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TheInfoList

Structural geology is the study of the three-dimensional distribution of
rock Rock most often refers to: * Rock (geology), a naturally occurring solid aggregate of minerals or mineraloids * Rock music, a genre of popular music Rock or Rocks may also refer to: Places United Kingdom * Rock, Caerphilly, a location in Wales ...
units with respect to their deformational histories. The primary goal of structural geology is to use measurements of present-day rock geometries to uncover information about the history of deformation ( strain) in the rocks, and ultimately, to understand the
stress field A stress field is the distribution of internal forces in a body that balance a given set of external forces. Stress fields are widely used in fluid dynamics and materials science. Consider that one can picture the stress fields as the stress create ...
that resulted in the observed strain and geometries. This understanding of the dynamics of the stress field can be linked to important events in the geologic past; a common goal is to understand the structural evolution of a particular area with respect to regionally widespread patterns of rock deformation (e.g.,
mountain building A mountain is an elevated portion of the Earth's crust, generally with steep sides that show significant exposed bedrock. A mountain differs from a plateau in having a limited summit area, and is larger than a hill, typically rising at least ...
,
rift 200px, Gulf of Suez Rift showing main extensional faults">extensional_fault.html" style="text-decoration: none;"class="mw-redirect" title="Gulf of Suez Rift showing main extensional fault">Gulf of Suez Rift showing main extensional faults I ...
ing) due to
plate tectonics upright=1.35, Diagram of the internal layering of Earth showing the lithosphere above the asthenosphere (not to scale) Plate tectonics (from the la, label=Late Latin, tectonicus, from the grc, τεκτονικός, lit=pertaining to building) is a ...
.

Use and importance

The study of geologic structures has been of prime importance in
economic geology Economic geology is concerned with earth materials that can be used for economic and/or industrial purposes. These materials include precious and base metals, nonmetallic minerals and construction-grade stone. Economic geology is a subdiscipline ...
, both
petroleum geology Petroleum geology is the study of origin, occurrence, movement, accumulation, and exploration of hydrocarbon fuels. It refers to the specific set of geological disciplines that are applied to the search for hydrocarbons (oil exploration). Sediment ...
and
mining geology Mining is the extraction of valuable minerals or other geological materials from the Earth, usually from an ore body, lode, vein, seam, reef, or placer deposit. These deposits form a mineralized commodity that is of economic interest to the min ...
. Folded and faulted rock
strata (Argentina). , Canada. These are Middle Cambrian marine sediments. This formation covers over half of Nova Scotia and is recorded as being 8,800 m (29,000 ft) thick in some areas. In geology and related fields, a stratum (plural: strata) is a l ...
commonly form traps that accumulate and concentrate fluids such as
petroleum Petroleum (), also known as crude oil and oil, is a naturally occurring, yellowish-black liquid found in geological formations beneath the Earth's surface. It is commonly refined into various types of fuels. Components of petroleum are separate ...
and
natural gas Natural gas (also called fossil gas; sometimes just gas) is a naturally occurring hydrocarbon gas mixture consisting primarily of methane, but commonly including varying amounts of other higher alkanes, and sometimes a small percentage of carb ...
. Similarly, faulted and structurally complex areas are notable as permeable zones for
hydrothermal Hydrothermal circulation in its most general sense is the circulation of hot water (Ancient Greek ὕδωρ, ''water'',Liddell, H.G. & Scott, R. (1940). ''A Greek-English Lexicon. revised and augmented throughout by Sir Henry Stuart Jones. with the ...
fluids, resulting in concentrated areas of base and precious metal ore deposits. Veins of minerals containing various metals commonly occupy faults and fractures in structurally complex areas. These structurally fractured and faulted zones often occur in association with intrusive
igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main rock types, the others being sedimentary and metamorphic. Igneous rock is formed through the cooling and solidification of magma or lav ...
s. They often also occur around geologic
reef A reef is a ridge or shoal of rock, coral or similar relatively stable material, lying beneath the surface of a natural body of water. Many reefs result from natural, abiotic processes—deposition of sand, wave erosion planing down rock outcr ...

complexes and collapse features such as ancient
sinkhole A sinkhole, also known as a cenote, sink, sink-hole, swallet, swallow hole, or doline (the different terms for sinkholes are often used interchangeably), is a depression or hole in the ground caused by some form of collapse of the surface lay ...

s. Deposits of
gold Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79, making it one of the higher atomic number elements that occur naturally. In a pure form, it is a bright, slightly reddish yellow, dense, soft, malleable, and ...
,
silver Silver is a chemical element with the symbol Ag (from the Latin ', derived from the Proto-Indo-European ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, white, lustrous transition metal, it exhibits the highest electrical cond ...
,
copper Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkish-orang ...

,
lead Lead is a chemical element with the symbol Pb (from the Latin ) and atomic number 82. It is a heavy metal that is denser than most common materials. Lead is soft and malleable, and also has a relatively low melting point. When freshly cut, lead ...
,
zinc Zinc is a chemical element with the symbol Zn and atomic number 30. Zinc is a slightly brittle metal at room temperature and has a silvery-greyish appearance when oxidation is removed. It is the first element in group 12 (IIB) of the periodic t ...
, and other metals, are commonly located in structurally complex areas. Structural geology is a critical part of
engineering geology Engineering geology is the application of geology to engineering study for the purpose of assuring that the geological factors regarding the location, design, construction, operation and maintenance of engineering works are recognized and account ...
, which is concerned with the physical and mechanical properties of natural rocks. Structural fabrics and defects such as faults, folds, foliations and
joints A joint or articulation (or articular surface) is the connection made between bones in the body which link the skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw-Hill Connect. Webp.274/ref> They are construc ...
are internal weaknesses of rocks which may affect the stability of human engineered structures such as
dam A dam is a barrier that stops or restricts the flow of water or underground streams. Reservoirs created by dams not only suppress floods but also provide water for activities such as irrigation, human consumption, industrial use, aquacultur ...
open pit The giant bucket-wheel excavators in the German Rhineland coal mines are among the world's biggest land vehicles. Open-pit mining, also known as open-cast or open cut mining, is a surface mining technique of extracting rock or minerals from the ea ...
mines and underground mines or road
tunnel A tunnel is an underground passageway, dug through the surrounding soil/earth/rock and enclosed except for entrance and exit, commonly at each end. A pipeline is not a tunnel, though some recent tunnels have used immersed tube construction ...

s.
Geotechnical#REDIRECT geotechnical engineering {{Redirect category shell, 1= {{R from other capitalisation ...
risk, including
earthquake An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in size from those that ...
risk can only be investigated by inspecting a combination of structural geology and
geomorphology incised into shale at the foot of the North Caineville Plateau, Utah, within the pass carved by the Fremont River and known as the Blue Gate. GK Gilbert studied the landscapes of this area in great detail, forming the observational foundation for ...
karst Karst is a topography formed from the dissolution of soluble rocks such as limestone, dolomite, and gypsum. It is characterized by underground drainage systems with sinkholes and caves. It has also been documented for more weathering-resistant r ...
landscapes which reside atop caverns, potential sinkholes, or other collapse features are of particular importance for these scientists. In addition, areas of steep slopes are potential collapse or landslide hazards. Environmental geologists and hydrogeologists need to apply the tenets of structural geology to understand how geologic sites impact (or are impacted by)
groundwater Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. A unit of rock or an unconsolidated deposit is called an aquifer when it can yield a usable quantity of water. The de ...
flow and penetration. For instance, a hydrogeologist may need to determine if seepage of toxic substances from waste dumps is occurring in a residential area or if salty water is seeping into an
aquifer An aquifer is an underground layer of water-bearing permeable rock, rock fractures or unconsolidated materials (gravel, sand, or silt). Groundwater can be extracted using a water well. The study of water flow in aquifers and the characterization ...
.
Plate tectonic upright=1.35, Diagram of the internal layering of Earth showing the lithosphere above the asthenosphere (not to scale) Plate tectonics (from the la, label=Late Latin, tectonicus, from the grc, τεκτονικός, lit=pertaining to building) is a ...
s is a theory developed during the 1960s which describes the movement of continents by way of the separation and collision of crustal plates. It is in a sense structural geology on a planet scale, and is used throughout structural geology as a framework to analyze and understand global, regional, and local scale features.

Methods

Structural geologists use a variety of methods to (first) measure rock geometries, (second) reconstruct their deformational histories, and (third) estimate the stress field that resulted in that deformation.

