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# What is True About Machine Learning?

In this Blog, we are going to explore about machine learning and the different key points What is True About Machine Learning? in detail.

Machine learning is a subfield of artificial intelligence (AI) that allows computer systems to learn and make predictions or decisions without being explicitly programmed to do so.

The future growth of machine learning is expected to be substantial, and it will continue to have a profound impact on the way we live and work.

## What is Machine learning?

Machine learning is a subfield of artificial intelligence (AI) that involves the development of algorithms and statistical models that allow computers to improve their performance in tasks over time through experience, without being explicitly programmed.

It involves feeding large amounts of data into these models and algorithms, and then using that data to train the system to make predictions or decisions.

There are different types of machine learning, including supervised learning, unsupervised learning, semi-supervised learning, and reinforcement learning.

>> In Supervised learning, the algorithm is provided with labeled data and learns to make predictions based on that data.

>> In unsupervised learning, the algorithm must find patterns in the data on its own, without any specific labeling.

>> Semi-supervised learning is a combination of supervised and unsupervised learning, where some of the data is labeled and some is not.

>> Reinforcement learning involves learning through trial-and-error and rewards, as in the case of teaching a machine to play a game.

Machine learning has been used in a wide range of applications, such as image and speech recognition, natural language processing, and predictive analytics.

It has the potential to transform various industries and has significant implications for the way we live and work.

## What is True About Machine Learning?

The following are some key points that are true about machine learning:

1. Data-driven: Machine learning algorithms rely on large amounts of data to train and improve their performance over time.
2. Predictive modeling: Machine learning models can be used to make predictions based on historical data, such as predicting the likelihood of a customer churning or the price of a stock.
3. Supervised and unsupervised learning: Machine learning algorithms can be divided into supervised learning, where the algorithm is provided with labeled data, or unsupervised learning, where the algorithm must find patterns in the data on its own.
4. Continuous improvement: Machine learning algorithms can continue to learn and improve their performance as they are fed more data.
5. Limitations: Machine learning algorithms are not perfect and can make errors or biased predictions, particularly if the training data is not representative of the real-world problem being solved.
6. Broad Applications: Machine learning has a wide range of applications, including image recognition, natural language processing, speech recognition, and predictive maintenance, among others.

It is important to note that machine learning is a rapidly evolving field, and new developments are being made all the time.

## Future Growth of Machine Learning

The future growth of machine learning is expected to be significant, with widespread applications across a variety of industries and fields. Here are a few trends and predictions for the future growth of machine learning:

Machine learning is expected to become more widely adopted and integrated into everyday processes, especially as the technology becomes more accessible and easier to use.

#### 2. Advancements in Natural Language Processing (NLP):

NLP is a rapidly growing area of machine learning, and is expected to play a major role in shaping the future of machine learning.

This will include advancements in speech recognition and text-to-speech synthesis, as well as improved machine translation and conversational AI.

#### 3. Expansion into New Industries:

Machine learning is already being used in a variety of industries, but is expected to expand even further into new areas, such as finance, healthcare, and education.

#### 4. Edge Computing:

The trend towards edge computing, where data is processed closer to the source of the data, is expected to continue to grow, and machine learning algorithms will play a crucial role in this trend.

#### 5. Increased Focus on Explainability and Bias:

As machine learning is used in more critical decision-making applications, there will be an increased focus on ensuring that the algorithms are transparent and fair, and that any biases in the data or algorithms are addressed.

#### 6. Collaboration between Humans and AI:

Machine learning is expected to play a larger role in augmenting human capabilities, rather than replacing them, by assisting and collaborating with humans to make better decisions.

## Why Choose Machine learning As a Career or Study?

Machine learning is a rapidly growing field that offers a wide range of career and study opportunities for individuals with diverse backgrounds and interests.

Here are some reasons why you might choose machine learning as a career or study:

#### 1. High Demand:

Machine learning is in high demand, and the demand for professionals with machine learning skills is expected to grow significantly in the coming years. This makes it a lucrative and stable career choice.

#### 2. Wide Range of Applications:

Machine learning has a wide range of applications, from natural language processing and speech recognition to computer vision and predictive analytics.

This means that there are many areas within machine learning to specialize in, depending on your interests.

#### 3. Interdisciplinary Nature:

Machine learning draws on a variety of disciplines, including mathematics, statistics, computer science, and engineering.

This interdisciplinary nature makes it a fascinating and challenging field to work in.

#### 4. Impactful Work:

Machine learning has the potential to have a profound impact on society and improve people’s lives, from improving healthcare outcomes to making our lives easier through intelligent personal assistants.

#### 5. Continuous Learning and Development:

Machine learning is a rapidly evolving field, and there is always more to learn and discover. This makes it a dynamic and rewarding field to work in, as you will be continuously challenged and have opportunities for professional growth.

Whether you are just starting out in your career or looking for a change, machine learning offers a wide range of opportunities for individuals with a variety of skills and interests.

Whether you want to work in academia, research, industry, or start your own company, there is a place for you in the field of machine learning.

## Conclusion

In conclusion, machine learning is a rapidly growing and dynamic field that offers a wide range of career and study opportunities, this all keypoints are True About Machine Learning.

The demand for machine learning professionals is expected to continue to grow, and the field has the potential to have a profound impact on society.

Whether you have a background in mathematics, statistics, computer science, or another field, there is a place for you in the field of machine learning.

With its interdisciplinary nature and opportunities for continuous learning and development, machine learning is a challenging and rewarding field to work in.

If you are interested in the field, consider exploring it further to see if it is the right fit for you.


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# Lines of reflection in a Dihedral group, Why is this paradox happening?

So, we know that in a Dihedral Group $D_n$, if $r$ represents counterclockwise rotation by $2\pi/n$ radians and $s$ is any axis of reflection, then the elements of reflection stand as follows:

{$s, rs,r^2s, ...... , r^{n-1}s$} .

In the image attached, there is an octagon . Suppose i consider the reflection axis $'s'$ as $ab$. Now, since $r$ stands for rotation by a degree of $360/8 = 45^o$, $rs$ must rotate $s$ by $45^o$ to obtain the $rs$ as $cd$. But, $rs$ is shown to be the axis passing through vertices $3$ and $7$ . Why is this paradox happening?

Instead of us calculating where vertices 3 and 4 go individually considering $r$ and $s$ , lets consider the cumulative effect of $rs$. Since, we have fixed the axis s, now, rs means, rotate this $s$ axis by $r$ radians in counter clockwise direction and then, since $rs$ is a reflection, reflect the original figure about the new axis described by $rs$. Thank you

• I take it $r$ is rotation clockwise through $\pi/4$. $r$ takes 3 to 4, then $s$ takes 4 to 3, so $rs$ fixes 3, so it's reflection in 3-7. – Gerry Myerson Mar 29 '14 at 8:14
• So, if it's clockwise rotation, r should rotate s ( $ab$ here ) through $45^o$ and produce $rs$ as the axis joining the mid points of $4-5$ and $8-7$ . But, textbooks in such cases show $rs$ as $4-8$ – MathMan Mar 29 '14 at 8:18
• $rs$ is $r$ followed by $s$. $r$ takes ab to mid45 - mid78, then $s$ takes that to cd, so it's reflection in 3-7, as you said earlier. But now you say the textbook says 4-8. That's consistent with $r$ being rotation counterclockwise. So: does the book say $rs$ is reflection in 3-7? or does it say it's reflection in 4-8? – Gerry Myerson Mar 29 '14 at 8:28
• No. Why should $rs$ "yield a line of reflection"? $rs$ is a reflection, and the question is, in what axis is it a reflection. And the answer is, since it takes ab to cd, it can only be the reflection in 3-7. – Gerry Myerson Mar 29 '14 at 12:15
• No. The cumulative effect of a composition of functions is the effect of doing the first function and then doing the second. And I have worked out for you what that cumulative effect is. $rs$ doesn't "give a new line of reflection"; $rs$ is a reflection, a reflection in 3-7. But we are repeating ourselves. – Gerry Myerson Mar 29 '14 at 22:25

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Subscribe
Issue No.03 - May-June (2008 vol.25)
pp: 284-285
Grant Martin , Tensilica
ABSTRACT
This is a review of Creating Assertion-Based IP (by Harry D. Foster and Adam C. Krolnik)—a tutorial on creating assertion-based verification IP for real designs, applicable to various classes of design blocks found in today's systems. This book uses the specific syntax of System-Verilog Assertions (SVA) and the verification environment of the Advanced Verification Methodology (AVM). However, the same principles apply to other assertion specification languages and within the context of other verification environments. Perhaps most importantly, this book teaches a way of thinking about creating and using assertion-based verification IP.

Reviewed in this issue
Creating Assertion-Based IP, by Harry D. Foster and Adam C. Krolnik (Springer, 2008, ISBN: 978-0-387-36641-8, 318 pp., $129). A variation on an old saying goes something like this: If you give an engineer a set of verification IP components, they can verify one design, but if you teach them to develop their own verification IP they can verify designs for the rest of their lives! Creating Assertion-Based IP, written by two of the foremost experts in the field, Harry Foster and Adam Krolnik, is such a tutorial. If you open this book without knowing very much about creating assertion-based verification IP for real designs, you will close it with a reasonable understanding of how to do so for various classes of design blocks found in today's systems. This book uses the specific syntax of SystemVerilog Assertions (SVA) and the verification environment of the Advanced Verification Methodology (AVM), which is a subset of the Open Verification Methodology (OVM). However, the same principles apply to other assertion specification languages and within the context of other verification environments. Perhaps most importantly, this book teaches a way of thinking about creating and using assertion-based verification IP. Another aspect of the book that I particularly like is that it is extremely well-written. It has a natural, flowing style that moves the reader through a topic and between topics in a way that's easy to follow. The book is organized into a logical sequence. The first chapter gently introduces the concept of assertion-based IP; properties and assertions; and background on abstractions, reuse, languages, libraries, and use models. The second chapter is well-placed, as it clearly defines the notations and terminology used throughout the rest of the book. Chapter 3, the last introductory chapter, defines a verification process and the assertion-based IP architecture discussed in the remaining chapters. Thus, this chapter develops the general approach of interface-based assertion IP and associated analysis ports that are fundamental to this book. The next set of chapters are based on a modern bus-based digital design consisting of various types of components: bus interfaces, arbiters, controllers, and datapath components. Chapter 4, which is only three pages long, gives a short overview of the components discussed in Chapters 5 through 8. All the remaining chapters then follow a common structure: First, the authors introduce components in each class and give an overview description, including well-drawn block diagrams. This is followed by a simple set of properties described in a natural-language format. This property set serves as a typical example of the component class under discussion, although it is not a complete set of properties, and each example set covers an interesting variety of the important properties to be verified. Next, the authors provide assertion examples in SVA based on the use of the AVM infrastructure. Finally, they show how to encapsulate the properties inside a module. This approach is both tutorial in nature and deliberately repetitive. By using this common style and repeating some of the same language from one component type to another, the authors make it easier for readers to move between sections or to dive into a later section without having to read the entire book from cover to cover. Each example is extensively worked out with code, tables, and timing diagrams, as well as textual explanations. The explanations go into detail when required without repeating the relatively obvious parts. This approach lets readers concentrate on the more complex pieces of each example. Two appendices—one a tutorial overview of SVA, the other a complete OVM-AVM testbench example—complement the main part of the book. The first appendix is a quick guide to SVA; the second one enables readers to apply the examples more quickly, especially since OVM and AVM are freely downloadable. Despite being specific to a language and an environment, the principles and concepts of Chapters 5 through 8 are applicable to other languages and contexts. Any team wishing to learn how to develop assertion-based verification IP would benefit from this book as a central resource for the team and as a basis for their own experimentation. The authors' emphasis on following common processes and their consistency in keeping definitions of properties, assertions, and portions of the testbenches relatively orthogonal and as simple as possible (rather than as intertwined complex pieces of IP that try to do everything in one module) is very helpful. This book could also be used as a text or reference for university-level verification courses. However, this brings me to my major problem with the book: its price. At$129, it is way too expensive for individual engineers or students to buy. Given its tutorial nature, it is more of a "teach-me" book to be used for a relatively short time and then occasionally as a reference, rather than something to use on a continual basis or as a deep reference. Being a hardcover book makes it more expensive than, say, a more natural format in soft cover. If it were priced well below \$60, it would be easier to see more people being able to buy an individual copy.
But price notwithstanding, if a course or design group finds a way to share this book's valuable tutorial content, they will certainly derive value from it. Although it is not aimed at the experienced assertion-based verification IP creator, it can nevertheless be a good vehicle for helping to develop the next generation of such engineers.


