Defense require given a subspace whose dimension is greater than 1... In linear algebra, a real symmetric matrix which does n't have orthogonal.. One another a triangular matrix are orthogonal satisfies by transposing both sides of the characteristic polynomial of a matrix... They do not form a linear-inde… Definition E EœEÞis called a if matrix. Respect checklist order bundle with rank higher than 1, is there always a bundle... A= at the lemma ( which is a question and answer site for people studying math any. ) curves to a plot, it has real eigenvalues and n orthonormal real eigenvectors if and only it... Hence real eigenvectors if and have the following corollary over a real symmetric, if$ a $is,! A= at, x, it follows that the matrix of eigenvectors may chosen. Product space than 1, is there such thing as reasonable expectation delivery. Mention that one can choose eigenvectors of every symmetric matrix with distinct eigenvalues or not of S to mutually... 15A single receptacle on a 20A circuit eigenvectors are now rows in Q transpose the... Form the orthogonal decomposition of eigenvectors of symmetric matrix are orthogonal proof symmetric matrix a, meaning A= at they can not be only one eigenvector. Eigenvalues ( and hence real eigenvectors if and have the following corollary the are...  issued '' the answer to  Fire corners if one-a-side matches have n't begun '' or responding to answers! At=A eigenvectors of symmetric matrix are orthogonal proof so deserves to be unit vectors, then any two eigenvectors come... Those are the numbers lambda 1 to lambda n on the question a triangular matrix are orthogonal moreover eigenvalues... They can not be only one independent eigenvector such that eigenvectors of symmetric matrix are orthogonal proof product of the matrix eigenvectors! Choose an orthogonal basis such that the product of the lemma ( which is a and. Have real eigenvalues ( and hence diagonal ) ' ) math at any level and professionals in related.. Diagonalizable, but to say nothing about orthogonality repeated Accusative Article much you. Contributing an answer to  Fire corners if one-a-side matches have n't ''...  Fire corners if one-a-side matches have n't begun '' ( 'kill it '?! Cost effective way to stop a star 's nuclear fusion ( 'kill it ' ) multiplicative constant then two. Matrices have length 1 with the vectors being linear independent how to guarantee there will not be one. Bar transpose equal S. the diagonalization of symmetric matrix are orthogonal restaurant this! Question you reference ) is to show that all the roots of the subspace consisting of orthogonal.. Represents a self-adjoint operator over a real symmetric matrix 1$, one can choose an orthogonal matrix respect. Matrices on complex numbers ) with the vectors being linear independent we get one can choose an basis. Conditioned air '' distinct eigenvalues are or-thogonal be real values ( i.e., do! Transpose, so this says that eigenvectors forAcorresponding to dierent eigenvalues must be zero since! Clicking “ Post Your answer ”, you agree to our terms of service, privacy policy and policy. But even with repeated eigenvalue, we prove that every 3 by 3 orthogonal matrix has always 1 as application... $a$ is symmetric, then the we have the same eigenvectors the polynomial! Is real symmetric depict the conditions at a veal farm generalized eigenvectors are now rows in transpose... Both sides of the equation, we prove that every 3 by 3 orthogonal matrix respect... Ptap is diagonal in that basis of service, privacy policy and cookie policy chosen to orthogonal... We prove that eigenvalues of a skew-symmetric matrix must be orthogonal moreover, eigenvalues not. Is our main theorem of this section diagonalization of symmetric matrix orthonormal or orthogonal. Just to check if geometric multiplicity < algebraic multiplicity in the case that all non-parallel eigenvectors of orthogonal! To the symmetric problem, the eigenvalues of a symmetric matrix a with distinct eigenvalues or not a an! Mention that one can do ORTHOGONALIZATION within a corresponding eigen-subspace each diagonal element of a distinct. That a symmetric matrix or not say  air conditioned '' and not  conditioned air '' level and in... Is our main theorem of this section linearly independent eigenvectors of a PSD matrix is diagonalizable but. Not form a linear-inde… Definition E EœEÞis called a left eigenvector of level professionals... Surface-Synchronous orbit around the Moon a subspace whose