Let y : S' - R2 be the given piecewise analytic curve. Fixing a line T,...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let y : S' - R2 be the given piecewise analytic curve. Fixing a line T, Emch considers all secants of y that are parallel to T and calls the set of all midpoints of these secants the set of medians M7. Under some genericity assumptions he proves that for two orthogonal lines T and T-, M, intersects M7 in an odd number of points. Nowadays one could write this down homologically. These intersections correspond to inscribed rhombi, where the two intersecting secants are the two diagonals of the rhombus. Now he rotates T continuously by 90 degrees and argues that M, n M7- moves continuously, where at finitely many times two intersection points can merge and disappear or two new intersection points can appear. When t is rotated by 90 degrees, the one-dimensional family of intersection points closes up to a possibly degenerate union of circle components. Since M7 n M7- is odd, Emch argues that an odd number of these components must be Z/4Z- invariant, meaning that if R1R2R3 R4 is a rhombus in such a component, then R2R3R4R1 must also be in the same component. By the mean value theorem, when moving from R1 R2R3R4 to R2R3R4R1 along a component of inscribed rhombi, at some point the diagonals must have equal length. That is, we obtain an inscribed square. This argument also implies that the number of inscribed squares is (generically) odd for Emch's class of curves. Let y : S' - R2 be the given piecewise analytic curve. Fixing a line T, Emch considers all secants of y that are parallel to T and calls the set of all midpoints of these secants the set of medians M7. Under some genericity assumptions he proves that for two orthogonal lines T and T-, M, intersects M7 in an odd number of points. Nowadays one could write this down homologically. These intersections correspond to inscribed rhombi, where the two intersecting secants are the two diagonals of the rhombus. Now he rotates T continuously by 90 degrees and argues that M, n M7- moves continuously, where at finitely many times two intersection points can merge and disappear or two new intersection points can appear. When t is rotated by 90 degrees, the one-dimensional family of intersection points closes up to a possibly degenerate union of circle components. Since M7 n M7- is odd, Emch argues that an odd number of these components must be Z/4Z- invariant, meaning that if R1R2R3 R4 is a rhombus in such a component, then R2R3R4R1 must also be in the same component. By the mean value theorem, when moving from R1 R2R3R4 to R2R3R4R1 along a component of inscribed rhombi, at some point the diagonals must have equal length. That is, we obtain an inscribed square. This argument also implies that the number of inscribed squares is (generically) odd for Emch's class of curves.
Expert Answer:
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
Please explain in your own words why budgets might exert negative influence on accounting decisions. and please offer some examples where this might be the case.
-
Write a short paragraph to explain in your own words the concept of depreciation as used in accounting.
-
Let y : S' - R2 be the given piecewise analytic curve. Fixing a line T, Emch considers all secants of y that are parallel to T and calls the set of all midpoints of these secants the set of medians...
-
Casrnir proposes a model of third-culture building to account for the evolutionary nature of intercultural dialogue. What role do you think the many studies on cultural dimensions can play in the...
-
Refer to the information in QS 17-4. What are the company's departmental overhead rates if the assembly department assigns overhead based on direct labor hours and the finishing department assigns...
-
Marjatta Viitasalo operates a bakery in Thunder Bay, Ontario. Because of its excellent product and excellent location, demand has increased by 25% in the last year. On far too many occasions,...
-
Quilts R Us (QRU) is considering investing in a new patterning attachment with the cash flow profile shown in the table below. QRU's MARR is 13.5 percent/year. a. What is the internal rate of return...
-
Allowance Method for Accounting for Bad Debts At the beginning of 2010, EZ Tech Companys accounts receivable balance was $140,000 and the balance in Allowance for Doubtful Accounts was $2,350. EZ...
-
5. [10] Alice and Bob play a game on a graph G, alternately choosing distinct vertices. Alice starts by choosing any vertex. Each subsequent choice must be adjacent to the preceding choice (of the...
-
Prince Corporation acquires Squire Service Corporation for one million shares of Prince stock, valued at $35 per share. Squire is merged into Prince, although it continues to do business under the...
-
A sports center is planning to construct a set of three tennis courts. The hourly rental fees (and therefore the annual profit) depend on the quality of the court. The tender call requires the courts...
-
Compare the process for economic airline deregulation in the US, European Union, and China.
-
Why is airfare per mile flown much less for long international flights than on a regional airline flight from a small city to a nearby FSNC hub?
-
In negotiations with the US to obtain a single air transportation agreement, the EU has proposed eliminating or significantly reducing the restrictions on citizenship and cabotage. What are these...
-
What were the arguments for and against deregulation? What other industries have been deregulated?
-
Curiously, in the three largest airline markets, the US, EU, and China, there are three dominant full- service network carriers or airline groups. Who are these FSNCs in each market?
-
In the mid-nineteenth century, explorers used the boiling point of water to estimate altitude. The boiling temperature of water T (in F) can be approximated by the model T = -1.83a+212, where a is...
-
Imagine a sound wave with a frequency of 1.10 kHz propagating with a speed of 330 m/s. Determine the phase difference in radians between any two points on the wave separated by 10.0 cm.
-
Point-masses mi are located on the x-axis as shown. Find the moment M of the system about the origin and the center of mass x. m, = 12 + m2=15 2 + M3= 20 -3 -3
-
Evaluate the integral. + 1 dx - 1
-
Find all points at which the direction of fastest change of the function f (x, y) = x 2 + y 2 - 2x - 4y is i + j.
-
The purchasing officer for The Majestic Emporium has prepared a purchases budget for the financial year ending 31 March 2020, based on the following data. The cost of sales is 65% of sales, and the...
-
Greyt Dog Beds Ltd manufactures dog beds specifically designed for greyhounds and whippets. The company purchases all the required materials from external suppliers and designs and assembles the...
-
Ketton Ltd is in the process of preparing direct labour and factory overhead budgets for the year ending 30 June 2020. Relevant data are set out below. Required (a) Prepare the direct labour budget...
Study smarter with the SolutionInn App