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# Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5

Up to topology, the 5D homogeneous space $$G_2/P$$ of the (real form of the) 14D exceptional Lie group $G_2$ is the 5D jet space $$M:=J^1(2,1)=\{(x,y,u,p,q)\}$$ of scalar functions in two independent variables.

WHY? Because $G_2/P=S^2\times S^3\approx\mathbb{P}T^*S^3\approx\mathbb{P}T^*\mathbb{R}^3=\mathbb{R}^2\times(\mathbb{R}\times\mathbb{R}^{2\ast})=J^2(2,1)$, where $\mathbb{R}^2$ is the plane of independent variables, $\mathbb{R}$ the line of the dependent one, and $\mathbb{R}^{2\ast}$ the plane of the derivatives of the latter w.r.t. the former.

Now $M$ comes equipped with a flag of distributions, viz $$(M,V,C)\, ,$$ where $V=\langle\partial_p,\partial_q\rangle$ is the 2D vertical and $C=V\oplus\langle \partial_x+p\partial_u, \partial_y+q\partial_u\rangle$ the 4D contact one.

Now there is a result (I've learned it from P. Nurowski) saying that

(THEOREM) There is a one-to-one correspondence between $(G_2,P)$-type Cartan geometries and generic 2D distributions in dimension 5.

Since - as textbooks recite - a $(G_2,P)$-type Cartan geometry is obtained by rolling without slipping the homogeneous space $G_2/P\approx M$, I guess that a generic 2D distribution in dimension five is obtained by "rolling without slipping" the vertical distribution $V$ of the homogeneous model $M$. This should be the essence of the theorem.

My point is simple: if I can "roll" the vertical distribution $V$, why I cannot "roll" the contact distribution $C$ as well?

QUESTION: Is it true that a $(G_2,P)$-type Cartan geometry in dimension five is the same as a manifold equipped with a $(2,4)$-type flag of distributions $(\widetilde{V},\widetilde{C})$? Under which circumstances $\widetilde{C}$ is contact? (This makes sense, being so at least for the flat case $M$.)

Maybe the question is ill-posed, due to my still poor grasp on Cartan geometries, but I hope that its essence will be clear: the "flat model", i.e., $M$, is a contact manifold equipped with a distinguished 2D contact-subdistribution (no doubts about that), and I'm just wondering how much of this structure descends to the "curved cases".

"I'm just a simple man trying to make my way in the universe of Cartan geometries" - J. Fett.

• A $(G_2,P)$-geometry with torsion can fail to induce a generic 2-dimensional distribution. In one of my papers (I am so old!) I worked out the translation invariant parabolic geometries. There are many, for every model, and all of the distributions they induce are bracket closed. – Ben McKay Mar 6 '16 at 8:52

Let $$\Bbb V$$ be the standard (i.e., $$7$$-dimensional irreducible) representation of $$\mathfrak{g}_2$$ (either the split real or the complex form); recall that $$G_2$$ is the stabilizer of cross product map $$\times : \Bbb V \times \Bbb V \to \Bbb V$$. The inclusion $$G_2 \hookrightarrow SO(3, 4)$$ ($$G_2 \hookrightarrow SO(7, \Bbb C)$$) determines an indefinite, nondegenerate, symmetric bilinear form $$H$$ on $$\Bbb V$$.

The "first" parabolic subgroup $$P_1$$ (corresponding to a cross on the first node of the Dynkin diagram of $$G_2$$ in the usual Bourbaki ordering) is the stabilizer of an isotropic $$1$$-dimensional subspace of $$\Bbb V$$. The cone $$\mathcal C$$ of nonzero isotropic vectors inherits an invariant filtration of tangent distributions, whose fibers at $$Y \in \mathcal C$$ are $$\ker (Z \mapsto Y \times Z) \subset \operatorname{im} (Z \mapsto Y \times Z) \subset T_Y \mathcal C$$ of dimensions $$3, 4, 6$$ (to apply the cross product, we implicitly use here the identifications determined by the canonical isomorphism $$T_Y \Bbb V \leftrightarrow \Bbb V$$). By linearity, this descends to a filtration $$\bfD \subset \bfD' \subset T\Bbb Q_5,$$ of dimensions $$2, 3, 5$$, on the null quadric $$\Bbb{Q}_5 := \Bbb P(\mathcal C) \subset \Bbb P(\Bbb V)$$, which is diffeomorphic to $$(\Bbb S^2 \times \Bbb S^3) / \Bbb Z_2$$, where $$\Bbb Z_2$$ acts by the antipodal map on both factors. Since the ingredients are $$G_2$$-invariant, so is $$\bfD$$ under the induced action on $$T\Bbb Q_5$$, and as one expects, it turns out that $$[\bfD, \bfD] = \bfD'$$ and $$[\bfD', \bfD] = T\Bbb Q_5$$. (NB there are no $$G_2$$-invariant linear or hyperplane distributions on $$T\Bbb Q_5$$.)

On the other hand, consider differential equations of the form $$z' = F(x, y, y', y'', z)$$. Any function $$F(x, y, p, q, z)$$ determines a total derivative $$D_x := \partial_x + p \partial_y + q \partial_p + F \partial_z$$ on the corresponding partial jet space $$J^{2, 0}(\Bbb R, \Bbb R) \cong \Bbb R^5_{xypqz}$$. Suitably regarded, the vertical fibers of the jet truncation map $$J^{2, 0}(\Bbb R, \Bbb R) \to J^{1, 0}(\Bbb R, \Bbb R)$$ are spanned by $$\partial_q$$. Computing directly shows that the distribution $$\bfD_F := \langle D_x, \partial_q \rangle$$ is generic iff $$F_{qq}$$ vanishes nowhere; conversely, a theorem of (I believe) Monge states than any generic $$2$$-plane distribution on a $$5$$-manifold is locally equivalent to $$\bfD_F$$ for some function $$F$$. If $$F(x, y, p, q, z) := q^2$$, then the resulting distribution has infinitesimal symmetry algebra isomorphic to $$\mathfrak{g}_2$$, so by a general fact about parabolic geometries the distribution $$(J^{2, 0}(\Bbb R, \Bbb R), \bfD_F)$$ corresponding to the differential equation $$z' = (y'')^2$$ is everywhere locally diffeomorphic to the homogeneous model distribution $$(\Bbb Q_5, \bfD)$$ above. In particular, it follows from this that neither of the distributions $$V$$ and $$\widetilde{C}$$ are invariant under the action of the infinitesimal symmetry algebra $$\mathfrak{g}_2$$ of $$(J^{2, 0}(\Bbb R, \Bbb R), \bfD_F)$$.

The correct statement of the theorem you mention is that there is an equivalence of categories between generic $$2$$-plane distributions on $$5$$-manifolds and normal, regular parabolic geometries of type $$(G_2, P_1)$$. (See the end of Subsubsection 4.3.2 in Cap & Slovak's book, Parabolic Geometries.)

On the other hand, we can consider the action of $$G_2$$ on the space of isotropic $$2$$-planes in $$\Bbb V$$. This action has two orbits, according to whether the cross product $$\times$$ restricts to the zero map on each $$2$$-plane. (Bryant calls the $$2$$-planes on which the restriction is zero special in his highly enjoyable lecture notes Elie Cartan and Geometric Duality [pdf], which treats the correspondence space construction for $$G_2 / P_1 \leftarrow G_2 / (P_1 \cap P_2) \to G_2 / P_2$$, as well as the analogous construction for $$A_2$$ and $$B_2 \cong C_2$$. NB that this article seems contains a few typos, replacing $$\Bbb N_5$$, introduced in a moment, with $$\Bbb Q_5$$, which is the essential apparent confusion in the question here.) The isotropy subgroup of a point in the $$5$$-dimensional space $$\Bbb N_5$$ of special $$2$$-planes is the "second" parabolic $$P_2 \subset G_2$$. Analogously to the situation for the first parabolic, we can view $$\Bbb N_5$$ as a subset of $$\Bbb P^{13} = \Bbb P(\mathfrak{g}_2)$$, but its geometry is apparently much more complicated than that of $$\Bbb Q_5$$: In the complex case, $$\Bbb N_5$$ is a variety of degree $$18$$, and its complete intersection with three hyperplanes in a general configuration is a K3 surface of genus $$10$$, but NB other geometric descriptions of this space (which look less daunting to non-algebraic geometers like myself) are available, too. Apparently this is worked out in the paper of Borcea cited below, but I can't find an ungated copy. See also the accessible historical survey paper of Agricola, also cited below.

Now, $$\Bbb N_5$$ inherits an invariant contact distribution $$\bfH$$. (Surely this can be written down with some much effort in terms of the cross product on $$\Bbb V$$, but to my knowledge this hasn't been done anywhere.) Moreover, the representation $$P_2$$ induces on each fiber of $$\bfH$$ turns out to be a trivial extension of a representation of $$GL(2, \Bbb F) \subset P_2$$, and this representation is isomorphic to $$S^3 \Bbb F^2$$ (this representation is conformally symplectic and so determines equivalently a nondegenerate cone in each fiber of $$\bfH$$). All of the $$G_2$$-invariant structure on $$\Bbb N_5$$ can be recovered from these objects, corresponding to the fact that a (normal, regular) parabolic geometry of type $$(G_2, P_2)$$ is a $$5$$-manifold $$M$$ equipped with a "$$G_2$$ contact structure", which is a contact structure $$\bfH \subset TM$$ together with an auxiliary rank-$$2$$ vector bundle $$\bfE \to M$$ and a vector bundle isomorphism $$S^3 \bfE \stackrel{\cong}{\to} \bfH$$ such that the Levi bracket $$\bfH \times \bfH \to TM / \bfH$$ (the tensorial map induced by the Lie bracket) is invariant under the induced action of $$\mathfrak{sl}(\bfE)$$. See Subsubsection 4.2.8 of Cap & Slovak's book. I don't know of any sensible analog of the Monge (quasi-)normal form $$z' = F(\cdots)$$ for $$G_2$$ contact structures, but I would be pleased to hear about one.

