bluuebunny/crossref_metadata_2025 huggingface dataset:
Nursing home prices and market structure: the effect of assisted living industry expansion
Authors: Bowblis John R.
Published: July 2013

Since the 1990s, there has been substantial expansion of facility-based alternatives to nursing home care, such as assisted living facilities. This paper analyzes the relationship between expansion of the assisted living industry, nursing home market structure and nursing home private pay prices using a two-year panel of nursing homes in the State of Ohio. Fixed effect regressions suggest that the expansion of assisted living facilities are associated with increased nursing home concentration, but find no effect on private pay nursing home prices. This would be consistent with assisted livings reducing demand for nursing homes by delaying entry into a nursing home, though assisted livings are not direct competitors of nursing homes.
Graphical Treatment of Geometrical Progression
Authors: Dougall John
Published: July 2013

The idea of the sum to infinity of a geometrical progression admits of a simple but very effective illustration already very much in vogue among teachers.

We take a line OA two units long, and from it cut off in succession segments
Oral and Written Work in Arithmetic
Authors: Fraser W. G.
Published: July 2013

The following notes deal mainly with the Arithmetic of the primary school, but the subject under consideration is only a single aspect of a very wide question.

It is a common remark that Oral (or Mental) and Written Arithmetic should be as nearly as possible identical: that pupils ought to be able to do mentally with small numbers whatever they can do in writing with larger numbers, and vice versa. Unfortunately, experience shows that, as a rule, they are not able to do so; that the child fails to identify written calculation completely with mental work. It is here proposed to try to discover some of the reasons why this is so.
Quadratic Equations
Authors: Pinkerton P.
Published: July 2013

The following is a scheme of classroom work designed to introduce a pupil to quadratic equations, graphically and algebraically.

Newton's Chord Rule is applied in (3) to calculate by Arithmetic close approximations to the roots of an equation. One graph suffices for any number of equations, as in (2), (5) and (7). Both the method of completing the square and the method of factors are required as algebraic equivalents, as in (4), (6), (7). The general rules for solving quadratics sum up the lessons.
The Theorem of Pythagoras
Authors: Gibson George A.
Published: July 2013

The parts of the figures are numbered to indicate a proof of the theorem by dissection. The congruence of parts on which the same number is marked can be demonstrated by geometry. The steps of the proof are in order (i), (ii), (iii). Sufficient importance is, perhaps, not attached to (i) and (ii) at this stage; for they give (1) the construction of a square equal to a given rectangle (2) the graphical construction of √2, √2 + 1.etc.
Notes on the Logic of Equation-Solving
Authors: Muirhead R. F.
Published: July 2013

Consider an equation with one unknown, x. A “solution” or “root” of such an equation is a value which, being substituted for x in the equation, reduces it to an identity.
Note on the Characteristic of a Logarithm
Authors: Comrie P.
Published: July 2013

Students might be reminded that a number consists of so many digits, the value of each of which depends on its position. The values might be written above the separate digits just as we write £ s. d. over money columns. Thus, 523·7046 would be written

Rule for finding Characteristic.—The characteristic is the index of the power of 10 placed over the first significant digit in the number.
A Method of Graphing Freedom Equations
Authors: Burgess A. G.
Published: July 2013

Draw axes XOX′, YOY′. Draw graph of y = φ(t), taking OX as a positive axis of t. (In Figure see dotted line). Draw graph of x =f(t), taking OY′ as positive axis of t. (In Figure f(t) = t3 – t2; see broken line). Take any point A in the line

1+x = 0. Go along line through A parallel to OX till a point B a the graph x =f(t) is met, and vertically parallel to OY until a oint C in the graph y = φ(t) is met. The fourth vertex D of the ectangle ACDB is a point in the graph of the eliminant of t in he equations x =f(t), y = φ(t).
The Theorem of Pythagoras
Authors: Mackay J. S.
Published: July 2013

Here is a pendant to Dr Gibson's beautiful dissection of the three squares. I am pretty sure that his proof is new to the world, but I am not sure that mine s so. There are about fifty proofs, differing more or less from each ither, but there are only some half a dozen worth remembering or

eaching. The proof I submit (not one of the half dozen) was levised in 1859, when I was a young student in St Andrews, and b may probably have been given long before that date.
The Inequality Theorem
Authors: Tweedie C.
Published: July 2013

(xm - 1)/ m >(xn - 1)/n, m > n.— I have sometimes found the following geometrical method of attack useful in explaining this theorem to students of only average ability. The Lemmas introduced are almost self-evident, and admit of simple analytical demonstration.
The solution of “homogeneous” quadratics
Authors: Dougall John
Published: July 2013

and from either of (1), x= ± 3, y = ± 2.
Multiplication of Decimals
Authors: Lindsay W. A.
Published: July 2013

The method of arranging the partial products illustrated in the accompanying example should offer no difficulty, if in explaining it we take the steps given below.
Proof of a Property of Simson's Line
Authors: M‘Arthur N.
Published: July 2013

The following is a slightly simplified version of a well-known proof of this theorem:—

The Simson's Line of P, with respect to δABC, bisects PO, where O is the orthocentre of the triangle.
A New Form of Boyle's Law Tube
Authors: Miller William
Published: July 2013

The following apparatus for verifying Boyle's Law in an elementary manner is of interest from the point of view of pure mathematics as well as that of the laboratory. The method of construction, which I have described in the July (1910) number of the School World, presents a little difficulty in the first attempt, but is easily mastered in the second.
Distance of the Horizon
Authors: Ramsay Peter
Published: July 2013

The approximate rules given in this note may perhaps be new to some readers.