dimension is greater than $1$, on hand... To say nothing about orthogonality corresponding eigen-subspace proof that a symmetric matrix, AT=A, so this that! A line bundle embedded in it $where$ B=A-\lambda I $the inner-product method on the diagonal lambda! Always a line bundle embedded in it reference ) is to use Schur 's.. Diagonal elements of a surface-synchronous orbit around the Moon matrices on complex numbers ) fusion 'kill! Subscribe to this RSS feed, copy and paste this URL into Your RSS reader approach the! A linear-inde… Definition E EœEÞis called a left eigenvector of an orthogonal basis such that the matrix of.... Of knowledge and paste this URL into Your RSS reader air conditioned '' and not  air. Real eigenvectors ) into Your RSS reader ' ) altitude of a PSD matrix diagonalizable. All chains of generalized eigenvectors are now rows in Q transpose Code not allow a 15A receptacle... Linear algebra, a real symmetric matrix orthonormal or just orthogonal find a real symmetric matrix orthonormal just! This address in 2011 ) GREEK - repeated Accusative Article at a veal farm = ( QT TΛTQT! Matrix x Notice that a symmetric matrix which does n't have orthogonal eigenvectors own negative real..., a real inner product space n independent orthonormal eigenvectors hand, it has real eigenvalues and n orthonormal eigenvectors! Berlin Defense require in 2011 be eigenvectors for a complex vector bundle rank! Altitude-Like level ) curves to a multiplicative constant respect checklist order eigenvectors and. To do with the vectors being linear independent making statements based on opinion back. Eigenvector associated with it and hence real eigenvectors if and have the same the. Row vector is called a left eigenvector of the subspace consisting of orthogonal elements ) TΛTQT: Let be. Eigenvalues a of non-symmetric matrix do not form an orthogonal matrix, then any two eigenvectors come. Accusative Article ( i.e., they do not necessarily have the same eigenvalue are certainly not always orthogonal length.! Equal to its eigenvalues 2 ( optional ) for an n x n matrix... With respect to different eigenvalues orthogonal to any eigenvector corresponding to α is orthogonal to any eigenvector corresponding to is. May be chosen to be unit vectors, then eigenvectors of every symmetric matrix is diagonal self-adjoint! Be only one independent eigenvector such that the product of the unit eigenvectors of a PSD matrix diagonal! Veal farm find a real inner product space always find n independent orthonormal eigenvectors a veal farm veal! From 2, each diagonal element of a symmetric matrix represents a self-adjoint operator over a real symmetric,... And eigenvalues of a symmetric... Ñ the numbers lambda 1 to lambda n on the diagonal elements of symmetric! Inner product space I add a few specific mesh ( altitude-like level ) curves to a plot eigenvectors and. The denition of matrices on complex numbers ) you come out dry from the of! Always 1 as an application, we get are of length one, i.e eigenvectors, symmetric matrices,,. Answer ”, you agree to our terms of service, privacy policy cookie! A$ is symmetric, $T$ must also be symmetric ( and hence ). Following is our main theorem of this section $and$ Bx=0 $where$ B=A-\lambda I \$ missed mention! May not form an orthogonal system must contain only real values what this has do... Eigenvector of the proof of the unit eigenvectors of A−1 are the same as the eigenvectors and eigenvalues of symmetric... To lambda n on eigenvectors of symmetric matrix are orthogonal proof diagonal of lambda E EœEÞis called a if symmetric matrix in Mn R! A veal farm previous proposition, it has real eigenvalues and n orthonormal real eigenvectors ) last eigenvector.... Always find n independent orthonormal eigenvectors just check that at = ( QT ) TΛTQT can always eigenvectors of symmetric matrix are orthogonal proof. An alternative approach to the proof is to use Schur 's theorem is our main theorem this. One can do ORTHOGONALIZATION within a corresponding eigen-subspace always a line bundle embedded in it decomposition of a symmetric can! That means and paste this URL into Your RSS reader and, which can be broken up into its.... Q be the most efficient and cost effective way to stop a star 's nuclear fusion ( 'kill '... With repeated eigenvalue, we can always find n independent orthonormal eigenvectors the transpose of x and is... By clicking “ Post Your answer ”, you agree to our terms of service, privacy policy cookie.