There are connections between these two types of parabolic geometry beyond the mentioned correspondence space construction. See this preprint of Leistner, Nurowski, & Sagerschnig.

I. Agricola, Old and New on the Exceptional Group $$G_2$$ [pdf], Notices Amer. Math. Soc. 55(8) (2008), 922-929.

C. Borcea, Smooth global complete intersections in certain compact homogeneous complex manifolds, J. Reine Angew. Math. 344 (1983), 65–70.

A. Cap, J. Slovak, Parabolic geometries I: Background and general theory. Math. Surveys Monogr. 154, Amer. Math. Soc., Providence, RI, 628pp.

T. Leistner, P. Nurowski, K. Sagerschnig, New relations between $$G_2$$-geometries in dimensions $$5$$ and $$7$$, Internat. J. Math. 28(13) (2017). arXiv:1601.03979

• You gave a sophisticated answer to a lousy question: chapeau! – Giovanni Moreno Mar 13 '16 at 22:06
• You're welcome, Giovanni, I hope you found it useful, too! It was good to meet you in Vienna earlier this year. – Travis Mar 13 '16 at 22:12

You are confusing (me and) the two different 5-dimensional homogeneous spaces of $G_2$, both of which are $X=G_2/P$ but for different parabolic subgroups $P \subset G_2$. One has a $G_2$-invariant nondegenerate rank 2 distribution, contained in a rank 3. The other has a $G_2$-invariant contact structure. See my paper on characteristic classes for lots of examples of invariant distributions on the models: http://arxiv.org/abs/0704.2555. The only $G_2$-invariant distributions on $G_2/P_2$ (in the notation of that paper) are the rank 2 and rank 3 (and the tangent bundle and its zero section). The only $G_2$-invariant distribution on $G_2/P_1$ is the contact structure.

A little more about rolling: take two surfaces $S_1, S_2$ embedded into $\mathbb{R}^3$. Draw a curve along one of them, and roll the other along that curve. (We need some geometric hypothesis to keep them from getting stuck'', but we ignore this.) At each moment in time, they are tangent at some pair of points, giving an isometry of their tangent planes. So the rolling is described as a curve in the manifold $M$ of tuples $(x_1,x_2,u)$ where $x_i \in S_i$ and $u \colon T_{x_1} S_1 \to T_{x_2} S_2$ is a linear isometry. Conversely, there is a 2-plane field $V$ on $M$ (i.e. a rank 2 subbundle $V \subset TM$) so that each path $x_1(t)$ with given initial values of $x_2(0), u(0)$ determines a unique path $(x_1(t),x_2(t),u(t))$ tangent to $V$, and this path is the rolling of one surface on the other. This $V$ depends only on the Riemannian metrics of the two surfaces, not on how they are embedded into $\mathbb{R}^3$. So we have a map $(S_1,S_2) \mapsto (M,V)$ from pairs of surfaces with Riemannian metric to 5-manifolds with 2-plane field. A 2-plane field $V$ on a 5-manifold is nondegenerate if any two local linearly independent sections $X,Y$ have $X, Y, [X,Y], [X,[X,Y]], [Y,[X,Y]]$ linearly independent. The 2-plane field $V$ on $M$ we have constructed by rolling is nodegenerate just at the points $(x_1,x_2,u)$ so that $S_1$ and $S_2$ have distinct Gauss curvature at $x_1$ and $x_2$. There is no rolling of $M$ involved in this picture.

So we have a map $(S_1,S_2) \mapsto (M,V)$ to nondegenerate 2-plane fields on 5-manifolds (cutting out points $(x_1,x_2,u)$ of $M$ where Gauss curvatures agree at $x_1$ and $x_2$). Two great unsolved questions: (1) what is the image and (2) what are the fibers? Well known methods should suffice to solve both problems. Not every nondegenerate 2-plane field $V$ arises, even locally, this way, as you can see by counting, once you know that Cartan proved that every nondegenerate 2-plane field has finite dimensional symmetry group. There are examples of pairs of pairs, i.e. $(S_1,S_2)$ and $(S_1',S_2')$ pairs of surfaces, so that the associated $(M,V)$ nondegenerate 2-plane fields are isomorphic, even though $(S_1',S_2')$ are not isometric to $(S_1,S_2)$ even after constant rescaling. But it seems likely that a typical choice of $(S_1,S_2)$ pair gives rise to an $(M,V)$ which is not obtained from any other pairs of surfaces $(S_1',S_2')$ except for rescaling the Riemannian metrics of both $S_1$ and $S_2$ by the same positive constant.

• Every complex simple Lie group has a unique homogeneous contact manifold, the orbit of a highest weight vector in the adjoint form. (See the papers of Landsberg and Manivel.) So $G_2$ doesn't preserve any contact structure on any but one of its homogeneous spaces, at least after you complexify. – Ben McKay Mar 6 '16 at 9:00
• I confess that after "A little more about rolling" I had a hard time understanding - though it is interesting and surely we'll get back to it in September. Concerning my specific question: if I got it, $G_2$ has two 5D homogeneous spaces, one is equipped with a contact distribution, and another with a $(2,3)$-type flag of distributions (the 2D one being "nondegenerate" in the sense of yours, or $(2,3,5)$ as others like to call it). Question: the former is $S^3\times S^2$; how the latter looks like? What is a good and quick reference to learn about $G_2$ and its homogeneous spaces? – Giovanni Moreno Mar 8 '16 at 6:56
• The second one is the projectivized null cone in the irreducible 7-dimensional representation of the split form of $G_2$, so a 5-sphere. It is also the adjoint variety (projectivized orbit of a highest weight vector in the adjoint form). Its complex form is a quadric hypersurface in projective space, so a 5-sphere. I don't know a good place to look this up. Maybe when I get to Poland, we can write a book. – Ben McKay Mar 8 '16 at 14:56
• Book-writing is a good entertainment for people over sixties - so let's discuss it in two decades - but a review paper why not? So, the two homogeneous 5-folds of $G_2$ are $S^2\times S^3$ and $S^5$, which are both contact, but only the former is contact homogeneous. The latter should be equipped with a $(2,3)$-type flag of distributions, but I fail to see it: is it evident? In fact, Sagershnig's paper says that the projectivised null cone in the imaginary octonions is $S^2\times S^3$, i.e., the opposite of what you say! That's why I'd like to see how a 2D and a 3D distribution on $S^5$ arise. – Giovanni Moreno Mar 8 '16 at 17:44
• An & Nurowski have given explicit examples of nonhomothetic pairs $(S_1, S_2)$ and $(S_1', S_2)$ of surfaces for which the corresponding plane fields $(M, V)$ are distinct: The usual choice rolling surface realization of the flat model of the geometry (i.e., model with local symmetry group $G_2$) is a pair of spheres, one of which has radius three times that of the other. On the other hand, there are exactly three surfaces $S$ up to homothety that admit a Killing vector such that $(\Bbb R^2, S)$ is also flat, and hence induce a flat distribution; see arxiv.org/pdf/1210.3536.pdf . – Travis Mar 12 '16 at 20:25


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# Natural-Gradient Variational Inference 2: ImageNet-scale

By Siddharth Swaroop

In our previous post, we derived a natural-gradient variational inference (NGVI) algorithm for neural networks, detailing all our approximations and providing intuition. We saw it converge faster than more naive variational inference algorithms on relatively small-scale data. But a couple of key questions remain:

1. Can we scale such algorithms to very large datasets and architectures?
2. Did we gain anything from having additional Bayesian principles, or put differently, do we have better performance than SGD or Adam?

We tackle these questions in this second part of the natural-gradient for variational inference series. We show that we can get good performance at large scales with Bayesian principles, while maintaining reasonable uncertainties. We start by focussing on question (i): the issue of scalability. We notice similarities between our NGVI algorithm and Adam, and exploit this to borrow tricks that the community has developed for Adam over many years. This allows us to scale up to very large datasets/architectures. We then turn our focus to question (ii): have we improved on neural networks’ poorly-calibrated uncertainties thanks to our Bayesian thinking? We will see some benefits. Along the way, we will discuss the price we pay for them.

We start with the equations for the VOGN algorithm, derived in our previous blog post. This also serves as a quick summary of notation: please look at the previous blog post if anything is unclear! (Colours are purely for illustrative purposes.)

\begin{align} \label{eq:VOGN_mu} \vmu_{t+1} &= \vmu_t - \alpha_t \frac{ {\color{purple}\hat{\vg}(\vparam_t)} + {\color{blue}\tilde{\delta}}\vmu_t}{\vs_{t+1} + {\color{blue}\tilde{\delta}}}, \newline \label{eq:VOGN_Sigma} \vs_{t+1} &= (1-\beta_t)\vs_t + \beta_t \frac{1}{M} \sum_{i\in\mathcal{M}_t}\left( {\color{purple}\vg_i(\vparam_t)} \right)^2. \end{align}

Remember, we are updating the parameters of our (mean-field Gaussian) approximate posterior $q_t(\vparam)=\gauss(\vparam; \vmu_t, \vSigma_t)$, where $\vparam$ are the parameters of a neural network. We do this by iteratively updating two vectors, $\vmu_t$ and $\vs_t$, where $t$ indexes the iteration. We have a zero-mean prior $p(\vparam) = \gauss(\vparam; \color{blue}\boldsymbol{0}, {\color{blue}\delta^{-1}} \vI)$, and ${\color{blue}\tilde{\delta}} = {\color{blue}{\delta}} / N$. Our dataset consists of $N$ data examples, and we are taking per-example gradients of the negative log-likelihood $\color{purple}\vg_i(\vparam_t)$ at a sample from our current approximate posterior, $\vparam_t \sim q_t(\vparam)$. For a randomly-sampled minibatch $\mathcal{M}_t$ of size $M$, we have defined the average gradient ${\color{purple}\hat{\vg}(\vparam_t)} = \frac{1}{M}\sum_{i\in\mathcal{M}_t} {\color{purple}\vg_i(\vparam_t)}$. There is a simple relation between $\vSigma_t$ and $\vs_t$, $\vSigma_t^{-1} = N\vs_t + {\color{blue}\delta \vI}$. Finally, $\alpha_t>0$ and $0<\beta_t<1$ are learning rates, and all operations are element-wise.

