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• ### Spins of primordial black holes formed in the matter-dominated phase of the Universe(1707.03595)

Angular momentum plays very important roles in the formation of PBHs in the matter-dominated phase if it lasts sufficiently long. In fact, most collapsing masses are bounced back due to centrifugal force, since angular momentum significantly grows before collapse. As a consequence, most of the formed PBHs are rapidly rotating near the extreme value $a_{*}=1$, where $a_{*}$ is the nondimensional Kerr parameter at their formation. The smaller the density fluctuation $\sigma_{H}$ at horizon entry is, the stronger the tendency towards the extreme rotation. Combining the effect of angular momentum with that of anisotropy, we estimate the black hole production rate. We find that the production rate suffers from suppression dominantly due to angular momentum for a smaller value of $\sigma_{H}$, while due to anisotrpopy for a larger value of $\sigma_{H}$. We argue that matter domination significantly enhances the production of PBHs despite the suppression. If the matter-dominated phase does not last so long, the effect of the finite duration significantly suppresses PBH formation and weakens the tendency towards large spins. (abridged)
• ### Non-linear collisional Penrose process: How large energy can a black hole release?(1708.04003)

Aug. 14, 2017 gr-qc
Energy extraction from a rotating or charged black hole is one of fascinating issues in general relativity. The collisional Penrose process is one of such extraction mechanisms and has been reconsidered intensively since Banados, Silk and West pointed out the physical importance of very high energy collisions around a maximally rotating black hole. In order to get results analytically, the test particle approximation has been adopted so far. Successive works based on this approximation scheme have not yet revealed the upper bound on the efficiency of the energy extraction because of lack of the back reaction. In the Reissner-Nordstrom spacetime, by fully taking into account the self-gravity of the shells, we find that there is an upper bound on the extracted energy, which is consistent with the area law of a black hole. We also show one particular scenario in which the almost maximum energy extraction is achieved even without the Banados-Silk-West collision.
• ### On the stability of a superspinar(1707.07242)

July 23, 2017 gr-qc
The superspinar proposed by Gimon and Horava is a rapidly rotating compact entity whose exterior is described by the over-spinning Kerr geometry. The compact entity itself is expected to be governed by superstringy effects, and in astrophysical scenarios it can give rise to interesting observable phenomena. Earlier it was suggested that the superspinar may not be stable but we point out here that this does not necessarily follow from earlier studies. We show, by analytically treating the Teukolsky equations by Detwiler's method, that in fact there are infinitely many boundary conditions that make the superspinar stable, and that the modes will decay in time. It follows that we need to know more on the physical nature of the superspinar in order to decide on its stability in physical reality.
• ### Non-linear stability of a brane wormhole(1704.02747)

April 27, 2017 gr-qc
We analytically study the non-linear stability of a spherically symmetric wormhole supported by an infinitesimally thin brane of negative tension, which has been devised by Barcelo and Visser. We consider a situation in which a thin spherical shell composed of dust falls into an initially static wormhole; The dust shell plays a role of the non-linear disturbance. The self-gravity of the falling dust shell is completely taken into account through Israel's formalism of the metric junction. When the dust shell goes through the wormhole, it necessarily collides with the brane supporting the wormhole. We assume the interaction between these shells is only gravity and show the condition under which the wormhole stably persists after the dust shell goes through it.
• ### Can we remove the systematic error due to isotropic inhomogeneities?(1607.02221)

March 3, 2017 gr-qc, astro-ph.CO
Usually, we assume that there is no inhomogeneity isotropic in terms of our location in our uni- verse. This assumption has not been observationally confirmed yet in sufficient accuracy, and we need to consider the possibility that there are non-negligible large-scale isotropic inhomogeneities in our universe. The existence of large-scale isotropic inhomogeneities affects the determination of the cosmological parameters. In particular, from only the distance-redshift relation, we can not dis- tinguish the inhomogeneous isotropic universe model from the homogeneous isotropic one, because of the ambiguity in the cosmological parameters. In this paper, in order to avoid such ambiguity, we consider three observables, the distance-redshift relation, the fluctuation spectrum of the cosmic microwave background radiation(CMBR) and the scale of the baryon acoustic oscillation(BAO), and compare these observables in two universe models; One is the inhomogeneous isotropic uni- verse model with the cosmological constant and the other is the homogeneous isotropic universe model with the dark energy other than the cosmological constant. We show that these two universe models can not predict the same observational data of all three observables but the same ones of only two of three, as long as the perturbations are adiabatic. In principle, we can distinguish the inhomogeneous isotropic universe from the homogeneous isotropic one through appropriate three observables, if the perturbations are adiabatic.
• ### Primordial black hole formation in the matter-dominated phase of the Universe(1609.01588)

Nov. 7, 2016 hep-th, gr-qc, astro-ph.CO
We investigate primordial black hole formation in the matter-dominated phase of the Universe, where nonspherical effects in gravitational collapse play a crucial role. This is in contrast to the black hole formation in a radiation-dominated era. We apply the Zel'dovich approximation, Thorne's hoop conjecture, and Doroshkevich's probability distribution and subsequently derive the production probability $\beta_{0}$ of primordial black holes. The numerical result obtained is applicable even if the density fluctuation $\sigma$ at horizon entry is of the order of unity. For $\sigma\ll 1$, we find a semi-analytic formula $\beta_{0}\simeq 0.05556 \sigma^{5}$, which is comparable with the Khlopov-Polnarev formula. We find that the production probability in the matter-dominated era is much larger than that in the radiation-dominated era for $\sigma\lesssim 0.05$, while they are comparable with each other for $\sigma\gtrsim 0.05$. We also discuss how $\sigma$ can be written in terms of primordial curvature perturbations.
• ### Orbital angular momentum of scalar field generated by gravitational scatterings(1606.09217)

June 30, 2016 gr-qc
It has been expected that astronomical observations to detect the orbital angular momenta of electromagnetic waves may give us a new insight into astrophysics. Previous works pointed out the possibility that a rotating black hole can produce orbital angular momenta of electromagnetic waves through gravitational scattering, and the spin parameter of the black hole can be measured by observing them. However, the mechanism how the orbital angular momentum of the electromagnetic wave is generated by the gravitational scattering has not been clarified sufficiently. In this paper, in order to understand it from a point of view of gravitational lensing effects, we consider an emitter which radiates a spherical wave of the real massless scalar field and study the deformation of the scalar wave by the gravitational scattering due to a black hole by invoking the geometrical optics approximation. We show that the frame dragging caused by the rotating black hole is not a necessary condition for generating the orbital angular momentum of the scalar wave. However, its components parallel to the direction cosines of images appear only if the black hole is rotating.
• ### Infinite efficiency of collisional Penrose process: Can over-spinning Kerr geometry be the source of ultra-high-energy cosmic rays and neutrinos ?(1510.08205)

May 10, 2016 hep-th, hep-ph, gr-qc, astro-ph.HE
The origin of the ultra-high-energy particles we receive on the Earth from the outer space such as EeV cosmic rays and PeV neutrinos remains an enigma. All mechanisms known to us currently make use of electromagnetic interaction to accelerate charged particles. In this paper we propose a mechanism exclusively based on gravity rather than electromagnetic interaction. We show that it is possible to generate ultra-high-energy particles starting from particles with moderate energies using the collisional Penrose process in an overspinning Kerr spacetime transcending the Kerr bound only by an infinitesimal amount, i.e., with the Kerr parameter $a=M(1+\epsilon)$, where we take the limit $\epsilon \rightarrow 0^+$. We consider two massive particles starting from rest at infinity that collide at $r=M$ with divergent center-of-mass energy and produce two massless particles. We show that massless particles produced in the collision can escape to infinity with the ultra-high energies exploiting the collisional Penrose process with the divergent efficiency $\eta \sim {1}/{\sqrt{\epsilon}} \rightarrow \infty$. Assuming the isotropic emission of massless particles in the center-of-mass frame of the colliding particles, we show that half of the particles created in the collisions escape to infinity with the divergent energies. To a distant observer, ultra-high-energy particles appear to originate from a bright spot which is at the angular location $\xi \sim {2M}/{r_{obs}}$ with respect to the singularity on the side which is rotating towards the observer. We show that the anisotropy in emission in the center-of-mass frame, which is dictated by the differential cross-section of underlying particle physics process, leaves a district signature on the spectrum of ultra-high-energy massless particles. Thus, it provides a unique probe into fundamental particle physics.
• ### Timescale for trans-Planckian collisions in Kerr spacetime(1503.08331)

May 15, 2015 hep-th, gr-qc, astro-ph.HE
We make a critical comparison between ultra-high energy particle collisions around an extremal Kerr black hole and that around an over-spinning Kerr singularity, mainly focusing on the issue of the timescale of collisions. We show that the time required for two massive particles with the proton mass or two massless particles of GeV energies to collide around the Kerr black hole with Planck energy is several orders of magnitude longer than the age of the Universe for astro-physically relevant masses of black holes, whereas time required in the over-spinning case is of the order of ten million years which is much shorter than the age of the Universe. Thus from the point of view of observation of Planck scale collisions, the over-spinning Kerr geometry, subject to their occurrence, has distinct advantage over their black hole counterparts.
• ### Systematic error due to isotropic inhomogeneities(1505.02472)

May 11, 2015 gr-qc, astro-ph.CO
Usually the effects of isotropic inhomogeneities are not seriously taken into account in the determination of the cosmological parameters because of Copernican principle whose statement is that we do not live in the privileged domain in the universe. But Copernican principle has not been observationally confirmed yet in sufficient accuracy, and there is the possibility that there are non-negligible large-scale isotropic inhomogeneities in our universe. In this paper, we study the effects of the isotropic inhomogeneities on the determination of the cosmological parameters and show the probability that non-Copernican isotropic inhomogeneities mislead us into believing, for example, the phantom energy of the equation of state, $p=w\rho$ with $w<-1$, even in case that $w=-1$ is the true value.
• ### How small can an over-spinning body be in general relativity?(1406.6798)

