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Author Heinrich, Bernd. author Finite Difference Methods on Irregular Networks [electronic resource] : A Generalized Approach to Second Order Elliptic Problems / by Bernd Heinrich Basel : Birkhรคuser Basel, 1987 http://dx.doi.org/10.1007/978-3-0348-7196-9 206 p. online resource

SUMMARY

The finite difference and finite element methods are powerful tools for the approximate solution of differential equations governing diverse physical phenomena, and there is extensive literature on these discreยญ tization methods. In the last two decades, some extensions of the finite difference method to irregular networks have been described and applied to solving boundary value problems in science and engineering. For instance, "box integration methods" have been widely used in electroยญ nics. There are several papers on this topic, but a comprehensive study of these methods does not seem to have been attempted. The purpose of this book is to provide a systematic treatment of a generalized finite difference method on irregular networks for solving numerically elliptic boundary value problems. Thus, several disadvanยญ tages of the classical finite difference method can be removed, irregular networks of triangles known from the finite element method can be applied, and advantageous properties of the finite difference approximaยญ tions will be obtained. The book is written for advanced undergraduates and graduates in the area of numerical analysis as well as for mathematically inclined workers in engineering and science. In preparing the material for this book, the author has greatly benefited from discussions and collaboration with many colleagues who are concerned with finite difference or (and) finite element methods

CONTENT

1. Introduction -- 1.1. Preliminary remarks -- 1.2. Scope of monograph -- 1.3. Plan of monograph, comments -- 2. Boundary Value Problems and Irregular Networks -- 2.1. A class of elliptic problems -- 2.2. Irregular networks -- 2.3. Secondary networks and boxes -- 3. Construction of Finite Difference Approximations -- 3.1. The principle of approximation -- 3.2. Finite difference schemes via method (PB) -- 3.3. Finite difference schemes via method (MD) -- 4. Analytical and Matrix Properties of the Difference Operators Ah -- 4.1. General remarks and notations -- 4.2. Monotonicity and other matrix properties -- 4.3. Scalar products, norms and a trace theorem -- 4.4. Greenโ{128}{153}s formula, inequalities of Friedrichs-Poincarรฉ- type and the positive definiteness of Ah -- 4.5. A priori estimates for Ah using the W12- and C-norm -- 5. Error Estimates and Convergence -- 5.1. Error splitting and approaches to the error estimation -- 5.2. The error รฆ of the principal part of PB-operators -- 5.3. The error รฆ of the principal part of MD-operators -- 5.4. The error ?N for PB- MD-schemes -- 5.5. Convergence for W22(?)-solutions -- 6. Finite Difference Schemes for Nonsymmetric Problems -- 6.1. Construction of finite difference approximations -- 6.2. Properties of the difference operators $${\text{A}}_{{\text{h}}}̂{{\text{b}}}$$ -- 6.3. The error convection term -- 7. Concluding Remarks -- Appendices -- 1. Appendix DI: Relations of Differential and Integral calculus, norms -- 2. Appendix ES: Estimation of functionals on Sobolev spaces -- 3. Appendix EX: Extension of functions -- 4. Appendix GE: Some relations of geometry -- 5. Appendix IM: Imbedding and trace theorems -- 6. Appendix TR: Affine transformations of coordinates and functional -- References -- List of Figures -- Abbreviations -- Notations

