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## Convex hull of the union of scaled polytopes create few new edges

Let $$P$$ and $$Q$$ be two polygons in $$mathbb{R}^2$$. Given $$a > 0$$, denote by $$aP$$ its image under the dilation by $$a$$ centered around the origin (i.e. the polygon obtained by replacing each vertex $$(p_0,p_1)$$ by the vertex $$(ap_0, ap_1)$$). Let $$C(P,Q)$$ denote the convex hull of the union $$P cup Q$$, and let $$N(P,Q)$$ denote the number of new edges created (i.e. edges in $$C(P, Q)$$ which do not appear in $$P$$ or $$Q$$). Let $$P+Q$$ denote the Minkowski sum of the two polygons.

Based on some empirical results, I was led to the below heuristic (if $$X$$ is a Gaussian centered at $$0$$, then $$|X|$$ follows the half-normal distribution).

Heuristic: Let $$a,b$$ be chosen from the half-normal distribution, and $$P, Q$$ be two fixed convex polygons in $$mathbb{R}^2$$ that are “sufficiently generic”. Then:
$$mathbb{E}(N(aP, bQ))<10$$

Unfortunately it turns out that this does not hold for all choices of polygons $$P$$ and $$Q$$, and finding a counterexample is not difficult (see below for details). Above $$10$$ is an arbitrarily chosen constant, the key point is that it does not depend on the number of vertices in the polygons $$P$$ and $$Q$$. Let a Gaussian random polygon $$P$$ be the convex hull of $$n$$ random, independent points in $$mathbb{R}^2$$ sampled according to the standard normal distribution, for some $$n$$; here $$n$$ is the “size” of the polygon. So this leads me to my real question:

Question: What are some conditions which ensure that the above inequality holds? These conditions should be “generic” in some sense, and should be satisfied (with high probability) when $$P$$ and $$Q$$ are Gaussian random polygon of fixed size.

In particular, an answer to the above question should rigorously prove that the heuristic holds (with high probability) when $$P$$ and $$Q$$ are Gaussian random polygons. The next question is a more general version, with Minkowski sums.

Question 2: Let $$a_1, cdots, a_m, b_1, cdots, b_n$$ be chosen from a half-normal distribution, and $$P_1, cdots, P_m, Q_1, cdots, Q_n$$ be polygons in $$mathbb{R}^2$$. Consider the following statement:
$$mathbb{E}(N(a_1P_1+cdots+a_mP_m, b_1Q_1+cdots+b_nQ_n)) < 10$$
What are some conditions on these polygons which ensure that the above inequality holds? These conditions should be “generic” in some sense, and should be satisfied (with high probability) when the $$P_i$$ and the $$Q_j$$ are Gaussian random polygon of fixed sizes.

If the phrasing of the above questions are too vague, I’d be happy with a proof that the above inequalities hold, with high probability, when the polygons are random Gaussian polygons. One special case I’m interested in is when the sizes of the Gaussian polygons is 2 (i.e. they are line segments).

The counterexample for Q1 can be constructed as follows. Suppose $$O P_1 P_2 cdots P_n$$ is a convex polygon, with the vertices in clockwise order. Let $$P_i P_{i+2}$$ intersect $$O P_{i+1}$$ at the point $$Q_{i+1}$$. The points should be chosen carefully so that the ratio $$frac{OQ_{i+1}}{OP_{i+1}} < epsilon$$, where $$epsilon>0$$ is small (this can be done inductively). Let $$P$$ be the polygon $$OP_2 P_4 …$$ and $$Q$$ the polygon $$OP_1 P_3 …$$. With this choice, it is easy to see that $$mathbb{E}(N(aP, bQ))$$ grows linearly with $$n$$.

## mg.metric geometry – Which polytopes can be deformed while keeping their edge-lengths?

Let $$PsubsetBbb R^d$$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while

• keeping its combinatorial type, and
• keeping its edge-lengths.

I know that the $$d$$-cube is flexible in this sense.
More generally, most (but not all) zonotopes are flexible (see the comments). Also all polygons are flexible. But are there any others?

$$quadquad$$

I also know that there are polytopes having several realizations with matching edge-lengths (e.g. see the image here), but these realizations cannot be continuously deformed into each other while preserving all edge-lengths.

## polygons – Integer Linear (ILP / MILP) Formulation for Collision Avoidance of Convex Polytopes / Polyhedra

I am looking for a possibility to avoid the collision of two convex polytopes using (mixed integer) linear programming. I know how I can detect a collision (Akgunduz, A., Banerjee, P., and Mehrotra, S. (March 14, 2005). “A Linear Programming Solution for Exact Collision Detection .” ASME. J. Comput. Inf. Sci. Eng. March 2005; 5(1): 48–55. https://doi.org/10.1115/1.1846053):

Vertices of polytopes are given: vj, wj

Weights are variables: αj ⩾ 0, βj ⩾ 0

∑vj αj − ∑wj βj = 0 (1)

∑αj = 1 (2)

∑βj = 1 (3)

Equation (1) applies to each of the x, y, and z coordinates of the vertices for P1 and P2. Here the presence of a feasible solution indicates a collision. If no solution is found, no collision exists. What I am now looking for is the opposite. I am looking for a feasible solution for no-collision and a not feasible solution if the polytopes collide. How can I change the equations to get this? It is also important for me, that the collision avoidance is not included in the objective function, since the objective parameters are others. The equations should just create a feasible solution space for non-collsion solutions.

## convex polytopes – Distance to the “boundary” of a polyhedral complex

Suppose I have a polyhedral complex $${P_1, ldots, P_k}$$ and let $$S := cup_{i = 1}^k P_i$$. I am interested in a function which measures the distance from a point $$x in S$$ to the “boundary” of my polyhedral complex, in other words the lower dimensional faces. More precisely, if $$x in text{int}(P_i)$$ for some $$i$$ then my function should return the distance from $$x$$ to the boundary of $$P_i$$. Otherwise, $$x$$ lies on a lower dimensional face and the function should return $$0$$.

Is there a name for this function? Has it appeared in the literature and been studied before? I’m also wondering if it has any special properties beyond simply being a distance function to a non-convex set. Thanks in advance!

## polytopes – Convex helmet of a convex curve as an infinite intersection of the convex helmet of triangles

Leave $$f (x)$$ be a univariate convex curve (say $$f (x) = x ^ 2$$) and let the domain be limited. The objective is to demonstrate that the convex hull of this curve in its domain can be expressed as an infinite intersection of the convex hull of the triangles. The convex hull sequence is constructed as follows:

(i) Partition the domain evenly in $$k$$ partitions, (ii) construct a triangle in each partition using the two tangents at the end points of the partition and the secant that joins the end points, (iii) take the convex hull of the result $$k$$ triangles Let the convex helmet form in the iteration $$k$$ be denoted by $$A_k$$.

Now we want to demonstrate $$cap_k A_k = operatorname {conv} (y = f (x))$$.
Any help is appreciated.

<img src = "https://i.stack.imgur.com/oDkRt.png" alt = "An example is shown for \$ k = 2 \$ y $$f (x) = x ^ 2$$ in the domain $$(- 2, 2)$$">

## Ag algebraic geometry: tests of Euler's formula for n-dimensional dimensioned convex polytopes

Twenty tests of Euler's formula V – E + F = 2 are presented in the geometry scrap yard.

1] Which ones can extend beyond three dimensions to any n-dimensional dimensioned convex polytope?

2] Do you have references to other tests for undefined dimensions?

## convex polytopes: deciding whether a subspace intersects with an open polyhedral cone

### I am trying to answer the following question:

Leave $$V = text {span} (v_1, …, v_k) subseteq mathbb {R} ^ n$$, where $$v_1, …, v_k$$ It can be assumed orthonormal. Leave $$C = {x in mathbb {R} ^ n: x ^ Tu_i <0, i = 1, …, N }$$ be an open polyhedral cone where $$u_1, …, u_N in mathbb {R} ^ n$$ They are given. Suppose we have $$v_1, …, v_k$$ Y $$u_1, …, u_N$$ in hand, how do we decide if $$V cap C = varnothing$$ or not?

### This is what I tried:

My first approach is to use the separation theorem to try to transform the problem into a standard linear viability problem:
$$V cap C = varnothing$$ If and only if $$exist y in mathbb {R} ^ n$$ such that
$$y ^ T (r_1v_1 + cdots + r_kv_k) geq 0, forall (r_1, …, r_k) in mathbb {R} ^ k text {y}$$

$$y ^ Tx <0, forall x: x ^ Tu_i <0, i = 1, …, N.$$

Written more compactly, $$V cap C = varnothing$$ If and only if $$exist y in mathbb {R} ^ n$$ such that
$$(i) y ^ TA_V z geq 0, forall z = (z_1, …, z_k) ^ T in mathbb {R} ^ k text {, y}$$

$$(ii) y ^ Tx <0, forall x in mathbb {R} ^ n: Ux <0, i = 1, …, N$$
where $$A_V$$ is the $$n times k$$ matrix with $$v_1, …, v_k$$ like its columns and $$U$$ is the $$N times n$$ matrix with $$v_1, …, u_N$$ as its rows (i) can be treated "sufficiently" considering only $$z = e_1, …, e_k, e_1 + … + e_k$$ since they cover positively $$mathbb {R} ^ k$$. About (ii), to replace the infinitely many $$x$$ finely many $$x$$, I would need the extreme rays of $$C$$I don't know how to find. But, even if I find these extreme rays, the problem will only become "similar" to the linear feasibility problem. "With some $$<$$ conditions ", which I still don't know how to treat completely.

My second approach is simply consider $$V ^ perp$$, the orthogonal complement of $$V$$. We can use the Gram-Schmidt process to generate an orthonormal basis for $$V ^ perp$$tell $$w_1, …, w_ {n-k}$$. Leave $$W$$ be the $$(n-k) times n$$ matrix with $$w_1, …, w_ {n-k}$$ as their ranks and $$U$$ be the $$N times n$$ matrix with $$u_1, …, u_N$$ like their ranks then $$V cap C = varnothing$$ yes and only if the system
$$Wx = 0$$
$$Ux <0$$
$$s.t. x in mathbb {R} ^ n$$
It has no solution. Again, I don't know how to deal with "$$<$$"in $$Ux <0$$.

### Sorry for my long paragraph that describes my attempts. Here are my questions:

(1) For a polyhedral cone described by $$C = {x in mathbb {R} ^ n: x ^ Tu_i <0, i = 1, …, N }$$ with $$u_i$$ known, $$i = 1, …, N$$Are there simple algorithms or analytical formulas to find their extreme rays?
(2) In the restrictions of a "linear program", if there are any restrictions with "$$<$$" instead of "$$leq$$"How do I deal with them?

