Rideau Hall

Rideau Hall is, since 1867, the official residence in Ottawa of both the Canadian monarch and his or her representative, the Governor General of Canada. It stands in Canada's capital on a 0.36-square-kilometre estate at 1 Sussex Drive, with the main building consisting of 175 rooms across 9,500 square metres, 27 outbuildings around the grounds. Rideau Hall's site lies outside the centre of Ottawa, it is one of two official royal residences maintained by the federal Crown, the other being the Citadelle of Quebec. Most of Rideau Hall is used for state affairs, only 500 square metres of its area being dedicated to private living quarters, while additional areas serve as the offices of the Canadian Heraldic Authority and the principal workplace of the governor general and his or her staff. Received at the palace are foreign heads of state, both incoming and outgoing ambassadors and high commissioners to Canada, Canadian Crown ministers for audiences with either the viceroy or the sovereign, should the latter be in residence.

Rideau Hall is the location of many Canadian award presentations and investitures, where prime ministers and other members of the federal Cabinet are sworn in, where federal writs of election are "dropped", among other ceremonial and constitutional functions. Rideau Hall and the surrounding grounds were designated as a National Historic Site of Canada in 1977; the house is open to the public for guided tours throughout the year. Since 1934, the Federal District Commission has managed the grounds; the site of Rideau Hall and the original structure were chosen and built by stonemason Thomas McKay, who immigrated from Perth, Scotland, to Montreal, Lower Canada, in 1817 and became the main contractor involved in the construction of the Rideau Canal. Following the completion of the canal, McKay built mills at Rideau Falls, making him the founder of New Edinburgh, the original settlement of Ottawa. With his newly acquired wealth, McKay purchased the 100 acre site overlooking both the Ottawa and Rideau Rivers and built a stone villa where he and his family lived until 1855 and which became the root of the present day Rideau Hall.

Locals referred to the structure as McKay's Castle. Before the building became a royal residence, the hall received noted visitors, including three Governors General of the Province of Canada: the Lord Sydenham, the Earl of Elgin, Sir Edmund Head, it was said that the watercolours of Barrack Hill painted by the latter governor's wife, Lady Head, while she was visiting Rideau Hall, had influenced Queen Victoria to choose Bytown as the national capital. On 2 September 1860, the day after he laid the cornerstone of the parliament buildings, Prince Edward, Prince of Wales, drove through the grounds of Rideau Hall as part of his tour of the region. After Bytown was chosen as the capital of the Province of Canada, a design competition was launched in 1859 for a new parliamentary campus; the Centre Block, departmental buildings, a residence for the governor general were each awarded separately. The winning scheme for Government House was a Second Empire design by Toronto architects Cumberland & Storm.

However, it was never built. In 1864, Rideau Hall was leased by the Crown from the McKay family for $4,000 per year and was intended to serve only as a temporary home for the viceroy until a proper government house could be constructed; the next year, Frederick Preston Rubidge oversaw the refinishing of the original villa and designed additions to accommodate the new functions. It was enlarged to three or four times the original size by way of a new 49 room wing, once complete, the first Governor General of Canada, the Viscount Monck, took residence; these additions were opposed by George Brown, who claimed that "the governor general's residence is a miserable little house, the grounds those of an ambitious country squire." Prime Minister John A. Macdonald agreed, complaining that more had been spent on patching up Rideau Hall than could have been used to construct a new royal palace. Nonetheless, the gatehouse was enhanced by Rubidge and the entire property purchased outright in 1868 for the sum of $82,000.

