text:
Show Summary Details
Page of

# (p. 15) Lung Ventilation: Natural and Mechanical

Lung Ventilation: Natural and Mechanical
Chapter:
Lung Ventilation: Natural and Mechanical
DOI:
10.1093/med/9780198784975.003.0003
Page of

PRINTED FROM OXFORD MEDICINE ONLINE (www.oxfordmedicine.com). © Oxford University Press, 2021. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a title in Oxford Medicine Online for personal use (for details see Privacy Policy and Legal Notice).

date: 27 October 2021

## 3.1 Introduction

This chapter describes the processes of respiration and lung ventilation. It focuses on the issues directly related to mechanical ventilation, especially its pneumatic process. It touches just briefly on general respiratory anatomy and physiology; please refer to relevant text or reference books for more details.

## 3.2 An overview of respiration

### 3.2.1 Anatomy of the respiratory system

The respiratory system refers to the six functional parts required to complete the vital process of gas exchange: the airway, lungs, chest wall, respiratory muscles, phrenic nerve, and respiratory centre. These parts can be divided roughly into two groups: (a) the anatomic foundation for gas exchange, and (b) the driving force and regulation of that gas exchange (Fig. 3.1).

Fig. 3.1 Six key parts of the respiratory system.

#### Airway

The airway, also known as the pulmonary airway or respiratory tract, refers to those parts of the respiratory system through which air flows, beginning at the nose and mouth, and ending at the alveoli (Fig. 3.2). As these names imply, the airway is the gas passageway between the atmosphere and alveoli. It is not involved in gas exchange between the alveoli and blood.

Fig. 3.2 Airway and the lungs.

The airway consists of the upper airway and the lower airway. Typically, we think of the upper airway as including the nose, nasal cavity, mouth, pharynx, and larynx. The lower airway includes everything from the trachea to the small bronchioles.

The airway poses resistance to gas flow in both directions. Airway resistance is one of the most important properties in lung mechanics. Several respiratory diseases, such as asthma, upper airway obstruction, and bronchospasm, result from abnormally high airway resistance.

The airway normally contains a certain amount of gas, which is always an inevitable part of tidal volume. This volume is called dead space or anatomical dead space because the volume is not involved in gas exchange. We need to pay particular attention to dead space, particularly when the tidal volume is tiny.

#### Lungs

Humans have two lungs located in the rib cage. They are sponge-like organs. The trachea divides into two main stem bronchi for the respective lungs. Each main stem bronchus then branches into smaller and smaller bronchi like a tree. The tiniest branch is called a bronchiole. At the end of each bronchiole is a cluster of tiny air sacs called alveoli.

Each alveolus has a mesh of tiny blood vessels called capillaries. The very thin walls of the alveoli and bronchioles provide a moist, extremely thin, and large surface for gas exchange to occur. The gradients of partial pressure of O2 and CO2 drive the gases to diffuse (Fig. 3.3). Inhaled O2 diffuses from the alveoli into the capillaries, while CO2 from the blood diffuses into the alveoli. The waste CO2 in the alveoli is then expired through lung ventilation.

Fig. 3.3 Gas exchange at alveolar wall.

Because energy demand for all living cells and tissues is continuous, respiration must also be continuous. For an individual cell as well as for the whole body, death is inevitable if respiration stops for a certain length of time.

The lungs and the chest wall are elastic. During quiet breathing, inspiration is an active process, meaning that contraction of respiratory muscles, especially the diaphragm, causes the total lung volume to increase from when the lungs were in their resting position. Expiration is normally a passive process, meaning that the inspiratory muscles relax, and the loaded elastic recoil force of the chest wall and lungs brings the lungs back to their resting position. This is similar to a stretched rubber band retracting when the applied force is removed. The elasticity of the lungs and chest wall is the basis of lung compliance, which is another key property of respiratory mechanics. The elasticity may be higher than normal (causing ‘stiff lungs’, as in patients with acute respiratory distress syndrome (ARDS)), or lower than normal (causing ‘soft lungs’, as in patients with chronic obstructive pulmonary disease (COPD)). In the case of pneumothorax, the elasticity causes the affected lung to collapse partially or completely.

The effectiveness of alveolar gas exchange is determined by: (a) the total area and thickness of the diffusion membrane, (b) alveolar ventilation, and (c) pulmonary capillary circulation. We will discuss these further in section 3.2.2.

#### Chest wall

The chest or thorax is a part of the human trunk between the neck and the abdomen (Fig. 3.4). The chest wall is made up of bones and muscles. The bones (primarily ribs, sternum, and vertebrae) form a protective cage for the internal structures of the thorax. The main muscles of the chest wall are the external and internal intercostals. The contraction of external intercostals enlarge the thoracic cavity by drawing the ribs together and elevating the rib cage, while the internal intercostals decrease the dimensions of the thoracic cavity.

Fig. 3.4 Anatomic structure of the chest wall and the organs inside.

There are three subdivisions inside the thorax. The two lateral subdivisions hold the lungs. Between the lungs is the mediastinum, which contains the heart, the great vessels, parts of the trachea and oesophagus, and other structures.

The lung surface and the inner wall of the chest cage are actually not attached directly to each other. Instead, the lung literally floats in the thoracic cavity, surrounded by a very thin layer of pleural fluid. This potential space is called the pleural cavity. The cavity normally contains a small amount of serous liquid for lubrication of lung movement during breathing. Although normally the pleural cavity is a potential space, under abnormal conditions it can contain a large amount of air (pneumothorax) or liquid (pleural effusion). If so, the affected lung collapses partially or totally, preventing it from performing its function.

The thorax contains several vital soft organs, including the heart, lungs, and large blood vessels. A high positive end-expiratory pressure (PEEP) to expand the lungs compresses the neighbouring organs, and disturbs haemodynamics to a certain extent.

As we mentioned earlier, lung compliance is a key property of a pulmonary system. It is the measurement of elasticity of the lungs and chest wall together. Sometimes respiratory compliance is used to indicate the total or sum of lung compliance and chest wall compliance.

The chest and abdomen are separated by the soft diaphragm, which permits thoracic pressure to be easily transmitted to the abdomen, and vice versa. For this reason, a high PEEP can lead to high abdominal tension. On the other hand, a high abdominal tension can decrease respiratory compliance.

#### Respiratory muscles

The contraction and relaxation of respiratory muscles increases or decreases the volume of the thoracic cavity, resulting in corresponding changes in alveolar pressure. Air is sucked into the lungs when alveolar pressure is lower than ambient pressure. Gas is pushed out of the lungs when alveolar pressure is higher than ambient pressure.

##### During inspiration

The principal muscles involved in inspiration are the diaphragm and the external intercostal muscles. The thoracic cavity is enlarged in two ways: (a) contraction of the diaphragm increases the vertical dimensions of the thoracic cavity, or (b) contraction of the external intercostal muscle increases the width of the thoracic cavity. During intensive breathing, accessory respiratory muscles also participate in inspiration. Typical accessory muscles are the sternocleidomastoid and the scalene muscles.

##### During expiration

During peaceful breathing, expiration is a passive process. When the inspiratory muscles relax, the elastic recoil force of lungs and chest wall brings the lung volume to its resting position, generating a positive alveolar pressure. The resultant pressure gradient pushes out a certain amount of the gas inside the lungs. When expiration is active, the abdominal muscles and the internal and innermost intercostal muscles help expel the gas.

Alternating contraction and relaxation of respiratory muscles provide the ultimate driving force for lung ventilation. If the muscle activities are weakened or suppressed by disease, fatigue, general anaesthesia, or trauma, lung ventilation deteriorates. In this case, mechanical ventilation is indicated.

The activation of accessory respiratory muscles is a strong indication of respiratory distress or ‘air hunger’.

#### Respiratory nerves

The diaphragm is innervated by the left and right phrenic nerves, which arise from the cervical spinal cord (C3–C5) in humans. Innervation of respiratory intercostal and abdominal muscles comes from the thoracolumbar spinal cord, T1–T11 and T7–L2, respectively.

Drawing on this fact, Swedish ventilator manufacturer Maquet developed a technical feature called NAVA (neurally adjusted ventilatory assist). Neurally adjusted ventilatory assist uses a special catheter sensor positioned at the lower oesophagus to detect phrenic nerve impulses. The detected signals in turn are used to guide the ventilator operation.

Spinal cord injury at C4 or higher can disrupt nerve impulses from the brain to the phrenic nerve. Such injuries can paralyse the diaphragm, requiring the injured person to be on a ventilator. A spinal cord injury below C5 does not involve the phrenic nerve; thus, a person with such an injury can still breathe despite possible paralysis of the lower limbs.

#### Respiratory centre

The respiratory centre refers to a group of nerve cells in the medulla oblongata and pons of the brain that: (a) receive sensory signals about the level of O2, CO2, and pH in the blood and cerebrospinal fluid; (b) determine whether and how to change the breathing pattern; and (c) send the signals to the respiratory muscles to execute this change in breathing pattern.