Any ideas, reference books or documents are welcome and appreciated. Thank you.

## ag geometry algebraic: polynomial inlays of toric varieties of polytopes?

Background: Leave $$P$$ be an integral polytope, and $$X_P$$ The toric variety associated with the normal range.

$$X_P$$ It is always projective, because the collection of characters corresponding to the points $$mathbb {Z} ^ n cap P$$ together they give an inlay of $$X_P$$ in projective space.

However, the dimension of this embedding is the number of integer points, which is generally exponentially large in a reasonable description of $$P$$.

Questions:

• Suppose that $$P$$ is given by $$Ax leq b$$ in $$mathbb {Q} ^ n$$, with $$A in M_ {n times m} ( mathbb {Q})$$ Y $$b in mathbb {Q} ^ n$$and that they promised us that $$P$$ is integral is there a projective incorporation of $$X_P$$ that only requires $$POLY (| A |, | b |)$$ bits to specify?
• There is a family $$A_n, b_n$$ such that the minimum dimension of an inlay grows exponentially in $$| A |, | b |$$?
• Are there any polytope parameters (in the sense of parametrized complexity) that control the size of a minimum (and efficiently computable) insert?

I'm deliberately lazy about whether the coding of $$A$$ is in binary or unary; Inlays of polynomial or pseudopolinomial size would be interesting for me.

Motivation: I am curious about whether there are polytope parameters that are evident through simple additions of the corresponding toric variety, and that could help with computational problems on the side of the polytope.

For example, if we know that $$X_P$$ It is a smooth complete intersection and we have the equations that cut it, we can calculate its Euler characteristic using the formula on page 146 of "About Chern's classes and Euler's characteristic for complete non-singular intersections" by Vicente Navarro Aznar. This would count the number of vertices of $$P$$, which is usually a $$# P$$ Difficult problem Of course, most polytopes will not provide a smooth toric variety or a complete intersection, and it is very likely that calculating the scale is difficult, so this observation is of limited use.

Anyway, I'm curious to know if we can measure the complexity of the politoe by the complexity of the toric variety as a projective variety. The basic question is whether or not we can efficiently find small inlays in general, hence this question.

## Linear programming – Formulation of mixed integers of polytopes?

Dice $$t$$ different unlimited polyhedra $$P_1: A ^ {(1)} x ^ {(1)} leq b ^ {(1)}, dots, P_t: A ^ {(t)} x ^ {(t)} leq b ^ {(t)}$$ we are looking for the representation of $$bigcup_ {i = 1} ^ tP_i$$ (not his convex helmet) with mixed integer programming.

1. What is the standard way of doing it by programming mixed integers with the least number of additional integer variables?

2. When it is possible to do it with only $$O (t ^ alpha)$$ additional whole variables where $$alpha in (0,1)$$?

I found a review where they say we can do this with $$O ( log t)$$ Whole variables if the polyhedra have a common recession cone.

## Are the isomorphic polytopes having the same extension complexity?

Is there a simple proof of the fact that isomorphic polytopes have a complexity of similar extent?


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# zbMATH — the first resource for mathematics

Monotone solutions of iterative fractional equations found by modified Darbo-type fixed-point theorems. (English) Zbl 1376.34011
Summary: There are many forms of solutions that occur obviously, and in various situations one cannot just switch to one of them and establish solutions in that sense. It is valuable to study solutions with strong differentiability properties, but it might be difficult to confirm their existence. Therefore, one essentially considers solutions in a weaker sense. In this note, we establish the existence of monotone solutions of the dynamic system of the bacterial growth of iterative fractional differential equations, found by a new coupled and triple fixed point theorem in the sense of Darbo processing. We propose these theorems (Darbo type) for a new contraction by using a measure of non-compactness in Banach spaces. The new outcomes show some special well-known recent results.
##### MSC:
 34A08 Fractional ordinary differential equations and fractional differential inclusions 47H10 Fixed-point theorems
Full Text:
##### References:
 [1] Aghajani, A; Banas, J; Sabzali, N, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin., 20, 345-358, (2013) · Zbl 1290.47053 [2] Aghajani, A; Sabzali, N, A coupled fixed point theorem for condensing operators with application to system of integral equations, J. Nonlinear Convex Anal., 15, 941-952, (2014) · Zbl 1418.47003 [3] Appell, J, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl., 83, 251-263, (1981) · Zbl 0495.45007 [4] Appell, J, Measure of noncompactness, condensing operators and fixed points: an application oriented survey, Fixed Point Theory, 6, 157-229, (2005) · Zbl 1102.47041 [5] Arab, R, Some fixed point theorems in generalized Darbo fixed point theorem and the existence of solutions for system of integral equations, J. Korean Math. Soc., 52, 125-139, (2015) · Zbl 1305.47038 [6] Arab, R, The existence of fixed points via the measure of noncompactness and its application to functional-integral equations, Mediterr. J. Math., 13, 759-773, (2016) · Zbl 1348.47041 [7] Arab, R.: Some generalizations of Darbo fixed point theorem and its application. Miskolc Math. Notes (in press) · Zbl 1399.54082 [8] Banas, J, Measures of noncompactness in the space of continuous tempered functions, Demonstr. Math., 14, 127-133, (1981) · Zbl 0462.47035 [9] Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics. Dekker, New York (1980) · Zbl 0441.47056 [10] Berinde, V; Borcut, M, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. Theory Methods Appl., 74, 4889-4897, (2011) · Zbl 1225.54014 [11] Burton, TA, Krasnoselskii’s inversion principle and fixed points, Nonlinear Anal., 30, 3975-3986, (1997) · Zbl 0894.47041 [12] Burton, TA; Kirk, C, A fixed point theorem of Krasnoselskii-Schaefer type, Mathematische Nachrichten, 189, 23-31, (1998) · Zbl 0896.47042 [13] Darbo, G, Punti uniti i transformazion a condominio non compatto, Rend. Sem. Math. Univ. Padova, 4, 84-92, (1995) [14] Eder, E, The functional differential equation $$x^{′ }(t) = x(x(t)),$$, J. Differ. Equ., 54, 390-400, (1984) · Zbl 0497.34050 [15] Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Volume 373 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (1996) · Zbl 0866.45004 [16] Ibrahim, RW, Existence of deviating fractional differential equation, CUBO A Math. J., 14, 127-140, (2012) [17] Ibrahim, RW, Existence of iterative Cauchy fractional differential equation, J. Math., 2013, 1-8, (2013) · Zbl 1268.34017 [18] Ibrahim, R.W., Darus, M.: Infective disease processes based on fractional differential equation. In: Proceedings of the 3rd International Conference on Mathematical Sciences, vol. 1602, pp. 696-703 (2014) · Zbl 1385.47021 [19] Ibrahim, RW; Kilicman, A; Damag, FH, Existence and uniqueness for a class of iterative fractional differential equations, Adv. Differ. Equ., 2015.1, 1-13, (2015) · Zbl 1351.34006 [20] Ibrahim, R.W.: Fractional Calculus of Multi-objective Functions and Multi-agent Systems. Lambert Academic Publishing, Saarbrucken (2017) [21] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland, Mathematics Studies. Elsevier, Amsterdam (2006) [22] Kuratowski, K, Sur LES espaces completes, Fund. Math., 15, 301-309, (1930) · JFM 56.1124.04 [23] Mursaleen, M; Mohiuddine, SA, Applications of measures of noncompactness to the infinite system of differential equations in $$l_p$$ spaces, Nonlinear Anal. Theory Methods Appl., 75, 2111-2115, (2012) · Zbl 1256.47060 [24] Mursaleen, M; Rizvi, SMH, Solvability of infinite systems of second order differential equations in $$c_0$$ and $$ℓ _1$$ by Meir-Keeler condensing operators, Proc. Am. Math. Soc., 144, 4279-4289, (2016) · Zbl 1385.47021 [25] Reich, S, Fixed points of condensing functions, J. Math. Anal. Appl., 41, 460-467, (1973) · Zbl 0252.47062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

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# Does a Riemannian manifold have a triangulation with quantitative bounds?

Suppose that $M$ is a closed Riemannian manifold with bounded geometry, i.e., curvature between $-1$ and $1$ and injectivity radius at least $1$. Since $M$ is a smooth manifold, it has a triangulation. Does it necessarily have a triangulation that is "nice" with respect to the metric?

For instance, is there an $\epsilon>0$ such that for any such $M$, there is a triangulation of $M$ whose simplices are all homeomorphic to the standard simplex by a map $f$ such that $f$ and $f^{-1}$ are both $\epsilon^{-1}$-Lipschitz?

The method I'm familiar with for constructing a triangulation is to embed $M$ in $\mathbb{R}^n$, construct a fine net of points in $M$, construct the Delaunay triangulation of those points, then project back to the manifold to get a triangulation. But this isn't very quantitative -- it depends on the embedding, and even if the embedding is nice, an unlucky choice of points will lead to some bad simplices. Is there a better way?

• Robert: There are two references which come close to what you want, but neither one does it. One is in Epstein et al book on word processing in groups (but only a sketch) and another is W. Breslin "Thick triangulations of hyperbolic n-manifolds". With extra work Breslin's arguments do generalize to the variable curvature case. Lastly, for most purposes, instead of a triangulation one can use "map to nerve", which is explained with all details in K. Grove and P. Petersen, Bounding homotopy types by geometry, Annals (1988). – Misha Jan 22 '16 at 0:02
• One more sketch is in O. Attie, Quasi-isometry classification of some manifolds of bounded geometry, Math. Z. 216 (1994), 501–527. – Misha Jan 22 '16 at 0:03
• Thank you for the references! I think I can see how to complete them. Usually, I would use a map to a nerve, but I ran into a problem where a true partition was necessary. – Robert Young Jan 22 '16 at 2:41
• Robert: Let me know if and when you have a complete proof. In the GGT book we currently list this as an open problem, but with a reasonable path to a solution. – Misha Jan 22 '16 at 18:54
• Could the almost-linear co-ordinates introduced by Jost and Karcher provide a means to proving this? You can find an English explanation of these co-ordinates in the following lecture notes: Jost, Jürgen Harmonic mappings between Riemannian manifolds. Proceedings of the Centre for Mathematical Analysis, Australian National University, 4. (I might be able to find my copy of it, if you think this would be useful) – Deane Yang Jan 22 '16 at 19:55

• If $(M^m,g)$ is of bounded geometry (in the $C^\infty$-sense, say), then there exists $\epsilon_0 > 0$ such that for that for any $0<\epsilon <\epsilon_)$ there is a countable cover of $M$ by geodesic balls $B_{\epsilon}(x_j)$, $\bigcup B_{\epsilon}(x_j) = M$, such that the cover of $M$ by the balls $B_{2\epsilon}(x_j)$ with double radius and the same centers is still uniformly locally finite.
Now choose points $y_k$ such that in each non-empty intersection of $B_{\epsilon}(x_j)$ they span an $m$-simplex, which is diffeomorphic to the standard $m$-simplex by any of the relevant exponential maps $\exp_{x_j}$. Then the maps you are looking for are just Riemannian exponential mappings which satisfy your assumptions by the properties of bounded geometry.
• This is good, but I think there are some constraints that need to be dealt with. For instance, if $m+1$ points are nearly coplanar in one chart, then in another chart, the orientation of that simplex could flip. So you need some bounds to make sure that the triangulation stays a triangulation between overlapping patches. I think this is roughly what Breslin does in his paper; he shows that, if you take small enough patches, then there is some set of points with well-defined Delaunay triangulation and limited "badness". – Robert Young Jan 22 '16 at 16:46


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Definition:Limit of Real Function/Intuition

Definition

Though the founders of Calculus viewed the limit:

$\ds \lim_{x \mathop \to c} \map f x$

as the behavior of $f$ as it gets infinitely close to $x = c$, the real number system as defined in modern mathematics does not allow for the existence of infinitely small distances.

