# Linear Sentence Examples

linear
• Given certain linear and angular measurements, the area must be so and so.

• The linear width of the band (e) is the increment of which alters p by 27r, so that e =27r /tr.

• The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.

• This variation, however, is not always linear.

• This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general.

• One of the most important questions in investigating a linear algebra is to decide the necessary relations between a and b in order that this product may be zero.

• If 2R be the diameter of the objectglass and D the distance of the object, the angle subtended by AP is E/D, and the angular resolving power is given by X/2 D sin a = X/2 R (3) This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.

• In a somewhat similar way a dark linear interruption in a bright ground may be visible, although its actual width is much inferior to the half wave-length.

• A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating.

• The vapour tension may approximate to a linear function of the composition, and the curve will then be practically a straight line.

• If we assume that s is a linear function of 0, s= so(I +aO), the adiabatic equation takes the form, s 0 log e OW +aso(0 - Oo) +R loge(v/vo) =o

• Other favourite types' of equation for approximate work are (I) p=RO/v±f(v), which makes p a linear function of 0 at constant volume, as in van der Waal's equation; (2) v=RO/p+f(p), which makes v a linear function of 0 at constant pressure.

• The body is held fixed, and the reactio of the mechanism and the resultant of the impulsive pressure on th surface are a measure of the impulse, linear,, , and angula A, µ, v, required to start the circulation.

• The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by the resultant axial linear momentum of the circulation.

• Reynolds, in his investigation, introducing no new form of law of distribution of velocities, uses a linear quantity, proportional to the mean free path of the gaseous molecules, which he takes to represent (somewhat roughly) the average distance from which molecules directly affect, by their convection, the state of the medium; the gas not being uniform on account of the gradient of temperature, the change going on at each point is calculated from the elements contributed by the parts at this particular distance in all directions.

• The word is also used as a unit of linear measurement of the magnifying power of a lens or microscope.

• In this instrument a considerable linear relative movement of the divided lens corresponds with a comparatively small separation of the double image, so that simple verniers reading to 6 1 0 in.

• Apart from speculation, the first definite evidence for the molecular structure of matter occurs when it is found that certain physical phenomena change their whole nature as soon as we deal with matter of which the linear dimensions are less than a certain amount.

• It follows that the average distance apart of the molecules in the gaseous state is roughly ten times as great as in the solid or liquid state, and hence that in the gaseous state the molecules are at distances apart which are large compared with their linear dimensions.

• The genus is represented in Europe, north Asia, North America and Australia, and is characterized by oblong or linear stem-leaves, flowers in.

• The failure seems (§ 2) to be due to difficulty in realizing the numerical expression of an area or a solid in terms of a specified unit, while the same difficulty does not arise in the case of linear measure or liquid measure, where the number of units can be ascertained by direct counting.

• The ideas of moment and of centroid are extended to geometrical figures, whether solid, superficial or linear.

• The simplest case is that in which u is constant or is a linear function of x, i.e.

• Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1.

• The velocity of propagation of a torsional disturbance along a wire of circular section may be found by the transfer of momentum method, remembering that we must now replace linear momentum by angular momentum.

• The unit of linear measure is the wall, which is subdivided into wah or sauk, a wah or kup, and into 9 1 6 wah or niew.

• Let E be the effective elasticity of the aether; then E = pc t, where p is its density, and c the velocity of light which is 3 X 10 10 cm./sec. If = A cos" (t - x/c) is the linear vibration, the stress is E dE/dx; and the total energy, which is twice the kinetic energy Zp(d/dt) 2 dx, is 2pn2A2 per cm., which is thus equal to 1.8 ergs as above.

• It has in fact been found, with the very great precision of which optical experiment is capable, that all terrestrial optical phenomenareflexion, refraction, polarization linear and circular, diffraction - are entirely unaffected by the direction of the earth's motion, while the same result has recently been extended to electrostatic forces; and this is our main experimental clue.

• Another powerful reason for taking the aether to be stationary is afforded by the character of the equations of electrodynamics; they are all of linear type, and superposition of effects is possible.

• Now the kinetics of a medium in which the parts can have finite relative motions will lead to equations which are not linear - as, for example, those of hydrodynamics - and the phenomena will be far more complexly involved.

• The polarization itself is determined from the electric force (P,Q,R) by the usual statical formula of linear type which becomes tor an isotropic medium (.f',g',h') = c2(P,Q,R), because any change of the dielectric constant K arising from the convection of the material through the aether must be independent of the sign of v and therefore be of the second order.

• In these the furcal branches are linear or rudimentary, the shell is without rostral sinus, and, besides distinguishing characters of the second 2ntennae, they have always a branchial plate well developed on the first maxillae, which is inconstant in the other tribe.

• For engineering and manufacturing purposes the more important linear gauges are, however, now used, adjusted to some fundamental unit of measure as the inch; although in certain trades, as for wires and flat metals, gauges continue to be used of arbitrary scales and of merely numerical sizes, having no reference to a legal unit of measure; and such are rarely accurate.