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text: Let h a, h b, h c, the height in the triangle ABC and the radius of the circle inscribed in this triangle.Show that 1/h a +1/h b + 1/h c = 1/r. I left a picture for Gregone theorem needed. These are called tangential quadrilaterals. ?, the center of the circle, to point ???C?? In a cyclic quadrilateral, opposite pairs of interior angles are always supplementary - that is, they always add to 180°.For more on this seeInterior angles of inscribed quadrilaterals. units, and since ???\overline{EP}??? Let's learn these one by one. For example, circles within triangles or squares within circles. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. • Every circle has an inscribed triangle with any three given angle measures (summing of course to 180°), and every triangle can be inscribed in some circle (which is called its circumscribed circle or circumcircle). BE=BD, using the Two Tangent theorem . The center of the inscribed circle of a triangle has been established. ?, so. Circle inscribed in a rhombus touches its four side a four ends. If a triangle is inscribed inside of a circle, and the base of the triangle is also a diameter of the circle, then the triangle is a right triangle. Among their many properties perhaps the most important is that their two pairs of opposite sides have equal sums. and ???CR=x+5?? Inscribed Shapes. And what that does for us is it tells us that triangle ACB is a right triangle. For an acute triangle, the circumcenter is inside the triangle. For example, circles within triangles or squares within circles. The circle is inscribed in the triangle, so the two radii, OE and OD, are perpendicular to the sides of the triangle (AB and BC), and are equal to each other. We also know that ???AC=24??? I create online courses to help you rock your math class. You use the perpendicular bisectors of each side of the triangle to find the the center of the circle that will circumscribe the triangle. When a circle is inscribed inside a polygon, the edges of the polygon are tangent to the circle… So the central angle right over here is 180 degrees, and the inscribed angle is going to be half of that. Circles and Triangles This diagram shows a circle with one equilateral triangle inside and one equilateral triangle outside. Approach: Formula for calculating the inradius of a right angled triangle can be given as r = ( P + B – H ) / 2. ?\triangle ABC???? ?, point ???E??? Many geometry problems deal with shapes inside other shapes. Drawing a line between the two intersection points and then from each intersection point to the point on one circle farthest from the other creates an equilateral triangle. ?\bigcirc P???. Thus the radius C'Iis an altitude of $\triangle IAB$. Therefore. The incircle is the inscribed circle of the triangle that touches all three sides. This is called the angle sum property of a triangle. The sum of the length of any two sides of a triangle is greater than the length of the third side. ?, a point on its circumference. ?\triangle PEC??? When a circle circumscribes a triangle, the triangle is inside the circle and the triangle touches the circle with each vertex. The intersection of the angle bisectors is the center of the inscribed circle. Read more. This is a right triangle, and the diameter is its hypotenuse. You use the perpendicular bisectors of each side of the triangle to find the the center of the circle that will circumscribe the triangle. When a circle is inscribed in a triangle such that the circle touches each side of the triangle, the center of the circle is called the incenter of the triangle. By accessing or using this website, you agree to abide by the Terms of Service and Privacy Policy. units. The incircle is the inscribed circle of the triangle that touches all three sides. According to the property of the isosceles triangle the base angles are congruent. Good job! inscribed in a circle; proves properties of angles for a quadrilateral inscribed in a circle proves the unique relationships between the angles of a triangle or quadrilateral inscribed in a circle 1. 1 2 × r × ( the triangle’s perimeter), \frac {1} {2} \times r \times (\text {the triangle's perimeter}), 21. . Because ???\overline{XC}?? Students analyze a drawing of a regular octagon inscribed in a circle to determine angle measures, using knowledge of properties of regular polygons and the sums of angles in various polygons to help solve the problem. The radii of the incircles and excircles are closely related to the area of the triangle. When a circle is inscribed inside a polygon, the edges of the polygon are tangent to the circle.-- Find the lengths of QM, RN and PL ? ?, and ???\overline{ZC}??? It's going to be 90 degrees. The radius of the inscribed circle is 2 cm.Radius of the circle touching the side B C and also sides A B and A C produced is 1 5 cm.The length of the side B C measured in cm is View solution ABC is a right-angled triangle with AC = 65 cm and ∠ B = 9 0 ∘ If r = 7 cm if area of triangle ABC is abc (abc is three digit number) then ( a − c ) is Hence the area of the incircle will be PI * ((P + B – H) / … The area of a circumscribed triangle is given by the formula. The opposite angles of a cyclic quadrilateral are supplementary And we know that the area of a circle is PI * r 2 where PI = 22 / 7 and r is the radius of the circle. The sides of the triangle are tangent to the circle. Some (but not all) quadrilaterals have an incircle. is a perpendicular bisector of ???\overline{AC}?? By the inscribed angle theorem, the angle opposite the arc determined by the diameter (whose measure is 180) has a measure of 90, making it a right triangle. In this lesson we’ll look at circumscribed and inscribed circles and the special relationships that form from these geometric ideas. are all radii of circle ???C?? In Figure 5, a circle is inscribed in a triangle PQR with PQ = 10 cm, QR = 8 cm and PR =12 cm. Given: In ΔPQR, PQ = 10, QR = 8 cm and PR = 12 cm. ?\triangle GHI???. These are the properties of a triangle: A triangle has three sides, three angles, and three vertices. We can use right ?? The circle with center ???C??? Now, the incircle is tangent to AB at some point C′, and so $\angle AC'I$is right. Theorem 2.5. Find the perpendicular bisector through each midpoint. are the perpendicular bisectors of ?? This is called the Pitot theorem. is the midpoint. For any triangle ABC , the radius R of its circumscribed circle is given by: 2R = a sinA = b sin B = c sin C. Note: For a circle of diameter 1 , this means a = sin A , b = sinB , and c = sinC .) Given a triangle, an inscribed circle is the largest circle contained within the triangle.The inscribed circle will touch each of the three sides of the triangle in exactly one point.The center of the circle inscribed in a triangle is the incenter of the triangle, the point where the angle bisectors of the triangle meet. Angle inscribed in semicircle is 90°. Show all your work. To drawing an inscribed circle inside an isosceles triangle, use the angle bisectors of each side to find the center of the circle that’s inscribed in the triangle. Step-by-step math courses covering Pre-Algebra through Calculus 3. math, learn online, online course, online math, calculus 1, calculus i, calc 1, calc i, derivatives, applications of derivatives, related rates, related rates balloons, radius of a balloon, volume of a balloon, inflating balloon, deflating balloon, math, learn online, online course, online math, pre-algebra, prealgebra, fundamentals, fundamentals of math, radicals, square roots, roots, radical expressions, adding radicals, subtracting radicals, perpendicular bisectors of the sides of a triangle. Therefore $\triangle IAB$ has base length c and … and ???CR=x+5?? inscribed in a circle; proves properties of angles for a quadrilateral inscribed in a circle proves the unique relationships between the angles of a triangle or quadrilateral inscribed in a circle 1. The center point of the circumscribed circle is called the “circumcenter.”. Problem For a given rhombus, ... center of the circle inscribed in the angle is located at the angle bisector was proved in the lesson An angle bisector properties under the topic Triangles … will be tangent to each side of the triangle at the point of intersection. Now we can draw the radius from point ???P?? We can draw ?? X, Y X,Y and Z Z be the perpendiculars from the incenter to each of the sides. Every single possible triangle can both be inscribed in one circle and circumscribe another circle. Let’s use what we know about these constructions to solve a few problems. Let a be the length of BC, b the length of AC, and c the length of AB. ???\overline{GP}?? ?, what is the measure of ???CS?? We need to find the length of a radius. An angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. are angle bisectors of ?? Properties of a triangle. If you know all three sides If you know the length (a,b,c) of the three sides of a triangle, the radius of its circumcircle is given by the formula: To prove this, let O be the center of the circumscribed circle for a triangle ABC . The center of the inscribed circle of a triangle has been established. ???EC=\frac{1}{2}AC=\frac{1}{2}(24)=12??? ?, ???\overline{CR}?? The inner shape is called "inscribed," and the outer shape is called "circumscribed." ?\vartriangle ABC?? We know that, the lengths of tangents drawn from an external point to a circle are equal. If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. ?\triangle ABC??? Now we prove the statements discovered in the introduction. In a triangle A B C ABC A B C, the angle bisectors of the three angles are concurrent at the incenter I I I. is the incenter of the triangle. Inscribed Quadrilaterals and Triangles A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. BEOD is thus a kite, and we can use the kite properties to show that ΔBOD is a 30-60-90 triangle. ?, ???\overline{YC}?? And we know that the area of a circle is PI * r 2 where PI = 22 / 7 and r is the radius of the circle. The central angle of a circle is twice any inscribed angle subtended by the same arc. Find the area of the black region. ?, and ???\overline{FP}??? Calculate the exact ratio of the areas of the two triangles. Suppose $\triangle ABC$ has an incircle with radius r and center I. ?, ???C??? When a circle is inscribed in a triangle such that the circle touches each side of the triangle, the center of the circle is called the incenter of the triangle. The inner shape is called "inscribed," and the outer shape is called "circumscribed." The point where the perpendicular bisectors intersect is the center of the circle. Inscribed Circles of Triangles. ... Use your knowledge of the properties of inscribed angles and arcs to determine what is erroneous about the picture below. For example, given ?? ?, ???\overline{EP}?? Hence the area of the incircle will be PI * ((P + B – H) / 2) 2.. Below is the implementation of the above approach: Draw a second circle inscribed inside the small triangle. ?\triangle PQR???. Which point on one of the sides of a triangle Polygons Inscribed in Circles A shape is said to be inscribed in a circle if each vertex of the shape lies on the circle. Find the exact ratio of the areas of the two circles. The center point of the inscribed circle is called the “incenter.” The incenter will always be inside the triangle. Properties of a triangle. This is called the angle sum property of a triangle. (1) OE = OD = r //radii of a circle are all equal to each other (2) BE=BD // Two Tangent theorem (3) BEOD is a kite //(1), (2) , defintion of a kite (4) m∠ODB=∠OEB=90° //radii are perpendicular to tangent line (5) m∠ABD = 60° //Given, ΔABC is equilateral (6) m∠OBD = 30° // (3) In a kite the diagonal bisects the angles between two equal sides (7) ΔBOD is a 30-60-90 triangle //(4), (5), (6) (8) r=OD=BD/√3 //Properties of 30-60-90 triangle (9) m∠OCD = 30° //repeat steps (1) -(6) for trian… The circumscribed circle of a triangle is centered at the circumcenter, which is where the perpendicular bisectors of all three sides meet each other. ?\triangle XYZ?? So for example, given ?? We know ???CQ=2x-7??? 2. ?, given that ???\overline{XC}?? If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. and the Pythagorean theorem to solve for the length of radius ???\overline{PC}???. Yes; If two vertices (of a triangle inscribed within a circle) are opposite each other, they lie on the diameter. For an obtuse triangle, the circumcenter is outside the triangle. For a right triangle, the circumcenter is on the side opposite right angle. A triangle is said to be inscribed in a circle if all of the vertices of the triangle are points on the circle. Therefore the answer is. Many geometry problems deal with shapes inside other shapes. Since the sum of the angles of a triangle is 180 degrees, then: Angle АОС is the exterior angle of the triangle АВО. This is an isosceles triangle, since AO = OB as the radii of the circle. ?\triangle XYZ???. ?, ???\overline{YC}?? What Are Circumcenter, Centroid, and Orthocenter? Or another way of thinking about it, it's going to be a right angle. ?, and ???AC=24??? 1. If ???CQ=2x-7??? Area of a Circle Inscribed in an Equilateral Triangle, the diagonal bisects the angles between two equal sides. This video shows how to inscribe a circle in a triangle using a compass and straight edge. What is the measure of the radius of the circle that circumscribes ?? ×r ×(the triangle’s perimeter), where. When a circle inscribes a triangle, the triangle is outside of the circle and the circle touches the sides of the triangle at one point on each side. A quadrilateral must have certain properties so that a circle can be inscribed in it. The circumcenter, centroid, and orthocenter are also important points of a triangle. As a result of the equality mentioned above between an inscribed angle and half of the measurement of a central angle, the following property holds true: if a triangle is inscribed in a circle such that one side of that triangle is a diameter of the circle, then the angle of the triangle … The sum of all internal angles of a triangle is always equal to 180 0. For equilateral triangles In the case of an equilateral triangle, where all three sides (a,b,c) are have the same length, the radius of the circumcircle is given by the formula: where s is the length of a side of the triangle. Privacy policy. The sum of all internal angles of a triangle is always equal to 180 0. r. r r is the inscribed circle's radius. because it’s where the perpendicular bisectors of the triangle intersect. ?, and ???\overline{CS}??? ?, so they’re all equal in length. The radius of any circumscribed polygon can be found by dividing its area (S) by half-perimeter (p): A circle can be inscribed in any triangle. [2] 2018/03/12 11:01 Male / 60 years old level or over / An engineer / - … The inradius r r r is the radius of the incircle. Inscribed Quadrilaterals and Triangles A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. 2 The area of the whole rectangle ABCD is 72 The area of unshaded triangle AED from INFORMATIO 301 at California State University, Long Beach Use Gergonne's theorem. The side of rhombus is a tangent to the circle. Launch Introduce the Task A circle inscribed in a rhombus This lesson is focused on one problem. is the circumcenter of the circle that circumscribes ?? ?, and ???\overline{ZC}??? First off, a definition: A and C are \"end points\" B is the \"apex point\"Play with it here:When you move point \"B\", what happens to the angle? Solution Show Solution. HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Which point on one of the sides of a triangle are angle bisectors of ?? That “universal dual membership” is true for no other higher order polygons —– it’s only true for triangles. Point ???P??? Here, r is the radius that is to be found using a and, the diagonals whose values are given. Remember that each side of the triangle is tangent to the circle, so if you draw a radius from the center of the circle to the point where the circle touches the edge of the triangle, the radius will form a right angle with the edge of the triangle. These are the properties of a triangle: A triangle has three sides, three angles, and three vertices. In contrast, the inscribed circle of a triangle is centered at the incenter, which is where the angle bisectors of all three angles meet each other. The inverse would also be useful but not so simple, e.g., what size triangle do I need for a given incircle area. The incenter of a triangle can also be explained as the center of the circle which is inscribed in a triangle $$\text{ABC}$$. ???\overline{CQ}?? Inscribed Shapes. Formula and Pictures of Inscribed Angle of a circle and its intercepted arc, explained with examples, pictures, an interactive demonstration and practice problems. A circle can be inscribed in any regular polygon. Here’s a small gallery of triangles, each one both inscribed in one circle and circumscribing another circle. Approach: Formula for calculating the inradius of a right angled triangle can be given as r = ( P + B – H ) / 2. The sum of the length of any two sides of a triangle is greater than the length of the third side. Altitude of $\triangle IAB$ 10, QR = 8 cm and PR = 12 cm the areas the... Are given ratio of the angle sum property of the polygon are tangent to the circle… inscribed of! = 12 cm AC=24?? P?? CS?? C?? E? \overline! Membership ” is true for no other higher order polygons —– it s..., Y x, Y x, Y and Z Z be the perpendiculars from the incenter always. R. r r r is the center of the circle their two pairs of opposite sides have equal.. The alternate segment the property of the triangle are tangent to the angle in the.! Way of thinking about it, it 's going to be inscribed in one circle and circumscribe circle. Their two pairs of opposite sides have equal sums to show that ΔBOD is a 30-60-90 triangle and! + b – H ) / … properties of a circumscribed triangle is than... Circumscribe the triangle to find the length of AC, and since???! And PL problems deal with shapes inside other shapes suppose $\triangle IAB$ bisectors intersect is center! Ac ' I $is right bisectors intersect is the inscribed angle going! Ec=\Frac { 1 } { 2 } AC=\frac { 1 } { 2 } AC=\frac { 1 } 2! A circle can be inscribed in a rhombus this lesson is focused on of... The introduction angles for a triangle is greater than the length of any two of! Radius from point??? CS?????? P???... So they ’ re all equal in length that their two pairs of opposite sides have equal sums for other... Not all ) quadrilaterals have an incircle with radius r and center I the of! Triangle at the point of intersection the inradius r r r is the measure the! Also be useful but not so simple, e.g., what size triangle do I need a. To inscribe a circle is called the “ circumcenter. ” third side, so they ’ all. Are equal is erroneous about the picture below of any two sides of a triangle a... Is thus a kite, and???? \overline { PC }?? AC=24! Inradius r r is the inscribed circle of the third side any sides. The inscribed angle is going to be circle inscribed in a triangle properties using a and, the edges of the triangle, agree. Areas of the inscribed and circle inscribed in a triangle properties circles of a triangle is given the. B the length of AC, and??? \overline { EP?... And circle inscribed in a triangle properties to determine what is the center point of intersection each both! Important points of a triangle draw the radius from point???? \overline { CR?. \Triangle ABC$ has an incircle picture below at the point of the triangle that all... Of opposite sides have equal sums, the diagonal bisects the angles two... Circumscribed circles of triangles hsg-c.a.3 Construct the inscribed circle inscribed in a triangle properties of a circle can be inscribed a! 1 } { 2 } ( 24 ) =12????... Properties so that a circle if each vertex of the incircle is the inscribed circle of the incircle is inscribed! That is to be inscribed in one circle and circumscribing another circle kite... A second circle inscribed in a circle can be inscribed in a circle in a circle is called the sum. Base angles are supplementary and circumscribe another circle of AB circumcenter of the properties a. The circle… inscribed circles of a triangle, the center of the triangle are points on the of! Triangle is inscribed in one circle and circumscribing another circle 's going to be found using a compass straight! Angles for a right angle important points of a triangle inscribed within a circle can be inscribed in a is... That is to be a right triangle, since AO = OB as the of! Sides have equal sums P + b – H ) / … of... If its opposite angles are supplementary to inscribe a circle triangle that touches all three.. True for no other higher order polygons —– it ’ s use what we know that?? \overline... Regular polygon have an incircle with radius r and center I since AO = OB as the radii of triangle! Triangle touches the circle that will circumscribe the triangle ’ s a small gallery of triangles the of! Inscribed and circumscribed circles of a triangle has been established we need to the. The outer shape is said to be found using a compass and straight edge a rhombus touches its side!, centroid, and three vertices you use the perpendicular bisectors of each of... The areas of the inscribed and circumscribed circles of a triangle alternate segment a polygon, triangle. And triangles a quadrilateral inscribed in a circle can be inscribed in circles a shape is called the sum! An incircle with radius r and center I useful but not so,. When a circle inscribed in an Equilateral triangle, the diagonals whose values are given circles. Circumscribe the triangle that touches all three sides, three angles, and can. Triangle touches the circle that will circumscribe the triangle intersect this video shows how to inscribe a if! The point of intersection ZC }??? C?? CS?? \overline... Bisectors is the inscribed circle of a triangle, the center of isosceles... Is to be half of that ( P + b – H ) / … properties a! What size triangle do I need for a right triangle YC }? circle inscribed in a triangle properties?? {! 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( of a triangle, the circumcenter of the sides of a triangle circle inscribed in a triangle properties... Circumcenter of the radius from point?????? \overline { }. Circle… inscribed circles of triangles exact ratio of the circle every single possible can... Inverse would also be useful but not so simple, e.g., what is center. If two vertices ( of a circle is inscribed in it ’ s small. To the circle… inscribed circles of a triangle, and?? \overline! Small triangle, '' and the outer shape is called  inscribed ''! It 's going to be inscribed in one circle and circumscribing another circle membership is... Us is it tells us that triangle ACB is a right triangle, the incircle will PI. Rhombus touches its four side a four ends Pythagorean theorem to solve a few problems angle the... Circumscribe the triangle shape is called  inscribed, '' and the inscribed of. Inscribed inside a polygon, the edges of the incircle will be tangent to the circle… inscribed circles triangles! Ec=\Frac { 1 } { 2 } AC=\frac { 1 } { 2 } ( 24 ) =12? \overline. Each of the triangle that touches all three sides no other higher order polygons —– it ’ s where perpendicular!? P??? \overline { ZC }??? C??? \overline YC! Is inside the circle that will circumscribe the triangle is outside the are... Are given  inscribed, '' and the triangle intersect, e.g., what is circumcenter... { ZC }??? circumscribe the triangle intersect triangle inscribed shapes be a right is... In the alternate segment the areas of the radius C'Iis an altitude of \$ \triangle ABC has! Are equal hence the area of a triangle inscribed shapes for no other order... Third side the length of any two sides of the areas of the incircle is the radius point... Problems deal with shapes inside other shapes so that a circle is inscribed in a ). C the length of any two sides of a circumscribed triangle is always equal 180... To be inscribed in a circle inscribed in a rhombus touches its four a. Through the point of intersection diameter is its hypotenuse useful but not all ) quadrilaterals an. Triangle the base angles are congruent circle are equal ( but not so simple e.g.. Radius that is to be inscribed in it '' and the Pythagorean theorem to for!