Geometries

Primary data sets for structural geology are collected in the field. Structural geologists measure a variety of planar features ( bedding planes, foliation planes, fold axial planes, fault planes, and joints), and linear features (stretching lineations, in which minerals are ductilely extended; fold axes; and intersection lineations, the trace of a planar feature on another planar surface). upright=1.3, Illustration of measurement conventions for planar and linear structures

Measurement conventions

The inclination of a planar structure in geology is measured by ''
strike and dip Strike and dip refer to the orientation or ''attitude'' of a geologic feature. The ''strike line'' of a bed, fault, or other planar feature, is a line representing the intersection of that feature with a horizontal plane. On a geologic map, thi ...
''. The strike is the line of intersection between the planar feature and a horizontal plane, taken according to the right hand convention, and the dip is the magnitude of the inclination, below horizontal, at right angles to strike. For example; striking 25 degrees East of North, dipping 45 degrees Southeast, recorded as N25E,45SE.
Alternatively, dip and dip direction may be used as this is absolute. Dip direction is measured in 360 degrees, generally clockwise from North. For example, a dip of 45 degrees towards 115 degrees azimuth, recorded as 45/115. Note that this is the same as above. The term ''hade'' is occasionally used and is the deviation of a plane from vertical i.e. (90°-dip). Fold axis plunge is measured in dip and dip direction (strictly, plunge and azimuth of plunge). The orientation of a fold axial plane is measured in strike and dip or dip and dip direction. Lineations are measured in terms of dip and dip direction, if possible. Often lineations occur expressed on a planar surface and can be difficult to measure directly. In this case, the lineation may be measured from the horizontal as a ''rake'' or ''pitch'' upon the surface. Rake is measured by placing a protractor flat on the planar surface, with the flat edge horizontal and measuring the angle of the lineation clockwise from horizontal. The orientation of the lineation can then be calculated from the rake and strike-dip information of the plane it was measured from, using a
stereographic projection In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smoot ...
. If a fault has lineations formed by movement on the plane, e.g.;
slickenside227px, Slickensides on a sample of sandstone of the Rt 322 northeast of State College, Pennsylvania">U.S. Route 322">Rt 322 northeast of State College, Pennsylvania In geology, a slickenside is a smoothly polished surface caused by frictional moveme ...
s, this is recorded as a lineation, with a rake, and annotated as to the indication of throw on the fault. Generally it is easier to record strike and dip information of planar structures in dip/dip direction format as this will match all the other structural information you may be recording about folds, lineations, etc., although there is an advantage to using different formats that discriminate between planar and linear data.

Plane, fabric, fold and deformation conventions

The convention for analysing structural geology is to identify the planar structures, often called ''planar fabrics'' because this implies a textural formation, the linear structures and, from analysis of these, unravel deformations. Planar structures are named according to their order of formation, with original sedimentary layering the lowest at S0. Often it is impossible to identify S0 in highly deformed rocks, so numbering may be started at an arbitrary number or given a letter (SA, for instance). In cases where there is a bedding-plane foliation caused by burial metamorphism or
diagenesis Diagenesis () is the process that describes physical and chemical changes in sediments first caused by water-rock interactions, microbial activity and compaction after their deposition. The increase of pressure and temperature only starts to play ...
this may be enumerated as S0a. If there are folds, these are numbered as F1, F2, etc. Generally the axial plane foliation or cleavage of a fold is created during folding, and the number convention should match. For example, an F2 fold should have an S2 axial foliation. Deformations are numbered according to their order of formation with the letter D denoting a deformation event. For example, D1, D2, D3. Folds and foliations, because they are formed by deformation events, should correlate with these events. For example, an F2 fold, with an S2 axial plane foliation would be the result of a D2 deformation. Metamorphic events may span multiple deformations. Sometimes it is useful to identify them similarly to the structural features for which they are responsible, e.g.; M2. This may be possible by observing
porphyroblast-garnet growing as a porphyroblast in a quartzitic gneiss. The garnet measures 3 cm. Location: Paraíba, Brazil. Image:Garnet Mica Schist Syros Greece.jpg, Dark-coloured porphyroblasts of garnet in mica schist at Syros, Greece. A porphyroblast is a la ...
formation in cleavages of known deformation age, by identifying metamorphic mineral assemblages created by different events, or via
geochronology 300px, An artistic depiction of the major events in the history of Earth Geochronology is the science of determining the age of rocks, fossils, and sediments using signatures inherent in the rocks themselves. Absolute geochronology can be accomplish ...
. Intersection lineations in rocks, as they are the product of the intersection of two planar structures, are named according to the two planar structures from which they are formed. For instance, the intersection lineation of a S1 cleavage and bedding is the L1-0 intersection lineation (also known as the cleavage-bedding lineation). Stretching lineations may be difficult to quantify, especially in highly stretched ductile rocks where minimal foliation information is preserved. Where possible, when correlated with deformations (as few are formed in folds, and many are not strictly associated with planar foliations), they may be identified similar to planar surfaces and folds, e.g.; L1, L2. For convenience some geologists prefer to annotate them with a subscript S, for example Ls1 to differentiate them from intersection lineations, though this is generally redundant.

Stereographic projections

Stereographic projection In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smoot ...
is a method for analyzing the nature and orientation of deformation stresses, lithological units and penetrative fabrics wherein linear and planar features (structural strike and dip readings, typically taken using a ) passing through an imagined sphere are plotted on a two-dimensional grid projection, facilitating more holistic analysis of a set of measurements.

Rock macro-structures

On a large scale, structural geology is the study of the three-dimensional interaction and relationships of stratigraphic units within
terrane A terrane in geology, in full a tectonostratigraphic terrane, is a fragment of crustal material formed on, or broken off from, one tectonic plate and accreted or "sutured" to crust lying on another plate. The crustal block or fragment preserves ...
s of rock or geological regions. This branch of structural geology deals mainly with the orientation, deformation and relationships of stratigraphy (bedding), which may have been faulted, folded or given a foliation by some tectonic event. This is mainly a geometric science, from which ''cross sections'' and three-dimensional ''block models'' of rocks, regions, terranes and parts of the Earth's crust can be generated. Study of regional structure is important in understanding
orogeny An orogeny is an event that leads to both structural deformation and compositional differentiation of the Earth's lithosphere (crust and uppermost mantle) at convergent plate margins. An orogen or orogenic belt develops when a continental plate ...
,
plate tectonics upright=1.35, Diagram of the internal layering of Earth showing the lithosphere above the asthenosphere (not to scale) Plate tectonics (from the la, label=Late Latin, tectonicus, from the grc, τεκτονικός, lit=pertaining to building) is a ...
and more specifically in the oil,
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or compo ...
and
mineral In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure, that occurs naturally in pure form.John P. Rafferty, ed. (2 ...

exploration industries as structures such as faults, folds and
unconformities An unconformity is a buried erosional or non-depositional surface separating two rock masses or strata of different ages, indicating that sediment deposition was not continuous. In general, the older layer was exposed to erosion for an interval of t ...
are primary controls on ore mineralisation and oil traps. Modern regional structure is being investigated using
seismic tomography Seismic tomography is a technique for imaging the subsurface of the Earth with seismic waves produced by earthquakes or explosions. P-, S-, and surface waves can be used for tomographic models of different resolutions based on seismic wavelength, wa ...
and
seismic Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other pl ...
reflection in three dimensions, providing unrivaled images of the Earth's interior, its faults and the deep crust. Further information from
geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' somet ...
such as
gravity Gravity (), or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or ''gravitate'' toward) one another. On Earth, gravity gives weight to p ...
and airborne magnetics can provide information on the nature of rocks imaged to be in the deep crust.

Rock microstructures

Rock microstructure or ''texture'' of rocks is studied by structural geologists on a small scale to provide detailed information mainly about
metamorphic rocks#REDIRECT Metamorphic rock {{Redirect category shell, 1= {{R from other capitalisation ...
and some features of
sedimentary rocks Sedimentary rocks are types of rock that are formed by the accumulation or deposition of mineral or organic particles at the Earth's surface, followed by cementation. Sedimentation is the collective name for processes that cause these particles to ...
, most often if they have been folded.
Textural study involves measurement and characterisation of foliations,
crenulation In a geological context, crenulation or crenulation cleavage is a fabric formed in metamorphic rocks such as phyllite, schist and some gneiss by two or more stress directions causing the formation of the superimposed foliations. Formation Crenulati ...
s, metamorphic minerals, and timing relationships between these structural features and mineralogical features.
Usually this involves collection of hand specimens, which may be cut to provide
petrographic Petrography is a branch of petrology that focuses on detailed descriptions of rocks. Someone who studies petrography is called a petrographer. The mineral content and the textural relationships within the rock are described in detail. The classifi ...
thin sections which are analysed under a
petrographic microscope Leica DMRX incident light microscope with mechanical stage and Swift F automated point counter for analysis of organic composition of coal and rock samples A petrographic microscope is a type of optical microscope used in petrology and opti ...
.
Microstructural analysis finds application also in multi-scale statistical analysis, aimed to analyze some rock features showing scale invariance.

Kinematics

Geologists use rock geometry measurements to understand the history of strain in rocks. Strain can take the form of brittle faulting and ductile folding and shearing. Brittle deformation takes place in the shallow crust, and ductile deformation takes place in the deeper crust, where temperatures and pressures are higher.

Stress fields

By understanding the constitutive relationships between stress and strain in rocks, geologists can translate the observed patterns of rock deformation into a stress field during the geologic past. The following list of features are typically used to determine stress fields from deformational structures. *In perfectly brittle rocks, faulting occurs at 30° to the greatest compressional stress. (Byerlee's Law) *The greatest compressive stress is normal to fold axial planes.

Characterization of the mechanical properties of rock

The mechanical properties of rock play a vital role in the structures that form during deformation deep below the earth's crust. The conditions in which a rock is present will result in different structures that geologists observe above ground in the field. The field of structural geology tries to relate the formations that humans see to the changes the rock went through to get to that final structure. Knowing the conditions of deformation that lead to such structures can illuminate the history of the deformation of the rock. Temperature and pressure play a huge role in the deformation of rock. At the conditions under the earth's crust of extreme high temperature and pressure, rocks are ductile. They can bend, fold or break. Other vital conditions that contribute to the formation of structure of rock under the earth are the stress and strain fields.