But:

$\exists L: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$

can be interpreted this way:

You want to get very close to the value $c$ on the $\map f x$ axis.
This degree of closeness is the positive real number $\epsilon$.
If the limit exists, I can guarantee you that I can give you a value on the $x$ axis that will satisfy your request.
This value on the $x$ axis is the positive real number $\delta$.


text: - Strong duality in conic linear programming: facial reduction and extended duals Gabor Pataki(gaborunc.edu) Abstract: The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program (P) \sup { | Ax \leq_K b} in the absence of any constraint qualification. The facial reduction algorithm solves a sequence of auxiliary optimization problems to obtain such a dual. Ramana's dual is applicable when (P) is a semidefinite program (SDP) and is an explicit SDP itself. Ramana, Tuncel, and Wolkowicz showed that these approaches are closely related; in particular, they proved the correctness of Ramana's dual using certificates from a facial reduction algorithm. Here we give a clear and self-contained exposition of facial reduction, of extended duals, and generalize Ramana's dual: -- we state a simple facial reduction algorithm and prove its correctness; and -- building on this algorithm we construct a family of extended duals when $K$ is a {\em nice} cone. This class of cones includes the semidefinite cone and other important cones. Keywords: Conic linear programming; minimal cone; semidefinite progr amming; facial reduction; extended duals; nice cones Category 1: Linear, Cone and Semidefinite Programming Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Category 3: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Download: [PDF]Entry Submitted: 02/13/2013Entry Accepted: 02/13/2013Entry Last Modified: 02/13/2013Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.


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# MA3G7 Functional Analysis I

Lecturer: Professor Vassili Gelfreich

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 lectures

Assessment: 3 hour examination

Prerequisites: MA244 Analysis III (or MA258 Mathematical Analysis III); MA259 Multivariable Calculus and MA260 Norms, Metrics and Topoology or MA222 Metric Spaces would be useful but not essential; MA359 Measure Theory would be a natural course to take in parallel.

Content: This is essentially a module about infinite-dimensional Hilbert spaces, which arise naturally in many areas of applied mathematics. The ideas presented here allow for a rigorous understanding of Fourier series and more generally the theory of Sturm-Liouville boundary value problems. They also form the cornerstone of the modern theory of partial differential equations.

Hilbert spaces retain many of the familiar properties of finite-dimensional Euclidean spaces ( $\mathbb{R}^n$) - in particular the inner product and the derived notions of length and distance - while requiring an infinite number of basis elements. The fact that the spaces are infinite-dimensional introduces new possibilities, and much of the theory is devoted to reasserting control over these under suitable conditions.

The module falls, roughly, into three parts. In the first we will introduce Hilbert spaces via a number of canonical examples, and investigate the geometric parallels with Euclidean spaces (inner product, expansion in terms of basis elements, etc.). We will then consider various different notions of convergence in a Hilbert space, which although equivalent in finite-dimensional spaces differ in this context. Finally we consider properties of linear operators between Hilbert spaces (corresponding to the theory of matrices between finite-dimensional spaces), in particular recovering for a special class of such operators (compact self-adjoint operators) very similar results to those available in the finite-dimensional setting.

Throughout the abstract theory will be motivated and illustrated by more concrete examples.

Books: A useful book to use as an accompanying reference is:
BP Rynne & MA Youngson, Linear Functional Analysis, Springer-Verlag, London, 2000.

Archived Pages: 2011 2013 2014 2015 2016 2017 2018

Year 1 regs and modules
G100 G103 GL11 G1NC

Year 2 regs and modules
G100 G103 GL11 G1NC

Year 3 regs and modules
G100 G103

Year 4 regs and modules
G103

Past Exams
Core module averages


text:
CBSEtips.in

Thursday, 28 January 2021

CBSE Class 10 Maths - MCQ and Online Tests - Unit 2 - Polynomials

CBSE Class 10 Maths – MCQ and Online Tests – Unit 2 – Polynomials

Every year CBSE conducts board exams for 10th standard. These exams are very competitive to all the students. So our website provides online tests for all the 10th subjects. These tests are also very effective and useful for those who preparing for competitive exams like NEET, JEE, CA etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.

These online tests are based on latest CBSE Class 10 syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps to revise the NCERT textbooks thoroughly.

CBSE Class 10 Maths – MCQ and Online Tests – Unit 2 – Polynomials

Question 1.
The maximum number of zeroes that a polynomial of degree 4 can have is
(a) One
(b) Two
(c) Three
(d) Four

Question 2.
The graph of the polynomial p(x) = 3x – 2 is a straight line which intersects the x-axis at exactly one point namely
(a) ($$\frac{-2}{3}$$, 0)
(b) (0, $$\frac{-2}{3}$$)
(c) ($$\frac{2}{3}$$, 0)
(d) $$\frac{2}{3}$$, $$\frac{-2}{3}$$

Answer: (c) ($$\frac{2}{3}$$, 0)

Question 3.
In fig. given below, the number of zeroes of the polynomial f(x) is

(a) 1
(b) 2
(c) 3
(d) None

Question 4.
The graph of the polynomial ax² + bx + c is an upward parabola if
(a) a > 0
(b) a < 0
(b) a = 0
(d) None

Question 5.
The graph of the polynomial ax² + bx + c is a downward parabola if
(a) a > 0
(b) a < 0
(c) a = 0
(d) a = 1

Question 6.
A polynomial of degree 3 is called
(a) a linear polynomial
(c) a cubic polynomial

Question 7.
If α, β are the zeroes of the polynomial x² – 16, then αβ(α + β) is
(a) 0
(b) 4
(c) -4
(d) 16

Question 8.
If α and $$\frac{1}{α}$$ are the zeroes of the polynomial ax² + bx + c, then value of c is
(a) 0
(b) a
(c) -a
(d) 1

Question 9.
Zeroes of the polynomial x² – 11 are
(a) ±$$\sqrt{17}$$
(b) ±$$\sqrt{3}$$
(c) 0
(d) None

Answer: (a) ±$$\sqrt{17}$$

Question 10.
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then α + β + γ is equal
(a) $$\frac{-b}{a}$$
(b) $$\frac{b}{a}$$
(c) $$\frac{c}{a}$$
(d) $$\frac{d}{a}$$

Answer: (a) $$\frac{-b}{a}$$

Question 11.
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then αβ + βγ + αγ is equal to
(a) $$\frac{-b}{a}$$
(b) $$\frac{b}{a}$$
(c) $$\frac{c}{a}$$
(d) $$\frac{d}{a}$$

Answer: (c) $$\frac{c}{a}$$

Question 12.
If the zeroes of the polynomial x³ – 3x² + x – 1 are $$\frac{s}{t}$$, s and st then value of s is
(a) 1
(b) -1
(c) 2
(d) -3

Question 13.
If the sum of the zeroes of the polynomial f(x) = 2x³ – 3kx² + 4x – 5 is 6, then the value of k is
(a) 2
(b) 4
(c) -2
(d) -4

Question 14.
If a polynomial of degree 4 is divided by quadratic polynomial, the degree of the remainder is
(a) ≤ 1
(b) ≥ 1
(c) 2
(d) 4

Question 15.
If a – b, a and a + b are zeroes of the polynomial fix) = 2x³ – 6x² + 5x – 7, then value of a is
(a) 1
(b) 2
(c) -5
(d) 7

Question 16.
Dividend is equal to
(a) divisor × quotient + remainder
(b) divisior × quotient
(c) divisior × quotient – remainder
(d) divisor × quotient × remainder

Answer: (a) divisor × quotient + remainder

Question 17.
A quadratic polynomial whose sum of the zeroes is 2 and product is 1 is given by
(a) x² – 2x + 1
(b) x² + 2x + 1
(c) x² + 2x – 1
(d) x² – 2x – 1

Answer: (a) x² – 2x + 1

Question 18.
If one of the zeroes of a quadratic polynomial ax² + bx + c is 0, then the other zero is
(a) $$\frac{-b}{a}$$
(b) 0
(c) $$\frac{b}{a}$$
(d) $$\frac{-c}{a}$$

Answer: (a) $$\frac{-b}{a}$$

Question 19.
The sum and the product of the zeroes of polynomial 6x² – 5 respectively are
(a) 0, $$\frac{-6}{5}$$
(b) 0, $$\frac{6}{5}$$
(c) 0, $$\frac{5}{6}$$
(d) 0, $$\frac{-5}{6}$$

Answer: (d) 0, $$\frac{-5}{6}$$

Question 20.
What should be subtracted from x³ – 2x² + 4x + 1 to get 1?
(a) x³ – 2x² + 4x
(b) x³ – 2x² + 4 + 1
(c) -1
(d) 1

Answer: (a) x³ – 2x² + 4x

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text: The font itself defines if it's monospaced or variable width, so no, you cannot select that on Writer or any other software. There are fonts that offers monospaced numerals as an OpenType feature (the +tnum tag), but nothing else. You may try switching to a good monospaced font like Source Code Pro, DejaVu Sans Mono, Fira Code (if you like fancy ligatures on a monospaced font), etc.