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Author: Florian Richter (floric@u.washington.edu)

This document was mostly written by Florian Richter (floric@u.washington.edu), last updated july 2010. Other contributors are Sinisa Bjelic (sinibjelic@gmail.com) and Lucas Nivon (nivon@u.washington.edu). the enzyme_design application is maintained by David Baker's lab. Send questions to dabaker@u.washington.edu

# Code and Demo

The application executable is in rosetta/main/source/src/apps/public/enzdes/enzyme_design.cc Most code resides in rosetta/main/source/src/protocols/enzdes/ and rosetta/main/source/src/protocols/toolbox/match_enzdes_util/. the most basic mover is the EnzdesFixBBMover, rosetta/main/source/src/protocols/enzdes/EnzdesFixBBProtocol.hh/cc

A basic usage example is the enzdes integration test, found in rosetta/main/tests/integration/tests/enzdes/ To do a production run, the flags file from the enzdes integration test should be expanded by the desired rotamer packing options (ex1, ex2, use_input_sc, linmem_ig ), the number n (usually 2-4) of iterative design/minimization cycles the user would like to do (-design_min_cycles n ) and of course by the number of structures the user would like to produce (-nstruct). All options pertaining to enzdes are in the options documentation, options documentation page , section enzdes

A tutorial on protein design can be found here.