Dec. 15, 2014 hep-th, gr-qc
The angular momentum of the Kerr singularity should not be larger than a threshold value so that it is enclosed by an event horizon: The Kerr singularity with the angular momentum exceeding the threshold value is naked. This fact suggests that if the cosmic censorship exists in our Universe, an over-spinning body without releasing its angular momentum cannot collapse to spacetime singularities. A simple kinematical estimate of two particles approaching each other supports this expectation and suggests the existence of a minimum size of an over-spinning body. But this does not imply that the geometry near the naked singularity cannot appear. By analyzing initial data, i.e., a snapshot of a spinning body, we see that an over-spinning body may produce a geometry close to the Kerr naked singularity around itself at least as a transient configuration.
• ### Comparison of two approximation schemes for solving perturbations in a LTB cosmological model(1407.4899)

Oct. 8, 2014 gr-qc, astro-ph.CO
Recently, the present authors studied perturbations in the Lemaitre-Tolman-Bondi cosmological model by applying the second-order perturbation theory in the dust Friedmann-Lemaitre-Robertson-Walker universe model. Before this work, the same subject was studied in some papers by analyzing linear perturbations in the Lemaitre-Tolman-Bondi cosmological model under the assumption proposed by Clarkson, Clifton and February, in which two of perturbation variables are negligible. However, it is a non-trivial issue in what situation the Clarkson-Clifton-February assumption is valid. In this paper, we investigate differences between these two approaches. It is shown that, in general, these two approaches are not compatible with each other. That is, in our perturbative procedure, the Clarkson-Clifton-February assumption is not valid at the order of our interest.
• ### Newtonian self-gravitating system in a relativistic huge void universe model(1409.2099)

Oct. 1, 2014 gr-qc, astro-ph.CO
We consider a test of the Copernican Principle through observations of the large-scale structures, and for this purpose we study the self-gravitating system in a relativistic huge void universe model which does not invoke the Copernican Principle. If we focus on the the weakly self-gravitating and slowly evolving system whose spatial extent is much smaller than the scale of the cosmological horizon in the homogeneous and isotropic background universe model, the cosmological Newtonian approximation is available. Also in the huge void universe model, the same kind of approximation as the cosmological Newtonian approximation is available for the analysis of the perturbations contained in a region whose spatial size is much smaller than the scale of the huge void: the effects of the huge void are taken into account in a perturbative manner by using the Fermi-normal coordinates. By using this approximation, we derive the equations of motion for the weakly self-gravitating perturbations whose elements have relative velocities much smaller than the speed of light, and show the derived equations can be significantly different from those in the homogeneous and isotropic universe model, due to the anisotropic volume expansion in the huge void. We linearize the derived equations of motion and solve them. The solutions show that the behaviors of linear density perturbations are very different from those in the homogeneous and isotropic universe model.
• ### Black Hole Universe: Time Evolution(1306.1389)

Sept. 26, 2013 gr-qc, astro-ph.CO
Time evolution of a black hole lattice universe is simulated. The vacuum Einstein equations in a cubic box with a black hole at the origin are numerically solved with periodic boundary conditions on all pairs of opposite faces. Defining effective scale factors by using the area of a surface and the length of an edge of the cubic box, we compare them with that in the Einstein-deSitter universe. It is found that the behaviour of the effective scale factors is well approximated by that in the Einstein-deSitter universe. Our result suggests that local inhomogeneities do not significantly affect the global expansion law of the universe even if the inhomogeneity is extremely nonlinear.
• ### What wormhole is traversable?: A case of a wormhole supported by a spherical thin shell(1306.6917)

June 28, 2013 gr-qc
We analytically explore the effect of falling matter on a spherically symmetric wormhole supported by a spherical shell composed of exotic matter located at its throat. The falling matter is assumed to be also a thin spherical shell concentric with the shell supporting the wormhole, and its self-gravity is completely taken into account. We treat these spherical thin shells by Israel's formalism of metric junction. When the falling spherical shell goes through the wormhole, it necessarily collides with the shell supporting the wormhole. To treat this collision, we assume the interaction between these shells is only gravity. We show the conditions on the parameters that characterize this model in which the wormhole persists after the spherical shell goes through it.
• ### Two-point correlation function of density perturbations in a large void universe(1306.5131)

June 21, 2013 gr-qc, astro-ph.CO
We study the two-point correlation function of density perturbations in a spherically symmetric void universe model which does not employ the Copernican principle. First we solve perturbation equations in the inhomogeneous universe model and obtain density fluctuations by using a method of non-linear perturbation theory which was adopted in our previous paper. From the obtained solutions, we calculate the two-point correlation function and show that it has a local anisotropy at the off-center position differently from those in homogeneous and isotropic universes. This anisotropy is caused by the tidal force in the off-center region of the spherical void. Since no tidal force exists in homogeneous and isotropic universes, we may test the inhomogeneous universe by observing statistical distortion of the two-point galaxy correlation function.
• ### Ultra-high energy collision with neither black hole nor naked singularity(1301.4618)

March 23, 2013 gr-qc
We explore the collision between two concentric spherical thin shells. The inner shell is charged, whereas the outer one is either neutral or charged. In the situation we consider, the charge of the inner shell is larger than its gravitational mass, and the inside of it is empty and regular. Hence the domain just outside it is described by the overcharged Reissner-Nordstrom geometry whereas the inside of it is Minkowski. First, the inner shell starts to shrink form infinity with finite kinetic energy, and then the outer shell starts to shrink from infinity with vanishing kinetic energy. The inner shell bounces on the potential wall and collides with the ingoing outer shell. The energy of collision between these shells at "their center of mass frame" does not exceed the total energy of the system. By contrast, by virtue of the very large gamma factor of the relative velocity of the shells, the energy of collision between two of the constituent particles of these shells at their center of mass frame can be much larger than the Planck scale. This result suggests that the black hole or naked singularity is not necessary for ultra-high energy collision of particles.
• ### Acceleration of particles and shells by Reissner-Nordstr\"{o}m naked singularities(1108.0288)

Sept. 20, 2012 hep-ph, gr-qc, astro-ph.HE
We explore the Reissner-Nordstr\"{o}m naked singularities with a charge $Q$ larger than its mass $M$ from the perspective of the particle acceleration. We first consider a collision between two test particles following the radial geodesics in the Reissner-Nordstr\"{o}m naked singular geometry. An initially radially ingoing particle turns back due to the repulsive effect of gravity in the vicinity of naked singularity. Such a particle then collides with an another radially ingoing particle. We show that the center of mass energy of collision taking place at $r \approx M$ is unbound, in the limit where the charge transcends the mass by arbitrarily small amount $0<1-M/Q\ll1$.The acceleration process we described avoids fine tuning of the parameters of the particle geodesics for the unbound center of mass energy of collisions and the proper time required for the process is also finite. We show that the coordinate time required for the trans-Plankian collision to occur around one solar mass naked singularity is around million years while it is many orders of magnitude larger than Hubble time in the black hole case. We then study the collision of the neutral spherically symmetric shells made up of dust particles. In this case, it is possible to treat the situation by exactly taking into account the gravity due to the shells using Israel`s thin shell formalism, and thus this treatment allows us to go beyond the test particle approximation. The center of mass energy of collision of the shells is then calculated in a situation analogous to the test particle case and is shown to be bounded above. However, we find thatthe energy of a collision between two of constituent particles of the shells at the center of mass frame can exceed the Planck energy.
• ### Black Hole Universe: Construction and Analysis of Initial Data(1204.2411)

June 27, 2012 gr-qc, astro-ph.CO
We numerically construct an one-parameter family of initial data of an expanding inhomogeneous universe model which is composed of regularly aligned black holes with an identical mass. They are initial data for vacuum solutions of the Einstein equations. We call this universe model the "black hole universe" and analyze the structure of these initial data. We study the relation between the mean expansion rate of the 3-space, which corresponds to the Hubble parameter, and the mass density of black holes. The result implies that the same relation as that of the Einstein-de Sitter universe is realized in the limit of the large separation between neighboring black holes. The applicability of the cosmological Newtonian $N$-body simulation to the dark matter composed of black holes is also discussed. The deviation of the spatial metric of the cosmological Newtonian $N$-body system from that of the black hole universe is found to be smaller than about 1% in a region distant from the particles, if the separation length between neighboring particles is 20 times larger than their gravitational radius. By contrast, the deviation of the square of the Hubble parameter of the cosmological Newtonian $N$-body system from that of the black hole universe is about 20% for the same separation length.
• ### Evolution of density perturbations in large void universe(1202.1582)

June 6, 2012 gr-qc, astro-ph.CO
We study the evolution of linear density perturbations in a large spherical void universe which accounts for the acceleration of the cosmic volume expansion without introducing dark energy. The density contrast of this void is not large within the light cone of an observer at the center of the void. Therefore, we describe the void structure as a perturbation with a dimensionless small parameter $\kappa$ in a homogeneous and isotropic universe within the region observable for the observer. We introduce additional anisotropic perturbations with a dimensionless small parameter $\epsilon$, whose evolution is of interest. Then, we solve perturbation equations up to order $\kappa \epsilon$ by applying second-order perturbation theory in the homogeneous and isotropic universe model. By this method, we can know the evolution of anisotropic perturbations affected by the void structure. We show that the growth rate of the anisotropic density perturbations in the large void universe is significantly different from that in the homogeneous and isotropic universe. This result suggests that the observation of the distribution of galaxies may give a strong constraint on the large void universe model.
• ### Dynamical instability in a relativistic cylindrical shell composed of counter rotating particles(1112.4252)

Dec. 19, 2011 gr-qc
We give a perturbative analysis for an infinitesimally thin cylindrical shell composed of counter rotating collisionless particles, originally devised by Apostolatos and Thorne. They found a static solution of the shell and concluded by C-energy argument that it is stable. Recently, the present authors and Ida reanalyzed this system by evaluating the C-energy on the future null infinity and found that the system has an instability, though it was not shown how the system is unstable. In this paper, it is shown in the framework of the linear perturbation theory that, if the constituent particles move slowly, the static shell is unstable in the sense that the perturbation of its circumferential radius oscillates with exponentially growing amplitude, whereas if the speed of the constituent particle exceeds a critical value, the shell just expands or contracts exponentially with time.
• ### Q-tubes and Q-crusts(1011.4828)

