International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 1 ISSN 2250-3153
MEASUREMENT OF QUANTUM STATE: ADEQUATIO INTELLECTUS NOSTRI CUM RE -A ZEITGEIST MODEL
Dr K N Prasanna Kumar, Prof B S Kiranagi, Prof C S Bagewadi
ABSTRACT: Serge Haroche and David Wineland were awarded Nobel Prize in Physics for being able to resolve the conflict of Quantum measurement. We give a concomitant model for the same. Study of concatenated equation is done in the next paper.
INTRODUCTION: Believe nothing on the faith of traditions, even though they have been held in honor for many generations and in diverse places. Do not believe a thing because many people speak of it. Do not believe on the faith of the sages of the past.
Do not believe what you yourself have imagined, persuading yourself that a God inspires you. Believe nothing on the sole authority of your masters and priests. After examination, believe what you yourself have tested and found to be reasonable, and conform your conduct thereto.
As the Buddha was dying, Ananda asked who would be their teacher after death. He replied to his disciple -
"Be lamps unto yourselves. Be refuges unto yourselves. Take yourself no external refuge. Hold fast to the truth as a lamp. Hold fast to the truth as a refuge. Look not for a refuge in anyone besides yourselves. And those, Ananda, who either now or after I am dead, Shall be a lamp unto themselves, Shall betake themselves as no external refuge, But holding fast to the truth as their lamp, Holding fast to the truth as their refuge, Shall not look for refuge to anyone else besides themselves,
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It is they who shall reach to the very topmost height; But they must be anxious to learn."
Quoted in Joseph Goldstein, The Experience of Insight
Buddha
Unbounded by time and space. If I look for you, I will always find you. But when I am not looking, You may be somewhere else entirely. For you are a pigment of my perception, As I am of yours. Together we form The brilliant and harmonious Rainbow of God's love.
-CONNEE
Acknowledgements: It is to be stated in unmistakable and unequivocal terms that the literary expatiation, predicational integrity, Introductory remarks, character constitution, ontological consonance, primordial exactitude, accolytish representation, atrophied asseveration, anamensial alienisms, anchorite aperitif, Arcadian Atticism reflective in passages, essential predications, primary focus on homologues receptiveness of the subject matter in question, differentiated instrumental and dynamism of the projective development of the topics consummate abstractions, rational representation, conferential extrinsicness, manifestation of histories, standard remarks, professed developments(Google Search) interfacial interference and syncopated justices, are taken from various sources Such as Wikipedia, author’s Home Page, ask a Physicist Column, Abstracts of articles of various authors, papers of various authors, Web Graphs, Google search photographs, Google search results and other sources which included literally dialectic deliberation, polemical argumentation, conjugatory confatalia with fellow Professors, and scholars of repute. I have to state that I have put all concerted efforts , sustained struggles, and protracted endeavor to mention each and every source at the cross reference or at the reference list at the end. In the eventuality of any act of omission or commission it is to be stated that such an eventuality has occurred attributable to inadvertence and in deliberation and I beg professedly and profusely and assiduously and avidly with all fervor and can dour the persons concerned. I am not presenting any panacea for all the ills despite the penance done for therefor, and it is attributable and ascribable to the fact that many highly esteemed and eminent persons allowed me to piggy ride on their backs I have been able to write summarily and expressly this paper. Explanation and deliberation of concatenation equations are done in the next paper. Towards the end of consummation, consolidations, concretization, reinforcement, revitalization, rejuvenation, resurrection, and consubstantiation of this mammoth project, singlehandedly I have gone through millions of pages and drafted and typed myself, and if by chance there are any repetition, I make a sincere entreat, earnest beseech, and fervent appeal and obsequesial dedication and consecration to kindly pardon me on that score
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(1) An ion trap is a combination of electric or magnetic fields that captures ions in a region of a vacuum system or tube. Ion traps have a number of scientific uses such as mass spectrometery and trapping ions while the ion's quantum state is manipulated. The two most common types of ion traps are the Penning trap and the Paul trap (quadrupole ion trap). (2) When using ion traps for scientific studies of quantum state manipulation, the Paul trap is most often used. This work may lead to a trapped ion quantum computer and has already been used to create the world's most accurate atomic clocks. (3) An ion trap mass spectrometer may incorporate a Penning trap (Fourier transform ion cyclotron resonance), Paul trap or the Kingdon trap. The Orbitrap, introduced in 2005, is based on the Kingdon trap. Other types of mass spectrometers may also use a linear quadrupole ion trap as a selective mass filter. In an electron gun (a device emitting high-speed electrons, such as those in CRTs), an ion trap may be implemented above the cathode (using an extra, positively-charged electrode between the cathode and the extraction electrode) to prevent its degradation by positive ions accelerated backward by the fields intended to pull electrons away from the cathode. Penning traps are devices for the storage of charged particles using a homogeneous static magnetic field and a spatially inhomogeneous static electric field. This kind of trap is particularly well suited to precision measurements of properties of ions and stable subatomic particles which have a non-zero electric charge. Recently this trap has been used in the physical realization of quantum computation and quantum information processing as well. The Penning trap has also been used in the realization of what is known as a geonium atom. Currently Penning traps are used in many laboratories worldwide. For example, they are used at CERN to
store antiprotons.
(4) Penning traps use a strong homogeneous axial magnetic field to confine particles radially and a quadrupole electric field to confine the particles axially. The static electric potential can be generated using a set of three electrodes: a ring and two end caps. In an ideal Penning trap the ring and end caps are hyperboloids of revolution. For trapping of positive (negative) ions, the end cap electrodes are kept at a positive (negative) potential relative to the ring. This potential produces a saddle point in the centre of the trap, which traps ions along
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the axial direction.
(5) The electric field causes ions to oscillate (harmonically in the case of an ideal penning trap) along the trap axis. The magnetic field in combination with the electric field causes charged particles to move in the radial plane with a motion which traces out an epitrochoid. The orbital motion of ions in the radial plane is composed of two modes at frequencies which are called the magnetron and the modified cyclotron frequencies. These motions are similar to the deferent and epicycle, respectively, of the Ptolemaic model of the solar system.
A classical trajectory in the radial plane for (Courtesy: Google Search: Wikipedia)
(6) The sum of these two frequencies is the cyclotron frequency, which depends only on the ratio of electric charge to mass and on the strength of the magnetic field. This frequency can be measured very accurately and can be used to measure the masses of charged particles. (7) Many of the highest-precision mass measurements (masses of the electron, proton, 2H, 20Ne and 28Si) come from Penning traps. Buffer gas cooling, resistive cooling and laser cooling are techniques to remove energy from ions in a Penning trap. Buffer gas cooling relies on collisions between the ions and neutral gas molecules that bring the ion energy closer the energy of the gas molecules. (8) In resistive cooling, moving image charges in the electrodes are made to do work through an external resistor, effectively removing energy from the ions. Laser cooling can be used to remove energy from some kinds of ions in Penning traps. This technique requires ions with an appropriate electronic structure. Radiative cooling is the process by which the ions lose energy by creating electromagnetic waves by virtue of their acceleration in the magnetic field. This process dominates the cooling of electrons in Penning traps, but is very small and usually negligible for heavier particles. (9) Using the Penning trap can have advantages over the radio frequency trap (Paul trap). Firstly, in the Penning trap only static fields are applied and therefore
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there is no micro-motion and resultant heating of the ion due to the dynamic fields. Also, the Penning trap can be made larger whilst maintaining strong trapping. The trapped ion can then be held further away from the electrode surfaces. Interaction with patch potentials on the electrode surfaces can be responsible for heating and decoherence effects and these effects scale as a high power of the inverse distance between the ion and the electrode. (10) Fourier transform ion cyclotron resonance mass spectrometry (also known as Fourier transform mass spectrometry), is a type of mass spectrometry used for determining the mass-to-charge ratio (m/z) of ions based on the cyclotron frequency of the ions in a fixed magnetic field. The ions are trapped in a Penning trap where they are excited to a larger cyclotron radius by an oscillating electric field perpendicular to the magnetic field. The excitation also results in the ions moving in phase (in a packet). The signal is detected as an image current on a pair of plates which the packet of ions passes close to as they cyclotron. The resulting signal is called a free induction decay (fid), transient or interferogram that consists of a superposition of sine waves. The useful signal is extracted from this data by performing a Fourier transform to give a mass spectrum. Single ions can be investigated in a Penning trap held at a temperature of 4 K. For this the ring electrode is segmented and opposite electrodes is connected to a superconducting coil and the source and the gate of a field effect transistor. The coil and the parasitic capacitances of the circuit form a LC circuit with a Q of about 50 000. The LC circuit is excited by an external electric pulse. The segmented electrodes couple the motion of the single electron to the LC circuit. Thus the energy in the LC circuit in resonance with the ion slowly oscillates between the many electrons (10000) in the gate of the field effect transistor and the single electron. This can be detected in the signal at the drain of the field effect transistor.
Ions trapped by optical fields
Ion sees the light: charged particle is trapped by laser(Courtesy: Google Search : Wikipedia)
(11) Physicists in Germany claimed to have trapped single ions using lasers for the first time – an achievement that could open the door to advanced simulations of quantum systems.
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(12) In the past, the trapping of atomic particles has followed a basic rule: use radio-frequency (RF) electromagnetic fields for ions, and optical lasers for neutral particles, such as atoms. (13) This is because RF fields can only exert electric forces on charges; try to use them on neutral particles and there's little effect. (14) A laser, on the other hand, can draw the dipole moments of neutral particles towards the centre of its beam. But the resultant optical trap is relatively weak, and so ions – which are sensitive to stray electric fields – easily, escape. Now, Tobias Schaetz and colleagues at the Max-Planck Institute for Quantum Optics in Garching, Germany, claim to have got around this problem. they describe an "experimental proof-of-principle" of how stronger, more focused lasers can optically trap ions for over a millisecond.(Courtesy: Physics World.com) (15) In the experiment, Schaetz's group cooled a single 24Mg+ ion in a standard RF trap before superimposing it with the field from a strong laser. The physicists then gradually reduced the RF field to zero so the ion was contained by the optical field alone. Finally, they observed the fluorescence light of the ion through a CCD camera to check it had been trapped successfully. Over several experiments, they calculated the trapping lifetime as about 1.8 ms. "The lifetime of the ion in the optical dipole trap is limited by photon scattering and is thus expected to be improvable by state of the art techniques," they note. (16) Optical traps for ions could benefit simulations on quantum systems. If optical traps are superimposed on RF traps, there comes the ability to simulate in two or three dimensions, or collide particles in new ways. (17) "The biggest advantage of being able to trap an ion optically, instead of [using] an RF electrode, is that one can then study the collision between a neutral atom and an ion at low energy," explains Yu-Ju Lin, a researcher at the Joint Quantum Institute at the University of Maryland, US. "If an ion is still trapped by an RF electrode, it always produces an RF-induced micro-motion – i.e. kinetic energy – which limits how low the energy of the collisions [can go]." (18) Andrew Steane, a quantum physicist at the University of Oxford, UK, calls the research "excellent experimental physics", although he notes that it is an extension of previous ideas and experiments. "Such [an optical trap] is of course already widely used in experiments on neutral atoms," he adds. "Applying that idea to ions extends the toolbox available to experiments in basic atomic physics and quantum mechanics." (From the reports of Jon Cartwright is a freelance journalist based in Bristol) (19) A quadrupole ion trap or quadrupole ion storage trap (QUISTOR) exists in both linear and 3D (Paul Trap, QIT) varieties and refers to an ion trap that uses constant DC and radio frequency (RF) oscillating AC electric fields to trap ions. It is commonly used as a component of a mass spectrometer. The invention of the 3D quadrupole ion trap itself is attributed to Wolfgang Paul who shared the Nobel Prize in Physics in 1989 for this work.
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Theory
The 3D trap itself generally consists of two hyperbolic metal electrodes with their foci facing each other and a hyperbolic ring electrode halfway between the other two electrodes. The ions are trapped in the space between these three electrodes by AC (oscillating, non-static) and DC (non-oscillating, static) electric fields. The AC radio frequency voltage oscillates between the two hyperbolic metal end cap electrodes if ion excitation is desired; the driving AC voltage is applied to the ring electrode. The ions are first pulled up and down axially while being pushed in radially. The ions are then pulled out radially and pushed in axially (from the top and bottom). In this way the ions move in a complex motion that generally involves the cloud of ions being long and narrow and then short and wide, back and forth, oscillating between the two states. Since the mid-1980s most 3D traps (Paul traps) have used ~1 mtorr of helium. The use of damping gas and the mass-selective instability mode developed by Stafford et al. led to the first commercial 3D ion traps.
Linear Ion Trap at the University of Calgary(Courtesy Google Search)
The quadrupole ion trap has two configurations: the three dimensional form described above and the linear form made of 4 parallel electrodes. A simplified rectilinear configuration has also been used. The advantage of the linear design is in its simplicity, but this leaves a particular constraint on its modeling. To understand how this originates, it is helpful to visualize the linear form. The Paul trap is designed to create a saddle-shaped field to trap a charged ion, but with a quadrupole, this saddle-shaped electric field cannot be rotated about an ion in the centre. It can only 'flap' the field up and down. For this reason, the motions of a single ion in the trap are described by the Mathieu Equations. These equations can only be solved numerically or equivalently by computer simulations.
The intuitive explanation and lowest order approximation is the same as strong focusing in accelerator physics. Since the field affects the acceleration, the position lags behind (to lowest order by half a period). So the particles are at defocused positions when the field is focusing and vice versa. Being farther from center, they experience a
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International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 8 ISSN 2250-3153 stronger field when the field is focusing than when it is defocusing.
Equations of motion
Further ,
And
The trapping of ions can be understood in terms of stability regions in and space.
Linear ion trap
LTQ (Linear trap quadrupole)
The linear ion trap uses a set of quadrupole rods to confine ions radially and static electrical potential on-end electrodes to confine the ions axially. The linear form of the trap can be used as a selective mass filter, or as an actual trap by creating a potential well for the ions along the axis of the electrodes Advantages of the linear trap design are increased ion storage capacity, faster scan times, and simplicity of construction (although quadrupole rod alignment is critical, adding a quality control constraint to their production. This constraint is additionally present in the machining requirements of the 3D trap).