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0

IN THIS ISSUE

Research Papers

J. Mechanisms Robotics. 2017;9(3):031001-031001-13. doi:10.1115/1.4035797.

In this work, the kinematics of a large size tunnel digging machine is investigated. The closed-loop mechanism is made by 13 links and 13 class 1 couplings, seven of which are actuated. This kind of machines are commonly used to perform ground drilling for the placement of reinforcement elements during the construction of tunnels. The direct kinematic solution is obtained using three methods: the first two are based on the numerical solution of the closure equation written using the Denavit–Hartenberg convention, while the third is based on the definition and solution in closed form of an equivalent spherical mechanism. The procedures have been tested and implemented with reference to a real commercial tunnel digging machine. The use of the proposed method for the closed-form solution of direct kinematics allows to obtain a major reduction of the computation time in comparison with the standard numerical solution of the closure equation.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031002-031002-8. doi:10.1115/1.4035985.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031003-031003-13. doi:10.1115/1.4035798.

This paper presents a general algorithm for solving the dynamic of tree structure robots with rigid and flexible links, active and passive joints, and with a fixed or floating base. The algorithm encompasses in a unified approach both the inverse and direct dynamics. It addresses also the hybrid case where each active joint is considered with known joint torque as in the direct dynamic case, or with known joint acceleration as in the inverse dynamic case. To achieve this goal, we propose an efficient recursive approach based on the generalized Newton–Euler equations of flexible tree-structure systems. This new general hybrid algorithm is easy to program either numerically or using efficient customized symbolic techniques. It is of great interest for studying floating base systems with soft appendages as those currently investigated in soft bio-inspired robotics or when a robotic system has to modify its structure for some particular tasks, such as transforming an active joint into a compliant flexible one, or modifying a task with a floating base into one with fixed base.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031004-031004-11. doi:10.1115/1.4035681.

Adaptive cable-driven parallel robots are a special subclass of cable-driven systems in which the locations of the pulley blocks are modified as a function of the end-effector pose to obtain optimal values of given performance indices within a target workspace. Due to their augmented kinematic redundancy, such systems enable larger workspace volume and higher performance compared to traditional designs featuring the same number of cables. Previous studies have introduced a systematic method to optimize design and trajectory planning of the moving pulley-blocks for a given performance index. In this paper, we study the motions of the pulley blocks that optimize two performance indices simultaneously: stiffness and dexterity. Specifically, we present a method to determine the pulley blocks motions that guarantee ideal dexterity with the best feasible elastic stiffness, as well as those that guarantee isotropic elastic stiffness with the best feasible dexterity. We demonstrate the proposed approach on some practical cases of planar adaptive cable-driven parallel robots.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031005-031005-8. doi:10.1115/1.4035986.

The pseudo-rigid-body model (PRBM) used to simulate compliant beams without inflection point had been well developed. In this paper, two types of PRBMs are proposed to simulate the large deflection of flexible beam with an inflection point in different configurations. These models are composed of five rigid links connected by three joints added with torsional springs and one hinge without spring representing the inflection point in the flexural beam. The characteristic radius factors of the PRBMs are determined by solving the objective function established according to the relative angular displacement of the two rigid links jointed by the hinge via genetic algorithm. The spring stiffness coefficients are obtained using a linear regression technique. The effective ranges of these two models are determined by the load index. The numerical result shows that both the tip locus and inflection point of the flexural beam with single inflection can be precisely simulated using the model proposed in this paper.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031006-031006-10. doi:10.1115/1.4035990.

Radiotherapy (RT) enables a selective destruction of tumor cells, although the treatment area is limited to the irradiated volume. Any RT technique comes along with multiple sources of error, which can lead to a deviation of the dose that is applied to the patient. Phantoms—structures that replicate a human and include measurement technology to assess the applied dosage—are used to make such errors observable. Past RT-technologies assumed static tumors. Correspondingly, most existing phantoms comprise only static components. Nowadays, RT is at a transition stage toward techniques which explicitly account for physiological motion. These techniques require phantoms generating such motion. Consequentially, a demand for new kinds of manipulators, which operate with a RT-phantom, has come up and will further increase in the future. Key demands of such manipulators are among others, the generation of full rigid body motion, high acceleration, high stiffness, compactness, little weight, and easy portability. Another indispensable feature is the spatial separation of mechatronic components and phantom structure to ensure human equivalency of the latter. In this work, a new kind of parallel kinematic manipulator (PKM), which is tailored to the requirements of RT-phantom technology, is presented. The PKM consists of low cost standardized mechanical components and sets the target structures, which are located inside a human-equivalent area, into translational and rotational motion in three degrees-of-freedom (DOFs). Only a part of the end-effector is located within the human-equivalent area. All the remaining parts of the PKM are located outside that area. Two versions of the manipulator are presented in detail: their kinematics are derived and their kinetostatic properties are compared. This includes a workspace analysis and the analysis of the transmission behavior in general, meaning the influence of the most important design parameters on the performance. It can be shown that practical differences of both kinematics are negligible, while the modified version provides significant mechanical advantages. In conclusion, a first special purpose manipulator for application in the evolving field of RT-phantom technology is presented. The PKM, which employs a novel kinematic structure, provides higher suitability for its purpose than any other robotic system employed so far for the same purpose.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031007-031007-10. doi:10.1115/1.4035542.

In this paper, we examine two spherical parallel manipulators (SPMs) constructed with legs that include planar and spherical subchains that combine to impose constraints equivalent to hidden revolute joints. The first has supporting serial chain legs constructed from three revolute joints with parallel axes, denoted R$∥$R$∥$R, followed by two revolute joints that have intersecting axes, denoted $RR̂$. The leg has five degrees-of-freedom and is denoted R$∥$R$∥$R-$RR̂$. Three of these legs can be assembled so the spherical chains all share the same point of intersection to obtain a spherical parallel manipulator denoted as 3-R$∥$R$∥$R-$RR̂$. The second spherical parallel manipulator has legs constructed from three revolute joints that share one point of intersection, denoted $RRR̂$, and a second pair of revolute joints with axes that intersect in a different point. This five-degree-of-freedom leg is denoted $RRR̂$-$RR̂$. The spherical parallel manipulator constructed from these legs is 3-$RRR̂$-$RR̂$. We show that the internal constraints of these two types of legs combine to create hidden revolute joints that can be used to analyze the kinematics and singularities of these spherical parallel manipulators. A quaternion formulation provides equations for the quartic singularity varieties some of which decompose into pairs of quadric surfaces which we use to classify these spherical parallel manipulators.

Topics: Manipulators , Chain
Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031008-031008-17. doi:10.1115/1.4035682.

Kinematic chains are obtained from the helicoid–helicoid intersections applying the method of surfaces generated by kinematic dyads. Some local properties of the helicoids are used to obtain the bifurcation points in the configuration space of the obtained kinematic chains. It is proven that certain relationships between the two helicoids lead to a periodic behavior of these bifurcations, which suggest that, if the kinematic pairs (P and H) could move without a limit, the kinematic chain would theoretically feature an infinity of operation modes. Finally, a mechanism which is able to change the helicoid–helicoid intersection curve during its motion is proven to change its finite mobility in one of its operation modes.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031009-031009-7. doi:10.1115/1.4035881.

Recently, there has been an increased interest in origami art from a mechanism design perspective. The deployable nature and the planar fabrication method inherent to origami provide potential for space and cost-efficient mechanisms. In this paper, a novel type of origami mechanisms is proposed in which the compliance of the facets is used to incorporate the spring behavior: compliant facet origami mechanisms (COFOMs). A simple model that computes the moment characteristic of a single vertex COFOM has been proposed, using a semispatial version of the pseudo-rigid-body (PRB) theory to model bending of the facets. The PRB model has been evaluated numerically and experimentally, showing good performance. The PRB model is a potential starting point for a design tool which would provide an intuitive way of designing this type of mechanisms including their spring behavior, with very low computational cost.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031010-031010-10. doi:10.1115/1.4035373.

This paper studies the collision problem of a robot manipulator and presents a method to minimize the impact force by pre-impact configuration designing. First, a general dynamic model of a robot manipulator capturing a target is established by spatial operator algebra (SOA) and a simple analytical formula of the impact force is obtained. Compared with former models proposed in literatures, this model has simpler form, wider range of applications, O(n) computation complexity, and the system Jacobian matrix can be provided as a production of the configuration matrix and the joint matrix. Second, this work utilizes the impulse ellipsoid to analyze the influence of the pre-impact configuration and the impact direction on the impact force. To illustrate the inertia message of each body in the joint space, a new concept of inertia quasi-ellipsoid (IQE) is introduced. We find that the impulse ellipsoid is constituted of the inertia ellipsoids of the robot manipulator and the target, while each inertia ellipsoid is composed of a series of inertia quasi-ellipsoids. When all inertia quasi-ellipsoids exhibit maximum (minimum) coupling, the impulse ellipsoid should be the flattest (roundest). Finally, this paper provides the analytical expression of the impulse ellipsoid, and the eigenvalues and eigenvectors are used as measurements to illustrate the size and direction of the impulse ellipsoid. With this measurement, the desired pre-impact configuration and the impact direction with minimum impact force can be easily solved. The validity and efficiency of this method are verified by a PUMA robot and a spatial robot.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031011-031011-16. doi:10.1115/1.4035988.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031012-031012-9. doi:10.1115/1.4035989.

Remote center-of-motion (RCM) parallel manipulators (PMs) are fit for robotized minimally invasive surgery (MIS). RCM PMs with fixed linear actuators have the advantages of high stiffness, reduced moving mass, and higher rigidity and load capacity. However, there are very few available architectures of these types of PMs. Using the Lie group algebraic properties of the set of rigid-body displacements, this paper proposes a new family of RCM PMs with fixed linear actuators for MIS. The general motion with a remote center has four degrees-of-freedom (DOF) and is produced by the in-series concatenation of a spherical S pair and a prismatic P pair and, therefore, is said to be SP equivalent. The SP-equivalent PMs can be used in minimally invasive surgery. First, the kinematic bonds of limb chains and their mechanical generators for SP-equivalent RCM PMs are presented. Limb chains with fixed linear actuators are then derived using the closure of products in subgroups. Structural conditions for constructing an SP-equivalent RCM PM with linear fixed actuators are revealed. Helical pairs are introduced to remove a local rotation and yield a 360-deg-rotation capability of the moving platform. Numerous new architectures with practical potential are presented.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031013-031013-9. doi:10.1115/1.4035991.