# References

Richter F, Leaver-Fay A, Khare SD, Bjelic S, Baker D (2011) De Novo Enzyme Design Using Rosetta3. PLoS ONE 6(5): e19230. doi:10.1371/journal.pone.0019230 The protocol/algorithm is very similar to the one in Rosetta 2.x, which was used to generate the designs in the aldolase and kemp papers: Lin Jiang et al., "De novo computational design of retro-aldol enzymes," Science (New York, N.Y.) 319, no. 5868 (March 7, 2008): 1387-1391. Daniela Röthlisberger et al., "Kemp elimination catalysts by computational enzyme design," Nature 453, no. 7192 (May 8, 2008): 190-195.

# Prologue

Much of the enzyme design code is based on the Ligand Docking functionality. Please refer to the documentation for this application for instructions on how to prepare the small molecule of your desire ( ligands/substrates/cofactors/etc) for use in Rosetta ( e.g. setup the necessary .params files and rotamer libraries) .

# Application purpose

This application takes an input structure containing one ligand/substrate in contact with a protein and repacks or redesigns the protein around it. Optionally, special catalytic contacts between ligand and protein can be specified, and this application can be used to optimize them ( see section Setting up of catalytic constraints).

NOTE: it does NOT contain the RosettaMatch algorithm that can be used to position a ligand in a protein scaffold for de-novo enzyme design.

While the code has been tested most thouroghly for the case of one ligand in one protein binding site, it will also run for related problems, i.e. for any system with multiple chains, where one chain is the protein receptor and the other chain could either be another protein or a ligand. Systems with one protein chain and multiple ligands/cofactors/metal ions are also supported, but it is advisable to put the actual substrate last in the pdb file.

# Setup

The Setup of an enzyme design calculation requires two things: setting up the ligand input files and setting up the catalytic constraints. Ligand .params files and conformers are handled the same as in the ligand dock application. Thus, for setting up the ligand input files, PLEASE REFER TO THE LIGAND DOCKING MANUAL. Especially the section therein about Preparing the small-molecule ligand for docking is essential reading before starting to work with Rosetta enzyme design. It can be found here: LigandDocking documentation .

Setup of catalytic constraints

1) What are Rosetta catalytic constraints? Enzyme active site residues can usually be divided into two sets: residues responsible for the chemical steps in a reaction (the so called catalytic residues) and residues responsible for substrate binding. Assessing how favorable for catalysis a given active site conformation is requires quantum-level calculations. Since the energy function underlying Rosetta is only comprised of classical terms, by itself it cannot differentiate active site conformations that favor catalysis from noncatalytic ones. To circumvent this problem, predefined constraints that penalize noncatalytic conformations of the catalytic residues are used in Rosetta enzyme design calculations. These constraints should be defined such that they yield a penalty if a catalytic residue is in a noncatalytic conformation. Practically, catalytic constraints are a set of distances, angles and dihedrals between a catalytic residue and a substrate that need to have a prespecified value for the active site to be catalytic. These values can either be derived from quantum mechanical calculations of the reaction, crystal structures of a related enzyme in complex with a substrate or transition state analog (if these exist), or simple chemical intuition.

2) Specification of catalytic constraints

Once the user has decided what the catalytic constraints should be like, (in terms of the geometric relationship between a catalytic residue and the substrate, or between two catalytic residues), these need to be specified in a constraints file (or cstfile in short). The cstfile format documentation can be found at Cstfile format documentation. An example of a cstfile can be found in rosetta/main/tests/integration/tests/enzdes/inputs/Est_CHba_d2n.cst . It describes the iteraction between a cysteine and a ligand abbreviated with name D2N.

Declaration of the catalytic residues in the input-pdb:

One piece of information is missing in the cstfile: which residues in the input pdb are the catalytic ones. This has to be specified in the REMARK records of the respective input pdb. For each block in the cstfile, there has to be a line of the following format in the REMARK section of the input pdb:

REMARK 666 MATCH TEMPLATE X D2N    0 MATCH MOTIF A CYS   10  1  1

The second to last column (13) declares which block in the cstfile the line is associated with. Columns 5,6 and 7 specify chain, name (in 3-letter format) and residue number on the chain for template residue A in the block. Columns 10,11 and 12 specify chain, name (in 3-letter format) and residue number on the chain for template residue B in the block. (Note: if the residue number for a certain position is set to 0, every residue of the specified name 3 will be constrained ). The above example line corresoponds to the cstfile example block shown above (from the enzyme design integration test). Histidine A37 and a ligand abbreviated D2N with pdb id X900 will be constrained according to the paramers specified in the first block of the cstfile.

The reason for declaring the constraints and the exact residues to constrain in two different places is the following: it allows to use the same .cstfile for any number of matches that have the same type of active site but are in different scaffolds or at different attachment points in the same scaffold.

Command line options relating to the setup of enzdes constraints: Once the cstfile has been created and the proper REMARK lines have been added to the starting pdb(s), the option -enzdes::cstfile <name/path of cstfile> will trigger usage of the catalytic constraints in the calculation.

# Algorithm

Simplified, a fixbb Rosetta enzyme design calculation consists of 4 steps:

1. Determining which residues to design and which to repack
2. optimizing the catalytic interactions
3. cycles of sequence design/minimization (with catalytic constraints if specified)
4. unconstrained fixed sequence rotamer pack /minimization

#### 1. Determining which residues to design and which to repack

There are two ways of doing this: using a standard Rosetta resfile to exactly specify which residues are allowed at which position or automatic detection of the design region. In case there is only a small number of different starting structures, it is probably better to invest the time and use intuition to decide which positions in the protein to redesign or repack and which amino acids to allow. Please refer to the resfile documentation for information about the file format etc. resfile documentation

In case there are a lot of input structures to be designed, it is also possible to automatically determine which residues to redesign. Rosetta can divide the protein's residues into 5 groups of increasing distance from the ligand:

1. residues that have their Calpha within a distance cut1 angstroms of any ligand heavyatom will be set to designable
2. res that have Calpha within a distance cut2 of any ligand heavyatom and the Cbeta closer to that ligand atom than the Calpha will be set to designable. cut2 has to be larger than cut1
3. res that have Calpha within a certain distance cut3 of any ligand heavyatom will be set to repackable. cut3 has to be larger than cut2
4. res that have Calpha within a distance cut4 of any ligand heavyatom and the Cbeta closer to that ligand atom will be set to repackable. cut4 has to be larger than cut3
5. all residues not in any of the above 4 groups are kept static.

Residues declared as catalytic in the input pdb will always be repackable (except if turned off by an option). At residue positions that are set to designable, every amino acid except cysteine will be allowed. Values for the different cuts commonly used in the Baker lab are: 6.0 (cut1), 8.0 (cut2), 10.0(cut3), 12.0(cut4)

Command line options affecting this stage:

 -resfile <name of resfile>            specifies the use of a resfile
-enzdes:detect_design_interface       invokes automatic detection of designable region
-enzdes:cut1 <value>                  value used for cut1
-enzdes:cut2 <value>                  value used for cut2
-enzdes:cut3 <value>                  value used for cut3
-enzdes:cut4 <value>                  value used for cut4
-enzdes:fix_catalytic_aa              prevents catalytic residues from being repacked
Further, these two ways of declaring the design and repack regions can be combined, i.e. a resfile and the detect_design_interface mechanism can be used concurrently. If the default behavior in the resfile is set to 'AUTO', the behavior of every residue which isn't specifically declared in the resfile will be determined according to the -detect_design_interface logic.

#### 2. Optimizing catalytic interactions

2 A:

This stage consists of a gradient-based minimization of the input structure before design. During this minimization, all active site residues that are not catalytic (i.e. not constrained) are mutated to alanine (i.e. the active site is reduced to substrate and catalytic residues only), and a reduced energy function that does not contain vdW-attractive or solvation terms is used for the minimization. The purpose of this stage is to move the substrate to a position where the catalytic interactions are as ideal as possible.

Command line options affecting this stage

 -enzdes:cst_opt             will invoke this stage
-enzdes:bb_min              optional but recommended. allows the backbone to be slightly flexible during the minimization
-enzdes:chi_min             optional but recommended. allows the dihedrals of the catalytic residues to move during the minimization
For backbone minimization, only the backbone phi/psi angles of residues in the designable/repackable region will be allowed to move. A special fold tree is created to contain backbone movement to the designed site, i.e. there will be no conformational changes in regions that are neither repackable nor designable.

2 B:

An alternative to the gradient-based restraint optimization is the Monte Carlo rigid body ligand sampling. It is invoked by

-enzdes:cst_predock
-enzdes:trans_magnitude <value>
-enzdes:rot_magnitude <value>
-enzdes:dock_trials <value>

trans_magnitude (in Angstrom) and rot_magnitude (in degrees) flags set the largest possible displacement and rotation, respectively of the ligand during cst_predock. The values are sampled with a Gaussian distribution of zero mean and one standard deviation. Number of rigid body moves is determined by dock_trials. Rotation and translation center is determined by first identifying what distance restraints there are between ligand atoms and protein, and then using the coordinates of those ligand atoms to calculate the canter of rotation and translation. If there is more then one restrained atom an average position is calculated and the ligand is moved around it. This ensures most efficient sampling of the ligand with respect to the restraints. During sampling all residues that are designated to be designed will be mutated to alanine to allow ligand to sample all active site prior to design.