Cylindrical ion trap
Cylindrical ion traps have a cylindrical rather than a hyperbolic ring electrode. This configuration has been used in miniature arrays of traps.
The term Ion trapping is used to describe the build-up of a higher concentration of a chemical across a cell membrane due to the pKa value of the chemical and difference of pH across the cell membrane. Generally speaking, these results in basic chemicals accumulate in acidic bodily fluids such as the cytosol, and acidic chemicals accumulating in basic fluids such as mastitic milk.
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Many cells have other mechanisms to pump a molecule inside or outside the cell against the concentration gradient, but these processes are active ones, meaning that they require enzymes and consume cellular energy. In contrast, ion trapping does not require any enzyme or energy. It is similar to osmosis in that they both involve the semipermeable nature of the cell membrane.
Cells have a more acidic pH inside the cell than outside (gastric mucosal cells being an exception). Therefore basic drugs (like bupivacaine, pyrimethamine) are more charged inside the cell than outside. The cell membrane is permeable to non-ionized (fat- soluble) molecules; ionized (water-soluble) molecules cannot cross it easily. Once a non-charged molecule of a basic chemical crosses the cell membrane to enter the cell, it becomes charged due to gaining a hydrogen ion because of the lower pH inside the cell, and thus becomes unable to cross back. Because transmembrane equilibrium must be maintained, another unionized molecule must diffuse into the cell to repeat the process. Thus its concentration inside the cell increases many times that of the outside. The non- charged molecules of the drug remain in equal concentration on either side of the cell membrane.
The charge of a molecule depends upon the pH of its solution. In an acidic medium, basic drugs are more charged and acidic drugs are less charged. The converse is true in a basic medium. For example, Naproxen is a non-steroidal anti-inflammatory drug that is a weak acid (its pKa value is 5.0). The gastric juice has a pH of 2.0. It is a three-fold difference (due to log scale) between its pH and its pKa; therefore there is a 1000× difference between the charged and uncharged concentrations. So, in this case, for every one molecule of charged Naproxen, there are 1000 molecules of uncharged Naproxen at a pH of 2. This is why weak acids are better absorbed from the stomach and weak bases from intestine where the pH is alkaline. When pH of a solution is equal to pKa of dissolved drug, then 50% of the drug is ionized, another 50% is unionized. Ion trapping is the reason why basic (alkaline) drugs are secreted into the stomach (for example morphine) where pH is acidic, and acidic drugs are excreted in urine when it is alkaline. Similarly, ingesting sodium bicarbonate with amphetamine, a weak base, causes better absorption of amphetamine (in stomach) and its lesser excretion (in urine), thus prolonging its actions. Ion trapping can cause partial failure of certain anti-cancer chemotherapies. Ion trapping is also important outside of pharmacology. For example it causes weakly acidic hormones to accumulate in the cytosol of cells. This is important in keeping the external concentration of the hormone low in the extracellular environment where many hormones are sensed. Examples of plant hormones that are subjected to ion trapping areabscisic acid, gibberellic acid and retinoic acid. Examples of animal hormones subjected to ion trapping include Prostacyclin and Leukotrienes.
Ions are generally categorized into the following groups based on mobility values and dimensions.
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(1) Free-floating electron
An free-floating electron exists by itself and weights only about 1/1800 of the hydrogen atom. Its mobility values are as large as beta rays generated by cathode rays or radiant substances. It is generally found at high altitudes where the air is rarefied, or in highly purified nitrogen, helium and argon.
(2) Ionized atom
An atom, which has lost an electron, is a positively ionized atom. An electronically neutral atom, which has obtained an electron, is a negatively ionized atom. Both types of ions along with electrons exist only in the upper layers of the atmosphere.
(3) Small ion
Most ions found in the atmosphere belong to this group (also known as Lightweight or Normal ion). As soon as an electron or ionized atom shows up in the atmosphere, it attracts gaseous molecules and combines with them to form a small ion molecule while positioning itself in the center. A small ion molecule consists of 2 to 30 molecules. Generally, positive ions weigh more than negatively charged ions, and mobility values are larger than 0.4-0.8 (cm2/Vs).
(4) Large ion
A large ion (also known as Heavy ion) is a negative or positive small ion (molecule) absorbed by dust, mist or another tiny particle. While having the same structure as small ions, it can weight 1,000 times more. Mobility values range from 0.0005 to 0.01 (cm2/Vs). Many exist in polluted air.
(5) Middle ion
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This group of ions was discovered by Pollock and exists only in low humidity conditions, and does not exist near the earth's surface. Mobility values range from 0.01 to 0.1 (cm2/Vs).
Ion channels are pore-forming proteins that help establish and control the voltage gradient across the plasma membrane of cells (see membrane potential) by allowing the flow of ions down their electrochemical gradient. They are present in the membranes that surround all biological cells. The study of ion channels involves many scientific techniques such as voltage clamp electrophysiology (in particular patch clamp), immunohistochemistry, and RT-PCR.
Basic features
Ion channels regulate the flow of ions across the membrane in all cells. Ion channels are integral membrane proteins; or, more typically, an assembly of several proteins. They are present on all membranes of cell (plasma membrane) and intracellular organelles (nucleus, mitochondria, endoplasmic reticulum, golgi apparatus and so on). Such "multi-subunit" assemblies usually involve a circular arrangement of identical or homologous proteins closely packed around a water-filled pore through the plane of the membrane or lipid bilayer. For most voltage-gated ion channels, the pore-forming subunit(s) are called the α subunit, while the auxiliary subunits are denoted β, γ, and so on. Some channels permit the passage of ions based solely on their charge of positive (cation) or negative (anion). However, the archetypal channel pore is just one or two atoms wide at its narrowest point and is selective for specific species of ion, such as sodium or potassium. These ions move through the channel pore single file nearly as quickly as the ions move through free fluid. In some ion channels, passage through the pore is governed by a "gate," which may be opened or closed by chemical or electrical signals, temperature, or mechanical force, depending on the variety of channel.
Biological role
Because channels underlie the nerve impulse and because "transmitter-activated" channels mediate conduction across the synapses, channels are especially prominent components of the nervous system. Indeed, most of the offensive and defensive toxins that organisms have evolved for shutting down the nervous systems of predators and prey (e.g., the venoms produced by spiders, scorpions, snakes, fish, bees, sea snails and others) work by modulating ion channel conductance and/or kinetics. In addition, ion channels are key components in a wide variety of biological processes that involve rapid changes in cells, such as cardiac, skeletal, and smooth muscle contraction, epithelial transport of nutrients and ions, T-cell activation and pancreatic beta-cell insulin release. In the search for new drugs, ion channels are a frequent target.
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Diversity
There are over 300 types of ion channels in a living cell. Ion channels may be classified by the nature of their gating, the species of ions passing through those gates, the number of gates (pores) and localization of proteins. Further heterogeneity of ion channels arises when channels with different constitutive subunits give rise to a specific kind of current. Absence or mutation of one or more of the contributing types of channel subunits can result in loss of function and, potentially, underlie neurologic diseases.
Classification by gating
Ion channels may be classified by gating, i.e. what opens and closes the channels. Voltage-gated ion channels open or close depending on the voltage gradient across the plasma membrane, while ligand-gated ion channels open or close depending on binding of ligands to the channel.
Voltage-gated ion channel
Voltage-gated ion channels open and close in response to membrane potential.
Voltage-gated sodium channels: This family contains at least 9 members and is largely responsible for action potential creation and propagation. The pore-forming α subunits are very large (up to 4,000 amino acids) and consist of four homologous repeat domains (I-IV) each comprising six transmembrane segments (S1-S6) for a total of 24 transmembrane segments. The members of this family also coassemble with auxiliary β subunits, each spanning the membrane once. Both α and β subunits are extensively glycosylated.
Voltage-gated calcium channels: This family contains 10 members, though these members are known to coassemble with α2δ, β, and γ subunits. These channels play an important role in both linking muscle excitation with contraction as well as neuronal excitation with transmitter release. The α subunits have an overall structural resemblance to those of the sodium channels and are equally large.
Cation channels of sperm: This small family of channels, normally referred to as Catsper channels, is related to the two-pore channels and distantly related to TRP channels.
Voltage-gated potassium channels (KV): This family contains almost 40 members, which are further divided into 12 subfamilies. These channels are known mainly for their role in repolarizing the cell membrane following action potentials. The α subunits have six transmembrane segments, homologous to a single domain of the sodium channels. Correspondingly, they assemble as tetramers to produce a functioning channel.
Some transient receptor potential channels: This group of channels, normally referred to simply as TRP channels, is named after their role in Drosophila phototransduction. This
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International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 13 ISSN 2250-3153 family, containing at least 28 members, is incredibly diverse in its method of activation. Some TRP channels seem to be constitutively open, while others are gated by voltage, intracellular Ca2+, pH, redox state, osmolarity, and mechanical stretch. These channels also vary according to the ion(s) they pass, some being selective for Ca2+ while others are less selective, acting as cation channels. This family is subdivided into 6 subfamilies based on homology: classical (TRPC), vanilloid receptors (TRPV), melastatin (TRPM), polycystins (TRPP), mucolipins (TRPML), and ankyrin transmembrane protein 1 (TRPA).
Hyperpolarization-activated cyclic nucleotide-gated channels: The opening of these channels is due to Hyperpolarization rather than the depolarization required for other cyclic nucleotide-gated channels. These channels are also sensitive to the cyclic nucleotides cAMP and cGMP, which alter the voltage sensitivity of the channel’s opening. These channels are permeable to the monovalent cations K+ and Na+. There are 4 members of this family, all of which form tetramers of six-transmembrane α subunits. As these channels open under hyperpolarizing conditions, they function as pacemaking channels in the heart, particularly the SA node.
Voltage-gated proton channels: Voltage-gated proton channels open with depolarization, but in a strongly pH-sensitive manner. The result is that these channels open only when the electrochemical gradient is outward, such that their opening will only allow protons to leave cells. Their function thus appears to be acid extrusion from cells. Another important function occurs in phagocytes (e.g. eosinophils, neutrophils, macrophages) during the "respiratory burst." When bacteria or other microbes are engulfed by phagocytes, the enzyme NADPH oxidase assembles in the membrane and begins to produce reactive oxygen species (ROS) that help kill bacteria. NADPH oxidase is electrogenic, moving electrons across the membrane, and proton channels open to allow proton flux to balance the electron movement electrically.
Ligand-gated
Also known as ionotropic receptors, this group of channels open in response to specific ligand molecules binding to the extracellular domain of the receptor protein. Ligand binding causes a conformational change in the structure of the channel protein that ultimately leads to the opening of the channel gate and subsequent ion flux across the plasma membrane. Examples of such channels include the cation-permeable "nicotinic" Acetylcholine receptor, ion tropic glutamate-gated receptors and ATP-gated P2X receptors, and the anion-permeable γ-amino butyric acid-gated GABAA receptor.
Ion channels activated by second messengers may also be categorized in this group, although ligands and second messengers are otherwise distinguished from each other.
Other gating
Other gating include activation/inactivation by e.g. second messengers from the inside of the cell membrane, rather as from outside, as in the case for ligands. Ions may count to
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International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 14 ISSN 2250-3153 such second messengers, and then causes direct activation, rather than indirect, as in the case were the electric potential of ions cause activation/inactivation of voltage-gated ion channels.
Some potassium channels
Inward-rectifier potassium channels: These channels allow potassium to flow into the cell in an inwardly rectifying manner, i.e., potassium flows effectively into, but not out of, the cell. This family is composed of 15 official and 1 unofficial members and is further subdivided into 7 subfamilies based on homology. These channels are affected by intracellular ATP, PIP2, and G-protein βγ subunits. They are involved in important physiological processes such as the pacemaker activity in the heart, insulin release, and potassium uptake in glial cells. They contain only two transmembrane segments, corresponding to the core pore-forming segments of the KV and KCa channels. Their α subunits form tetramers.
Calcium-activated potassium channels: This family of channels is, for the most part, activated by intracellular Ca2+ and contains 8 members.
Two-pore-domain potassium channels: This family of 15 members forms what is known as leak channels, and they follow Goldman-Hodgkin-Katz (open) rectification.
Light-gated channels like channelrhodopsin are directly opened by the action of light.
Mechanosensitive ion channels are opening under the influence of stretch, pressure, shear, displacement.
Cyclic nucleotide-gated channels: This superfamily of channels contains two families: the cyclic nucleotide-gated (CNG) channels and the Hyperpolarization-activated, cyclic nucleotide-gated (HCN) channels. It should be noted that this grouping is functional rather than evolutionary.
Cyclic nucleotide-gated channels: This family of channels is characterized by activation due to the binding of intracellular cAMP or cGMP, with specificity varying by member. These channels are primarily permeable to monovalent cations such as K+ and Na+. They are also permeable to Ca2+, though it acts to close them. There are 6 members of this family, which is divided into 2 subfamilies.
Hyperpolarization-activated cyclic nucleotide-gated channels
Temperature Gated Channels: Members of the Transient Receptor Potential ion channel superfamily, such as TRPV1 or TRPM8 are opened either by hot or cold temperatures.
Classification by type of ions
Chloride channels: This superfamily of poorly-understood channels consists of approximately 13 members. They include ClCs, CLICs, Bestrophins and CFTRs. These
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International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 15 ISSN 2250-3153 channels are non-selective for small anions; however chloride is the most abundant anion, and hence they are known as chloride channels.
Potassium channels
Voltage-gated potassium channels e.g., Kvs, Kirs etc.
Calcium-activated potassium channels e.g., BKCa or MaxiK, SK, etc.
Inward-rectifier potassium channels
Two-pore-domain potassium channels: This family of 15 members forms what is known as leak channels, and they follow Goldman-Hodgkin-Katz (open) rectification.
Sodium channels
Voltage-gated sodium channels NaVs
Epithelial sodium channels (ENaC)
Calcium channels CaVs
Proton channels
Voltage-gated proton channels
Non-selective cation channels: These let many types of cations, mainly Na+, K+ and Ca2+ through the channel.
Most Transient receptor potential channels
Other classifications
There are other types of ion channel classifications that are based on less normal characteristics, e.g. multiple pores and transient potentials.