The functioning of the respiratory centre is critical to proper respiration. In most ventilated patients, the respiratory centre is intact, that is, an active patient has a normal respiratory response to changes in blood O2, CO2, and pH. A normal respiratory centre is required for several newer ventilator features, such as the proportional assist ventilation (PAV) mode, tube resistance compensation (TRC, or automatic tube compensation (ATC)), and NAVA. Nevertheless, the respiratory centre may not function properly in neurological or neurosurgical patients.

### 3.2.2 Physiology of respiration

#### Two essential questions about respiration

##### Why is respiration necessary?

All living cells need an energy supply to survive and execute their physiological functions. Energy is produced in the cells through the biochemical process of metabolism (Fig. 3.5). The metabolic chemical process consumes oxygen (O2) and glucose, and produces water, carbon dioxide (CO2), and adenosine triphosphate (ATP), which is the major ‘currency’ of energy in the body.

Fig. 3.5 Diagram of metabolic process.

Metabolism is a continuous process, continuously consuming O2 and producing CO2. To keep local O2 and CO2 concentrations within proper ranges, new O2 must be continuously brought to the cells and the waste CO2 removed. This is the task of respiration.

##### What is respiration?

In short, respiration is the process of transporting O2 from atmospheric air to the cells within tissues, and transporting CO2 from the cells to the air. In general, respiration has three major parts: gas exchange in the lungs, blood circulation, and gas exchange in tissues and cells (Fig. 3.6).

Fig. 3.6 Diagram of the entire respiration process.

Oxygen and CO2 are transported in blood as it circulates. If the blood supply to a tissue is drastically reduced or even stopped, the local O2 concentration falls, and the CO2 concentration rises rapidly. The tissue will die if the normal blood supply does not resume quickly. A typical example is a heart infarct.

#### (p. 20) Gas transport

Oxygen and CO2 are transported in three ways: (1) gas diffusion, (2) lung ventilation, and (3) blood circulation.

##### Gas diffusion

Gas diffusion is a natural process in which gas molecules move from an area of high concentration to a neighbouring area of low concentration. The two areas share a common diffusion membrane. Such gas diffusion takes place mainly in three areas: (a) alveolar walls, (b) blood capillary walls, and (c) tissues and cell membranes.

The speed of gas diffusion depends on: (a) the difference in gas molecular concentrations, (b) properties of the diffusion membrane, including its total area and thickness, and (c) the solubility and molecular weight of the gas involved. Carbon dioxide diffuses 20 times as fast as O2.

##### Blood transport of O2 and CO2

Oxygen is transported in two ways within the blood. Red blood cells or erythrocytes carry 97% of all O2 molecules in chemical combination with haemoglobin. The remaining 3% are dissolved in plasma.

Haemoglobin (Hb), a globular protein, is the primary vehicle for O2 transport in blood. At the alveolar capillary where the O2 concentration is high, O2 binds readily to the haemoglobin present. At tissue blood capillaries where the O2 concentration is low, the haemoglobin releases the O2 into the tissue. The oxygen-haemoglobin dissociation curve is used to express the relationship between the O2 concentration and whether the haemoglobin is acquiring or releasing O2 molecules.

CO2 is transported in blood in three ways. Most CO2 molecules are transported in the form of bicarbonate ions (HCO3-), about 10% are bound to haemoglobin and plasma proteins, and the remaining 5% are dissolved in plasma.

##### Lung ventilation

Lung ventilation is an essential part of respiration, responsible for the gas exchange between alveoli and the atmospheric air. It involves regularly replacing stale gases in the lungs with fresh gases from the atmosphere.

A simple physical model can help us better understand lung ventilation (Fig. 3.7). A suitable one is a modified pair of fireplace bellows with ‘extendable thoracic walls,’ an ‘airway,’ and a ‘total lung volume.’ A spring is added between the two handles of the bellow to mimic ‘elastic recoil force’. Another modification is that the model does not have a one-way valve, so the air enters and exits exclusively through the nozzle.

Fig. 3.7 The airway-lung-chest wall assembly can be mimicked by modified fireplace bellows.

The respiratory system always has two opposite forces, one for lung expansion and the other for lung retraction. The lung volume is determined by the balance of the two forces. The lungs are inflated if the expansion force is greater than the retraction force, and they are deflated if the opposite occurs. The lung volume is unchanged if both forces are equal. At the end of expiration, the lung volume is stable at the resting position. This volume is called the functional residual capacity (FRC) (Fig. 3.8). FRC is critical for alveolar gas exchange.

Fig. 3.8 Functional residual capacity (FRC), tidal volume, and dead space.

Te: expiratory time: Ti; inspiratory time: VT; tidal volume.

During natural inspiration, the contraction of respiratory muscles (mainly the diaphragm) increases the chest volume, generating a temporary negative alveolar pressure (Palv). Air is sucked into the lungs, and is mixed with the gases present there. This inhaled gas volume is called the inspiratory tidal volume. During inspiration the elastic recoil force (shown as the stretched spring) is loaded.

During expiration, the respiratory muscles relax, and the elastic recoil force pulls the chest and lungs back to their resting position, generating a temporary positive Palv. A certain amount of gas is pushed out of the lungs. This expelled gas volume is called expiratory tidal volume. A breath must include both an inspiratory action and an expiratory action. Inspiratory and expiratory tidal volumes are roughly equal.

The tidal volume of every breath contains two parts. The part that participates in alveolar gas exchange is alveolar tidal volume. The other part that does not participate in the gas exchange is (anatomical) dead space. Dead space volume is always moved in or out first.

Dead space is inevitable. Do not forget it when setting and interpreting tidal volume or minute volume. During mechanical ventilation, the dead space usually increases due to the presence of the artificial airway. The effective alveolar ventilation is determined by the difference between the tidal volume and the total dead space. If the tidal volume is very close to, or equal to, the dead space volume, the alveolar ventilation is (nearly) zero, i.e. CO2 removal is (nearly) zero. This unwanted situation is known as dead space ventilation.

Note that after every breath, only a part of the alveolar gas is replaced.

In addition to defining ventilation in terms of a single breath, we can also define it over a minute interval (Fig. 3.9). When we talk about minute ventilation or minute volume, we need to define a few common respiratory terms:

• Respiratory rate: The number of breaths occurring per minute;

• Minute volume: The sum of the tidal volume (inspiratory or expiratory) of all breaths occurring per minute;

• Alveolar ventilation: The sum of the alveolar tidal volume (inspiratory or expiratory) of all breaths occurring per minute.

Fig. 3.9 The relationship between minute volume, tidal volume, rate, and dead space.

The relationship can be expressed with a simple equation:

$Display mathematics$

##### Regulation of respiration

Even under normal conditions, a human’s energy demand varies widely. Think about how much energy you need during sleep compared to during physical exercise. Biochemically these activities differ greatly in metabolic rate, O2 consumption, and CO2 production. There is no such thing as a normal value for energy demand.

On the other hand, it is physiologically important to maintain the arterial partial pressure of oxygen and carbon dioxide (PaO2, PaCO2), and pH within relatively narrow normal ranges even when energy demand changes. This is achieved through a control mechanism that automatically and precisely adapts the breathing pattern (i.e. rate and depth of breathing) to the current levels of PaO2, PaCO2, and pH. To a limited extent, we can freely change our breathing pattern.

This control mechanism uses a three-part sequence:

1. a. Central and peripheral chemoreceptors detect the current O2, CO2, and pH in the blood and cerebrospinal fluid.

2. b. The controller (respiratory centre) at the medulla and pons receives signals from the receptors, decides how to respond, and then sends the instruction to the effectors.

3. c. The effectors (respiratory muscles) execute the commands received.

The mechanism responds to changes in PaCO2, PaO2, and arterial pH. Of these, PaCO2 is the primary stimulant. As Fig. 3.10 shows, increased PaCO2 results in sharply increased alveolar ventilation, and vice versa. In this manner, all three stimulants are normally maintained within their normal ranges even when O2 consumption and/or CO2 production changes drastically.

Fig. 3.10 Effect of increased arterial PCO2 and decreased arterial pH on the rate of alveolar ventilation.

Reprinted with permission from Textbook of Medical Physiology, 8th edition, Guyton A.C., p447. Copyright (1990) with permission from Harcourt College Publishers and Elsevier Inc.

In most ventilated patients, this respiratory control mechanism remains intact. The mechanism plays a key role in respiratory distress syndrome, patient-ventilator asynchrony, and weaning. It may be abnormal in some neurological and neurosurgical patients.

### 3.2.3 Respiratory failure

In summary, respiration is a mechanism to maintain PaO2 and PaCO2 within their normal ranges even when energy demand fluctuates.

Respiratory failure refers to the syndrome where the respiratory system fails to maintain PaO2 or PaCO2 within normal ranges, that is PaO2 = 80–100 mmHg and PaCO2 = 35–45 mmHg (Table 3.1). Respiratory failure can occur due to severe functional impairment of the airway, lungs, chest wall, respiratory centre, respiratory nerves, and respiratory muscles for a variety of clinical reasons.