• Both as regards structure and habits, the leopard may be reckoned as one of the more typical representatives of the genus Felis, belonging to that section in which the hyoid bone is loosely connected with the skull, owing to imperfect ossification of its anterior arch, and the pupil of the eye when contracted under the influence of light is circular, not linear as in the smaller cats.

• The twigs are densely clothed with flat spreading linear leaves of a fine glossy green above and glaucous beneath; in the old trees they become shorter and more rigid and partly lose their distichous habit.

• This frame or tube is so constructed of iron and brass (one-third iron and two-thirds brass) that its temperature coefficient of linear expansion is the same as that of the platinumsilver alloy.

• As the frame has the same linear expansion as the wire, external changes of the temperature will not affect their relative length, but if the fine wire is heated by the passage of an electric current, its expansion will move the indicating needle over the scale, the motion being multiplied by the gear.

• The linear ridges of this mIddle section are often called the Alleghany Mountains.

• They agree with primitive nephridia in being of ectodermic origin, in consisting of perforated cells in linear series, and in having no communication with the coelom.

• The rice plant is an annual grass with long linear glabrous leaves, each provided with a long sharply pointed ligule.

• The slit of the collimator confines the light to a nearly linear source, the beam diverging from each point of the source being subsequently made parallel by means of a lens.

• In order to record line spectra it is by no means necessary that the receiving instrument (bolometer or radiometer) should be linear in shape, for the separation of adjacent lines may be obtained if the linear receiver be replaced by a narrow slit in a screen placed at the focus of the condensing lens.

• If the linear velocity of the cups in feet a second is V 1, and the linear velocity of the jet is V2, then the velocity of the jet relative to the cup is V2 - V1 feet a second, and if the whole energy of the water is to be given up to the cups, the water must leave the cup with zero absolute velocity.

• This will become zero if V 1 is 2V 2, that is, if the linear velocity of the cupcentres is one-half that of the jet of water impinging upon them.

• The form of the leaf shows a very great variety ranging from the narrow linear form with parallel sides, as in grasses or the needle-like leaves of pines and firs to more or less rounded or orbicular - descriptions of these will be found in works on descriptive botany - FIG.

• The first leaves produced by the young plant are linear,the second are sagittate and hastate, the third are rounded-cordate and the next are orbicular.

• The flow of heat may still be linear if the horizontal layers of the soil are of uniform composition, but the quantity flowing through each layer is no longer the same.

• This gives an average value of the conductivity over the range, but it is better to observe the temperatures at three distances, and to assume k to be a linear function of the temperature, in which case the solution of the equation is still very simple, namely, 0+Ze6 2 =a log r+b, (3) where e is the temperature-coefficient of the conductivity.

• The chief uncertainty of this method is the area from which the heat is collected, which probably exceeds that of the central column, owing to the disturbance of the linear flow by the projecting bulb of the calorimeter.

• As in the case of steady-flow methods, by far the simplest example to consider is that of the linear flow of heat in an infinite solid, which is most nearly realized in nature in the propagation of temperature waves in the surface of the soil.

• The equation of the method is the same as that for the linear flow with the addition of a small term representing the radiation loss.

• Although this correction should be made if the definition were strictly followed, it is more convenient in practice to include the small effect of linear expansion in the temperature-coefficient in the case of solid bodies.

• Smoothing them, Carlheim-Gyllenskold gives f = too' -7.3c as the most probable linear relation between c, the amount of cloud, and f, the frequency, assuming the latter to be loo when there is no cloud.

• Those of the earliest period, the lower limit of which is put about 150o B.C., are aeneolithic, metal being, however, rare and only found in the form of small ornaments; pottery with linear decoration is abundant.

• The earlier Choripetalae embraces the Polypetalae and Monochlamydae of the French systems. It includes 21 series, and is an attempt to arrange as far as possible in a linear series those orders which are characterized by absence or freedom of petals.

• They range from subjects of the homeliest and most mirthful realism to others serious and devout, and from literal or almost literal transcripts of natural form to the most whimsically abstract combinations of linear pattern and tendril .and flourish.

• The two carried out some of the earliest thermochemical investigations, devised apparatus for measuring linear and cubical expansions, and employed a modification of Joseph Black's ice calorimeter in a series of determinations of specific heats.

• Hottonia (water violet) is a floating water plant with submerged leaves cut into fine linear segments.

• In some cases these transitions may be unsatisfactory and forced; it is apparent that the linear development from " being " to the " idea " is got by transforming into a logical order the sequence that has roughly prevailed in philosophy from the Eleatics; cases might be quoted where the reasoning seems a play upon words; and it may often be doubted whether certain ideas do not involve extra-logical considerations.

• When the parallax of a star is known, we are able to infer from its proper motion its actual linear speed in miles per hour, in so far as the motion is transverse to the line of sight.

• Regarded as a linear velocity, the parallactic motion is the same for all stars, being exactly equal and opposite to the solar motion; but its amount, as measured by the corresponding angular displacement of the star, is inversely proportional to the distance of the star from the earth, and foreshortening causes it to vary as the sine of the angular distance from the apex.