Almost all ion channels have one single pore. However, there are also those with two:
Two-pore channels: This small family of 2 members putatively forms cation-selective ion channels. They are predicted to contain two KV-style six-transmembrane domains, suggesting they form a dimer in the membrane. These channels are related to catsper channels channels and, more distantly, TRP channels.
There are channels that are classified by the duration of the response to stimuli:
Transient receptor potential channels: This group of channels, normally referred to simply as TRP channels, is named after their role in Drosophila photo transduction. This family, containing at least 28 members, is incredibly diverse in its method of activation. Some TRP channels seem to be constitutively open, while others are gated by voltage, intracellular Ca2+, pH, redox state, Cosmolarity, and mechanical stretch. These channels also vary according to the ion(s) they pass, some being selective for
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Ca2+ while others are less selective, acting as cation channels. This family is subdivided into 6 subfamilies based on homology: canonical (TRPC), vanilloid receptors (TRPV), melastatin (TRPM), polycystins (TRPP), mucolipins (TRPML), and ankyrin transmembrane protein 1 (TRPA).
Detailed structure
Channels differ with respect to the ion they let pass (for example, Na+, K+, Cl−), the ways in which they may be regulated, the number of subunits of which they are composed and other aspects of structure. A channel belonging to the largest class, which includes the voltage-gated channels that underlie the nerve impulse, consists of four subunits with six transmembrane helices each. On activation, these helices move about and open the pore. Two of these six helices are separated by a loop that lines the pore and is the primary determinant of ion selectivity and conductance in this channel class and some others. The existence and mechanism for ion selectivity was first postulated in the 1960s by Clay Armstrong. He suggested that the pore lining could efficiently replace the water molecules that normally shield potassium ions, but that sodium ions were too small to allow such shielding, and therefore could not pass through. This mechanism was finally confirmed when the structure of the channel was elucidated. The channel subunits of one such other class, for example, consist of just this "P" loop and two transmembrane helices. The determination of their molecular structure by Roderick MacKinnon using X-ray crystallography won a share of the 2003 Nobel Prize in Chemistry.
Because of their small size and the difficulty of crystallizing integral membrane proteins for X-ray analysis, it is only very recently that scientists have been able to directly examine what channels "look like." Particularly in cases where the crystallography required removing channels from their membranes with detergent, many researchers regard images that have been obtained as tentative. An example is the long-awaited crystal structure of a voltage-gated potassium channel, which was reported in May 2003. One inevitable ambiguity about these structures relates to the strong evidence that channels change conformation as they operate (they open and close, for example), such that the structure in the crystal could represent any one of these operational states. Most of what researchers have deduced about channel operation so far they have established through electrophysiology, biochemistry, gene sequence comparison and mutagenesis.
Channels can have single (CLICs) to multiple transmembrane (K channels, P2X receptors, Na channels) domains which span plasma membrane to form pores. Pore can determine the selectivity of the channel. Gate can be formed either inside or outside the pore region.
Diseases of ion channels
There are a number of chemicals and genetic disorders which disrupt normal functioning of ion channels and have disastrous consequences for the organism. Genetic
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Chemicals
Tetrodotoxin (TTX), used by puffer fish and some types of newts for defense. It blocks sodium channels.
Saxitoxin is produced by a dinoflagellate also known as "red tide". It blocks voltage dependent sodium channels.
Conotoxin, is used by cone snails to hunt prey.
Lidocaine and Novocaine belong to a class of local anesthetics which block sodium ion channels.
Dendrotoxin is produced by mamba snakes, and blocks potassium channels.
Iberiotoxin is produced by the Buthus tamulus (Eastern Indian scorpion) and blocks potassium channels.
Heteropodatoxin is produced by Heteropoda venatoria (brown huntsman spider or laya) and blocks potassium channels.
Genetic
Shaker gene mutations cause a defect in the voltage gated ion channels, slowing down the repolarization of the cell.
Equine hyperkalaemic periodic paralysis as well as Human hyperkalaemic periodic paralysis (HyperPP) is caused by a defect in voltage dependent sodium channels.
Paramyotonia congenita (PC) and potassium aggravated myotonias (PAM)
Generalized epilepsy with febrile seizures plus (GEFS+)
Episodic Ataxia (EA), characterized by sporadic bouts of severe dis coordination with or without myokymia, and can be provoked by stress, startle, or heavy exertion such as exercise.
Familial hemiplegic migraine (FHM)
Spinocerebellar ataxia type 13
Long QT syndrome is a ventricular arrhythmia syndrome caused by mutations in one or more of presently ten different genes, most of which are potassium channels and all of which affect cardiac repolarization.
Brugada syndrome is another ventricular arrhythmia caused by voltage-gated sodium channel gene mutations.
Cystic fibrosis is caused by mutations in the CFTR gene, which is a chloride channel.
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Mucolipidosis type IV is caused by mutations in the gene encoding the TRPML1 channel
References
Neural networks were trained using whole ion mobility spectra from a standardized database of 3137 spectra for 204 chemicals at various concentrations. Performance of the network was measured by the success of classification into ten chemical classes. Eleven stages for evaluation of spectra and of spectral pre-processing were employed and minimums established for response thresholds and spectral purity. After optimization of the database, network, and pre-processing routines, the fraction of successful classifications by functional group was 0.91 throughout a range of concentrations. Network classification relied on a combination of features, including drift times, number of peaks, relative intensities, and other factors apparently including peak shape. The network was opportunistic, exploiting different features within different chemical classes. Application of neural networks in a two-tier design where chemicals were first identified by class and then individually eliminated all but one false positive out of 161 test spectra. These findings establish that ion mobility spectra, even with low resolution instrumentation, contain sufficient detail to permit the development of automated identification systems.
(Adapted from lectures by Dr. Richard F.W. Bader Professor of Chemistry / McMaster University / Hamilton, Ontario)
Classification of Chemical Bonds
To make a quantitative assessment of the type of binding present in a particular molecule it is necessary to have a measure of the extent of charge transfer present in the molecule relative to the charge distributions of the separated atoms. This information is contained in the density difference or bond density distribution, the distribution obtained by subtracting the atomic densities from the molecular charge distribution. Such a distribution provides a detailed measure of the net reorganization of the charge densities of the separated atoms accompanying the formation of the molecule.
The density distribution resulting from the overlap of the undistorted atomic densities (the distribution which is subtracted from the molecular distribution) does not place sufficient charge density in the binding region to balance the nuclear forces of repulsion. The regions of charge increase in a bond density map are, therefore, the regions to which charge is transferred relative to the separated atoms to obtain a state of electrostatic equilibrium and hence a chemical bond. Thus we may use the location of this charge increase relative to the positions of the nuclei to characterize the bond and to obtain an explanation for its electrostatic stability.
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In covalent binding we shall find that the forces binding the nuclei are exerted by an increase in the charge density which is shared mutually between them. In ionic binding both nuclei are bound by a charge increase which is localized in the region of a single nucleus.
Covalent Binding
The bond density map of the nitrogen molecule (Fig. 7-2) is illustrative of the characteristics of covalent binding.
Fig. 7-2. Bond density (or density difference) maps and their profiles along the internuclear axis for N2 and LiF. The solid and dashed lines represent an increase and a decrease respectively in the molecular charge density relative to the overlapped atomic distributions. These maps contrast the two possible extremes of the manner in which the original atomic charge densities may be redistributed to obtain a chemical bond. Click here for contour values.
The principal feature of this map is a large accumulation of charge density in the binding region, corresponding in this case to a total increase of one quarter of an electronic charge. As noted in the study of the total charge distribution, charge density is also transferred to the antibinding regions of the nuclei but the amount transferred to either region, 0.13 e-, is less than is accumulated in the binding region. The charge density of the original atoms is decreased in regions perpendicular to the bond at the positions of the nuclei. In three dimensions, the regions of charge deficit correspond to two continuous rings or roughly doughnut-shaped regions encircling the bond axis.
The increase in charge density in the antibinding regions and the removal of charge density from the immediate regions of the nuclei result in an increase in the forces of repulsion exerted on the nuclei, forces resulting from the close approach of the two atoms and from the partial overlap of their density distributions. The repulsive forces
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International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 20 ISSN 2250-3153 are obviously balanced by the forces exerted on the nuclei by the shared increase in charge density located in the binding region. A bond is classified as covalent when the bond density distribution indicates that the charge increase responsible for the binding of the nuclei is shared by both nuclei. It is not necessary for covalent binding that the density increase in the binding region be shared equally as in the completely symmetrical case of N2. We shall encounter heteronuclear molecules (molecules with different nuclei) in which the net force binding the nuclei is exerted by a density increase which, while shared, is not shared equally between the two nuclei. The pattern of charge rearrangement in the bond density map for N2 is, aside from the accumulation of charge density in the binding region, quite distinct from that found for H2 , another but simpler example of covalent binding. The pattern observed for nitrogen, a charge increase concentrated along the bond axis in both the binding and antibinding regions and a removal of charge density from a region perpendicular to the axis, is characteristic of atoms which in the orbital model of bonding employ p atomic orbitals in forming the bond. Since a p orbital concentrates charge density on opposite sides of a nucleus, the large buildup of charge density in the antibinding regions is to be expected. In the orbital theory of the hydrogen molecule, the bond is the result of the overlap of s orbitals. The bond density map in this case is characterized by a simple transfer of charge from the antibinding to the binding region since s orbitals do not possess the strong directional or nodal properties of p orbitals. Further examples of both types of charge rearrangements or polarizations will be illustrated below.
Ionic Binding
We shall preface our discussion of the bond density map for ionic binding with a calculation of the change in energy associated with the formation of the bond in LiF. While the calculation will be relatively crude and based on a very simple model, it will illustrate that the complete transfer of valence charge density from one atom to another in forming a molecule is in certain cases energetically possible.
Lithium possesses the electronic configuration 1s22s1 and is from group IA of the periodic table. It possesses a very low ionization potential and an electron affinity which is zero for all practical purposes. Fluorine is from group VIIA and has a configuration 1s22s22p5. It possesses a high ionization potential and a high electron affinity. The following calculation will illustrate that the 2s electron of Li could conceivably be transferred completely to the 2p shell of orbitals on F in which there is a single vacancy. This would result in the formation of a molecule best described as Li+F-, and in the electron configurations 1s2 for Li+ and 1s22s22p6 for F-.
(2)
The two ions are oppositely charged and will attract one another. The energy released when the two ions approach one another from infinity to form the LiF molecule is easily estimated. To a first approximation it is simply -e2/R where R is the final equilibrium distance between the two ions in the molecule:
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The transfer of charge density from lithium to fluorine is very evident in the bond density map for LiF (Fig. 7-2). The charge density of the 2s electron on the lithium atom is a very diffuse distribution and consequently the negative contours in the bond density map denoting its removal are of large spatial extent but small in magnitude. The principal charge increase is nearly symmetrically arranged about the fluorine nucleus and is completely encompassed by a single nodal surface. The total charge increase on fluorine amounts to approximately one electronic charge. The charge increase in the antibinding region of the lithium nucleus corresponds to only 0.01 electronic charges. (The great disparity in the magnitudes of the charge increases on lithium and fluorine are most strikingly portrayed in the profile of the bond density map, also shown in Fig. 7-2) It is equally important to realize that the charge increase on lithium occurs within the region of the 1s inner shell or core density and not in the region of the valence density. Thus the slight charge increase on lithium is primarily a result of a polarization of its core density and not of an accumulation of valence density.
The pattern of charge increase and charge removal in the region of the fluorine, while similar to that for a nitrogen nucleus in N2, is much more symmetrical, and the charge density corresponds very closely to the distribution obtained from a single 2p electron. Thus the simple orbital model of the bond in LiF which describes the bond as a transfer of the 2s electron on lithium to the single 2p vacancy on fluorine is a remarkably good one.
While the bond density map for LiF substantiates the concept of charge transfer and the formation of Li+ and F- ions it also indicates that the charge distributions of both ions are polarized. The charge increase in the binding region of fluorine exceeds slightly that in its antibinding region (the F- ion is polarized towards the Li+ ion) and the charge distribution of the Li+ ion is polarized away from the fluorine. A consideration of the forces exerted on the nuclei in this case will demonstrate that these polarizations are a necessary requirement for the attainment of electrostatic equilibrium in the face of a complete charge transfer from lithium to fluorine.
Consider first the forces acting on the nuclei in the simple model of the ionic bond, the model which ignores the polarizations of the ions and pictures the molecule as two closed-shell spherical ions in mutual contact. If the charge density of the Li+ ion is spherical it will exert no net force on the lithium nucleus. The F- ion possesses ten electrons and, since the charge density on the F- ion is also considered to be spherical, the attractive force this density exerts on the Li nucleus is the same as that obtained for all ten electrons concentrated at the fluorine nucleus. Nine of these electrons will screen the nine positive nuclear charges on fluorine from the lithium nucleus. The net force on the lithium nucleus is, therefore, one of attraction because of the one excess negative charge on F.
For the molecule to be stable, the final force on the lithium nucleus must be zero. This can be achieved by a distortion of the spherical charge distribution of the Li+ ion. If a small amount of the 1s charge density on lithium is removed from the region adjacent to
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International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 22 ISSN 2250-3153 fluorine and placed on the side of the lithium nucleus away from the fluorine, i.e., the charge distribution is polarized away from the fluorine, it will exert a force on the lithium nucleus in a direction away from the fluorine. Thus the force on the lithium nucleus in an ionic bond can be zero only if the charge density of the Li+ ion is polarized away from the negative end of the molecule.
A similar consideration of the forces exerted on the fluorine nucleus demonstrates that the F- ion density must also be polarized. The fluorine nucleus experiences a net force of repulsion because of the presence of the lithium ion. The two negative charges centred on lithium screen only two of its three nuclear charges. Therefore, the charge density of the F- ion must be polarized towards the lithium in order to exert an attractive force on the fluorine nucleus which will balance the repulsive force arising from the presence of the Li+ ion. Thus both nuclei in the LiF molecule are bound by the increase in charge density localized in the region of the fluorine.
The charge distribution of a molecule with an ionic bond will necessarily be characterized not only by the transfer of electronic charge from one atom to another, but also by a polarization of each of the resulting ions in a direction counter to the transfer of charge, as indicated in the bond density map for LiF.