It turns out that this update equation is very similar to Adam (Kingba & Ba, 2015). To see this, let’s write down the form that commonly-used optimisers take, such as SGD, RMSProp (Hinton, 2012), and Adam:

\begin{align} \label{eq:Adam_mu} \vmu_{t+1} &= \vmu_t - \alpha_t \frac{ {\color{purple}\hat{\vg}(\vmu_t)} + {\color{blue}\delta}\vmu_t} {\sqrt{\vs_{t+1}} + \epsilon}, \newline \label{eq:Adam_Sigma} \vs_{t+1} &= (1-\beta_t)\vs_t + \beta_t \left( \frac{1}{M} \sum_{i\in\mathcal{M}_t} {\color{purple}\vg_i(\vmu_t)} + {\color{blue}\delta} \vmu_t \right) ^2, \end{align}

where $\delta>0$ is our weight-decay regulariser, and $\epsilon>0$ is a small scalar constant. Immediately we can see striking similarities in the overall form of the equations! Let’s take a closer look at the similarities and differences:

1. Similarity: Both updates for $\vmu_t$ are similar, of the form $\vmu_{t+1} = \vmu_t - \alpha_t (\hat{\vg} + \delta \vmu_t) / \mathrm{function}{(\vs_{t+1})}$.
2. Difference: The denominator in the update for the means is slightly different. VOGN uses $(\vs_{t+1} + \tilde{\delta})$, while Adam has a square root, $\sqrt{\vs_{t+1}}$.
3. Difference: VOGN calculates gradients at a sample $\vparam_t \sim q_t(\vparam)$, while Adam calculates gradients just at the mean $\vmu_t$. In fact, when we remove this difference, we get a deterministic version of VOGN, which we call OGN.
4. Similarity: Both updates for $\vs_t$ take the form of a moving average update.
5. Difference: VOGN uses a Gauss-Newton approximation, requiring $\sum_i (\vg_i)^2$, while Adam (and other SGD-based algorithms) use a gradient-magnitude, $\left( \sum_i \vg_i \right) ^2$. Note that in VOGN, the sum is outside the square, while in SGD-based algorithms, the sum is inside the square.

The Gauss-Newton approximation (Difference 5) is a better approximation to second-order information (Hessian) than the gradient-magnitude approach. This better approximation is likely why VOGN does not require a square root over $\vs_{t+1}$ in the update for the means (Difference 2). However, calculating the Gauss-Newton approximation requires additional computation in frameworks such as PyTorch, leading to VOGN being slower (per epoch) compared to Adam. This is despite using speed-up tricks (Goodfellow, 2015).

The similarities in the equations indicate that we might be able to take techniques people use to scale Adam up to large datasets and architectures, and apply similar ideas to scale VOGN up. We can use batch normalisation, momentum, clever initialisations, data augmentation, learning rate scheduling, and so on.

## Borrowing existing deep-learning techniques

Let’s go over a list of each of the changes we make, providing some intuition for them. Please see Osawa et al. (2019) for further details. Using all these techniques, we are able to scale VOGN to datasets like CIFAR-10 and ImageNet, and architectures such as ResNets.

### 1. Batch normalisation and momentum

Batch normalisation (Ioffe & Szegedy, 2015) empirically speeds up and stabilises training of neural networks. We can use BatchNorm layers as is normal in deep learning. In fact, in our VOGN implementation, we found that we do not have to maintain uncertainty over BatchNorm parameters.

We can also use momentum for VOGN in a similar way to Adam: we introduce momentum on $\vmu_t$, along with a momentum rate.

### 2. Initialisation

Over many years of training neural networks with SGD and Adam, the community has found tricks to speed up training using clever initialisation. We can get these same benefits by changing VOGN to look more like Adam at initialisation, before slowly relaxing our algorithm to become the full VOGN algorithm later in training.

This is achieved by introducing a tempering parameter $\tau$ in front of the KL term in the ELBO, which propagates its way through to the VOGN equations. To see exactly where $\tau$ crops up, please look at Equation 4 from Osawa et al. (2019), or see Algorithm 1 below. As $\tau\rightarrow 0$, we (loosely speaking) get more similar to Adam. So, at the beginning of training, we initialise $\tau$ at something small (like $0.1$) and increase to $1$ during the first few optimisation steps.

Other initialisations are the same as Adam: $\vmu_t$ is initialised using init.xavier_normal from PyTorch (Glorot & Bengio, 2010) and the momentum term is initialised to zero, like in Adam. VOGN’s $\vs_t$ is initialised using an additional forward pass through the first minibatch of data.

### 3. Learning rate scheduling

We can use learning rate scheduling for $\alpha_t$ exactly like is used for Adam and SGD at a large scale. We regularly decay the learning rate by a factor (typically a factor of 10).

### 4. Data augmentation

When training on image datasets, data augmentation (DA) can improve performance drastically. For example, we can use random cropping and/or random horizontal flipping of images.

Unfortunately, directly applying DA to VOGN does not lead to improvements, and also negatively affects uncertainty calibration. But we note that DA can be viewed as increasing the effective size of our dataset: remember that our dataset size $N$ affects VOGN (as opposed to Adam and SGD, where $N$ does not appear, as it unidentifiable with the weight-decay factor). So, we view DA as increasing $N$ by some factor depending on the exact DA technique: for example, if we horizontally flip each image with a probability of 50%, we increase $N$ by a factor of 2.

This is still a heuristic, and not mathematically rigorous. It seems to work quite well in our experiments, but requires further theoretical investigation. It is also closely related to KL-annealing in variational inference, as well as the recently-termed ‘cold posterior effect’ (Wenzel et al., 2020; Loo et al., 2021; Aitchison, 2021).

### 5. Distributed training

We would like to use multiple GPUs in parallel to perform large experiments quickly. Typically, we would just parallelise data, splitting up large minibatch sizes by sending different data to different GPUs. With VOGN, we can also parallelise computation over Monte-Carlo samples $\vparam_t \sim q_t(\vparam)$. Every GPU can use a different sample $\vparam_t$. This reduces variance during training, and we empirically find it leads to quicker convergence.

### 6. External damping factor

We introduce an external damping factor $\gamma$, added to $\vs_{t+1}$ in the denominator of Equation \eqref{eq:VOGN_mu} (Zhang et al., 2018). This increases the lower bound of the eigenvalues on the diagonal covariance $\vSigma_t$, preventing the step size and noise from becoming too large. However, this also detracts from the principled variational inference equations, and there is currently no theoretical justification for this (beyond the intuition we just provided).

### Final algorithm

Let’s recap. We derived the VOGN equations (Equations \eqref{eq:VOGN_mu} and \eqref{eq:VOGN_Sigma}) in the previous blog post. We started this post by comparing the equations to Adam, noting key similarities and differences. One key difference was based off the Gauss-Newton approximation, which slows VOGN down (per epoch) compared to SGD-based algorithms like Adam. Based on the similarities, we borrowed tricks to scale Adam to large data settings, and applied them to VOGN.

All of these tricks are important to get VOGN’s results on ImageNet. The final algorithm is summarised in Algorithm 1 below. One downside of VOGN when compared to Adam is the additional hyperparameters that require tuning. At the end of this blog post, we provide best practices for tuning these hyperparameters.

Algorithm 1: Final algorithm, ready for running on ImageNet. Additional notes explaining key steps are in red. The vanilla VOGN equations (Equations \eqref{eq:VOGN_mu} and \eqref{eq:VOGN_Sigma}) are in Steps 8–12 & 18–19. The final list of hyperparameters are summarised in the bottom right. The final four hyperparameters are specific to VOGN, and we provide best practices for tuning them at the end of the blog post.

## Results on ImageNet

We are finally in a place to run VOGN on ImageNet and analyse results. We take Algorithm 1and run it on ImageNet.

Figure 1: Results on ImageNet. VOGN converges in about as many epochs as Adam and SGD (top left plot), but is almost twice as slow per epoch (top middle plot). VOGN's calibration is better (top right plot). Overall, VOGN gets good accuracy and uncertainty metrics. See the paper for standard deviations over many runs.

Let’s go through these results slowly.

• Top left plot: VOGN’s convergence rate (per epoch) is very similar to Adam’s. The step increases (at epochs 30 and 60) are due to learning rate scheduling, which is best practice for training algorithms on ImageNet.
• Top middle plot: VOGN is about twice as slow (total time) compared to SGD and Adam. This is impressive compared to other approaches like Bayes-By-Backprop (Blundell et al., 2015), which currently can’t scale to ImageNet even if given an order of magnitude more time!
• Top right plot: In this calibration curve, the red line is closer to the diagonal than the other lines, showing better calibration. This plot is summarised in the ECE (Expected Calibration Error) column in the Table, where VOGN is better than SGD and Adam.
• Turning our attention to the Table, MC-dropout gets very good ECE, but this is at the cost of validation accuracy, and only achieved after a fine-grain sweep of hyperparameters (specifically the dropout rate, see Appendix G in the paper).
• OGN is a deterministic version of VOGN, where we do not use the reparameterisation trick to sample $\vparam_t$ during training (Steps 8 & 9 in Algorithm 1), and instead just use the mean $\vmu_t$.
• K-FAC has a Kronecker-factored structure as in Osawa et al. (2018), where they impressively trained on ImageNet in very few iterations. This blog post provides an introduction to K-FAC at a large scale.
• OGN and K-FAC perform well, but their metrics (particularly validation accuracy, validation negative-log-likelihood and ECE) are worse than VOGN.
• Noisy K-FAC (Zhang et al., 2018) takes a similar algorithm to VOGN and adds K-FAC structure to the covariance matrix. It is therefore more computationally expensive than VOGN (slower per epoch and total training time), but requires fewer epochs. It performs decently, but not as well as VOGN in this experiment.

There are many more results in Osawa et al. (2019), such as CIFAR-10 with a variety of architectures. We tend to see a similar story, where VOGN performs comparably on validation accuracy, and well on uncertainty metrics.

Due to the Bayesian nature of VOGN, we see some interesting trade-offs (see the paper for figures).

1. Reducing the prior precision $\delta$ results in higher validation accuracy, but also a larger train-test gap, corresponding to more overfitting. With very small prior precisions, performance is similar to non-Bayesian methods like Adam.

2. Increasing the number of training MC samples ($K$ in Algorithm 1) improves VOGN’s convergence rate and stability, as it reduces gradient variance during training. But this is at the cost of increased computation. Increasing the number of MC samples during testing improves generalisation.

## Downstream uncertainty performance

If you are like me, metrics such as negative-log-likelihood and expected calibration error do not mean much when it comes to analysing if your algorithm has ‘better uncertainty’. We should also test on downstream tasks to see how reliable our methods are, and more and more papers are starting to do so (see also this year’s NeurIPS Bayesian Deep Learning workshop, which makes this a priority). The VOGN paper tested on two downstream tasks: continual learning and out-of-distribution performance.

Continual Learning: I personally think continual learning is a very good way to test approximate Bayesian inference algorithms, particularly variational deep-learning algorithms. We tested VOGN on Permuted MNIST, finding it performs as well as VCL (Nguyen et al., 2018; Swaroop et al., 2019), but trained more than an order of magnitude quicker. More recently, VOGN has also achieved good results on a bigger Split CIFAR benchmark (see Section 4.5 of Eschenhagen (2019)), which VCL struggles to scale to.

Out-of-distribution performance: We also tested VOGN on standard out-of-distribution benchmarks, such as training on CIFAR-10 and testing on SVHN and LSUN. Figure 5 in the paper shows results (histograms of predictive entropy), where we qualitatively see VOGN performing well.

