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To each isolated singularity of a hypersurface of dimensionn, one associates the local fundamental groupG of the moduli space minus the discriminant locus, and a representation σ:G→Aut(H), whereH is then-homology group, with integer coefficients, of the non singular fibre. Although, in general it is very difficult to determine even a presentation ofG, we show that the image of σ can be computed rather easily, by exploiting some relations in a first aproximate presentation ofG, in the case of Brieskorn polynomials namely, polynomials of the type $$x_0^{a_0 } + \cdot \cdot \cdot + x_0^{a_n } $$ . In this way we solve an open problem stated by Brieskorn [1] and Pham [8].

The equation of motion for a general structure can be cast as a system of second order ordinary differential equations in discretized form. The locus of the roots for the homogeneous form of these equations is analyzed and related to the stability region for general, stiffly stable, linear multistep methods. The problem of recasting the second order system as a first order system is analyzed, noting that the final system to be solved is no larger than the original system. A discussion of error control is included and some specific formulas are given. Finally, a new family of third order formulas with a number of advantages over previously published formulas is given.

SynopsisThe sharing of a common self-polar simplex by n−1 quadrics in [n] confers special features on their curve of intersection Γn. The three-dimensional locus Mn of chords of Γn. has certain singularities, some of which are described. In conclusion, a few comments refer to the case n = 4 when Mn is defined by a single equation.

It is shown that plastic analysis is a suitable method for analyzing structures composed of precast concrete panel walls mounted on a steel frame. The panels are bolted on the frame using connecting bolts embedded in the panels. These bolts are assumed to be elastic perfectly plastic. A yield locus for the panel-bolt-frame system is developed. With the aid of the yield locus the collapse load and mode of collapse for a panel frame system can be determined. Of particular interest is the use of panel walls as wind bracing and as partial bearing walls. Two illustrative examples of this type are described. With this method of analysis, full advantage can be taken of the panel's ability to transmit forces between points in the frame.

The Pontryagin/Lawden scalar product of the deviation phase vector and the adjoint phase vector may be identified with Lagrange's reciprocal formula for two variant motions if the acceleration field is conservative. Hence for two slightly different trajectories with common end-points, the terminal velocity differences must have equal scalar product with Lawden's primer vectors. The final velocity difference is orthogonal to the constant time of flight locus for isoenergetic motions from a common initial point. A Keplerian trajectory behaves like a rigid curve as far as radial deviation is concerned in the case of a small change of direction of the initial velocity vector.

The c-isochrone has been defined by Fraps and Juhn1 as the locus of points in the regenerated feather representing a given level of cell division around the annular collar of the feather germ. This locus appears as a constant configuration in each vane-half of the grown feather and is symmetrical with respect to a common transverse shaft level. If the barbs of either vane-half are in parallel array (at any uniform angle with the shaft), the c-isochrone is a straight line from shaft to margin, equidistant on shaft and a given barb from the point of union of that barb with the shaft. In Fig. 1, for example, barbs are laid out at right angles to the shaft, AB (apex of the feather at A), and the c-isochrone must form in this particular construction an angle of 45° with the shaft. Since the apex of the feather forms first, an apical-most c-isochrone, Am of Fig. 1, drawn from apices of the two barbs uniting at their bases to form the shaft (when these are in alignment with the shaft) identifies elements represe...

A simple proof is given of the following observed property of alternating-current networks: In a network consisting of any number of linear bilateral impedances, connected in any manner whatsoever, and any number of generators of the same frequency, the locus of any voltage or current in the system will be a circle as any one self-impedance is varied so that its locus is a straight line in the complex plane. The proof given is based upon the application of the fundamental theorems of network analysis to a general alternating-current network and upon the identification of a circular form of the linear fractional transformation of function theory. A simple analysis of alternating-current networks is frequently possible by virtue of this property. Reference is made to examples of circuit analysis based upon this property. It is further shown that as a result of the observed property the impedance looking into any two terminals of a general passive network of linear bilateral elements has a circular locus as any impedance in the system is varied so that its locus is a straight line.