• To arrive at some estimate of the speed of the solar motion, we may consider the motions of those stars whose parallaxes have been measured, and whose actual linear speed is accordingly known (disregarding motion in the line of sight).

• Somewhere, apparently, in the 4th millennium B.C., we begin to find inscriptions written on clay, in an almost linear script, in the Sumerian tongue.

• The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary.

• In its essential nature a set is a linear function of any number of " distinct " units of the same species.

• Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.

• Again a unit point p. may be regarded as by multiplication changing (a) from octonion to point-plane-scalar, (b) from point-plane-scalar to octonion, (c) from plane-scalar to linear element, (d) from linear element to plane-scalar.

• Perry (Steam Engine, p. 580), assuming a characteristic equation similar to Zeuner's (which makes v a linear function of the temperature at constant pressure, and S independent of the pressure), calculates S as a function of the temperature to satisfy Regnault's formula (10) for the total heat.

• Under these conditions both S and s may be regarded as approximately constant, so that L is a linear function of the temperature.

• A formula of the same type was given by Athenase Dupre (Theorie de chaleur, p. 96, Paris, 1869), on the assumption that the latent heat was a linear function of the temperature, taking the instance of Regnault's formula (io) for steam.

• For the conditions of equilibrium of the forces on each pin furnish vi equations, viz, two for each point, which are linear in respect of the stresses and the extraneous forces.

• The translation parallel to this axis is lox + mly + nhz (Xf + uv + vi) Ic. (8) The linear magnitude which measures the ratio of translation to rotation in a screw is called the pitch.

• The complex is therefore of the type called linear (in relation to the degree of this locus).

• It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element s must satisfy the conditions of equilibrium laid down in I.

• The first of these may be called the Principle of Linear Momentum.

• If there are no extraneous forces, the resultant linear momentum is constant in every respect.

• If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric additiofi of the total impulse of the extraneous forces.

• On this basis we can predicate the principles of linear and angular momentum, as in 15.

• If we integrate with respect to I between fixed limits, we obtain the principles of linear and angular momentum in the form previously given, Hence, whichever form of postulate we adopt, we are led to the principles of linear and angular momentum, which form in fact the basis of all our subsequent work.

• It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9).

• This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the qs) executes a simple vibration of period 21r/u.

• The series of equations of the type (3) is then replaced by a single linear partial differential equation, or by a set of two or three such equations, according to the number of dependent variables.

• Let the angular velocity of the rotation be denoted by a=dO/dt, then the linear velocity of any point A at the distance r from the axis is or; and the path of that point is a circle of the radius r described about the axis.

• Let yr be the linear velocity of the point E fixed in the plane of axes AOB.

• Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone.

• That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.

• Let a small body of the weight w undergo translation in a circulai path of the radius p, with the angular velocity of deflexion a, so that the common linear velocity of all its particles is v=ap. Then the actual energy of that body is WV2/2g = Waip2/2g.

• The principles according to which variations of angular velocity store and restore energy are the same as those of 117, only substituting moment of inertia for mass, and angular for linear velocity.

• These investigations led him to the announcement of the fundamental law of action between elements of current, or currents in infinitely short lengths of linear conductors, upon one another at a distance; summed up in compact expression this law states that the action is proportional to the product of the current strengths of the two elements, and the lengths of the two elements, and inversely proportional to the square of the distance between the two elements, and also directly proportional to a function of the angles which the line joining the elements makes with the directions of the two elements respectively.

• For infinitely distant objects the radius of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (vide supra," Monochromatic Aberration of the Axis Point "); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is proportional to the ratio of the aperture to the focal length, i.e.

• If r be the number of quotients in the recurring cycle, we can by writing down the relations connectin g the successive p's and q's obtain a linear relation connecting p nr +m, t'(n-1)r +m, +m in which the coefficients are all constants.

• We Will Assume That When, As In Most Cases, Viscosity Maybe Neglected, The Mass (M) Of A Drop Depends Only Upon The Density (V), The Capillary Tension (T), The Acceleration Of Gravity (G), And The Linear Dimension Of The Tube (A).

• Its specific heat is 0.05701 (Regnault) or 0.0559 (Bunsen); its coefficient of linear expansion is 0.0000-1921.

• It is usually the case that the secondary caustic is easier to determine than the caustic, and hence, when determined, it affords a ready means for deducing, the primary caustic. It may be shown by geometrical considerations that the secondary caustic is a curve similar to the first positive pedal of the reflecting curve, of twice the linear dimensions, with respect to the luminous point.

• In this work he introduced the use of linear functions in place of the ordinary co-ordinates; he also made the fullest use of the principles of collineation and reciprocity.

• Lynxes are found in the northern and temperate regions of both the Old and New World; they are smaller than leopards, and larger than true wild cats, with long limbs, short stumpy tail, ears tufted at the tip, and pupil of the eye linear when contracted.

• They are hypogynous, and have long and very delicate filaments, and large, linear or oblong two-celled anthers, dorsifixed and ultimately very versatile, deeply indented at each end, and commonly exserted and pendulous.