Stress-strain curve

Stress is a pressure, defined as a directional force over area. When a rock is subjected to stresses, it changes shape. When the stress is released, the rock may or may not return to its original shape. That change in shape is quantified by strain, the change in length over the original length of the material in one dimension. Stress induces strain which ultimately results in a changed structure. Elastic deformation refers to a reversible deformation. In other words, when stress on the rock is released, the rock returns to its original shape. Reversible, linear, elasticity involves the stretching, compressing, or distortion of atomic bonds. Because there is no breaking of bonds, the material springs back when the force is released. This type of deformation is modeled using a linear relationship between stress and strain, i.e. a Hookean relationship. :$\epsilon = \frac$ Where σ denotes stress, $\epsilon$ denotes strain, and E is the elastic modulus, which is material dependent. The elastic modulus is, in effect, a measure of the strength of atomic bonds. Plastic deformation refers to non-reversible deformation. The relationship between stress and strain for permanent deformation is nonlinear. Stress has caused permanent change of shape in the material by involving the breaking of bonds. One mechanism of plastic deformation is the movement of dislocations by an applied stress. Because rocks are essentially aggregates of minerals, we can think of them as poly-crystalline materials. Dislocations are a type of crystallographic defect which consists of an extra or missing half plane of atoms in the periodic array of atoms that make up a crystal lattice. Dislocations are present in all real crystallographic materials.

Hardness

Hardness is difficult to quantify. It is a measure of resistance to deformation, specifically permanent deformation. There is precedent for hardness as a surface quality, a measure of the abrasiveness or surface-scratching resistance of a material. If the material being tested, however, is uniform in composition and structure, then the surface of the material is only a few atomic layers thick, and measurements are of the bulk material. Thus, simple surface measurements yield information about the bulk properties. Ways to measure hardness include: *
Mohs Scale The Mohs scale of mineral hardness () is a qualitative ordinal scale characterizing scratch resistance of various minerals through the ability of harder material to scratch softer material. Created in 1822 by German geologist and mineralogist Frie ...
* Dorry abrasion test * Deval abrasion test * Indentation hardness Indentation hardness is used often in metallurgy and materials science and can be thought of as resistance to penetration by an indenter.

Toughness

Toughness can be described best by a material's resistance to cracking. During plastic deformation, a material absorbs energy until fracture occurs. The area under the stress-strain curve is the work required to fracture the material. The toughness modulus is defined as: :$M_t = \frac \sigma_ \; \epsilon_f$ Where $\sigma_$ is the ultimate tensile strength, and $\epsilon_$ is the strain at failure. The modulus is the maximum amount of energy per unit volume a material can absorb without fracturing. From the equation for modulus, for large toughness, high strength and high ductility are needed. These two properties are usually mutually exclusive. Brittle materials have low toughness because low plastic deformation decreases the strain (low ductility). Ways to measure toughness include: Page impact machine and Charpy Impact Test

Resilience

Resilience is a measure of the elastic energy absorbed of a material under stress. In other words, the external work performed on a material during deformation. The area under the elastic portion of the stress-strain curve is the strain energy absorbed per unit volume. The resilience modulus is defined as: :$M_R = \frac$ where $\sigma_y$ is the yield strength of the material and E is the elastic modulus of the material. To increase resilience, one needs increased elastic yield strength and decreased modulus of elasticity.


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# diophantus

Hello, this is beta version of diophantus. If you want to report about a mistake, please, write to hello@diophantus.org

#### The Mass and Radius of the Unseen M-Dwarf Companion in the Single-Lined Eclipsing Binary HAT-TR-205-013

31 Mar 2007 astro-ph arxiv.org/abs/0704.0059
Abstract. We derive masses and radii for both components in the single-lined eclipsing binary HAT-TR-205-013, which consists of a F7V primary and a late M-dwarf secondary. The system's period is short, $P=2.230736 \pm 0.000010$ days, with an orbit indistinguishable from circular, $e=0.012 \pm 0.021$. We demonstrate generally that the surface gravity of the secondary star in a single-lined binary undergoing total eclipses can be derived from characteristics of the light curve and spectroscopic orbit. This constrains the secondary to a unique line in the mass-radius diagram with $M/R^2$ = constant. For HAT-TR-205-013, we assume the orbit has been tidally circularized, and that the primary's rotation has been synchronized and aligned with the orbital axis. Our observed line broadening, $V_{\rm rot} \sin i_{\rm rot} = 28.9 \pm 1.0$ \kms, gives a primary radius of $R_{\rm A} = 1.28 \pm 0.04$ \rsun. Our light curve analysis leads to the radius of the secondary, $R_{\rm B} = 0.167 \pm 0.006$ \rsun, and the semimajor axis of the orbit, $a = 7.54 \pm 0.30 \rsun = 0.0351 \pm 0.0014$ AU. Our single-lined spectroscopic orbit and the semimajor axis then yield the individual masses, $M_{\rm B} = 0.124 \pm 0.010$ \msun and $M_{\rm A} = 1.04 \pm 0.13$ \msun. Our result for HAT-TR-205-013 B lies above the theoretical mass-radius models from the Lyon group, consistent with results from double-lined eclipsing binaries. The method we describe offers the opportunity to study the very low end of the stellar mass-radius relation.

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# Gauss Elimination Method

Gauss elimination method is used to solve a system of linear equations. Let’s recall the definition of these systems of equations. A system of linear equations is a group of linear equations with various unknown factors. As we know, unknown factors exist in multiple equations. Solving a system involves finding the value for the unknown factors to verify all the equations that make up the system.

If there is a single solution that means one value for each unknown factor, then we can say that the given system is a consistent independent system. If multiple solutions exist, the system has infinitely many solutions; then we say that it is a consistent dependent system. If there is no solution for unknown factors, and this will happen if there are two or more equations that can’t be verified simultaneously, then we say that it’s an inconsistent system.

This can be summarized in a table as given below:

 Name of the system of equations Number of solutions Consistent independent system 1 Consistent dependent system Multiple or Infinitely many Inconsistent system 0

Now, let’s have a look at the method that can be used to find the solution(s) of the given system of equations.

## What is the gauss elimination method?

In mathematics, the Gaussian elimination method is known as the row reduction algorithm for solving linear equations systems. It consists of a sequence of operations performed on the corresponding matrix of coefficients. We can also use this method to estimate either of the following:

• The rank of the given matrix
• The determinant of a square matrix
• The inverse of an invertible matrix

To perform row reduction on a matrix, we have to complete a sequence of elementary row operations to transform the matrix till we get 0s (i.e., zeros) on the lower left-hand corner of the matrix as much as possible. That means the obtained matrix should be an upper triangular matrix. There are three types of elementary row operations; they are:

• Swapping two rows and this can be expressed using the notation ↔ , for example, R2 ↔ R3
• Multiplying a row by a nonzero number, for example, R1 → kR2 where k is some nonzero number
• Adding a multiple of one row to another row, for example, R2 → R2 + 3R1

The obtained matrix will be in row echelon form. The matrix is said to be in reduced row-echelon form when all of the leading coefficients equal 1, and every column containing a leading coefficient has zeros elsewhere. This final form is unique; that means it is independent of the sequence of row operations used. We can understand this in a better way with the help of an example given below.

## Gauss Elimination Method with Example

Let’s have a look at the gauss elimination method example with a solution.

Question:

Solve the following system of equations:

x + y + z = 2

x + 2y + 3z = 5

2x + 3y + 4z = 11

Solution:

Given system of equations are:

x + y + z = 2

x + 2y + 3z = 5

2x + 3y + 4z = 11

Let us write these equations in matrix form.

Subtracting R1 from R2 to get the new elements of R2, i.e. R2 → R2 – R1.

From this we get,

Let us make another operation as R3 → R3 – 2R1

Subtract R2 from R1 to get the new elements of R1, i.e. R1 → R1 – R2.

Now, subtract R2 from R3 to get the new elements of R3, i.e. R3 → R3 – R2.

Here,

x – z = -1

y + 2z = 3

0 = 4

That means, there is no solution for the given system of equations.

### Gauss Elimination Method Problems

1. Solve the following system of equations using Gauss elimination method.
2. x + y + z = 9

2x + 5y + 7z = 52

2x + y – z = 0

1. Solve the following linear system using Gaussian elimination method.
2. 4x – 5y = -6

2x – 2y = 1

1. Using Gauss elimination method, solve:
2. 2x – y + 3z = 9

x + y + z = 6

x – y + z = 2


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# Multiple of Divisor Divides Multiple/Proof 2

## Theorem

Let $a, b, c \in \Z$.

Let:

$a \divides b$

where $\divides$ denotes divisibility.

Then:

$a c \divides b c$

## Proof

By definition, if $a \divides b$ then $\exists d \in \Z: a d = b$.

Then:

$\paren {a d} c = b c$

that is:

$\paren {a c} d = b c$

which follows because Integer Multiplication is Commutative and Integer Multiplication is Associative.

Hence the result.