In the first blog post, we derived VOGN, our natural-gradient variational inference algorithm. In this blog post, we scaled it all the way to ImageNet. We made approximations along the way, but by being clever about when and where to make approximations, we have ended up with a practical algorithm that has Bayesian principles. Our final algorithm therefore performs reasonably well in downstream tasks.

It has been two years since publishing VOGN’s performance on ImageNet, and the field continues to move at break-neck pace. More algorithms and more benchmarks continue to be published, as well as more insight into VI.

• If you are interested in the maths of improving natural-gradient variational inference algorithms, I particularly recommend work by Wu Lin and co-authors. They looked at improving VON (same as VOGN but without the Gauss-Newton matrix approximation), deriving another quick NGVI algorithm that can perform well (Lin et al., 2020). They have also expanded to mixtures of exponential family distributions (Lin et al., 2019), and looked at structured natural gradient descent, drawing links to Newton-like algorithms (Lin et al., 2021).
• There is also interesting work looking at pathologies of mean-field VI on neural networks (VOGN is a mean-field VI algorithm). There are problems in the single-hidden-layer setting (Foong et al., 2020), and problems in the wide limit (Coker et al., 2021).

## Acknowledgements

Firstly, many thanks to my co-authors Kazuki Osawa, Anirudh Jain, Runa Eschenhagen, Richard Turner, Rio Yokota and Emtiyaz Khan, many of whom also provided valuable feedback on these blog posts. I would also like to thank Andrew Foong, Wessel Bruinsma and Stratis Markou for their comments during drafting of these blog posts.

## Post-scipt: A guide on how to tune VOGN

As we saw in Algorithm 1, there are many hyperparameters that need tuning for VOGN (and generally for VI at a large-scale). Here we briefly summarise how we did this in Osawa et al. (2019), following the guidelines presented in Section 3 of the paper. The key idea is to make sure VOGN training closely follows Adam’s trajectory in the beginning of training.

1. First, we tune hyperparameters for OGN, which is the same as VOGN except setting $\vparam_t=\vmu_t$ (no MC sampling). OGN is more stable than VOGN and is a convenient stepping stone as we move from Adam to VOGN. So, we initialise OGN’s hyperparameters at Adam’s values, and tune until OGN is competitive with Adam. This requires tuning learning rates, prior precision $\delta$, and setting a suitable value for the data augmentation factor (if using data augmentation).

2. Then, we move to VOGN. Now, we (fine-)tune the prior precision $\delta$, warm-start the tempering parameter $\tau$ (such as increasing $\tau$ from $0.1\rightarrow1$ during the first few optimisation steps), and the number of MC samples $K$ for VOGN (more samples means more stable training, but more computation cost). We also now consider adding an external damping factor $\gamma$ if required.

Published on 24 November 2021.


text:
# Recommended root canal

Algebra Level 4

Find the largest possible product $n\cdot m$ of two distinct positive integers $n$ and $m,$ such that the equations $(x+1)^n=x^n+1$ and $(x+1)^m=x^m+1$ have the same set of complex roots.

Details and assumptions

The multiplicities of the roots can, of course, be different, but each root of one equation is a root of the other.

×


text:
# Continuously stirred tank reactor system

## Model description:

A schematic of the CSTR plant is shown in the attached image. The process dynamics are described by

\begin{align*} \dot{C}_{a} &=\frac{q}{V}(C_{a0}-c_{a})-a_{0}C_{a}e^{-\frac{E}{RT_{a}}} \\ \dot{T}_{a} &=\frac{q}{V}(T_{f}-T_{a})+a_{1}C_{a}e^{-\frac{E}{RT_{a}}}+a_{3}q_{c}\left(1-e^{\frac{a_{2}}{q_{c}}}\right)(T_{cf}-T_{a}), \end{align*}

where the variables $C_a$ and $T_a$ are the concentration and temperature of the tank, respectively; the coolant flow rate $q_c$ is the control input and the parameters of the plant are defined in the attached table. Within the tank reactor, two chemicals are mixed and react to produce a product compound $A$ at a concentration $C_a(t)$ with the temperature of the mixture being $T(t)$. The reaction is both irreversible and exothermic.

In the paper, authors assumed that plant parameters $q, C_{a0}, T_f$ and $V$ are at the nominal values given in the attached table. The activation energy $E/R = 1 \times 10^4K$ is assumed to be known. The state variables the input and the output are defined as $x=[x_1,x_2]^{\mathrm T}=[C_a,T_a]^{\mathrm T},u=q_c,y=C_a$. Using this notation, the CSTR plant can be re-expressed as

\begin{align*} \dot{x}_{1} &=1-x_{1}-a_{0}x_{1}e^{-\frac{10^4}{{\rm a}_2}} \\ \dot{x}_{2} &=T_{f}-x_{2}+a_{1}x_{1}e^{-\frac{10^4}{{\rm a}_2}}+a_{3}u\left(1-e^{-\frac{a_2}{u}}\right)(T_{cf}-x_{2}) \\ y &= x_{1}, \end{align*}

where the unknown constant parameters are $a_0, a_1, a_2$ and $a_3$.

## Publication details:

 Title Adaptive nonlinear control of continuously stirred tank reactor systems Publication Type Conference Paper Authors Zhang, T., and Guay M.

# Discrete bilinear plant

## Model description:

The plant is

$$y(k)=1.2y(k-1)-0.8y(k-2)+0.2y(k-1)u(k-1)+u(k-1)+0.6u(k-2) + d(k),$$

where $d(k)$ is a disturbance.

2

## Publication details:

 Title Adaptive Bilinear Model Predictive Control Publication Type Conference Paper Authors Yeo, Y.K., and Williams D.C.

# SISO NLTI plant

## Model description:

Consider the following unknown discrete nonlinear dynamic system:

\begin{align*} y(k+1)&=p[{\bf q}(k), u(k)]=0.2\cos[0.8(y(k)+y(k-1))] \\ & +0.4\sin[0.8(y(k)+y(k-1))+2u(k)+u(k-1)] \\ &+0.1[9+y(k)+y(k-1)]+\left[{2(u(k)+u(k-1))\over 1+\cos(y(k))}\right] \end{align*}

for $k=0,1,2,\ldots$ with $y(k)=0,u(k)=0$, for $k=0,-1,-2,\ldots$, $\Delta t := t(k+1)-t(k)=0.02sec$, for $k=0,1,2,\ldots$.

2

## Publication details:

 Title Robust nonlinear adaptive control using neural networks Publication Type Conference Paper Authors Adetona, O., Sathananthan S., and Keel L.H.

# 2-input 2-output nonlinear system

## Model description:

The suggested tracker scheme is tested with a 2-input 2-output nonlinear system given by:

\begin{align*} y_{1} (k) & = 0.21y_{1} (k-1)-0.12y_{2} (k-2) \\ & + 0.3y_{1} (k-1)u_{2} (k-1)-1.6u_{2} (k-1) \\ & + 1.2u_{1} (k-1), \\ y_{2} (k) & = 0.25y_{2} (k-1)-0.1y_{1} (k-2) \\ &- 0.2 y_{2} (k-1)u_{1} (k-1)-2.6u_{1} (k-1) \\ &-1.2u_{2} (k-1). \end{align*}

## Publication details:

 Title U-model Based Adaptive Tracking Scheme for Unknown MIMO Bilinear Systems Publication Type Conference Paper Authors Azhar, A.S.S., Al-Sunni F.M., and Shafiq M.

# Kinematic Car Model

## Model description:

The attached image depicts the kinematic car in the horizontal plane. Let us suppose that the Ackermann steering assumptions hold true, hence all wheels turn around the same point (denoted by P) which lies on the line of the rear axle. It follows that the kinematics of the car can be fully described by the kinematics of a bicycle fitted in the middle of the car (see attached image. The coordinates of the rear axle midpoint are given by $x$ and $y$. The orientation of the car with respect to the axis of $x$ is denoted by 9. The angle of the front wheel of the bicycle with respect to the longitudinal symmetry axis of the car is denoted by $φ$ . One may consider $φ$ or its time derivative $u_2=\dot{φ}$ as input. The longitudinal velocity of the rear axle midpoint is denoted by $u_1$ if it is a control input (two input case) and by $v_{car}$ if not (one input case). All lengths involved in the kinematic calculations, and in particular $l$, equal to one.

\begin{align*} \dot{x} &= u_1 \cos{\theta},\\ \dot{y} &= u_1 \sin{\theta},\\ \dot{\theta} &= u_1 \tan{\varphi},\\ \dot{\varphi} &= u_2. \end{align*}

4

## Publication details:

 Title On-line time-scaling control of a kinematic car with one input Publication Type Conference Paper Authors Kiss, B., and Szadeczky-Kardoss E.


text:
Codeforces celebrates 10 years! We are pleased to announce the crowdfunding-campaign. Congratulate us by the link https://codeforces.com/10years. ×

A. Expression
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

Petya studies in a school and he adores Maths. His class has been studying arithmetic expressions. On the last class the teacher wrote three positive integers a, b, c on the blackboard. The task was to insert signs of operations '+' and '*', and probably brackets between the numbers so that the value of the resulting expression is as large as possible. Let's consider an example: assume that the teacher wrote numbers 1, 2 and 3 on the blackboard. Here are some ways of placing signs and brackets:

• 1+2*3=7
• 1*(2+3)=5
• 1*2*3=6
• (1+2)*3=9

Note that you can insert operation signs only between a and b, and between b and c, that is, you cannot swap integers. For instance, in the given sample you cannot get expression (1+3)*2.

It's easy to see that the maximum value that you can obtain is 9.

Your task is: given a, b and c print the maximum value that you can get.

Input

The input contains three integers a, b and c, each on a single line (1 ≤ a, b, c ≤ 10).

Output

Print the maximum value of the expression that you can obtain.

Examples
Input
123
Output
9
Input
2103
Output
60


text:
##### I need help with this problem asap

label Management
account_circle Unassigned
schedule 1 Day
account_balance_wallet \$5

Jul 20th, 2015

--------------------------------------------------------------------------

$\bar{\bar{x}}$ is the average of averages.

$\bar{\bar{x}}=\frac{1}{k}\sum\bar{x}_i$

where k is the number of groups

and $\bar{x}_i$ is the average/mean of the i th group.

$\\ \bar{\bar{x}}=\frac{156.9+153.2+155.6+157.5+156.6}{5}\\ \\ \bar{\bar{x}}=155.96\hspace{5}mm$

--------------------------------------------------------------------------

Jul 20th, 2015

yes it says round that problem to two decimal places

Jul 20th, 2015

## What are the UCLx and LCLx, using 3s? Plot the data. What are the UCLR and LCLR, using 3s? Plot the data.If the true diameter mean should be 155 mm and you want this as your center ( nominal) line, what are the new UCLx andLCLx?