Minimally invasive surgery (MIS) requires four degrees-of-freedom (DOFs) (pitch, translation, yaw, and roll) at the incision point, but the widely used planar remote center of motion (RCM) mechanisms only provide one degree-of-freedom. The remaining three DOFs are achieved through external means (such as cable-pulleys or actuators mounted directly on the distal-end) which adversely affect the performance and design complexity of a surgical manipulator. This paper presents a new RCM mechanism which provides the two most important DOFs (pitch and translation) by virtue of its mechanical design. Kinematics of the new mechanism is developed and its singularities are analyzed. To achieve maximum performance in the desired workspace region, an optimal configuration is also evaluated. The design is optimized to yield maximum manipulability and tool translation with smallest size of the mechanism. Unlike the traditional planar RCM mechanisms, the proposed design does not rely on external means to achieve translation DOF, and therefore, offers potential advantages. The mechanism can be a suitable choice for surgical applications demanding a compact distal-end or requiring multiple manipulators to operate in close proximity.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031014-031014-10. doi:10.1115/1.4035802.

This paper presents a novel passive redundant spherical joint with a very large range of motion. A kinematic model is first developed in order to provide a framework for the analysis. The principle of the redundant joint is then introduced. The proposed joint does not require any active component since the redundancy is passively handled using springs. A static model of the joint is developed in order to clearly demonstrate how all singularities or jamming configurations can be avoided. Two possible arrangements are presented, one using linear springs and one using a torsional spring. Finally, experimental prototypes are demonstrated that can attain a range of tilt angle of ±150 deg.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031015-031015-5. doi:10.1115/1.4035532.

A configuration of a mechanical linkage is defined as regular if there exists a subset of actuators with their corresponding Jacobian columns spans the gripper's velocity space. All other configurations are defined in the literature as singular configurations. Consider mechanisms with grippers' velocity space $ℝm$. We focus our attention on the case where m Jacobian columns of such mechanism span $ℝm$, while all the rest are linearly dependent. These are obviously an undesirable configuration, although formally they are defined as regular. We define an optimal-regular configuration as such that any subset of m actuators spans an m-dimensional velocity space. Since this densely constraints the work space, a more relaxed definition is needed. We therefore introduce the notion of k-singularity of a redundant mechanism which means that rigidifying k actuators will result in an optimal-regularity. We introduce an efficient algorithm to detect a k-singularity, give some examples for cases where m = 2, 3, and demonstrate our algorithm efficiency.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031016-031016-12. doi:10.1115/1.4035987.

Torque control of small-scale robotic devices such as hand exoskeletons is challenging due to the unavailability of miniature and compact bidirectional torque actuators. In this work, we present a miniature Bowden-cable-based series elastic actuator (SEA) using helical torsion springs. The three-dimensional (3D) printed SEA is 38 mm × 38 mm × 24 mm in dimension and weighs 30 g, excluding motor which is located remotely. We carry out a thorough experimental testing of our previously presented linear compression spring SEA (LC-SEA) (Agarwal et al. 2015, “An Index Finger Exoskeleton With Series Elastic Actuation for Rehabilitation: Design, Control and Performance Characterization,” Int. J. Rob. Res., 34(14), pp. 1747–1772) and helical torsion spring SEA (HT-SEA) and compare the performance of the two designs. Performance characterization on a test rig shows that the two SEAs have adequate torque source quality (RMSE < 12% of peak torque) with high torque fidelities (>97% at 0.5 Hz torque sinusoid) and force tracking bandwidths of 2.5 Hz and 4.5 Hz (0.2 N·m), respectively, which make these SEAs suitable for our application of a hand exoskeleton.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031017-031017-13. doi:10.1115/1.4035800.
FREE TO VIEW

In recent years, applications in industrial assemblies within a size range from 0.5 mm to 100 mm are increasing due to the large demands for new products, especially those associated with digital multimedia. Research on grippers or robotic hands within the mesoscopic scale of this range has not been explored in any great detail. This paper outlines the development of a gripper to bridge the gap between microgrippers and macrogrippers by extending the gripping range to the mesoscopic scale, particularly without the need to switch grippers during industrial assembly. The mesoscopic scale gripper (meso-gripper) researched in this work has two modes of operation: passive adjusting mode and angled gripping mode, adapting its configuration to the shape of object automatically. This form of gripping and the associated mechanism are both novel in their implementation and operation. First, the concept of mesoscopic scale in robotic gripping is presented and contextualized around the background of inefficient hand switching processes and applications for assembly. The passive adjusting and angled gripping modes are then analyzed and a dual functional mechanism design proposed. A geometric constraint method is then demonstrated which facilitates task-based dimensional synthesis after which the kinematics of synthesized mechanism is investigated. The modified synthesized mechanism gripper is then investigated according to stiffness and layout. Finally, a 3D printed prototype is successfully tested, and the two integrated gripping modes for universal gripping verified.

Commentary by Dr. Valentin Fuster
J. Mechanisms Robotics. 2017;9(3):031018-031018-12. doi:10.1115/1.4035993.

This paper introduces a general method for analyzing flexure systems of any configuration, including those that cannot be broken into parallel and serial subsystems. Such flexure systems are called interconnected hybrid flexure systems because they possess limbs with intermediate bodies that are connected by flexure systems or elements. Specifically, the method introduced utilizes screw algebra and graph theory to help designers determine the freedom spaces (i.e., the geometric shapes that represent all the ways a body is permitted to move) for all the bodies joined together by compliant flexure elements within interconnected hybrid flexure systems (i.e., perform mobility analysis of general flexure systems). This method also allows designers to determine (i) whether such systems are under-constrained or not and (ii) whether such systems are over-constrained or exactly constrained (i.e., perform constraint analysis of general flexure systems). Although many flexure-based precision motion stages, compliant mechanisms, and microarchitectured materials possess topologies that are highly interconnected, the theory for performing the mobility and constraint analysis of such interconnected flexure systems using traditional screw theory does not currently exist. The theory introduced here lays the foundation for an automated tool that can rapidly generate the freedom spaces of every rigid body within a general flexure system without having to perform traditional computationally expensive finite element analysis. Case studies are provided to demonstrate the utility of the proposed theory.

Commentary by Dr. Valentin Fuster


text:
+0

HELP QUICK!

+1
96
1
+108

One of the roots of 2x^3-9x^2+13x+k=0 is x=2.Find the other 2 roots. Explain how you got the answer.

Feb 17, 2021

#1
+11086
+5

One of the roots of 2x^3-9x^2+13x+k=0 is $$x_1=2$$.Find the other 2 roots. Explain how you got the answer.

Hello whatdoiputhere!

$$(2x ^ 3-9x ^ 2 + 13x + k)$$  $$:(x-2)=$$  $$2x^2-5x+3$$

$$\underline{2x^3-4x^2}$$

$$-5x^2+13x$$

$$\underline{-5x^2+10x}$$

$$3x+k$$

$$\underline{3x-6}$$

$$0$$

$$k=-6$$

$$2x^2-5x+3=0\\ x^2-2.5x+1.5=0\\ x=1.25\pm \sqrt{1.25^2-1.5}\\ \color{blue}x=1.25\pm0.25$$

$$x\in\{1,1.5,2\}$$

!

Feb 17, 2021
edited by asinus  Feb 17, 2021

#1
+11086
+5

One of the roots of 2x^3-9x^2+13x+k=0 is $$x_1=2$$.Find the other 2 roots. Explain how you got the answer.

Hello whatdoiputhere!

$$(2x ^ 3-9x ^ 2 + 13x + k)$$  $$:(x-2)=$$  $$2x^2-5x+3$$

$$\underline{2x^3-4x^2}$$

$$-5x^2+13x$$

$$\underline{-5x^2+10x}$$

$$3x+k$$

$$\underline{3x-6}$$

$$0$$

$$k=-6$$

$$2x^2-5x+3=0\\ x^2-2.5x+1.5=0\\ x=1.25\pm \sqrt{1.25^2-1.5}\\ \color{blue}x=1.25\pm0.25$$

$$x\in\{1,1.5,2\}$$

!

asinus Feb 17, 2021
edited by asinus  Feb 17, 2021


text:
# Help needed in task involving structures and models

The task is as follows:
Let $\Sigma$ be a set of closed formulas such that for any closed formula $\phi$, either $\Sigma\models\phi$ or $\Sigma \models \neg\phi$.
Now let structure $A$ be a model of the set $\Sigma$. Show that for any closed formula $\psi$,

$$A\models \psi \iff \Sigma \models \psi$$

(Closed formula means here a formula such that it's variables are bound with quantifier)

I know that since structure $A$ is model of a $\Sigma=\{\phi_1,\phi_2\dots\}$, it follows that $\Sigma\models\psi$, since all formulas are sharing the same model. Not sure whether this is correct or formal enough.

Also don't know how to prove it to the other direction.

Any tips?

For the first part, we have to use the definition of $Σ \vDash \psi$:
$\text { for every structure } A, \text { if } A \vDash \phi_i, \text { for every } \phi_i \in \Sigma, \text { then } A \vDash \psi$.
Thus, we have that $\Sigma \vDash \psi$ implies $A \vDash \psi$, for a structure $A$ that is a model of $\Sigma$.
For the other part: assume $Σ \nvDash \psi$.
This means $Σ⊨¬ψ$, by property of $Σ$, and this implies that in every structure $A$ that is a model of $Σ$ we have that: $A⊨¬ψ$, contradicting the fact that $A \vDash \psi$.


text:
# boot.crq

0th

Percentile

##### Bootstrapping Censored Quantile Regression

Functions used to estimated standard errors, confidence intervals and tests of hypotheses for censored quantile regression models using the Portnoy and Peng-Huang methods.

Keywords
regression
##### Usage
boot.crq(x, y, c, taus, method, ctype = "right", R = 100, mboot, bmethod = "jack", ...)
##### Arguments
x

The regression design matrix

y

The regression response vector

c

The censoring indicator

taus

The quantiles of interest

method

The fitting method: either "P" for Portnoy or "PH" for Peng and Huang.

ctype

Either "right" or "left"

R

The number of bootstrap replications

bmethod

The bootstrap method to be employed. There are (as yet) three options: method = "jack" uses the delete-d jackknife method described by Portnoy (2013), method = "xy-pair" uses the xy-pair method, that is the usual multinomial resampling of xy-pairs, while method = "Bose" uses the Bose and Chatterjee (2003) weighted resampling method with exponential weights. The "jack" method is now the default.

mboot

optional argument for the bootstrap method: for bmethod = "jack" it specifies the number, d, of the delete-d jackknife, for method = "xy-pair" it specifies the size of the bootstrap samples, that permits subsampling (m out of n) bootstrap. By default in the former case it is set to 2 [sqrt(n)], for the latter the default is n. Obviously mboot should be substantially larger than the column dimension of x, and should be less than the sample size in both cases.

...

Optional further arguments to control bootstrapping

##### Details

There are several refinements that are still unimplemented. Percentile methods should be incorporated, and extensions of the methods to be used in anova.rq should be made. Note that bootstrapping for the Powell method "Powell" is done via boot.rq. For problems with n > 3000 a message is printed indicated progress in the resampling.