Table 3.1 Definition of normal and abnormal pH, PaO2, and PaCO2

Below normal

Normal ranges

Above normal

• pH < 7.35

• Acidosis

pH 7.35–7.45

• pH > 7.45

• Alkalosis

• PaO2 < 80 mmHg

• Hypoxaemia

PaO2 80–100 mmHg

• PaO2 > 100 mmHg

• Hyperoxaemia

• PaCO2 < 35 mmHg

• Hypocapnia

PaCO2 35–45 mmHg

• PaCO2 > 45 mmHg

• Hypercapnia

At this point, it is necessary to introduce two key terms. Hypoxia means that PaO2 is below 80 mmHg, while hypercapnia means that PaCO2 is above 45 mmHg.

Respiratory failure can be classified roughly into two types, type 1 and type 2.

Type 1 respiratory failure is also known as hypoxic respiratory failure or lung failure. Its primary feature is abnormally low PaO2 (<60 mmHg) but nearly normal PaCO2. Type 1 respiratory failure is typically caused by inadequate oxygenation when blood passes through the lungs due to: (a) ventilation/perfusion mismatch, (b) arteriovenous shunt, or (c) gas diffusion impairment.

Type 2 respiratory failure is also known as hypercapnic respiratory failure or pump failure. Its primary feature is abnormally high PaCO2 (>50 mmHg) and abnormally low PaO2 (<60 mmHg). Type 2 respiratory failure is typically caused by inadequate lung ventilation due to: (a) excessive airway resistance, (b) decreased respiratory drive, (c) respiratory muscle fatigue or failure, or (d) abnormal status of the lungs and chest wall.

The clinical signs of respiratory failure include tachypnoea, tachycardia, cyanosis, sweating, intercostal retractions, grunting, and nose flaring. Pulse oximetry and blood gas analysis can help diagnose respiratory failure. Note that these clinical signs are non-specific.

For simplicity, we can think of the pathophysiologic process of respiratory failure as having several steps (Fig. 3.11):

1. a. The underlying diseases lead to deterioration in the efficiency and effectiveness of respiratory function.

2. b. PaO2 tends to decrease and/or PaCO2 tends to increase.

3. c. The patient strengthens their breathing efforts with the intention to maintain normal PaO2 and/or PaCO2.

4. d. Increased breathing efforts further increase the energy demand.

Fig. 3.11 Respiratory failure is a downward spiral.

If the patient can maintain normal PaO2 and PaCO2 with these intensified breathing efforts, the compensation is successful. If not, however, respiratory failure is inevitable.

Depending on the underlying diseases, both types of respiratory failure can be acute, with symptoms occurring rapidly; as in near drowning, asthma attack, respiratory arrest, drug overdose, upper airway obstruction, or chest and lung injury. Respiratory failure can also be progressive (chronic), as in emphysema, chronic bronchitis, or neuromuscular disease. For purposes of clinical treatment, it is important to differentiate between types 1 and 2, as shown in Table 3.2.

Table 3.2 Summary of respiratory failure

Classification

Type 1 respiratory failure

Type 2 respiratory failure

Other names

• Hypoxic respiratory failure

• Lung failure

• Hypercapnic respiratory failure

• Pump failure

Main feature

Hypoxia and normal PaCO2

Hypercapnia and hypoxia

Typical causes

• Ventilation/perfusion mismatch

• Arteriovenous shunt

• Gas diffusion impairment

• Excessive airway resistance

• Decreased ventilator drive

• Respiratory muscle fatigue or failure

• Abnormal status of the lungs and chest wall

The treatment of respiratory failure typically involves: (a) oxygen therapy, (b) ventilatory support with a ventilator system or a continuous positive airway pressure (CPAP) system, (c) treatment of the underlying cause, and (d) other supporting measures, such as administration of fluids and nutrition. Acute respiratory failure is usually treated in an intensive care unit, while chronic respiratory failure is usually treated at home or in a long-term care facility.

## 3.3 Mechanical ventilation

### 3.3.1 What is mechanical ventilation?

Today, mechanical ventilation is the principal therapy used to treat severe respiratory failure caused by a serious disease or injury of any of the six key parts of respiratory system (i.e. the lungs, chest wall, airway, respiratory centre, respiratory nerves, and respiratory muscles).

If applied appropriately, this therapy effectively assists, supports, or replaces compromised natural lung ventilation, artificially satisfying the vital demands of respiration. This gives the clinician valuable time to treat the underlying diseases and improve the general condition of the patient.

In most cases, mechanical ventilation therapy is temporary, lasting several hours, days, or weeks. The therapy should be terminated as soon as the patient can breathe adequately on their own. The only exception is the patient with permanently damaged pulmonary function, who may be ventilator dependent for their entire life.

Is there a limit to what mechanical ventilation can accomplish?

This is a critical but seldom asked question. The answer is yes. Mechanical ventilation depends entirely on the residual functioning of the patient’s injured lungs. When the functionality of a patient’s lungs falls below a certain threshold, mechanical ventilation cannot help. In this case, ECMO (extracorporeal membrane oxygenation) should be used.

### 3.3.2 Three operating principles

As we noted in Chapter 1, mechanical ventilation can be realized with one of three principles: intermittent positive pressure ventilation (IPPV), intermittent negative pressure ventilation (INPV), and high-frequency ventilation (HFV). Let’s take a close look at each of these operating principles.

#### Intermittent positive pressure ventilation (IPPV)

With the IPPV principle, the patient’s respiratory system is integrated into the ventilator system. A positive pressure is applied intermittently to the patient’s airway. When the airway pressure is temporarily higher than the alveolar pressure, fresh gas is pushed into the lungs, the process of inspiration. When the airway pressure is lower than alveolar pressure, the gas is expelled out of the lungs, the process of expiration. Both inspiration and expiration are regulated by the operator’s settings. Fig. 3.12 shows a typical pressure waveform in IPPV.

Fig. 3.12 Pressure waveform in intermittent positive pressure ventilation (IPPV).

IPPV is the common principle of most modern ventilators, whether invasive or non-invasive. Some popular IPPV ventilator models are shown in Fig 3.13.

Fig. 3.13 Some popular ICU ventilators based on the intermittent positive pressure ventilation (IPPV) principle.

#### (p. 27) Intermittent negative pressure ventilation (INPV)

With the INPV principle, the ventilated patient’s mouth and nose are open to the atmosphere so that gas can move in or out when alveolar pressure changes relative to atmospheric pressure. During inspiration, a negative pressure is applied to the surface of the chest wall, temporarily reducing the alveolar pressure. Fresh air is now sucked into the lungs. During expiration, the applied negative pressure is removed. The elastic recoil force temporarily generates a positive alveolar pressure, squeezing the stale gas out of the lungs. The driving pressure changes opposite to the way it does in IPPV. Fig. 3.14 shows a typical pressure waveform in INPV.

Fig. 3.14 Pressure waveform in intermittent negative pressure ventilation (INPV).

Two common ventilator designs are based on INPV (Fig. 3.15). The first is the famous iron lung, where a patient’s body, except for the head, is placed in a sealed gas container with rigid walls. A negative pressure is generated intermittently inside the container, resulting in inspiration and expiration. Different variations of the iron lung were developed during the 1920s and 1930s. They were used widely during the polio outbreaks in the 1940s and 1950s. The second type is the cuirass ventilator, in which a rigid shell or cuirass fits over the thoracic area only. An intermittent negative pressure is applied locally to change the thoracic volume.

Fig. 3.15 a) An iron lung, b) A cuirass ventilator.

(a) Poumon artificiel, Wikimedia Commons, accessed 6 October 2016, https://commons.wikimedia.org/wiki/File:Poumon_artificiel.jpg This image is in the public domain and thus free of any copyright restrictions. This media comes from the Centers for Disease Control and Prevention’s Public Health Image Library.

Today, INPV ventilators are used primarily for non-invasive ventilatory assistance.

#### High-frequency ventilation (HFV)

Both IPPV and INPV are regarded as ‘conventional’, because the tidal volume and respiratory rate are similar to physiological ones.

The third principle of mechanical ventilation is high-frequency ventilation. HFV uses a much higher respiratory rate (150 b/min or higher) than we see with the other types (Fig. 3.16). The tidal volume is much smaller than the physiological range, often smaller than dead space.

Fig. 3.16 During high-frequency ventilation (HFV), positive pressure or positive-negative pressure is applied to the airway opening at a very high rate.

The mechanisms of gas transport and exchange in HFV are very different from IPPV and INPV, and are not well understood.

Due to the differences in deployed technologies, respiratory rate, ranges of pressure swings, and baseline pressure, HFV has evolved into five forms: (a) high-frequency positive pressure ventilation (HFPPV), (b) high-frequency jet ventilation (HFJV), (c) high-frequency flow interruption (HFFI), (d) high-frequency percussive ventilation (HFPV), and (e) high-frequency oscillatory ventilation (HFOV). HFJV and HFOV are the commonly used forms.

Clinically HFV is often applied in neonates. Sometimes it is also used to treat patients with ARDS, especially those who require very high positive airway pressures. HFV can be combined with conventional IPPV as well as surfactant therapy. Fig. 3.17 shows a typical high-frequency ventilator.