We explore equilibrium solutions of non-topological solitons in a general class of scalar field theories which include global U(1) symmetry. We find new types of solutions, tube-shaped and crust-shaped objects, and investigate their stability. Like Q-balls, the new solitons can exist in supersymmetric extensions of the Standard Model, which may responsible for baryon asymmetry and dark matter. Therefore, observational signals of the new solitons would give us more informations on the early universe and supersymmetric theories.
• ### Is super-Planckian physics visible? -- Scattering of black holes in 5 dimensions(1105.3331)

May 17, 2011 gr-qc
It may be widely believed that probing short-distance physics is limited by the presence of the Planck energy scale above which scale any information is cloaked behind a horizon. If this hypothesis is correct, we could observe quantum behavior of gravity only through a black hole of Planck mass. We numerically show that in a scattering of two black holes in the 5-dimensional spacetime, a visible domain, whose curvature radius is much shorter than the Planck length, can be formed. Our result indicates that super-Planckian phenomena may be observed without an obstruction by horizon formation in particle accelerators.
• ### Acceleration of colliding shells around a black hole: Validity of the test particle approximation in the Banados-Silk-West process(1010.5438)

Feb. 5, 2011 hep-th, gr-qc, astro-ph.CO
Recently, Banados, Silk and West (BSW) showed that the total energy of two colliding test particles has no upper limit in their center of mass frame in the neighborhood of an extreme Kerr black hole, even if these particles were at rest at infinity in the infinite past. We call this mechanism the BSW mechanism or BSW process. The large energy of such particles would generate strong gravity, although this has not been taken into account in the BSW analysis. A similar mechanism is seen in the collision of two spherical test shells in the neighborhood of an extreme Reissner-Nordstr\"om black hole. In this paper, in order to draw some implications concerning the effects of gravity generated by colliding particles in the BSW process, we study a collision of two spherical dust shells, since their gravity can be exactly treated. We show that the energy of two colliding shells in the center of mass frame observable from infinity has an upper limit due to their own gravity. Our result suggests that an upper limit also exists for the total energy of colliding particles in the center of mass frame in the observable domain in the BSW process due the gravity of the particles.
• ### Perturbative Analysis of a Stationary Magnetosphere in an Extreme Black Hole Spacetime : On the Meissner-like Effect of an Extreme Black Hole(1010.4104)

Oct. 20, 2010 hep-th, gr-qc, astro-ph.CO
It is known that the Meissner-like effect is seen in a magnetosphere without an electric current in black hole spacetime: no non-monopole component of magnetic flux penetrates the event horizon if the black hole is extreme. In this paper, in order to see how an electric current affects the Meissner-like effect, we study a force-free electromagnetic system in a static and spherically symmetric extreme black hole spacetime. By assuming that the rotational angular velocity of the magnetic field is very small, we construct a perturbative solution for the Grad-Shafranov equation, which is the basic equation to determine a stationary, axisymmetric electromagnetic field with a force-free electric current. Our perturbation analysis reveals that, if an electric current exists, higher multipole components may be superposed upon the monopole component on the event horizon, even if the black hole is extreme.


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• Abstract

# A Segmental Bus-Invert Coding Method for Instruction Memory Data Bus Power Efficiency

This paper presents a bus coding methodology for instruction memory data-bus switching reduction. Compared to the existing state of the art multi-way partial bus-invert coding (MPBI) which relies on data bit-correlation, our approach is very effective in reducing switching activity for bus data, since little correlation can be observed on instruction data buses. Our experiments demonstrate the proposed encoding can reduce up to 42% of switching activity, with an average of 30% reduction, while MPBI achieves just 17.6% reduction in switching activity.

SECTION I

## Introduction

DESIGNS of portable consumer electronic devices such as mobile phones, PDAs, video games and other embedded systems are increasingly demanding low power consumption to maximize the battery life, reduce weight and improve reliability. These types of power sensitive devices are usually equipped with microprocessors as the processing elements and memories as the storage units. With current CMOS technology, a large portion of power consumption is consumed in the form of dynamic power1, which in turn is determined by the bit switching and the switched load capacitance. Since the microprocessor fetches instructions over the memory bus every clock cycle and bus lines to memory are often much longer than buses within the processor, the power consumed by the bus due to instruction fetch is significant.

In this paper, we target a system consisting of a processor with Harvard architecture, where instruction memory and data memory are separated and each memory has different buses for address and data transmission. We want to reduce switching on the instruction data bus.

Research for instruction data bus switching reduction has generally concentrated on code compression. The compressed code causes less memory access thus reducing the bus activity. Compression requires complicated compression/decompression units, which reside in the critical path and can considerably affect the overall system performance. In this paper, we investigate a different approach—bus encoding.

Most of existing bus encoding schemes are effective for address or data memory buses, and mainly utilize correlations of transferred data. For example, T0 [2] and Gray encodings [3] use the temporal correlation of data on address buses, while the bus-invert [4] exploits the spatial transition correlation among the data bits.

We investigated the data on the instruction data buses and found that the bit switching behavior of the instruction data bus is different from the other types of buses. Fig. 1 shows an experimental result of the bit transition probability for three different memory buses: instruction memory address bus (imab), instruction memory data bus (imdb) and data memory data bus (dmdb) (all over the 32-bit bus space). The SimpleScalar PISA [5] and the application of audio encoding [6] have been used for our investigation.

Fig. 1. Bit Switching Probability of Different Buses.

As can be seen from Fig. 1, switching activity on the instruction address bus concentrates on the low section of bits, largely due to the sequential access of instruction memory. For the data memory data bus, the switching activity spreads over all bus bits with almost 50% switching probability. But for the instruction data bus, the switching probability is not evenly distributed. Some bits show very low switching activity. Therefore, most of existing encodings for address buses and data memory data buses do not suit for encoding of instruction data buses.

Since there are some bits on the instruction buses with high switching frequency, it is possible to use segmental bus-invert encoding—a set of bus segments are selected and to each segment the traditional bus-invert (BI) encoding is performed such that the bus switching activity can be reduced.

We further investigated the bit correlation of the instruction data and found that there is little correlation in the instruction data, as is illustrated by our experimental results shown in Fig. 2, which gives the percentage of bit pairs of the instruction data buses (and the address buses for comparison) in different correlation coefficient ranges. The bigger the coefficient, the higher the correlation of a two-bit pair has. The figure shows that over 80% of bits pairs on the instruction data bus are hardly correlated, with the correlation coefficient below 0.3. In comparison, the address bus data are highly correlated, with about 60% of the bus bit pairs having correlation coefficient over 0.3. Therefore, approaches that are based on the correlation of bit pairs are not effective for the instruction data bus switching reduction.

Fig. 2. Bus Bit Correlation: Instruction Data Buses vs Address Buses. X-axis: correlation coefficients range, Y-axis: percentage of bus bit pairs.

In this paper, we develop a segmental bus-invert coding (SBI) and a fast segment searching algorithm to effectively reduce instruction data bus switching with as small hardware overhead as possible. Our main contributions are:

1. an analytical model of bus switching reduction for bus segments with the bus-invert encoding, and

2. a fast segment search algorithm using the instruction-field based search space partition and the Hamming distance (HD) of bus segments.

The rest of the paper is organized as follows. Section II reviews some existing bus coding schemes for low power system design. Section III analyzes the effect of bus-invert encoding on switching reduction and area cost, based on which we propose the segmental bus encoding design in Section IV. Section V presents the experimental setup, followed by the simulation results and related discussions. And finally the paper is concluded in Section VI.

SECTION II

## Related Work

Bus encoding techniques for low power consumption have been studied in the last couple of decades.

The Gray encoding [3] was proposed for the instruction address bus where binary addresses are converted into Gray code for bus transmission. When instructions are sequentially executed, the address bus has only one bit flip per instruction.

Another approach [2] for address bus encoding is the asymptotic zero-transition activity encoding, known as T0. For the instructions of a program to be executed sequentially without any branches, T0 can ideally achieve zero bus switching.

In [7], Henkel and Lekatsas presented an adaptive address bus encoding (A2 BC) for low power address buses in the deep sub-micron design, where the coupling effects of bus lines were considered.

Stan and Burleson [4] proposed the bus-invert (BI) coding for bus switching activity reduction. This method uses either the original or the inverted value to encode the data bus. If the current value to be sent over the bus causes more than half of the bus bits to switch, its inverted value will be transferred on the bus. An extra invert control line is required to indicate whether the data is inverted or not. This approach achieves a good switching reduction if the transferred data are random and evenly distributed over the whole data range. For the wider data bus without evenly distributed random distributed data, the same authors proposed a partitioned bus-invert coding, partitioning the wider bus into several narrower sub-buses and applying the BI encoding to each sub-bus. This partitioning approach improves the switching reduction at the cost of extra invert control lines.

The partitioned bus-invert approach has been modified and proposed as partial bus-invert (PBI) coding [8] for the address bus. The approach selects and encodes a subgroup of bus lines that are correlated and frequently switched. In the same paper, they extended this approach to multi-way partial bus-invert (MPBI), where highly correlated bus lines were clustered into multiple sub-buses and each of them was encoded independently.

A dictionary-based approach to reduce the data bus power consumption has been introduced in [9]. This approach exploits frequent data patterns detected from the application trace and uses two synchronized dictionaries on both sides of the bus. The dictionaries cache recently transferred data so that the same data that can be accessed in the local dictionary will not be transferred on the bus to reduce bus switching activity.

For the instruction bus power reduction, most previous researchers have focused on code compression. The pioneer work by Wolfe and Chainin [10], mainly aimed for program memory reduction. With their approach, the total bus switching activity can be reduced via compressed code that are transferred over the bus. A decompression unit is required to restore each instruction before execution.

Scheme in [11] also compresses instructions and compacts more compressed instructions into one bus word to reduce the total number of memory access, hence the total number of bus switches. This code compression scheme was recently extended in [12] to further reduce switching between consecutive instruction words.

Petrov and Orailoglu [13] introduced an instruction bus encoding, where the major loops are encoded and stored in the memory so that when they are fetched, the switching activity on the bus is minimized. This approach can achieve good switching reduction but requires a complex code transformation and control in the decoding logic.

The bus encoding problem was even generalized by Ramprasad et al. in [14]. In this paper, the authors present an encoding framework where an encoding can be abstracted as a two-step process: decorrelating and encoding. Data to be transferred over the bus is first decorrelated for high entropy, which then leads to small encoding code and minimal bus bit switchings.