Jul 20th, 2015

I need help with these problems that I just post up

Jul 20th, 2015

Jul 20th, 2015

i hope so let me try it

Jul 20th, 2015

I am not sure whether these are correct. These are based on my understanding of the problem solved in

$UCL_x = \bar{\bar{x}} + A_2\bar{R}=155.96+0.308\times4.52=157.35\hspace{5}mm$

$LCL_x = \bar{\bar{x}} - A_2\bar{R}=155.96-0.308\times4.52=154.57\hspace{5}mm$

The value for $A_2$ is obtained from the table in

$\\ UCL_R = D_4\bar{R}=1.777\times4.52=8.03\\ \\ LCL_R = D_3\bar{R}=0.223\times4.52=1.01\\$

The values for $D_3\hspace{5}and\hspace{5}D_4$ are obtained from the table in

Jul 20th, 2015

thank you thats correct

Jul 20th, 2015

and this question here

## If the true diameter mean should be 155 mm and you want this as your center ( nominal) line, what are the new UCLx andLCLx?

Jul 20th, 2015

Sorry I don't know how to solve this question.

Jul 20th, 2015

its okay you did great on the other ones thank you

Jul 20th, 2015

:)

Jul 20th, 2015

...
Jul 20th, 2015
...
Jul 20th, 2015
Oct 23rd, 2017
check_circle


text:
# zbMATH — the first resource for mathematics

A theory of may testing for asynchronous calculi with locality and no name matching. (English) Zbl 1275.68106
Kirchner, Hélène (ed.) et al., Algebraic methodology and software technology. 9th international conference, AMAST 2002, Saint-Gilles-les-Bains, Reunion Island, France, September 9–13, 2002. Prodeedings. Berlin: Springer (ISBN 3-540-44144-1/pbk). Lecture Notes in Computer Science 2422, 223-238 (2002).
Summary: We present a theory of may testing for asynchronous calculi with locality and no name matching. Locality is a non-interference property that is common in systems based on object-paradigm. Concurrent languages such as Join and Pict disallow name matching, which is akin to pointer comparison in imperative languages, to provide for an abstract semantics that would allow useful program transformations. May testing is widely acknowledged to be an effective notion for reasoning about safety properties. We provide a trace-based characterization of may testing for versions of asynchronous $$\pi$$-calculus with locality and no name matching, which greatly simplifies establishing equivalences between processes. We also exploit the characterization to provide a complete axiomatization for the finitary fragment of the calculi.
For the entire collection see [Zbl 1045.68011].

##### MSC:
 68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
Full Text:


text:
# The Ace of Base(-three numbers) trick

This trick was shown to me by one of the excellent MathsBusking people. They didn’t explain why it works or how to generalise it to other numbers of cards, so I did a bit of thinking and then I made this video.

Great, well done!

Slight niggle: I disagree “there is no slight of hand”. You have to order the piles differently without people noticing. There is no trickery in the sense that you aren’t hiding the chosen card or forcing a choice.

When we did this at Nottingham MathsJam I also showed the 21 cards trick which is similar except you always put it in the middle with 10 cards either side. I have a note that we thought the trick would work in base $b$ for $x<b^n$ cards for odd $x$ if you always wanted to put it in the middle.

Thanks.

The 21-card version leads much more quickly to an understanding of what’s going on. Actually, I think the condition is that if you are dealing out $p=2m+3$ piles, $m \geq 0$, then decks of $(2n+1)p$ cards, $n \geq 0$, will work in $n+1$ goes. A proof by induction should work.

Actually, I’d better clear this up to make sure I’ve understood: your 21-card version doesn’t ask the mark for a number, and just puts their card in the middle of the deck after a certain number of goes?

Sorry, $n+1$ goes is way too generous. Something like $\lceil \log_p (2n+1)p \rceil$ goes.

Actually, I’d better clear this up to make sure I’ve understood: your 21-card version doesn’t ask the mark for a number, and just puts their card in the middle of the deck after a certain number of goes?

Correct. I was shown this at the same time. Actually, I think it was through copying the 21 version that I worked out how to repeat the 27 version.


text:
## Tag: proofs

15 Why doesn't Q-learning converge when using function approximation? 2019-04-05T18:23:46.233

11 Proof that Artificial General Intelligence is possible 2017-07-03T16:55:46.917

11 Where can I find the proof of the universal approximation theorem? 2019-07-11T08:40:14.007

10 What are the implications of the "No Free Lunch" theorem for machine learning? 2019-09-27T13:52:15.007

9 Why is baseline conditional on state at some timestep unbiased? 2018-09-09T20:31:07.373

9 How do we prove the n-step return error reduction property? 2018-12-08T05:24:56.380

8 Why is A* optimal if the heuristic function is admissible? 2018-04-13T02:25:29.013

6 Can two admissable heuristics not dominate each other? 2019-10-03T00:55:04.350

5 Can deep learning be used to help mathematical research? 2020-05-21T15:40:18.533

4 How do I show that uniform-cost search is a special case of A*? 2018-11-26T14:59:20.600

4 How are the reward functions $R(s)$, $R(s, a)$ and $R(s, a, s')$ equivalent? 2019-02-07T15:38:58.217

4 What is the proof that the branch and bound algorithm always finds optimal path in a graph? 2019-05-19T19:52:06.153

4 How is G(z) related to x in GAN proof? 2019-06-15T23:16:55.513

4 How to show temporal difference methods converge to MLE? 2019-08-14T16:15:30.013

4 Why does estimation error increase with $|H|$ and decrease with $m$ in PAC learning? 2019-09-16T10:51:21.577

4 If an heuristic is not admissible, can it be consistent? 2019-11-09T11:18:09.290

4 Why are the Bellman operators contractions? 2020-07-31T02:48:34.320

3 Why is the max a non-expansive operator? 2019-03-14T20:50:46.340

3 How do I find whether this heuristic is or not admissible and consistent? 2019-03-26T10:33:47.713

3 Is unsupervised disentanglement really impossible? 2019-08-12T00:38:56.740

3 Is the summation of consistent heuristic functions also consistent? 2020-02-28T15:25:48.877

3 What is the proof that policy evaluation converges to the optimal solution? 2020-04-16T06:44:00.997

3 Why does (not) the distribution of states depend on the policy parameters that induce it? 2020-08-27T10:36:32.770

2 Why is exact inference in a Bayesian network both NP-hard and P-hard? 2018-07-05T10:36:42.557

2 Understanding lemma 2 of the "Trust Region Policy Optimization" paper 2018-11-27T16:52:12.273

2 Why Nilsson's Sequence Score isn't an admissible heuristic? 2019-02-24T06:24:58.143

2 Understanding the proof that A* search is optimal 2019-06-27T07:16:51.563

2 How to show Monte Carlo methods converge to an estimate which minimizes mean squared error? 2019-08-14T16:53:35.113

2 What happens to the optimal value function if the reward is multiplied by a constant? 2019-09-15T20:58:34.960

2 Convert a PAC-learning algorithm into another one which requires no knowledge of the parameter 2020-01-16T04:57:48.913

2 A problem about the relation between 1-oracle and 2-oracle PAC model 2020-01-16T12:44:11.397

2 Why does KL divergence not satisfy the triangle inequality? 2020-02-14T08:23:15.227

2 Is there a mathematical theory behind why MLP can classify handwritten digits? 2020-02-14T19:47:27.067

2 Is an oracle that answers only with a "yes" or "no" dangerous? 2020-03-06T14:53:13.503

2 How to prove $\mathcal H$ with VC dimension $d$ shatter all subsets with size less than $d-1$? 2020-03-28T21:12:24.790

2 Does Rice's theorem prove safe AI is undecidable? 2020-04-02T20:39:10.230

2 Is the derivative of the loss wrt a single scalar parameter proportional to the loss? 2020-04-03T12:01:55.227

2 Proof of Maximization Bias in Q-learning? 2020-06-15T18:56:54.630

2 Do we assume the policy to be deterministic when proving the optimality? 2020-08-18T09:32:53.853

1 Why does Q-learning converges to optimal policy even if I am acting suboptimally? 2018-11-17T22:06:42.443

1 Proof of Correctness of Monte Carlo Tree Search 2019-06-10T18:54:18.250

1 Is there a rigorous proof for finding Hopfield minima? 2019-07-22T17:05:02.367

1 Is there a simple proof of the convergence of TD(0)? 2020-02-22T22:59:51.977

1 Does TD(0) prediction require Robbins-Monro conditions to converge to the value function? 2020-02-24T18:00:33.417

1 What are the conditions for the convergence of SARSA to the optimal value function? 2020-02-27T12:53:48.450

1 How to prove that gradient descent doesn't necessarily find the global optimum? 2020-03-16T02:36:18.900

1 Why is it hard to prove the convergence of the deep Q-learning algorithm? 2020-05-10T16:01:31.993

1 How do you prove that minimax algorithm outputs a subgame-perfect Nash equilibrium? 2020-05-28T14:56:25.563

1 What is the proof that "reward-to-go" reduces variance of policy gradient? 2020-06-10T13:38:53.023

1 When to use AND and when to use Implies in first-order logic? 2020-06-18T18:39:43.483

0 Can a computer make a proof by induction? 2020-08-09T03:14:05.647

-3 Is there a proof that states that an AI can become smarter than its creator? 2016-12-29T23:51:07.190


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# Implement Pearson Correlation Coefficient Loss in TensorFlow – TensorFlow Tutorial

By | March 3, 2021

Pearson Correlation Coefficient can measure the strength of the relationship between two variables. Here is a tutorial:

A Beginner Guide to Pearson Correlation Coefficient – Machine Learning Tutorial

We can use it as a loss to measure the correlation between two distributions in deep learning model. In this tutorial, we will create this loss function using tensorflow.

## Preliminary

We will create two distributions in tensorflow.

import numpy as np
import tensorflow as tf
a = np.array([[0.15, 0.16, 0.9], [0.8, 4.15, 0.15]])
b = np.array([[0.7, 0.12, 0.1], [0.15, 0.19, 0.05]])

aa = tf.convert_to_tensor(a, tf.float32)
bb = tf.convert_to_tensor(b, tf.float32)

$$aa$$ and $$bb$$ are two distributions, we will compute their pearson correlation coefficient loss.

## Pearson Correlation Coefficient Loss

Similar to cosine distance loss, pearson correlation coefficient loss is defined as:

$$loss = 1 – p$$

$$p$$ is pearson correlation coefficient.

## How to compute pearson correlation coefficient loss in tensorflow?

We will create a function to calculate. Here is an example:

def pearson_r(y_true, y_pred):
x = y_true
y = y_pred
mx = tf.reduce_mean(x, axis=1, keepdims=True)
my = tf.reduce_mean(y, axis=1, keepdims=True)
xm, ym = x - mx, y - my
t1_norm = tf.nn.l2_normalize(xm, axis = 1)
t2_norm = tf.nn.l2_normalize(ym, axis = 1)
cosine = tf.losses.cosine_distance(t1_norm, t2_norm, axis = 1)
return cosine

In this example, we will use cosine distance loss to compute pearson correlation coefficient loss. Here is the reason:

Understand the Relationship Between Pearson Correlation Coefficient and Cosine Similarity – Machine Learning Tutorial

Then we can compute the pearson loss between $$aa$$ and $$bb$$.

a_s = pearson_r(aa, bb)

init = tf.global_variables_initializer()
init_local = tf.local_variables_initializer()
with tf.Session() as sess:
sess.run([init, init_local])
np.set_printoptions(precision=4, suppress=True)

a = (sess.run(a_s))

print('a=')
print(a)

Run this code, we will get the loss:

0.85890067

## Evaluate our pearson correlation coefficient loss function

In order to make sure our function is correct, we will use scipy.stats.pearsonr() to evaluate our function.