$\blacksquare$


text:
Communication System – JEE Main Previous Year Questions with Solutions
JEE Main Previous Year Question of Physics with Solutions are available here. Practicing JEE Main Previous Year Papers Questions of Physics will help all the JEE aspirants in realizing the question pattern as well as help in analyzing their weak & strong areas. Get detailed Class 11th &12th Physics Notes to prepare for Boards as well as competitive exams like IIT JEE, NEET etc. eSaral helps the students in clearing and understanding each topic in a better way. eSaral is providing complete chapter-wise notes of Class 11th and 12th both for all subjects. Besides this, eSaral also offers NCERT Solutions, Previous year questions for JEE Main and Advance, Practice questions, Test Series for JEE Main, JEE Advanced and NEET, Important questions of Physics, Chemistry, Math, and Biology and many more. Download eSaral app for free study material and video tutorials.   Simulator Previous Years AIEEE/JEE Mains Questions
Q. This questions has Statement-1 and Statement-2. Of the four choice given after the statements, choose the one that best describes the two statements. Statement-1 : Sky wave signals are used for long distance radio communication. These signals are in general, less stable than ground wave signals. Statement-2 : The state of ionosphere varies from hour to hour, day to day and season to season. (1) Statement-1 is true, Statement-2 is true, Statement-2 is the correct explanation of statement-1 (2) Statement-1 is true, Statement-2 is true, Statrment-2 is not the correct explanation of Statement-1 (3) Statement-1 is false, Statement-2 is true (4) Statement-1 is true, Statement-2 is false [AIEEE-2011]

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Sol. (2)

Q. A radar has a power of 1 kW and is operating at a frequency of 10 GHz. It is located on a mountain top of height 500 m. The maximum distance up to which it can detect object located on the surface of the earth (Radius of earth $\left.=6.4 \times 10^{6} \mathrm{m}\right)$ is : (1) 40 km (2) 64 (3) 80 km (4) 16km [AIEEE-2012]

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Sol. (3) $\mathrm{d}=\sqrt{2 \mathrm{Rh}}=\sqrt{2 \times 6400 \times 0.5}=80 \mathrm{km}$

Q. A diode detector is used to detect an amplitude modulated wave of 60% modulation by using a condenser of capacity 250 pico farad in parallel with a load resistance 100 kilo ohm. Find the maximum modulated frequency which could be detected by it. (1) 10.62kHz (2) 5.31 MHz (3) 5.31 kHz (4) 10.62MHz [JEE Main-2013]

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Sol. (3) $\mathrm{f}_{\mathrm{max}} \leq \frac{\sqrt{\frac{1}{\mathrm{m}^{2}}-1}}{2 \pi \mathrm{RC}}$ $\mathrm{f}_{\max } \leq \frac{8}{6 \times 2 \pi \times 100 \times 10^{3} \times 250 \times 10^{-12}}$ $\mathrm{f}_{\max } \leq \frac{8 \times 10^{6}}{12 \pi \times 25}$ $\mathrm{f}_{\max } \leq 8.4925 \mathrm{kHz}$

Q. A single of 5 kHz frequency is amplitude modulated on a carrier wave of frequency 2 MHz. The frequencies of the resultant signal is/are – (1) 2005 kHz, 2000 kHz and 1995 kHz (2) 2000 kHz and 1995 kHz (3) 2 MHz only (4) 2005 kHz and 1995 kHz [JEE Main-2015]

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Sol. (1) Frequency present after modulation $\mathrm{f}_{\mathrm{c}}, \mathrm{f}_{\mathrm{c}} \pm \mathrm{f}_{\mathrm{s}}$ $\Rightarrow 2000 \mathrm{KHz}, 2005 \mathrm{KHz}$ and $1995 \mathrm{KHz}$

Q. Choose the correct statement : (1) In frequency modulation the amplitude of the high frequency carrier wave is made to vary in proportion to the frequency of the audio signal. (2) In amplitude modulation the amplitude of the high frequency carrier wave is made to vary in proportion to the amplitude of the audio signal. (3) In amplitude modulation the frequency of the high frequency carrier wave is made to vary in proportion to the amplitude of the audio signal. (4) In frequency modulation the amplitude of the high frequency carrier wave is made to vary in proportion to the amplitude of the audio signal. [JEE Main-2016]

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Sol. (2)

Q. In amplitude modulation, sinusoidal carrier frequency used is denoted by $\omega_{\mathrm{c}}$ and the signal frequency is denoted by $\omega_{\mathrm{m}}$. The bandwidth $\left(\Delta \omega_{\mathrm{m}}\right)$ of the signal is such that $\Delta \omega_{\mathrm{m}}<<\omega_{\mathrm{c}}$. Which of the following frequencies is not contained in the modulated wave ? [JEE Main-2017]

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Sol. (3) Refer NCERT Page No. 526 Three frequencies are contained $\omega_{\mathrm{m}}+\omega_{\mathrm{c}}, \omega_{\mathrm{c}}-\omega_{\mathrm{m}} \& \omega_{\mathrm{c}}$

Q. A telephonic communication service is working at carrier frequency of 10 GHz. Only 10% of it is utilized for transmission. How many telephonic channels can be transmitted simultaneously if each channel requires a bandwidth of 5 kHz ? [JEE Main-2018]

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Sol. (2) Since the carrier frequency is distributed as band width frequency, so 10% of 10 GHz = n × 5 kHz where n = no of channels $\frac{10}{100} \times 10 \times 10^{9}=n \times 5 \times 10^{3} \mathrm{n}=2 \times 10^{5}$ telephonic channels

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1
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1
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3
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1
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0
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11
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8
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35
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15
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6
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3
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Sukriya

0
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2
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1
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1
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1
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1
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2
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More concept &formulas on communication systems

3
• March 22, 2020 at 5:38 pm


text:
# Sharp quantum states

1. Sep 15, 2015

### IxRxPhysicist

Hey all,
I'm reading through an anecdotal work about the philosophical foundations of quantum field theory and the authors keep referring to states having the ability to be "sharp." As in it's possible for P to be sharp if the system is mixed, where P is some property of the system. Thanks! IR

2. Sep 15, 2015

### BvU

Hello,

probably they refer to the Heisenberg uncertainty relation (which google).

3. Sep 15, 2015

### vanhees71

Well, philosophers tend to be not very sharp usually (some are, because they have a proper education in math, logics, and other hard sciences). That's why you cannot make sense of the phrase "a quantum state is sharp".

I've a vague clue what they might mean. A state in quantum theory is represented by a normalized vector in Hilbert space (for the purpose of discussion we can be a bit sloppy here; more precisely a pure state is represented by a ray in Hilbert space and not by a single vector). Observables are represented by self-adjoint operators in Hilbert space. These have (generalized) eigenvectors and eigenvalues, and the eigenvalues are the possible outcomes of exact measurements of this observable. If you know the pure state of the quantum system (and you can only know it, if the system has been prepared such as to be in that state), then the probability (density) to find a possible eigenvalue $a$ of the observable $A$, represented by the self-adjoint operator $\hat{A}$ is given by
$$P(a)=|\langle a|\psi \rangle|^2,$$
where I have assumed that the eigenvalues are not degenerate, i.e., that for each (generalized) eigenvalue $a$ there's exactly one (generalized) eigenvector $|a \rangle$. This is known as "Born's rule" and is one of the basic postulates of quantum theory.

What I mean with "generalized eigenvector and eigenvalue" is the following. Many operators/observables in quantum theory have eigenvalues which build a continuum. Examples are a component of a position or momentum vector of a massive, which have the entire real axis as a spectrum, i.e., generalized eigenvalues. Sometimes you also have observables which can only take discrete values. An example is one component of angular momentum like $\hat{J}_z$, which can take only either half-integer or integer values (for half-integer and integer spin particles respectively) or you can have the case that an operator has both a discrete and a continuous spectrum like the Hamiltonian of a hydrogen atom.

The generalized eigenvector for an eigenvalue in the continuous part of the spectrum is never normalizable to 1 and thus can never represent a state, but you can always normalized these generalized states in the sense of a $\delta$ distribution,
$$\langle a |a' \rangle=\delta(a-a').$$
After setting up this mathematics (which however is very rough and not really accurate; a mathematician would cry, when reading this ;-)) we can discuss, what might be meant by this phrase of a state being sharp.

If you have an observable with a discrete spectrum. Then if you take an eigenvalue of the operator in this discrete part of the spectrum, you have a true normalizable eigenvector $|a \rangle$, which you can normalize such that $\langle a|a \rangle=1$, you can prepare your system to be in this state. For a theorist the most simple way to think of such a preparation procedure is that you measure somehow the observable in a way such that you can through all particles away which have not the value $a$ for this observable and keep those which have this value (this is known as a von Neumann filter measurement).

Now there's a theorem saying that eigenvectors of a self-adjoint operator with different eigenvalues (let's for simplicity take the example for an operator that has only discrete nondegenerate eigenvalues) are orthogonal to each other and that they are complete, i.e., you can span the entire Hilbert space of states as a series
$$|\psi \rangle=\sum_{a'} \psi_{a'} |a' \rangle.$$
Then if the particle is prepared in the state $|\psi \rangle=|a \rangle$ the probability to measure the eigenvalue $a'$ is given by
$$P(a')=|\langle a|a' \rangle|^2=\delta_{a,a'}.$$
So you have $P(a)=1$ and $P(a')=0$ for all $a' \neq a$. This means that you measure the value $a$ with certainty, if the particle is prepared in an eigenstate of the operator that represents this measured observable. Then you say that due to the preparation procedure the value of the observable is determined or you also say it has the "sharp value" $a$. In this sense you can say that a true normalizable eigenstate is "sharp" with the meaning that then this value of the considered observable is "sharp".

It is also immediately clear that this can never work with generalized eigenvalues in the continuous spectrum of the operator. Formally you can argue very much in the same way, but then you have $\langle{a'}{a}=\delta(a-a')$, but this implies that you cannot square this to get a probability. The reason is that the generalized eigenvector $|a \rangle$ is not a proper state. You can only prepare the particle to take the value $a$ within a "small interval around $a$" but not to be really sharply $a$.