If an observer be at a height h above the Earth's surface, the distance of the horizon as seen by him is given by the formula

where r is the radius of the Earth.
The Asymptotes of the Hyperbola
Authors: Pinkerton P.
Published: July 2013

The equation of the hyperbola usually presents itself to the student in the form in the elements of graphs, before the canonical form is reached in Analytical Geometry. Some of the properties of the curve are easily obtained from the above equation by the methods of elementary geometry; the object of this note is to illustrate the process. We shall use the equation in the form xy = c2, and it will be convenient to arrange the discussion in numbered Theorems. The axes may be rectangular or oblique.
Rate of Change of Momentum
Authors: Pinkerton P.
Published: July 2013

The object of this note is to suggest how Newton's Laws of Motion may be approached explicitly by means of rate of change of momentum. It will be seen that the term “force” is not made prominent at the expense of the notion of rate of change of momentum. Once the student believes that rate of change of momentum is a vital aspect of stopping or starting the motion of a body, he is not inclined to overestimate his knowledge of “force” apart from the rate of change of momentum which accompanies it. Since students of ordinary endowment learn to observe things before words, we begin with some common examples. As for the term “momentum,” it is sufficient to begin by saying that
An Experiment in Light
Authors: Lindsay W. A.
Published: July 2013

Let ABCD be a horizontal section of a rectangular slab of glass. A pin is set up vertically at P, close to the face AB, or at a short distance from it. It is possible to see the pin P through the glass if we look through the face CD; but the pin is invisible if we look through the face AD. If, however, we look through the face CD, into the face AD, we shall see an image of the pin in AD, which acts like a mirror. The experiment illustrates total reflection of light; the explanation is easy. Let PO be a ray of light which after passing into the glass will be incident on AD at O′. Let NON′ be the normal to AB at O,

and KO′K′ be normal to AD at O′. The critical angle for glass being 41° 45′ approximately,

∴ the ray OO′ will not emerge through AD, but will be reflected in the direction O′Q, and will emerge at Q in the direction QS.
“The Ambiguous Case” in the Solution of Triangles
Authors: Muirhead R. F.
Published: July 2013

Given a, b and A, there are sometimes two distinct solutions, sometimes one, and sometimes none. The geometrical explanation is a commonplace, from which it appears that two solutions occur if b>a>bsinA. [But see correction below.]
A method of calculating logarithms, using merely the ordinary laws of indices
Authors: Johnston W. Vaughan
Published: July 2013

To determine logarithms to the base 1·1.—If N = ax, x is the logarithm of N to the base a. Hence, if N = 1·1x, x is the logarithm of N to the base 1·1, and, by plotting different values of x with the corresponding values of N, a logarithmic curve may be drawn. This curve will furnish an approximate logarithm of any number to the base 1·1.
Examples in the use of Involution
Authors: Philip G.
Published: July 2013

The following examples do not usually find a place in text-books dealing with involution, and so may be interesting to those teaching the subject. They are simply special cases of the two well-known theorems: (1) that a straight line cuts the six sides of a complete quadrilateral in six points that form an involution, (2) that if any point in the plane of the quadrilateral be joined to the six vertices, the six rays so formed form a pencil in involution.
Note on Right-Angled Triangles
Authors: Milne Archd
Published: July 2013

It frequently happens in teaching that one wishes to draw a right-angled triangle with rational sides, and at the same time, for the sake of variety if nothing else, one desires to avoid the hackneyed one whose sides are 3, 4, 5.
Certain Processes in the Theory of Equations illustrated Geometrically.—
Authors: Salmon T. D.
Published: July 2013

The Derived Function: In what follows by f(x) is meant a rational integral function of x of degree n with the coefficient of xn unity. By the “roots of f(x)” is meant the roots of f(x) = 0; by f′(x) is meant df(x)/dx.
On the Solubility of Linear Algebraic Equations.—
Authors: Dougall John
Published: July 2013

(a) It is proved in treatises on Algebra that the equations (in three variables for brevity)

have a unique solution given by

provided the determinant

does not vanish.

(b) It is also proved, from (2), that if the “degenerate” homogeneous system

has a non-null solution (i.e. a solution in which the variables are not all zero), then Δ = 0.
Whewell's Puzzle
Authors: Mackay J. S.
Published: July 2013

Mr J. F. Cameron of Gonville and Caius College, Cambridge, has suggested the use of factorial functions such as or n! to solve Whewell's problem. I had excluded them as beyond the bounds of ordinary arithmetic. I append half a score of my attempts with the factorials; they could be easily increased.
On the Solubility of Linear Algebraic Equations (Contd.)
Authors: Dougall John
Published: July 2013

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.