##### Value

A matrix of dimension R by p is returned with the R resampled estimates of the vector of quantile regression parameters. When mofn < n for the "xy" method this matrix has been deflated by the factor sqrt(m/n)

##### References

Bose, A. and S. Chatterjee, (2003) Generalized bootstrap for estimators of minimizers of convex functions, J. Stat. Planning and Inf, 117, 225-239. Portnoy, S. (2013) The Jackknife's Edge: Inference for Censored Quantile Regression, CSDA, forthcoming.

summary.crq


text:
# Tag Info

In the context of multiple linear regression, "adjusting" for a covariate simply means including it as an explanatory variable. There is an equivalent way of understanding multiple linear regression that provides insight into this question. To regress, say, $Z$ on $X$ and $Y$, we may (arbitrarily) select one of the explanatory variables (let it be $X$) and ...


text:
# Equalize histogram

Equalizes a histogram by making the gray level distribution as uniform as possible. An image can be equalized by connecting the image and cumulativeDistribution output of this tool to map gray levels.

## Inputs

histogram
Histogram vector of size 1-by-N or N-by-1.

## Outputs

histogram
Equalized histogram. The sum of all entries equals the sum of entries of the input histogram.
cumulativeDistribution
Cumulative distribution of the histogram output.


text:
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# Algebra + Exponents cheat sheet !

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Algebra + Exponents cheat sheet !  [#permalink]

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08 Oct 2018, 19:20
7

Algebra

For a quadratic equation $$ax^2 +bx +c =0$$

a= coefficient of $$x^2$$
b= coefficient of x
c= constant value of the equation

x = $$\frac{-b ∓ \sqrt{b^{2}- 4ac}}{2a}$$

Let the two roots of the equation be $$\alpha$$ & $$\beta$$ then ,

$$\alpha$$ + $$\beta$$ = $$\frac{-b}{a}$$

$$\alpha*\beta$$ = $$\frac{c}{a}$$

Exponents

$$0^{n}$$ = 0

$$1^{n}$$ = 1

$$x^{0}$$ = 1

$$x^{1}$$ = x

$$x^{-n}$$ = $$\frac{1}{x}$$

$$x^{m}*x^{n}$$ = $$x^{m+n}$$

$$\frac{x^{m}}{x^{n}}$$ = $$x^{m-n}$$

$$(x^{m})^{n}$$ = $$x^{m*n}$$

$$(\frac{x}{y})^{n}$$ = $$\frac{x^{n}}{y^{n}}$$
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11 Oct 2018, 04:50
chirayuvyas97

Thanks for sharing.
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11 Oct 2018, 21:02
Thanks!! Looks good, keep up the good work.

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Re: Algebra + Exponents cheat sheet !   [#permalink] 11 Oct 2018, 21:02
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text:
# rational expressions #5

• July 24th 2013, 03:26 PM
Mathnood768
rational expressions #5
http://puu.sh/3KWlL.jpg
i dont know what those trinomals factor into to become binomals
• July 24th 2013, 04:01 PM
BobP
Re: rational expressions #5
The first step has to be to combine those two terms on the LHs as a single term.
• July 24th 2013, 04:34 PM
Mathnood768
Re: rational expressions #5
no i need them as binomals but i dont know what they would be, what would x^2 + x + 2 be factored out as to be in a (x ... )(x .... ) form as well as x^2 - 1
• July 24th 2013, 04:48 PM
Plato
Re: rational expressions #5
Quote:

Originally Posted by Mathnood768
http://puu.sh/3KWlL.jpg
i dont know what those trinomals factor into to become binomals

@Mathnood768, you have no idea how any of this actually works.

If $x\ne\pm 1$ then multiply both sides by $x^2-1$.
We get:
$\\(x-1)(x^2+x+2)-x(x^2-1)=x^2-5\\x^3+x-2-x^3+x=x^2-5\\2x-2=x^2-5$


text:
# zbMATH — the first resource for mathematics

Dequantization, statistical mechanics and econophysics. (English) Zbl 1180.82064
Litvinov, G. L. (ed.) et al., Tropical and idempotent mathematics. International workshop TROPICAL-07, Moscow, Russia, August 25–30, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4782-4/pbk). Contemporary Mathematics 495, 239-279 (2009).
Summary: Using a rigorous statement and rigorous proof of the Maxwell distribution as an example, we establish estimates of the distribution depending on the parameter $$N$$, the number of particles. Further, we consider the problem of the occurrence of dimers in a classical gas as an analog of the Bose condensation and establish estimates of the lower level of the analog of the Bose condensation. Using dequantization principles, we find the relationship of this level to “capture” theory in the scattering problem corresponding to an interaction of the form of the Lennard-Jones potential. This also solves the problem of the Gibbs paradox.
We derive the equation of state for a non-ideal gas as a result of pair interactions of particles in Lennard Jones models and, for classical gases, discuss the $$\lambda$$ transition to the condensed state (the state in which $$V_{\text{sp}}$$ does not vary with increasing pressure; for heat capacity, this is the $$\lambda$$ point).
We use econophysics to explain the nature of a financial crisis.
For the entire collection see [Zbl 1172.00019].

##### MSC:
 82B30 Statistical thermodynamics 91B02 Fundamental topics (basic mathematics, methodology; applicable to economics in general) 81S99 General quantum mechanics and problems of quantization 82C22 Interacting particle systems in time-dependent statistical mechanics 60C05 Combinatorial probability 46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics 46N55 Applications of functional analysis in statistical physics 91B80 Applications of statistical and quantum mechanics to economics (econophysics)


text:
Searches for Supersymmetry using the MT2 Variable in Hadronic Events Produced in $pp$ Collisions at 8 TeV

CMS Collaboration; Khachatryan, V; Sirunyan, A M; Tumasyan, A; Amsler, C; Canelli, F; Chiochia, V; De Cosa, A; Hinzmann, A; Hreus, T; Kilminster, B; Lange, C; Mejias, B M; Ngadiuba, J; Pinna, D; Robmann, P; Ronga, F J; Taroni, S; Verzetti, M; Yang, Y; et al (2015). Searches for Supersymmetry using the MT2 Variable in Hadronic Events Produced in $pp$ Collisions at 8 TeV. Journal of High Energy Physics, 2015(5):78.

Abstract

Searches for supersymmetry (SUSY) are performed using a sample of hadronic events produced in 8 TeV pp collisions at the CERN LHC. The searches are based on the MT2 variable, which is a measure of the transverse momentum imbalance in an event. The data were collected with the CMS detector and correspond to an integrated luminosity of 19.5 fb−1. Two related searches are performed. The first is an inclusive search based on signal regions defined by the value of the MT2 variable, the hadronic energy in the event, the jet multiplicity, and the number of jets identified as originating from bottom quarks. The second is a search for a mass peak corresponding to a Higgs boson decaying to a bottom quark-antiquark pair, where the Higgs boson is produced as a decay product of a SUSY particle. For both searches, the principal backgrounds are evaluated with data control samples. No significant excess over the expected number of background events is observed, and exclusion limits on various SUSY models are derived.

Abstract

Searches for supersymmetry (SUSY) are performed using a sample of hadronic events produced in 8 TeV pp collisions at the CERN LHC. The searches are based on the MT2 variable, which is a measure of the transverse momentum imbalance in an event. The data were collected with the CMS detector and correspond to an integrated luminosity of 19.5 fb−1. Two related searches are performed. The first is an inclusive search based on signal regions defined by the value of the MT2 variable, the hadronic energy in the event, the jet multiplicity, and the number of jets identified as originating from bottom quarks. The second is a search for a mass peak corresponding to a Higgs boson decaying to a bottom quark-antiquark pair, where the Higgs boson is produced as a decay product of a SUSY particle. For both searches, the principal backgrounds are evaluated with data control samples. No significant excess over the expected number of background events is observed, and exclusion limits on various SUSY models are derived.

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Title:Dynamics of exponential maps

Author:Lasse Rempe

URN:NBN:urn:nbn:de:gbv:8-diss-7816

Faculty:Faculty of Mathematics and Natural Sciences
DDC:510 Mathematics

Date of thesis defense:2003-07-08

Referent:Prof. Dr. Walter Bergweiler
second reviewer(s) :Prof. Dr. Alexandre Eremenko, Prof. Dr. Dierk Schleicher

Description:This thesis contains several new results about the dynamics of exponential maps $z\mapsto \exp(z)+\kappa$. In particular, we prove that periodic external rays of exponential maps with nonescaping singular value always land. This is an analog of a theorem of Douady and Hubbard for polynomials. We also answer a question of Herman, Baker and Rippon by showing that the boundary of an unbounded exponential Siegel disk always contains the singular value. In addition to the presentation of new results, the thesis also aims to give an overview of the current state of knowledge on the dynamics of exponential maps.

Keywords:Exponentialabbildung , dynamical systems, dynamische Systeme, function theory, Funktionentheorie, holomorphic dynamics, holomorphe Dynamik, exponential maps, Exponentialabbildungen, Julia set, Juliamenge

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1887

## Frequency Transformations

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## Thank you

In this chapter we discuss the realization of low-pass (LP), high-pass (HP), bandpass (BP), and bandstop (BS) filters by applying suitable frequency transformations to the normalized LP filter. These transformations generally preserve the LP magnitude response (attenuation); other LP characteristics are often retained, however, especially in the case of the narrow-band BP filter. We also give the necessary equations (where possible) for denormalizing the LP responses, allowing them to characterize the appropriate filter type. When the transformation preserves the important attributes of the LP response, the tedious and time consuming approximation step in the design sequence is eliminated. This is the big advantage of the frequency-transformation approach. Furthermore, if the transformation function is of the same form as a reactance function, the filter element values are easily determined. Unfortunately certain LP characteristics are distorted by the transformation, and to avoid these distortions we either re-solve the approximation problem or use a different transformation. We present an example of each approach. The first results in the wide-band constant-delay filter and the second results in the low -transient HP filter. This chapter also includes narrow-band BP filter design, which is applicable not only to lumped-constant filters but also to transmission line and waveguide filters.