Thereafter, the house became the social centre of Ottawa—even Canada—hosting foreign visitors, swearing-in ceremonies, dinners, garden parties, children's parties, theatrical productions in the ballroom, in which members of the household and viceregal family would participate. The largest event held in the ballroom was a fancy dress ball hosted by the Dufferins that took place on the evening of 23 February 1876 and which saw 1,500 guests attending. Still, despite the popularity of the events that took place in the building, negative first impressions of Rideau Hall itself were a theme until the early part of the 20th century. Upon arrival there in 1872, the Countess of Dufferin said in her journal: "We have been so enthusiastic about everything hitherto that the first sight of Rideau Hall did lower our spirits just a little!" In 1893, Lady Stanley, wife of Governor General the Lord Stanley of Preston, said "you will find the furniture in the rooms old-fashioned & not pretty... The red drawing room... had no furniture except chairs & tables

Four-vertex theorem

The classical four-vertex theorem states that the curvature function of a simple, smooth plane curve has at least four local extrema. The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex; this theorem has many generalizations, including a version for space curves where a vertex is defined as a point of vanishing torsion. An ellipse has four vertices: two local maxima of curvature where it is crossed by the major axis of the ellipse, two local minima of curvature where it is crossed by the minor axis. In a circle, every point is both a local maximum and a local minimum of curvature, so there are infinitely many vertices; the four-vertex theorem was first proved for convex curves in 1909 by Syamadas Mukhopadhyaya. His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has 4th-order contact with the curve; the four-vertex theorem was proved in general by Adolf Kneser in 1912 using a projective argument.

For many years the proof of the four-vertex theorem remained difficult, but a simple and conceptual proof was given by Osserman, based on the idea of the minimum enclosing circle. This is a circle that has the smallest possible radius. If the curve includes an arc of the circle, it has infinitely many vertices. Otherwise, the curve and circle must be tangent at at least two points. At each tangency, the curvature of the curve is greater than that of the circle. However, between each pair of tangencies, the curvature must decrease to less than that of the circle, for instance at a point obtained by translating the circle until it no longer contains any part of the curve between the two points of tangency and considering the last point of contact between the translated circle and the curve. Therefore, there is a local minimum of curvature between each pair of tangencies, giving two of the four vertices. There must be a local maximum of curvature between each pair of local minima, giving the other two vertices.

The converse to the four-vertex theorem states that any continuous, real-valued function of the circle that has at least two local maxima and two local minima is the curvature function of a simple, closed plane curve. The converse was proved for positive functions in 1971 by Herman Gluck as a special case of a general theorem on pre-assigning the curvature of n-spheres; the full converse to the four-vertex theorem was proved by Björn Dahlberg shortly before his death in January 1998, published posthumously. Dahlberg's proof uses a winding number argument, in some ways reminiscent of the standard topological proof of the Fundamental Theorem of Algebra. One corollary of the theorem is that a homogeneous, planar disk rolling on a horizontal surface under gravity has at least 4 balance points. A discrete version of this is. However, in three dimensions there do exist monostatic polyhedra, there exists a convex, homogeneous object with 2 balance points, the Gömböc. There are several discrete versions of the four-vertex theorem, both for convex and non-convex polygons.

Here are some of them: The sequence of angles of a convex equilateral polygon with at least four vertices has at least four extrema. The sequence of side lengths of a convex equiangular polygon with at least four sides has at least four extrema. A circle circumscribed around three consecutive vertices of a polygon with at least four vertices is called extremal if it contains all remaining vertices of the polygon, or has none of them in its interior; such a convex polygon is generic. Every generic convex polygon with at least four vertices has at least four extremal circles. Two convex n-gons with equal corresponding side length have either zero or at least 4 sign changes in the cyclic sequence of the corresponding angle differences. Two convex n-gons with parallel corresponding sides and equal area have either zero or at least 4 sign changes in the cyclic sequence of the corresponding side lengths differences; some of these variations are stronger than the other, all of them imply the four-vertex theorem by a limit argument.

The stereographic projection from the sphere to the plane preserves critical points of geodesic curvature. Thus simple closed spherical curves have four vertices. Furthermore, on the sphere vertices of a curve correspond to points. So for space curves a vertex is defined as a point of vanishing torsion. In 1994 V. D. Sedykh showed that every simple closed space curve which lies on the boundary of a convex body has four vertices. In 2015 Mohammad Ghomi generalized Sedykh's theorem to all curves. Last geometric statement of Jacobi Tennis ball theorem The Four Vertex Theorem and Its Converse—An expository article which explains Robert Osserman's simple proof of the Four-vertex theorem and Dahlberg's proof of its converse, offers a brief overview of extensions and generalizations, gives biographical sketches of Mukhopadhyaya and Dahlberg