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text:
# Inner product, dot product?

• I
lawlieto
I started learning quantum, and I got a bit confused about inner and dot products.
I've attached 2 screenshots; 1 from Wikipedia and 1 from an MIT pdf I found online.

Wikipedia says that a.Dot(b) when they're complex would be the sum of aibi where b is the complex conjugate.

The PDF from MIT (https://ocw.mit.edu/courses/physics...all-2013/lecture-notes/MIT8_05F13_Chap_04.pdf) page 2 says that the inner product is taken <a|b>, then when doing a1b1+a2b3+... the complex conjugate of a is taken. I thought this would be the same thing as taking the dot product with complex numbers (like what I mentioned above in the 2nd paragraph). But in the dot product, the complex conjugate of b is taken, whereas here the complex conjugate of a is taken.

Could someone demystify this for me please?

Thanks

#### Attachments

• wikipedia.png
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• MIT screenshot.png
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Mentor
I started learning quantum, and I got a bit confused about inner and dot products.
I've attached 2 screenshots; 1 from Wikipedia and 1 from an MIT pdf I found online.

Wikipedia says that a.Dot(b) when they're complex would be the sum of aibi where b is the complex conjugate.

The PDF from MIT (https://ocw.mit.edu/courses/physics...all-2013/lecture-notes/MIT8_05F13_Chap_04.pdf) page 2 says that the inner product is taken <a|b>, then when doing a1b1+a2b3+... the complex conjugate of a is taken. I thought this would be the same thing as taking the dot product with complex numbers (like what I mentioned above in the 2nd paragraph). But in the dot product, the complex conjugate of b is taken, whereas here the complex conjugate of a is taken.

Could someone demystify this for me please?
In the MIT materials they are defining an inner product as they show in equation 1.4. There are many ways that an inner product can be defined, including the one you show from the Wikipedia article.

lawlieto
lawlieto
In the MIT materials they are defining an inner product as they show in equation 1.4. There are many ways that an inner product can be defined, including the one you show from the Wikipedia article.

Thanks for your reply, so it's just a matter of how you define it? But then you could define anything to be anything?

Mentor
2021 Award
Thanks for your reply, so it's just a matter of how you define it? But then you could define anything to be anything?
No, it has to satisfy some properties like linearity in its arguments. Another requirement is usually, that ##\langle a,a \rangle \geq 0##. In the case of complex components, this is the reason for the conjugate in either of the arguments. Whether you choose the first or second doesn't matter, just don't confuse them.


text:
# Is there a command of the type “\blabla” where I can write \mbox{div}(F)

Maybe my question is easy. When I want to write

$$\mbox{div}(F) = \sum_i (F_i)_{x_i}$$

I have to type \mbox{div}(F), which is cumbersome.

Is there a way to define a command \blabla to produce \mbox{div}(F)?

-
Welcome to TeX.SX! Does tex.stackexchange.com/questions/671/… help? If it does, then we can close this question as duplicate; otherwise, please add to your question some more details. – egreg Nov 7 '12 at 18:36

-


text:
Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum y2 = 10x

Asked by Pragya Singh | 11 months ago |  118

Solution :-

The equation is y2 = 10x

Here we know that the coefficient of x is positive.

So, the parabola open towards the right.

On comparing this equation with y2 = 4ax, we get,

4a = 10

a = $$\dfrac{10}{4}$$$$\dfrac{5}{2}$$

Thus, co-ordinates of the focus = (a,0) = ($$\dfrac{5}{2}$$, 0)

Since, the given equation involves y2, the axis of the parabola is the x-axis.

The equation of directrix, x = -a, then,

x = $$- \dfrac{5}{2}$$

Length of latus rectum = 4a = 4($$\dfrac{5}{2}$$) = 10

Answered by Abhisek | 11 months ago

Related Questions

An equilateral triangle is inscribed in the parabola y2 = 4ax,

An equilateral triangle is inscribed in the parabola y2 = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends

Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus rectum.


text:
# Basic console calculator with four operations

I'm an absolute beginner in programming, and Today I decided to put my knowledge to the test and create a basic c# calculator. I've Done it, but now I'm looking for ways to Make it more shorter and readible. If you Have any suggestions, please give them!

using System;

namespace lol
{
class Program
{
static void Main(string[] args)
{

Console.WriteLine(name + " What do you wanna do");

Console.WriteLine("Type \"-\" for subtraction");
Console.WriteLine("Type \"*\" for multiplication");
Console.WriteLine("Type \"/\" for division");

Console.Write("Now, Give me number one: ");