Chapter Contents:

• 4.1 Normalized Parameters
• 4.2 Low-Pass Filter
• 4.2.1 Transient Responses
• 4.2.2 Element Values
• 4.2.3 Example of Low-Pass Calculation
• 4.3 High-Pass Filter
• 4.3.1 Conventional Transformation
• 4.3.2 Transient Responses
• 4.3.3 Relationship Between LP and HP Transient Responses
• 4.3.4 Element Values
• 4.3.5 Example of High-Pass Calculation
• 4.3.6 Preservation of LP Transient Characteristics
• 4.3.7 Application of New High-Pass Transformation
• 4.4 Bandpass Filter
• 4.4.1 Conventional Transformation
• 4.4.2 Group Delay Behavior
• 4.4.3 Pole-Zero Locations
• 4.4.4 Element Values
• 4.4.5 Example of Bandpass Calculation
• 4.5 Narrow-Band Bandpass Filter
• 4.5.1 Basic Definitions
• 4.5.2 Example of Attenuation Calculation
• 4.5.3 Pole-Zero Locations
• 4.5.4 Group Delay Behavior
• 4.5.5 Example of Group Delay Calculation
• 4.5.6 Example of Wide-Band Group Delay Calculation
• 4.5.7 Transient Responses
• 4.5.8 Realizations with Nodal and Mesh Networks
• 4.5.9 Example of Network Design
• 4.5.10 Features of Nodal and Mesh Networks
• 4.6 Wide-Band Constant-Delay Bandpass Filter
• 4.6.1 Transfer Function and Least-Squares Approximation
• 4.6.2 Tables of Optimum Parameters
• 4.6.3 Normalized Attenuation and Group Delay Curves
• 4.6.4 Element Values
• 4.7 Bandstop Filter
• 4.7.1 Conventional Transformation
• 4.7.2 Example of Bandstop Calculation
• 4.7.3 Narrow-Band Transformation
• References
• Problems

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# Expressing Integers As A Difference Of Squares

Let $$f(n)$$ be defined as the number of ways a positive integer $$n$$ can be expressed as $$a^2-b^2$$, where $$a$$ and $$b$$ are positive integers.

Let $$g(n)$$ be defined as the smallest possible positive integer $$y$$ such that $$f(y)=n$$.

If $$M$$ is the least common multiple of $$g(1)$$, $$g(2)$$, $$g(3)$$, $$g(4)$$, and $$g(5)$$, then what is $$f(M)$$?

×


text:
## new power spectrum new parameters

Use of Healpix, camb, CLASS, cosmomc, compilers, etc.
moslem zarei
Posts: 28
Joined: March 11 2008
Affiliation: Isfahan University of Technology
Contact:

### new power spectrum new parameters

Dear all
I have calculated this new power spectrum:

$P=A_{s}\big(\frac{k}{k_{\star}}\big)^{n_{s}-1}(1+\sigma(\frac{k}{k_{\star}})^{-\varepsilon}\cos(2/\sigma_{0} (\frac{k}{k_{\star}})^{\varepsilon}))$

where $\sigma$ and $\varepsilon$ are the new parameters. I want to use cosmomc for fitting data. I hope someone would reply to that which files must be changed?
I have also read the previous posts but I want to be sure about my work.


text:
# The Three Gap Theorem: why should it be true?

Fix $\alpha\in\mathbb R$ and $N\in \mathbb N$, consider the set $S(\alpha,N)$ of $\{k\alpha\},k=1,\dots,N$, where $\{x\}$ denotes the fractional part of $x$. Let $a_1,\ldots a_N$ be the elements of $S(\alpha, N)$ arranged in increasing order, and consider the gaps sizes $$|a_{i+1}-a_i|, i=1,\dots,N-1.$$ The well-known Three Gap Theorem tells us that the number of distinct gap sizes is at most three for any $\alpha$ and $N$.

There are many proofs and generalisations of this theorem, but I haven't gained any intuition regarding the statement. In particular, is there any heuristic/intuitive reason as to why the theorem should be true?

• Do you have intuition why it should be true when $\alpha$ is very small? Imagine walking round the circle marking off successive multiples of $\alpha$. Apr 16, 2018 at 15:06
• Could you say more? I'm not asking for a proof of the statement—I know a couple off hand. I don't see how looping around in multiples of $\alpha$, let's say small and irrational, explains why the differences shoud be finite, and in fact exactly three. Apr 16, 2018 at 17:16
• there is an article in the March maa Monthly, pages 264-266, called A Footnote to the Three Gaps Theorem. Some references. Apr 16, 2018 at 17:39
• I agree with Ben Barber's implication: actually working through a specific example by hand—say with $\alpha=0.11$ or $\alpha=0.13$—will increase your intuition quite a bit. Apr 16, 2018 at 18:45
• One can view this theorem as an approximate version of the One Gap Theorem: in a subgroup of the unit circle, the number of distinct gap sizes is 1. From the perspective of modern additive combinatorics, an arithmetic progression such as $\{ \{k\alpha\}: k=1,\dots,N\}$ should be thought of as an approximate version of a subgroup of the unit circle (it is approximately closed under addition). May 25, 2021 at 16:15

I'm in something of the opposite situation to you. I'd never heard of the three distances/gaps theorem a week ago, since when I've come across it three times. I think the first person to introduce it to me was trying to convey how surprising it was, but as soon as I understood the statement I thought "isn't that obvious?". I'm now convinced that it isn't obvious, in the sense that I'm unable to make up a proof off the top of my head that I know will work, but perhaps my initial thought process will help your intuition.

Fix some moderately small $\alpha$. It's clear that things work well for the first few steps. In fact, we get by with only two distances until we're about to overtake the start point.

At the very next step we only create one new distance. We cut an interval in half, but we know by symmetry that one of the distances created will agree with the new distance we made at the previous step.

The next step cuts another interval, but into the same parts that we've just seen. This carries on for the whole of our second lap round the circle, until we're about to cross the start point again, at which point we would have to stop and think again about what is going to happen.

For other $\alpha$ the picture might look different, but (presumably) the same principles apply: we keep cutting old gaps in two, always eliminating an old distance before we need to use a new one.

Its all about modular arithmetic. Draw $$n$$ points $$P_0 = 0, P_1, \ldots, P_{n-1}$$, with the largest $$P_{max}$$, and smallest (non-zero) $$P_{min}$$, with $$min, max \in \{1, \dots, n-1\}$$. Now, the initial gap, $$a = P_{min} - P_0$$, is repeated for every point with at least $$min$$ successors, i.e. the first $$n - min$$ points. Similarly, the final gap, $$b = P_0 - P_{max}$$, is repeated for every point with at least $$max$$ predecessors, i.e. the final $$n - max$$ points.

There is also a middle group of $$min + max - n$$ points/gaps that get bisected into an $$a$$ and $$b$$ every time $$n$$ increments until $$n = min + max$$, and no more mid-points remain. The next time that $$n$$ increments, the new point $$P_{min + max} = a - b$$ bisects the larger of the initial $$a$$ and final $$b$$, and it all starts over.

There are usually three gaps, $$a$$, $$b$$ and $$a + b$$, that reduce to two, $$a$$ and $$b$$, every time $$n = min + max$$.


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# Thread: Find minimum AND-OR expression using a Karnaugh Map?

1. ## Find minimum AND-OR expression using a Karnaugh Map?

Use a Karnaugh Map to find the minimum AND-OR expression for x(a,b,c):

For part a the answer is: x(a, b, c) = ac + b'c'

How do I get that answer? I know from the problem that x=1 at rows 0, 4, 5, and 7 of the respective truth table (which is not given), however I thought x can only equal 1 if a, b, and c are all 1 by the definition of AND? I'm just confused how truth tables work for inputs I suppose.

2. ## Re: Find minimum AND-OR expression using a Karnaugh Map?

Originally Posted by lamentofking
I know from the problem that x=1 at rows 0, 4, 5, and 7 of the respective truth table (which is not given), however I thought x can only equal 1 if a, b, and c are all 1 by the definition of AND?
That would be true if x(a, b, c) = a AND b AND c, but nobody said that x has this form.

Any function with a finite number of inputs can be specified by explicitly listing outputs corresponding to those inputs. This is what a truth table does. The notation $\Sigma(0,4,5,7)$ specifies a function that equals 1 for the following values of inputs: $a=0,b=0,c=0$; $a=1,b=0,c=0$; $a=1,b=0,c=1$; $a=1,b=1,c=1$. The values of inputs occur in rows 0, 4, 5 and 7 of the truth table if rows are counted from 0. Note also that the values of inputs: 000, 100, 101 and 111, are 0, 4, 5 and 7 in binary. For other inputs x(a,b,c) = 0 by definition.

Describing how to use Karnaugh maps is beyond the scope of this post. You can read about it, for example, in Wikipedia. There is also an online Karnaugh map generator here.

3. ## Re: Find minimum AND-OR expression using a Karnaugh Map?

Originally Posted by emakarov
That would be true if x(a, b, c) = a AND b AND c, but nobody said that x has this form.

Any function with a finite number of inputs can be specified by explicitly listing outputs corresponding to those inputs. This is what a truth table does. The notation $\Sigma(0,4,5,7)$ specifies a function that equals 1 for the following values of inputs: $a=0,b=0,c=0$; $a=1,b=0,c=0$; $a=1,b=0,c=1$; $a=1,b=1,c=1$. The values of inputs occur in rows 0, 4, 5 and 7 of the truth table if rows are counted from 0. Note also that the values of inputs: 000, 100, 101 and 111, are 0, 4, 5 and 7 in binary. For other inputs x(a,b,c) = 0 by definition.

Describing how to use Karnaugh maps is beyond the scope of this post. You can read about it, for example, in Wikipedia. There is also an online Karnaugh map generator here.
Is there more than one answer in reducing the AND-OR expression? Because when I did it I got ac + b'c' + ab' . My adjacent terms were a'b'c' and ab'c', ab'c' and ab'c, ab'c and abc.

4. ## Re: Find minimum AND-OR expression using a Karnaugh Map?

Originally Posted by lamentofking
Is there more than one answer in reducing the AND-OR expression?
Yes, but usually one expression is shorter than the rest. I am not sure right now whether the minimal expression is unique. Also there are different ways to measure the size of an expression: total number of symbols, the number of logical operations, the number of variables and so on.

Originally Posted by lamentofking
Because when I did it I got ac + b'c' + ab'.
Note that having ab' does not add anything to the function. All four outputs of 1 are already given by ac + b'c'.

5. ## Re: Find minimum AND-OR expression using a Karnaugh Map?

Originally Posted by emakarov
Yes, but usually one expression is shorter than the rest. I am not sure right now whether the minimal expression is unique. Also there are different ways to measure the size of an expression: total number of symbols, the number of logical operations, the number of variables and so on.

Note that having ab' does not add anything to the function. All four outputs of 1 are already given by ac + b'c'.
Just to make sure this is a correct way of doing this...