This is all consistent due to the Heisenberg-Robertson uncertainty relation between incompatible observables, and there are always pairs of incompatible observables. This means the following: You can only prepare a particle in a state such that two observables $A$ and $B$ take always sharp values, if there is a complete set of (generalized) simultaneous eigenvectors of both operators, representing these observables. One can show that under this condition the two operators commute (and also the opposite is true, i.e., if the operators commute you always find a common complete set of (generalized) eigenvalues). So one calls observables compatible, if the representing operators commute. If they do not commute, you have the uncertainty relation, which states that for any (true normalizable) state $|\psi \rangle$ the standard deviations of the observables $\Delta A$ and $\Delta B$ fullfil
$$\Delta A \Delta B \geq \frac{1}{2} |\langle{\psi}|[\hat{A},\hat{B}]\psi \rangle|.$$
The most famous uncertainty relation is that between position and momentum components in the same direction,
$$[\hat{x],\hat{p}_x]=\frac{\hbar}{2} \hat{1},$$
which implies
$$\Delta x \Delta p_x \geq{\hbar}{2}.$$
This in turn implies that there's no proper (normalizable) state in which you can prepare the particle to have either a sharp position or a sharp momentum (because position and momenum operators have only a continuous spectrum). You can prepare the particle with as sharp a position as you like, i.e., you can make the standard deviation $\Delta x$ as small as you like (but not 0!), but then the Heisenberg-Robertson uncertainty relation tells you that the momentum must be quite undetermined, i.e., it's standard deviation $\Delta p_x$ becomes pretty large.


text:
Randall, Dana J.

Compute Distance To:
 Author ID: randall.dana-j Published as: Randall, Dana; Randall, Dana J.; Randall, D. more...less
 Documents Indexed: 62 Publications since 1987 3 Contributions as Editor Co-Authors: 40 Co-Authors with 57 Joint Publications 1,295 Co-Co-Authors
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Co-Authors

 6 single-authored 9 Miracle, Sarah 8 Bhatnagar, Nayantara 7 Greenberg, Sam 7 Pascoe Streib, Amanda 6 Cannon, Sarah M. 6 Sinclair, Alistair 6 Tetali, Prasad 5 Bhakta, Prateek 4 Galvin, David J. 4 Kenyon, Claire M. 3 Daymude, Joshua J. 3 Richa, Andrea Werneck 2 Bezáková, Ivona 2 Blanca, Antonio 2 Chayes, Jennifer Tour 2 Cryan, Mary 2 Dyer, Martin E. 2 Fahrbach, Matthew 2 Karlin, Anna R. 2 Luby, Michael G. 2 Martin, Russell A. 2 Vazirani, Vijay V. 2 Vigoda, Eric 1 Andrés Arroyo, Marta 1 Bernstein, Megan 1 Borgs, Christian 1 Chawla, Shuchi 1 Chen, Yuxuan 1 Cousins, Ben 1 Doyle, Peter G. 1 Gokmen, Cem 1 Janson, Svante 1 Kahn, Jeff D. 1 Lagarias, Jeffrey C. 1 Madras, Neal 1 Odlyzko, Andrew M. 1 Raghavendra, Prasad 1 Sorkin, Gregory B. 1 Spencer, Joel H. 1 Wilson, David Bruce 1 Winkler, Peter M. 1 Yngve, Gary
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Serials

 3 Algorithmica 3 SIAM Journal on Discrete Mathematics 3 Random Structures & Algorithms 3 Combinatorics, Probability and Computing 2 Journal of Mathematical Physics 2 Journal of Statistical Physics 2 SIAM Journal on Computing 2 Theoretical Computer Science 1 Discrete Applied Mathematics 1 Discrete & Computational Geometry 1 Complex Systems 1 The Annals of Applied Probability 1 Notices of the American Mathematical Society 1 Journal of Combinatorial Optimization 1 ACM Transactions on Algorithms
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Fields

 35 Probability theory and stochastic processes (60-XX) 31 Computer science (68-XX) 20 Combinatorics (05-XX) 20 Statistical mechanics, structure of matter (82-XX) 9 Numerical analysis (65-XX) 5 Convex and discrete geometry (52-XX) 3 General and overarching topics; collections (00-XX) 3 Statistics (62-XX) 2 Operations research, mathematical programming (90-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Biology and other natural sciences (92-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Functions of a complex variable (30-XX)