Here is the example code:

from scipy.stats import pearsonr
p1, _ = pearsonr(a[0,:], b[0,:])
p2, _ = pearsonr(a[1,:], b[1,:])
print(p1)
print(p2)
print(p1+p2)

d = 1-(p1+p2)/2
print(d)

Run this code, $$d$$ is:

0.8589005906554071

It is almost same to $$a_s$$ in tensorflow, which means our function is correct.


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# On Natural Coalgebra Decompositions of Tensor Algebras and Loop Suspensions - Memoirs of the American Mathematical Society (Paperback)

£51.50
Paperback 109 Pages / Published: 30/10/2000
• We can order this

Usually dispatched within 2 weeks

Abstract. We consider functorial decompositions of $\Omega\Sigma X$ in the case where $X$ is a $p$-torsion suspension. By means of a geometric realization theorem, we show that the problem can be reduced to the one obtained by applying homology: that of finding natural coalgebra decompositions of tensor algebras. We solve the algebraic problem and give properties of the piece $A^{\mathrm {min}} (V)$ of the decomposition of $T(V)$ which contains $V$ itself, including verification of the Cohen conjecture that in characteristic $p$ the primitives of $A^{\mathrm {min}} (V)$ are concentrated in degrees of the form $p^t$. The results tie in with the representation theory of the symmetric group and in particular produce the maximum projective submodule of the important $S_n$-module $\mathrm {Lie} (n)$.

Publisher: American Mathematical Society
ISBN: 9780821821107
Number of pages: 109
Weight: 227 g
Dimensions: 234 x 156 mm

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text:
Within the field of explainable AI (XAI), the technique of counterfactual explainability has progressed rapidly, with many exciting developments just in the past couple years. To help crystallize and understand the major development areas, we’ll be presenting a new paper at the NeurIPS Workshop on ML Retrospectives, Surveys, and Meta-Analyses. This post will provide a high-level summary, but if you’re interested in getting started in this area, we encourage you to check out the full paper.

## What are counterfactual explanations?

Counterfactual explanations (CFEs) are an emerging technique for local, example-based post-hoc explanations methods. Given a datapoint A and its prediction P from a model, a counterfactual is a datapoint close to A, such that the model predicts it to be in a different class Q (P ≠ Q). A close datapoint is considered a minimal change that needs to be made to get a different prediction. This can help in scenarios like rejection of a loan or credit card request, where the applicant is willing to know about the smallest change in the feature set that can lead to acceptance of the request.

Explicitly, consider a person who applies for a loan and is rejected. In our simple example, say that the person is represented by a length-3 feature vector x: liquid assets of $10K; outstanding debt of$50K; and annual income of $45K. So, x = ($10K, $50K,$45K). Our fictitious loan agency uses a pre-trained binary “loan acceptance” classifier, f, that takes as input length-3 feature representations of applications and returns one of two labels: y = 0 (reject), or y = 1 (accept). Here, then, the applicant is rejected because f( x = ($10K,$50K, $45K) ) = reject. Roughly speaking, a counterfactual explanation for this decision (reject) would describe changes, call them x’, to the applicant’s initial feature vector x such that f( x’ ) = accept. There may be many possible counterfactual explanations: for example, lowering outstanding debt from$50K to $25K to form x’ = ($10K, $25K,$45K); or, increasing liquid assets from $10K to$20K to form x’’ = ($20K,$50K, \$45K). Some may be easier or harder to attain for the applicant -- and some may be completely impossible to achieve -- motivating research into the creation of “the best” counterfactual explanations for a particular use case, which we discuss in greater depth below.

Fig1: Illustrative diagram counterfactual explanations. The datapoint labeled X (blue) got classified in the negative class. CF1 (red) and CF2 (green) are two counterfactuals for X, which the model classifies in the positive class. Several counterfactuals can be generated for a datapoint, which differ in closeness to the original datapoint and other desirable properties.

## Themes of research in CFEs

Much of the literature in counterfactual explanations have proposed algorithms to address additional aspects of the problem. We categorize recent research into the following major themes:

Actionability: A CFE is only useful if it prescribes changes to features that can actually change. It would be unhelpful if I were told to change my birthplace in order to receive a loan.

Sparsity: A useful CFE should modify only a few features in order to be simple and easy to use.

Proximity: A useful CFE should be the smallest possible change that achieves the desired outcome.

Causality: A useful CFE must be able to adhere to any causal constraints that a domain expert specifies. For example, I should not have to decrease my age in order to get a loan.

Data Manifold: A useful CFE should result in a datapoint that is similar to other datapoints seen in the training data. It would be less trustworthy if the resulting datapoint is utterly unlike anything the classifier has ever seen.

Speed: CFEs should be generated quickly for new, incoming datapoints.

Model Access: Some CFE approaches require detailed knowledge of model internals and gradients. Others can work in a black-box fashion and are model-agnostic.

In our survey paper, we collect, review, and categorize 39 recent papers that propose algorithms to solve the counterfactual explanation problem. We design a rubric with desirable properties of counterfactual explanation algorithms and comprehensively evaluate all currently-proposed algorithms against that rubric. This provides easy comparison and comprehension of the advantages and disadvantages of different approaches and serves as an introduction to major research themes in this field. We also identify gaps and discuss promising research directions in the space of counterfactual explainability.

## Conclusions

CFEs present a compelling form of XAI, providing users with understandable and actionable feedback. The additional constraints and desiderata explored in recent years seek to ensure that these explanations are always reasonable and useful. Many exciting open questions remain, and we close our paper by proposing research challenges for the community to tackle in the coming years. We firmly believe that CFEs will form a long-lasting part of the ML explainability toolkit.


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# Ricci tensor for Schwarzschild metric

1. Jan 1, 2014

### CarlosLara

Good morning. I have calculated the components of the Ricci tensor for the Schwarzschild metric and they are all 0, except the last component R33. I have calculated it over and over again, and I keep getting -2(sinΘ)^2. Does anyone know how it vanishes? I accounted for every index, and my Christoffel symbols are correct, but it's not vanishing. Also, for the ø component of the Schwarzschild metric, is it (r^2)(sinΘ)^2 or (r^2)(cosΘ)^2 ? Leonard Susskind uses cosine, but everywhere else I read it's sine. Which is it?

Thank you for your time. Happy New Year

2. Jan 1, 2014

### CarlosLara

I'm sorry, one of my Christoffel symbols was missing a minus sign. Now the component is 0. Thanks anyway.


text:
Document Preview

## Factorizations of diffeomorphisms of compact surfaces with boundary

Wand, Andrew.  University of California, Berkeley. ProQuest Dissertations Publishing, 2010. 3413516.


text:
# Quark Matter 2019 - the XXVIIIth International Conference on Ultra-relativistic Nucleus-Nucleus Collisions

3-9 November 2019
Wanda Reign Wuhan Hotel
Asia/Shanghai timezone

## First direct measurement of the dead-cone effect at colliders using iterative declustering techniques in the Lund plane, with D meson jets in pp collisions at $\sqrt{s}$ = 13 TeV with ALICE

5 Nov 2019, 09:20
20m
Ball Room 2 (Wanda Reign Wuhan Hotel)

### Ball Room 2

#### Wanda Reign Wuhan Hotel

Oral Presentation Jet modifications and medium response

### Speaker

Nima Zardoshti for the ALICE Collaboration (CERN)

### Description

We report the first direct measurement of the dead-cone effect at colliders, using iterative jet declustering techniques in pp collisions at $\sqrt{s}=13$ TeV. The procedure detailed in, exposes the splittings of D mesons in the jet shower, by iteratively declustering the angular ordered C/A tree. The splitting history of the D meson initiated jet is mapped onto the Lund plane, where appropriate cuts can be made to suppress hadronisation effects. The reported variable is the splitting angle with respect to the D meson axis, which is updated after each splitting. Track-based jet finding, along with the low $p_{\rm T}^{\rm ch}$ reach of the ALICE detector, allow for an accurate reconstruction of the splitting angle in the phase-space where the dead-cone effect is expected to be largest. The results are compared to those of inclusive jets and theoretical models.


text:
Partial Derivative Definition. Partial derivative examples. Partial Derivatives – In this section we will look at the idea of partial derivatives. Then we say that the function f partially depends on x and y. In this section we will the idea of partial derivatives. On the page Definition of the Derivative, we have found the expression for the derivative of the natural logarithm function $$y = \ln x:$$ $\left( {\ln x} \right)^\prime = \frac{1}{x}.$ Now we consider the logarithmic function with arbitrary base and obtain a formula for its derivative. For example, if ˙ = is chosen, the time derivative becomes equal to the partial time derivative, which agrees with the definition of a partial derivative: a derivative taken with respect to some variable (time in this case) holding other variables constant (space in this case). Partial derivative is a method for finding derivatives of multiple variables. This definition shows two differences already. Computationally, partial differentiation works the same way as single-variable differentiation with all other variables treated as constant. If you know how to take a derivative, then you can take partial derivatives. Now, if we calculate the derivative of f, then that derivative is known as the partial derivative of f. As you will see if you can do derivatives of functions of one variable you won’t have much of an issue with partial derivatives. First, the notation changes, in the sense that we still use a version of Leibniz notation, but the in the original notation is replaced with the symbol (This rounded is usually called “partial,” so is spoken as the “partial of with respect to This is the first hint that we are dealing with partial derivatives. (1) The above partial derivative is sometimes denoted for brevity. The Definition of the Partial Derivative. More information about video. without the use of the definition). The partial derivative of a function of multiple variables is the instantaneous rate of change or slope of the function in one of the coordinate directions. We will give the formal definition of the partial derivative as well as the standard notations and how to compute them in practice (i.e. Once you understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn't difficult. Learn More at BYJU’S. Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation. Partial Derivative Definition: Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest is held fixed during the differentiation.. Let f(x,y) be a function with two variables. without the use of the definition). Partial derivative definition is - the derivative of a function of several variables with respect to one of them and with the remaining variables treated as constants. We will give the formal definition of the partial derivative as well as the standard notations and how to compute them in practice (i.e. (Unfortunately, there are special cases where calculating the partial derivatives is hard.) Get an idea on partial derivatives-definition, rules and solved examples. If we remove the limit from the definition of the partial derivative with respect to $$x$$, the difference quotient remains: $\dfrac{f(x+h,y)−f(x,y)}{h}.$ This resembles the difference quotient for the derivative of a function of one variable, except for the presence of the $$y$$ variable. Suppose, we have a function f(x,y), which depends on two variables x and y, where x and y are independent of each other.

O autorovi


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# What is the empirical price elasticity of demand for insulin?