In this paper, we propose a bus encoding for instruction data buses. Our approach is similar to the PBI/MPBI approach in that we both apply the bus invert (BI) encoding to a set of sub-buses. But there exists a major difference: their approach to finding bus sub-sets for BI application is based on data bit-pair correlations. We found that there is very little bit-pair correlation on instruction data; therefore, their approach is not effective for instruction data bus switching reduction. We propose a segment search algorithm based on Hamming distance to achieve a better result, as will be demonstrated in our results in Section V.

SECTION III

## Bus Invert Encoding

The effectiveness of our approach is closely related to the segments selected for the bus encoding. Therefore, we first study the effect of BI encoding on switching reduction and the hardware overhead, which leads to a search criteria for our design space exploration.

### A. Switching Reduction Rate With BI Encoding

For a sequence of w-bit code words, assume their Hamming distances are {h1, h2,…,hn}, TeX Source $$h_i =\sum^w_{j=1} s_{(i-1)_j}\oplus s_{ij},$$where n is the length of the code sequence, sij the jth bit of the word i (denoted by si) in the sequence. The number of bit switches (SA) if the sequence is transferred on the bus, is TeX Source $$SA = \sum^n_{i=1}h_i.\eqno{\hbox{(1)}}$$When BI is applied to this sequence, some words will be bit-inverted, if their Hamming distances are larger than w/2, the half of word width. The associated Hamming distances will be changed accordingly. For a word, si, if si−1 has been inverted, the new Hamming distance is2 TeX Source $$\sum^w_{j=1}\overline{s_{(i-1)_j}}\oplus s_{ij} = w - h_i.$$Therefore the Hamming distance of word si can be generalized as TeX Source $$H_i = c_{i-1}(w - h_i) + (1-c_{i-1})h_i,\eqno{\hbox{(2)}}$$where ci−1 is the invert control of the previous transfer; when it equals 1, the previous transferred value has been bit inverted.

The total bit switching saving (SAsave) is TeX Source $$SA_{save} = \sum^n_{i=1}((2h_i -w)c_i-c_i).\eqno{\hbox{(3)}}$$The switching reduction rate (r) is TeX Source $$r = SA_{save}/SA {\sum^n_{i=1}((2h_i -w)c_i-c_i) \over \sum^n_{i=1}h_i}.\eqno{\hbox{(4)}}$$where ci = 1, when hi > w/2, otherwise, ci = 0.

As can be see from Formula 4, when the Hamming distance of each word in the sequence is close to the maximum value, w, the reduction rate is close to 100%. If the average HD, E(HD), is around w/2, a larger deviation of HD, Dev(HD), can achieve a better reduction. If the average HD is small and E(HD) + Dev(HD) ≤ w/2, the reduction rate becomes very small. Therefore, we use TeX Source $$\delta = E(HD) + Dev(HD),\eqno{\hbox{(5)}}$$as a criterion parameter in searching instruction word segments for BI encoding. For a segment to be selected for BI encoding, we want δ > w/2 and δ to be as big as possible.

### B. Bus-Invert Control Logic

For each segment to be applied with bus-invert encoding, there needs to be some control logic for bus-invert operation as illustrated in Fig. 3, where w bit lines are applied with the bus-invert encoding.

Fig. 3. Bus Invert Logic.

The logic checks whether the Hamming distance of current data value is larger than half of the segment size and determines the actual bus value to be transferred. The logic circuit contains several computing components: a w-bit inverter (INV) to invert the input data value; a w-bit register, made of D flip-flops, to store previously transferred data; a w-bit logic xor (⊕) to find bit transitions; an adder (+) to calculate Hamming distance of data transition on the w bit segment; a w-bit comparator ( > ) to compare the Hamming distance with half of the segment size and a w-bit multiplexor (Mux) to choose between the inverted and un-inverted data values.

The area of each component, except for the adder that has wlog(w) area complexity, is linearly proportional to the number of bits of the input data, w. Since the area of the adder increases dramatically when its input bit size becomes large, we want the segment size to be small. This will be used as a guide in our segment search algorithm.

SECTION IV

## Approach for Segmental Bus-Invert Encoding

Full space search of multiple segments for optimal switching reduction is a time consuming process since there are a large number of possibilities. Just for a single segment in an n-bit instruction space, the number of solutions is ∑i = 2n Cni (at least 2 bits are required for BI encoding). These solutions will form a huge group of sub sets if n is large. To speed up the search process, we propose to partition the search space into several bit divisions and perform the search on each of the divisions.

#### Search Space Partition

We investigated the percentage of transferred segment words whose Hamming distance is greater than half of the segment size for three different cases: 1) no partition, 2) evenly partitioned, and 3) partition based on the instruction fields of PISA [5]. Fig. 4 shows the percentage of transitions that have more than half of the segment bus bits switches for an application. For the case without partition (hence only one segment), just 5% bus transmissions have more than half of bus bits switching. In the case of the even partition, the bus is partitioned into four segments of an equal size, the percentage value is below 20%. With the instruction field-based partition, the segment size varies, but all four segments have a higher percentage than other two cases, which allows for more bit switching reduction if BI encoding is applied. Thus we base our bit space partition on the instruction fields.

Fig. 4. HD Distribution of Different Partition Methods.

For an application, its execution can be represented with a basic block diagram. Instructions within a basic block are executed sequentially. Often, the switching activity is mainly determined by the frequently executed loop blocks (also named as dominant block in this paper).

To find a partition, we use instruction types in the dominant blocks. Fields that are sensitive to the input are grouped as one division and the other fields are each treated as a separate division.

#### BI Segment Search

For each bit space division, we investigate all bus segments of different sizes and locations. We use the leftmost bit of the segment to mark the segment location. For each location, we start from the smallest segment of a two-bit window, then increase the window rightward by 1 bit to form a new segment. We compare the new segment with the one currently deemed as the best. If the following condition is satisfied, TeX Source $$\delta_{new} - \delta_{best} > = (w_{new} - w_{best})/2\eqno{\hbox{(6)}}$$namely, the extra wnewwbest bits of the new segment will statistically increase the switching reduction, the new segment is recorded as the best segment; otherwise, the new segment is discarded. After all possible window sizes are explored for a location, we continue with another segment location starting from the two-bit window again. This time, it is possible that δnew > δbest but wnew < wbest holds. In this case, the new segment with small size w but large δ is always recorded as the best segment. This process is repeated until all possible cases are finished.

#### BI Segment Merge

Some BI segments locally generated can be merged to save the bus invert control lines with same or improved switching reduction.

Fig. 5 shows an example of merging two code segment sequences. With BI encoding, no bit switching is saved for segment one, and for segment two, five bit switching can be saved. If the two segments are merged, eight bit switchings can be saved; If we apply the sub-set of bus bits in the newly merged segment, as illustrated in the shaded area, a further 1 bit switching can be saved. Therefore, for each merge attempt, we re-run the segment search.

Fig. 5. Merge Example.

Since large segment may result in large invert control logic, we start from small segments for merge operation so that after merge we have as small number of segments as possible while with each segment being not expensive.

SECTION V

## Experimental Results

To examine the efficiency of our segmental bus-invert coding, we applied this approach to a set of applications from MiBench [6] and compared our approach with the most related encodings: traditional Bus-Invert, Partitioned Bus-Invert, Partial Bus-Invert, and Multi-way Partial Bus Invert. We used ASIPMeister [15] to generate the processor VHDL model as the experimental platform for the applications. The SimpleScalar PISA [5] has been chosen as the target processor instruction set architecture. The instruction format of this architecture can be extended to 64 bits, but 40 bits are actually used in normal designs. Therefore, our simulations adopted the 40-bit instruction format.

The experiment starts with a given application written in C, which is compiled by the SimpleScalar tool and simulated on the processor VHDL model generated by ASIPMeister. The instruction trace over the instruction data bus is extracted during the simulation. This trace is used to determine the bus segments for BI encoding based on our encoding design approach proposed in Section IV. The related BI encoding/decoding and control logic is implemented in the processor VHDL model, which is then synthesized by Synopsys Design Compiler for area and power evaluation based on the Tower 0.18-micron standard cells [16].

#### Bus Switching Reduction and Overheads

Table I gives the simulation results obtained from different bus encoding approaches for each application listed in Column 1. Columns 2 & 3 provide the number of total bit switches and the average switching bits per instruction for each application without any bus encoding. The percentage of the switching reduced with the traditional bus-invert encoding [4] (BI) is presented in Column 4. The switching reduction data and related overheads from the Partial Bus-Invert encoding (PBI) and Multiple Partial Bus-Invert (MPBI) encoding are shown in Columns 5–8, and Columns 9–13, respectively; where %Red stands for the switching reduction rate, I the number of invert control bus lines, S the total number of bus lines for bus-invert encoding, and Area and Power each correspondingly for the area (in μm2) and power (in mW) overhead of the encoding/decoding logic. Columns 14–18 give the simulation results from our encoding approach (SBI).

TABLE I Results for Bus Switchings Reduction

From the table, we can see the traditional BI encoding achieves very little switching reduction (on average, only 4.1%). This ineffectiveness can also be seen from the other encodings: with average reduction rates from PBI and MPBI being 11.2% and 17.6%, respectively; for some application, the reduction rate is as small as just 7.1%. By using our segmental bus-invert encoding approach, however, we can achieve from 22.3 % up to 42.1% switching reduction. On average, 30.3% bus switching can be reduced with SBI.

Switching reduction is achieved at the cost of extra control lines and the associated control logic for each BI segment, thus incurring the overhead of area and power. As BI has extremely low switching reduction efficiency, not suitable for the instruction data bus switching reduction, we only compare PBI, MPBI with our approach for encoding/decoding logic overhead. As can been seen from Table I, SBI achieves considerable switching reduction at a lower cost than MPBI. Among them, PBI is the cheapest to encode, but it is also the least effective.