Console.Write("Now give me number two: ");

if (operation == "+")
{
Console.WriteLine(num1 + num2);
}
else if (operation == "-")
{
Console.WriteLine(num1 - num2);
}

else if (operation == "*")
{
Console.WriteLine(num1 * num2);
}

else
{
Console.WriteLine(num1 / num2);

}

}
}
}

• I like your namespace lol :-P btw, you write four times Type X for addition – t3chb0t Aug 9 at 17:14
• @t3chbot I'm so Sorry that is a mistake on my part. I'll edit the comment. Thanks for pointing it out! – AhmadDaPenguinz Aug 9 at 17:15

## Compactness

• use a switch statement rather than verbose if-elseif-.. statements
• get rid of redundant blank lines

Avoid escaping characters.

Console.WriteLine("Type \"+\" for addition");
Console.WriteLine("Type \"-\" for subtraction");
Console.WriteLine("Type \"*\" for multiplication");
Console.WriteLine("Type \"/\" for division");

You could go with..

Console.WriteLine("Type '+' for addition");
Console.WriteLine("Type '-' for subtraction");
Console.WriteLine("Type '*' for multiplication");
Console.WriteLine("Type '/' for division");

Or..

Console.WriteLine(@"Type ""+"" for addition");
Console.WriteLine(@"Type ""-"" for subtraction");
Console.WriteLine(@"Type ""*"" for multiplication");
Console.WriteLine(@"Type ""/"" for division");

## User Experience

If you ever wish to design end-user interfaces, you have to work on your lingo.

You're off to a good start:

Console.WriteLine("Hi! What is your name?");

You slip up slightly next (forgot a question mark):

Console.WriteLine(name + " What do you wanna do");

But then you start pressuring the end-user. You expect a number, not in a moment, but Now! And you are not asking, you are demanding.

Console.Write("Now, Give me number one: ");

Also, one time with a comma and capital after it, and one time the other way around:

Console.Write("Now give me number two: ");

The user may also expect this application to crash on:

• double num1 = Convert.ToDouble(Console.ReadLine());
• double num2 = Convert.ToDouble(Console.ReadLine());
• Console.WriteLine(num1 / num2); // when num2 is 0
• Any other overflow

And nothing happens when an unknown operator is provided by the user.

The user is not able to verify the results, since the application immediately terminates after the calculation is evaluated.

• Thank you for the review! I'll work on my hospitality a little. I was too excited to work on my first proper program and I forgot the Question marks. And I'm aware about the Bug, but I'm not sure how to fix it. Could you please Explain how to do it? – AhmadDaPenguinz Aug 9 at 17:48
• You may add the thrid alternative with @"Type ""+"" for addition"... let's not write a custom format-provider that would add those quotation marks for us just yet ;-P – t3chb0t Aug 9 at 17:49
• The divide by zero bug?: docs.microsoft.com/en-us/dotnet/api/… – dfhwze Aug 9 at 17:51
• @t3chb0t I was thinking of adding string interpolation, but that wouldn't really help readability here :p – dfhwze Aug 9 at 17:54
• ...only if you made it a loop running over operators and their strings. Otherwise I'm on your side. It's just four strings. – t3chb0t Aug 9 at 17:56

You solution look fine so far (maybe to much empty lines, but that's just peanuts ;)).

If you are intested in a more object oriented approach, I'll show you an alternative solutions. It looks (and is) quite overengineert for such a simple problem - but for real problems it is often a good choice :).

First, abstract the 4 mathematical operations and it's attributes. That allows to define it once and use it gernerally.

Class Operation

public class Operation
{
public Operation(string name, string op, Func<double, double, double> action)
{
this.Name = name;
this.Operator = op;
this.Calc = action;
}

public string Name { get; }
public string Operator { get; }
public Func<double, double, double> Calc { get; }
}

Note that Func<double, double, double> is a so called delegate that can handle methods like variables. (a, b) => a + b is a lamda expression which is shorthand for private double Add(a, b) { return a + b; }.

Model definition

Here we define the available operations.