• On the other, posterior, side of the grain is a more or less evident, sometimes punctiform, sometimes elongated or linear mark, the hilum, the place where the ovule was fastened to the wall of the ovary.

• He has used all his acquired science of linear and aerial perspective to create an almost complete illusion to the eye, but an illusion that has in it nothing trivial, and in heightening our sense of the material reality of the scene only heightens its profound spiritual impressiveness and gravity.

• Leonardo sought to achieve that conquest and at the same time to carry the old Florentine excellences of linear drawing and psychological expression to a perfection of which other men had not dreamed.

• Leaves simple, small, linear or short and scale-like, usually persisting for more than one year.

• The long linear leaves of some species of Podocarpus, in which the lamina is traversed by a single vein, recall the pinnae of Cycas; the branches of some Dacrydiums and other forms closely resemble those of lycopods; these superficial resemblances, both between different genera of conifers and between conifers and other plants, coupled with the usual occurrence of fossil coniferous twigs without cones attached to them, render the determination of extinct types a very unsatisfactory and frequently an impossible task.

• From this first theca originates a second, opening in the same direction, and from the second a third, and so on, in a continuous linear series until the polypary is complete.

• In particular if D =o, that is, if the given curve be unicursal, the transformed curve is a line, 4 is a mere linear function of 0, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 0; it is easy to see that for a given curve of the order m, these functions of 0 must be of the same order m.

• The long, sparingly branched stem bore at the somewhat swollen nodes whorls of six to eighteen wedge-shaped or linear leaves, which did not alternate in successive whorls.

• What is known at present, while it does not indicate the phylogeny of the Lycopodiales, at least shows that such living orders as Lycopodiaceae and Selaginellaceae cannot be regarded as forming a linear series.

• The leaves are simple and linear in Pilularia, but in Marsilia bear a pinnate four-lobed lamina.

• It further appears that a determinant is a linear function' of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or - according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs.

• By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations.

• Reverting to the system of linear equations written down at the beginning of this article, consider the determinant ax+by+cz - d,b,c a' x+b' y+c'z - d', b', c" a"x+b"y+c"z - d", b", c" it appears that this is viz.

• To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding theorem for the multiplication of two matrices.

• The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear equations.

• Similar phenomena occur in the case of many other couples, and it is found that the thermoelectric power p is not in general a constant, and that the simple linear formula (I) is applicable only for small differences of temperature.

• If we suppose that s is not quite constant, but increases or diminishes slightly with change of temperature according to a linear formula s=50-1-2cT (in which so represents the constant part of s, and c may have either sign), we obtain a more general formula which is evidently the sum of the two previous solutions and may be made to cover a greater variety of cases.

• According to this formula, the Peltier effect is a linear function of the temperature.

• This might be appropriately represented, as already suggested, by a linear formula s=so - cT.

• They are generally of a more or less oval or elliptical form, or they may be globular, as in Mercurialis annua; at other times linear or clavate, curved, flexuose, or sinuose, as in bryony and gourd.

• The free lamina, however, was always considerably more developed than in the recent family; in form it was usually linear or narrowly lanceolate.

• In Asterophyllites., the generic distinction of which from Annularia is not always clear, the narrow linear leaves are in crowded whorls, and the ultimate branches distichously arranged; in the Calamocladus of Grand' Eury - characteristic of the Upper Coal Measures - the whorls are more remote, and the twigs polystichous in arrangement.

• As examples of these doubtful forms may be mentioned Thinnfeldia, characteristic of Rhaetic and Lower Jurassic rocks; Dichopteris, represented by some exceptionally fine Jurassic specimens, described by Zigno, from Italy; and Ctenis, a genus chiefly from Jurassic beds, founded on pinnate fronds like those of Zamia and other Cycads, with linear pinnae characterized by anastomosing veins.

• A number of hairy linear bracts enclose the whole; internal to these occur 12 to 20 crowded pinnate leaves (sporophylls), with their apical portions bent over towards the axis of the flower, the bases of the petioles being fused laterally into a disk surrounding the base of the conical receptacle.

• I simply did not want to engage with a linear narrative.

• In 2006 a further three linear accelerators will be bought to bring the total at the new West of Scotland Cancer Center to 11.

• Figure 4. [below] Linear regression of plasma ACTH following fasting stress on daily fatigue severity rating.

• Study the mode of CO 2 adsorption on the hydrotalcite surface, to determine whether linear or bridged adsorption on the hydrotalcite surface, to determine whether linear or bridged adsorption occurs.

• While linear methods have proven invaluable over many years in generating first-order approximations, they are severely limited in both qualitative and quantitative predictions.

• A simple artifice is used to represent any spanwise twist distribution by a combination of linear segments.

• Sharaf al-Din al-Tusi (born 1201) invented the linear astrolabe.

• Users of industrial automation are more likely to specify linear motor systems rather than separate linear components.

• Parametric analyzes showed a significant linear relationship between increasing proportions of horizontally averted gaze and increased rCBF in the MPF cortex.

