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Reviews (5)
Jul 18, 2018
3M Kind Removal Tape 2 inches wide
3M Kind Removal Tape 2 inches wide is significantly more expensive than 3M Nexcare Gentle Paper Tape 2" wide that is available at Walgreens but it has better adhesion and can be peeled off the skin to inspect the wound underneath the bandage and then re-adhered to the skin several times before it needs to be replaced. 3M Kind Removal Tape 2 inches wide is what the Stanford Outpatient Clinic in Redwood City, CA uses.
Jul 17, 2016
Review of the Vittoria Corsa Elite Tubular Tire - 700 x 23 mm
I only have 40 miles on my 700 x 23 mm Vittoria Corsa Elite tubular tire mounted on the rear wheel and the ride is very precise in that it picks up every feature of the road and transmits the feel to me through my stiff Klein aluminum frame. Like any tire review I am comparing it to other tires in its class and not ones that literally cost twice as much. The main reason I am submitting this review is to pass along information I have learned from the one tire I have ridden. The Vittoria website is really lame in that it does not give enough technical information to differentiate one tire from another in their product line. Every tire company’s website is also lame, which doesn’t help.
Summary:
My Vittoria Corsa Elite is the 700 x 23 mm wide (advertised width, 22.3 to 22.8 mm measured width in 8 places), 270 grams (advertised weight, 294 grams measured weight) tubular (i.e., sew-up) tire that has a butyl inner tube with a 34 mm long non-removable valve stem with a removable valve core. It is a handmade racing tire that is a significant step above the Vittoria Rally in terms of construction technology. I am using the term hand made in the sense that the tread is molded separately and then glued onto the sewn up casing instead of being molded and vulcanized on to the casing in one step. You can identify tires with vulcanized on treads by the way they fold up neatly and are usually sold folded up in a box. Handmade tires do not fold up neatly and are usually displayed in stores in their circular form hanging from a rack.
Tires that have a butyl inner tube and a removable valve core allow for insertion of fibrous clogging mechanism sealant such as Slime to fix small punctures. I have not used the latex sealant from Vittoria which comes in a pressurized can and does not require a removable valve core (and is also chemically compatible with latex inner tubes). The down side to sealants is that it makes the inner tube harder to patch if needed in the future.
This tire is rated for 115 to 200 psi inflation pressure, has an Aramid (generic for Kevlar) puncture protection strip and is made in Thailand.
Each tire company seems to have their own method of determining the thread count of the tire casing. So the TPI should only be used as a relative rating for tires made by the same company. See the You Tube video entitled "How A Vittoria Tubular Is Made". The Kenda Super Domestique tubular tires that I have used and repaired have two layers of bias ply cords (which is the typical construction) with each layer having 100 threads per inch (that I measured perpendicular to the thread direction). Kenda advertises these tires as having 300 TPI. How you add two layers of 100 TPI cord material together to get a 300 TPI casing is totally beyond me.
It is definitely worth the money to step up from the training tubular tires with 220 threads per inch casings and vulcanized on (i.e., molded in place) treads to tires such as the Vittoria Corsa Elite with 290 TPI casings and glued on treads.
Aug 24, 2021
. “Introduction to Bessel Functions” by Frank Bowman (1958) is NOT a good introductory book on Bessel Functions.
“Introduction to Bessel Functions” by Frank Bowman (1958) is NOT a good introductory book on Bessel Functions because most of the book is devoted to advanced topics and the basic topics are not covered adequately enough for a student to fully understand them if they don't already know them. There is a much better introductory book that I will discuss later.
The first chapter entitled Bessel Function of Zero Order is completely incomprehensible unless you are well versed in the subject. Just throwing the basic concepts at the uninitiated reader without adequate explanation doesn't create understanding of the subject.
The second chapter (Applications) attempts to develop the basic concepts and apply them to analyze a swinging vertical chain suspended from one end and the vibration of a circular membrane of a drum head as well as several other problems. There is also a comprehensible geometric development of the Laplacian in polar coordinates which is useful in understanding where the Bessell differential equation comes from that the Bessell functions are solutions to. With enough study of this derivation based on a circular drum head you can understand where all of the terms in the Laplacian come from. Extending the concept to heat flow is a minor variation. Using a drum head as an example creates a direct geometric correlation with the mathematical statement. The same geometric approach using temperature gradients and heat flow can also be used to understand the Laplacian in spherical coordinates. The 3 pages devoted to the Lapalcian development justifies buying this book.
A much better book for learning about Bessel functions is the “Schaum's Outline Series of Fourier Analysis with Applications to Boundary Value Problems” (Including 205 Solved Problems) by Murray R. Spiegel (1974). Unfortunately, this book does not present a development of the Laplacian in polar coordinates that gives rise to the second order differential equations that Bessel functions are the solution to. It also comes only frustratingly close to explaining orthogonality of the Bessel functions that make up an infinite series of them but in the end just pulls some equations out of midair which prevent an in depth understanding of that concept. But the development of the basic and some advanced concepts is very good. It also covers Fourier Series solutions for problems described in Cartesian coordinates and Legender polynomial solutions for problems described in spherical coordinates. This is the book that I was able to learn the subject from because of its step-by-step comprehensible derivations and applications to real problems like circular drum heads and heat conduction in cylindrical objects.
The classic book “Higher Mathematics for Engineers and Physicist” by I.S. and E.S. Sokolnikoff (1942) has a very rigorous algebraic development of the Laplacian in polar coordinates that is interesting but does not give any insight into the physical origins like the geometric based derivation in the Bowman book which does state that an algebraic derivation is also possible but does not present it. The Sokolnikoff book has a section on Bessel function solutions along with many other topics but is not a substitute for the Schaum's Outline for the beginning student.
The swinging chain problem is covered more completely in the UConn class notes entitled "Oscillations of a Hanging Chain" by Roman. This paper's only weaknesses are 2 obvious mistakes stating the boundary conditions and the diagram near the end that shows the mode shapes incorrectly.
The chapters in “Introduction to Bessel Function” entitled Modified Bessel Functions, Asymptotic Expansions and Bessel functions of Any Real Order are very advanced mathematics. In the final chapter also entitled Applications a drum head with both radial and angular modes, a non-linear deflection of a vertical column and an astronomy problem involving Kepler's second law are solved in a fashion comprehensible to the intermediate level reader.
So, in conclusion “Introduction to Bessel Function” is NOT an introductory book but may be of some interest to a student who has mastered the material in the Schaum's Outline book but it seems more directed toward university level mathematics professors.