Fig. 3.17 3000A high-frequency oscillatory ventilator from CareFusion.

Image of 3100A Ventilator © 2014, CareFusion Corporation; Used with permission.

Of all the three principles, IPPV currently dominates. Unless otherwise specified, the term ‘mechanical ventilator’ in this book refers to a ventilator based on the IPPV principle.

### 3.3.3 IPPV pneumatic process

To recap, the objective of mechanical ventilation is to restore or maintain adequate lung ventilation by using artificial means to intermittently change lung volume. Mechanical ventilation requires a ventilator system, which will be described in depth in Chapters 4 and 5.

Pneumatically, a ventilator system can be regarded as having two pressure areas connected by an airway (Fig. 3.18). The pressure on the device side is called the airway opening pressure (Pao), and the pressure on the lung side is called the alveolar pressure (Palv). Both Pao and Palv fluctuate regularly during mechanical ventilation. If the ventilated patient is passive, Pao is actively changed, while Palv follows. If the patient is active, both Pao and Palv can change actively.

Fig. 3.18 A ventilator system has two pressure areas that are connected by an airway.

It is necessary to mention that a patient’s lungs are a dead-end structure. During inspiration, alveolar pressure rises as more and more gas enters the lungs. In a pressure-based breath, inspiratory flow drops to zero if Pao and Palv are equal at any level.

Under normal conditions of mechanical ventilation, at any time point Pao, Palv, and lung volume are determined by these forces (Fig. 3.19):

Fig. 3.19 The forces to inflate and deflate the lungs.

• The inherent force to keep the lungs open, including that contributed by the chest wall, pleural negative pressure, and surfactant;

• The inherent force to retract the lungs, mainly the elastic recoil force of the lungs and chest wall;

• The external force to inflate the lungs, including contraction of inspiratory muscles and the application of a positive Pao;

• Additional recoil force, which is generated when the lungs and chest wall are stretched. This is the force to pull the lung and chest wall back to the resting position. This situation is comparable to a stretched rubber band. Whenever the stretching force is removed, the recoil force causes the band to retract.

#### The lungs at their resting position

The lungs are at their resting position typically at the end of an adequate expiration. This is a static state with zero airway flow, because the inherent forces to open and retract the lungs are equal and there is no externally applied force. Note that in this position, the lungs are not totally empty, but instead contain a volume of gas known as the FRC. The FRC is physiologically important. Some lung diseases can cause FRC to abnormally increase (e.g. COPD) or decrease (e.g. ARDS).

#### The lungs at their inflated position

The inflated position of the lungs represents another static state with zero airway flow. Here the forces to inflate the lungs and the forces to deflate the lungs are equal. The lungs are at their inflated position in these two cases: (a) at the end of an adequate inspiration in a pressure-based breath, and (b) at the end of inspiration in a volume breath with inspiratory pause.

#### Inspiration

Inspiration is the process to increase lung volume. Typically it begins at the resting position and may or may not end at the inflated position. The process differs in pressure breaths and volume breaths.

In a pressure breath, Pao rises quickly to and stays at a preset level (Fig. 3.20). The pressure gradient is the greatest at the beginning, resulting in the maximum inspiratory flow. Over time, Palv increases as more and more gas enters the lungs. The pressure gradient diminishes, causing a corresponding drop in inspiratory flow. If the inspiration is sufficiently long, the lungs reach the inflated position.

Fig. 3.20 Typical Pao (orange), Palv (dotted white), and airway flow change during a pressure breath (a), and volume breath (b), in a passive patient.

In a volume breath with constant flow, the most common inspiratory flow pattern, the applied positive Pao pushes gas into the lungs at a constant, defined inspiratory flow. The applied Pao must increase steadily to maintain the required pressure gradient. At the end of inspiration, the lungs do not reach their inflated position unless an inspiratory pause is imposed.

#### Expiration

Expiration is the process to decrease lung volume. The applied positive Pao drops suddenly to the baseline. The additional recoil force causes the lungs to retract. The pressure gradient (Palv > Pao) pushes the gas out. Over time, Palv and the resultant expiratory flow decrease. The lungs return to their resting position if sufficient expiratory time is allowed. The expiration process is the same in both pressure and volume breaths.

#### In spontaneously breathing patients

So far, we have discussed the four driving forces, the lung resting and inflated positions, and the inspiration and expiration processes. All of them share the same condition: the patient is passive. In reality, however, many ventilated patients are actively breathing. But the situation is more complicated than that. We know there are two external forces to inflate the lungs, the applied positive Pao and the negative Palv caused by contraction of inspiratory muscles. A similar situation may also be present at expiration, as the contraction of expiratory muscles generates additional positive Palv. This gives us six forces altogether (Table 3.3).

Table 3.3 The six physical forces involved in mechanical ventilation

Situation

Lung inflating forces

Lung deflating forces

Lung at resting position (with FRC)

Inherent force to keep the lungs open

Inherent recoil force to retract the lungs

Lung tidal volume change

Applied positive Pao to expand the lungs

Additional recoil force to bring the lungs back to the resting position

Active patients only

Contraction of inspiratory muscles to enlarge the chest cavity and lower Palv

Contraction of expiratory muscles to reduce the chest cavity and raise Palv

Ideally, the patient’s breathing efforts synchronize with the intermittently applied positive Pao for inspiration and expiration. If not, a troubling phenomenon called patient-ventilator asynchrony or ‘patient-ventilator fight’ is inevitable.

### 3.3.4 Comparison of natural and mechanical ventilation (IPPV)

#### Similarities

The ultimate goal of lung ventilation, for both natural and mechanical ventilation, is to alternately increase and decrease the lung volume.

Both natural and mechanical ventilation are realized through the natural pulmonary system, although the functioning of this system is often deteriorated in mechanically ventilated patients. The direction of gas movement during inspiration and expiration is the same for both forms of lung ventilation.

Natural and mechanical expiration are similar. In both cases, the elastic recoil force generates a positive Palv, and the gradient (Palv > Pao) drives the gas out of lungs.

#### Differences

With natural ventilation, gas exchange occurs directly between the human’s lungs and the atmosphere. With mechanical ventilation, the patient’s airway and lungs are integrated into a ventilator system. The patient breathes exclusively through the connected ventilator, isolated from atmospheric air.

With natural inspiration, the contraction of inspiratory muscles generates a negative Palv. The pressure gradient (Palv < Pao) sucks air into the lungs. With mechanical inspiration, a positive Pao is applied, and the pressure gradient (Pao > Palv) pushes the gas into the lungs.

Normally, the FRC is related to the balance of the inherent lung inflation/deflation forces. During mechanical ventilation, a moderate level of PEEP (3–5 cmH2O) is frequently used. It can be regarded as an addition to the inherent force to inflate the lungs, leading to an increased FRC.

With natural ventilation, the respiratory centre automatically and precisely regulates respiratory rate and breath intensity to satisfy current physiological demand. With mechanical ventilation, however, an operator must set the mechanical breath rate and intensity. Ventilator settings often require readjustment to meet the patient’s changing metabolic demands and lung conditions. Actively breathing patients may refuse to accept the imposed ventilation, including the rate, tidal volume, and inspiratory pressure, causing patient-ventilator asynchrony.


text:
## The Annals of Probability

### Universal Almost Sure Data Compression

#### Abstract

An $n$-code is a mapping $c_n$ from the set $A^n$ of sequences of length $n$ drawn from a finite set $A$ into the set of finite length binary sequences $B^\ast$. A decoder with distortion $D$ is a map from $B^\ast$ back into $A^n$ that sends $c_n(a^n_1)$ into a sequence that agrees with $a^n_1$ in all but at most $Dn$ places. We describe a sequence of codes and associated decoders of distortion $D$ such that, for almost every sequence from an ergodic process, the number of bits per $A$-symbol converges almost surely to $R(D)$, the optimal compression attainable for the process. The codes are universal in that the statistics of the process need not be known in advance. Expected value results of this type were first obtained by Davisson and, independently, Fittinghof; almost sure results for the invertible case $(D = 0)$ are implicitly contained in the Ziv-Lempel algorithm. Our results also apply, virtually without change in proof, to random fields.

#### Article information

Source
Ann. Probab., Volume 18, Number 2 (1990), 441-452.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990840

Digital Object Identifier
doi:10.1214/aop/1176990840

Mathematical Reviews number (MathSciNet)
MR1055415

Zentralblatt MATH identifier
0699.94004

JSTOR

format: plain


text:
## Monday, October 01, 2012

### Pushing the Boundaries in Compressive Sensing

I did not pay too much attention to it when it came out but I should have. Andrea Montanari commented on the recent From Compressive Sensing to Machine Learning entry the following:

Hi Igor,
I wanted to point out that the paper http://arxiv.org/abs/1111.1041
also describe an algorithm that achieves rho =1 for block sparse signals and random sensing matrices (see sections 3.2, 3.3 and 6.5).
A
and indeed, more specifically, an AMP solver using the block James-Stein shrinkage produces the following phase transition (rho =1 is the line equivalent to \delta =\epsilon.)