#### Power Savings Estimation

We use following formula to estimate the net power savings of SBI, PBI and MPBI encodings: TeX Source $$P_{save} = 0.5 \ast C_{bus} \ast V_{dd}^2 \ast f \ast (switch./insn.) \ast Red\hbox{\%} - P_{logic},$$where Cbus is the bus load capacitance, Vdd the supply voltage, f the frequency, (switch./insn.) the switched bus bits per instruction, Red%, the switching reduction rate, and Plogic the encoding/decoding logic power consumption estimated with the Design Compiler.

The bus capacitance varies with the system architecture and low level implementation. The load capacitance of the off-chip bus is normally multiple orders of magnitude higher than that of standard cells. Based on the 0.3pF standard cell capacitance, the supply voltage (1.8v) and clock frequency (100MHZ) used in Synopsys DesignPower, as well as the average (Switch./insn.) and %Red from our experiment (last row in Table I), we calculate the power savings with different bus capacitances, ranging from 3–300pF. The results are plotted in Fig. 6. It can be seen, from the plots, SBI brings higher savings than other two encodings. With increase of the bus capacitance, the power saving of each encoding reaches to their switching reduction rate, as depicted by the horizontal lines in the figure. On the other hand, when the bus capacitance is decreased to a certain value (e.g., 3pF or 10 times of the cell capacitance, for PBI), the power overhead of the encoding/decoding logic will cancel out the power saving from the bus switching reduction.

Fig. 6. Estimated Power Saving over Different Bus Capacitances.
SECTION VI

## Conclusion

In this paper, we have discussed the switching reduction of the instruction memory data bus for lower power processor-based systems with the Harvard architecture.

We found that the data on the instruction data bus have little temporal correlation, and the randomness of the data is also hardly exploited by the existing bus encodings due to its unevenly bit switching distribution. We proposed a segmental bus-invert encoding that can take the simplicity of the encoding approach and at the same time effectively reduce bus switching activity.

Our encoding idea is similar to the existing multi-way partial bus invert (MPBI) approach. But we use a different search algorithm for bus segments so that by applying the bus invert encoding to each of the segments, we can achieve an average 30% switching reduction on a set of benchmarks, in contrast to the 17.6% obtained by MPBI. In addition, compared to the traditional bus invert encoding, our approach comes with the reduced area for encoding/decoding logic, with an average of two more extra control lines. In contrast, MPBI requires three additional lines.

## Footnotes

Ji Gu and Hui Guo are with the School of Computer Science and Engineering, The University of New South Wales, Australia (jigu@cse.unsw.edu.au, huig@cse.unsw.edu.au).

1. Leakage power becomes unneglectable in nanoscaled devices. However, leakage power optimization achieves better in low-leakage component design at the physical level, for which paper [1] can be a good reference. In this paper we mainly focus on dynamic power reduction at the system level of the offchip bus.

2. Due to the space limit, equation derivations are omitted in this paper.

## References

1. Gate leakage reduction for scaled devices using transistor stacking

S. Mukhopadhyay, C. Neau, R. T. Cakici, A. Agarwal, C. H. Kim, K. Roy

IEEE Transactions on VLSI Systems, issue (4), pp. 716–730, 2003-08

2. Asymptotic zero-transition activity encoding for address busses in low-power microprocessor-based systems

L. Benini, G. de Micheli, E. Macii, D. Sciuto, C. Silvano

Proceedings of IEEE 7th Great Lakes Symposium on VLSI, 1997, 77–82

3. Saving power in the control path of embedded processors

C.-L. Su, C.-Y. Tsui, A. M. Despain

IEEE Design and Test of Computers, vol. 11, p. 24–30, 1994

4. Bus-invert coding for low power i/o

M. R. Stan, W. P. Burleson

IEEE Transactions on VLSI Systems, vol. 3, issue (1), p. 49–58, 1995

5. The Simplescalar Tool Set, Version 2.0

D. Burger, T. Austin

Dept. of Computer Science, Univ. of Wisconsin, Madison, Tech. Report CS-TR-1997-1342, 1997

6. Mibench: A free, commercially representative embedded benchmark suite

M. R. Guthaus, J. S. Ringenberg, D. Ernst, T. M. Austin, T. Mudge, R. B. Brown

IEEE 4th Annual Workshop on Workload Characterization, 2001, 83–94

7. A2 BC: Adaptive address bus coding for low power deep sub-micron designs

J. Henkel, H. Lekatsas

Proceedings of DAC, 2001, 744–749

8. Partial bus-invert coding for power optimization of application-specific systems

Y. Shin, S.-I. Chae, K. Choi

IEEE Transactions on VLSI Systems, Vol. 9, issue (2), 2001-04

9. A dictionary-based en/decoding scheme for low-power data bus

T. Lv, J. Henkel, H. Lekatsas, W. Wolf

IEEE Transactions on VLSI Systems, vol. 11, issue (5), p. 943–951, 2003

10. Executing compressed programs on an embedded risc architecture

A. Wolfe, A. Chanin

Proceedings of the 25th Annual International Symposium on Microarchitecture, 1992, 81–91

11. Code compression for low power embedded system design

H. Lekatsas, J. Henkel, W. Wolf

Proceedings of DAC, 2000, 294–299

12. Approximate arithmetic coding for bus transition reduction in low power designs

H. Lekatsas, J. Henkel, W. Wolf

IEEE Transactions on VLSI Systems, vol. 13, issue (6), p. 696–707, 2005

13. Low-power instruction bus encoding for embedded processors

P. Petrov, A. Orailoglu

IEEE Transactions on VLSI Systems, vol. 12, issue (8), p. 812–826, 2004

14. A coding framework for low-power address and data busses

S. Ramprasad, N. R. Shanbhag, I. N. Hajj

IEEE Transactions on VLSI Systems, p. 212–221, 1999

15. Peas-iii: An asip design environment

M. Itoh, S. Higaki, Y. Takeuchi, A. Kitajima, M. Imai, J. Sato, A. Shiomi

Proceedings of the 2000 IEEE ICCD, 2000, 430–436

16. Tower Semiconductor Ltd.

http://www.towersemi.com

## Cited By

No Citations Available

## Keywords

### IEEE Keywords

Encoding, Switches, Bismuth, Correlation, Capacitance, Hamming distance, Decoding

### INSPEC: Controlled Indexing

microprocessor chips, encoding, instruction sets, low-power electronics

### INSPEC: Non-Controlled Indexing

No Keywords Available

### Authors Keywords

No Keywords Available

### More Keywords

No Keywords Available

No Corrections

## Media

No Content Available
This paper appears in:
International Symposium on Circuits and Systems
Issue Date:
2009
On page(s):
137 - 140
ISBN:
N/A
Print ISBN:
978-1-4244-3827-3
INSPEC Accession Number:
10760365
Digital Object Identifier:
10.1109/ISCAS.2009.5117704
Date of Current Version:
26 Jun, 2009

format: plain


text:
## Homogeneous Drift

(This post is part of Polymath5; see Gowers’ weblog to understand what I’m talking about.)

The following trees represent ways to assign primes in a completely multiplicative sequence. If a leaf is labeled {N,k}, it means that the corresponding assignment has $\sum_{i=N-k+1}^N x_i$ with absolute value “drift”.

drift = 2

drift = 3

drift = 4

drift = 5


text:
# beamer box around equations

How can I make a fancy box around equations like the one in the left? I tried many time on my own but nothing satisfactory obtained!

• Just start with a block: \begin{block}{} equation here \end{block}. Then you can add some block styles. – Sigur Feb 2 '15 at 13:41

Here are some options with empheq and tcolorbox adopting code from this answer.

\documentclass{article}
\usepackage[svgnames,hyperref]{xcolor}
\usepackage{empheq}
\usepackage[many]{tcolorbox}
%\tcbuselibrary{skins}

\tcbset{highlight math style={enhanced,
colframe=red!60!black,colback=yellow!50!white,arc=4pt,boxrule=1pt,
}}

\newtcbox{\mybox}[1][]{nobeforeafter,math upper,tcbox raise base,
enhanced,frame hidden,boxrule=0pt,interior style={top color=green!10!white,
bottom color=green!10!white,middle color=green!50!yellow},
fuzzy halo=1pt with green,drop large lifted shadow,#1}

\begin{document}

\begin{empheq}[box=\tcbhighmath]{align}
a&=b\\
E&=mc^2 + \int_a^a x\, dx
\end{empheq}

a&=b\\
E&=mc^2 + \int_a^a x\, dx
\end{empheq}

\begin{empheq}[box=\mybox]{align}
a&=b\\
E&=mc^2 + \int_a^a x\, dx
\end{empheq}

\end{document}

• I cannot find drop large lifted shadow in the documentation of tcolorbox. Do you use a development version or is it just undocumented? – Henri Menke Feb 2 '15 at 15:07
• @HenriMenke It is there in documentation, page 155, of tcolorbox version 3.36 (9-1-2015) – user11232 Feb 2 '15 at 15:09
• Ah, I see. Then texdoc.net is not up-to-date. – Henri Menke Feb 2 '15 at 15:44
• @HenriMenke I tried to implement your codes but I got an error message: "File 'tcolorbox.sty' not found". I found it by search and then I got another error message: "I can't find file tcbskins.code.tex". I also found this file by search but I do not know what should I do. Do you know how to solve this issues? – Moody Feb 2 '15 at 16:55

This should be a starting point. I used tcolorbox for the fancy equation box. The font is biolinum.