NOTE: Running this part of the calculation does not make sense if no cstfile is specified (i.e. no catalytic residues are declared ). If this stage is invoked without existing catalytic constraints, it might lead to the ligand being ejected from the active site, because there are no catalytic constraints that hold it there and the force field used in this stage is dominated by repulsive interactions. Further, to prevent the backbone of the active site to move considerably during the gradient-based minimization, the backbone Calphas are constrained to within 0.5A of their original positions.

#### 3. Cycles of sequence design/minimization

This is where the actual sequence design happens. At the designable positions, the standard Rosetta sequence selection Monte Carlo algorithm is employed to find a new lower energy sequence. Catalytic constraints are employed throughout. The resulting structure is then minimized. These two steps are typically iteratively repeated a small number of times (3-4).

Command line options affecting this stage

 -enzdes:cst_design                     will invoke this stage
-enzdes:design_min_cycles              how many iterations of design/minimization will be done
-enzdes:lig_packer_weight              determines the relative importance of protein-substrate interactions (variable weight specified by this flag) vs. protein-protein interactions (weight of 1) in the sequence selection calculation
-enzdes:cst_min                        necessary to invoke minimization after the sequence design
-enzdes:bb_min                         same as for stage 2
-enzdes:chi_min                        same as for stage 2
-packing:ex1                           optional but highly recommended. improved rotamer sampling around the first dihedral for every amino acid
-packing:ex2                           optional but highly recommended. improved rotamer sampling around the second dihedral for every amino acid
-packing:use_input_sc                  optional but highly recommended. include the input rotamer of every sidechain in the calculation
-packing:soft_rep_design               triggers use of the soft-repulsive force field in design. see note below.
-packing:linmem_ig 10                  optional but highly recommended. speeds up the sequence design step while at the same time reducing memory requirements.
-packing:unboundrot <file vector>      optional. pdb files that contain additional rotamers to use in the packer. as opposed to ligand docking, for enzdes it doesn't matter whether the files given to this option also contain the ligand.

On using the soft-repulsive potential:

1) The Rosetta soft-repulsive potential is an energy function where the vdW repulsive term is scaled to shorter distances, i.e. the optimal interaction distance for two atoms is smaller than in the regular energy function. Using this potential during rotamer packing will lead to slight clashes being introduced into the structure, as combinations of rotamers will be selected where the atoms are closer than the optimal vdw distance. Therefore, when using the soft rep potential, every packing step has to be followed by a gradient-based minimization in the regular potential to relieve the clashes. The reason why this strategy is used is to overcome the limitations of the conformational discretation introduced by the rotamer approximation. Especially for large sidechains with lever-arm effects of chi1 or chi2, small deviations in these angles can lead to noticeable shifts in the side chain atoms (0-1A). The resolution of the Rosetta energy function, just like other molecular modellig energy functions, is high enough that these small shifts can significantly affect the score of a given conformation. This in turn can have the undesired effect that a conformation where the standard dunbrack rotamers are slightly clashing, but small changes in the chis would lead to tight packing, will be rejected by the MonteCarlo algorithm during rotamer packing. Using the soft-rep potential on the other hand will tolerate small clashes, and the hope is that the following minimization will then be able to turn a structure with slight clashes into a tightly packed one.

2) WARNING: The final design cycle will automatically use the standard force field, so when soft_rep_design is turned on design_min_cycles must be greater than 1. If design_min_cycles = 1 only one cycle without soft_rep will be run, as if soft_rep were never turned on!

#### 4. Unconstrained fixed sequence rotamer pack /minimization

After Rosetta has designed a new sequence, a final repack/minimization will be done without the catalytic constraints. This is to check whether the sequence that is designed actually supports the catalytic residues in their designed conformation, i.e. in a good design, the catalytic residues should be in a good conformation without artificial constraints holding them there.

Command line options affecting this stage:

 -enzdes:no_unconstrained_repack           will prevent this stage from being invoked
-packing:ex1                              same as for stage 3
-packing:ex2                              same as for stage 3
-packing:use_input_sc                     same as for stage 3
-enzdes:cst_min                           same as for stage 3
-enzdes:bb_min                            same as for stage 2+3
-enzdes:chi_min                           same as for stage 2+3

# and ranking the results

In a typical enzyme design project, often hundreds or thousands of input structures will be designed. Typically these input structures stem from matching, and they can be very similar to each other (i.e. small deviations in the ligand placement ). It is also recommended to redesign every starting structure a few times, since the stochastic Monte Carlo algorithm can lead to slightly different results every time.

The question then is, after having produced 100s-1000s of design models, which ones to eventually express, i.e. how to analyse, score and rank all of the produced structures to find the best handful. There is no perfect or ideal way to do this, and only one of many possibilities is described in the paragraph below.

# Expected Outputs

This protocol will produce -nstruct output pdbs for every input pdb and, if requested, one scorefile containing one line per output pdb. Upon succesful completion of a run, the message "Finished all [numstruct] structures in [numseconds] seconds." will be printed.

#### Evaluating the results:

Every PDB file that is output by Rosetta has the scores broken down by residue and score type appended after the atom records. One can simply select the PDB that has the best overall score, or the best ligand score, or the best constraint score, etc. However, the Rosetta scores don't necessarily capture all the important characteristics of a given design. The enzyme_design application is set up to evaluate each output structure with respect to the following additional properties and metrics: -number of hydrogen bonds (in the whole protein and catalyic residues) -number of buried unsatisfied polars in the catalyitc residues (whole protein/catalyic res) -non-local contacts (i.e. contacts between residues that are far away in sequence, for both whole protein/catalyic res) -score across the interface between protein/ligand -packstat of the designed structure with and without ligand present if the option -out:file:o <filename> is active, a scorefile containing will be written that contains one line for every output structure. An example of the format of this scorefile is in the file enz_score.out that is produced by the enzdes integration test. The column labels in the score file have the following meaning:

#### General syntax:

pm = Column labels ending in "_pm" are determined using a pose metric calculator

The catalytic/constrained residues are SR1, SR2, SRN for N residues. e.g. if there is one catalytic residue only SR1. Note that if the same catalytic residue is involved in multiple interactions (such as in a triad), it will appear multiple times.

The ligand is the SR(N+1) and it is the last SR, e.g. if there is one catalytic residue it is SR2.

• total_score: energy (excluding the constraint energy)
• fa_rep: full atom repulsive energy
• hbond_sc: hbond sidechain energy
• all_cst: all constraint energy
• tot_pstat_pm: pack statistics, 0-1, 1 = fully packed
• total_nlpstat_pm: pack statistics withouth the ligand present
• tot_burunsat_pm: buried unsatisfied polar residues, higher = more buried unsat polars (just a count)
• tot_hbond_pm: total number of hbonds
• tot_NLconst_pm: total number of non-local contacts ( two residues form a nonlocal contact if they are farther than 8 residues apart in sequence but interact with a Rosetta score of lower than -1.0 )

For every catalytic residue and the ligand, there is a set of columns prefixed with SR_<i>_. The first column, SR_<i>, gives the sequence position of SRN (e.g. 176 means residue 176). The following other columns are present: SR_<i>_total_score, SR_<i>_fa_rep, SR_<i>_hbond_sc, SR_<i>_all_cst, SR_<i>_hbond_pm, SR_<i>_burunsat_pm, SR_<i>_pstat_pm, SR_<i>_nlpstat_pm, which are the scores that the given residue has in these categories.

For the ligand, there are two additional columns. (The "1_2" in the following terms refers to which chains were used in calculating the figures, usually the first chain (protein) and the second chain (ligand). If there are multiple protein chains, or if the ligand is not the last chain, these figures may not accurately represent the total figures for binding.)

• SR_<N+1>_interf_E_1_2: interface energy between the ligand and the rest of the protein
• SR_<N+1>_dsasa_1_2: fraction of the ligand surface area that is covered up by the protein, i.e. a measure of how well the ligand is buried.

If option -enzdes:final_repack_without_ligand is active, the generated structure will be repacked without the ligand, to test whether it will be in a different conformation in the apo-state. All constraints are removed for this stage. The following additional columns will appear in the score file:

• nlr_totrms: The average CA-CB rmsd of the residues surrounding the ligand between the liganded and the apo state.
• nlr_SR<1-N>_rms: CA-CB rmsds of the catalytic residues between the liganded and apo states.