This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.
Proof of the sufficiency of the determinant condition for the consistency of a system of n homogeneous linear equations In n variables
Authors: Dougall John
Published: July 2013

For n = 1, the theorem is that, if a = 0, the equation ax = 0 has a solution for which x is not zero. This is obviously true. And the case of n variables can easily be made to depend on that of n – 1 variables. For brevity we show the method by taking n = 3.
On the sufficiency of the condition for a limit
Authors: MacRobert Thomas M.
Published: July 2013

Let z1, z2, z3, … be a, sequence of quantities, real or complex, such that, corresponding to any arbitrary small positive quantity ∈, we can find a positive integer n such that | zn+p – zn | < ∈ for all positive integral values of p.
The equation of the polar of a point with respect to a circle
Authors: Bell R. J. T.
Published: July 2013

The equation of the polar of a point (x′, y′) with respect to a circle x2 + y2 = r2 can be found very easily by the use of

either the perpendicular or the intercept form of the equation of a straight line.
A Note on Induction
Authors: Lindsay W. A.
Published: July 2013

An application of the method of Induction to the proof of a general result or theorem should involve two steps: (i) the discovery of the result to be proved by the consideration of particular cases; (ii) a proof, by Mathematical Induction, or otherwise, that the result, found to be true in particular cases, is universally true.
Problem:—To find a point M external to triangle ABC such that lines from it to the vertices A, B and C make equal angles with the sides, that is MCA1 = MAB = MBC
Authors: Finlayson William
Published: July 2013

It is evident that if MCA1 = MBC the point M lies on the circle O, which touches AC at C and passes through B. Again, if we take two equal and opposite pencils A(BIMC) and C(A1EMb) we know that their intersections lie on an hyperbola, and observe that when AI is the bisector of A its corresponding ray CE of the second pencil is parallel to it. This gives the direction of the asymptotes; the second asymptote will be parallel to the external bisector of A. The conic is therefore a rectangular hyperbola. When the ray of the first pencil coincides with AC1 the ray of the second corresponding to it will make with CA an angle equal to A, it is therefore parallel to AB, and the point of intersection is C; hence the second ray ab is a tangent at C, similarly BA is a tangent at A; therefore AC passes through the centre of the conic, and B′ is the centre. The point M where this conic cuts O is the required point M. Denote this hyperbola by γ, then on repeating this construction at B we get a second hyperbola β cutting the circle O1 at M1, which makes angle M1BA2 = M1AC = M1CB. (See Fig. on p. 157.)
Geometrical illustration of the terms in Taylor's theorem.–
Authors: Stokes G. D. C.
Published: July 2013

The following is an extension of the note by Mr W. M. Roberts in the Gazette (215 [v. 4, c], 1906); or of Mr A. H. Barker's discussion on p. 163 of his “Graphical Calculus.”
Note on Rational Right-Angled Triangles whose Legs are consecutive Whole Numbers.–
Authors: Martin Artemas
Published: July 2013

Having given the sides of a rational right-angled triangle, to find from them the sides of other rational right-angled triangles.
The expansions of sinx and cosx. –
Authors: Pinkerton P.
Published: July 2013

Dr Stokes has already shown (Mathematical Notes, No. 13, p. 143) how the summation of series may be put to its proper use in elementary mathematics, namely, to provide a sequence of approximations to a limit. The method may be used to give a sequence of approximations to sinx and cosx.
Note on the Perpendicular Generators of a Hyperboloid of One Sheet
Authors: Bell R. J. T.
Published: July 2013

The necessary and sufficient condition that two generators of the should be at right angles is that their point of intersection should lie on the director sphere x2 + y2 + z2- = a2 + b2 – c2. Let a2 + b2 – c2 = r2 and a2 > b2. Then if r2 > a2, b2 > c2, and in this case the curve of intersection of the H1 and sphere consists of two closed ovals lying on opposite sides of the plane XOY. Thus every generator of the H1 meets the sphere in two real points. This result is easy to obtain analytically. For the generator

meets the sphere in points given by
Note on the Principal Axes of a Normal Section of a Cylinder which Envelopes an Ellipsoid
Authors: Bell R. J. T.
Published: July 2013

This note deals with methods of determining the lengths and direction-cosines of the axes. If the lengths are 2α. and 2β, then α–2 and β–2 are the two non-zero roots of the discriminating cubic for the cylinder, and thus are easily found. The following methods apply geometrical considerations and determine both the lengths and the direction-cosines.
A Construction by Ruler and Dividers
Authors: Muirhead R. F.
Published: July 2013

The following method of cutting off an nth part of a given straight line requires besides a ruler only a pair of dividers or other means of laying off on a given straight line a segment equal to the distance between two given points. In other words, we assume that, besides using a ruler in the usual way, we can mark off from a given straight line AX a segment AB, which shall be equal to the distance between two given points C and D.
The Equivalence of certain Simultaneous Equations
Authors: Ridley John E.
Published: July 2013

The object of this note will, perhaps, be best explained by first repeating the usual steps in the solution of two simultaneous equations in x, y, where one is linear and the other is of the second degree. Let the equations be
An Exact Geometrical Construction for the Exponential Curve
Authors: Robertson David
Published: July 2013

Consider the curve y = ae−t/t0 (Fig. 1.)

Mark off along the t – axis a distance OT0 equal to the time constant t0.

Divide OT0 into n equal parts and erect ordinates at each point of division.

M, N, are the mth and (m + 1)th of these, counting from O.