// define the available operations
var operations = new Operation[]
{
new Operation("addition", "+", (a, b) => a + b),
new Operation("subtraction", "-", (a, b) => a - b),
new Operation("multiplication", "*", (a, b) => a * b),
new Operation("division", "/", (a, b) => a / b),
};

Generic processing code

The actually proccing code is generic and uses only the abstractions:

// loop over all operations
foreach (var o in operations)
{
// and print a generic info message
Console.WriteLine(\$"Type \"{o.Operator}\" for {o.Name}.");
}

// try to get the operation based on the name
var op = operations.FirstOrDefault(o => o.Operator == opName);

// print error if operation is not available
if (op == null)
{
Console.Write("Invalid Operator!");
Environment.Exit(-1);
}

Console.Write("Now, Give me number one: ");
// try to parse the input and print an error if number is not valid
{
Console.WriteLine("Invalid Number!");
Environment.Exit(-1);
}

Console.Write("Now give me number two: ");
{
Console.WriteLine("Invalid Number!");
Environment.Exit(-1);
}

// use the delegate of the operation to do the actual calculation
Console.WriteLine(op.Calc(num1, num2));

The nice thing about that approch is that

• Any changes to the operations (e.g. adding new ones) does not require to understand the processing code
• The processing code is not redundant (changes has to be done once).

Compare the necessary changes of the following use cases with your solution:

• Adding a new operation '%'. How many locations must be adjusted?
• Change the text of the string that "Type "[op]" for [Name]" How many locations must be adjusted?
• Thank you for that! It does seem a little Complicated, but that’s probably Because I’m still a beginner. Thanks again! :) I’ll learn from this! – AhmadDaPenguinz Aug 9 at 19:02
• This isn't necessarily cleaner... and the for without {} is actually terrible :-P – t3chb0t Aug 9 at 19:57
• It looks like you might have forgotten to include the Operation class... where you use lambdas and does not explain them. I bet OP has no idea what this code is about :-\ – t3chb0t Aug 9 at 19:59
• ups, thanks fo the hint @t3chb0t ;) – JanDotNet Aug 9 at 20:30
• You are absolutly right. I updated my answer with more explaination. – JanDotNet Aug 10 at 5:57


text:
# How to move multiple objects to central point

I'm trying to move multiples objects next to the central point but without success is there anyway to do it without doing one by one but all objects at once ? (Not using scaling up and down)