In other words, while the results of Krzakala et al  gets to rho = 1 for sparse vectors, their approach requires dedicated sensing matrices. Here, rho =1 is reached with random sensing matrices but with a requirement of structured sparsity. The results of Zhilin while reaching rho = 1 also requires a block sparsity argument but show that qualitatively, some signals are still well represented (not exactly) when rho is above 1.

Which leads me to the following point:

• Given that compressive sensing is based on an argument of sparsity
• Given that sparsity is all around us because most data seem to be sparse or compressible in a wavelet basis
• Given that wavelet decomposition not only shows not just simple compressibility but structured compressibility of most datasets
• Given that we now have some results showing better compressive sensing with a structured sparsity argument
Isn't it time we stopped talking about RIP ? the Donoho-Tanner phase transition ? L_1 recovery ?

(side question: has anybody implemented this AMP solver with the block James-Stein shrinkage?)

Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula precisely delineating the allowable degree of of undersampling of generalized sparse objects. The formula applies to Approximate Message Passing (AMP) algorithms for compressed sensing, which are here generalized to employ denoising operators besides the traditional scalar shrinkers (soft thresholding, positive soft thresholding and capping). This paper gives several examples including scalar shrinkers not derivable from convex optimization -- the firm shrinkage nonlinearity and the minimax} nonlinearity -- and also nonscalar denoisers -- block thresholding (both block soft and block James-Stein), monotone regression, and total variation minimization. Let the variables \epsilon = k/N and \delta = n/N denote the generalized sparsity and undersampling fractions for sampling the k-generalized-sparse N-vector x_0 according to y=Ax_0. Here A is an n\times N measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve \delta = \delta(\epsilon) separating successful from unsuccessful reconstruction of x_0 by AMP is given by: \delta = M(\epsilon| Denoiser), where M(\epsilon| Denoiser) denotes the per-coordinate minimax mean squared error (MSE) of the specified, optimally-tuned denoiser in the directly observed problem y = x + z. In short, the phase transition of a noiseless undersampling problem is identical to the minimax MSE in a denoising problem.
Of related note: Universality in Polytope Phase Transitions and Message Passing Algorithms by Mohsen Bayati, Marc Lelarge and  Andrea Montanari. The abstract reads:
We consider a class of nonlinear mappings FA;N in R N indexed by symmetric random matrices A 2 R N N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are known as `approximate message passing' algorithms. We study the high-dimensional (large N) behavior of the iterates of F for polynomial functions F, and prove that it is universal, i.e. it depends only on the rst two moments of the entries of A, under a subgaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves {for a broad class of random projections{ a conjecture by David Donoho and Jared Tanner.

Thanks Andrea.

Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.


text:
# Define an infinite subset of primes such that the sum of reciprocals converges

How can we define an infinite subset of primes such that the sum of reciprocals converges?

$S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and some condition on}\ p\}$ s.t. $\sum\limits_{p\in{S}}\frac{1}{p}\neq\infty$

A few options that come to mind for the condition on $p$ are:

• $\log_2(p+1)\in\mathbb{N}$
• $\log_2(p-1)\in\mathbb{N}$

But it has not been proved that there are infinite many such primes for either one of these options.

Any ideas?

-

For every $n\in \mathbb{N}$ define prime $p_n$ as the smallest of the primes which are greater than $2^n$. Then the set $\{p_n \mid n\in \mathbb{N}\}$ is infinite and

$$\sum_n \frac{1}{p_n} \leq \sum_n \frac{1}{2^n} = 1\text{.}$$

-
Nice! I was actually hoping for something a little more "concrete" (like, some sort of condition with which you could compute the exact values of the elements in the subset), but this answer is generally correct. –  barak manos Aug 29 at 8:08
To be a bit more precise: Define $p_n$ to be a (or: the smallest) prime greater than $2^n$ and greater than $p_0, p_1, \dots, p_{n-1}$. This'll make sure that you'll definitely end up with an infinite collection of primes as your sequence will be well-defined and injective. –  Frunobulax Aug 29 at 8:09
@Frunobulax My set is infinite for sure. For example, a subset $A$ of set $\mathbb{N}$ is infinite $\Leftrightarrow$ For every $n\in \mathbb{N}$ there exists $a\in A$ such that $a>n$ ... –  JanezNovak Aug 29 at 8:14
@Jaka: I wasn't meaning to say that your set isn't infinite. I just think the construction is more obvious and explicit this way. –  Frunobulax Aug 29 at 8:19
@Frunobulax you can remove the condition "and greater than $p_0,p_1,\ldots,p_{n-1}$" I think because of Bertrand's postulate. There always exists a prime between $2^n$ and $2^{n+1}$ and so the least such can be chosen for each $p_n$ without worrying about them coinciding. –  Daniel Rust Aug 29 at 13:24

Let $p_n$ be a prime divisor of $(n^2)! + 1$. Then we have $p_n > n^2$, which means that $$\sum_{n \geq 1} \frac{1}{p_n} < \sum_{n \geq 1} \frac{1}{n^2} = \frac{\pi^2}{6} < \infty.$$

-
Nice!!! Thanks. –  barak manos Aug 29 at 12:09
You should probably say "the smallest prime divisor" or the largest or something, or else this doesn't uniquely define a sequence $(p_n)_{n\in\Bbb N}$. –  Mario Carneiro Aug 30 at 0:29

Take the smallest prime divisor of Fermat numbers $F_n=2^{2^n}+1$
It is wel-known that they are pairwise coprime and that every prime divisor of these numbers is of the form $p=k\cdot 2^{n+2}+1>2^n \Rightarrow \Sigma \frac{1}{p}<\Sigma\frac{1}{2^n}=1$ which proves that the sum of their reciprocals converges.

-
See also OEIS: A093179. –  Jeppe Stig Nielsen Aug 29 at 11:08
Nice!!! Thanks. –  barak manos Aug 29 at 12:10
This is still way too much machinery for the original question, which doesn't even require using the fact that the numbers are prime. –  AAA Aug 30 at 1:48
@djechlin Because it does not require the fact that the numbers are prime it is not too much machinery –  Konstantinos Gaitanas Aug 30 at 8:57

$S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and some condition on}\ p\}$ s.t. $\sum\limits_{p\in{S}}\frac{1}{p}\neq\infty$

Here's an example with a simple explicit condition on $p$.

$$S= \{p \in \mathbb{Z}^+ | \text{p is prime and there is an n\in\mathbb{Z} such that 2^{n^2-1} \leq p \leq 2^{n^2}} \}.$$

It is known that there is always a prime between $k$ and $2k$ for each $k\in\mathbb{Z}$. Thus the set is infinite. [Wikipedia]

It is also known that there is a constant $C$ such that the number of primes smaller than an integer $x$, denoted by $\pi(x)$, is less than $C\frac{x}{\ln x}$. [Wikipedia]

Now we can get a very crude upper bound: Because the number of primes between $2^{n^2-1}$ and $2^{n^2}$ is less than $\pi(2^{n^2})$ and their reciprocals are less than $\frac{1}{2^{n^2-1}}$, we get $$\sum_{p\in{S}}\frac{1}{p} \leq \sum_{n\in\mathbb{Z}^+} \frac{1}{2^{n^2-1}} \pi(2^{n^2}) \leq \sum_{n\in\mathbb{Z}^+} \frac{1}{2^{n^2-1}} C\frac{2^{n^2}}{\ln 2^{n^2}} \leq \sum_{n\in\mathbb{Z}^+} \frac{2 C}{\ln 2} \frac{1}{n^2} = \frac{2 C}{\ln 2} \frac{\pi^2}{6}.$$

-
Nice!!! Thanks. –  barak manos Aug 29 at 12:11

Let $f:\Bbb N\rightarrow\Bbb N$ be such that $\sum_{n=0}^{\infty}1/\!f(n)$ converges (e.g. $f(n)=2^n$). Then the sum of the reciprocals of the $f(n)$th primes ($n=0,1,...$) converges.

-
That's an original one. Thank you. –  barak manos Aug 30 at 7:13

I see lots of answers giving subsets of primes yielding different upper bounds on $\sum{\frac{1}{p_n}}$, but none stating explicitly what the bound is. So I thought I'd give a subset for which the exact sum is known.

Let $x$ be any positive real number. Now define $p_n$ to be the smallest prime such that $\sum_{i=1}^{n}{\frac{1}{p_i}} < x$ and $\forall i<n: p_n \neq p_i$. Then it holds that $\sum_{i=1}^{\infty}{\frac{1}{p_i}} = x$

-
Interesting, thanks... You've managed to define a bound, but it doesn't really provide a concise method for generating the sequence itself. –  barak manos Aug 30 at 12:11
@barakmanos I don't believe there could be a concise method for generating such a sequence with a specific bound. The remainder of the sequence has to depend on prior numbers. Coming up with a concise method for producing an infinite sequence of primes is a difficult problem even if there are no constraints on what primes you may chose. –  kasperd Aug 30 at 12:23
Yep, I agree on that. I am currently aware of only one simple method for generating primes, which is Rowland's recurring relation (en.wikipedia.org/wiki/Formula_for_primes#Recurrence_relation)... And even that, as far as I know, doesn't guarantee "when the next prime is going to be found". –  barak manos Aug 30 at 12:32
@barakmanos Interesting sequence, but doesn't look like you could compute $a_n$ without first computing all predecessors. –  kasperd Aug 30 at 12:37

Define the set of primes as $\{p_n \mid n\in \mathbb{N}\}$ ,where $p_n$ is the smallest prime such that $n^2<p_n$.