The bond density maps for N2 and LiF are shown side by side to provide a contrast of the changes in the atomic charge densities responsible for the two extremes of chemical binding. In a covalent bond the increase in charge density which binds both nuclei is shared between them. In an ionic bond both nuclei are bound by the forces exerted by the charge density localized on a single nucleus. It must be stressed that there is no fundamental difference between the forces responsible for a covalent or an ionic bond. They are electrostatic in each case.
FORMULATION OF THE PROBLEM:
NOTATION :
퐺13 : Category one of Field (We are classifying based on the characteristics of the systems to which the ion separation is executed)
퐺14 : Category Two of Field
퐺15 : Category Three of Field
푇13 :Category one of Ions (Again classification is based on systems under investigation)
푇14 : Category Two of Ions
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푇15 : Category Three of Ions
퐺16 : Category one of Laser (We are again classifying based on the different laser types used in the investigatory process)
퐺17 : Category Two of Laser
퐺18 : Category Three of Laser
푇16 :Category one: Suppression of thermal conduction of ions and hence decrease in temperature (cooling) and study of properties of ions WITH photons
푇17 : Category Two: Suppression of thermal conduction of ions and hence decrease in temperature (cooling) and study of properties of ions WITH photons
푇18 : Category Three: Suppression of thermal conduction of ions and hence decrease in temperature (cooling) and study of properties of ions WITH photons
1 1 1 1 1 1 2 2 2 푎13 , 푎14 , 푎15 , 푏13 , 푏14 , 푏15 푎16 , 푎17 , 푎18 2 2 2 푏16 , 푏17 , 푏18 : are Accentuation coefficients
′ 1 ′ 1 ′ 1 ′ 1 ′ 1 ′ 1 ′ 2 ′ 2 ′ 2 푎13 , 푎14 , 푎15 , 푏13 , 푏14 , 푏15 , 푎16 , 푎17 , 푎18 , ′ 2 ′ 2 ′ 2 푏16 , 푏17 , 푏18 are Dissipation coefficients
FIELD-ION SYSTEM:
GOVERNING EQUATIONS:
The differential system of this model is now
푑퐺 13 = 푎 1 퐺 − 푎′ 1 + 푎′′ 1 푇 , 푡 퐺 1 푑푡 13 14 13 13 14 13
푑퐺 14 = 푎 1 퐺 − 푎′ 1 + 푎′′ 1 푇 , 푡 퐺 2 푑푡 14 13 14 14 14 14
푑퐺 15 = 푎 1 퐺 − 푎′ 1 + 푎′′ 1 푇 , 푡 퐺 3 푑푡 15 14 15 15 14 15
푑푇 13 = 푏 1 푇 − 푏′ 1 − 푏′′ 1 퐺, 푡 푇 4 푑푡 13 14 13 13 13
푑푇 14 = 푏 1 푇 − 푏′ 1 − 푏′′ 1 퐺, 푡 푇 5 푑푡 14 13 14 14 14
푑푇 15 = 푏 1 푇 − 푏′ 1 − 푏′′ 1 퐺, 푡 푇 6 푑푡 15 14 15 15 15
′′ 1 + 푎13 푇14 , 푡 = First augmentation factor 7
′′ 1 − 푏13 퐺, 푡 = First detrition factor 8
LASER AND ION COOLOING SYSTEM AND STUDY OF CONCOMITANT PROPERTIES: 9
GOVERNING EQUATIONS:
The differential system of this model is now
푑퐺 16 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 퐺 10 푑푡 16 17 16 16 17 16
푑퐺 17 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 퐺 11 푑푡 17 16 17 17 17 17
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푑퐺 18 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 퐺 12 푑푡 18 17 18 18 17 18
푑푇 16 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 푇 13 푑푡 16 17 16 16 19 16
푑푇 17 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 푇 14 푑푡 17 16 17 17 19 17
푑푇 18 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 푇 15 푑푡 18 17 18 18 19 18
′′ 2 + 푎16 푇17 , 푡 = First augmentation factor 16
′′ 2 − 푏16 퐺19 , 푡 = First detrition factor 17
CONCATENATED EQUATIONS: 18
푑퐺 13 = 푎 1 퐺 − 푎′ 1 + 푎′′ 1 푇 , 푡 + 푎′′ 2,2 푇 , 푡 퐺 19 푑푡 13 14 13 13 14 16 17 13
푑퐺 14 = 푎 1 퐺 − 푎′ 1 + 푎′′ 1 푇 , 푡 + 푎′′ 2,2 푇 , 푡 퐺 20 푑푡 14 13 14 14 14 17 17 14
푑퐺 15 = 푎 1 퐺 − 푎′ 1 + 푎′′ 1 푇 , 푡 + 푎′′ 2,2 푇 , 푡 퐺 21 푑푡 15 14 15 15 14 18 17 15
′′ 1 ′′ 1 ′′ 1 22 Where 푎13 푇14 , 푡 , 푎14 푇14, 푡 , 푎15 푇14 , 푡 are first augmentation coefficients for category 1, 2 and 3 23
′′ 2,2 ′′ 2,2 ′′ 2,2 + 푎16 푇17 , 푡 , + 푎17 푇17, 푡 , + 푎18 푇17, 푡 are second augmentation coefficients for category 1, 2 and 3
24
푑푇 13 = 푏 1 푇 − 푏′ 1 − 푏′′ 1 퐺, 푡 + 푏′′ 2,2 퐺 , 푡 푇 25 푑푡 13 14 13 13 16 19 13
푑푇 14 = 푏 1 푇 − 푏′ 1 − 푏′′ 1 퐺, 푡 + 푏′′ 2,2 퐺 , 푡 푇 26 푑푡 14 13 14 14 17 19 14
푑푇 15 = 푏 1 푇 − 푏′ 1 − 푏′′ 1 퐺, 푡 + 푏′′ 2,2 퐺 , 푡 푇 27 푑푡 15 14 15 15 18 19 15
′′ 1 ′′ 1 ′′ 1 28 Where − 푏13 퐺, 푡 , − 푏14 퐺, 푡 , − 푏15 퐺, 푡 are first detrition coefficients for category 1, 2 and 3 ′′ 2,2 ′′ 2,2 ′′ 2,2 + 푏16 퐺19, 푡 , + 푏17 퐺19, 푡 , + 푏18 퐺19, 푡 are second augmentation coefficients for category 1, 2 and 3
29
푑퐺 16 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 + 푎′′ 1,1 푇 , 푡 퐺 30 푑푡 16 17 16 16 17 13 14 16
푑퐺 17 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 + 푎′′ 1,1 푇 , 푡 퐺 31 푑푡 17 16 17 17 17 14 14 17
푑퐺 18 = 푎 2 퐺 − 푎′ 2 + 푎′′ 2 푇 , 푡 + 푎′′ 1,1 푇 , 푡 퐺 32 푑푡 18 17 18 18 17 15 14 18
′′ 2 ′′ 2 ′′ 2 33 Where + 푎16 푇17, 푡 , + 푎17 푇17 , 푡 , + 푎18 푇17 , 푡 are first augmentation coefficients for category 1, 2 and 3
′′ 1,1 ′′ 1,1 ′′ 1,1 + 푎13 푇14 , 푡 , + 푎14 푇14 , 푡 , + 푎15 푇14, 푡 are second detrition coefficients for
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34
푑푇 16 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 − 푏′′ 1,1 퐺, 푡 푇 35 푑푡 16 17 16 16 19 13 16
푑푇 17 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 − 푏′′ 1,1 퐺, 푡 푇 36 푑푡 17 16 17 17 19 14 17
푑푇 18 = 푏 2 푇 − 푏′ 2 − 푏′′ 2 퐺 , 푡 − 푏′′ 1,1 퐺, 푡 푇 37 푑푡 18 17 18 18 19 15 18
′′ 2 ′′ 2 ′′ 2 38 Where − 푏16 퐺19, 푡 , − 푏17 퐺19, 푡 , − 푏18 퐺19, 푡 are first detrition coefficients for category 1, 2 and 3 ′′ 1,1 ′′ 1,1 ′′ 1,1 − 푏13 퐺, 푡 , − 푏14 퐺, 푡 , − 푏15 퐺, 푡 are second detrition coefficients for category 1, 2 and 3 39
40
Where we suppose
1 ′ 1 ′′ 1 1 ′ 1 ′′ 1 (A) 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 41
푖, 푗 = 13,14,15
′′ 1 ′′ 1 42 (B) The functions 푎푖 , 푏푖 are positive continuous increasing and bounded.
1 1 Definition of (푝푖 ) , (푟푖 ) :
′′ 1 1 (1) 푎푖 (푇14 , 푡) ≤ (푝푖 ) ≤ ( 퐴13 )
′′ 1 1 ′ 1 (1) 푏푖 (퐺, 푡) ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵 13 ) ′′ 1 1 43 (C) 푙푖푚푇2→∞ 푎푖 푇14 , 푡 = (푝푖 )
′′ 1 1 limG→∞ 푏푖 퐺, 푡 = (푟푖 )
(1) (1) Definition of ( 퐴 13 ) , ( 퐵 13 ) :
(1) (1) 1 1 Where ( 퐴 13 ) , ( 퐵 13 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 13,14,15
They satisfy Lipschitz condition: 44
(1) ′′ 1 ′ ′′ 1 (1) ′ −( 푀 13 ) 푡 45 |(푎푖 ) 푇14 , 푡 − (푎푖 ) 푇14 , 푡 | ≤ ( 푘13 ) |푇14 − 푇14 |푒
(1) 46 ′′ 1 ′ ′′ 1 (1) ′ −( 푀 13 ) 푡 |(푏푖 ) 퐺 , 푡 − (푏푖 ) 퐺, 푡 | < ( 푘13 ) ||퐺 − 퐺 ||푒
With the Lipschitz condition, we place a restriction on the behavior of functions 47 ′′ 1 ′ ′′ 1 ′ (푎푖 ) 푇14 , 푡 and(푎푖 ) 푇14 , 푡 . 푇14 , 푡 and 푇14 , 푡 are points belonging to the interval (1) (1) ′′ 1 ( 푘13 ) , ( 푀 13 ) . It is to be noted that (푎푖 ) 푇14 , 푡 is uniformly continuous. In the eventuality of (1) ′′ 1 the fact, that if ( 푀 13 ) = 1 then the function (푎푖 ) 푇14 , 푡 , the first augmentation coefficient attributable to terrestrial organisms, would be absolutely continuous.
(1) (1) Definition of ( 푀 13 ) , ( 푘13 ) : 48
(1) (1) (D) ( 푀 13 ) , ( 푘13 ) , are positive constants
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1 1 (푎푖) (푏푖) (1) , (1) < 1 ( 푀 13 ) ( 푀 13 )
(1) (1) Definition of ( 푃 13 ) , ( 푄 13 ) : 49
(1) (1) (E) There exists two constants ( 푃 13 ) and ( 푄 13 ) which together with 50 (1) (1) (1) (1) ( 푀 13 ) , ( 푘13 ) , (퐴 13) 푎푛푑 ( 퐵 13 ) and the constants 1 ′ 1 1 ′ 1 1 1 51 (푎푖 ) , (푎푖 ) , (푏푖 ) , (푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 13,14,15, 52 satisfy the inequalities 53 1 1 ′ 1 (1) (1) (1) (1) [ (푎푖 ) + (푎푖 ) + ( 퐴13 ) + ( 푃13 ) ( 푘13 ) ] < 1 ( 푀 13 )
1 1 ′ 1 (1) (1) (1) (1) [ (푏푖 ) + (푏푖 ) + ( 퐵13 ) + ( 푄13 ) ( 푘13 ) ] < 1 ( 푀 13 )
Where we suppose 54
2 ′ 2 ′′ 2 2 ′ 2 ′′ 2 (F) 푎푖 , 푎푖 , 푎푖 , 푏푖 , 푏푖 , 푏푖 > 0, 푖, 푗 = 16,17,18 55
′′ 2 ′′ 2 (G) The functions 푎푖 , 푏푖 are positive continuous increasing and bounded. 56
2 2 Definition of (pi) , (ri) : 57
′′ 2 2 2 58 푎푖 푇17 , 푡 ≤ (푝푖 ) ≤ 퐴16
′′ 2 2 ′ 2 (2) 푏푖 (퐺, 푡) ≤ (푟푖 ) ≤ (푏푖 ) ≤ ( 퐵 16 ) 59
′′ 2 2 60 (H) lim푇2→∞ 푎푖 푇17, 푡 = (푝푖 )
′′ 2 2 lim퐺→∞ 푏푖 퐺19 , 푡 = (푟푖 ) 61
(2) (2) Definition of ( 퐴 16 ) , ( 퐵 16 ) : 62
(2) (2) 2 2 Where ( 퐴 16 ) , ( 퐵 16 ) , (푝푖 ) , (푟푖 ) are positive constants and 푖 = 16,17,18
They satisfy Lipschitz condition: 63
(2) ′′ 2 ′ ′′ 2 (2) ′ −( 푀 16 ) 푡 64 |(푎푖 ) 푇17 , 푡 − (푎푖 ) 푇17, 푡 | ≤ ( 푘16 ) |푇17 − 푇17 |푒
(2) ′′ 2 ′ ′′ 2 (2) ′ −( 푀 16 ) 푡 65 |(푏푖 ) 퐺19 , 푡 − (푏푖 ) 퐺19 , 푡 | < ( 푘16 ) || 퐺19 − 퐺19 ||푒
′′ 2 ′ With the Lipschitz condition, we place a restriction on the behavior of functions (푎푖 ) 푇17 , 푡 66 ′′ 2 ′ (2) (2) and(푎푖 ) 푇17 , 푡 . 푇17 , 푡 and 푇17 , 푡 are points belonging to the interval ( 푘16 ) , ( 푀 16 ) . It is ′′ 2 (2) to be noted that (푎푖 ) 푇17 , 푡 is uniformly continuous. In the eventuality of the fact, that if ( 푀 16 ) = ′′ 2 1 then the function (푎푖 ) 푇17 , 푡 , the first augmentation coefficient attributable to terrestrial organisms, would be absolutely continuous.