• They grow in the upper axils of the linear or lance shaped, deep green arching leaves.

• Designs include linear phase filters, a wideband phase shifter, and chirp filters with fractional bandwidths exceeding 100% .

• It never falls into a linear plot, which remains the bane of the script writer.

• Whether the force is fighting in a linear or a nonlinear battlefield, the factors of METTT will determine location of the FARP.

• The corrugated cloth surround is coated with a damping material to make diaphragm movement more linear and discourage cone breakup.

• Thus, linear causality is the bedrock of technology and social contracts.

• Following a section on normed linear spaces, we will consider periodic approximation using classical Fourier series.

• We therefore suggest that the linear attenuation coefficient be treated as a separate unit.

• In the longitudinal cohorts, the rate of change was determined by linear regression.

• There are two main design proposals for the future linear collider.

• A major component of modern linear accelerators which assists this technique is the multi-leaf collimator (MLC ).

• Mechanical coolers used in space are generally based upon a linear compressor with clearance seals.

• This is done by coding DTDs and integrity constraints with linear constraints on the integers.

• Should we think in terms of a linear expansion of western consumerism ending in global convergence?

• It's a gameplay contrivance that keeps you on a linear path through each area.

• Modern power recorders tend to have linear convertors but with a high bit number.

• Two voxel-based similarity measures, the linear correlation coefficient and the entropy correlation coefficient, are used.

• This approach has been proven to be highly resistant against many attacks such as differential and linear cryptanalysis.

• History is linear and not cyclical and going on for ever.

• The series of linear earthworks, just north of the fort, also merit further investigation.

• Most likely, Wansdyke was a defensive linear earthwork.

• The program tests for linear dependence by calculating the eigenvalues of the overlap matrix.

• Course Description Solving linear systems Ax = b or finding the eigenvalues of a matrix may appear to be trivial tasks.

• The poled microstructure determines the linear elastic, dielectric and piezoelectric moduli.

• The current work assumes linear elasticity which is a weak model for real soft tissue.

• Von Koch's first results were on infinitely many linear equations in infinitely many unknowns.

• Identifying strategies by some kind of behavioral equivalence we expect to receive a linear category.

• Simple methods define an arbitrary law, such as an inverse square relationship, or a linear falloff.

• The problem being that Flash downloads in a linear fashion from frame 1 through to the last frame.

• Whence the concept of linear karma dominates the foreground of ethics.

• Biographical data, treatment patterns, and clinical impressions were analyzed by the generalized linear model and generalized estimating equations method.

• The most obvious effect is that it stimulates linear growth of the skeleton.

• In addition there are numerous stone circles and several possible henges and linear stone alignments.

• Also focuses on the decipherment and decoding of cuneiform, Egyptian hieroglyphs, Linear B, and Mayan Glyphs.

• The free layer is now being swept in a hard axis, and shows reduced hysteresis, and a more linear response.

• In turn point counting gave lower values than linear intercepts.

• Such metabolic pathways are not linear but often intersect to form a complex network.

• Invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariants of finite groups which come from a linear group action.

• The set of linear equations is then solved by means of a matrix inversion.

• It also shows how causation distinguishes time from space, makes time linear, gives it its direction and enables us to perceive it.

• As individual elements we choose piecewise linear or smooth maps, and compare results obtained in both cases.

• Agents of the Enlightenment thought history was strictly linear, he says.

• First, they have tended to see the global in terms of an essentially linear process, or linear processes, of globalization.

• Memory usage is approximately linear in RE size, and largely insensitive to RE complexity, except for bounded repetitions.

• This fall in heart rate is fairly linear, decreasing by approximately 1 beat per minute per year.

• Therefore, these algorithms rely on a locally linear internal model of the black box.

• To rectify this I relocated both handles to the bottom edge in which the two points remain linear.

• Have you established that the relationship is linear or can be transformed to become linear?

• If you place the dots with even spacing from frame to frame, the motion will look linear.

• In order to deliver the radiation to the patient we use devices called medical linear accelerators, or " linacs " .

• The Radiotherapy Department has eight linear accelerators and a Pantak superficial x-ray therapy unit.

• Szilard's ideas included the linear accelerator, cyclotron, electron microscope, and nuclear chain reaction.

• Oxford Radcliffe Hospitals NHS Trust is to get two new linear accelerators to cut delays and improve treatment for cancer patients in Oxfordshire.

• Even where new linear accelerators have been purchased, often with Lottery money, there are not always the staff to operate them.

• This implied that there should be 48 additional linear accelerators in the UK, of which 22 would replace old cobalt machines.

• Core modules include mathematical ecology, mathematical biology and medicine and applied linear algebra.

• Areas covered include linear algebra, optimization, quadrature, differential equations, regression analysis, and time series analysis.

• Knowledge of Maple as a tool for doing linear algebra.

• Here's an example describing *one* type of process associated with matrices, that I've used in teaching linear algebra.

• In numerical linear algebra he developed backward error analysis methods.