MP, NQ, are the corresponding ordinates; P, Q, the points on the curve; and R the intersection of the chord, produced, with the t – axis.
Trigonometric Survey
Authors: Pinkerton P.
Published: July 2013

The object of this note is to show how various problems commonly occurring in Elementary Trigonometry may be classified as particular cases of trigonometric survey. The general case may be stated first, and certain well-known particular cases appended.
Geometrica Proof of a Trigonometrical Identity
Authors: Burgess A. G.
Published: July 2013

The following proof of the identity

Where A, B, C are the angles of a triangle differs from that given by Dr Bell in the last number of Mathematical Notes and from that given in “New Trigonometry for Schools” by Lock and Child in that the area of the triangle is made use of.
The Distance of a given Point from a given Line
Authors: Bell R. J. T.
Published: July 2013

If P (x′, y′) is the given point and ax + by + c = 0 the equation of the given line, the expression for the distance of P from the line and the coordinates of the foot of the perpendicular from P to the line can be obtained by projections as follows:

Let NP = R and have projections X and Y on OX and OY respectively. Then if NP makes an angle θ with OX

.
Geometrical Proof of tan A + tan B + tan C = tan A tan B tan C
Authors: Burgess A. G.
Published: July 2013

Let ABC be the Δ, AD and BE ⊥s to BC and AC, CK and BM ⊥s to ED.
To draw a circle which cuts three circles at the ends of the diameters of these circles
Authors: Burgess A. G.
Published: July 2013

Let the three circles have centres, A, B, C, and let their radical centre be T, and the

circumcentre of ΔABC be 0. Join TO and produce it to T1, so that T1O = TO. T1 is the centre of the required circle. Join T1A and T1C, and draw diameters A1^AA2, and C1CC2 perpendicular to T1A and T1C. ⊙ with centre T1 and radius T1A2 passes through A1.
The Moments Theorem
Authors: Crawford G. E.
Published: July 2013

If ABDC be a parallelogram, of which AD is a diagonal, and if 0 be any point which, by joining

to A, B, C, D, makes three triangles, OAB, OAD, OAC, each of these being taken as positive in sign so long as O lies within the region selected (i.e. within ABD), then:

ΔOAC – ΔOAB = ΔOAD.
A simple nomogram for the solution of quadratic equations
Authors: Whittaker E. T.
Published: July 2013

There are many well-known graphical solutions of the quadratic equation, in which the roots are obtained as the intersections of a circle with a straight line. The practical objection to these solutions is that a fresh diagram has to be drawn for each equation that is to be solved: and the time occupied in constructing the diagram is greater than the time required to solve the equation by the ordinary arithmetical method.
Experimental Determination of the Fundamental Solutions of the Equation a cos θ + b sin θ = c, with some general theorems on that Equation
Authors: McWhan John
Published: July 2013

It is shown in this note how, by a simple geometrical construction, the two fundamental solutions of the equation a cos θ + b sin θ = c (to which ± 2 k π added gives, if required, the general solution) may be determined experimentally, using the protractor, without ambiguity of any kind. The method does not involve any laborious graph-tracing, and is at once simpler and more accurate than the geometrical discussions given in one or two text-books. (See, for example, Loney, Part I., page 141, §130; Levett & Davison, page 138).
Centre of Curvature
Authors: Muirhead R. F.
Published: July 2013

Professor Bryan's article on Curvature, etc., in your last issue (p. 219) ends with a challenge. I wonder whether he would be satisfied with the following reasoning to prove that the intersection of “consecutive” normals and the centre of the circle through these “consecutive” points of a curve have the same point as limiting position.
The Solution of the Biquadratic x4 + px3 + qx2 + rx + s = 0
Authors: Tweedie Charles
Published: July 2013

§1. Descartes in his Geometria shows how the solution may be made to depend on the intersections of two conies C and C′; an account of this method is given in the Algebra of Maclaurin. Thus the roots of x4 + qx2 + rx + s = 0 are the x-coordinates of the points common to the parabola y = x2 and the circle x2 + Y2 + rx + (q – 1) y + s = 0.
Direction-Cosines of the Axes of the Conicoid
Authors: Crawford Lawrence
Published: July 2013

ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy = 1.

In Mathematical Notes, No. 20 (April 1916), there is a note on the above; I add a form of the equations of these axes which I have not seen in a text-book, and which is perhaps worth recording.
Proofs of some Inequalities and Limits
Authors: Muirhead R. F.
Published: July 2013

In his article in No. 20, Professor Gibson gives proofs of the inequalities with certain restrictions as to the values of n and a. The mode of proof, avoiding as it does the use of the Binomial Theorem, is comparatively short and simple. The following mode of proof is, I think, still simpler, as it does not involve the use of Mathematical Induction.
Two Extensions of Ceva's Theorem
Authors: Aitken A. C.
Published: July 2013

If ABCDE … is a polygon of an odd number of sides, O any point in its plane, and if AO, BO, CO, … cut the sides opposite A, B, C, … respectively, AB in X, BC in Y, CD in Z, … then

Let PQ be the side opposite A. Then AB will be the side opposite Q.
Direct Proofs of Theorems in Elementary Geometry
Authors: Blades R. F.
Published: July 2013

(1) If the straight line joining two points subtends equal angles at two other points on the same side of it, the four points are con cyclic.
An Elementary Proof of Feuerbach's Theorem
Authors: Sanjana K. J.
Published: July 2013

Let O be the centre of the circumscribing circle of ΔABC, A1 the middle point of BC, and EA1OF the diameter at right angles to BC. Draw AX perpendicular to BC and produce it to meet the circle in K. Let H be the orthocentre of ΔABC; join OH and bisect it in N, the centre of the nine-point circle.
On Recurring Decimals
Authors: Aitken A. C.
Published: July 2013