-
this doesn't define a unique set of primes so you can't say "the set of primes ..." –  mercio Aug 29 at 9:00
ohh. Thanks. I did not see that. I'll edit my answer –  Nilan Aug 29 at 9:04
Is there always a prime between $n^2$ and $(n+1)^2$? –  Najib Idrissi Aug 29 at 9:15
@NajibIdrissi: Thank sir. One of my friend says that it is still an open problem. So once again I have to edit my answer. I have to learn lot of things from you guys. Again thanks. –  Nilan Sep 1 at 5:48


text:
# Weighted norm estimates for the Semyanistyi fractional integrals and Radon transforms

Weighted norm estimates for the Semyanistyi fractional integrals and Radon transforms - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Descargar gratis o leer online en formato PDF el libro: Weighted norm estimates for the Semyanistyi fractional integrals and Radon transforms
Semyanistyis fractional integrals have come to analysis from integral geometry. They take functions on \$R^n\$ to functions on hyperplanes, commute with rotations, and have a nice behavior with respect to dilations. We obtain sharp inequalities for these integrals and the corresponding Radon transforms acting on \$L^p\$ spaces with a radial power weight. The operator norms are explicitly evaluated. Similar results are obtained for fractional integrals associated to \$k\$-plane transforms for any \$1\le k

Autor: Boris Rubin

Fuente: https://archive.org/


text:
If you find any mistakes, please make a comment! Thank you.

## Count the number of cyclic subgroups in an elementary abelian p-group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.11 Solution: By definition, every nonidentity element of $E_{p^n}$ has order $p$. Thus every nonidentity element generates…

## The characteristic of an integral domain is prime or zero

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.28 Solution: Suppose the characteristic $n$ of $R$ is composite, and that $n = ab$ where $a$…

## n divides the totient of $p^n-1$ when p is prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.15 Let $p$ be a prime and let $n$ be a positive integer. Find the order of…

## Alternate proof of Cauchy’s Theorem

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.9 Solution: (1) We prove this equality by attempting to choose an arbitrary element of $\mathcal{S}$. Note…

## Examples of nilpotent elements

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.13 An element $x \in R$, $R$ a ring, is called nilpotent if $x^m = 0$ for some…


text:
# Strobl18

Harmonic Analysis and Applications

June 4-8, 2018

Strobl, AUSTRIA

### "Analyis of Shearlet Coorbit Spaces as Decomposition Spaces"

#### Koch, Rene

In order to analyze anisotropic information of signals, the shearlet transform has been introduced as class of directionally selective wavelet transform. One way of describing the approximation-theoretic properties of such generalized wavelet systems relies on {\em coorbit spaces}, i.e., spaces defined in terms of sparsity properties with respect to the system. In higher dimensions, there are several distinct possibilities for the definition of shearlet systems, and their approximation-theoretic properties are currently not well-understood. In this talk we investigate shearlet systems in higher dimensions derived from two particular classes of shearlet groups, the standard shearlet group and the Toeplitz shearlet group. We want to show that different groups lead to different approximation theories. The analysis of the associated coorbit spaces relies on an alternative description via {\em decomposition spaces} that was recently established by F\"uhr and Voigtlaender. For a shearlet group $H\subset \mathrm{GL}(d,\mathbb{R})$, this identification is based on a covering of the dual orbit $H^{-t}\xi=:\mathcal{O}\subset \mathbb{R}^d$ ($\xi\in \mathbb{R}^d$ suitable) induced by the shearlet group. The geometry of the sets in this covering is the determining factor for the associated decomposition space. We will see that $\mathcal{O}$ can be equipped with a metric structure that encodes essential properties of this covering. The orbit map $p_\xi:H\to \mathcal{O}, h\mapsto h^{-t}\xi$ then allows to compare the geometric properties of coverings induced by different groups without the need to explicitly compute the respective coverings, which gets increasingly difficult for higher dimensions. This argument relies on a rigidity theorem which states that \textit{geometrically incompatible} coverings lead to different decomposition spaces in almost all cases.

« back


text:
# Python program for Complex Numbers

PythonServer Side ProgrammingProgramming

There are always two real roots of a positive number. For example, if x2 is 25, x is ±5. However, if x2 is -25 real roots do not exist. The square root of any negative number is the square root of its absolute value multiplied by an imaginary unit j = √−1.

Hence √−25 = √25 𝑋−1 = √25 × √−1 = 5j

A Complex number consists of real and imaginary component. It is represented as x+yj. Both x and y are real numbers. Y multiplied by imaginary unit forms an imaginary part of complex number.

Examples: 3+2j, 10-5.5J, 9.55+2.3j, 5.11e-6+4j

Python has a built-in complex data type. A complex number object can be created by literal representation as follows −

>>> x = 2+3j
>>> type(x)

The complex number object has two attributes real (returns the real component) and imag (returns imaginary component excluding imaginary unit j)

>>> x.real
2.0
>>> x.imag
3.0

It also has conjugate() method. Conjugate of a complex number has the same real component and imaginary component with the opposite sign. Hence conjugate of 2+3j is 2-3j

>>> x.conjugate()
(2-3j)

Python also has built-in complex() function which returns a complex number object. The function takes two parameters, one each for real and imaginary component. They can be of any numeric type (int, float or complex)

>>> complex(9,5)
(9+5j)
>>> complex(-6, -2.5)
(-6-2.5j)
>>> complex(1.5j, 2.5j)
(-2.5+1.5j)

If only one parameter is given it is treated as real component, imaginary component is assumed to be zero.

>>> complex(15)
(15+0j)

The function can also take a string as argument provided it contains numeric representation.

>>> complex('51')
(51+0j)
>>> complex('1.5')
(1.5+0j)

Addition and subtraction of complex numbers is similar to that of integers or floats. Real and imaginary parts are added / subtracted separately.

>>> a = 6+4j
>>> b = 3+6j
>>> a+b
(9+10j)
>>> a-b
(3-2j)

For multiplication consider complex number as a binomial and multiply each term in the first number by each term in the second number.

a = 6+4j
b = 3+2j
c = a*b
c = (6+4j)*(3+2j)
c = (18+12j+12j+8*-1)
c = 10+24j

In Python console the result verifies this −

>>> a = 6+4j
>>> b = 3+2j
>>> a*b
(10+24j)

Division of complex numbers is done as follows −

Let the two numbers be

a = 2+4j

b = 1-2j

and we want to calculate a/b.

Obtain conjugate of denominator which is 1+2j

Multiply numerator and denominator by the conjugate of the denominator to get result of division

c = a/b
c = (2+4j)*(1+2j)/(1-2j)(1+2j)
c = (2+4j+4j+8*-1)/(1+2j-2j-4*-1)
c = (-6+8j)/5
c = -1.2+1.6j

Following Python console session verifies above treatment.

>>> a = 2+4j
>>> b = 1-2j
>>> a/b
(-1.2+1.6j)

## The cmath module

Mathematical functions defined in math module of Python’s standard library process floating point numbers. For complex numbers, Python library contains cmath module.

Complex number z = x+yj is a Cartesian representation. It is internally represented in polar coordinates with its modulus r (as returned by built-in abs() function) and the phase angle Φ (pronounced as phi) which is counterclockwise angle in radians, between the x axis and line joining x with the origin. Following diagram illustrates polar representation of complex number −

Functions in cmath module allow conversion of Cartesian representation to polar representation and vice versa.

polar() − This function returns polar representation of a Cartesian notation of complex number. The return value is a tuple consisting of modulus and phase.

>>> import cmath
>>> a = 2+4j
>>> cmath.polar(a)
(4.47213595499958, 1.1071487177940904)

Note that the modulus is returned by abs() function

>>> abs(a)
4.47213595499958

phase() − This function returns counterclockwise angle between x axis and segment joining a with origin. The angle is represented in radians and is between π and -π

>>> cmath.phase(a)
1.1071487177940904
z = x+yj
Φ

rect() − This function returns Cartesian representation of complex number represented in polar form i.e. in modulus and phase

>>> cmath.rect(4.47213595499958, 1.1071487177940904)
(2.0000000000000004+4j)

The cmath module contains alternatives for all mathematical functions defined in math module. There are trigonometric and logarithmic functions as explained below −

cmath.sin() − This function returns the sine trigonometric ratio for phase angle represented in radians.

>>> import cmath
>>> a = 2+4j
>>> p = cmath.phase(a)
>>> cmath.sin(p)
(0.8944271909999159+0j)

Similarly, functions for other ratios cos(), tan(), asin(), acos() and atan() are defined in cmath module.

cmath.exp() − Similar to math.exp(), this function returns ex where x is a complex number and e is 2.71828

>>> cmath.exp(a)
(-1.1312043837568135+2.4717266720048188j)

cmath.log10() − This function calculates log value of complex number taking base as 10

>>> a = 1+2j
>>> cmath.log10(a)
(0.3494850021680094+0.480828578784234j)

cmath.sqrt() − This function returns square root of complex number.