\documentclass[9pt]{beamer}
\usefonttheme[onlymath]{serif}
\setbeamertemplate{itemize item}[circle]
\defbeamertemplate{footline}{sky blue}
{
\leavevmode%
\hbox{\begin{beamercolorbox}[wd=\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=1em,rightskip=1em]{frametitle}%
\end{beamercolorbox}}%
\vskip0pt%
}
\setbeamertemplate{footline}[sky blue]

\usepackage{tcolorbox,mathtools,biolinum}
\tcbuselibrary{skins,theorems}

\definecolor{sblue}{HTML}{0049A9}
\definecolor{scyan}{HTML}{CBEAFC}
\definecolor{sred}{HTML}{B5595C}
\definecolor{sgreen}{HTML}{609B57}
\definecolor{spink}{HTML}{FFB0FF}
\setbeamercolor{frametitle}{fg=sblue,bg=scyan}

\newcommand\markit[1]{\textcolor{sblue}{#1}}
\newtcolorbox{fancyeqbox}[1][]{
boxrule=0pt,arc=0pt,boxsep=0pt,
left=.5em,right=.5em,top=1ex,bottom=1ex,
colback=white,#1
}
\setlength{\leftmargini}{0pt}
\begin{document}

\title{Controllo di Processo e dei Sistemi di Produzione -- A.a.\ 2009/09}

\begin{frame}{Receding horizon philosophy}
\begin{columns}[totalwidth=\linewidth]
\column{.5\linewidth}
\begin{itemize}
\item \underline{At time $t$:} solve and \markit{optimal control}
problem over a finite future horizon of \markit{$N$} steps:
\begin{fancyeqbox}[ams align*,width=\linewidth+.7cm,enlarge left by=-.7cm]
\min_{u_t,\ldots,u_{t+N-1}}
&\left\{ \sum_{k=0}^{N-1} \| {\color{sred} y_{t+k} - r(t)} \|^ 2 + \right. \\
&\quad\left. \rho \| {\color{sred} u_{t+k} - u_r(t)} \|^2 \right\} \\
& {\color{sblue} x_{t+k+1} = f(x_{t+k},u_{t+k})} \\
& {\color{sblue} y_{t+k} = g(x_{t+k},u_{t+k})} \\
& {\color{sgreen} u_{\mathrm{min}} \leq u_{t+k} \leq u_{\mathrm{max}}} \\
& {\color{sgreen} y_{\mathrm{min}} \leq y_{t+k} \leq y_{\mathrm{max}}} \\
& x_{t} = x(t), k=0,\ldots,N-1
\end{fancyeqbox}
\end{itemize}

\column{.5\linewidth}
\centering
\includegraphics[width=.7\linewidth]{example-image-golden-upright}
\end{columns}
\begin{itemize}
\item Only apply the firt optimal move \markit{$u^*(t)$}
\item \underline{At time $t+1$:} \markit{Get new measurements},
repeat the optimization. And so on\ldots
\end{itemize}
\begin{center}
\colorbox{spink}{Advantage of repeated on-line optimization: \textcolor{red}{FEEDBACK!}}
\end{center}
\end{frame}
\end{document}


text:
# Coil’s wire should connected again

#### yiwotxxx

Joined Nov 15, 2016
17
Hi, I have a coil like that:

It got two cuts; so there are extra 4 endings. When I measure it I take 8 ohms. When I connect two endings I take 76 ohms. When I connect other ends I take 7.3 ohms.

So what do you think?

#### AlbertHall

Joined Jun 4, 2014
12,190
I think one winding has about ten times as many turns as the other if they are both wound with the same wire.


text:
Recurrence of breast cancer from 198 breast cancer patients, all of which exhibited no evidence of distant metastases at the time of diagnosis. The first 30 features of the data describe characteristics of the cell nuclei present in the digitized image of a fine needle aspirate (FNA) of the breast mass.

## Source

The data were obtained from the UCI machine learning repository, see http://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+(Prognostic).

## Examples

# \donttest{
## ------------------------------------------------------------
## Standard analysis
## ------------------------------------------------------------

data(breast, package = "randomForestSRC")
breast <- na.omit(breast)
o <- rfsrc(status ~ ., data = breast, nsplit = 10)
print(o)

# }


text:
# Tree Boosted Varying Coefficient Models

1 Apr 2019  ·  Yichen Zhou, Giles Hooker ·

This paper investigates the integration of gradient boosted decision trees and varying coefficient models. We introduce the tree boosted varying coefficient framework which justifies the implementation of decision tree boosting as the nonparametric effect modifiers in varying coefficient models. This framework requires no structural assumptions in the space containing the varying coefficient covariates, is easy to implement, and keeps a balance between model complexity and interpretability. To provide statistical guarantees, we prove the asymptotic consistency of the proposed method under the regression settings with $L^2$ loss. We further conduct a thorough empirical study to show that the proposed method is capable of providing accurate predictions as well as intelligible visual explanations.

PDF Abstract

Methodology

## Datasets

Add Datasets introduced or used in this paper


text:
# Using shared library in LibraryLink

Recently I've asked a question somewhat related to this[*], but I consider this an actual new problem. So I hope it is indeed appropriate to ask you again.

I have created a shared C++ library (with extern "C") and want to use it in a code for CreateLibrary in Mathematica.

First of all, this is my code for CreateLibrary (reduced to what is relevant):

#include "WolframLibrary.h"

DLLEXPORT mint WolframLibrary_getVersion(){
return WolframLibraryVersion;
}

DLLEXPORT int WolframLibrary_initialize( WolframLibraryData \
libData) {
return 0;
}

DLLEXPORT void WolframLibrary_uninitialize( WolframLibraryData \
libData) {
return;
}

#include "test.h"
//int test(){return 6;}

DLLEXPORT int cwvTimeEvolution(WolframLibraryData libData,
mint Argc, MArgument *Args, MArgument Res) {
int err;

err=test();

MArgument_setInteger(Res, err);
return LIBRARY_NO _ERROR;
}

DLLEXPORT int testfunc(WolframLibraryData libData,
mint Argc, MArgument *Args, MArgument Res) {
MArgument_setInteger(Res, minipotato+POTATOCOUNT);
return LIBRARY_NO _ERROR;
}

It shall use a shared library 'test'. I have the two files 'test.h' and 'libtest.so' (compiled with g++ as c++ code). They are placed at 'usr/local/lib'.

test.h has the form

//test.h
#define POTATOCOUNT 8

#ifdef __cplusplus
extern "C" {
#endif

int minipotato;
int test();

#ifdef __cplusplus
}
#endif

and the code basis for test.so is

//test.c
#include "test.h"

int minipotato = 26;

int test(){
return 123;
}

I compiled the shared library via

g++ -c -Wall -Werror -fPIC ./test.c
$g++ -shared -o libtest.so test.o  Now I tell you what happens when I try to use the library in Mathematica. The long C code (the first code I posted here) is stored as a string named 'cwvsrc' in Mathematica. Now I want to create a library out of it and call it afterwards (all in Mathematica). cwvtwolib = CreateLibrary[cwvsrc, "libtestfunc", "LibraryDirectories" -> {"/usr/local/lib"}, "Libraries" -> {"test"}, "IncludeDirectories" -> {"/usr/local/lib"}] Testfunc = LibraryFunctionLoad[cwvtwolib, "testfunc", {}, {Integer}]  Creating it works, and it tells me where it stored 'libtestfunc.so'. But the second command only returns LibraryFunction::libload : The function testfunc was not loaded from the file (don't want to post the path)/libtestfunc.so.  I assume the reason is my shared library. A motivation: If I do not plug in the flags "LibraryDirectories" -> {"/usr/local/lib"}, "Libraries" -> {"test"}  it compiles properly as well, AND I get a function I can call. But if I call Testfunc[], it returns 8. Let me explain why. Testfunc returns the sum of two values from my test library. It returns POTATOCOUNT+minipotato. 'POTATOCOUNT' was defined in test.h to be 8. 'minipotato' was only defined to be an integer. In test.c (and hence in libtest.so) I set minipotato to be equal 26. Thus, Mathematica actually used the header, but not the corresponding library. (not surprising since I did not include the library at all, but I wanted to say that this works) Also, if I call cwvTimeEvolution from c code above, my Mathematica kernel crashes. (also not surprising since it's lacking a definition) I thus assume that in my first described attempt, I did link the shared library correctly, but Mathematica doesn't know where to find it anymore as soon as it calls the function itself. I have tried several things to work around this. I have tried (but apparently failed) to follow a guide from here[**]. Also I showed Mathematica how to find the library: $Path = Append[$Path, "/usr/local/lib"];$LibraryPath = Append[\$LibraryPath, "/usr/local/lib"];
FindFile["libtest.so"]
/usr/local/lib/libtest.so

Still it doesn't work for some reason. There were several other potential reasons I have already covered, but I do not consider it sensible to list them all. (for example I tried to cover every problem between C and C++, although I'm not certain of whether I did it properly)

Hence, I would be really grateful if anyone could help me solve this problem. By now I have done so much and read up so much that I consider this a problem likely to occur to other people as well.

Maybe here are some names not perfectly correct. The original functions do not have 'cwv' in their names. Instead their names are ridiculously long, and I tried to replace everything in a convenient way. So if there is any strange combination of words beginning with c, please tell me so I can edit it.

• If you are linking to test dynamically (not static linking), it may be necessary to explicitly load it into Mathematica first: LibraryLoad[".../libtest.so"], before using LibraryFunctionLoad. Try this first. Use the full path to the library. Do not use LibraryLoad on the library you create with CreateLibrary, as LibraryFunctionLoad suffices for that. – Szabolcs Nov 1 '16 at 15:40
• Another comment: not sure what cwvsrc is here, but if you can avoid it, don't pass a source code string to CreateLibrary. Put the source code in a file and pass the file name as CreateLibrary[{"file.c", ...}, ...]. This will save you trouble later and will allow using C++. – Szabolcs Nov 1 '16 at 15:41
• Thanks for the answers! This sounds like it might be the solution. I'll write whether it has worked as soon as I find time to test it. – Fred Nov 2 '16 at 21:43
• It took me quite a while, because I was knocked out by a flu, but now I was able to test it. LibraryLoad indeed solves the problem that Testfunc only returned 8. Now it returns the correct value 34. Thank you very much so far! Yet, a function call inside the C code still doesn't work. If I use MArgument_setInteger(Res, test()); and then try to call the edited Testfunc, my kernel crashes. Do you know why? – Fred Nov 15 '16 at 11:47
• Thank you very much again! I have tried the cast, but it still does not work. The kernel crashes without a warning. Also I have just realised something. I have tried to use my code on my own PC (we're using an afs system, and I'm usually logged into someone else's PC). Here the error LibraryFunction::libload : The function testfunc was not loaded... is still there. I have no clue why. The files on both PCs seem to be the same... But that's just for your information. I'll figure this one out on my own. I just wanted to inform you that my code seems to be not the only problem...^^ – Fred Nov 15 '16 at 12:48


text:
In algebraic number theory, a quadratic field is an algebraic number field of degree two over, the rational numbers.