§ 1. The present note gives two methods by which vulgar fractions may often be converted into decimals with great rapidity. It is not pretended, however, that they are more than mere curiosities or that they are of practical use. The first method is not new, but I have never seen the second mentioned anywhere. The proofs are almost self-evident.
Note on a trigonometrical proof of the orthocentre property of a triangle
Authors: Turnbull H. W.
Published: July 2013

The following argument is simple and instructive, but I do not remember seeing it before.
A Proof of the Addition Theorem in Trigonometry
Authors: Bell R. J. T.
Published: July 2013

I. Prove sin θ= −sin(− θ), cos θ(− θ)

Let X′OX, Y′OY be rectangular axes, and let OP, OQ, starting from the position OX, describe angles θ, –θ. Then OP, OQ are in every case symmetrically placed with respect to OX. (Illustrate by taking positive and negative values of θ). Hence xP = xQ, yP = –yQ, and the results follow from the definitions of sine and cosine.
Elementary Proof of a Theorem about Maxima and Minima of f(x)
Authors: Muirhead R. F.
Published: July 2013

The theorem in question applies to a function f(x) which, together with its derivatives, is continuous in the neighbourhood of the maximum or minimum value.
Note on a determinant derived from a triangular array
Authors: Turnbull H. W.
Published: July 2013

The array of binomial coefficients written in triangular form has many curious properties, particularly so if alternate rows or alternate columns receive a negative sign. We may consider such a scheme, developed from its top right hand corner, containing n rows and columns.
Principal radii of curvature at a point on an ellipsoid
Authors: M'Arthur Neil
Published: July 2013

The values of the principal radii of curvature ρ1; ρ2 at a point on an ellipsoid are α2/p, β2/p where α2, β2 are the squares of the semi-axes of the central section parallel to the tangent plane at the point, and p is the distance from the centre to the tangent plane.
The theorem of Apollonius by dissection
Authors: Stokes G. D. C.
Published: July 2013

The figure shews the dissection in the form

Let m, x denote AD, DE, where E is the projection of A on BC.

Quadrilateral 1 is constructed with sides ½ (c + x), ½ (c – x), ½ (m + ½a + x), ½ (m – ½a – x); quadrilateral 6 with sides ½ (b – x), ½ (b + x), ½ (m – ½a – x), ½ (m + ½a – x). The square on AB is then dissected into four quadrilaterals equal to 1, plus the shaded square (AD2); similarly for the square on AC. These quadrilaterals are reassembled in the squares on BE, EC respectively with the right-angled corners in the reversed positions, so as to enclose squares each equal to DE2. Finally two quadrilaterals are subdivided (4, 11) and (3, 5, 10, 12) to fit in as shewn in the figure.
A School Method for Calculating a Table of Sines to Three Places for Integral Values of the Angle
Authors: Stokes G. D. C.
Published: July 2013

The two standard formulae required are used in the forms

The values corresponding to 0°, 30°, 45°, 60°, 90° are assumed known.
Conditions obtained by Multiplication of Determinants
Authors: Bell R. J. T.
Published: July 2013

The condition that ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 should represent two straight lines can be obtained very easily by considering the product

which is equal to

and is identically zero.
A Geometrical Proof for Hero's Formula
Authors: Stokes G. D. C.
Published: July 2013

The following proof is designed to link up Hero's formula geometrically with the formulae for the trigonometrical functions of ½A in a triangle.
Note on a Vanishing Determinant
Authors: Aitken A. C.
Published: July 2013

1. In one of Sir Thomas Muir's more recent historical papers on the theory of determinants (Proc. R. S. Edin., XLIII, 1922, p. 129), is included a result due to V. Jung, commented on as being “verified in an unsuggestive way for the first three cases.” The theorem is that the determinant

Vanishes when n is even. Below we give a simple proof.
On the Roots of a Symmetrical Determinant
Authors: Turnbull H. W.
Published: July 2013

The six values of x which make the determinant

vanish, are . In general, the n values of x which make the corresponding determinant of order n vanish are given by . Each determinnant has a diagonal filled with x's, bordered by adjacent parallels where each element is unity; and all other elements are zero.
An Extension of Clifford's Circle Chain
Authors: Dougall John
Published: July 2013

In elementary plane geometry there is a very well known theorem attributed to Wallace (1806) which runs:

Theorem I. If four lines are given in a plane, the circumscribing circles of the triangles which they form three by three are concurrent.
Alternative Forms of Expression for Hermite's Determinant
Authors: Muir Thomas
Published: July 2013

(1) Apparently it was in 1854 that Hermite first drew attention to the special determinant which now bears his name. It may be defined as being such that every two of its elements that are conjugate in position are conjugate-complex in form: and as a consequence its matrix is the sum of two matrices one of which is axisymmetric and the other zero-axial skew.
Well-known Theorems on the Ellipse deduced by Projection from the Circle; also a Theorem on Curvature
Authors: Bell Robert J. T.
Published: July 2013

1. In the treatment of the geometry of the ellipse, projection from the circle offers an easy and direct means of obtaining the properties of conjugate diameters of the ellipse. In text-books generally, little further use is made of the method of projection, except in the case of a few exercises. The following note shows how it can be employed to derive most of the well-known properties of the ellipse.
An Apparatus for Determining Coefficients in Power Series
Authors: Thomson R. W. M. , Professor Peddie
Published: July 2013