(2) (2) Definition of ( 푀 16 ) , ( 푘16 ) : 67
(2) (2) (I) ( 푀 16 ) , ( 푘16 ) , are positive constants 68
2 2 (푎푖) (푏푖) (2) , (2) < 1 ( 푀 16 ) ( 푀 16 )
(2) (2) Definition of ( 푃 13 ) , ( 푄 13 ) : 69
(2) (2) There exists two constants ( 푃 16 ) and ( 푄 16 ) which together (2) (2) (2) (2) with ( 푀 16 ) , ( 푘16 ) , (퐴 16) 푎푛푑 ( 퐵 16 ) and the constants
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2 ′ 2 2 ′ 2 2 2 (푎푖 ) , (푎푖 ) , (푏푖 ) , (푏푖 ) , (푝푖 ) , (푟푖 ) , 푖 = 16,17,18,
satisfy the inequalities
1 2 ′ 2 (2) (2) (2) 70 (2) [ (ai) + (ai) + ( A16 ) + ( P16 ) ( k16 ) ] < 1 ( M 16 )
1 2 ′ 2 (2) (2) (2) 71 (2) [ (푏푖 ) + (푏푖 ) + ( 퐵16 ) + ( 푄16 ) ( 푘16 ) ] < 1 ( 푀 16 )
72
Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions 73
Definition of 퐺푖 0 , 푇푖 0 :
1 1 푀 13 푡 0 퐺푖 푡 ≤ 푃13 푒 , 퐺푖 0 = 퐺푖 > 0
(1) (1) ( 푀 13 ) 푡 0 푇푖 (푡) ≤ ( 푄13 ) 푒 , 푇푖 0 = 푇푖 > 0
74
Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions 75
Definition of 퐺푖 0 , 푇푖 0
(2) (2) ( 푀 16 ) 푡 0 퐺푖 푡 ≤ ( 푃16 ) 푒 , 퐺푖 0 = 퐺푖 > 0
(2) (2) ( 푀 16 ) 푡 0 푇푖 (푡) ≤ ( 푄16 ) 푒 , 푇푖 0 = 푇푖 > 0 76
PROOF: 77
(1) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which satisfy
0 0 0 (1) 0 (1) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃13 ) , 푇푖 ≤ ( 푄13 ) , 78
(1) 0 (1) ( 푀 13 ) 푡 79 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃13 ) 푒
(1) 0 (1) ( 푀 13 ) 푡 80 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄13 ) 푒
By 81
0 푡 1 ′ 1 ′′ 1 퐺13 푡 = 퐺13 + 0 (푎13) 퐺14 푠 13 − (푎13 ) + 푎13 ) 푇14 푠 13 , 푠 13 퐺13 푠 13 푑푠 13
0 푡 1 ′ 1 ′′ 1 82 퐺14 푡 = 퐺14 + 0 (푎14 ) 퐺13 푠 13 − (푎14) + (푎14) 푇14 푠 13 , 푠 13 퐺14 푠 13 푑푠 13
0 푡 1 ′ 1 ′′ 1 83 퐺15 푡 = 퐺15 + 0 (푎15) 퐺14 푠 13 − (푎15 ) + (푎15) 푇14 푠 13 , 푠 13 퐺15 푠 13 푑푠 13
0 푡 1 ′ 1 ′′ 1 84 푇13 푡 = 푇13 + 0 (푏13 ) 푇14 푠 13 − (푏13) − (푏13) 퐺 푠 13 , 푠 13 푇13 푠 13 푑푠 13
0 푡 1 ′ 1 ′′ 1 85 푇14 푡 = 푇14 + 0 (푏14 ) 푇13 푠 13 − (푏14) − (푏14) 퐺 푠 13 , 푠 13 푇14 푠 13 푑푠 13
0 푡 1 ′ 1 ′′ 1 86 T15 t = T15 + 0 (푏15) 푇14 푠 13 − (푏15) − (푏15) 퐺 푠 13 , 푠 13 푇15 푠 13 푑푠 13
Where 푠 13 is the integrand that is integrated over an interval 0, 푡
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87
PROOF: 88
(2) Consider operator 풜 defined on the space of sextuples of continuous functions 퐺푖 , 푇푖 : ℝ+ → ℝ+ which satisfy
0 0 0 (2) 0 (2) 퐺푖 0 = 퐺푖 , 푇푖 0 = 푇푖 , 퐺푖 ≤ ( 푃16 ) , 푇푖 ≤ ( 푄16 ) , 89
(2) 0 (2) ( 푀 16 ) 푡 90 0 ≤ 퐺푖 푡 − 퐺푖 ≤ ( 푃16 ) 푒
(2) 0 (2) ( 푀 16 ) 푡 91 0 ≤ 푇푖 푡 − 푇푖 ≤ ( 푄16 ) 푒
By 92
0 푡 2 ′ 2 ′′ 2 퐺16 푡 = 퐺16 + 0 (푎16) 퐺17 푠 16 − (푎16 ) + 푎16 ) 푇17 푠 16 , 푠 16 퐺16 푠 16 푑푠 16
0 푡 2 ′ 2 ′′ 2 93 퐺17 푡 = 퐺17 + 0 (푎17 ) 퐺16 푠 16 − (푎17) + (푎17 ) 푇17 푠 16 , 푠 17 퐺17 푠 16 푑푠 16
0 푡 2 ′ 2 ′′ 2 94 퐺18 푡 = 퐺18 + 0 (푎18 ) 퐺17 푠 16 − (푎18) + (푎18 ) 푇17 푠 16 , 푠 16 퐺18 푠 16 푑푠 16
0 푡 2 ′ 2 ′′ 2 95 푇16 푡 = 푇16 + 0 (푏16 ) 푇17 푠 16 − (푏16) − (푏16) 퐺 푠 16 , 푠 16 푇16 푠 16 푑푠 16
0 푡 2 ′ 2 ′′ 2 96 푇17 푡 = 푇17 + 0 (푏17 ) 푇16 푠 16 − (푏17) − (푏17) 퐺 푠 16 , 푠 16 푇17 푠 16 푑푠 16
0 푡 2 ′ 2 ′′ 2 97 푇18 푡 = 푇18 + 0 (푏18 ) 푇17 푠 16 − (푏18) − (푏18) 퐺 푠 16 , 푠 16 푇18 푠 16 푑푠 16
Where 푠 16 is the integrand that is integrated over an interval 0, 푡
(a) The operator 풜(1) maps the space of functions satisfying 34,35,36 into itself .Indeed it is obvious 98 that
푡 (1) 0 1 0 (1) ( 푀13 ) 푠 13 퐺13 푡 ≤ 퐺13 + 0 (푎13) 퐺14+( 푃13 ) 푒 푑푠 13 =
(푎 ) 1 ( 푃 )(1) (1) 1 0 13 13 ( 푀 13 ) 푡 1 + (푎13 ) 푡 퐺14 + (1) 푒 − 1 ( 푀 13 )
From which it follows that 99
( 푃 )(1)+퐺0 1 − 13 14 (1) (푎13 ) 0 0 −( 푀13 ) 푡 (1) 0 퐺14 (1) 퐺13 푡 − 퐺13 푒 ≤ (1) ( 푃13 ) + 퐺14 푒 + ( 푃13 ) ( 푀 13 )
0 퐺푖 is as defined in the statement of theorem 1
Analogous inequalities hold also for 퐺14 , 퐺15, 푇13 , 푇14 , 푇15 100
(b) The operator 풜(2) maps the space of functions satisfying 34,35,36 into itself .Indeed it is obvious 101 that
푡 (2) 0 2 0 (6) ( 푀16 ) 푠 16 102 퐺16 푡 ≤ 퐺16 + 0 (푎16) 퐺17+( 푃16 ) 푒 푑푠 16 =
(푎 ) 2 ( 푃 )(2) (2) 2 0 16 16 ( 푀 16 ) 푡 1 + (푎16 ) 푡 퐺17 + (2) 푒 − 1 ( 푀 16 )
From which it follows that 103
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( 푃 )(2)+퐺0 2 − 16 17 (2) (푎16 ) 0 0 −( 푀16 ) 푡 (2) 0 퐺17 (2) 퐺16 푡 − 퐺16 푒 ≤ (2) ( 푃16 ) + 퐺17 푒 + ( 푃16 ) ( 푀 16 )
Analogous inequalities hold also for 퐺17 , 퐺18, 푇16 , 푇17 , 푇18 104
1 1 (푎푖) (푏푖) 105 It is now sufficient to take (1) , (1) < 1 and to choose ( 푀 13 ) ( 푀 13 )
(1) (1) ( P 13 ) and ( Q 13 ) large to have
(1) 0 ( 푃13 ) +퐺푗 106 − (푎 ) 1 0 푖 1 (1) 0 퐺푗 (1) 1 ( 푃13) + ( 푃13 ) + 퐺푗 푒 ≤ ( 푃13 ) (푀 13)
107
(1) 0 ( 푄13 ) +푇푗 − (푏 ) 1 0 푖 (1) 0 푇푗 (1) (1) 1 ( 푄13 ) + 푇푗 푒 + ( 푄13 ) ≤ ( 푄13 ) (푀 13)
(1) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 satisfying 34,35,36 108 into itself
The operator 풜(1) is a contraction with respect to the metric 109
푑 퐺 1 , 푇 1 , 퐺 2 , 푇 2 = 110
1 1 1 2 −(푀 13) 푡 1 2 −(푀 13) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+
Indeed if we denote 111
Definition of 퐺 , 푇 : 퐺 , 푇 = 풜(1)(퐺, 푇)
It results
푡 1 1 1 2 1 1 2 −( 푀13) 푠 13 ( 푀13) 푠 13 퐺13 − 퐺푖 ≤ 0 (푎13) 퐺14 − 퐺14 푒 푒 푑푠 13 +
푡 1 1 ′ 1 1 2 −( 푀13) 푠 13 −( 푀13) 푠 13 0 {(푎13 ) 퐺13 − 퐺13 푒 푒 +
1 1 ′′ 1 1 1 2 −( 푀13) 푠 13 ( 푀13) 푠 13 (푎13) 푇14 , 푠 13 퐺13 − 퐺13 푒 푒 +
1 1 2 ′′ 1 1 ′′ 1 2 −( 푀13) 푠 13 ( 푀13) 푠 13 퐺13 |(푎13 ) 푇14 , 푠 13 − (푎13 ) 푇14 , 푠 13 | 푒 푒 }푑푠 13
Where 푠 13 represents integrand that is integrated over the interval 0, t
From the hypotheses on 25,26,27,28 and 29 it follows
1 퐺 1 − 퐺 2 푒−( 푀13) 푡 ≤ 112 1 1 ′ 1 1 1 1 1 1 2 2 1 (푎13) + (푎13) + ( 퐴13) + ( 푃13) ( 푘13) 푑 퐺 , 푇 ; 퐺 , 푇 ( 푀13)
And analogous inequalities for 퐺푖 푎푛푑 푇푖 . Taking into account the hypothesis (34,35,36) the result follows
′′ 1 ′′ 1 Remark 1: The fact that we supposed (푎13) and (푏13 ) depending also on t can be considered as not 113
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International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 30 ISSN 2250-3153 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 1 1 1 ( 푀13) 푡 1 ( 푀13) 푡 necessary to prove the uniqueness of the solution bounded by ( 푃13) 푒 푎푛푑 ( 푄13) 푒 respectively of ℝ+.
If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it ′′ 1 ′′ 1 suffices to consider that (푎푖 ) and (푏푖 ) , 푖 = 13,14,15 depend only on T14 and respectively on 퐺(푎푛푑 푛표푡 표푛 푡) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any 푡 where 퐺푖 푡 = 0 푎푛푑 푇푖 푡 = 0 114
From 19 to 24 it results
푡 ′ 1 ′′ 1 0 − 0 (푎푖 ) −(푎푖 ) 푇14 푠 13 ,푠 13 푑푠 13 퐺푖 푡 ≥ 퐺푖 푒 ≥ 0
′ 1 0 −(푏푖 ) 푡 푇푖 푡 ≥ 푇푖 푒 > 0 for t > 0
Definition of ( 푀 ) 1 , ( 푀 ) 1 푎푛푑 ( 푀 ) 1 : 115 13 1 13 2 13 3
Remark 3: if 퐺13 is bounded, the same property have also 퐺14 푎푛푑 퐺15 . indeed if
푑퐺 퐺 < ( 푀 ) 1 it follows 14 ≤ ( 푀 ) 1 − (푎′ ) 1 퐺 and by integrating 13 13 푑푡 13 1 14 14
퐺 ≤ ( 푀 ) 1 = 퐺0 + 2(푎 ) 1 ( 푀 ) 1 /(푎′ ) 1 14 13 2 14 14 13 1 14
In the same way , one can obtain
퐺 ≤ ( 푀 ) 1 = 퐺0 + 2(푎 ) 1 ( 푀 ) 1 /(푎′ ) 1 15 13 3 15 15 13 2 15 116
If 퐺14 표푟 퐺15 is bounded, the same property follows for 퐺13 , 퐺15 and 퐺13 , 퐺14 respectively.
Remark 4: If 퐺13 푖푠 bounded, from below, the same property holds for 퐺14 푎푛푑 퐺15 . The proof is 117 analogous with the preceding one. An analogous property is true if 퐺14 is bounded from below.