Citations contained in zbMATH Open

43 Publications have been cited 328 times in 247 Documents Cited by Year
Markov chain algorithms for planar lattice structures. Zbl 0992.82013
Luby, Michael; Randall, Dana; Sinclair, Alistair
2001
Markov chain decomposition for convergence rate analysis. Zbl 1017.60080
Madras, Neal; Randall, Dana
2002
Analyzing Glauber dynamics by comparison of Markov chains. Zbl 0974.60052
Randall, Dana; Tetali, Prasad
2000
Approximating the number of monomer-dimer coverings of a lattice. Zbl 1081.82523
Kenyon, Claire; Randall, Dana; Sinclair, Alistair
1996
Dynamic TCP acknowledgment and other stories about $$e/(e-1)$$. Zbl 1045.68155
Karlin, Anna R.; Kenyon, Claire; Randall, Dana
2003
Sampling spin configurations of an Ising system. Zbl 1052.82525
Randall, Dana; Wilson, David
1999
Torpid mixing of simulated tempering on the Potts model. Zbl 1318.82023
Bhatnagar, Nayantara; Randall, Dana
2004
Self-packing of centrally symmetric convex bodies in $$\mathbb{R}{}^ 2$$. Zbl 0756.52016
Doyle, P. G.; Lagarias, J. C.; Randall, D.
1992
Markov chain algorithms for planar lattice structures. (Extended abstract). Zbl 0938.68927
Luby, Michael; Randall, Dana; Sinclair, Alistair
1995
Sampling stable marriages: why spouse-swapping won’t work. Zbl 1192.91157
Bhatnagar, Nayantara; Greenberg, Sam; Randall, Dana
2008
Slow mixing of Glauber dynamics via topological obstructions. Zbl 1192.82058
Randall, Dana
2006
Clustering and mixing times for segregation models on $$\mathbb{Z}^2$$. Zbl 1426.91187
Bhakta, Prateek; Miracle, Sarah; Randall, Dana
2014
The van den Berg-Kesten-Reimer inequality: a review. Zbl 1115.82320
Borgs, C.; Chayes, J. T.; Randall, D.
1999
Torpid mixing of local Markov chains on 3-colorings of the discrete torus. Zbl 1302.05052
Galvin, David; Randall, Dana
2007
Mixing points on a circle. Zbl 1142.68624
Randall, Dana; Winkler, Peter
2005
Disjoint decomposition of Markov chains and sampling circuits in Cayley graphs. Zbl 1103.65012
Martin, Russell; Randall, Dana
2006
Phase coexistence and slow mixing for the hard-core model on $$\mathbb Z^{2}$$. Zbl 1405.68224
Blanca, Antonio; Galvin, David; Randall, Dana; Tetali, Prasad
2013
Random three-dimensional tilings of Aztec octahedra and tetrahedra: An extension of domino tilings. Zbl 0951.05023
Randall, Dana; Yngve, Gary
2000
Random dyadic tilings of the unit square. Zbl 1022.52008
Janson, Svante; Randall, Dana; Spencer, Joel
2002
Dynamic TCP acknowledgement and other stories about $$e/(e-1)$$. Zbl 1323.68300
Karlin, Anna R.; Kenyon, Claire; Randall, Dana
2001
Phase coexistence and torpid mixing in the 3-coloring model on $${\mathbb Z}^d$$. Zbl 1321.82012
Galvin, David; Kahn, Jeff; Randall, Dana; Sorkin, Gregory B.
2015
Approximately counting integral flows and cell-bounded contingency tables. Zbl 1217.68245
Cryan, Mary; Dyer, Martin; Randall, Dana
2010
Approximately counting integral flows and cell-bounded contingency tables. Zbl 1192.68887
Cryan, Mary; Dyer, Martin; Randall, Dana
2005
Phase transitions in random dyadic tilings and rectangular dissections. Zbl 1371.60129
Cannon, Sarah; Miracle, Sarah; Randall, Dana
2015
Sampling biased lattice configurations using exponential metrics. Zbl 1421.68175
Greenberg, Sam; Pascoe, Amanda; Randall, Dana
2009
Mixing times of Markov chains for self-organizing lists and biased permutations. Zbl 1423.60109
Bhakta, Prateek; Miracle, Sarah; Randall, Dana; Streib, Amanda Pascoe
2013
Mixing times of Markov chains on 3-orientations of planar triangulations. Zbl 1296.05057
Miracle, Sarah; Randall, Dana; Streib, Amanda Pascoe; Tetali, Prasad
2012
Slow mixing of Markov chains using fault lines and fat contours. Zbl 1204.60065
Greenberg, Sam; Randall, Dana
2010
Self-testing algorithms for self-avoiding walks. Zbl 0977.82020
Randall, Dana; Sinclair, Alistair
2000
Phase transitions in random dyadic tilings and rectangular dissections. Zbl 1394.60078
Cannon, Sarah; Miracle, Sarah; Randall, Dana
2018
A Markov chain algorithm for compression in self-organizing particle systems. Zbl 1375.68193
Cannon, Sarah; Daymude, Joshua J.; Randall, Dana; Richa, Andréa W.
2016
Efficient generation of random nonsingular matrices. Zbl 0793.65032
Randall, Dana
1993
On the Diaconis-Gangolli Markov chain for sampling contingency tables with cell-bounded entries. Zbl 1279.62030
Bezáková, Ivona; Bhatnagar, Nayantara; Randall, Dana
2011
Phase coexistence for the hard-core model on $$\mathbb{Z}^2$$. Zbl 1434.60034
Blanca, Antonio; Chen, Yuxuan; Galvin, David; Randall, Dana; Tetali, Prasad
2019
On the periods of some graph transformations. Zbl 0672.05080
Odlyzko, Andrew M.; Randall, Dana J.
1987
Simulated tempering and swapping on mean-field models. Zbl 1348.82042
Bhatnagar, Nayantara; Randall, Dana
2016
Proceedings of the 22nd annual ACM-SIAM symposium on discrete algorithms, SODA 2011, San Francisco, CA, USA, January 23–25, 2011. Zbl 1258.68015
2011
Decomposition methods and sampling circuits in the Cartesian lattice. Zbl 1004.60076
Randall, Dana
2001
Sampling on lattices with free boundary conditions using randomized extensions. Zbl 1419.82009
Cannon, Sarah; Randall, Dana
2016
Sampling and counting 3-orientations of planar triangulations. Zbl 1337.60174
Miracle, Sarah; Randall, Dana; Streib, Amanda Pascoe; Tetali, Prasad
2016
Matchings in lattice graphs. Zbl 1310.68242
Kenyon, Claire; Randall, Dana; Sinclair, Alistair
1993
The effect of boundary conditions on mixing rates of Markov chains. Zbl 1155.82307
Bhatnagar, Nayantara; Greenberg, Sam; Randall, Dana
2006
Random bichromatic matchings. Zbl 1147.05054
Bhatnagar, Nayantara; Randall, Dana; Vazirani, Vijay V.; Vigoda, Eric
2008
Phase coexistence for the hard-core model on $$\mathbb{Z}^2$$. Zbl 1434.60034
Blanca, Antonio; Chen, Yuxuan; Galvin, David; Randall, Dana; Tetali, Prasad
2019
Phase transitions in random dyadic tilings and rectangular dissections. Zbl 1394.60078
Cannon, Sarah; Miracle, Sarah; Randall, Dana
2018
A Markov chain algorithm for compression in self-organizing particle systems. Zbl 1375.68193
Cannon, Sarah; Daymude, Joshua J.; Randall, Dana; Richa, Andréa W.
2016
Simulated tempering and swapping on mean-field models. Zbl 1348.82042
Bhatnagar, Nayantara; Randall, Dana
2016
Sampling on lattices with free boundary conditions using randomized extensions. Zbl 1419.82009
Cannon, Sarah; Randall, Dana
2016
Sampling and counting 3-orientations of planar triangulations. Zbl 1337.60174
Miracle, Sarah; Randall, Dana; Streib, Amanda Pascoe; Tetali, Prasad
2016
Phase coexistence and torpid mixing in the 3-coloring model on $${\mathbb Z}^d$$. Zbl 1321.82012
Galvin, David; Kahn, Jeff; Randall, Dana; Sorkin, Gregory B.
2015
Phase transitions in random dyadic tilings and rectangular dissections. Zbl 1371.60129
Cannon, Sarah; Miracle, Sarah; Randall, Dana
2015
Clustering and mixing times for segregation models on $$\mathbb{Z}^2$$. Zbl 1426.91187
Bhakta, Prateek; Miracle, Sarah; Randall, Dana
2014
Phase coexistence and slow mixing for the hard-core model on $$\mathbb Z^{2}$$. Zbl 1405.68224
Blanca, Antonio; Galvin, David; Randall, Dana; Tetali, Prasad
2013
Mixing times of Markov chains for self-organizing lists and biased permutations. Zbl 1423.60109
Bhakta, Prateek; Miracle, Sarah; Randall, Dana; Streib, Amanda Pascoe
2013
Mixing times of Markov chains on 3-orientations of planar triangulations. Zbl 1296.05057
Miracle, Sarah; Randall, Dana; Streib, Amanda Pascoe; Tetali, Prasad
2012
On the Diaconis-Gangolli Markov chain for sampling contingency tables with cell-bounded entries. Zbl 1279.62030
Bezáková, Ivona; Bhatnagar, Nayantara; Randall, Dana
2011
Proceedings of the 22nd annual ACM-SIAM symposium on discrete algorithms, SODA 2011, San Francisco, CA, USA, January 23–25, 2011. Zbl 1258.68015
2011
Approximately counting integral flows and cell-bounded contingency tables. Zbl 1217.68245
Cryan, Mary; Dyer, Martin; Randall, Dana
2010
Slow mixing of Markov chains using fault lines and fat contours. Zbl 1204.60065
Greenberg, Sam; Randall, Dana
2010
Sampling biased lattice configurations using exponential metrics. Zbl 1421.68175
Greenberg, Sam; Pascoe, Amanda; Randall, Dana
2009
Sampling stable marriages: why spouse-swapping won’t work. Zbl 1192.91157
Bhatnagar, Nayantara; Greenberg, Sam; Randall, Dana
2008
Random bichromatic matchings. Zbl 1147.05054
Bhatnagar, Nayantara; Randall, Dana; Vazirani, Vijay V.; Vigoda, Eric
2008
Torpid mixing of local Markov chains on 3-colorings of the discrete torus. Zbl 1302.05052
Galvin, David; Randall, Dana
2007
Slow mixing of Glauber dynamics via topological obstructions. Zbl 1192.82058
Randall, Dana
2006
Disjoint decomposition of Markov chains and sampling circuits in Cayley graphs. Zbl 1103.65012
Martin, Russell; Randall, Dana
2006
The effect of boundary conditions on mixing rates of Markov chains. Zbl 1155.82307
Bhatnagar, Nayantara; Greenberg, Sam; Randall, Dana
2006
Mixing points on a circle. Zbl 1142.68624
Randall, Dana; Winkler, Peter
2005
Approximately counting integral flows and cell-bounded contingency tables. Zbl 1192.68887
Cryan, Mary; Dyer, Martin; Randall, Dana
2005
Torpid mixing of simulated tempering on the Potts model. Zbl 1318.82023
Bhatnagar, Nayantara; Randall, Dana
2004
Dynamic TCP acknowledgment and other stories about $$e/(e-1)$$. Zbl 1045.68155
Karlin, Anna R.; Kenyon, Claire; Randall, Dana
2003
Markov chain decomposition for convergence rate analysis. Zbl 1017.60080
Madras, Neal; Randall, Dana
2002
Random dyadic tilings of the unit square. Zbl 1022.52008
Janson, Svante; Randall, Dana; Spencer, Joel
2002
Markov chain algorithms for planar lattice structures. Zbl 0992.82013
Luby, Michael; Randall, Dana; Sinclair, Alistair
2001
Dynamic TCP acknowledgement and other stories about $$e/(e-1)$$. Zbl 1323.68300
Karlin, Anna R.; Kenyon, Claire; Randall, Dana
2001
Decomposition methods and sampling circuits in the Cartesian lattice. Zbl 1004.60076
Randall, Dana
2001
Analyzing Glauber dynamics by comparison of Markov chains. Zbl 0974.60052
Randall, Dana; Tetali, Prasad
2000
Random three-dimensional tilings of Aztec octahedra and tetrahedra: An extension of domino tilings. Zbl 0951.05023
Randall, Dana; Yngve, Gary
2000
Self-testing algorithms for self-avoiding walks. Zbl 0977.82020
Randall, Dana; Sinclair, Alistair
2000
Sampling spin configurations of an Ising system. Zbl 1052.82525
Randall, Dana; Wilson, David
1999
The van den Berg-Kesten-Reimer inequality: a review. Zbl 1115.82320
Borgs, C.; Chayes, J. T.; Randall, D.
1999
Approximating the number of monomer-dimer coverings of a lattice. Zbl 1081.82523
Kenyon, Claire; Randall, Dana; Sinclair, Alistair
1996
Markov chain algorithms for planar lattice structures. (Extended abstract). Zbl 0938.68927
Luby, Michael; Randall, Dana; Sinclair, Alistair
1995
Efficient generation of random nonsingular matrices. Zbl 0793.65032
Randall, Dana
1993
Matchings in lattice graphs. Zbl 1310.68242
Kenyon, Claire; Randall, Dana; Sinclair, Alistair
1993
Self-packing of centrally symmetric convex bodies in $$\mathbb{R}{}^ 2$$. Zbl 0756.52016
Doyle, P. G.; Lagarias, J. C.; Randall, D.
1992
On the periods of some graph transformations. Zbl 0672.05080
Odlyzko, Andrew M.; Randall, Dana J.
1987
all top 5

Cited by 432 Authors

 17 Randall, Dana J. 9 Goldberg, Leslie Ann 8 Vigoda, Eric 6 Jerrum, Mark R. 6 Rémila, Eric 6 Toninelli, Fabio Lucio 5 Bezáková, Ivona 5 Kayibi, Koko Kalambay 5 Pak, Igor 5 Pirzada, Shariefuddin 5 Tetali, Prasad 4 Bhatnagar, Nayantara 4 Dyer, Martin E. 4 Laslier, Benoît 4 Miracle, Sarah 4 Nachmias, Asaf 4 Sinclair, Alistair 4 Tassy, Martin 3 Blanca, Antonio 3 Borodin, Alexei 3 Cai, Jin-Yi 3 Caputo, Pietro 3 Chhita, Sunil 3 Desreux, Sébastien 3 Dürr, Christoph 3 Epstein, Leah 3 Galvin, David J. 3 Greenberg, Sam 3 Jalsenius, Markus 3 Johansson, Kurt 3 Khan, Muhammad Ali 3 Kozma, Gady 3 Lu, Pinyan 3 Martin, Russell A. 3 Martinelli, Fabio 3 Mertzios, George B. 3 Molitor, Louise 3 Pascoe Streib, Amanda 3 Patt-Shamir, Boaz 3 Perkins, Will 3 Pillai, Natesh S. 3 Sly, Allan 3 Stauffer, Alexandre O. 3 Voudouris, Alexandros A. 2 Afek, Yehuda 2 Ardila, Federico 2 Bhakta, Prateek 2 Bienkowski, Marcin 2 Bodini, Olivier 2 Böhm, Martin 2 Bordewich, Magnus 2 Byrka, Jarosław 2 Cannon, Sarah M. 2 Caraceni, Alessandra 2 Chebolu, Prasad 2 Chen, Zongchen 2 Chrobak, Marek 2 Díaz, Josep 2 Emek, Yuval 2 Erdős, Péter L. 2 Feldheim, Ohad Noy 2 Folwarczný, Lukáš 2 Franceschetti, Massimo 2 Friedland, Shmuel 2 Friedrich, Tobias 2 Goles Chacc, Eric 2 Gorin, Vadim 2 Hermon, Jonathan 2 Holroyd, Alexander E. 2 Huber, Mark L. 2 Jeż, Łukasz 2 Kijima, Shuji 2 Kolikant, Noa 2 Labbé, Cyril 2 Lacoin, Hubert 2 Lenzner, Pascal 2 Levi, Amir 2 Levin, David A. 2 Liu, Tianyu 2 Löwe, Matthias 2 Lubetzky, Eyal 2 Madras, Neal 2 Meeks, Kitty 2 Menz, Georg 2 Miklós, István 2 Milet, Pedro H. 2 Möller, Daniel 2 Moore, Cristopher 2 Nardi, Francesca Romana 2 Omidvar, Hamed 2 Paturi, Ramamohan 2 Peled, Ron 2 Peres, Yuval 2 Rapaport, Ivan 2 Rastegari, Baharak 2 Richerby, David M. 2 Rosenthal, Jeffrey S. 2 Rudolf, Daniel 2 Saldanha, Nicolau Corção 2 Schneider, Stefan ...and 332 more Authors
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Cited in 86 Serials