The way I learned it was that after finding the adjacent terms I got this: ac + b’c’ + ab’ so in order to find minimal expression, I looked at the expression and crossed out each letter that showed up at least twice in the expression. Kind of like counting pairs. If a total pair (xy) has been crossed out then it won't be a part of the final expression. So in this example I start left to right and look at ac. Is 'a' repeated anywhere else in the expression? Yes in ab' so I cross all the a's out. Then I look at c. c is not repeated anywhere else so it stays. So now I know that the factor ac will be in the final expression. b' is repeated so I cross all of them out. I can stop at this point because ab' has been totally eliminated so the final expression will be ac + b'c'.

6. ## Re: Find minimum AND-OR expression using a Karnaugh Map?

I am not familiar with the algorithm you described, which starts with writing adjacent terms and proceeds by crossing out pairs. I just try to break the cells of the Karnaugh map marked with 1 into rectangular groups of $2^k$ cells (k = 1, 2, ...). The idea is to select rectangles as large as possible because they correspond to terms with fewer variables. Rectangles can overlap. Then for each rectangle I write the corresponding term.

7. ## Re: Find minimum AND-OR expression using a Karnaugh Map?

Originally Posted by emakarov
I am not familiar with the algorithm you described, which starts with writing adjacent terms and proceeds by crossing out pairs. I just try to break the cells of the Karnaugh map marked with 1 into rectangular groups of $2^k$ cells (k = 1, 2, ...). The idea is to select rectangles as large as possible because they correspond to terms with fewer variables. Rectangles can overlap. Then for each rectangle I write the corresponding term.
Well what I'm doing is effectively writing the inputs (which in this case are: a’b’c’ + ab’c’ + ab’c + abc) and grouping together the terms that differ by one term (eg. a'b'c' and ab'c') and then keeping the common factors (b'c') for my answer. That's how I got ac + b'c' + ab' . The algorithm was just one I stumbled on accidentally. It could be a coincidence that it works. Well lets see then: is the answer for 31b: a'b + bc ?


text:
Surface of Revolution
Get Surface of Revolution essential facts below. View Videos or join the Surface of Revolution discussion. Add Surface of Revolution to your PopFlock.com topic list for future reference or share this resource on social media.
Surface of Revolution
A portion of the curve x = 2 + cos z rotated around the z-axis

A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.[1]

Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus).

Properties

The sections of the surface of revolution made by planes through the axis are called meridional sections. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis.[2]

The sections of the surface of revolution made by planes that are perpendicular to the axis are circles.

Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular.

Area formula

If the curve is described by the parametric functions x(t), y(t), with t ranging over some interval [a,b], and the axis of revolution is the y-axis, then the area Ay is given by the integral

${\displaystyle A_{y}=2\pi \int _{a}^{b}x(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,}$

provided that x(t) is never negative between the endpoints a and b. This formula is the calculus equivalent of Pappus's centroid theorem.[3] The quantity

${\displaystyle {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}}$

comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity 2?x(t) is the path of (the centroid of) this small segment, as required by Pappus' theorem.

Likewise, when the axis of rotation is the x-axis and provided that y(t) is never negative, the area is given by[4]

${\displaystyle A_{x}=2\pi \int _{a}^{b}y(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt.}$

If the continuous curve is described by the function y = f(x), a x b, then the integral becomes

${\displaystyle A_{x}=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx=2\pi \int _{a}^{b}f(x){\sqrt {1+{\big (}f'(x){\big )}^{2}}}\,dx}$

for revolution around the x-axis, and

${\displaystyle A_{y}=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx}$

for revolution around the y-axis (provided a >= 0). These come from the above formula.[5]

For example, the spherical surface with unit radius is generated by the curve y(t) = sin(t), x(t) = cos(t), when t ranges over [0,?]. Its area is therefore

{\displaystyle {\begin{aligned}A&{}=2\pi \int _{0}^{\pi }\sin(t){\sqrt {{\big (}\cos(t){\big )}^{2}+{\big (}\sin(t){\big )}^{2}}}\,dt\\&{}=2\pi \int _{0}^{\pi }\sin(t)\,dt\\&{}=4\pi .\end{aligned}}}

For the case of the spherical curve with radius r, y(x) = rotated about the x-axis

{\displaystyle {\begin{aligned}A&{}=2\pi \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,dx\\&{}=2\pi r\int _{-r}^{r}\,{\sqrt {r^{2}-x^{2}}}\,{\sqrt {\frac {1}{r^{2}-x^{2}}}}\,dx\\&{}=2\pi r\int _{-r}^{r}\,dx\\&{}=4\pi r^{2}\,\end{aligned}}}

A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area.[6] A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.[6]

There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.[7]

Coordinate expressions

A surface of revolution given by rotating a curve described by ${\displaystyle y=f(x)}$ around the x-axis may be most simply described in cylindrical coordinates by ${\displaystyle r=f(z)}$. In Cartesian coordinates, this yields the parametrization in terms of ${\displaystyle z}$ and ${\displaystyle \theta }$ as ${\displaystyle (f(z)\cos(\theta ),f(z)\sin(\theta ),z)}$. If instead we revolve the curve around the y-axis, then the curve is described in cylindrical coordinates by ${\displaystyle z=f(r)}$, yielding the expression ${\displaystyle (r\cos(\theta ),r\sin(\theta ),f(r))}$ in terms of the parameters ${\displaystyle r}$ and ${\displaystyle \theta }$.

If x and y are defined in terms of a parameter ${\displaystyle t}$, then we obtain a parametrization in terms of ${\displaystyle t}$ and ${\displaystyle \theta }$. If ${\displaystyle x}$ and ${\displaystyle y}$ are functions of ${\displaystyle t}$, then the surface of revolution obtained by revolving the curve around the x-axis is described in cylindrical coordinates by the parametric equation ${\displaystyle (r,\theta ,z)=(y(t),\theta ,x(t))}$, and the surface of revolution obtained by revolving the curve around the y-axis is described by ${\displaystyle (r,\theta ,z)=(x(t),\theta ,y(t))}$. In Cartesian coordinates, these (respectively) become ${\displaystyle (y(t)\cos(\theta ),y(t)\sin(\theta ),x(t))}$ and ${\displaystyle (x(t)\cos(\theta ),x(t)\sin(\theta ),y(t))}$. The above formulae for surface area then follow by taking the surface integral of the constant function 1 over the surface using these parametrizations.

Geodesics

Meridians are always geodesics on a surface of revolution. Other geodesics are governed by Clairaut's relation.[8]

Toroids

A toroid generated from a square

A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid.[9] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus.

Applications

The use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed.

References

1. ^ Middlemiss; Marks; Smart. "15-4. Surfaces of Revolution". Analytic Geometry (3rd ed.). p. 378. LCCN 68015472.
2. ^ Wilson, W.A.; Tracey, J.I. (1925), Analytic Geometry (Revised ed.), D.C. Heath and Co., p. 227
3. ^ Thomas, George B. "6.7: Area of a Surface of Revolution; 6.11: The Theorems of Pappus". Calculus (3rd ed.). pp. 206-209, 217-219. LCCN 69016407.
4. ^ Singh, R.R. (1993). Engineering Mathematics (6 ed.). Tata McGraw-Hill. p. 6.90. ISBN 0-07-014615-2.
5. ^ Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 617, ISBN 0-87150-341-7
6. ^ a b
7. ^
8. ^ Pressley, Andrew. "Chapter 9 - Geodesics." Elementary Differential Geometry, 2nd ed., Springer, London, 2012, pp. 227-230.
9. ^


text:
# COSMO19

2-6 September 2019
Europe/Zurich timezone

## Cosmology with Rayleigh Scattering

4 Sep 2019, 16:40
20m
Talk CMB, Universe Expansion, and Thermal History

### Speaker

Benjamin Beringue (University of Cambridge)

### Description

With the next generation of CMB surveys promising to map linear perturbation modes (in both temperature and E-mode polarization) down to the cosmic variance limit for $\ell$ below $\sim 3000$, cosmologists are turning to different avenues to further constrain the $\Lambda CDM$ model. Primordial gravitational waves, in the form of linear B-modes polarization, could be detected in the near future by upcoming ground-based experiments. Secondary anisotropies are another more recently studied potential source of cosmological information. CMB lensing for example, has been successfully measured and used to obtain stringent constraints on neutrino masses. The Sunayev – Zel’dovich (SZ) Effects, both kinetic and thermal, although they heavily depend on astrophysical processes, carry their own cosmological information.
Among these secondaries, Rayleigh scattering of the CMB is a less studied yet potentially powerful probe of the recombination history. Scattering of CMB photons off neutral species right after recombination presents a distinctive $\nu^4$ scaling with frequency as well as a strong correlation with the primary CMB. These unique features should guarantee its detection by the next generation of ground based CMB experiments. We will present detectability forecasts combining the Simons Observatory and CCAT-prime telescopes as well as more futuristic space missions. Finally, we will present potential cosmological implications of the detection of this signal by studying improvement of parameter constraints.

### Primary author

Benjamin Beringue (University of Cambridge)

### Co-authors

Prof. P.D. Meerburg (University of Groningen) Joel Meyers (University of Toronto)


text:
# Hugo has

Hugo has 12 apples, Julka has seven more than Hugo and six less than Anna. How many apples have Anna and how many Julka?

Correct result:

A =  25
J =  19

#### Solution:

$J=19$

We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!

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text:
#### Volume 13, issue 1 (2013)

 Recent Issues
Author Index
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
Complete intersections and mod $p$ cochains

### David J Benson, John P C Greenlees and Shoham Shamir

Algebraic & Geometric Topology 13 (2013) 61–114
##### Abstract

We give homotopy invariant definitions corresponding to three well-known properties of complete intersections, for the ring, the module theory and the endomorphisms of the residue field, and we investigate them for the mod $p$ cochains on a space, showing that suitable versions of the second and third are equivalent and that the first is stronger. We are particularly interested in classifying spaces of groups, and we give a number of examples. The case of rational homotopy theory is treated in [J. Pure Appl. Algebra 217 (2013) 636–663], and there are some interesting contrasts.

##### Keywords
complete intersection, commutative ring spectrum, derived category, group cohomology, mod $p$ cochains
##### Mathematical Subject Classification 2010
Primary: 13C40, 55P43, 13D99, 20J06, 55N99
Secondary: 14M10, 55P42, 55U35, 20J05
##### Publication
Received: 15 April 2011
Revised: 14 August 2012
Accepted: 14 August 2012
Published: 6 February 2013
##### Authors
 David J Benson Department of Mathematics University of Aberdeen Meston Building Aberdeen AB24 3UE UK http://homepages.abdn.ac.uk/mth192/pages/html/index.html John P C Greenlees School of Mathematics and Statistics University of Sheffield Hicks Building Sheffield S3 7RH UK http://greenlees.staff.shef.ac.uk Shoham Shamir Department of Mathematics University of Bergen 5008 Bergen Norway