• Such a differential structure, expressing the local versus global nature of brain structure, is lacking in adaptive linear algebra.

• We shall concentrate on a particularly nice class of codes called linear codes, a beautiful application of elementary linear algebra.

• The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive.

• Topics will include a review of multiple linear regression and will then focus on logistic regression and multinomial logistic regression.

• These include complete sets of curriculum material for various college and university level courses, including calculus, linear algebra and engineering mathematics.

• Analysis involved the use of a longitudinal linear model ANCOVA.

• As a general principle, off-line moorings are always preferable to creating new linear moorings.

• We work mostly with nucleic acids here, incorporating modified nucleosides into linear as well as small circular DNA/RNA chains.

• This powerful method is applicable to linear odes or systems of linear ODEs with constant coefficients.

• Topiramate exhibits linear pharmacokinetics with low levels of protein binding.

• This means that the action will be linear in beta, delta phi, and alpha.

• A new system has also been implemented for supporting the moving platen on the machine base, using linear bearings.

• The marked linear polarization of the emission from the condensate is also measured.

• Target materials are low and linear low density polyethylene, but the new masterbatches can also be used with HDPE and PP.

• Wheelchair users and buggy pushers may wish to return by the same route, making it into a linear walk.

• Unlike morphea, linear scleroderma tends to involve layers of tissue below the skin.

• Most asters have simple, linear to lance-shaped leaves, sometimes hairy and sometimes serrated.

• Linear programming problems are processed by sparse simplex (SSX) with both PRIMAL and DUAL variants.

• Ground survey at some of the linear monuments such as stone rows is comparatively sparse.

• The quadratic spline takes longer to render than a linear spline.

• Substantial experience has been accumulated in solving large-scale linear, integer programming problems, and recently stochastic integer programming.

• The linear subregions are then constructed in this new space instead of the high dimensional shape space.

• A vector with a linear subscript is processed as a continuous or constant stride vector.

• Those solutions belong (or asymptotically tend) to a certain invariant linear subspace - cluster manifold.

• The numerical substructure may be either linear or non-linear, the latter using a new, fast non-linear analysis algorithm which we have developed.

• It is pounded with an ivory or wood mallet which may be incised with cross-hatch or linear grooves to produce a subtle textured surface.

• Linear broadcast networks like ITV are in danger of becoming the equivalent of fixed-line telephony operators.

• The spatial linear model provides a powerful tool to model such behavior.

• The basic toolbox also allows you to access functions in Maple's linear algebra package.

• Similarly, where an old embankment has been plowed out leaving a zone of thinner topsoil, a linear negative feature can be detected.

• The Radiotherapy Department now boasts a total of ten linear accelerators.

• Compound Array linear transducer composed of multiple layers of arrays enabling electronic focusing in the elevation direction.

• To choose a basis with respect to which the matrix of a linear transformation has a particularly manageable form.

• Historically, these confounding effects have been controlled using linear models assuming a constant additive error variance.

• The components are moved to the pick-up position using a linear vibrator to assist the slope of the feeder.

• Linear systems are much easier to understand than nonlinear ones and the solutions are generally well-behaved.

• About the colour there can be no prima facie difficulty; for, as soon as the question is raised, it is seen that the standard of linear dimension, with reference to which the particles are called small, is the wave-length of light, and that a given set of particles would (on any conceivable view as to their mode of action) produce a continually increasing disturbance as we pass along the spectrum towards the more refrangible end.

• Robur, but in old age the boughs generally curve downwards, and the tree acquires a wide spreading head; the bark is dark brown, becoming grey and furrowed in large trees; the foliage varies much, but in the prevailing kinds the leaves are very deeply sinuated, with pointed, often irregular lobes, the footstalks short, and furnished at the base with long linear stipules that do not fall with the leaf, but remain attached to the bud till the following spring, giving a marked feature to the young shoots.

• For the few cases where data are available - data, however, belonging to engines representing standard practice in their construction and in the design of cylinders and steam ports and passages - the law connecting p and v is approximately linear and of the form p=c - bv (28) where b and c are constants.

• The surgeon should then wait until the "line of demarcation," linear ulceration, between the living and the dead part is evident,, and then, if the case permits, should amputate at a higher level..

• The allies were aware of his position, and still adhering to the old " linear " system, marched to turn his right flank (see Austerlitz).

• When a homogeneous polynomial is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables.

• But the present tendency is to regard the early association of arithmetic with linear measurement as important; and it seems to follow that we may properly (at any rate at an early stage of the subject) multiply a length by a length, and the product again by another length, the practice being dropped when it becomes necessary to give a strict definition of multiplication.

• The linear dimension of the diffraction pattern is inversely as that of the aperture, and the brightness at corresponding points is as the square of the area of aperture.

• He issued in 1719 an improved version of his work on perspective, with the title New Principles of Linear Perspective, revised by Colson in 1749, and printed again, with portrait and life of the author, in 1811, A French translation appeared in 1753 at Lyons.

• He did not obey the laws of I linear perspective as they are formulated in the Occident, nor did he show cast shadows, but his aerial perspective and his foreshortening left nothing to be desired.