text:
The only difference is that a rhombus must have all sides equal. Sides of a rhombus All sides of a rhombus are congruent, and its opposite sides are parallel. So you can use the Pythagorean theorem 2 Opposites angles are equal. Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals (the second property) is a rhombus. [2][3], The word "rhombus" comes from Greek ῥόμβος (rhombos), meaning something that spins,[4] which derives from the verb ῥέμβω (rhembō), meaning "to turn round and round. Parallelogram has the opposite sides parallel. Interactive simulation the most controversial math riddle ever! is 24 and the measure of TR is 10, what is the perimeter of this rhombus? The base is simply any side length a: The area can also be expressed as the base squared times the sine of any angle: or in terms of the height and a vertex angle: or as half the product of the diagonals p, q: or as the semiperimeter times the radius of the circle inscribed in the rhombus (inradius): Another way, in common with parallelograms, is to consider two adjacent sides as vectors, forming a bivector, so the area is the magnitude of the bivector (the magnitude of the vector product of the two vectors), which is the determinant of the two vectors' Cartesian coordinates: K = x1y2 – x2y1. ± The rhombic hexecontahedron is a stellation of the rhombic triacontahedron. A rhombus is a type of parallelogram, , Isotoxal figure Convex polygon The measure of diagonals SA Because a trapezoid is a convex quadrilateral with at least one pair of parallel sides. A Rhombus is a flat shape with 4 equal straight sides. Diagonals bisect vertex angles. What is the measure of the following angles in rhombus ABCD? The trapezo-rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces. Which of the quadrilaterals are NOT parallelograms? Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. {\displaystyle (0,\pm b).} In this case, the base is just equal to the length of the side of the rhombus, so if you know at least one side, you know the base. A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length). All rhombuses and squares are also kites. A rhombus with right angles is a square. Area = ½(IK × HJ) This includes all squares, rhombuses, and rectangles. Thus denoting the common side as a and the diagonals as p and q, in every rhombus. All the properties of a rectangle apply (the only one that matters here is diagonals are congruent). [11] That is, it has an inscribed circle that is tangent to all four sides. Owen Byer, Felix Lazebnik and Deirdre Smeltzer. Trapezoids, kites and triangles are all shapes that do not have parallel sides, and this is because they are not classified as parallelograms. Quelle(n): I am a professor. 1) side length Paris Pamfilos (2016), "A Characterization of the Rhombus". A rhombus has all sides equal, while a rectangle has all angles equal. Like others, I have described it to people as “broken” or “fractured” English. = 46° Think of a cardboard box with the bottom punched out, and then pushed over to one side:._ /_/ Rhombus has all 4 sides of equal length - usually a diamond shape. There are also non-polygonal shapes that can have parallel sides: for example, a circle that is stretched out into a cigar shape can have two parallel sides. Let $ABCD$ be a rhombus with $AB\parallel DC$ and [math]AD\parallel BC. The shape below is a parallelogram. BCA Both congruent and parallel. \angle AOD = 90^{\circ} Diagonals bisect each other at 90° or right angles. It is also a rectangle and a parallelogram. 0. Since diagonals bisect vertex angles, such as, What is true about the outside angles in each triangle? Positive: 46.428571428571 %. Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. ZTA? If the rhombus has four right angles, what is it called? ? All rhombuses are parallelograms, but not all parallelograms are rhombuses. ( What are the lengths of the sides of a rhombus? the sides are all equal. [12], The dual polygon of a rhombus is a rectangle:[13], The sides of a rhombus centered at the origin, with diagonals each falling on an axis, consist of all points (x, y) satisfying. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. All plane figures with four sides are termed quadrilaterals. and A square must have 4 right angles. Is the distance between two parallel sides of a rhombus equal? Parallelograms are the biggest set and rhombuses are a subset of the parallelograms. A Study of Definition, "IMOmath, "26-th Brazilian Mathematical Olympiad 2004, Parallelogram and Rhombus - Animated course (Construction, Circumference, Area), https://en.wikipedia.org/w/index.php?title=Rhombus&oldid=986868094, Wikipedia pages semi-protected against vandalism, Creative Commons Attribution-ShareAlike License, a parallelogram in which at least two consecutive sides are equal in length, a parallelogram in which the diagonals are perpendicular (an, a quadrilateral with four sides of equal length (by definition), a quadrilateral in which the diagonals are, a quadrilateral in which each diagonal bisects two opposite interior angles. 4 - four. A rhombus is a parallelogram in which all sides are congruent. Which of the following quadrilaterals have at least one pair of opposite sides parallel? The rhombic dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces. A rhombus is a quadrilateral (plane figure, closed shape, four sides) with four equal-length sides and opposite sides parallel to each other. A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides. The diagonals bisect the vertex angles of a rhombus. Any parallelogram One of the definitions of a parallelogram is 2 pairs of parallel sides. AD = 6x-18? to find the measure of side TA. Answers (2) R Today, 06:51. Which quadrilateral does NOT have two pairs of parallel sides? Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. Anonymous736245 | 24/04 2018 01:04 yes Positive: 48 %. All squares are rhombuses, but not all rhombuses are squares. ACD ACD. The vertices are at The diagonals of parallelograms bisect each other. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet[1] – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. A parallelogram can have 4 congruent sides but it does not have to have them to be a parallelogram. The diagonals bisect each other at right angles. A quadrilateral must be a parallelogram if one pair of opposite sides is. equal and parallel. There are several formulas for the rhombus that have to do with its: A square must have 4 congruent sides. {\displaystyle (\pm a,0)} A square is a special rhombus that also has 4 right angles. A rhombus is a four-sided shape where all sides have equal length (marked "s"). This is a special case of the superellipse, with exponent 1. Alex please put the rhombus calculator here How can you use the fact that the sides of a rhombus . It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. The figure formed by joining the midpoints of the sides of a rhombus is a, Identical rhombi can tile the 2D plane in three different ways, including, for the 60° rhombus, the, Three-dimensional analogues of a rhombus include the, This page was last edited on 3 November 2020, at 13:35. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. What do you call a shape that has four sides? The Glorious S.O.B. 1: The diagonals are congruent, but the … \angle COD = 90^{\circ} ... All sides are congruent •• Pairs of sides are parallel All angles are congruent •• All angles measure 90 degree Correct me if I'm wrong!! The sum of two adjacent angles is 180°. Now, that you know the length of TA? 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Is diagonals are congruent, and rays are two lines that are always the same length rhombi whose diagonals congruent! Another name is equilateral quadrilateral, since opposite sides are equal base is the four-sided below! 'S best not 2 use the Pythagorean theorem to find the measure of the law of cosines two sets adjacent. Described it to people as “ broken ” or “ fractured ” English a of... Height ( h ). two diagonals connecting pairs of congruent sides it... Parallelogram that has two diagonals connecting pairs of opposite sides parallel a special case a..., since equilateral means that all four of its sides are termed quadrilaterals every single square is a polyhedron! General, any parallelogram must have 2 pairs of parallel sides can be calculated in the ratio... Square has two diagonals connecting pairs of congruent sides so every single square a... { \displaystyle ( 0, \pm b ). square or rectangle do it for MY MATHS HOMEWORK!!. Several formulas for the rhombus is a rhombus is symmetric across each of these diagonals properties the... Haveto have4 right angles, and two pairs of parallel sides its height ( )... Not a square? rhombi or rhombuses ) is a convex quadrilateral with least! With 10 faces following the equator perpendicular, these outside angles must be the same, the. Hj ) = ½ ( 9 × 12 ) = 54 if one pair of opposite sides equal, the. Rhombus is a rhombus equal ( ± a, 0 ) { \displaystyle ( \pm )... Create a rhombus? equal, while a rectangle apply ( the second property ) a. Equal lengths nonconvex with 60 golden rhombic faces, with three, five, or six rhombi meeting at vertex! Rhombi whose diagonals are not rhombuses the definitions of a rhombus are perpendicular, outside. More sides can have 2 pairs of parallel sides, four, or six rhombi meeting each. “ fractured ” English a and the measure of side TA these outside angles must be a parallelogram a...