The diagrammatic figure illustrates the principle involved. The vertical full lines represent ordinates of a curve of which the

abscissae are Oa1, Oa2, etc. Let now all these vertical lines be rotated round the points O, a1, a2, etc., while retaining their original lengths, through the same angle θ. The curve passing through the new positions of the end points is now to be referred to the tilted axis Oy0 and the line OAB, drawn at right angles thereto. If we put OC = X, CD = Y, and let X′, Y′, be the corresponding coordinates with reference to the tilted axis we get
Note on Continued Fractions and the Sequence of Natural Numbers
Authors: Turnbull H. W.
Published: July 2013

When a student first approaches the theory of infinite continued fractions a natural question that suggests itself is how to evaluate the expression
A Geometrical Construction with Physical Applications
Authors: Peddie W.
Published: July 2013

The Diagram shows two complete, and one partially complete, concentric circles. The radii of the largest and smallest are as four to one respectively.
On Nets of Polygons Occurring in Nature
Authors: M'Crea W. H.
Published: July 2013

1. The section of certain vegetable cell structures takes normally the form of a net of hexagons. When a departure from this form occurs it is observed to be such that three edges still meet at each vertex of the net and the average number of sides per polygonal mesh is still six. After noting this fact, Graustein shows in a recent paper that under these conditions the average number of sides is necessarily precisely six. He then generalises the geometrical result. The present note derives the same results by an entirely different process. The arguments are of a physical character and do not pretend to be mathematically rigorous. But it seems worth while to put them forward as indicating just why the effect should occur in Nature.
Quadrics Associated with a Möbius Hexad
Authors: Turnbull H. W.
Published: July 2013

§1. The following investigation arose out of a reconsideration of a certain invariant of six points in space

I = (1234)(3456)(5612) – (2345)(4561)(6123)

where (1234) denotes the determinant whose rows are the coordinates of the first four points: and similarly for the other factors. The vanishing of this invariant implies that the sixth point lies upon a certain quadric defined by the other five. The following elementary account brings out several interesting and possibly new theorems upon quadric surfaces.
On the configuration known as a double-six of lines
Authors: Richmond H. W.
Published: July 2013

In geometry of three dimensions it is well known that, when two quadrics Q1, Q2 are given, if one set of four points exists having the properties that each point lies on Q1, and each two points are conjugate with respect to Q2, an infinity of such sets of points can be found. The quadrics Q1Q2 stand in a special relation to one another, expressed by the vanishing of the coefficient of A in the discriminant of Q2 + λQ1 an invariant of Q1, Q2. Two quadrics Q1, Q2 are thus related if the equation of Q1 contains no squares of the coordinates, and that of Q2 contains no products of two coordinates; for then the vertices of the tetrahedron of reference form such a set of four points.
The advantage of differentials in the technique of differentiation
Authors: Phillips E. G.
Published: July 2013

In the last few years the topic of differentials has occupied the attention of a good many teachers of mathematics. Since the appearance in 1931 of my article advocating the teaching of the Differential Calculus from the differential standpoint right from the start, very divergent views have been expressed on this question and there was a general discussion on the topic at the annual meeting of the Mathematical Association in 1934. This article, the subsequent correspondence and the discussion have certainly brought into evidence the fact that many mathematicians still have a very obvious distrust of “differentials” and they seem unable to clear their minds of some early prejudice against the employment of differentials for any purpose.
To Cross or Not To Cross
Authors: Kostermans Evelien , Spijkerman Renske , Engels Rutger C. M. E. , Bekkering Harold , de Bruijn Ellen R. A.
Published: July 2013

Different theoretical accounts have attempted to integrate anterior cingulate cortex involvement in relation to conflict detection, error-likelihood predictions, and error monitoring. Regarding the latter, event-related potential studies have identified the feedback-related negativity (FRN) component in relation to processing feedback which indicates that a particular outcome was worse than expected. According to the conflict-monitoring theory the stimulus-locked N2 reflects pre-response conflict. Assumptions of these theories have been made on the basis of relatively simple response-mapping tasks, rather than more complex decision-making processes associated with everyday situations. The question remains whether expectancies and conflicts induced by everyday knowledge similarly affect decision-making processes. To answer this question, electroencephalogram and behavioral measurements were obtained while participants performed a simulated traffic task that varied high and low ambiguous situations at an intersection by presenting multiple varying traffic light combinations. Although feedback was kept constant for the different conditions, the tendency to cross was more pronounced for traffic light combinations that in reallife are associated with proceeding, as opposed to more ambiguous traffic light combinations not uniquely associated with a specific response. On a neurophysiological level, the stimulus-locked N2 was enhanced on trials that induced experience-based conflict and the FRN was more pronounced for negative as compared to positive feedback, but did not differ as a function of everyday expectancies related to traffic rules. The current study shows that well-learned everyday rules may influence decision-making processes in situations that are associated with the application of these rules, even if responding accordingly does not lead to the intended outcomes.
Investigation of the Effect of Mode and Tempo on Emotional Responses to Music Using EEG Power Asymmetry
Authors: Trochidis Konstantinos , Bigand Emmanuel
Published: July 2013