• But the more credible explanation is that he merely followed Chinese example in this matter, as he did also in linear perspective, accepting without question the curious canon that lines converge as they approach the spectator.

• He made experiments, simultaneously with Wallis and Wren, on the collision of hard spherical bodies, and his statement of the results (1669) included a clear enunciation of the conservation of linear momentum, as demonstrated for these cases of collision, and apparently correct in certain other cases, mass being estimated by weight.

• He indulged in a great deal of speculation as to the existence of an extra-spatial unit, which was to furnish the raison d'etre of the numerical part, and render the quaternion homogeneous as well as linear.

• While all wings are graduated as explained, innumerable varieties occur as to their general contour, some being falcated or scythe-like, others oblong, others rounded or circular, some lanceolate and some linear.

• In the species on which the genus was founded the leaves, as the generic name implies, are cuneate and entire, or toothed on their anterior margina l in other cases they are deeply divided by dichotomy into narrow segments, or the whorl consists of a larger number (up to 30) of apparently simple, linear leaves, which may represent the segments of a smaller number.

• The response curve of PD excretion vs purine absorbed is linear.

• Values of 1 or 2 will cause linear or quadratic interpolation to be used.

• This paper considers a linear triangular simultaneous equations model with conditional quantile restrictions.

• Linear regression I have made my views known on the dangers of fitting straight lines.

• A second rolling circle event makes a long linear +ve strand which is again cleaved by the ribozyme activity.

• Each channel accepts digital input as an index input for a rotary encoder or as a home sensor input for a linear encoder.

• We apply the method to compressible and incompressible linear elasticity problems, including problems with stress singularities.

• In all cases, however, the non-linear optimization will reflect at least some improvement relative to the linear sinusoid fit.

• Fit Details The Add Sine Component Linear Fit Details adds a statistical breakdown of each sinusoid in the fit.

• Trials of the protocol resulted in considerably improved recovery on thawing of human spermatozoa compared with vapor freezing or conventional linear cooling.

• A few years ago, Colin Stirling and David Walker produced a tableau system for model-checking the linear time mu-calculus on finite systems.

• The CD-Rom adds non-linear, or differently linear, temporalities to this tradition.

• The linear, static paradigm of mechanistic science based on interactions between separate, independent parts is a travesty of organic reality.

• Linear focusing spot to flood beam with a twist of the wrist.

• This allows routine computation of undamped vibration of any linear structure, for example by Finite Element methods.

• Then there is a linear transformation A which takes x to y and u j to v j.

• While some missions occasionally veer into more linear territory, you 're always free to approach the levels exactly the way you wish.

• Energy goes to linear (vertical) generators.

• Too many rigid lines of straight-legged furniture, striped fabrics, linear wall décor, and even straight lamps repeated throughout the room can result in an uncomfortable room design.

• Now that you've realized your room needs an oval area rug to break up the repetitious linear design, you need to decide on the size, style, and color for this new centerpiece in your room.

• The notes in Bath are not as pronounced as those in Beach; here, they all seem to meld into one, resulting in a rather linear fragrance that smells primarily of Ivory soap.

• If the chronological approach to scrapbook organization is too linear for your tastes, you may wish to try organizing your albums by subject.

• The plant forms a rosette of linear sheathing leaves, from which columnar spikes 1 to 2 feet high, bearing bright yellow starry flowers in a dense raceme, and having the aspect of a miniature Eremurus, issue.

• The chief difference from the netted Iris is in the bulb and leaves, which are narrow, linear, deeply channelled on the inner face, with a central band or rib like a Crocus leaf, and pale green without the glaucous tint usual to this group.

• Linear fluorescents are big energy savers, too, reducing lighting energy output by nearly 50 percent.

• Figure the linear footage needed plus and additional 10 percent to cover for waste and mistakes.

• A typical kit includes all you need to complete the installation, including glue, decorative corner pieces, and enough linear feet of trim to cover a modestly sized room.

• Track lighting fixtures use a linear system that can make great use of limited electrical junction boxes, and some systems can even be plugged into wall outlets.

• Expect to spend as little as \$5 a square foot for field tiles and \$12 a linear foot for border.

• Divide the total number of inches by 12 and round up to get the number of linear feet of screen required to cover your gutters.

• Progression through the worlds is fairly linear.

• Not terribly hard puzzles because the gameplay is mostly linear.

• Final Fantasy X-2 is the first linear sequel to a Final Fantasy -- ever.

• The game is pretty linear and pretty much directs you through levels.

• Missions are no longer linear and can be strategically completed in any order you choose.

• The second are the more action-oriented American-developed titles such as Diablo that tend to be more subtle both graphically and in terms of plot, while offering larger worlds to explore and less linear gameplay.

• As you move through the somewhat linear levels, there seems to be no lag issues of any kind.

• Instead of providing a linear stage-to-stage obstacle course to follow, Super Mario 64 gave gamers a chance to make a few of their own decisions, emphasizing exploration and diversity.