Every such quadratic field is some where is a (uniquely defined) square-free integer different from and . If, the corresponding quadratic field is called a real quadratic field, and for an imaginary quadratic field or complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers.

Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important.

Discriminant

For a nonzero square free integer

d

, the discriminant of the quadratic field

K=Q(\sqrt{d})

is

d

if

d

is congruent to 1 modulo 4, and otherwise

4d

. For example, if

d

is -1, then

K

is the field of Gaussian rationals and the discriminant is -4. The reason for such a distinction is that the ring of integers of

K

is generated by

\tfrac{1}{2}(1+\sqrt{d})

in the first case, but by

\sqrt{d}

in the second case.

The set of discriminants of quadratic fields is exactly the set of fundamental discriminants.

Prime factorization into ideals

Any prime number p gives rise to an ideal pOK in the ring of integers OK of a quadratic field K. In line with general theory of splitting of prime ideals in Galois extensions, this may be[1]

p is inert: (p) is a prime ideal
• The quotient ring is the finite field with p2 elements: OK/pOK = Fp2
p splits: (p) is a product of two distinct prime ideals of OK.
• The quotient ring is the product OK/pOK = Fp × Fp.
p is ramified: (p) is the square of a prime ideal of OK.
• The quotient ring contains non-zero nilpotent elements.

The third case happens if and only if p divides the discriminant D. The first and second cases occur when the Kronecker symbol (D/p) equals -1 and +1, respectively. For example, if p is an odd prime not dividing D, then p splits if and only if D is congruent to a square modulo p. The first two cases are in a certain sense equally likely to occur as p runs through the primes, see Chebotarev density theorem.

The law of quadratic reciprocity implies that the splitting behaviour of a prime p in a quadratic field depends only on p modulo D, where D is the field discriminant.

Class group

Determining the class group of a quadratic field extension can be accomplished using Minkowski's bound and the Kronecker symbol because of the finiteness of the class group.[2] A quadratic field

K=Q(\sqrt{d})

has discriminant$\Delta_K = \begind & \text d \equiv 1 \pmod 4,\\4d & \text d \equiv 2,3 \pmod 4;\end$so the Minkowski bound is[3] $M_K = \begin\frac \sqrt$
& \text d < 0, \\\frac \sqrt
& \text d > 0 .\end

Then, the ideal class group is generated by the prime ideals whose norm is less than

MK

. This can be done by looking at the decomposition of the ideals

(p)

for

p\inZ

prime where

|p|<Mk.

page 72 These decompositions can be found using the Dedekind–Kummer theorem.

The quadratic subfield of the prime cyclotomic field

A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive p-th root of unity, with p a prime number > 2. The uniqueness is a consequence of Galois theory, there being a unique subgroup of index 2 in the Galois group over Q. As explained at Gaussian period, the discriminant of the quadratic field is p for p = 4n + 1 and -p for p = 4n + 3. This can also be predicted from enough ramification theory. In fact p is the only prime that ramifies in the cyclotomic field, so that p is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants -4p and 4p in the respective cases.

Other cyclotomic fields

If one takes the other cyclotomic fields, they have Galois groups with extra 2-torsion, and so contain at least three quadratic fields. In general a quadratic field of field discriminant D can be obtained as a subfield of a cyclotomic field of D-th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the conductor-discriminant formula.

Orders of quadratic number fields of small discriminant

The following table shows some orders of small discriminant of quadratic fields. The maximal order of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant of the field. The discriminant of a non-maximal order is the product of the discriminant of the corresponding maximal order by the square of the determinant of the matrix that expresses a basis of the non-maximal order over a basis of the maximal order. All these discriminants may be defined by the formula of .

For real quadratic integer rings, the ideal class number, which measures the failure of unique factorization, is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924.

Z[{{radic|−1}}]−41±1, ±i cyclic of order 4Gaussian integers

Some of these examples are listed in Artin, Algebra (2nd ed.), §13.8.


text:
# Trigonometry Practise [closed]

Prove: $\sin^2 3α - \sin^2α = \sin4α\sin2α$

Also, simplify: $\sqrt{\dfrac {1+\cos6α} {2}}$

Any help would be appreciated.

-

## closed as off-topic by Thomas, martini, Trevor Wilson, azimut, user1337Oct 9 '13 at 19:53

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Thomas, martini, Trevor Wilson, azimut, user1337
If this question can be reworded to fit the rules in the help center, please edit the question.

For the first use $$\sin(A+B)\sin(A-B)=\sin^2A-\sin^2B$$ (Proof)
For the second use $$\cos2A=2\cos^2A-1$$


text:
# Silicide Engineering to Boost Si Tunnel Transistor Drive Current

## Abstract

In this paper, we present for the first time a novel Si p-tunnel field effect transistor (pTFET) with high-$k$ dielectric and metal gate fabricated in a multiple gate technology. The device exhibits an on-state current of 7 μA/μm at $V_{\text{DD}}$ of 0.9 V and a high $I_{\text{ON}}/I_{\text{OFF}}$ ratio of ${\sim}10^{6}$ with a fin width of 10 nm. The high on-current is believed to be due to an enhanced electric field caused by silicide encroachment and dopant segregation. Low variability of the device performance is reported for the different fin widths. Temperature measurements also show that the current is due to different transport mechanisms at different gate biases. Temperature measurements and TCAD simulations both confirm the presence of trap-assisted tunneling (TAT) as main responsible for the degradation of the subthreshold swing.

## Journal

Jpn J Appl Phys

Jpn J Appl Phys 50(4), 04DC05-04DC05-4, 2011-04-25

The Japan Society of Applied Physics

## Codes

• NII Article ID (NAID) :
150000055532
• NII NACSIS-CAT ID (NCID) :
AA12295836
• Text Lang :
EN
• Article Type :
特集
• ISSN :
00214922
• NDL Article ID :
11075542
• NDL Source Classification :
ZM35(科学技術--物理学)
• NDL Call No. :
Z53-A375
• Databases :
NDL  JSAP/JPS


text:
# Definition of affine combination in Euclidean space

One can make Euclidean space an affine space by letting the points in the space be the points $p$ in your affine space $\Bbb A$ and letting addition by a vector $\mathbf v\in V$ (i.e. $p+\mathbf v$) represent a translation of $p$ a distance $\|\mathbf v\|$ in the direction of $\mathbf {\hat v}$. This is a nice way of getting to terms with what an affine space is.

But affine spaces can also be defined as a set of points closed under affine combinations. I.e. where the fundamental operation on points is affine combinations rather than addition of a point and a vector. I'm trying to figure out how to make Euclidean space an affine space in this sense, but the only way I can think to make sense of some affine combination $$q = a_1p_1 + a_2p_2 + \cdots + a_kp_k, \quad a_1 + \cdots + a_k = 1$$ is by converting back to point + vector form: $$q = p_1 + a_2(p_2-p_1) + \cdots + a_k(p_k-p_1)$$ Is there some intuitive way of geometrically defining an affine combination of points in a Euclidean space without having to go back to the other definition of an affine space?

This is easier to see if you introduce a third view of affine spaces: an affine space is closed under binary affine combinations $(x,y)\mapsto (1-t)x+ty$ for $t\in \mathbb{R}$. A binary affine combination has a very simple geometric description: $(1-t)x+ty$ is the point on the line from $x$ to $y$ that is "$t$ of the way from $x$ to $y$" (that is, its signed distance from $x$ is $t$ times the distance from $x$ to $y$).
General $n$-ary affine combinations can then be constructed by taking repeated binary affine combinations. For instance, to get $a_1p_1+a_2p_2+a_3p_3$, you can first take a binary combination $p_{12}=\frac{a_1}{a_1+a_2}p_1+\frac{a_2}{a_1+a_2}p_2$ of $p_1$ and $p_2$ and then take a binary combination $(a_1+a_2)p_{12}+a_3p_3$ of $p_{12}$ and $p_3$.


text:
# Not necesssarily locally compact and locally compact difference

A topological space is said to be locally compact if each point $$x\in X$$ has at least one neighbourhood which is compact. If $$f$$ is continuous open mapping of a locally compact space $$(X,\tau)$$ onto a topological space $$(Y,\tau_1)$$ then $$(Y,\tau_1)$$ is locally compact.

If $$y\in Y$$ then there exists a neighbourhood $$V$$ so that $$y\in V$$. Suppose there exists at least an $$x$$ such that $$f^{-1}(y)=x$$ then $$f^{-1}(V)$$ contains $$U$$ that is a compact neighbourhood of $$x$$. Then $$y\in f(U)\subset V$$.

In the previous question it was asked:

Prove continuous image of a locally compact space is not necessarily locally compact.

Questions:

1) What is the difference in proof from the present question and the previous one? Why does it change from "not necessarily compact" to "locally compact"?

2) Is my proof right?

• @Jakobian : Some people define a neighborhood of $x$ to be a set whose interior contains $x$, so their neighborhoods aren't necessarily open sets. I don't care for that myself, I like neighborhoods to be open sets. – MPW Dec 27 '18 at 14:52
• @Jakobian No, the definition I wrote is the one the book introduce me to. – Pedro Gomes Dec 27 '18 at 14:52
• Note that in the first question, $f$ is also assumed to be open, not just continuous. That's not the case in the second question. – MPW Dec 27 '18 at 14:53
• @MPW I know that. What I do not understand is why the fact the function being open changes to the fact that $(Y,\tau_1)$ must be locally continuous. – Pedro Gomes Dec 27 '18 at 14:55
• I guess you mean "locally compact", not "locally continuous"? Anyway, the big thing that openness of $f$ gives you is that if $U$ is a neighborhood of $x$, then $f(U)$ is a neighborhood of $f(x)$. If $f$ is not open, this isn't necessarily true. – MPW Dec 27 '18 at 14:59

As to 1, the difference is that $$f$$ is now also assumed to be open, when previously we only had continuity. Note that this immediately "kills" the counterexample to that first question, that uses the identity from a discrete space. This map is only open when the image space also has the discrete topology.

Your proof is sloppy and unclear, and does not explicitly use the extra assumption of openness, which is a bad sign for a proof.