′′ 1 ′ 1 Remark 5: If T13 is bounded from below and lim푡→∞ ((푏푖 ) (퐺 푡 , 푡)) = (푏14 ) then 푇14 → ∞. 118
1 Definition of 푚 and 휀1 :
Indeed let 푡1 be so that for 푡 > 푡1
1 ′′ 1 1 (푏14) − (푏푖 ) (퐺 푡 , 푡) < 휀1, 푇13 (푡) > 푚
푑푇 Then 14 ≥ (푎 ) 1 푚 1 − 휀 푇 which leads to 119 푑푡 14 1 14
(푎 ) 1 푚 1 1 14 −휀1푡 0 −휀1푡 −휀1푡 푇14 ≥ 1 − 푒 + 푇14 푒 If we take t such that 푒 = it results 휀1 2
1 1 (푎14 ) 푚 2 푇14 ≥ , 푡 = 푙표푔 By taking now 휀1 sufficiently small one sees that T14 is unbounded. 2 휀1 ′′ 1 ′ 1 The same property holds for 푇15 if lim푡→∞ (푏15 ) 퐺 푡 , 푡 = (푏15 )
We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to 42
120
2 2 (푎푖) (푏푖) 121 It is now sufficient to take (2) , (2) < 1 and to choose ( 푀 16 ) ( 푀 16 )
(2) (2) ( 푃 16 ) 푎푛푑 ( 푄 16 ) large to have
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(2) 0 ( 푃16 ) +퐺푗 122 − (푎 ) 2 0 푖 2 (2) 0 퐺푗 (2) 2 ( 푃16) + ( 푃16 ) + 퐺푗 푒 ≤ ( 푃16 ) (푀 16)
(2) 0 ( 푄16 ) +푇푗 123 − (푏 ) 2 0 푖 (2) 0 푇푗 (2) (2) 2 ( 푄16 ) + 푇푗 푒 + ( 푄16 ) ≤ ( 푄16 ) (푀 16)
(2) In order that the operator 풜 transforms the space of sextuples of functions 퐺푖 , 푇푖 satisfying 34,35,36 124 into itself
The operator 풜(2) is a contraction with respect to the metric 125
1 1 2 2 푑 퐺19 , 푇19 , 퐺19 , 푇19 =
2 2 1 2 −(푀 16) 푡 1 2 −(푀 16) 푡 푠푢푝{푚푎푥 퐺푖 푡 − 퐺푖 푡 푒 , 푚푎푥 푇푖 푡 − 푇푖 푡 푒 } 푖 푡∈ℝ+ 푡∈ℝ+
Indeed if we denote 126
(2) Definition of 퐺 19, 푇 19 : 퐺 19, 푇 19 = 풜 (퐺19, 푇19)
It results 127
푡 2 2 1 2 2 1 2 −( 푀16) 푠 16 ( 푀16) 푠 16 128 퐺16 − 퐺푖 ≤ 0 (푎16) 퐺17 − 퐺17 푒 푒 푑푠 16 +
푡 2 2 ′ 2 1 2 −( 푀16) 푠 16 −( 푀16) 푠 16 0 {(푎16 ) 퐺16 − 퐺16 푒 푒 +
2 2 ′′ 2 1 1 2 −( 푀16) 푠 16 ( 푀16) 푠 16 (푎16) 푇17 , 푠 16 퐺16 − 퐺16 푒 푒 +
2 2 2 ′′ 2 1 ′′ 2 2 −( 푀16) 푠 16 ( 푀16) 푠 16 퐺16 |(푎16 ) 푇17 , 푠 16 − (푎16 ) 푇17 , 푠 16 | 푒 푒 }푑푠 16
Where 푠 16 represents integrand that is integrated over the interval 0, 푡 129
From the hypotheses on 25,26,27,28 and 29 it follows
2 1 2 −( M16) t 130 퐺19 − 퐺19 e ≤ 1 2 ′ 2 2 2 2 1 1 2 2 2 (푎16 ) + (푎16 ) + ( A16) + ( P16) ( 푘16) d 퐺19 , 푇19 ; 퐺19 , 푇19 ( M16)
And analogous inequalities for G푖 and T푖 . Taking into account the hypothesis (34,35,36) the result follows 131
′′ 2 ′′ 2 Remark 1: The fact that we supposed (푎16) and (푏16 ) depending also on t can be considered as not 132 conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition 2 2 2 ( M16) t 2 ( M16) t necessary to prove the uniqueness of the solution bounded by ( P16) e and ( Q16) e respectively of ℝ+.
If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it ′′ 2 ′′ 2 suffices to consider that (푎푖 ) and (푏푖 ) , 푖 = 16,17,18 depend only on T17 and respectively on 퐺19 (and not on t) and hypothesis can replaced by a usual Lipschitz condition.
Remark 2: There does not exist any t where G푖 t = 0 and T푖 t = 0 133
From 19 to 24 it results
t ′ 2 ′′ 2 0 − 0 (푎푖 ) −(푎푖 ) T17 푠 16 ,푠 16 d푠 16 G푖 t ≥ G푖 e ≥ 0
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′ 2 0 −(푏푖 ) t T푖 t ≥ T푖 e > 0 for t > 0
Definition of ( M ) 2 , ( M ) 2 and ( M ) 2 : 134 16 1 16 2 16 3
Remark 3: if G16 is bounded, the same property have also G17 and G18 . indeed if
dG G < ( M ) 2 it follows 17 ≤ ( M ) 2 − (푎′ ) 2 G and by integrating 16 16 dt 16 1 17 17
G ≤ ( M ) 2 = G0 + 2(푎 ) 2 ( M ) 2 /(푎′ ) 2 17 16 2 17 17 16 1 17 In the same way , one can obtain
G ≤ ( M ) 2 = G0 + 2(푎 ) 2 ( M ) 2 /(푎′ ) 2 18 16 3 18 18 16 2 18
If G17 or G18 is bounded, the same property follows for G16 , G18 and G16 , G17 respectively.
Remark 4: If G16 is bounded, from below, the same property holds for G17 and G18 . The proof is 135 analogous with the preceding one. An analogous property is true if G17 is bounded from below.
′′ 2 ′ 2 Remark 5: If T16 is bounded from below and limt→∞ ((푏푖 ) ( 퐺19 t , t)) = (푏17) then T17 → ∞. 136
2 Definition of 푚 and ε2 :
Indeed let t2 be so that for t > t2
2 ′′ 2 2 (푏17) − (푏푖 ) ( 퐺19 t , t) < ε2, T16 (t) > 푚 137
dT Then 17 ≥ (푎 ) 2 푚 2 − ε T which leads to 138 dt 17 2 17
(푎 ) 2 푚 2 1 17 −ε2t 0 −ε2t −ε2t T17 ≥ 1 − e + T17e If we take t such that e = it results ε2 2
2 2 (푎17 ) 푚 2 139 T17 ≥ , 푡 = log By taking now ε2 sufficiently small one sees that T17 is unbounded. 2 ε2 ′′ 2 ′ 2 The same property holds for T18 if lim푡→∞ (푏18 ) 퐺19 t , t = (푏18)
We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to 42
140
Behavior of the solutions of equation 37 to 42 141
Theorem 2: If we denote and define
1 1 1 1 Definition of (휎1) , (휎2) , (휏1) , (휏2) :
1 1 1 1 (a) 휎1) , (휎2) , (휏1) , (휏2) four constants satisfying
1 ′ 1 ′ 1 ′′ 1 ′′ 1 1 −(휎2) ≤ −(푎13) + (푎14) − (푎13 ) 푇14 , 푡 + (푎14 ) 푇14 , 푡 ≤ −(휎1)
1 ′ 1 ′ 1 ′′ 1 ′′ 1 1 −(휏2) ≤ −(푏13 ) + (푏14) − (푏13 ) 퐺, 푡 − (푏14) 퐺, 푡 ≤ −(휏1)
142
1 1 1 1 1 1 Definition of (휈1) , (휈2) , (푢1) , (푢2) , 휈 , 푢 :
1 1 1 1 (b) By (휈1) > 0 , (휈2) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots of the equations
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1 1 2 1 1 1 1 1 2 1 1 1 (푎14) 휈 + (휎1) 휈 − (푎13 ) = 0 and (푏14) 푢 + (휏1) 푢 − (푏13 ) = 0
1 1 1 1 Definition of (휈1 ) , , (휈2 ) , (푢 1) , (푢 2) : 143
1 1 1 1 By (휈1 ) > 0 , (휈2 ) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the roots of the equations 1 1 2 1 1 1 1 1 2 1 1 1 (푎14) 휈 + (휎2) 휈 − (푎13) = 0 and (푏14) 푢 + (휏2) 푢 − (푏13) = 0
1 1 1 1 1 Definition of (푚1) , (푚2) , (휇1) , (휇2) , (휈0) :- 144
1 1 1 1 (c) If we define (푚1) , (푚2) , (휇1) , (휇2) by
1 1 1 1 1 1 (푚2) = (휈0) , (푚1) = (휈1) , 푖푓 (휈0) < (휈1)
1 1 1 1 1 1 1 (푚2) = (휈1) , (푚1) = (휈1 ) , 푖푓 (휈1) < (휈0) < (휈1 ) ,
0 1 퐺13 and (휈0) = 0 퐺14
1 1 1 1 1 1 ( 푚2) = (휈1) , (푚1) = (휈0) , 푖푓 (휈1 ) < (휈0) and analogously 145
1 1 1 1 1 1 (휇2) = (푢0) , (휇1) = (푢1) , 푖푓 (푢0) < (푢1)
1 1 1 1 1 1 1 (휇2) = (푢1) , (휇1) = (푢 1) , 푖푓 (푢1) < (푢0) < (푢 1) ,
푇0 and (푢 ) 1 = 13 0 푇0 14 146 1 1 1 1 1 1 1 1 ( 휇2) = (푢1) , (휇1) = (푢0) , 푖푓 (푢 1) < (푢0) where (푢1) , (푢 1)
are defined by 59 and 61 respectively
Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities 147
1 1 1 0 (푆1) −(푝13) 푡 0 (푆1) 푡 퐺13푒 ≤ 퐺13(푡) ≤ 퐺13푒
1 where (푝푖 ) is defined by equation 25
1 1 1 1 1 0 (푆1) −(푝13) 푡 0 (푆1) 푡 1 퐺13푒 ≤ 퐺14(푡) ≤ 1 퐺13푒 (푚1) (푚2)
(푎 ) 1 퐺0 1 1 1 1 15 13 (푆1) −(푝13) 푡 −(푆2) 푡 0 −(푆2) 푡 148 ( 1 1 1 1 푒 − 푒 + 퐺15푒 ≤ 퐺15 (푡) ≤ (푚1) (푆1) −(푝13) −(푆2) (푎 ) 1 퐺0 1 ′ 1 ′ 1 15 13 (푆1) 푡 −(푎15 ) 푡 0 −(푎15) 푡 1 1 ′ 1 [푒 − 푒 ] + 퐺15푒 ) (푚2) (푆1) −(푎15)
1 1 1 0 (푅1) 푡 0 (푅1) +(푟13) 푡 149 푇13 푒 ≤ 푇13 (푡) ≤ 푇13 푒
1 1 1 1 1 0 (푅1) 푡 0 (푅1) +(푟13) 푡 150 1 푇13 푒 ≤ 푇13 (푡) ≤ 1 푇13푒 (휇1) (휇2)
(푏 ) 1 푇0 1 ′ 1 ′ 1 15 13 (푅1) 푡 −(푏15) 푡 0 −(푏15) 푡 151 1 1 ′ 1 푒 − 푒 + 푇15푒 ≤ 푇15 (푡) ≤ (휇1) (푅1) −(푏15)
(푎 ) 1 푇0 1 1 1 1 15 13 (푅1) +(푟13) 푡 −(푅2) 푡 0 −(푅2) 푡 1 1 1 1 푒 − 푒 + 푇15 푒 (휇2) (푅1) +(푟13) +(푅2)
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1 1 1 1 Definition of (푆1) , (푆2) , (푅1) , (푅2) :- 152
1 1 1 ′ 1 Where (푆1) = (푎13) (푚2) − (푎13 )