 22 Theoretical Computer Science 17 Journal of Statistical Physics 13 The Annals of Applied Probability 11 Probability Theory and Related Fields 8 Communications in Mathematical Physics 7 Algorithmica 6 SIAM Journal on Discrete Mathematics 6 Stochastic Processes and their Applications 6 Combinatorics, Probability and Computing 5 The Annals of Probability 5 Transactions of the American Mathematical Society 5 Random Structures & Algorithms 5 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 4 Journal of Computer and System Sciences 4 Discrete & Computational Geometry 4 The Electronic Journal of Combinatorics 4 Theory of Computing Systems 3 Journal of Mathematical Physics 3 Advances in Mathematics 3 SIAM Journal on Computing 3 Journal of Discrete Algorithms 3 Journal of Statistical Mechanics: Theory and Experiment 2 Discrete Applied Mathematics 2 Discrete Mathematics 2 Journal of Combinatorial Theory. Series A 2 Distributed Computing 2 Electronic Journal of Probability 2 Journal of Combinatorial Optimization 2 LMS Journal of Computation and Mathematics 2 Journal of Machine Learning Research (JMLR) 2 Algorithms 2 Statistics and Computing 2 SIAM Journal on Mathematics of Data Science 1 Advances in Applied Probability 1 Artificial Intelligence 1 Communications on Pure and Applied Mathematics 1 Information Processing Letters 1 Israel Journal of Mathematics 1 Journal of Computational Physics 1 Linear and Multilinear Algebra 1 Metrika 1 Nuclear Physics. B 1 Annals of the Institute of Statistical Mathematics 1 The Annals of Statistics 1 Applied Mathematics and Computation 1 Dissertationes Mathematicae 1 Illinois Journal of Mathematics 1 Inventiones Mathematicae 1 Journal of Applied Probability 1 Journal of Combinatorial Theory. Series B 1 Journal of Computational and Applied Mathematics 1 Mathematica Scandinavica 1 Operations Research 1 Proceedings of the American Mathematical Society 1 SIAM Journal on Control and Optimization 1 Advances in Applied Mathematics 1 Statistics & Probability Letters 1 Graphs and Combinatorics 1 Journal of Complexity 1 Information and Computation 1 Computers & Operations Research 1 Journal of the American Mathematical Society 1 Journal of Cryptology 1 Japan Journal of Industrial and Applied Mathematics 1 Computational Geometry 1 International Journal of Foundations of Computer Science 1 Games and Economic Behavior 1 Journal of Statistical Computation and Simulation 1 Linear Algebra and its Applications 1 Selecta Mathematica. New Series 1 The Journal of Artificial Intelligence Research (JAIR) 1 Monte Carlo Methods and Applications 1 Bernoulli 1 Abstract and Applied Analysis 1 Optimization Methods & Software 1 Mathematical Physics, Analysis and Geometry 1 Journal of the European Mathematical Society (JEMS) 1 Acta Mathematica Sinica. English Series 1 Annales Henri Poincaré 1 Acta Numerica 1 Discrete Optimization 1 Mathematics in Computer Science 1 Journal of Statistical Theory and Practice 1 Acta Universitatis Sapientiae. Mathematica 1 Acta Universitatis Sapientiae. Informatica 1 Journal of Applied Mathematics & Informatics
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Cited in 28 Fields

 108 Probability theory and stochastic processes (60-XX) 86 Computer science (68-XX) 83 Combinatorics (05-XX) 78 Statistical mechanics, structure of matter (82-XX) 35 Numerical analysis (65-XX) 22 Convex and discrete geometry (52-XX) 20 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 19 Statistics (62-XX) 15 Operations research, mathematical programming (90-XX) 9 Dynamical systems and ergodic theory (37-XX) 6 Linear and multilinear algebra; matrix theory (15-XX) 4 Number theory (11-XX) 4 Biology and other natural sciences (92-XX) 3 Order, lattices, ordered algebraic structures (06-XX) 3 Partial differential equations (35-XX) 3 Information and communication theory, circuits (94-XX) 2 Commutative algebra (13-XX) 2 Algebraic geometry (14-XX) 2 Special functions (33-XX) 2 Functional analysis (46-XX) 1 Group theory and generalizations (20-XX) 1 Measure and integration (28-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Mechanics of deformable solids (74-XX) 1 Fluid mechanics (76-XX) 1 Quantum theory (81-XX)


text:
• Higher-dimensional fractional time-independent Schrödinger equation via fractional derivative with generalised pseudoharmonic potential

• # Fulltext

https://www.ias.ac.in/article/fulltext/pram/093/05/0076

• # Keywords

Fractional radial Schrödinger equation; generalised pseudoharmonic potential; bound-state solutions; Mittag–Leffler function

• # Abstract

In this paper, we obtain approximate bound-state solutions of $N$-dimensional time-independent fractional Schrödinger equation for the generalised pseudoharmonic potential which has the form $V(r^{\alpha}) = a_{1}r^{2\alpha} + (a_{2}/r^{2\alpha}) + a_{3}$. Here $\alpha$ (0 < $\alpha$ < 1) acts like a fractional parameter for the space variable $r$. The entire study consists of the Jumarie-type fractional derivative and the elegance of Laplace transform. As a result, we can successfully express the approximate bound-state solution in terms of Mittag–Leffler function and fractionally defined confluent hypergeometric function. Our study may be treated as a generalisation of all previous workscarried out on this topic when $\alpha = 1$ and $N$ arbitrary. We provide numerical result of energy eigenvalues and eigenfunctions for a typical diatomic molecule for different α close to unity. Finally, we try to correlate our work with a Cornell potential model which corresponds to $\alpha = 1/2$ with $a_{3} = 0$ and predicts the approximate mass spectra of quarkonia.

• # Author Affiliations

1. Kodalia Prasanna Banga High School (HS), South 24 Parganas, Kolkata 700 146, India
2. Department of Applied Mathematics, University of Calcutta, Kolkata 700 073, India
3. Reactor Control System Design Section (E & I Group), Bhabha Atomic Research Centre,Mumbai 400 085, India

• # Pramana – Journal of Physics

Volume 96, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
# Shell Service¶

The shell service is used to provide shell access and commanding from mission operations to the OBC. It may also be used between a developer’s system and the OBC when in a development or testing environment.

The shell service provides shell functionality by implementing the shell protocol. The shell protocol is UDP-based which means a connection is required between the OBC and ground segment capable of transferring UDP packets. This should be established using a standard network connection.

## Configuration¶

The shell service has a couple configuration options which may be defined in the system’s config.toml file:

• [shell-service.addr]

• ip - Specifies the service’s IP address
• port - Specifies the port on which the service will be listening for UDP packets

For example:

[shell-service.addr]
ip = "0.0.0.0"
port = 8010

## Running the Service from KubOS¶

The Kubos Linux distribution (as of v1.3.0) ships with the shell service installed and configured to run on boot. This can be verified by booting the KubOS system, running the ps command and looking for the shell-service process. If the service is not running then it can be started like so:

$/etc/init.d/S90shell-service start  ## Running the Service from Source¶ The shell service can also be run from source if required. The source is located in the folder kubos/services/shell-service in the KubOS source repo. The service can be started like so: $ cd kubos/services/shell-service
\$ cargo run -- -c config.toml

The service will look for the given config.toml file in order to get the needed configuration options.


text:
SCIENTIA SINICA Informationis, Volume 47 , Issue 4 : 401-415(2017) https://doi.org/10.1360/N112016-00179

## Developable surface reconstruction from noisy data with $\boldsymbol L_{\rm 0}$-norm minimization

• AcceptedOct 24, 2016
• PublishedMar 1, 2017
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### Funded by

• Figure 1

(Color online) Optimization of normal vector field preserving developability. (a) Original noisy model with normal vector field; (b) neighborhood of data points; (c) optimized normal vector field; (d) original normal vectors on Gaussian sphere (up) and optimized normal vectors on Gaussian sphere (down)

• Table 1   Computation time
 Model #Vertex $d$ in Eq. (eqn:noise) Time of normal optimization (s) Time of surface fitting (s) Figure 2 820 0.08 20 7 Figure 3 2720 0.08 61 13 Figure 4 1999 0.05 42 9 Figure 5 4769 0.08 133 25 Figure 6 5297 0.1 178 31 Figure 7 992 0.08 28 11

Citations

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# Jewelry Company Quality Test Failure Probability

## Problem 731

A jewelry company requires for its products to pass three tests before they are sold at stores. For gold rings, 90 % passes the first test, 85 % passes the second test, and 80 % passes the third test. If a product fails any test, the product is thrown away and it will not take the subsequent tests. If a gold ring failed to pass one of the tests, what is the probability that it failed the second test?