The heat capacity and magnetization at constant field versus temperature, as well as the adiabatic variation of temperature with magnetic field, have been determined in fields directed along the $c$ axis of a large spherical single crystal of Mn${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}4${\mathrm{H}}_{2}$O to virtually the limits of resolution of the dc methods used. In this study of the behavior of these quantities in the neighborhood of the antiferromagnetic-paramagnetic transition, one specific goal was to observe the (near) singularity in ${(\frac{\ensuremath{\partial}M}{\ensuremath{\partial}T})}_{H}$ at ${T}_{N}(H)$. In addition, we sought to test the predictions that isentropes cross the phase boundary (defined as the locus of maxima in ${C}_{H}$) tangentially, and that this crossing point should prove to be the point of inflection of the isentropes provided ${C}_{H}$ does not diverge too strongly. A test for determining the existence of a divergence without the necessity of measuring infinitely high values is outlined. The fact that the maximum in the zero-field adiabatic susceptibility occurs at a temperature ${T}_{max}g{T}_{N}(0)$ has been found to be reflected in the persistence of a minimum in plots of the isentropic variation of $T$ versus $H$ up to $T={T}_{max}$. This curious behavior has led us to speculate on a larger coexistence region of somewhat different character than has heretofore seemed reasonable.

After a brief description of the construction and connections of the condenser motor, the necessity of varying the capacitance is discussed and the performance at start and under load considered. Locus diagrams illustrating the operating performance in detail are given, followed by a more detailed discussion of starting torque. The Appendix gives the mathematical derivation of many of the formulas discussed.

A method for the design of time-limited and effectively band-limited waveforms for analog communication systems is developed. The design problem is first formulated in the L_2 (0, T) space. Nonlinear modulation threshold effects are incorporated directly into the design by introducing constraints in the form of minimum distance between points on the locus. It is shown that all pertinent information can be expressed in terms of the inner product function generated by the locus and its cross derivative. The design is then transformed into a constrained minimization problem in a space of inner product functions. The cases of stationary locus and N -dimensional locus are further investigated.

The theory of the plastic potential may be based on the principle of maximum rate of dissipation work represented by inequality (1.2). The second fundamental theorem of Thermodynamics yields another inequality, (1.1). Since a consistent theory must include both inequalities, the origin of stress space cannot lie outside the yield locus.

This paper shows a method for using describing functions in a control system with multiple nonlinearities. The linear system resulting after the proper describing functions have been inserted is analyzed using an "input dependent root locus" which is a modified root locus technique. The rules for construction of the input dependent locus and a proof of its applicability are given along with examples which illustrate the method. The technique shows the existence of limit cycles, their amplitude and frequency when they exist, and the stability of the limit cycle.

An analytical solution by successive approximation including the effect of end of end leakage from the oil film of short journal bearings has been presented because of the importance of endwise flow in journal bearings of the short length commonly used. The analytical approximation is based upon the short-bearing approximation originally proposed by Michell and Cardullo and practically developed by DuBois and Ocvirk, and this analytical solution is obtained by using the solution of DuBois and Ocvirk as the fiirst approximate solution and then applying successive approximation method. In this paper pressure distribution, load capacity, locus of journal center and fluid-film friction are discussed, and these results may be applied best to bearings having a length-diameter ratio up to 0.6.

It is the purpose of this paper to discuss and compare certain concepts bearing on the relationship to one another of two families of curves on a surface in a euclidean space of three dimensions. In the first place, there are set over against one another the concept of parallelism, in the sense of LeviCivita, of one family of curves with respect to another and the concept of equidistance of the first family with respect to the second. Secondly, a measure of the deviation from parallelism of the one family of curves with respect to the other, to which is given the name "angular spread," is compared with a measure of the deviation from equidistance of the first family with respect to the second, which is called "distantial spread." In addition to the introduction of the concepts in question, Part I contains the fundamental relations between the spreads of each of two families with respect to the other. These relations lead to quantitative results concerning the spreads, as well as to interesting conclusions concerning the connection between parallelism and equidistance. Part II deals more in detail with the properties of the spreads. In particular, it is found that the angular spread, at a point P, of a fixed family of curves with respect to a variable family of curves is represented geometrically by a variable vector at P the locus of whose terminal point is a circle. This fact leads directly to a generalization of Liouville's formula for geodesic curvature which is not only of interest in itself but tends to show that the original formula is essentially a relationship between angular spreads. Part III bears on the connection between directional derivatives and distantial spreads. It develops an invariant form of Bonnet's formula for geodesic curvature, finds the laws of transformation of the spreads of one system of curves into those of a second, and applies these laws to the study of Tchebycheff systems. In Part IV, the relationships between the spreads of a system of curves on a surface and the spreads of the corresponding system of curves on the spherical representation of the surface are discussed and application is made of the results to the comparison of parallelism and equidistance on the sur