1 1 1 (푆2) = (푎15) − (푝15 )
1 1 1 ′ 1 (푅1) = (푏13) (휇2) − (푏13)
1 ′ 1 1 (푅2) = (푏15) − (푟15 )
Behavior of the solutions of equation 37 to 42 153
Theorem 2: If we denote and define
2 2 2 2 Definition of (σ1) , (σ2) , (τ1) , (τ2) : 154
2 2 2 2 (d) σ1) , (σ2) , (τ1) , (τ2) four constants satisfying
2 ′ 2 ′ 2 ′′ 2 ′′ 2 2 −(σ2) ≤ −(푎16 ) + (푎17 ) − (푎16) T17 , 푡 + (푎17 ) T17 , 푡 ≤ −(σ1) 155
2 ′ 2 ′ 2 ′′ 2 ′′ 2 2 −(τ2) ≤ −(푏16) + (푏17 ) − (푏16) 퐺19 , 푡 − (푏17) 퐺19 , 푡 ≤ −(τ1) 156
2 2 2 2 Definition of (휈1) , (ν2) , (푢1) , (푢2) : 157
2 2 2 2 By (휈1) > 0 , (ν2) < 0 and respectively (푢1) > 0 , (푢2) < 0 the roots 158
2 2 2 2 2 2 159 (e) of the equations (푎17 ) 휈 + (σ1) 휈 − (푎16 ) = 0
2 2 2 2 2 2 160 and (푏14) 푢 + (τ1) 푢 − (푏16) = 0 and
2 2 2 2 Definition of (휈1 ) , , (휈2 ) , (푢 1) , (푢 2) : 161
2 2 2 2 By (휈1 ) > 0 , (ν 2) < 0 and respectively (푢 1) > 0 , (푢 2) < 0 the 162
2 2 2 2 2 2 163 roots of the equations (푎17 ) 휈 + (σ2) 휈 − (푎16 ) = 0
2 2 2 2 2 2 164 and (푏17) 푢 + (τ2) 푢 − (푏16 ) = 0
2 2 2 2 Definition of (푚1) , (푚2) , (휇1) , (휇2) :- 165
2 2 2 2 (f) If we define (푚1) , (푚2) , (휇1) , (휇2) by 166
2 2 2 2 2 2 (푚2) = (휈0) , (푚1) = (휈1) , 푖푓 (휈0) < (휈1) 167
2 2 2 2 2 2 2 (푚2) = (휈1) , (푚1) = (휈1 ) , 푖푓 (휈1) < (휈0) < (휈1 ) , 168
0 2 G16 and (휈0) = 0 G17
2 2 2 2 2 2 ( 푚2) = (휈1) , (푚1) = (휈0) , 푖푓 (휈1 ) < (휈0) 169 and analogously 170
2 2 2 2 2 2 (휇2) = (푢0) , (휇1) = (푢1) , 푖푓 (푢0) < (푢1)
2 2 2 2 2 2 2 (휇2) = (푢1) , (휇1) = (푢 1) , 푖푓 (푢1) < (푢0) < (푢 1) ,
0 2 T16 and (푢0) = 0 T17
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2 2 2 2 2 2 ( 휇2) = (푢1) , (휇1) = (푢0) , 푖푓 (푢 1) < (푢0) 171
Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities 172
2 2 2 0 (S1) −(푝16) t 0 (S1) t G16e ≤ 퐺16 푡 ≤ G16e
2 (푝푖 ) is defined by equation 25 173
1 2 2 1 2 0 (S1) −(푝16) t 0 (S1) t 174 2 G16e ≤ 퐺17(푡) ≤ 2 G16e (푚1) (푚2)
(푎 ) 2 G0 2 2 2 2 18 16 (S1) −(푝16) t −(S2) t 0 −(S2) t 175 ( 2 2 2 2 e − e + G18e ≤ G18(푡) ≤ (푚1) (S1) −(푝16) −(S2) (푎 ) 2 G0 2 ′ 2 ′ 2 18 16 (S1) t −(푎18 ) t 0 −(푎18) t 2 2 ′ 2 [e − e ] + G18e ) (푚2) (S1) −(푎18)
2 2 2 0 (R1) 푡 0 (R1) +(푟16) 푡 176 T16 e ≤ 푇16(푡) ≤ T16 e
1 2 1 2 2 0 (R1) 푡 0 (R1) +(푟16) 푡 177 2 T16 e ≤ 푇16 (푡) ≤ 2 T16 e (휇1) (휇2)
(푏 ) 2 T0 2 ′ 2 ′ 2 18 16 (R1) 푡 −(푏18) 푡 0 −(푏18) 푡 178 2 2 ′ 2 e − e + T18 e ≤ 푇18 (푡) ≤ (휇1) (R1) −(푏18)
(푎 ) 2 T0 2 2 2 2 18 16 (R1) +(푟16) 푡 −(R2) 푡 0 −(R2) 푡 2 2 2 2 e − e + T18e (휇2) (R1) +(푟16) +(R2)
2 2 2 2 Definition of (S1) , (S2) , (R1) , (R2) :- 179
2 2 2 ′ 2 Where (S1) = (푎16) (푚2) − (푎16 ) 180
2 2 2 (S2) = (푎18) − (푝18 )
2 2 1 ′ 2 (푅1) = (푏16) (휇2) − (푏16 ) 181
2 ′ 2 2 (R2) = (푏18) − (푟18 ) 182
PROOF : From 19,20,21,22,23,24 we obtain 183
푑휈 1 = (푎 ) 1 − (푎′ ) 1 − (푎′ ) 1 + (푎′′ ) 1 푇 , 푡 − (푎′′ ) 1 푇 , 푡 휈 1 − (푎 ) 1 휈 1 푑푡 13 13 14 13 14 14 14 14 퐺 Definition of 휈 1 :- 휈 1 = 13 퐺14
It follows
2 푑휈 1 2 − (푎 ) 1 휈 1 + (휎 ) 1 휈 1 − (푎 ) 1 ≤ ≤ − (푎 ) 1 휈 1 + (휎 ) 1 휈 1 − (푎 ) 1 14 2 13 푑푡 14 1 13
From which one obtains
1 1 Definition of (휈1 ) , (휈0) :-
0 1 퐺13 1 1 (a) For 0 < (휈0) = 0 < (휈1) < (휈1 ) 퐺14
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− 푎 1 (휈 ) 1 −(휈 ) 1 푡 (휈 ) 1 +(퐶) 1 (휈 ) 1 푒 14 1 0 (휈 ) 1 −(휈 ) 1 휈 1 (푡) ≥ 1 2 , (퐶) 1 = 1 0 − 푎 1 (휈 ) 1 −(휈 ) 1 푡 (휈 ) 1 −(휈 ) 1 1+(퐶) 1 푒 14 1 0 0 2
1 1 1 it follows (휈0) ≤ 휈 (푡) ≤ (휈1)
In the same manner , we get 184
− 푎 1 (휈 ) 1 −(휈 ) 1 푡 (휈 ) 1 +(퐶 ) 1 (휈 ) 1 푒 14 1 2 (휈 ) 1 −(휈 ) 1 휈 1 (푡) ≤ 1 2 , (퐶 ) 1 = 1 0 − 푎 1 (휈 ) 1 −(휈 ) 1 푡 (휈 ) 1 −(휈 ) 1 1+(퐶 ) 1 푒 14 1 2 0 2
1 1 1 From which we deduce (휈0) ≤ 휈 (푡) ≤ (휈1 )
0 1 1 퐺13 1 185 (b) If 0 < (휈1) < (휈0) = 0 < (휈1 ) we find like in the previous case, 퐺14
− 푎 1 (휈 ) 1 −(휈 ) 1 푡 (휈 ) 1 + 퐶 1 (휈 ) 1 푒 14 1 2 (휈 ) 1 ≤ 1 2 ≤ 휈 1 푡 ≤ 1 − 푎 1 (휈 ) 1 −(휈 ) 1 푡 1+ 퐶 1 푒 14 1 2
− 푎 1 (휈 ) 1 −(휈 ) 1 푡 (휈 ) 1 + 퐶 1 (휈 ) 1 푒 14 1 2 1 2 ≤ (휈 ) 1 − 푎 1 (휈 ) 1 −(휈 ) 1 푡 1 1+ 퐶 1 푒 14 1 2
0 1 1 1 퐺13 186 (c) If 0 < (휈1) ≤ (휈1 ) ≤ (휈0) = 0 , we obtain 퐺14
− 푎 1 (휈 ) 1 −(휈 ) 1 푡 (휈 ) 1 + 퐶 1 (휈 ) 1 푒 14 1 2 (휈 ) 1 ≤ 휈 1 푡 ≤ 1 2 ≤ (휈 ) 1 1 − 푎 1 (휈 ) 1 −(휈 ) 1 푡 0 1+ 퐶 1 푒 14 1 2 187 And so with the notation of the first part of condition (c) , we have
Definition of 휈 1 푡 :-
1 1 1 1 퐺13 푡 (푚2) ≤ 휈 푡 ≤ (푚1) , 휈 푡 = 퐺14 푡
In a completely analogous way, we obtain
Definition of 푢 1 푡 :-
1 1 1 1 푇13 푡 (휇2) ≤ 푢 푡 ≤ (휇1) , 푢 푡 = 푇14 푡
Now, using this result and replacing it in 19, 20,21,22,23, and 24 we get easily the result stated in the theorem.
Particular case :
′′ 1 ′′ 1 1 1 1 1 1 If (푎13) = (푎14) , 푡푕푒푛 (휎1) = (휎2) and in this case (휈1) = (휈1 ) if in addition (휈0) = 1 1 1 1 1 (휈1) then 휈 푡 = (휈0) and as a consequence 퐺13(푡) = (휈0) 퐺14(푡) this also defines (휈0) for the special case
′′ 1 ′′ 1 1 1 Analogously if (푏13) = (푏14) , 푡푕푒푛 (휏1) = (휏2) and then
1 1 1 1 1 (푢1) = (푢 1) if in addition (푢0) = (푢1) then 푇13(푡) = (푢0) 푇14 (푡) This is an important 1 1 1 consequence of the relation between (휈1) and (휈1 ) , and definition of (푢0) .
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PROOF : From 19,20,21,22,23,24 we obtain 188
d휈 2 = (푎 ) 2 − (푎′ ) 2 − (푎′ ) 2 + (푎′′ ) 2 T , t − (푎′′ ) 2 T , t 휈 2 − (푎 ) 2 휈 2 dt 16 16 17 16 17 17 17 17
G16 189 Definition of 휈 2 :- 휈 2 = G17
It follows 190
2 d휈 2 2 − (푎 ) 2 휈 2 + (σ ) 2 휈 2 − (푎 ) 2 ≤ ≤ − (푎 ) 2 휈 2 + (σ ) 2 휈 2 − (푎 ) 2 17 2 16 dt 17 1 16
From which one obtains 191
2 2 Definition of (휈1 ) , (휈0) :-
0 2 G16 2 2 (d) For 0 < (휈0) = 0 < (휈1) < (휈1 ) G17
− 푎 2 (휈 ) 2 −(휈 ) 2 푡 (휈 ) 2 +(C) 2 (휈 ) 2 푒 17 1 0 (휈 ) 2 −(휈 ) 2 휈 2 (푡) ≥ 1 2 , (C) 2 = 1 0 − 푎 2 (휈 ) 2 −(휈 ) 2 푡 (휈 ) 2 −(휈 ) 2 1+(C) 2 푒 17 1 0 0 2
2 2 2 it follows (휈0) ≤ 휈 (푡) ≤ (휈1)
In the same manner , we get 192
− 푎 2 (휈 ) 2 −(휈 ) 2 푡 (휈 ) 2 +(C ) 2 (휈 ) 2 푒 17 1 2 (휈 ) 2 −(휈 ) 2 휈 2 (푡) ≤ 1 2 , (C ) 2 = 1 0 − 푎 2 (휈 ) 2 −(휈 ) 2 푡 (휈 ) 2 −(휈 ) 2 1+(C ) 2 푒 17 1 2 0 2
2 2 2 From which we deduce (휈0) ≤ 휈 (푡) ≤ (휈1 ) 193
0 2 2 G16 2 194 (e) If 0 < (휈1) < (휈0) = 0 < (휈1 ) we find like in the previous case, G17 − 푎 2 (휈 ) 2 −(휈 ) 2 푡 (휈 ) 2 + C 2 (휈 ) 2 푒 17 1 2 (휈 ) 2 ≤ 1 2 ≤ 휈 2 푡 ≤ 1 − 푎 2 (휈 ) 2 −(휈 ) 2 푡 1+ C 2 푒 17 1 2
− 푎 2 (휈 ) 2 −(휈 ) 2 푡 (휈 ) 2 + C 2 (휈 ) 2 푒 17 1 2 1 2 ≤ (휈 ) 2 − 푎 2 (휈 ) 2 −(휈 ) 2 푡 1 1+ C 2 푒 17 1 2
0 2 2 2 G16 195 (f) If 0 < (휈1) ≤ (휈1 ) ≤ (휈0) = 0 , we obtain G17
− 푎 2 (휈 ) 2 −(휈 ) 2 푡 (휈 ) 2 + C 2 (휈 ) 2 푒 17 1 2 (휈 ) 2 ≤ 휈 2 푡 ≤ 1 2 ≤ (휈 ) 2 1 − 푎 2 (휈 ) 2 −(휈 ) 2 푡 0 1+ C 2 푒 17 1 2
And so with the notation of the first part of condition (c) , we have
Definition of 휈 2 푡 :- 196
2 2 2 2 퐺16 푡 (푚2) ≤ 휈 푡 ≤ (푚1) , 휈 푡 = 퐺17 푡
In a completely analogous way, we obtain 197
Definition of 푢 2 푡 :-
2 2 2 2 푇16 푡 (휇2) ≤ 푢 푡 ≤ (휇1) , 푢 푡 = 푇17 푡
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Now, using this result and replacing it in 19, 20,21,22,23, and 24 we get easily the result stated in the 198 theorem.