>>> cmath.sqrt(a)
(1.272019649514069+0.7861513777574233j)

In this article we learned important features of Python’s complex number data type and how arithmetic operations can be done on it. We also explored various functions defined in cmath module.

Published on 05-Apr-2019 10:07:18


text:
# Which of the following helps in the blood clotting?

1. Vitamin A
2. Vitamin D
3. Vitamin K
4. Folic acid

Option 3 : Vitamin K

## Detailed Solution

• Vitamin K is a vitamin found in leafy green vegetables, broccoli, and Brussels sprouts.
• In the body, vitamin K plays a major role in blood clotting. So it is used to reverse the effects of “blood-thinning” medications when too much is given; to prevent clotting problems in newborns who don’t have enough vitamin K, and to treat bleeding caused by medications.

Tricks:

# Which juice secreted by the organs in the alimentary canal plays an important role in the digestion of fats?

1. Pancreatic juice, saliva
2. Hydrochloric acid, mucus
3. Bile juice, Pancreatic juice
4. Saliva, hydrochloric acid

Option 3 : Bile juice, Pancreatic juice

## Detailed Solution

The correct answer is Bile juice, Pancreatic juice.

Key Points

• Bile juice, Pancreatic juice secreted by the organs plays an important role in the digestion of fats.
• Bile juice is secreted by the liver.
• It does not contain any types of enzymes.
• The bile juice helps to make the food alkaline and break down the fat molecules.
• Pancreatic juice is secreted by the pancreas.
• It contains enzymes like amylase, trypsin, pancreatic lipase, nucleases, amylase, and lipase.
• Secretion of the Pancreatic juice is regulated by the hormones secretin and cholecystokinin.
• Lipase is the digestive enzyme of fat.
• Ptyalin is the digestive enzyme of the Saliva.
• Hydrochloric acid is produced naturally in the human stomach to help the digestion of food.

# Ribosomes are sites for

1. Protein synthesis
2. Photosynthesis
3. Fat synthesis
4. Respiration

Option 1 : Protein synthesis

## Detailed Solution

The correct answer is Protein synthesis.

• Ribosomes are membranous granular structures present in the cytoplasm.
• They were first observed under an electron microscope as dense particles by George Palade in the year 1953.
• Ribosomes are the site for ''protein synthesis'' so they are also called the ''protein factory'' of the cell.
• There are two types of ribosomes
1. Eukaryotic ribosomes - 80s - occurs in the cytoplasm of eukaryotic cell
2. Prokaryotic ribosomes - 70s - occur in the cytoplasm as well as are associated with the cell membrane of prokaryotic cell

• Composition of the structure of ribosome:
• They are composed of ribonucleic acid (RNA) and proteins
 Type Composition 70s 60% rRNA + 40% proteins 80s 40% rRNA + 60% proteins

• Photosynthesis: It is the process by which green plants and some other organisms use sunlight to synthesize nutrients from carbon dioxide and water. In this process, plant the chlorophyll, carbon dioxide, water, sunlight, and release oxygen.
• Synthesis of Fatty acids occurs in the cytoplasm.

# Prawns and butterflies belong to the same phylum because of the presence of _______.

1. bilateral symmetry
2. jointed legs
3. antennae
4. segmented body

Option 2 : jointed legs

## Detailed Solution

The correct answer is jointed legs.

Arthropoda

• The word ‘arthropod’ means ‘jointed legs’.
• Ex: houseflies, spiders, scorpions, prawns, butterflies and crabs.

The Hierarchy of Classification of living organisms -

• Living organisms are classified into five kingdoms, by Whittaker, i.e.
• Kingdom Monera,
• Kingdom Protista,
• Kingdom Fungi,
• Kingdom Plantae,
• Kingdom Animalia.
• Further classification is done by the following scheme: Kingdom, Phylum (for animals) / Division (for plants), Class, Order, Family, Genus, Species.
• Kingdom Animalia is further classified based on the extent and type of the body design differentiation found, as below:

# The outer whorl is called the ________, and consists of the sepals.

1. Calyx
2. Corolla
3. Androecium
4. Gynaecium

Option 1 : Calyx

## Detailed Solution

Key Points

• Flowers contain the plant’s reproductive structures.
• A typical flower has four main parts - or whorls - known as the calyx, corolla, androecium, and gynoecium.
• The outermost whorl of the flower has green, leafy structures known as sepals.
• The sepals, collectively called the calyx, help to protect the unopened bud.

Important Points

• The second whorl is comprised of petals - usually, brightly coloured - collectively called the corolla.
• The number of sepals and petals varies depending on whether the plant is a monocot or dicot.
• In monocots, petals usually number three or multiples of three; in dicots, the number of petals is four or five, or multiples of four and five.
• Together, the calyx and corolla are known as the perianth.
• The third whorl contains the male reproductive structures and is known as the androecium.
• The androecium has stamens with anthers that contain the microsporangia.
• The innermost group of structures in the flower is the gynoecium, or the female reproductive component(s).
• The carpel is the individual unit of the gynoecium and has a stigma, style, and ovary.
• A flower may have one or multiple carpels.

# Which of the following aquatic animals does NOT have gills?

1. Octopus
2. Squid
3. Clown fish
4. Whale

Option 4 : Whale

## Detailed Solution

Whale does not have gills.

Key Points

• Gills are respiratory organs found in most of the aquatic organisms.
• Gills can extract dissolved oxygen from water and excretes carbon dioxide.
• Gills can be found in Octopus, Squid, Clownfish, Tadpole, Prawn, etc.
• Lungs are the breathing organ of Whales.

Respiratory organs of different Animals:
Animal

Respiratory Organ

Earthworm Skin.
Whale Lungs.
Spider, Scorpion Booklungs.
Cockroach Trachea.
Frog Skin, Lungs, Buccal cavity
Amphibians, mammals, and birds Lungs.

# A machine gun fires 60 bullets per minute with a velocity of 700 m/s. If each bullet has a mass of 50 g. the power developed by the gun is:

1. 250 W
2. 50 W
3. 12250 W
4. 2250 W

Option 3 : 12250 W

## Detailed Solution

The correct answer is 12250 W.

Concept:

• Power: Power is the amount of energy transferred or converted per unit time.
• The unit of power is watt or J/s.
• Its dimensional formula is: [M L2 T-3].

Calculation:

Given: v = 700 m/s, m = 50 g = 0.05 kg, n/t = 60 bullets/min ⇒  1 bullet/s

The power developed by the gun is given by formula -

Power = Work done / time

$$Power = n \times \frac{{\frac{1}{2}m{v^2}}}{t}$$

$$Power = \frac{n}{t}\; \times \frac{1}{2}m{v^2}$$

$$Power = 1 \times \frac{1}{2} \times 0.05 \times {700^2}$$

Power = 12250 watt

# At which point of the Earth is there no gravity?

1. At North and South Pole
2. At equator
3. On the ocean surface
4. At centre of the Earth

Option 4 : At centre of the Earth

## Detailed Solution

The correct answer is At the centre of the Earth.

• The centre of the Earth is such that if we are at that place, the mass around us can be considered to be condensed at the surface of the Earth itself, i.e considering the Earth as a spherical shell.
• Inside a spherical shell, there is no change in potential as one moves inside, and since only a change in potential implies a force there is no force.
• Hence the acceleration due to gravity is zero at the centre of the Earth.

# The value of one Angstrom is ________ (in micron)

1. 10-4 micron
2. 10-6 micron
3. 10-10 micron
4. 10-2 micron

Option 1 : 10-4 micron

## Detailed Solution

The correct answer is 10-4 micron.

Key Points

Read the question ,it is asking value  in micron not meter .

If asked in meters then the value will be 1 Angstroms  = 10-10 m.

• 1 mm = 10-3 m
• 1 micron = 10-3 mm
• 1 micron = 10-6 m
• 1 Angstrom = 10-10 m
• 1 Angstrom = 10-10 m = 10-4 × 10-6 m 10-4 micron.

Confusion Point

1 Angstrom = 10-10 meter

1 Angstrom =  10-4 micron

• An Angstrom is a unit of length used to measure very small distances.
• 1 Angstroms to = 0.0001 Microns
• It is named after Anders Jonas Ångström (Swedish physicist).
• A micron is one-millionth of a meter (10−6 m) and one Angstrom is 10-4 micron.