Solutions between the A apex of the compositional triangle and a BC ordered compound have been investigated, from the statistical mechanics point of view, for comparison with the phase section from pure iron to NiAl in which a miscibility gap separates a disordered iron-rich from an ordered NiAl-rich phase. The “zeroth” approximation was used with allowance for next nearest neighbour interactions. It was predicted that the critical temperature for ordering (second order transformation) and the locus of equal equilibrium values of the long range order parameter fell linearly with additions of the A atoms. The critical temperature and composition for phase separation depend on the ratio of the BC exchange interaction energies for first and second neighbours as well as on the exchange interaction energies of the atom pairs AB and AC. Suitable values caused phase separation to occur at higher temperatures in the ordered solution than in the disordered. The theory accounts for the general properties of the iron-NiAl phase section.

(For pt. I, see abstr. A5084 of 1973). The author determines the distribution of zeros of the partition function in the antiferromagnetic case. A principle to determine the limiting locus of zeros of the partition function in the complex fugacity plane has been proposed and the locus was obtained for the antiferromagnetic Husimi-Temperley model. The principle states that the locus is obtained as the place where the real parts of two branches of chi (z) ( identical to analytic continuation of lim 1/N ln Z, where Z is the partition function), whose real parts are the largest and the next largest among several branches, take the same value. The proof of the principle is given for the AHT model. The principle is intuitively plausible and it is suggested that it may work for quite a wide class of models.

With the end positions of the stroke and the radius of the crank specified, the locus of the fixed end of the crank in the plane of crank rotation can be graphically or analytically determined.

In Schubert's book Abzählende Geometrie(3) there are listed, with no formal verifications, eleven first-order degenerations of the twisted cubic curve in S3. The curve, in this connexion, was seen by Schubert as a union C = (CL, CE, CT) of three simply infinite element systems, the locus CL of its points, the envelope CE of its osculating planes and the system CT of its tangent lines, the complete twisted cubic so envisaged being regarded as a single geometric variable of freedom 12, and any degeneration (projective specialization) of it, C¯ = (C¯L, C¯E, C¯T), being said to be of the first order if it has freedom 11. Schubert used the symbols λ, λ′, k, k′ w, w′, θ, θ′, δ, δ′, η to denote the eleven first-order degenerations in question, the dashed symbol denoting always the dual of that denoted by the undashed. In this paper we exhibit analytically the existence of the above degenerations and indicate further how degenerations of higher order can be systematically investigated. We use for this purpose the analytical theory of complete collineations which has just recently been developed (1,4), and also some of the ideas contained in van der Waerden's earlier development of his general method of degenerate collineations (5).

There are simple examples in which a variety is irreducible at a point but has a reducible tangent cone. The following theorem is proved. If X p {X_p} is an irreducible analytic germ and if the Jacobian ideal beomes principal on the normalization then the tangent cone of X X at p p is irreducible. If, moreover, the singular set of X X is a manifold at p p then X X is Whitney a , b a,b -regular along the singular set at p p .

We treat the time-optimal problem of reaching the origin within linear autonomous control systems inn-space, with one-dimensional controls and standard constraints. A description (formula in closed form) of the switching locus is obtained, making it possible to construct the optimal feedback control; some further consequences are also presented.

An experiment on the behavior of a wet particle on a rotating disk was carried out with two kinds of spherical and non-spherical particles. The average diameter of them ranged from 0.261 to 0.620cm and the running speed of the a disk ranged from 92 to 880rpm.As the result, it was found that motion locus of a wet particle on the rotating disk followed to Archimedes' spiral irrespective of the particle shape.

A study of the response of a continuum to given loads is carried out assuming a piecewise linear yield locus, normality, non-interacting yield planes and linear strain bardening. Minimum properties are determined for the plastic strain rates in the incremental case and for plastic strains satisfying holonomic and non-holonomic stress-strain laws, thus extending to continua the minimum properties determined elsewhere for discrete structural linear hardening systems.