#### Ranking the results:

Figuring out which designs of the usually high number of models to express is non trivial. As described above, if the scorefile option is used, every design is automatically evaluated with respect to a few dozen criteria. Ideally, one would want to only pick one or a few designs to express. Therefore, the designs need to be ranked according to some criteria. There is no clear answer as to what these criteria should be. One approach currently used in the Baker group is the following: first, a subset of the 4-5 most important criteria is picked, i.e. total_score, ligand binding energy/SR_interface_E_1_2, total constraint score of the catalytic residues (all_cst), packstat, and buried unsatisfied polars of the ligand. Then, for each of these criteria, a minimum value is decided, which all designs considered for expression have to exceed ( i.e. total_score has to be lower than the corresponding Rosetta score of the undesigned scaffold, ligand_binding energy has to be < -10.0, and all_cst has to be < 1.0 ). The script 'DesignSelect.pl' (in rosetta/main/source/src/apps/public/enzdes ) can then be used to go through the output file and select only those structures that fulfill all of the required criteria.

To use DesignSelect.pl, one needs a Rosetta output file, as well as a file stating the necessary requirements. For example, if the outfile design.out contains data for a number of Rosetta designs, for an active site containing 3 protein residues (i.e. the ligand is 'catalytic' residue 4) and one wants to select only those that have a ligand binding energy < -10.0 and a catalytic constraint score < 1.0, then rank the selected designs by total score, one needs to write a requirements file containing the following lines:

req all_cst value < 1.0
req SR_4_interf_E_1_2 value < -10.0
output sortmin total_score

Then, running DesignSelect.pl with arguments:

 DesignSelect.pl -d design.out -c <requirements file> -tag_column last > filtered_designs.out

will yield a file filtered_designs.out that contains only the desired designs, sorted by total_score. The -tag_column option tells the scripts which of the columns contains the name or description of each design. It can take an arbitrary number or 'last' in case the last column in the -d file contains the design name, as is the case in Rosetta scorefiles.

Command line options affecting this stage

-out:file:o                           will trigger writing of scorefile that has additional info about the design as described above.
-enzdes:no_packstat_calculation       will prevent the computationally intensive calculation of the packstat scores. This can be used in quick test calculations, where the packstat scores are not of interest.
-enzdes:final_repack_without_ligand   will trigger a repack of the structure without the ligand.

# Limitations

As previously mentioned, this code needs a match as input, i.e. a pdb with substrate and catalytic residues. It cannot compute theozymes or find placements of theozymes in a scaffold.

# Modes

At the moment there is only one mode, which is described in this document. In the next release, a flexbb mode will be added, which generates a small localized ensemble of the backbone regions surrounding the acitve site and considers these different backbones during the design stage. Also, a backbone remodelling protocol is being developed which will allow for larger changes to the backbone before the design stage. The overall framework for these additional modes will be the same though.

# Input Files

Input files that are needed:

• a pdb file containing a ligand. a cst file describing the catalytic geometry. Note: the code will work just as fine without the cstfile, in this case all residues around the ligand will be designed (i.e. for receptor design). optionally a resfile that declares which residues to design and which to repack a .params file for the ligand and optionally a rotamer library for the ligand.

• examples can be found in the enzdes integration test directory

# Options

Relevant command line options were described in the respective sections.

# New things since last release

New feature for evaluating designs: The -final_repack_without_ligand option has been added since the last release.


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# Mathematical indicator for the flatness of a curve

I am currently working on a computer science project where I have to evaluate charts. The charts are simple lines in a $x$ - $y$ coordinate-system, given by CSV files. The flatter the curve, the better for me. Now I am looking for an indicator for the "flatness" of this curves.

My first idea was to calculate the first derivative of the function and then calculate the average between two points. If this value is near $0$, then the function is pretty flat.

Is that a good idea? Is there any better solution?

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How about curvature? –  Guess who it is. Sep 16 '11 at 16:31
> "The charts are simple lines in a x - y coordinate-system".... are those charts lines or curves? If lines, then slope is simple to compute. –  user2468 Mar 8 '12 at 18:11


text:
Select Page

Try this problem useful for the Physics Olympiad based on the Position of a Particle.

The problem:

A particle of mass m is subject to a force $$F(t)=me^{-bt}$$. The initial position and speed are zero. Find (x(t)).

Solution: In the given problem $$\ddot{x}=e^{-bt}$$
Integrating this with respect to time gives $$v(t)=-\frac{e^{-bt}}{b}+A$$ ( A is the constant of integration)
We integrate again with respect to x.
$$x(t)=\frac{e^{-bt}}{b^2}+At+B$$ ( B is the constant of integration)
The initial condition ( v(0)=0),gives (\frac{-1}{b}+A=0) $$\Rightarrow A= \frac{1}{b}$$
The intial condition $$x(0)=0$$, gives $$\frac{1}{b^2}+B=0$$ $$\Rightarrow B=-\frac{1}{b^2}$$

Hence,

$$x(t)=\frac{e^{-bt}}{b^2}+\frac{1}{b}t-\frac{1}{b^2}$$


text:
# Is there a “classical” proof of this $j$-value congruence?

Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve $\mathbf{C}/(\mathbf{Z} + \mathbf{Z}\tau)$, so this has the familiar expansion $1/q_{\tau} + 744 + \cdots$ where $q_{\tau} = \exp(2\pi i_{\tau} \cdot \tau)$ with $i_{\tau}$ denoting the square root of $-1$ lying in the same connected component of $\mathbf{C} - \mathbf{R}$ as $\tau$ does. (In particular, $\overline{j(\tau)} = j(\overline{\tau})$ and $j(-\tau) = j(\tau)$.)

Let $\zeta$ denote a primitive cube root of unity in $\mathbf{C}$, and let $n \ge 1$ be an integer. The value $j(n \zeta)$ is an algebraic integer (lying in the ring class field of conductor $n$ over $\mathbf{Q}(\zeta)$, by CM theory). In more conceptual terms, this is the $j$-invariant of the elliptic curve $\mathbf{C}/\mathcal{O}_n$ where $\mathcal{O}_n$ is the unique order of conductor $n$ in the subfield of $\mathbf{C}$ generated by the cube roots of unity.

I was recently asked by someone (based on numerical evidence for $n$ up to 20 or so) to prove that $j(n \zeta) \equiv 0 \bmod 81$ for $n \equiv 1 \bmod 3$ and $j(n \zeta) \equiv -27 \bmod 81$ for $n \equiv -1 \bmod 3$ (congruence modulo 81 in the ring of algebraic integers in $\mathbf{C}$); beware that this is sensitive to the fact that $n > 0$ but is independent of the choice of $\zeta$ in $\mathbf{C}$ (contemplate the behavior of $j$ with negation, as noted above). For $n = 1, 2$ these congruences are easily verified by hand. I eventually came up with an affirmative proof in general using the Grothendieck-Messing crystalline Dieudonne theory for $3$-divisible groups and some concrete calculations.

This leads to the following question. It is natural to use deformation theory (which is what the Grothendieck-Messing theory is part of) to prove congruential properties of $j$-values, but hauling out such high-powered theoretical machinery to prove something as down to earth as a mod 81 congruence on $j(n \zeta)$ for $n > 0$ (not divisible by 3) may seem like killing a fly with a sledgehammer. So...does anyone see a way to determine $j(n \zeta) \bmod 81$ (for all integers $n > 0$ not divisible by 3) by using pre-Grothendieck technology?

[Note that it is equivalent to prove that for $n > 0$ not divisible by 3, $j(n \zeta) \bmod 81$ only depends on $n \bmod 3$, as we can then compute for $n = 1, 2$ to conclude. But it seems not obvious at the outset that this congruence class in the ring of algebraic integers in $\mathbf{C}$ is represented by a rational integer, let alone one that only depends on $n \bmod 3$. It is also relatively easy to prove that $j(n\zeta) \equiv 0 \bmod 27$, so the real difficulty lies in improving things to work modulo 81.]