Particular case : 199
′′ 2 ′′ 2 2 2 2 2 2 If (푎16) = (푎17) , 푡푕푒푛 (σ1) = (σ2) and in this case (휈1) = (휈1 ) if in addition (휈0) = 2 2 2 2 (휈1) then 휈 푡 = (휈0) and as a consequence 퐺16(푡) = (휈0) 퐺17(푡)
′′ 2 ′′ 2 2 2 Analogously if (푏16) = (푏17) , 푡푕푒푛 (τ1) = (τ2) and then
2 2 2 2 2 (푢1) = (푢 1) if in addition (푢0) = (푢1) then 푇16(푡) = (푢0) 푇17 (푡) This is an important 2 2 consequence of the relation between (휈1) and (휈1 )
We can prove the following 200
′′ 1 ′′ 1 Theorem 3: If (푎푖 ) 푎푛푑 (푏푖 ) are independent on 푡 , and the conditions (with the notations 25,26,27,28)
′ 1 ′ 1 1 1 (푎13) (푎14) − 푎13 푎14 < 0
′ 1 ′ 1 1 1 1 1 ′ 1 1 1 1 (푎13) (푎14) − 푎13 푎14 + 푎13 푝13 + (푎14) 푝14 + 푝13 푝14 > 0
′ 1 ′ 1 1 1 (푏13) (푏14) − 푏13 푏14 > 0 ,
′ 1 ′ 1 1 1 ′ 1 1 ′ 1 1 1 1 (푏13) (푏14) − 푏13 푏14 − (푏13) 푟14 − (푏14) 푟14 + 푟13 푟14 < 0
1 1 푤푖푡푕 푝13 , 푟14 as defined by equation 25 are satisfied , then the system
′′ 2 ′′ 2 Theorem 3: If (푎푖 ) 푎푛푑 (푏푖 ) are independent on t , and the conditions (with the notations 201 25,26,27,28) 202
′ 2 ′ 2 2 2 (푎16) (푎17) − 푎16 푎17 < 0 203
′ 2 ′ 2 2 2 2 2 ′ 2 2 2 2 (푎16) (푎17) − 푎16 푎17 + 푎16 푝16 + (푎17) 푝17 + 푝16 푝17 > 0 204
′ 2 ′ 2 2 2 (푏16) (푏17) − 푏16 푏17 > 0 , 205
′ 2 ′ 2 2 2 ′ 2 2 ′ 2 2 2 2 (푏16) (푏17) − 푏16 푏17 − (푏16) 푟17 − (푏17) 푟17 + 푟16 푟17 < 0 206
2 2 푤푖푡푕 푝16 , 푟17 as defined by equation 25 are satisfied , then the system
207
1 ′ 1 ′′ 1 푎13 퐺14 − (푎13 ) + (푎13) 푇14 퐺13 = 0 208
1 ′ 1 ′′ 1 푎14 퐺13 − (푎14 ) + (푎14) 푇14 퐺14 = 0 209
1 ′ 1 ′′ 1 푎15 퐺14 − (푎15 ) + (푎15) 푇14 퐺15 = 0 210
1 ′ 1 ′′ 1 푏13 푇14 − [(푏13) − (푏13 ) 퐺 ]푇13 = 0 211
1 ′ 1 ′′ 1 푏14 푇13 − [(푏14) − (푏14 ) 퐺 ]푇14 = 0 212
1 ′ 1 ′′ 1 푏15 푇14 − [(푏15) − (푏15 ) 퐺 ]푇15 = 0 213 has a unique positive solution , which is an equilibrium solution for the system (19 to 24) 214
2 ′ 2 ′′ 2 푎16 퐺17 − (푎16 ) + (푎16) 푇17 퐺16 = 0 215
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2 ′ 2 ′′ 2 푎17 퐺16 − (푎17 ) + (푎17) 푇17 퐺17 = 0 216
2 ′ 2 ′′ 2 푎18 퐺17 − (푎18 ) + (푎18) 푇17 퐺18 = 0 217
2 ′ 2 ′′ 2 푏16 푇17 − [(푏16) − (푏16 ) 퐺19 ]푇16 = 0 218
2 ′ 2 ′′ 2 푏17 푇16 − [(푏17) − (푏17 ) 퐺19 ]푇17 = 0 219
2 ′ 2 ′′ 2 푏18 푇17 − [(푏18) − (푏18 ) 퐺19 ]푇18 = 0 220 has a unique positive solution , which is an equilibrium solution for (19 to 24) 221
222
Proof: 223
(a) Indeed the first two equations have a nontrivial solution 퐺13, 퐺14 if
′ 1 ′ 1 1 1 ′ 1 ′′ 1 ′ 1 ′′ 1 퐹 푇 = (푎13) (푎14 ) − 푎13 푎14 + (푎13) (푎14 ) 푇14 + (푎14) (푎13 ) 푇14 + ′′ 1 ′′ 1 (푎13) 푇14 (푎14) 푇14 = 0 Proof: 224
(a) Indeed the first two equations have a nontrivial solution 퐺16, 퐺17 if
′ 2 ′ 2 2 2 ′ 2 ′′ 2 ′ 2 ′′ 2 F 푇19 = (푎16 ) (푎17) − 푎16 푎17 + (푎16 ) (푎17) 푇17 + (푎17 ) (푎16 ) 푇17 + ′′ 2 ′′ 2 (푎16) 푇17 (푎17) 푇17 = 0
∗ Definition and uniqueness of T14 :- 225
′′ 1 After hypothesis 푓 0 < 0, 푓 ∞ > 0 and the functions (푎푖 ) 푇14 being increasing, it follows that 226 ∗ ∗ there exists a unique 푇14 for which 푓 푇14 = 0. With this value , we obtain from the three first equations
1 1 푎13 퐺14 푎15 퐺14 퐺13 = ′ 1 ′′ 1 ∗ , 퐺15 = ′ 1 ′′ 1 ∗ (푎13) +(푎13) 푇14 (푎15) +(푎15 ) 푇14
∗ Definition and uniqueness of T17 :- 227
′′ 2 After hypothesis 푓 0 < 0, 푓 ∞ > 0 and the functions (푎푖 ) 푇17 being increasing, it follows that ∗ ∗ there exists a unique T17 for which 푓 T17 = 0. With this value , we obtain from the three first equations
2 2 푎16 G17 푎18 G17 228 퐺16 = ′ 2 ′′ 2 ∗ , 퐺18 = ′ 2 ′′ 2 ∗ (푎16) +(푎16) T17 (푎18) +(푎18) T17
229
(c) By the same argument, the equations 92,93 admit solutions 퐺13, 퐺14 if 230
′ 1 ′ 1 1 1 휑 퐺 = (푏13) (푏14) − 푏13 푏14 −
′ 1 ′′ 1 ′ 1 ′′ 1 ′′ 1 ′′ 1 (푏13 ) (푏14 ) 퐺 + (푏14 ) (푏13 ) 퐺 +(푏13) 퐺 (푏14 ) 퐺 = 0
Where in 퐺 퐺13, 퐺14, 퐺15 , 퐺13, 퐺15 must be replaced by their values from 96. It is easy to see that φ is a decreasing function in 퐺14 taking into account the hypothesis 휑 0 > 0 , 휑 ∞ < 0 it follows that there ∗ ∗ exists a unique 퐺14 such that 휑 퐺 = 0
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(d) By the same argument, the equations 92,93 admit solutions 퐺16, 퐺17 if 231
′ 2 ′ 2 2 2 φ 퐺19 = (푏16 ) (푏17 ) − 푏16 푏17 −
′ 2 ′′ 2 ′ 2 ′′ 2 ′′ 2 ′′ 2 (푏16 ) (푏17 ) 퐺19 + (푏17) (푏16) 퐺19 +(푏16) 퐺19 (푏17) 퐺19 = 0
Where in 퐺19 퐺16, 퐺17, 퐺18 , 퐺16, 퐺18 must be replaced by their values from 96. It is easy to see that φ 232 is a decreasing function in 퐺17 taking into account the hypothesis φ 0 > 0 , 휑 ∞ < 0 it follows that ∗ ∗ there exists a unique G14 such that φ 퐺19 = 0
233
Finally we obtain the unique solution of 89 to 94 234
∗ ∗ ∗ ∗ 퐺14 given by 휑 퐺 = 0 , 푇14 given by 푓 푇14 = 0 and
1 ∗ 1 ∗ ∗ 푎13 퐺14 ∗ 푎15 퐺14 퐺13 = ′ 1 ′′ 1 ∗ , 퐺15 = ′ 1 ′′ 1 ∗ (푎13) +(푎13) 푇14 (푎15) +(푎15) 푇14
1 ∗ 1 ∗ ∗ 푏13 푇14 ∗ 푏15 푇14 푇13 = ′ 1 ′′ 1 ∗ , 푇15 = ′ 1 ′′ 1 ∗ (푏13) −(푏13) 퐺 (푏15) −(푏15) 퐺
Obviously, these values represent an equilibrium solution of 19,20,21,22,23,24
Finally we obtain the unique solution of 89 to 94 235
∗ ∗ ∗ ∗ G17 given by φ 퐺19 = 0 , T17 given by 푓 T17 = 0 and 236
2 ∗ 2 ∗ ∗ a16 G17 ∗ a18 G17 237 G16 = ′ 2 ′′ 2 ∗ , G18 = ′ 2 ′′ 2 ∗ (a16) +(a16) T17 (a18) +(a18) T17
2 ∗ 2 ∗ ∗ b16 T17 ∗ b18 T17 238 T16 = ′ 2 ′′ 2 ∗ , T18 = ′ 2 ′′ 2 ∗ (b16) −(b16) 퐺19 (b18) −(b18) 퐺19
Obviously, these values represent an equilibrium solution of 19,20,21,22,23,24 239
240
ASYMPTOTIC STABILITY ANALYSIS 241
Theorem 4: If the conditions of the previous theorem are satisfied and if the functions ′′ 1 ′′ 1 1 (푎푖 ) 푎푛푑 (푏푖 ) Belong to 퐶 ( ℝ+) then the above equilibrium point is asymptotically stable. Proof: Denote
Definition of 픾푖, 핋푖 :-
∗ ∗ 퐺푖 = 퐺푖 + 픾푖 , 푇푖 = 푇푖 + 핋푖
′′ 1 ′′ 1 휕(푎14 ) ∗ 1 휕(푏푖 ) ∗ 푇14 = 푞14 , 퐺 = 푠푖푗 휕푇14 휕퐺푗
Then taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain from 19 to 24 242
푑픾 13 = − (푎′ ) 1 + 푝 1 픾 + 푎 1 픾 − 푞 1 퐺∗ 핋 243 푑푡 13 13 13 13 14 13 13 14
푑픾 14 = − (푎′ ) 1 + 푝 1 픾 + 푎 1 픾 − 푞 1 퐺∗ 핋 244 푑푡 14 14 14 14 13 14 14 14
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푑픾 15 = − (푎′ ) 1 + 푝 1 픾 + 푎 1 픾 − 푞 1 퐺∗ 핋 245 푑푡 15 15 15 15 14 15 15 14
푑핋 13 = − (푏′ ) 1 − 푟 1 핋 + 푏 1 핋 + 15 푠 푇∗ 픾 246 푑푡 13 13 13 13 14 푗 =13 13 푗 13 푗
푑핋 14 = − (푏′ ) 1 − 푟 1 핋 + 푏 1 핋 + 15 푠 푇∗ 픾 247 푑푡 14 14 14 14 13 푗 =13 14 (푗 ) 14 푗
푑핋 15 = − (푏′ ) 1 − 푟 1 핋 + 푏 1 핋 + 15 푠 푇∗ 픾 248 푑푡 15 15 15 15 14 푗 =13 15 (푗 ) 15 푗
249
ASYMPTOTIC STABILITY ANALYSIS 250
Theorem 4: If the conditions of the previous theorem are satisfied and if the functions ′′ 2 ′′ 2 2 (a푖 ) and (b푖 ) Belong to C ( ℝ+) then the above equilibrium point is asymptotically stable
Proof: Denote 251
Definition of 픾푖, 핋푖 :-
∗ ∗ G푖 = G푖 + 픾푖 , T푖 = T푖 + 핋푖 252
′′ 2 ′′ 2 ∂(푎17) ∗ 2 ∂(푏푖 ) ∗ 253 T17 = 푞17 , 퐺19 = 푠푖푗 ∂T17 ∂G푗 taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain from 19 to 24 254 d픾 16 = − (푎′ ) 2 + 푝 2 픾 + 푎 2 픾 − 푞 2 G∗ 핋 255 dt 16 16 16 16 17 16 16 17 d픾 17 = − (푎′ ) 2 + 푝 2 픾 + 푎 2 픾 − 푞 2 G∗ 핋 256 dt 17 17 17 17 16 17 17 17 d픾 18 = − (푎′ ) 2 + 푝 2 픾 + 푎 2 픾 − 푞 2 G∗ 핋 257 dt 18 18 18 18 17 18 18 17 d핋 16 = − (푏′ ) 2 − 푟 2 핋 + 푏 2 핋 + 18 푠 T∗ 픾 258 dt 16 16 16 16 17 푗 =16 16 푗 16 푗 d핋 17 = − (푏′ ) 2 − 푟 2 핋 + 푏 2 핋 + 18 푠 T∗ 픾 259 dt 17 17 17 17 16 푗 =16 17 (푗 ) 17 푗 d핋 18 = − (푏′ ) 2 − 푟 2 핋 + 푏 2 핋 + 18 푠 T∗ 픾 260 dt 18 18 18 18 17 푗 =16 18 (푗 ) 18 푗
261
The characteristic equation of this system is 262
1 ′ 1 1 1 ′ 1 1 휆 + (푏15) − 푟15 { 휆 + (푎15 ) + 푝15
1 ′ 1 1 1 ∗ 1 1 ∗ 휆 + (푎13 ) + 푝13 푞14 퐺14 + 푎14 푞13 퐺13
1 ′ 1 1 ∗ 1 ∗ 휆 + (푏13) − 푟13 푠 14 , 14 푇14 + 푏14 푠 13 , 14 푇14
1 ′ 1 1 1 ∗ 1 1 ∗ + 휆 + (푎14) + 푝14 푞13 퐺13 + 푎13 푞14 퐺14
1 ′ 1 1 ∗ 1 ∗ 휆 + (푏13) − 푟13 푠 14 , 13 푇14 + 푏14 푠 13 , 13 푇13
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1 2 ′ 1 ′ 1 1 1 1 휆 + (푎13) + (푎14) + 푝13 + 푝14 휆
1 2 ′ 1 ′ 1 1 1 1 휆 + (푏13) + (푏14 ) − 푟13 + 푟14 휆
1 2 ′ 1 ′ 1 1 1 1 1 + 휆 + (푎13 ) + (푎14) + 푝13 + 푝14 휆 푞15 퐺15
1 ′ 1 1 1 1 ∗ 1 1 1 ∗ + 휆 + (푎13) + 푝13 푎15 푞14 퐺14 + 푎14 푎15 푞13 퐺13
1 ′ 1 1 ∗ 1 ∗ 휆 + (푏13) − 푟13 푠 14 , 15 푇14 + 푏14 푠 13 , 15 푇13 } = 0
+
2 ′ 2 2 2 ′ 2 2 휆 + (푏18) − 푟18 { 휆 + (푎18 ) + 푝18
2 ′ 2 2 2 ∗ 2 2 ∗ 휆 + (푎16 ) + 푝16 푞17 G17 + 푎17 푞16 G16
2 ′ 2 2 ∗ 2 ∗ 휆 + (푏16) − 푟16 푠 17 , 17 T17 + 푏17 푠 16 , 17 T17
2 ′ 2 2 2 ∗ 2 2 ∗ + 휆 + (푎17) + 푝17 푞16 G16 + 푎16 푞17 G17
2 ′ 2 2 ∗ 2 ∗ 휆 + (푏16) − 푟16 푠 17 , 16 T17 + 푏17 푠 16 , 16 T16
2 2 ′ 2 ′ 2 2 2 2 휆 + (푎16) + (푎17) + 푝16 + 푝17 휆
2 2 ′ 2 ′ 2 2 2 2 휆 + (푏16 ) + (푏17) − 푟16 + 푟17 휆
2 2 ′ 2 ′ 2 2 2 2 2 + 휆 + (푎16 ) + (푎17) + 푝16 + 푝17 휆 푞18 G18
2 ′ 2 2 2 2 ∗ 2 2 2 ∗ + 휆 + (푎16) + 푝16 푎18 푞17 G17 + 푎17 푎18 푞16 G16
2 ′ 2 2 ∗ 2 ∗ 휆 + (푏16) − 푟16 푠 17 , 18 T17 + 푏17 푠 16 , 18 T16 } = 0
And as one sees, all the coefficients are positive. It follows that all the roots have negative real part, and this proves the theorem.
Acknowledgments: The introduction is a collection of information from various articles, Books, News Paper reports, Home Pages Of authors, Journal Reviews, the internet including Wikipedia. We acknowledge all authors who have contributed to the same. In the eventuality of the fact that there has been any act of omission on the part of the authors, We regret with great deal of compunction, contrition, and remorse. As Newton said, it is only because erudite and eminent people allowed one to piggy ride on their backs; probably an attempt has been made to look slightly further. Once again, it is stated that the references are only illustrative and not comprehensive
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