The authors have constructed an apparatus with three vibrators and a halfmirror which indicates a spatial vector as a light spot in three-dimensional space. A new type of spatial vectorcardiograph was produced by attaching a vectorcardiograph to this apparatus. The lead system of Frank and a new coordinate system of the authors were employed, and the observation of the cardiac vector was performed. The cardiac vector was visualized as a locus of the light spot which forms three-dimensional loops in space. The observation and the analysis of the heart vector with determination of the quadrants of the loops were performed in normal individuals and in some clinical cases. It was possible to record the spatial vectorcardiogram by the usual photography or stereophotography and to perform objective analysis. This method may be of some value for clinical and educational purposes.

An explosive system is used to drive a strong shock wave into a plate of 24ST aluminum. This shock wave propagates through the 24ST aluminum into small test specimens which are in contact with the front surface of the plate. A photographic technique is used to measure velocities associated with the 24ST aluminum shock wave and with the shock wave in each specimen.The measured velocities are transformed, using the conservation relations, to pressure-compression points. Resulting pressure-compression curves are given for 27 metals. The range of data is different for each material but typically covers the pressure interval 150 to 400 kilobars; probable errors in reported experimental pressure-compression curves are 1 or 2% in compression for a given pressure.The experimental curves, which consist thermodynamically of a known $P$, $V$, $E$ locus for each material, are used to calculate a more complete high-pressure equation of state. This is done by means of a theoretical estimate of the volume variation of the Gr\"uneisen ratio $\ensuremath{\gamma}(V)=V{(\frac{\ensuremath{\partial}P}{\ensuremath{\partial}E})}_{V}$. Calculated $P$, $V$, $T$ states are listed for the various materials. For 24ST aluminum, quantities of application in shock-wave hydrodynamics are also tabulated.

Assuming a two component system for the muscle, a series elastic element and a contractile component, the analyses of the isotonic and isometric data points were related to obtain the series elastic stiffness, dP/dls, from the relation, See PDF for Equation From the isometric data, dP/dt was obtained and shortening velocity, v, was a result of the isotonic experiments. Substituting (P0 - P)/T for dP/dt and (P0 - P)/(P + a) times b for v, dP/dls = (P + a) /bT, where P < P0, and a, b are constants for any lengths l ≤ l0 (Matsumoto, 1965). If the isometric tension and the shortening velocity are recorded for a given muscle length, l0, although the series elastic, ls, and the contractile component, lc, are changing, the total muscle length, l0 remains fixed and therefore the time constant, T. Integrating, See PDF for Equation the stress-strain relation for the series elastic element, See PDF for Equation is obtained; lsc0 - ls + lc0where lco equals the contractile component length for a muscle exerting a tension of P0. For a given P/P0, ls is uniquely determined and must be the same whether on the isotonic or isometric length-tension-time curve. In fact, a locus on one surface curve can be associated with the corresponding locus on the other.

A simple model is presented to explain the marked behaviour of the de-excitation γ-pattern in certain recent inelastic α-particle scattering experiments from even nuclei (0+→2+ excitation). The model is a slight generalization of the well-known ring-locus model in that it is based on a ring locus of finite angular width. The dependence of the inelastic angular distribution and the reaction-plane γ-pattern on the parameters of the ring locus is discussed.

Let M be a compact and simply connected Riemannian n-manifold of class C% We shall deal with M having the same geodesic structure as projective spaces possess. Namely, each point p on M has such a special cut locus C(p) that the distance between p and any point on C(p) is a constant. Then every geodesic on M is simply closed and has the same length, the same index 2, where 2 = 1, 3, 7 or n 1, and n = m(2 + 1) (see [4]). Moreover for every point p on M, the cut locus coincides with the first conjugate locus Q(p), (see [4]), and hence C(p) is a compact submanifold of dimension (m t) (2 + 1). Furthermore for every p e M and every x e C (p), let N (p, x): = {y e M; d(p, y) + d(y, x) = d(p, x)}, where d denotes the distance function on M. Then the point set forms a (2 + 1)dimensional submanifold without boundary which is homeomorphic to S~ +l (see [4]). One might be interested in the differentiable structure of C(p) as well as the one of M. Now this strong situation enables us to consider a special differentiable free S i action on the standard sphere S "ta+l~-l. In this paper we shall restrict ourselves to consider the case 2 = 1. A differentiable S 1 action (S 2ra-1, ~D,S 1) is by definition: ~pOp(u, s), t) = ~(u, st) for s, t e S l, u e S 2m-i, ~p(u, 1) = u for 1 e S 1 (identity element of $1), tp:S 2"-1 x S l ---,S 2"-1 is smooth.