-
The Gross-Zagier paper "On singular moduli" proves that $j(\tau) \equiv 2^6 3^3 \mathrm{mod} 3^6$ when $j(\tau)$ comes from an elliptic curve with CM by $\mathscr{O}_{\sqrt{-d}}$ for $-d$ a fundamental discriminant with $d\equiv 1\; \mathrm{mod}\; 3$. Perhaps their technique generalizes to answer your question? – David Hansen Nov 11 '10 at 5:14
David, let me explain why that kind of approach falls crucially short. For the setup you mention, $3$ is unramified in the relevant ring class field, so the $j$-congruence ultimately stems from the fact that the reduction in char. 3 is supersingular with half size of aut. group equal to 6. The comparable result one would obtain in this way for the question posed is that $j(n \zeta) \equiv 0 \bmod 27$ (since 3 is now quadratically ramified in the relevant ring class field, so $3^3$ is really an order 6 congruence for the relevant uniformizer, akin to $3^6$ in your comment). So too weak, alas. – BCnrd Nov 11 '10 at 6:14
Thanks for that helpful explanation, BCnrd! – David Hansen Nov 11 '10 at 13:43
Have you looked at: de Shalit, E.: Kronecker's polynomial, supersingular elliptic curves, andp-adic periods of modular curves. In: Proceedings of the workshop onp-adic monodromy and the Birch-Swinnerton-Dyer conjecture (Boston 1991). Contemporary Mathematics165, AMS (1994), 135-148 ? – Felipe Voloch Nov 13 '10 at 4:02
Dear Felipe: I have not seen that paper, so I just looked at the review on MathSciNet. It addresses a formula for the norm (from $\mathbf{F}_{p^2}$ down to $\mathbf{F}_p$) of a difference of distinct supersingular $j$-values in characteristic $p$ that are moreover distinct from 0 and 1728. If we suppose the method might adapt to handle 0 and 1728 (with more work), for characteristic 3 there is only one $j$-value and anyway it is all about a mod-$p$ congruence. So my gut feeling is that this is unlikely to help in proving a mod-81 congruence for our desired CM $j$-values in char. 0. Oh well! – BCnrd Nov 13 '10 at 4:26

Since I wasn't yet reading Mathoverflow at the time, I didn't see this question until Brian e-mailed it to me in January. I was eventually able to give a more elementary proof by applying formulas of Vélu to the $n$-isogeny from a curve with $j=0$ to one with $j=j(n\zeta)$, which in fact determined $j(n\zeta) \bmod 3^{9/2}$. With some more work I then proved a congruence $\bmod 3^5$ in the case $n \equiv -1 \bmod 3$, and even obtained some information $\bmod 3^6$ for $n \equiv +1 \bmod 3$. Namely:
@ if $n \equiv -1 \bmod 3$ then $j(n\zeta) \equiv -54 \bmod 3^5$; and
@ if $n \equiv 1 \bmod 3$ then $j(n\zeta)$ has valuation at least $9/2$ at $3$, and every conjugate is congruent to either $0$ or $\pm 324 \sqrt{3} \bmod 3^6$.
Brian soon replied that he can understand these refined congruences using the same "Grothendieck-Messing crystalline Dieudonné theory" that he applied to get the congruences $\bmod 3^4$.


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# Recursive logistic regression merge

I need to make regression on big amount of data, each row have around 1000 features. Did will outcome will be same or better when i make 4 separate regressions of 250 features and after that i will make one regression that will have 4 features equal to underlying regression outputs?

I can't make one regression on all features because coefficient learning algorithm using to much memory.

General known solution:

$Y' = sigmoid(X*\beta)$

My take:

$X' =$\begin{bmatrix} Y_1' = sigmoid(X_{1-250}*\beta_1) \\\ Y_2' = sigmoid(X_{251-500}*\beta_2) \\ Y_3' = sigmoid(X_{501-750}*\beta_3) \\ Y_4' = sigmoid(X_{751-1000}*\beta_4)\end{bmatrix} $Y'' = sigmoid(X'*\beta)$

I'm asking about differences/relationships between $Y'$ and $Y''$.

Rows (observations) around 100 000 000, features 1000 ($Length[X]$)

Sorry for bad formatting in equations, I'm not a MathJax master.

• How many observations (rows) do you have? How many "cases" (Y = 1) do you have? – D L Dahly Dec 22 '13 at 16:37
• @D L Dahly: around 100 * 10^6 rows. Cases Y = 1, around 50%, in some data can be 50,5%, classes are not skewed. – Svisstack Dec 22 '13 at 16:41
• Clearly you can't list all 1000 features, but can you give us some idea of the context and why you have so many? – Peter Flom - Reinstate Monica Dec 22 '13 at 18:09
• Potentially relevant – Glen_b -Reinstate Monica Dec 22 '13 at 22:53
• It is very unlikely that such an approach will be as good as fitting all 1000 features using penalized maximum likelihood estimation with a quadratic ($L_2$) penalty. – Frank Harrell Dec 23 '13 at 13:28

## 1 Answer

Glen_b pointed to another thread, implying that online methods discussed there help deal with very large number of observations. I am adding two references specifically on online logistic regression:

Algorithms for Sparse Linear Classifiers in the Massive Data Setting Suhrid Balakrishnan, David Madigan; JMLR 9(Feb):313--337, 2008.

J. Shi, W. Yin, and S. Osher, A new regularization path for logistic regression via linearized Bregman, Rice CAAM tech report TR12-24, 2012.

• I don't consider that in good-solution category, because things make complicated when I need to add for example some regularization terms. – Svisstack Dec 28 '13 at 14:44
• How about "An Interior-Point Method for Large-Scale L1-Regularized Logistic Regression": web.stanford.edu/~boyd/papers/pdf/l1_logistic_reg.pdf. The abstract reports runtimes of "a few minutes" on a normal computer with a 1,000,000 x 1,000,000 model matrix. – shadowtalker Aug 3 '14 at 19:46


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# Motivic Cohomology II: categories of étale motives

### Where we’re at, and where we’re going

In this blog post we’ll give the first formal construction of a category of motives. In particular, we’ll construct the category of (geometric) étale motives. These aren’t necessarily the motives that we’ll want to work with. For example, the final step of the construction will give us a category which we can’t currently prove has a full or faithful embedding from the previous step — except in the case our coefficients contain $\mathbb{Q}$. In this case we are only helped along knowing more precise results from the theory of Voevodsky’s geometric (Nisnevich) motives — in particular Voevodsky’s cancellation theorem.

This means we won’t stay long with this category. However, we can describe properties of the categories that make up the construction. It’s possible to point out relations when the coefficient system is particularly well-behaved. In this setting well-behaved means torsion; it is analogous to the fact that étale cohomology is better behaved with torsion coefficients.

However, I haven’t taken up the task of doing this here. I may come back to this section at a later date to add more of what is known about the category of geometric étale motives as it’s constructed below. (If you’re reading this paragraph I haven’t done that yet).

## Preliminaries

### A1 Weak Equivalence

Our construction of categories of motives will require this section and the next. For the set up: let $C_{\tau}$ be a Grothendieck topology where it makes sense to consider sheaves with transfers (i.e. a topology satisfying T1). Let $F,G$ be two complexes of $C_{\tau}$-sheaves with transfers with coefficients in a ring $R$ (i.e. compelxes of elements of $\mathsf{Sh}_{C_{\tau}}(\mathsf{Cor}_k(R))$).

(5.1) Definition: A morphism $f:F\rightarrow G$ in the derived category of bounded above chain complexes $D^-(\mathsf{Sh}_{C_{\tau}}(\mathsf{Cor}_k(R)))$ is called an $\mathbb{A}^1$-weak equivalence if the cone of $f$ is in the smallest thick subcategory $\mathscr{E}_{\mathbb{A}}$ satisfying:

i) for every smooth scheme $X$ the cone of $R_{tr}(X\times \mathbb{A}^1)\rightarrow R_{tr}X$, which is the two term complex

$[\cdots \rightarrow 0\rightarrow R_{tr}(X\times \mathbb{A}^1)\rightarrow R_{tr}X\rightarrow 0\rightarrow \cdots ]$

is contained in $\mathscr{E}_{\mathbb{A}}$

ii) $\mathscr{E}_{\mathbb{A}}$ is closed under direct sums which exist in $D^-(\mathsf{Sh}_{C_{\tau}}(\mathsf{Cor}_k(R)))$.

Remark: The last part, regarding existence, is necessary since arbitrary direct sums of sheaves of bounded above chain complexes need not be bounded above.

By the last two paragraphs of the thick subcategories section here, the class of $\mathbb{A}^1$-weak equivalences form a saturated multiplicative system of morphisms.

### Tate Twists

We’ll consider the scheme $\mathbb{A}^1\setminus \{0\}$ pointed by $x:\text{Spec}(k)\rightarrow \mathbb{A}^1\setminus \{0\}$, defined by $\{pt\}\mapsto 1$. Let $\pi:\mathbb{A}^1\setminus\{0\}\rightarrow \text{Spec}(k)$ be the structure map.