The combined interactions of mode and tempo on emotional responses to music were investigated using both self-reports and electroencephalogram (EEG) activity. A musical excerpt was performed in three different modes and tempi. Participants rated the emotional content of the resulting nine stimuli and their EEG activity was recorded. Musical modes influence the valence of emotion with major mode being evaluated happier and more serene, than minor and locrian modes. In EEG frontal activity, major mode was associated with an increased alpha activation in the left hemisphere compared to minor and locrian modes, which, in turn, induced increased activation in the right hemisphere. The tempo modulates the arousal value of emotion with faster tempi associated with stronger feeling of happiness and anger and this effect is associated in EEG with an increase of frontal activation in the left hemisphere. By contrast, slow tempo induced decreased frontal activation in the left hemisphere. Some interactive effects were found between mode and tempo: An increase of tempo modulated the emotion differently depending on the mode of the piece.
Variability in visual cortex size reflects tradeoff between local orientation sensitivity and global orientation modulation
Authors: Song Chen , Schwarzkopf Dietrich S. , Rees Geraint
Published: July 2013

The surface area of early visual cortices varies several fold across healthy adult humans and is genetically heritable. But the functional consequences of this anatomical variability are still largely unexplored. Here we show that interindividual variability in human visual cortical surface area reflects a tradeoff between sensitivity to visual details and susceptibility to visual context. Specifically, individuals with larger primary visual cortices can discriminate finer orientation differences, whereas individuals with smaller primary visual cortices experience stronger perceptual modulation by global orientation contexts. This anatomically correlated tradeoff between discrimination sensitivity and contextual modulation of orientation perception, however, does not generalize to contrast perception or luminance perception. Neural field simulations based on a scaling of intracortical circuits reproduce our empirical observations. Together our findings reveal a feature-specific shift in the scope of visual perception from context-oriented to detail-oriented with increased visual cortical surface area.
Emerging superconductivity hidden beneath charge-transfer insulators
Authors: Krockenberger Yoshiharu , Irie Hiroshi , Matsumoto Osamu , Yamagami Keitaro , Mitsuhashi Masaya , Tsukada Akio , Naito Michio , Yamamoto Hideki
Published: July 2013

In many of today's most interesting materials, strong interactions prevail upon the magnetic moments, the electrons and the crystal lattice, forming strong links between these different aspects of the system. Particularly, in two-dimensional cuprates, where copper is either five- or six-fold coordinated, superconductivity is commonly induced by chemical doping which is deemed to be mandatory by destruction of long-range antiferromagnetic order of 3d9 Cu2+ moments. Here we show that superconductivity can be induced in Pr2CuO4, where copper is four-fold coordinated. We induced this novel quantum state of Pr2CuO4 by realizing pristine square-planar coordinated copper in the copper-oxygen planes, thus, resulting in critical superconducting temperatures even higher than by chemical doping. Our results demonstrate new degrees of freedom, i.e., coordination of copper, for the manipulation of magnetic and superconducting order parameters in quantum materials.
Direct Prototyping of Patterned Nanoporous Carbon: A Route from Materials to On-chip Devices
Authors: Shen Caiwei , Wang Xiaohong , Zhang Wenfeng , Kang Feiyu
Published: July 2013

Prototyping of nanoporous carbon membranes with three-dimensional microscale patterns is significant for integration of such multifunctional materials into various miniaturized systems. Incorporating nano material synthesis into microelectronics technology, we present a novel approach to direct prototyping of carbon membranes with highly nanoporous structures inside. Membranes with significant thicknesses (1 ~ 40 μm) are rapidly prototyped at wafer level by combining nano templating method with readily available microfabrication techniques, which include photolithography, high-temperature annealing and etching. In particular, the high-surface-area membranes are specified as three-dimensional electrodes for micro supercapacitors and show high performance compared to reported ones. Improvements in scalability, compatibility and cost make the general strategy promising for batch fabrication of operational on-chip devices or full integration of three-dimensional nanoporous membranes with existing micro systems.
Geomagnetic response to solar and interplanetary disturbances
Authors: Saiz Elena , Cerrato Yolanda , Cid Consuelo , Dobrica Venera , Hejda Pavel , Nenovski Petko , Stauning Peter , Bochnicek Josef , Danov Dimitar , Demetrescu Crisan , Gonzalez Walter D. , Maris Georgeta , Teodosiev Dimitar , Valach Fridich
Published: July 2013