There is given a necessary and sufficient (global) topological condition on a complex space X for the existence of a local finite analytic mapping Y→X with Y a manifold and branched locus of codimension>1.

The conformal solution theory (CST) is applied to the problem of gas–gas immiscibility. It is found that the occurrence of this phenomena can be predicted in the system He–Xe; in particular, the calculated critical locus is in satisfactory agreement with the experimental data of de Swaan Arons and Diepen, provided a suitable reference is chosen. Furthermore, using CST as a guide, it was found that criteria could be developed, related to Temkin's criteria, which permit the prediction and classification of the three types of gas–gas equilibria known to occur in mixtures of nonpolar molecules.

Up to this time, we have tacitly assumed in the theory to reduce the shape of Transition Curve that a particle as a center of gravity of a car runs along the center-line of railway track and at the same time contacts with the surface of it.However, actually, the car moves in plan along the locus which the middle point of the chord linking two bogie-centers of a car traces, and on the other hand runs in elevation along the curve which is fairly biased to the inside of the curve of the center-line of railway track, on account of the existence of a Cant in the Track. Therefore we must treat the center-line of railway track and the locus described by the center of gravity of a car as a different line and must investigate, in the research about the shape of Transition Curve, the shape of center-line of railway track which enables the center of gravity of a car to move smoothly.

The deformation systems of α-zirconium iodide have been studied in coarse-grained polycrystalline specimens deformed by upsetting. The orientation of the grains was determined from Laue patterns obtained in a special back-reflection camera using a small-diameter beam. The indices of the deformation systems were determined by the two-surface method and by the pole locus method.

This paper deals with the frequency domain specifications in the design of linear control systems in the parameter plane. Locus for constant phase margin and constant gain margin are plotted in the plane of the two adjustable parameters. The method is illustrated by two examples.

Two methods of recovering the equivalent dipole of variable locus and moment from voltages of 140 or fewer surface recording sites have been examined: (1) summation of all available elements (Gabor and Nelson 8 ) and (2) lead-vector search (Helm and Chou 11 ). Summation requires many recording sites but is rapidly done on a digital computer, whereas lead-vector search requires fewer recording sites but involves much longer computing times. Recovery of a single noise-free factitial input dipole by summating over 140 points was good as to locus and moment. Recovery by lead-vector search was also good whether 7 or 28 electrodes were employed. Increasing error of estimation—especially of locus—was noted, however, with the lead-vector search with (1) increasing the noise level, (2) diminishing the number of electrode sites (from 28 to 20 to 7), and (3) increasing the complexity of the generator (from single dipole input, to double dipole input, to "live data"). Clinical application awaits improved methodology—perhaps by compromise between slow-recording, fast-computing summation, and fast-recording, slow-computing lead-vector search.

The model developed here leads to some useful conclusions. If we consider an economy without trade where the supply of inputs varies in response to the level of current real consumption paid to inputs, then the relevant social input constraint for the planners is not a single production possibilities locus, but the planners feasibility constraint developed here. This constraint traces the locus of points on a family of production possibility surfaces that would be generated by each alternative quantity of inputs. The rate of product transformation along the planners' feasibility function will differ from the rate of product transformation along the production possibilities locus by an amount that reflects the marginal input supply.

Spatial coordinates determined optically from Richland, Washington, for 13 recent SAR arcs are transformed to L intercepts at 400 km. The resulting L values for the SAR arcs are mapped into the magnetic equatorial plane. The locus is found to be a segment of an asymmetric ring. The ring conforms approximately to the expected intercept of the plasmapause in the same plane. Proximity of the SAR-arc field lines to the plasmapause, the electron trough, and the ring current is discussed and the observed north-south drifts of SAR arcs are reviewed.