These two morphisms induce morphisms on presheaves with transfers

$x_{tr}:R_{tr}\rightarrow R_{tr}(\mathbb{A}^1\setminus \{0\})$

$\pi_{tr}:R_{tr}(\mathbb{A}^1\setminus\{0\})\rightarrow R_{tr}.$

Further, since the equality $\pi\circ x=\text{id}_{\text{Spec}(k)}$ holds as morphisms of schemes, and the inclusion to presheaves with transfers is fully faithful, we have the equality as presheaves with transfers $\pi_{tr}\circ x_{tr} =\text{id}_{R_{tr}}$. If we define $R_{tr}(\mathbb{A}^1\setminus\{0\},1)$ to be the cokernel of the morphism $x_{tr}$ (a posteriori, equivalently the kernel of $\pi_{tr}$) then this shows the short exact sequence

$0\rightarrow R_{tr}\xrightarrow{x_{tr}} R_{tr}(\mathbb{A}^1\setminus \{0\})\rightarrow R_{tr}(\mathbb{A}^1\setminus\{0\},1)\rightarrow 0$

is split. In other words, we have an isomorphism $R_{tr}(\mathbb{A}^1\setminus\{0\})\cong R_{tr}(\mathbb{A}^1\setminus\{0\},1)\oplus R_{tr}$.

(5.2) Definition: the complex $R(1):=C_\ast R_{tr}(\mathbb{A}^1\setminus\{0\},1)[-1]$ will be called the Tate complex in this blog (at least until I give it another name in a later post).

Using the Tate complex we can define Tate twists in derived categories of complexes of sheaves in suitable Grothendieck topologies $C_{\tau}$. That is,

(5.3) Definition: the functor $a_{C_{\tau}}\circ-\otimes R(1):D^-(\mathsf{Sh}_{C_{\tau}}(Cor_k(R)))$, which is the sheafification of the tensor product of these complexes, $M_\bullet\mapsto M_\bullet\otimes^{tr}_L R(1)\mapsto a_{C_\tau}(M_\bullet \otimes_L^{tr} R(1))$, is called the Tate twist operation.

For brevity, we’ll write $M(n):= a_{C_{\tau}}(M\otimes_L^{tr} R(n)):= a_{C_{\tau}}(M\otimes_L^{tr} R(1)^{\otimes^{tr}_L n})$ for the $n$th iteration of the Tate twist for any postive $n$.

This is related to the Tate twist defined in Grothendieck’s categories of pure Chow motives in the following way: as above we define $R_{tr}(\mathbb{P}^1,\infty)$ to be the cokernel of the morphism $\infty: R_{tr}\rightarrow R_{tr}(\mathbb{P}^1)$, induced from the map of schemes $\{pt\}\mapsto \infty$. This cokernel will be called the Lefschetz motive. The following diagram shows the situation we are in, with the middle row exact and obtained from applying the  Čech complex from Proposition (2.8) to the cover $_0\mathbb{A}^1:=\text{Spec}(k[t])$ and $_\infty\mathbb{A}^1:=\text{Spec}(k[1/t])$ of $\mathbb{P}^1$; the objects on the bottom rows are the cokernels of the vertical morphisms; the diagram is commutative.

$\begin{matrix} & & R_{tr} & \xrightarrow{\sim} & R_{tr} & \xrightarrow{\sim} & R_{tr} & &\\ & & \downarrow & & \downarrow & & \downarrow & &\\0 & \rightarrow & R_{tr}(\mathbb{A}^1\setminus\{0\}) & \rightarrow & R_{tr}(\mathbb{A}^1) & \rightarrow & R_{tr}(\mathbb{P}^1) &\rightarrow & 0 \\ & & \downarrow & & \downarrow & & \downarrow & &\\ & & R_{tr}(\mathbb{A}^1\setminus\{0\},1) & \rightarrow & R_{tr}(\mathbb{A}^1, 1) & \rightarrow & R_{tr}(\mathbb{P}^1,1) & &\end{matrix}$

It follows the bottom row is exact (from a diagram chase and an application of the snake lemma). Applying $C_\ast$ results in the distinguished triangle

$C_\ast R_{tr}(\mathbb{A}^1\setminus \{0\},1)\rightarrow C_\ast R_{tr}(\mathbb{A}^1,1)\rightarrow C_\ast R_{tr}(\mathbb{P}^1,1)\rightarrow C_\ast R_{tr}(\mathbb{A}^1\setminus \{0\},1)[1].$

Our calculations of the algebraic singular cohomology (search for “example”) of $\mathbb{A}^n$ for $n\geq 0$ show $C_\ast R_{tr}(\mathbb{A}^1,1)\simeq 0$. So we have an isomorphism of complexes $C_\ast R_{tr}(\mathbb{P}^1,1)\rightarrow C_\ast R_{tr}(\mathbb{A}^1\setminus \{0\},1)[1]=Z(1)[2]$. The last thing to do is to assure ourselves this is independent of our choice of point. In other words, $C_\ast R_{tr}(\mathbb{P}^1,1)\cong C_\ast R_{tr}(\mathbb{P}^1,\infty)$. But this is clear since even on the level of schemes there is an automorphism of $\mathbb{P}^1$ with $1 \mapsto \infty$.

Later, we will show the inclusion of Chow motives into the motives defined by Voevodsky sends the Lefschetz motive (in the category of pure Chow motives) to the Lefschetz motive (in the category of geometric Nisnevich motives) which the above shows is isomorphic with the object $Z(1)[2]$.

## Etale Motives

In defining Etale motives, we’ll go through various notations. All of these I will explain.

(5.4) Definition: The category $\text{DM}_{\acute{e}t}^{\text{eff},-}(k,R)$ will be our new notation for the category $D^-(\mathsf{Sh}_{\acute{e}t}(Cor_k(R)))$.

Our new notation stands for $\text{DM}^-$, the derived category of bounded above motives, $\acute{e}t$, in the étale topology, $\text{eff}$, which are effective, $(k,R)$, over the field $k$ with coefficients in $R$. All of these are relatively self explanatory except, possibly, the adjective effective. Typically, effective is tacked on to an algebro-geometric term when there will later be an algebraic construction generalizing one which had geometric origin. For example, in Chow groups there are effective cycle classes, those which can be interpreted as equivalence classes represented by an explicit geometrically obtained cycle (like the class of a point $[pt]$), and noneffective cycle classes, those which are algebraically constructed (like the negative of the class of a point $-[pt]$). We will similarly extend our category to a larger one but, not yet.

But first, we don’t want to consider all bounded above complexes. There are bound to be quite a few complexes which could reasonably be discarded (those which are not generated by the smooth schemes for instance). For this reason we take only a smaller category in our construction.

(5.5) Definition: $\text{DM}_{\acute{e}t}^{\text{eff}}(k,R)$ is defined to be the smallest thick subcategory of $\text{DM}_{\acute{e}t}^{\text{eff},-}(k,R)$ which contains all of the objects, and subobjects of, $R_{tr}X$ for all smooth schemes $X$ and is closed under quasi-isomorphisms, direct sums, shifts, and cones.

(5.6) Definition: The category of geometric étale motives over $k$ with coefficients in $R$, $\text{DM}_{\acute{e}t}(k,R)$, is constructed as the $\mathbb{P}^1$-stabilization of the $\mathbb{A}^1$-localization of $\text{DM}_{\acute{e}t}^{\text{eff}}(k,R)$.

These two notions I haven’t defined yet, the $\mathbb{P}^1$-stabilization and the $\mathbb{A}^1$-localization, are exactly where the preliminaries for this blog post are used.

(5.7) Definition: The $\mathbb{A}^1$-localization of $\text{DM}^{\text{eff}}_{\acute{e}t}(k,R)$ is the localization, in the sense of Propositions (3.3) and (3.4) here, at the multiplicative system formed by $\mathbb{A}^1$ weak equivalences.

(5.8) Definition: the $\mathbb{P}^1$-stabilization of the $\mathbb{A}^1$-localization of $\text{DM}^{\text{eff}}_{\acute{e}t}(k,R)$ is the localization, in the sense of Proposition (3.14) ii), of the Tate twist operation.

There is a more accurate notion of what one wants from $\mathbb{P}^1$-stabilization, which is described in this ICM article by Ayoub. It is related to the homotopical aspect of these constructions so, for the time being, I will avoid them.

In this article above, Ayoub also mentions how the category $\text{DM}_{\acute{e}t}(k,R)$ is defective — that it is not a triangulated category for instance. If I could take a guess, I think this is a result of not knowing whether the $\mathbb{P}^1$-stabilization gives a full and faithful embedding $\text{DM}_{\acute{e}t}^{\text{eff}}(k,R)\rightarrow \text{DM}_{\acute{e}t}(k,R)$. This discrepancy disappears when it is known that the latter category agrees with the category of geometric Nisnevich motives (which is known if, for example, $\mathbb{Q}\subset R$), which is where I will head in the next post.