• For linear games with a definite set of goals, you can actually continue playing after you are done with the game.

• Although each puzzle is fairly linear (as is the entire game), we owe our thanks to Ubisoft for adhering to this, as a completely open-ended Prince of Persia would be nearly impossible to complete...

• There are over 20 "chapters" in all to explore, fighting wave after endless wave of alien monstrosities, but by and large, the action is fairly linear.

• On the PS2, however, the Band World Tour is much more like the solo career mode; it's linear like the career mode in Guitar Hero III.

• It is not linear in nature, since the game world is completely open for you to take on whatever tasks you'd like.

• Instead of limiting players to a linear set of missions, the Superman Returns video game offers a more open-ended environment, not unlike the one popularized in the Grand Theft Auto franchise.

• Most of the levels are linear, but they just make sense.

• Other than that, cosmetic flaws like the less-than-stellar voice acting (with the exception of Patrick Stewart and Lou Diamond Phillips) are hardly intrusive to the joys of linear gameplay.

• Complete the look with large area rugs and lamps that utilize geometric shapes or linear lines in the design.

• The most obvious effect of hGH is on linear skeletal growth (height), but metabolic effects of hGH (the results of hGH activity in the body) on muscle, the liver, and fat cells are a critical part of its function.

• Types include linear, oblique, transverse, longitudinal, and spiral fractures.

• Linear fractures have a break that runs parallel to the bone's main axis or in the direction of the bone's shaft.

• For example, a linear fracture of the arm bone could extend the entire length of the bone.

• A longitudinal fracture is similar to a linear fracture.

• In the great majority of children of tall stature, genetics and nutrition are the cause of the greater-than-average height, and linear growth ceases with the end of puberty.

• In individuals with gigantism who are not treated, linear growth can continue unchecked for several decades.

• Linear growth of the long bones occurs by a different process.

• This process is stimulated by the hormones produced by the testes and ovaries, which provide the developmental signal that the linear growth of the long bones should reach completion or full development.

• Thus, poorly nourished boys and girls may be delayed in linear bone growth, diminished in all bone widths, later in the appearance of ossification centers, and delayed in epiphyseal union (completion of long-bone growth).

• If you want to explore the world of bobbed hair without the harsh or linear shape of the original bob, a choppy bob is a very wearable and versatile option.

• If you want to keep your look softer and go longer in between stylist visits, you'll want to avoid sharp linear and precision cuts and instead opt for a softer, blended shape.

• Women who tend to opt for feminine styles may feel out of place with an edgy, linear cut.

• If you've always been enamored by the bob yet intimidated by its linear shape, the following short layered bob hairstyles are for you!

• A linear timeline is especially good for elementary school students.

• This works best with older students who understand the concept of a linear timeline.

• Additionally, women also need to take into account bra size and a linear measurement from nipple to nipple.

• Breakups don't always happen in a clean, linear way.

• Good at logical thinking and pattern recognition, kids with high logical/mathematical intelligence may do especially well with activities involving linear thinking, scientific exploration and reasoning.

• From the moment people first set foot in a schoolroom, they are trained to think and reason in a linear way.

• If you are a water sign, perhaps your tribal piece can be fluid and waved rather than flat and linear?

• The nautical star is a clean and crisp tattoo image consisting of five linear points shaded in a pattern that resembles a compass.

• Due to their long and linear nature, orchids also work for unique placement areas such as full sleeves and leg tattoos.

• These watches come in digital and linear formats and often offer other features such as multiple time zone or city scales.

• In contrast, most weight training equipment, with the exception of cable systems, provide resistance in a linear fashion.

• In the past, most exercise programs involved linear movement.

• While there are many strength and flexibility exercises that make use of rotation, most aerobic machines still use linear movement.

• Most of the movements on these machines were linear.

• He later learns these aliens live inside the wormhole and have little understanding for linear, corporeal existence.

• Most people know time as linear - past, present and future.

• Bite patterns - Bed bugs appear in linear patterns on the skin surface.

• Most color charts are laid out like the shades in a paint store, with colors in a linear shading.

• Hence it is impossible to form a satisfactory linear series.

• There is evidence that the use in Crete of both linear and pictorial signs existed in the Early Minoan period, contemporary with the first Egyptian dynasties.

• During the Third Middle Minoan period, the lower limits of which approach 1600 B.C., this pictographic script finally gives way to a still more developed linear system - which is itself divided into an earlier and a later class.

• The later class (B) of the linear script is that used on the great bulk of the clay tablets of the Cnossian palace, amounting in number to nearly 2000.

• Foot narrow; dorsal papillae linear or fusiform, in several - series.

• It is hard to arrange the Exopterygota in a linear series, for some of the orders that are remarkably primitive in some respects are rather highly specialized in others.

• This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions.

• Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.

• The present article is merely concerned with algebraical linear transformation.

• Since (ab) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being aob l -a l b 0.

• This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put equal to one another and (ab) vanishes.

• A single linear form has, in fact, no invariant.

• If the forms be all linear and different, the function is an invariant, viz.