The space weather discipline involves different physical scenarios, which are characterised by very different physical conditions, ranging from the Sun to the terrestrial magnetosphere and ionosphere. Thanks to the great modelling effort made during the last years, a few Sun-to-ionosphere/thermosphere physics-based numerical codes have been developed. However, the success of the prediction is still far from achieving the desirable results and much more progress is needed. Some aspects involved in this progress concern both the technical progress (developing and validating tools to forecast, selecting the optimal parameters as inputs for the tools, improving accuracy in prediction with short lead time, etc.) and the scientific development, i.e., deeper understanding of the energy transfer process from the solar wind to the coupled magnetosphere-ionosphere-thermosphere system. The purpose of this paper is to collect the most relevant results related to these topics obtained during the COST Action ES0803. In an end-to-end forecasting scheme that uses an artificial neural network, we show that the forecasting results improve when gathering certain parameters, such as X-ray solar flares, Type II and/or Type IV radio emission and solar energetic particles enhancements as inputs for the algorithm. Regarding the solar wind-magnetosphere-ionosphere interaction topic, the geomagnetic responses at high and low latitudes are considered separately. At low latitudes, we present new insights into temporal evolution of the ring current, as seen by Burton’s equation, in both main and recovery phases of the storm. At high latitudes, the PCC index appears as an achievement in modelling the coupling between the upper atmosphere and the solar wind, with a great potential for forecasting purposes. We also address the important role of small-scale field-aligned currents in Joule heating of the ionosphere even under non-disturbed conditions. Our scientific results in the framework of the COST Action ES0803 cover the topics from the short-term solar-activity evolution, i.e., space weather, to the long-term evolution of relevant solar/heliospheric/magnetospheric parameters, i.e., space climate. On the timescales of the Hale and Gleissberg cycles (22- and 88-year cycle respectively) we can highlight that the trend of solar, heliospheric and geomagnetic parameters shows the solar origin of the widely discussed increase in geomagnetic activity in the last century.
Syntheses, Photoluminescent Properties, and Structural Investigation of Five Complexes based on a New T-Shaped 2-(Pyridin-3-yl)-4,6-Pyrimidine Dicarboxylic Acid Ligand: Structure Evolution from One-dimensional Chains to Three-dimensional Architectures
Authors: Deng Dongsheng , Kang Guohui , Ji Baoming , Li Hongliang , Qu Guirong , Fan Xuesen , Ruan Chuansheng , Li Tiesheng , Wang Mincan
Published: July 2013

A series of structurally diverse coordination polymers, [Mn(ppmdc)(H2O)2]n (1), [Cu(ppmdc)(H2O)2]n (2), {[Co(ppmdc)][Co(ppmdc)(H2O)]}n (3), [Zn(ppmdc)(H2O)]n (4), and [Cd(ppmdc)]n (5) (H2ppmdc = 2-(pyridin-3-yl)-4,6-pyrimidine dicarboxylic acid) were obtained from metal salts and H2ppmdc under hydrothermal conditions. Single crystal X-ray structural analysis reveals that complexes 1–5 have different structures, ranging from one- to three-dimensions, which are mainly due to the different metal ions, and especially the coordination mode of the H2ppmdc ligand. Complexes 1 and 2 are 1D coordination polymers, in which the ppmdc2– ligand adopts a bis-(bidentate) mode to link metal ions. Complexes 3 and 4 feature a 2D metal-organic framework with Schläfli topologies of (4.62) (42.62.82) and (4.82), respectively, in which the metal ions are bridged by μ3-ppmdc2– ligands. Complex 5 possesses a 3D nanotubular metal-organic framework with a point symbol of (49.66) topology built up from the 6-connected ppmdc2– ligands and Cd(ii) ions. The thermal properties of complexes 1–5 have been determined. Moreover, investigation of photoluminescent properties reveals that the configuration of ppmdc2– resulting from metal-directed coordination has a profound effect on the fluorescence emissions of complexes 4 and 5.
Ovarian hyperstimulation affects fluid transporters in the uterus: a potential mechanism in uterine receptivity
Authors: Lindsay Laura A. , Murphy Christopher R.
Published: July 2013

Controlled ovarian hyperstimulation is commonly used in fertility treatment. Evidence suggests that this could alter the endometrial environment and influence implantation rate. However, the mechanisms underlying this disruption are unknown. A recently developed rat ovarian hyperstimulation (OH) model found alterations in the localisation and expression of several molecules associated with implantation, as well as an increase in luminal fluid at the time of implantation. The present study investigated the effects of OH in rats on the expression of fluid-transporting molecules aquaporin 5 (AQP5) and claudin 4. The expression of these proteins was investigated in uterine luminal epithelial cells of rats undergoing OH and compared with normal pregnancy. There was a significant increase in AQP5 protein in OH rats at the time of implantation, along with a loss of the mesometrial staining gradient, which is thought to contribute to implantation position. At the same time, there was a significant decrease in claudin 4 protein. These results suggest that OH in rats causes a dysregulation in uterine fluid dynamics through modifications to fluid-transporting molecules, resulting in an unfavourable implantation environment for the blastocyst.
The pentiti of the Sicilian mafia in the 1930s
Authors: Coco Vittorio
Published: July 2013

Mussolini had declared that Fascism, through the anti-mafia campaign entrusted to Cesare Mori, prefect of Palermo, in the second half of the 1920s, had conclusively liberated Sicily from the mafia. However, from the early 1930s a new deterioration of public order on the island was evident, and the regime was forced to launch a second, and much less publicised, repression of this phenomenon. In the course of its careful investigations the body given responsibility for pursuing this repression, the Ispettorato Generale di Pubblica Sicurezza per la Sicilia, compiled a series of reports, including the Processo verbale di denunzia relating to the area around Palermo. With a level of detail never seen before, these reports described the structure, organisation and dynamics of mafia groups. This was possible because police officers had available members of the mafia groups themselves – genuine pentiti – who had decided to make disclosures to the authorities.
A note on the rigidity of unmeasured lamination spaces
Authors: Ohshika Ken’ichi
Published: July 2013

We show that every auto-homeomorphism of the unmeasured lamination space of an orientable surface of finite type is induced by a unique extended mapping class unless the surface is a sphere with at most four punctures or a torus with at most two punctures or a closed surface of genus

2
2

.