## Solution.

Let $F$ be the event that a gold ring fails one of the three tests. Let $F_2$ be the event that it fails the second test. Then what we need to compute is the conditional probability
$P(F_2 \mid F) = \frac{P(F_2 \cap F)}{P(F)}.$

The numerator is
$P(F_2 \cap F) = P(F_2) = 0.9 \cdot 0.15.$ (A gold ring passes the first test with probability $0.9$ and fails the second test with probability $1-0.85=0.15$.)

The complement $F^c$ of $F$ is the event that a gold ring passes all the tests. Thus
$P(F) = 1- P(F^c) = 1 – 0.9 \cdot 0.85 \cdot 0.8.$ It follows that the desired probability is
\begin{align*}
P(F_2 \mid F) &= \frac{0.9 \cdot 0.15}{1 – 0.9 \cdot 0.85 \cdot 0.8}
= \frac{135}{388} \approx 0.348
\end{align*}

Therefore, given that a gold ring failed to pass one of the tests, the probability that it failed the second test is about 34.8 %.


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# 14: Coupled Linear Oscillators

• 14.1: Introduction to Coupled Linear Oscillators
Coupled linear oscillators are ubiquitous in life.
• 14.2: Two Coupled Linear Oscillators
A basic two-coupled oscillator system.
• 14.3: Normal Modes
Independent modes of two coupled linear oscillators.
• 14.4: Center of Mass Oscillations
Spurious center of mass oscillations.
• 14.5: Weak Coupling
There are myriad examples involving weakly-coupled oscillators in many aspects of the natural world. There are many examples applied to musical instruments, acoustics, and engineering. Weakly coupled oscillators are a dominant theme throughout biology as illustrated by congregations of synchronously flashing fireflies, crickets that chirp in unison, an audience clapping at the end of a performance.
• 14.6: General Analytic Theory for Coupled Linear Oscillators
The development of a general analytic theory  of n coupled linear oscillators, that is capable of finding the normal modes and their eigenvalues and eigenvectors. The solution of many coupled linear oscillators is a classic eigenvalue problem where one has to rotate to the principal axis system to project out the normal modes. The following discussion presents a general approach to the problem of finding the normal coordinates for a system of n coupled linear oscillators.
• 14.7: Two-body coupled oscillator systems
Examples of two-body coupled oscillators.
• 14.8: Three-body coupled linear oscillator systems
Mean field and nearest neighbor coupling.
• 14.9: Molecular coupled oscillator systems
Linear and ring molecular systems.
• 14.10: Discrete Lattice Chain
A crystalline lattice comprises thousands of coupled oscillators in a three dimensional matrix. A classical treatment of lattice dynamics of is of interest since classical mechanics underlies many features of the motion of atoms in a crystalline lattice. The linear discrete lattice chain is the simplest example of many-body coupled oscillator systems that can illuminate the physics underlying a range of interesting phenomena in solid-state physics.
• 14.11: Damped Coupled Linear Oscillators
In general, dissipative forces are non linear which greatly complicates solving the equations of motion for damped coupled oscillator systems. However, for some systems the dissipative forces depend linearly on velocity which allows use of the Rayleigh dissipation function.
• 14.12: Collective Synchronization of Coupled Oscillators
Collective synchronization of coupled oscillators is a multifaceted phenomenon where large ensembles of coupled oscillators, with comparable natural frequencies, self synchronize leading to coherent collective modes of motion. Biological examples include congregations of synchronously flashing fireflies, crickets that chirp in unison, an audience clapping at the end of a performance, networks of pacemaker cells in the heart, as well as neural networks in the brain and spinal cord.
• 14.E: Coupled linear oscillators (Exercises)
• 14.S: Coupled linear oscillators (Summary)

Thumbnail: A double pendulum consists of two pendulums attached end to end. (CC BY-SA 3.0; 100Miezekatzen).

This page titled 14: Coupled Linear Oscillators is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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# Schrödinger Equation

I am reading up on the Schrödinger equation and I quote

Because the potential is symmetric under $x\to-x$, we expect that there will be solutions of definite parity.

Could someone kindly explain why this is true? And perhaps also what it means physically?

-

Good question! First you need to know that parity refers to the behavior of a physical system, or one of the mathematical functions that describe such a system, under reflection. There are two "kinds" of parity:

• If $f(x) = f(-x)$, we say the function $f$ has even parity
• If $f(x) = -f(-x)$, we say the function $f$ has odd parity

Of course, for most functions, neither of those conditions are true, and in that case we would say the function $f$ has indefinite parity.

Now, have a look at the time-independent Schrödinger equation in 1D:

$$-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2}\psi(x) + V(x)\psi(x) = E\psi(x)$$

and notice what happens when you reflect $x\to -x$:

$$-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2}\psi(-x) + V(-x)\psi(-x) = E\psi(-x)$$

If you have a symmetric (even) potential, $V(x) = V(-x)$, this is exactly the same as the original equation except that we've transformed $\psi(x) \to \psi(-x)$. Since the two functions $\psi(x)$ and $\psi(-x)$ satisfy the same equation, you should get the same solutions for them, except for an overall multiplicative constant; in other words,

$$\psi(x) = a\psi(-x)$$

Normalizing $\psi$ requires that $|a| = 1$, which leaves two possibilities: $a = +1$ (even parity) and $a = -1$ (odd parity).

As for what this means physically, it tells you that whenever you have a symmetric potential, you should be able to find a basis of eigenstates which have definite even or odd parity (though I haven't proved that here,* only made it seem reasonable). In practice, you get linear combinations of eigenstates with different parities, so the actual state may not actually be symmetric (or antisymmetric) around the origin, but it does at least tell you that if your potential is symmetric, you could construct a symmetric (or antisymmetric) state. That's not guaranteed otherwise. You'd probably have to get input from someone else as to what exactly definite-parity states are used for, though, since that's out of my area of expertise (unless you care about parity of elementary particles, which is rather weirder).

*There is a parity operator $P$ that reverses the orientation of the space: $Pf(x) = f(-x)$. Functions of definite parity are eigenfunctions of this operator. I believe you can demonstrate the existence of a definite-parity eigenbasis by showing that $[H,P] = 0$.

-
David, thank you so much! Your explanation is superb! :) – bra-ket Aug 26 '11 at 7:16
States with a definite parity are eigenstates possessing certain energy $E_n$. A perturbed system becomes a superposition with no certain energy and with no certain parity $\Psi(t)=\sum_n C_n \psi_n e^\frac{-iE_n t}{\hbar}$, i.e., it is time-dependent. A tme-independent superposition (without those oscillating exponentials) arise only if you want to represent some time-independent function $\Phi$ as a spectral sum $\sum_n A_n \psi_n$. – Vladimir Kalitvianski Aug 26 '11 at 7:49
Ooh, btw why can't $a\in\mathbb{C}$ and $|a|=1$? – bra-ket Aug 26 '11 at 7:52
Doing reflection twice, you get $a^2 =1$ so it is a real number. – Vladimir Kalitvianski Aug 26 '11 at 7:54
Aha! Thanks! :) – bra-ket Aug 26 '11 at 8:05


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# Block Projected up Plane with Spring atop

1. Oct 10, 2008

### PatrickR.

1. The problem statement, all variables and given/known data

A spring (k = 200 N/m) is fixed at the top of a frictionless plane inclined at angle θ = 40 °. A 1.6 kg block is projected up the plane, from an initial position that is distance d = 0.80 m from the end of the relaxed spring. With what kinetic energy must the block be projected up the plane if it is to stop momentarily when it has compressed the spring by 0.50 m?

2. Relevant equations

Ui + Ki = Uf + Kf

U(elastic)=.5kx2

U(gravitational)= mgy

3. The attempt at a solution

So I assumed that Ui=0 because I'm assuming the block starts at the bottom of the plane.

I also assumed that because the block is momentarily stopping that Kf=0

so that leaves me with

Ki=Uf and solving for Uf will give me Ki

Uf= U(elastic) + U(gravitational)

Uf = .5(200N/m)(.5m)2 + (1.6kg)(9.8m/s2)([.5m+.8m] sin 40)

Uf= 38.1026J

therefore Ki should equal 38.1026

However, that answer is refused so I would like to know what I am doing wrong. Thankyou.

2. Oct 10, 2008

### Rake-MC

You have a good start. Do you know why you took the sin(40) in the fourth last line?

This is because the gravitational potential is $$mg\Delta h$$

By taking this sine, you are making the change in height parallel to the force such that you get the right answer. However you also took the sine for spring potential energy. However the line of force of the spring is parallel to the plane.

3. Oct 10, 2008

### PatrickR.

Even more troubling, I still got the answer wrong and it was my last attempt so they just show you how to do it then and my setup actually was correct however the computer did not want to take my numbers (the numbers are individualized for everyone.

Rake I do actually think the right way to do it is what you said and not consider the .5 into the sin 40
so, it should look like this

Uf = .5(200N/m)(.5m)2 + (1.6kg)(9.8m/s2)([.5m]+.8 sin 40)

but this just makes me hate physics even more...