# What is pH of Lemon Juice?

1. 3.8
2. 5.5
3. 6.6
4. 2.5

Option 4 : 2.5

## Detailed Solution

pH of some common aqueous solutions

 Solution pH 1M HCl 0.0 Gastric juice 1.0 Lemon juice 2-3 Vinegar 3.0 Tomato juice 4.1 Pure water, sugar solution 7.0

# Sweet potato is an underground structure which stores food. Where is the food prepared in this plant?

1. Leaves
2. Fruit
3. Stem
4. Root

Option 1 : Leaves

## Detailed Solution

Photosynthesis - Food Making Process in Plants -

• In plants, leaves are the food factories.
• Hence, all the raw materials, like water and minerals, carbon dioxide must reach leaves.
• Chlorophyll, the green pigment present in leaves helps to capture the energy of the sunlight to synthesize (prepare) food from carbon dioxide and water.
• The process is called photosynthesis because the synthesis of food occurs in the presence of sunlight.

Sweet Potato stored food in their root.

But make their food at leaves.

# Who was the father of DNA fingerprinting?

1. James Watson
2. Hargobind Khurana
3. Alec Jeffreys
4. Nirenberg

Option 3 : Alec Jeffreys

## Detailed Solution

The correct answer is Alec Jeffreys.

• DNA fingerprinting is a technique that shows the genetic makeup of living things.
• Father of DNA fingerprinting in the world is Sir Alec John Jeffreys.
• Father of DNA fingerprinting in India is Dr Lalji Singh.

Some other important names:

Scientists Study
Carlous Linnaeus Father of Taxonomy
Francis Galton Father of Eugenics
Hippocrates Father of Medicine
Hugo de Vries Father of Mutation Theory
Leonardo Da Vinci Father of Palaeontology
William Harvey Father of blood circulation
Rudolph Virchow Father of Pathology
Karl Landsteiner Father of Blood Groups
Gregor Mendel Father of Modern Genetics
Robert Hooke Father of Cytology
W. M. Stanley Father of virology
Edward Jenner Father of Immunology
Leeuwenhoek Father of Microbiology
Eugene Odum Father of Ecology
Louis Pasteur Father of Bacteriology
Paul Berg Father of Genetic Engineering
Ivan Pavlov Father of Conditioned Reflex
Empedocles Father of Evolutionary Ideas
Marcello Malpighi Father of Microscopy
Micheli Father of Mycology
Christopher Hales Father of Plant Physiology
Charaka Father of Indian Medicine
Susruta Father of Indian surgery
Birbal Sahni Father of Indian Palaeobotany
R. Mishra Father of Indian Ecology

# Which blood group is universal acceptor?

1. O+
2. O–
3. AB–
4. AB+

Option 4 : AB+

## Detailed Solution

AB+ is the blood group which is a universal acceptor since it has both kinds of antigens, which are Antigen A and Antigen B. People having a blood group of AB+ can accept blood from any person of any blood group.

# Which of the following is NOT based on the heating effect of current?

1. Electric heater
2. Electric bulb (with filament)
3. Electric iron
4. Microwave

Option 4 : Microwave

## Detailed Solution

The correct answer is the Microwave.

Key Points

• A microwave oven uses microwaves to heat food.
• In the case of microwave ovens, the commonly used radio wave frequency is roughly 2,500 megahertz (2.5 gigahertz).
• Radio waves in this frequency range absorbed by water, fats, and sugars. When they are absorbed they are converted directly into atomic motion or vibration.
• Therefore heavy motion or vibration is converted into heat.

• Microwaves are not absorbed by most plastics, glass, or ceramics.
• The frequency of the Microwave equals the resonance frequency of water. Therefore foods could be heated up by the oscillation of water molecules.

• An Electric heater, Electric blub (with filament), Electric iron is the electrical device that converts an electric current into heat.
• They working on the principle of the Joule heating effect.
• An electric current passing through a resistor will convert that electrical energy into heat energy.

# Bile Juice is formed in the

1. Kidney
2. Salivary Gland
3. Liver
4. Lung

Option 3 : Liver

## Detailed Solution

Liver-
• The liver is the largest gland in the body.
• The liver mainly secretes 'Bile Juice' which is stored in a Gall bladder.
• Bile Juice and Pancreatic juice are released into the small intestine by a common duct.
• Unused glucose is stored in the liver in the form of glycogen.
• Heparin, Urea, and Bile Juice are produced in the liver.
• The major supply of blood to the liver is by 'portal vein'(75%) & remaining (25%) by Hepatic artery.
• That is why the liver is known to have a 'Dual blood supply'.

# In a uniform circular motion:

1. Velocity is constant
2. Distance is constant
3. Displacement is constant
4. Speed is constant

Option 4 : Speed is constant

## Detailed Solution

The correct answer is Speed is constant.

Key Points

• Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path.
• The distance and displacement are different at every instance at any particular point of time.

Confusion Points

• While the speed is a scalar quantity, in uniform motion, it remains constant.
• As, Velocity is a vector quantity, and the direction of movement changes continuously, velocity changes continuously.
• Due to this, the object is in accelerated motion.

Important Points

• As the direction of the velocity changes at every instant, it is an accelerated motion.
• During the motion, the particle repeats it's paths along the same trajectory.
• Thus the motion is periodic.

Uniform Circular Motion:

• During circular motion, if the speed of the period remains constant, it is a Uniform Circular Motion. Hence option 4 is correct.

# The flexibility in plants is due to a tissue called

1. Parenchyma
2. Collenchyma
3. Sclerenchyma
4. None of these

Option 2 : Collenchyma

## Detailed Solution

Key Points

• Collenchyma is a specialized simple permanent tissue that provides support and flexibility to plants.
• This tissue is made up of living cells having uneven thickenings of cellulose, pectin, and hemicellulose on the walls.
• Cells possess chloroplast, then it is involved in manufacturing sugar and starch.
• Provides tensile strength and flexibility to the plant body.

• Parenchyma is the most common and abundant tissue in plants which is made up of thin-walled living cells having a distinct nucleus.
• The main function of parenchyma is to store and assimilate food. Hence, they are referred to as food storage tissues.
• Serves as a packing tissue to fill the spaces between other tissues and maintain the shape and firmness of the plant.
• Stores waste products of plants.
• Sclerenchyma is a lignified supportive tissue made up of thick-walled and lignified cells.
• Provides mechanical strength to the plant.

# Which of the following is an example of a green algae?

1. Laminaria
2. Sargassum
3. Chlamydomonas
4. Fucus

Option 3 : Chlamydomonas

## Detailed Solution

• Chlamydomonas is found in stagnant water and on damp soil as well as freshwater, seawater, and even in snow.
• Class: Chlorophyceae
• Scientific name: Chlamydomonas
• Phylum: Chlorophyta
Sl no. Name Type Description
1. Laminaria Brown Algae It is also known as Kelp. It is generally found in the deep sea.
2. Sargassum Brown Algae It is a type of seaweed (Macroalgae) which is generally known for its planktonic species.
3.  Chlamydomona  Green Algae It is found almost everywhere and consists of 325 species.
4. Fucus Brown Algae It is basically a rockweed and it is found in intertidal zones.

# When 1 litre of water is cooled from 4°C to 0°C, its volume _____.

1. first decreases and then increases
2. remains the same
3. increases
4. decreases

Option 3 : increases

## Detailed Solution

Important Points

• In normal cases, the volume of substances increases on heating and decreases while cooling.
• When 1 Kg of water is cooled from 4°C to 0°C volume of water will start increasing this due to water's unique property known as 'Anomalous Expansion of Water'.
• Anomalous Expansion of Water occurs between 4°C to 0°C.
• The density of water is a maximum at 4 °C.
• When water is cooled from 4°C to 0°C, its density decreases.
• The anomalous expansion of water helps preserve aquatic life during very cold weather.

Explanation:

• When water reaches 4°C the molecules have been pushed as close to one another as possible and the density of water becomes precisely 1.00 g/cm³
• When water freezes at 0°C due to the crystal structure the molecules arranged in some structured fashion so a little far apart ended up less dense - 0.93 g/cm - and so floats due to buoyancy.

As density decreases the volume increases.

Volume = mass /density.

# Which one of the following statements is correct about camphor and ammonium chloride?

1. Both of them are inorganic compounds
2. Both of them are organic compounds
3. Both of them undergo sublimation
4. Both 2 and 3

Option 3 : Both of them undergo sublimation

## Detailed Solution

The correct answer is Both of them undergo sublimation.

Important Points

• Both camphor and ammonium chloride are solid at room temperature.
• But they are vaporised (gas phase) when heat is applied.
• This property is called Sublimation.

Key Points

• Sublimation - Direct conversion of a solid into the vapour state is called sublimation.
• The heat of sublimation - Heat required to change a unit mass of solid directly into vapour, at a given temperature, is called the heat of sublimation at that temperature.


text:
NOTES

# Belief Propogation

An approximation algorithm that allows us conduct inference on Bayesian networks and Markov random fields. Essentially we calculate the marginal distribution for each unobserved node, given the observed nodes. Marginal distribution of a given value X is the summation of $p()$ of all other variables.

Any Bayesian network or Markov random field can be represented as a factor graph. The algorithm passes real-valued functions, messages, between variables and factors. The messages carry the information of the influence of the variables on one another. The message differs whether it is a variable node sending a message to a factor node, or vice versa. The recipient sends as message the constant function equal to 1, the message is the product of all the messages from the neighboring factor nodes.

### Low Density Parity Check codes

Here is an example of belief propogation in action.