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1. SCHROETER'S t classic work on the general cubic leaves little to be desired in point of symmetry and generality. It is nevertheless interesting to build up the theory of the unicursal cubic, the curve being defined as the locus of the intersection of corresponding rays of two projective pencils, one of the first and the other of the second order. This has in fact been done by DRASCH. t The following discussion, based likewise on this definition, is materially simplified by the use of the properties of the point designated in ? 7 by E. Incidentally the investigation brings to light a remarkable one-to-one correspoindence between the points of the plane and the line elements on the cubic. 2. The locus described above has at least one and at inost three points in common with any line in the plane. We assume the truth of this theorem, a proof of which may be found in the eleventh chapter of REYE's Geometrie der Lage. Notations.-Throughout this paper we shall use the following notations: The pencil of the first order will be denoted by s, its center by S, and its rays by a, 6, c, etc. The pencil of the second order (and also the conic enveloped by it) will be denoted by K, and the rays by a, /3, 'y, etc. The cubic itself will be denoted by C. 3. THEOREM.-NO point of the cubic C lies within the conic K. 4. THEOREM.-The cubic C touches the conic K in at least one point, and at most in three. To prove this take S', a point on K, for the center of a pencil s' of the first order perspective to K. This pencil generates with s a conic which cuts KC in at least one and at most three other points besides S'. These are easily seen to be points on C.

The constitutive laws of the limit analysis of plates are formulated directly in terms of stress resultants and the corresponding kinematical quantities. General tensorial relations for two-dimensional continua are studied in order to disclose the form of a constitutive equation for plates at collapse. The derivation is based on the requirement that the constitutive equation be homogeneous of degree zero in time. Yield locus and the general form of flow rule are derived. The plastic potential flow rule as well as some other satisfying the material stability postulate follow as particular cases. Examples of compatible collapse modes are given.

Let V be an analytic subvariety of an open subset Ω \Omega of C n {{\text {C}}^n} of pure dimension r; for any p ∈ V p \in V , there exists an n − r n - r dim plane T such that π T : V → C r {\pi _T}:V \to {{\text {C}}^r} , the projection along T to C r {{\text {C}}^r} , is a branched covering of finite sheeting order μ ( V , p , T ) \mu (V,p,T) in some neighborhood of V about p. π T {\pi _T} is called a global parametrization of V if π T {\pi _T} has all discrete fibers, e.g. dim p V ∩ ( T + p ) = 0 {\dim _p}V \cap (T + p) = 0 for all p ∈ V p \in V . Theorem. B = { ( p , T ) ∈ V × G ( n − r , n ) | dim p V ∩ ( T + p ) > 0 } B = \{ (p,T) \in V \times G(n - r,n)|{\dim _p}V \cap (T + p) > 0\} is an analytic set. If π 2 : V × G → G {\pi _2}:V \times G \to G is the natural projection, then π 2 ( B ) {\pi _2}(B) is a negligible set in G. Theorem. { ( p , T ) ∈ V × G | μ ( V , p , T ) ≥ k } \{ (p,T) \in V \times G|\mu (V,p,T) \geq k\} is an analytic set. For each p ∈ V p \in V , there is a least μ ( V , p ) \mu (V,p) and greatest m ( V , p ) m(V,p) sheeting multiplicity over all T ∈ G T \in G . If Ω \Omega is Stein, V is the locus of finitely many holomorphic functions but its ideal in O ( Ω ) \mathcal {O}(\Omega ) is not necessarily finitely generated. Theorem. If μ ( V , p ) \mu (V,p) is bounded on V, then its ideal is finitely generated.

One of the calculation methods of the future food consumption pattern was developed. A n dimensional pattern vector was thought to exist and move on the surface of the unit sphere in n dimensional space. A calculation method was studied on the next 2 assumptions, (1) a pattern moves on a straight line in the sphere surface, (2) infinite time is necessary to move from a boundary to another boundary of the first quadrant area.Firstly, a locus of pattern moving is expressed as vectors using a vector OPin which OA represents the present pattern, OB represents the past pattern and OP is a vector which has a right angle with vector OA. Secondly, the position of a future pattern is decided using the distance (radian) from the starting boundary point, according to the logistic curve equation, in which y is a distance from the starting point, t is a time and K, ao and a1 are parameters. In this study, the distance from the starting point to the targent point is used as the K value.