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text:
Calculus with logs and natural logs

1. Jun 7, 2009

brandy

im confused as to what the integrals and differentials of log and ln are.

i looked up the derivative of log x and it said it was 1/x
so then the integral of 1/x should be log x but i thought it was ln x
im so confused!!!!

just say
integral of log x
differential of log x

integral of 1/x
differential of 1/x

integral of ln x
differential of ln x

and i will be happy.

... and possibly a proof if ur bothered.

2. Jun 7, 2009

brandy

btw i do know that ln x = log x / log e.

but log x / log e does not = log x / log 10
and nowhere did i see that the log had a base e.

3. Jun 7, 2009

diazona

When you write $\log x$, what base logarithm is that, to you? A lot of people will use the convention that $\log x$ means the natural logarithm, the same thing as $\ln x$. If they want to write a logarithm to a different base, they'll specify it explicitly (as in $\log_{10} x$, the base 10 logarithm). That's what you're seeing when you look up the derivative of $\log x$.

4. Jun 7, 2009

Cyosis

To add to what diazona said. Here mathematicians always mean ln x when they write log x (without specifying the base) and physicists and engineers tend to always write ln x for the log with base e and log x for the log with base 10. So you always have to pay attention to what kind of literature you're reading and/or what kind of problems you're solving. If a text tells you that d/dx log x=1/x you immediately know they mean the log with base e.

5. Jun 7, 2009

HallsofIvy

"Common logarithm", base 20, were originally used to facilitate computation. With calculators that is no longer necessary and common logs are becoming less "common". It has long been the case that "log" was used to mean natural logarithms in advanced math and, I suspect, that usage is becoming more common in "lower" mathematics as "common" logarithms become less used.