Better: let $$f: X \to Y$$ be continuous open and surjective and $$X$$ locally compact. Then $$Y$$ is locally compact: let $$y \in Y$$ be arbitrary. By ontoness we find $$x \in X$$ with $$f(x)=y$$. By local compactness of $$X$$ there is a compact set $$C \subseteq X$$ such that $$C$$ is a neighbourhood of $$x$$, i.e. there is an open subset $$O$$ of $$X$$ with $$x \in O \subseteq C$$.

Now $$y = f(x) \in f[O] \subseteq f[C]$$

which shows that $$f[C]$$, which is compact (as $$f$$ is continuous!) is a neighbourhood of $$y$$ (as $$f[O]$$ is open (by openness of $$f$$!) inside $$f[C]$$ containing $$y$$). So $$Y$$ is locally compact at $$y$$, and as $$y$$ was arbitary, everywhere.

Note that we now used all the assumptions on $$f$$.

1) The second question is asking you for an example of topological spaces $$X$$ and $$Y$$ and of a function $$f\colon X\longrightarrow Y$$ such that:

• $$f$$ is continuous;
• $$f$$ is surjective;
• $$X$$ is locally compact;
• $$Y$$ is not locally compact.

2) You did not justify the assertion that $$f(U)$$ is a neighborhood of $$y$$. And $$f$$ doesn't need to have an inverse; therefore, the equality $$f^{-1}(y)=x$$ makes no sense.

• How should I prove the first? How should I write a proof of the first question so that it does not apply to the second one? – Pedro Gomes Dec 27 '18 at 14:58
• I write instead $f(x)=y$? How should I justify the assertion $f(U)$ is a neighbourhood of $y$? – Pedro Gomes Dec 27 '18 at 15:14
• You write: take $x$ such that $f(x)=y$. And $f(U)$ is a neighborhood of $y$ because $U$ contains an open set $O$ such that $x\in O$ and $O\subset U$. Therefore, since $f$ is open (you did not use this hypothesis) $f(O)$ is open and $y\in f(O)\subset f(U)$. This proves that $f(U)$ is a neighborhood of $y$. – José Carlos Santos Dec 27 '18 at 15:18


text:
# Derivation of Distribution Function (CDF) of Gamma Distribution using Poisson Process

I found the following result on Wikipedia relating to the CDF of the Gamma Distribution when the shape parameter is an integer. (Note: there is a slight difference on how I have defined the scale parameter and how it is given on the Wikipedia page) $$X \sim Gamma(n, \lambda)\\ f_x(x; n, \lambda) = \frac{e^{-\lambda}x^{n-1}}{\lambda^n \Gamma(n)}\\ F(x;n,\lambda) = 1 - \sum_{i=0}^{n-1}{\frac {(\lambda x)^i} {i!}e^{-\lambda x}}$$

Can we prove the above result by considering a Poisson Process with parameter $\lambda$?

I reasoned this out as follows: (I believe this is not a very rigorous proof)

Let $X_t$ denote the Poisson Process with parameter $\lambda$. We know that: $$X_t \sim Poi(\lambda t)$$

Also, denote by $T_i$ the inter-arrival time between the ${(i-1)}^{th}$ and $i^{th}$ occurrence of the event. Now, since we are dealing with a Poisson Process with parameter $\lambda$ so, $T_i$'s are $iid$ $Exp(\lambda)$ distributed.

As a result, $$\sum_{i=1}^{n}{T_i} \sim Gamma(n,\lambda)$$ Now, \begin{align} \mathbb{P}[\sum_{i=1}^{n}{T_i} > t] &= Pr[by\ time\ t\ less\ than\ n\ events\ have\ occured]\\ &= \mathbb{P}[X_t < n]\\ &= \sum_{x=0}^{n-1}{\frac {(\lambda t)^x} {x!}e^{-\lambda t}} \end{align}

Finally, subtracting the above value from $1$ gives us the CDF in the required form.

Is this reasoning technically correct? I would highly appreciate if anyone could suggest a different method.

Yes, you are correct, except that you are using $X_i$ in the place of $T_i$.
If $\{X_t,t \geq 0\}$, is a Poisson process with intensity parameter $\lambda$, then the inter-arrival times $T_{1},T_{2},\cdots$ are are iid exponential random variables with mean $1/\lambda$.
If $W_n$ denotes the waiting time for the $n^{th}$ event to occur, then $$W_{n}=T_{1}+T_{2}+\cdots +T_{n}$$.
Then, we have, $P\{W_{n}>t\} \Leftrightarrow P\{X_t<n\}$.
$$P\{W_{n}>t\}=P\{X_t<n\} = \sum_{x=0}^{n-1}\dfrac{(\lambda t)^{x}}{x!}e^{\lambda t}$$


text:
# Coulombic charge of a battery pole

walt
Occasionally I need to undrstand electric fields and I usually find an equation that allows me to solve my problem without understanding the solution. Dangerous but... Anyway I've encountered another field problem and now I'd like to understand the solution to the problem.

Put in general terms I'd like to understand what the charge in Coulombs might be on the positive post of a 12 V battery that is fully charged but in which there is no external connection (other than air) between the positive and negative posts. Further, if the post can be treated as a point source is it reasonable to use Coulombs Law to calculate the electric field at a given distance (say 10 cm) directly above the post.

Intuitively it seems the post should bear a coulombic charge and thus act as a point source but I'm having trouble proving to myself that my intuition is in agreement with fact.

Homework Helper
There is no accumulation of charges on the the positive or negative posts. The electrolyte inside the battery contain positive and negative ions and they are equal in number. The posts are neutral when the circuit is open.

walt
Most sites say that there is an electric field associated with the posts of the battery. It is clear that the field 10 cm above a post is proportional to Q (charge on the post in Coulombs divided by square of the distance away from the post. What I can't arrive at is the relationship between the voltage at the pole and the Coulombic charge (Q) at the pole.

rbj
There is no accumulation of charges on the the positive or negative posts.

oh, there's a little bit, but very, very, very little.

think of the two posts as "plates" in a capacitor spaced about by about 25 cm and charged to 12 volts.

walt
That's right but I need to know what that little number is.

rbj
think of the two posts as "plates" in a capacitor spaced about by about 25 cm and charged to 12 volts.

That's right but I need to know what that little number is.

oh, okay. this is not very precise, but will get you in the ballpark.

for a parallel plate capacitor, where the spacing of the plates is small compared to the diameter of the plates, the capacitance is:

$$C = \frac{\epsilon_0 A}{d}$$

where C is capacitance, A is the plate area, d is the spacing.

now, the problem is that the posts are not exactly parallel plates, but you might want to approximate the plate area with the vertical cross-section area. the bigger problem is that the spacing is much larger than the sqrt of the "plate" area, but hell, it's just an approximation to an order of magnitude (i hope).

the other formula you need is the amount of charge on the plates of a capacitor, if the plates are charged to a known voltage:

$$Q = C V$$

where Q is the charge (you want), C is the capacitance, and V is the voltage.

so now you have to get your ruler out, measure how big the posts are and how widely they are separated.

walt
Thanks, I can see that this is an approximation but that is what I need at the moment. From the last equation it apears that if I know the capacitance in Farads and the voltage in volts I can directly calculate the charge (Q), right? One other thing, in the first equation epsilon subzero is the dielectric constant, correct? And if I had a conventional battery it would be the value for air?

I'll work on this problem some more on Sunday and get back to you if I encounter any additional problem. I do appreciate your help. Walt

Last edited:
rbj
Thanks, I can see that this is an approximation but that is what I need at the moment. From the last equation it apears that if I know the capacitance in Farads and the voltage in volts I can directly calculate the charge (Q), right?

yup, and given those units, the charge would be in coulombs.

One other thing, in the first equation epsilon subzero is the dielectric constant, correct?

learn to use LaTeX, okay? you can quote any of us to see how we do it. i'll answer your question when you do that.

And if I had a conventional battery it would be the value for air?

not if the conventional battery is sitting around in a tank of oil or something like that. so the dependency is not regarding the conventionalness of the battery, but what material is separating the posts (which you are calling the "plates" of a capacitor).


text:
# Tag Info

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The equation for adding sinusoidal functions which differ in phase resembles the equation for adding the x (or y or imaginary) components of vectors which are rotating in the xy or complex number plane. It is often more convenient to work with these (non-existent) vectors than with the functions. This especially true with vectors represented by imaginary ...

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The "real" part of wave function is no more real than the imaginary part. Both these parts are equally real or equally imaginary. None of them can independently describe the physical reality. Only when both these part are taken together then they represent the physical reality. Either one of them can be termed real or imaginary. Since complex numbers ...

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Well, first of all, a complex series RLC circuit can be analyzed using the following method. The total impedance of the circuit is given by: $$\underline{\text{Z}}_{\space\text{in}}=\text{R}+\text{j}\omega\text{L}+\frac{1}{\text{j}\omega\text{C}}\tag1$$ The input voltage can be written as: \underline{\text{V}}_{\space\text{in}}=\hat{\text{u}}\exp\left(\...

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text:
# How can you find the mean absolute deviation? afap

###### Question:

How can you find the mean absolute deviation? afap

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text:
## kayceeanddaltonsmommy Group Title A building lot in a city is shaped as a 30° -60° -90° triangle. The side opposite the 30° angle measures 41 feet. a. Find the length of the side of the lot opposite the 60° angle. b. Find the length of the hypotenuse of the triangular lot. c. Find the sine, cosine, and tangent of the 30° angle in the lot. Write your answers as decimals rounded to four decimal places. one year ago one year ago

• This Question is Open
1. Mertsj Group Title

Did you draw the picture?

2. satellite73 Group Title

rations of a 30 - 60- 90 right triangle are $$1:\sqrt{3}:2$$ for short side: long side : hypotenuse

3. kayceeanddaltonsmommy Group Title

|dw:1357877378978:dw|

4. kayceeanddaltonsmommy Group Title

I honestly have no clue how to do this

5. Mertsj Group Title

In this special triangle, the side opposite the 30 degree angle is 1/2 the hypotenuse. The long leg is the short leg multiplied by sqrt3

6. Mertsj Group Title

|dw:1357877629359:dw|

7. Mertsj Group Title

Now use the definitions of the trig functions to find sine, cosine, and tangent of the 30 degree angle.

8. kayceeanddaltonsmommy Group Title

I got the answers already for a and b just need help on c i dont know how to do that