Cash balance plan

A cash balance plan is a defined benefit retirement plan that maintains hypothetical individual employee accounts like a defined contribution plan. The hypothetical nature of the individual accounts was crucial in the early adoption of such plans because it enabled conversion of traditional plans without declaring a plan termination; the employees' accounts earn a fixed rate of return that can change over a period of time from year to year. Although it works much like a defined-contribution plan, it is a defined-benefit plan for legal purposes. In 2003, over 20% of US workers with defined benefit plans were in cash balance plans, according to Bureau of Labor Statistics data. Most of these plans resulted from conversions from traditional defined-benefit plans; the status of such plans was in legal limbo, the number of conversions slowed. However, legislation was passed that cleared the way for plan sponsors to adopt cash balance plans. Cash balance conversions have been controversial and have raised the ire of workers and their advocates.

In 2005 the Government Accountability Office released a report analyzing the effects of cash balance conversions on worker benefits. They found that in a typical conversion the cash balance plan would provide lower benefits for most workers than if the defined-benefit plan had remained unchanged and the worker had stayed in their job until retirement age; this decline in benefits tends to be largest for older workers. This is because in a traditional plan, where benefits are based on final average pay, the "value" of the benefits accrues much faster for older workers than for younger workers. In contrast, in a DC or cash balance plan, contributions are made at the same rate, a dollar contributed to a younger worker's account is more valuable because it has more time to compound before retirement, thus some argue. On the other hand, this may not be the relevant comparison. If the alternative to cash balance conversion is that the plan is frozen or terminated, all workers would be much worse off than in a cash balance conversion.

This is a realistic possibility. For the many employees who leave their job before retirement, many would be better off under the cash balance conversion than under the original defined-benefit plan. In addition, about half of cash balance conversions have grandfathered in some or all of the existing participants in the defined-benefit plan; the ubiquitous 401 plan is an example of a defined contribution plan because the Internal Revenue Code §414 states hat the term defined-contribution plan means any plan that provides retirement benefits to a worker based on the amount contributed to the account and any income, gains net of any expenses and losses. Under the definition of accrued benefit under Code §411 in the case of a plan, not a defined benefit plan, means the balance the employee's account. On the other hand, for defined-benefit plans, Section §411 states that "accrued benefit" means "the employee’s annual benefit" as it is "determined under the plan … expressed in the form of an … … commencing at normal retirement age."

The Code's definition for defined benefit plans are all plans that are not defined contribution plans. Cash balance plans are defined-benefit plans. A worker's right to a pension in a defined-benefit plan represents a contingent and hence uncertain financial obligation to the employer sponsoring the plan. Section 412 of the Code requires the employer to make annual contributions to the plan to ensure that the plan assets will be sufficient to pay the promised benefits at retirement; as part of this process the plan is required to have an actuary perform annual "actuarial valuations" in which the present value of each worker's "accrued benefit" is estimated and each present value for each worker covered by the plan is added up so that the minimum annual contribution can be determined. The "actuarial present values" for the "accrued benefit" for each worker is the lump sum dollar amount that represents the financial value of the employer's liability on the date of the valuation, it does not include the future accrual of pension benefits nor does it include the effect of projected future salary increases.

Thus the lump sum value for each worker is not based on that worker's projected final salary at retirement, but only the worker's salary on the date of valuation. Some cash balance plans communicate to workers that these "actuarial present values" are "hypothetical accounts" because upon termination of service, the employer will give the former worker the option to take "all his money" from the pension plan out. In reality, if both the worker and employer agree in a normal defined-benefit plan a former worker may take away "all his money" from the pension plan. There are no legal differences in this "portability" aspect between a traditional defined-benefit plan and a cash balance plan. A typical "design" for a cash balance plan would provide each worker a "hypothetical account" and pay credits in the current year of say 5% of current salary. In addition, the cash balance plan would provide an interest credit of say 6% of the prior year's balance in each worker's "hypothetical account" so that the current year's balance would be the sum of the prior year's balance and the