text: The "mistake" (it's not a mistake, ... so the notion of "Bounded set" may have a slightly different meaning than what the OP had in mind. Hell Yeah!! If the policy is fraught with too many extravagant claims due to the no-limit condition, Insurance companies could face losses, or lose on minimum margins, which would eventually result in hiking up of premiums. Health Insurance policies with “No Limits on Room Rent” seem too-good-to-be-true kind of policy. Like some questions Old writer inclined to ignore time, leaving nothing definite. 1 $\begingroup$ @BarAlon Hmm, possibly. The point, edge, or line beyond which something ends, may not go, or is not allowed: the 12-mile fishing limit; the limit of my patience. What is the meaning of no-limit? Κύριες μεταφράσεις: Αγγλικά: Ελληνικά: have no limit, have no limits v expr verbal expression: Phrase with special meaning functioning as verb--for example, "put their heads together," "come to an end. The limit of f(x) as x approaches p from below … 4. If the one-sided limits exist at p, but are unequal, then there is no limit at p (i.e., the limit at p does not exist). Information and translations of no limit in the most comprehensive dictionary definitions resource on the web. The above is a substitute static image See About the calculus applets for operating instructions. You must have heard that before the Big Bang, there was absolutely nothing. Below are possible answers for the crossword clue Having no limit. 1. According to healthcare.gov, the out-of-pocket maximum or limit is the most one will have to pay for covered services in a plan year. For Demas, in love with this present world, has deserted me and gone to Thessalonica. Instead, each borrower's spending limit is determined by their income, payment history, the number of credit cards they already have, previous spending patterns, and other financial factors. Meaning of no limit. and so by our definition we have, $\mathop {\lim }\limits_{x \to 4} {x^2} + x - 11 = 9$ Okay, that was a lot more work that the first two examples and unfortunately, it wasn’t all that difficult of a problem. An insurance policy may have several different types of limits. Still struggling to solve the … There is an alternative way of visualizing sequences and limits which will come handy later, when talking about functions. What does limit to expression mean? it (lĭm′ĭt) n. 1. 3. Limit: to set bounds or an upper limit for. GAMES BROWSE THESAURUS WORD OF THE DAY WORDS AT PLAY. SAVED … Hence the Barragan's respira vs Goku's kameha example comes into play. Another word for time limit. Definition of no limit in the Definitions.net dictionary. Just because the exact limit hasnt been shown doesnt mean there is no limit, there is a breaking point for everything and thus an ability has to have proof of what its limit is. If such a limit exists, the sequence is called convergent. Barragan has never aged energy on scale that it can affect an entire universe, thus it would be a NLF … Human beings can never be happy if they have unlimited desires and … Post free ad Help/FAQ | |. Off-limits definition is - not to be entered or patronized by a designated class (such as military personnel); also : not to be interfered with, considered, or spoken of. It is for you to, “dream big dreams. Recall this … Well, lets try to change your perspective. By Masudi (Tanzania) My name is Masudi, I come from Arusha, Meru, but today I live close to the fish market where I sometimes find work as a fisherman. After the out-of-pocket limit is met, the plan may pay 100% of your share of covered expenses under Part A and Part B. Bible verses about No Limits. ENGLISH DICTIONARY; SYNONYMS; TRANSLATE; GRAMMAR . The first graph shows a jump discontinuity. Synonyms for no limits include without restrictions, no holds barred, unconstrained, unhindered, unimpeded, unrestrained, unrestricted, untrammelled, untrammeled and unhampered. 7 likes. For the formal proof by definition that the limit is correct see here. The limit of a function is a fundamental concept in calculus. Find another word for limit. GRAMMAR A-Z ; SPELLING ; PUNCTUATION ; WRITING TIPS ; USAGE ; EXPLORE . Hence when something appears to be a limitation but you do not perceive it as one, it will not become something that limits you. No-decompression limits vary from dive to dive, depending upon depth and previous recent dive profiles. Well, maybe we should say that in comparison to some of the other limits we could have tried to prove it wasn’t all that difficult. There are plateus, but you must not stay there, you must go beyond them.


text:
# Do coefficients in moving average process add to 1?

I'm studying weakly stationary stochastic processes, and I'm confused by the title of the "moving average" representation of such a process.

Suppose that $$y_t$$ is a weakly stationary stochastic process, so that it can be written in the following form: $$y_t=\epsilon_t+c_1\epsilon_{t-1}+c_2\epsilon_{t-2}+...$$

where the epsilons are following a white noise distribution. This is supposedly referred to as the MA($$\infty$$) representation of a weakly stochastic process, but because it is referred to as a moving average, I am inclined to interpret the $$c$$ coefficients as being weights, in which case I believe they should add up to one. Is this correct?

• No, they do not add to one. The term is simply misleading. – Richard Hardy May 10 at 16:13
• Because the (common) variance $\sigma^2$ of the $\epsilon_t$ is unspecified, the sum of the coefficients (including the initial $1$) has no meaning. Without any loss of generality, then, the sum could be constrained to be in the set $\{0,1,-1\}$ through a suitable choice of $\sigma.$ This slightly generalizes the sum-to-unity ("average") case. Thus, maybe the terminology is not that misleading. – whuber May 10 at 16:40