A lot of textbooks say that demand for insulin is perfectly inelastic. I don’t think this is quite true (and I am a type 1 diabetic). It’s clearly very inelastic, but not perfectly so. There are things you can do to reduce your insulin intake, e.g. eat less carbohydrates.

I’m curious if any studies have actually gone out and measured this…

• Thanks! They write For chronic conditions prescription drugs, the price elasticity is -0.08 (-0.03) for those whose pre-retirement co-payment is 10% (less than 10%) That suggests that elasticity increases as the co-payment increases, meaning that when it hits 100% it's likely to be significantly larger (in magnitude) than -0.08. -0.2 seems plausible. Aug 20 '20 at 11:36


text:
# Is this right? chem heat of combustion

#### joejo

Hi guys can you please check if this is right....thanks in advance

The heat released by the reaction 2C + 2H2 = CH4 is 74.8kJ/mol. Using this information and a table of standard enthalpy of formation values, determine the heat of combustion for the burning of methane to form CO2 and H2O(g).
C02 = -393.3 kJ/mol
H20 = -286.2 kJ/mol

2C + 2H2 = CH4 H1 = 74.8
C + O2 = CO2 H 2= -393.3

H2 +1/2 O2 = H2O H3 = -286..2

CH4 +3O2 = CO2 + 2H2O -H1 + H2 +2H3 = -1040.5 Kj/mol

Related Introductory Physics Homework News on Phys.org

#### joejo

can anyone help me out?

#### The Bob

I may be incorrect but I got -754.3 kJ mol-1.

The formation of CH4 is 74.8 kJ mol-1.
The formation of CO2 and H2O is -679.5 kJ mol-1.
The formation of O is not counted due to it not changing.

This means that -74.8 kJ mol-1 + (-679.5 kJ mol-1) = -754.3 kJ mol-1.

#### VietDao29

Homework Helper
Bob, you are wrong.
joejo, the H1 is negative -74.8 kJ/mol.
It should be -890.9 kJ.
Viet Dao,

#### The Bob

VietDao29 said:
Bob, you are wrong.
joejo, the H1 is negative -74.8 kJ/mol.
It should be -890.9 kJ.
Viet Dao,
The Bob said:
-74.8 kJ mol-1 + (-679.5 kJ mol-1) = -754.3 kJ mol-1.
It was. See???

#### The Bob

Now I have had another look, I am going to agree with joejo. Why? because he is right. It is -1040.5 kJ mol-1.

Sorry

P.S.
joejo said:
-1040.5 Kj/mol
Just as a small hate of mine, can you please use the correct capitals?? Cheers.

Last edited:

#### VietDao29

Homework Helper
The heat released by the reaction 2C + 2H2 = CH4 is 74.8kJ/mol.
. That means $\Delta H_1 = -74.8 \mbox{ kJ/mol}$.
So $-\Delta H_1 +\Delta H_2 + 2 \Delta H_3 = -890.9 \mbox{ kJ}$.
Am I missing something?
Viet Dao,

Last edited:

#### The Bob

VietDao29 said:
Bob, you are wrong.
Although this did not affect what I thought or wrote, it normally gets people's defenses up and does not allow them to see errors. May I ask that you address the issue in a lighter-hearted way???

VietDao29 said:
Am I missing something?
Nothing is missed. You are correct. I did the missing. However, both of us have done something wrong. joejo has (perhaps) lost some experience here because we both told him what had to be done. Although I was wrong, as was joejo, our methods were correct. What joejo really needed was some aimed help, not the answer. Of this we are both guilty and here is a point where we must learn to help (not tell the answer and method), not to argue and to have a bit of fun along the way.

I will take responsibilty if joejo is confused (not insult intended) due to my incorrect posts.

#### VietDao29

Homework Helper
The Bob said:
May I ask that you address the issue in a lighter-hearted way???
I just want my posts to be neat, short, but also contain enough information. So others can finish it in a glance. . Anyway, I'll take your advice. Thanks
Viet Dao,

#### Oerg

Exothermic reactions should always be negative enthalpy.

Last edited:

#### The Bob

Oerg said:
Exothermic reactions should always be negative enthalpy.
Absolutely correct. So what is a reaction with a positive enthaply?

Endothermic!

#### The Bob

NewScientist said:
Endothermic!

#### joejo

i think its still wrong....

shouldn't it be.... -802.3kJ

#### joejo

By definition the burning of methane to make CO2 and H2O is...

CH4 + 2 O2 --> CO2 + 2 H2O

The heat for this reaction is 2(-286.3) + (-393.3) -(-74.8) - 0 = -891.1 kJ.

right??

#### joejo

can anyone help me out?

Which is it?

#### joejo

the bob....read above....i'll post it below just in case u never saw it.

By definition the burning of methane to make CO2 and H2O is...

CH4 + 2 O2 --> CO2 + 2 H2O

The heat for this reaction is 2(-286.3) + (-393.3) -(-74.8) - 0 = -891.1 kJ.

right??

#### The Bob

joejo said:
the bob....read above....i'll post it below just in case u never saw it.
I did not miss it. I saw it. I was commenting on this:

joejo said:
By definition the burning of methane to make CO2 and H2O is...

CH4 + 2 O2 --> CO2 + 2 H2O

The heat for this reaction is 2(-286.3) + (-393.3) -(-74.8) - 0 = -891.1 kJ.
and
joejo said:
i think its still wrong....

shouldn't it be.... -802.3kJ
These ARE two different answers, both posted by you.

The next problem is also by you. Your original post states:
joejo said:
H2 +1/2 O2 -> H2O is -286.2
yet you said in your last post:
joejo said:
the bob....read above....i'll post it below just in case u never saw it.

By definition the burning of methane to make CO2 and H2O is...

CH4 + 2 O2 --> CO2 + 2 H2O

The heat for this reaction is 2(-286.3) + (-393.3) -(-74.8) - 0 = -891.1 kJ.
You have got a difference for the formation of water. You said it was 286.2 and now you say it is 286.3. Please make up your mind.

If the value is 286.2 then me and VietDao29 are right. If it is 286.3 then you are right.

This is why I asked. Ok?????

P.S. May I ask you not to be sarcastic in future. It makes me, if no one else, very on edge and will explain all of the boldness in this post. I was annoyed due to your sarcasm.

#### joejo

i wasn't being sarcastic...sorry for the misunderstanding...so was my second answer right...

#### The Bob

joejo said:
i wasn't being sarcastic...sorry for the misunderstanding...so was my second answer right...
In which case I apologise. I did misunderstand.

Your answer is correct if the Enthalpy change of Formation for water is -286.3 kJ mol-1. If it is -286.2 kJ mol-1 then VietDao29 and me are correct.

P.S. Apologise again.

Last edited:

#### joejo

thank you...just to verify...my second answer (-891.1 kJ.) right?? sorry im getting confused and I can imagine you are too! thanks again

#### The Bob

joejo said:
thank you...just to verify...my second answer (-891.1 kJ.) right?? sorry im getting confused and I can imagine you are too! thanks again
Yes, if the enthaply change of formation of water is -286.3 kJ mol-1.

#### joejo

thanks again!! :)

### Physics Forums Values

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text:
# Comparing Bulletproofs+ and Bulletproofs - Part I

Bulletproofs+ is a recently proposed range proof which is similar in spirit to Bulletproofs. Both of these range proof protocols enable proof sizes logarithmic in the number of bits of the range using a recursive inner product protocol. Bulletproofs relies on the improved inner product protocol 1 while Bulletproofs+ uses a weighted inner product protocol 2.

### Improved Inner Product Argument

The recursive inner product argument introduced in [1] gives an argument of knowledge for the language given by

$\mathcal{L}_{\textsf{IP}}(\sigma) = \bigg\{ \underbrace{\textbf{g}, \textbf{h} \in \mathbb{G}^n, P \in \mathbb{G}, c \in \ \mathbb{Z}_q}_{\textsf{statement}}; \ \underbrace{\textbf{a}, \textbf{b} \in \mathbb{Z}_q^n}_{\textsf{witness}} \ \bigg| \ \underbrace{P = \textbf{g}^{\textbf{a}} \cdot \textbf{h}^{\textbf{b}} \cdot u^{c} \ \wedge \ c = \langle \textbf{a}, \textbf{b} \rangle}_{\textsf{relation}} \bigg\}$

where $\sigma = (\mathbb{G}, q = |\mathbb{G}|, u \in \mathbb{G})$ is the common reference string (crs). Note that $P$ is a Pedersen vector commitment to $\textbf{a}, \textbf{b} \in \mathbb{Z}_q^n$. We can write the witness as $\textbf{a} = (\textbf{a}_L, \textbf{a}_R) \in \mathbb{Z}_q^{\frac{n}{2} \times 2}$ and $\textbf{b} = (\textbf{b}_L, \textbf{b}_R) \in \mathbb{Z}_q^{\frac{n}{2} \times 2}$. Similarly, the base vectors can be written as $\textbf{g} = (\textbf{g}_L, \textbf{g}_R) \in \mathbb{G}^{\frac{n}{2} \times 2}$ and $\textbf{h} = (\textbf{h}_L, \textbf{h}_R) \in \mathbb{G}^{\frac{n}{2} \times 2}$. Now, we compute Pedersen vector commitments to the vector tuples $(\textbf{a}_L, \textbf{b}_R)$ and $\textbf{a}_R, \textbf{b}_L$ as

$L = \textbf{g}_R^{\textbf{a}_L} \cdot \textbf{h}_L^{\textbf{b}_R} \cdot u^{\langle \textbf{a}_L, \textbf{b}_R\rangle}, \quad R = \textbf{g}_L^{\textbf{a}_R} \cdot \textbf{h}_R^{\textbf{b}_L} \cdot u^{\langle \textbf{a}_R, \textbf{b}_L\rangle}.$

Given a challenge scalar $x \in \mathbb{Z}_q$, we can blind the original witness to get

$\hat{\textbf{a}} = x\textbf{a}_L + x^{-1}\textbf{a}_R, \quad \hat{\textbf{b}} = x^{-1}\textbf{b}_L + x\textbf{b}_R.$

The 4-tuple $(L,R,\hat{\textbf{a}}, \hat{\textbf{b}})$ becomes a valid proof of knowledge of the witness. This can be verified by checking the following

$L^{x^2} \cdot P \cdot R^{x^{-2}} = (\textbf{g}_L^{x^{-1}} \circ \textbf{g}_R^x)^{\hat{\textbf{a}}} \cdot (\textbf{h}_L^{x} \circ \textbf{h}_R^{x^{-1}})^{\hat{\textbf{b}}} \cdot u^{\langle \hat{\textbf{a}}, \hat{\textbf{b}}\rangle}$

Instead of publishing vectors $(\hat{\textbf{a}}, \hat{\textbf{b}})$, we can further repeat the same process of dividing $(\hat{\textbf{a}}, \hat{\textbf{b}})$ in halves and compute $(L_1,R_1,\hat{\textbf{a}}_1, \hat{\textbf{b}}_1)$ and so on until we are left with scalars $(\hat{a},\hat{b}) \in \Z_q$. This is the main idea of the log-sized inner product protocol and the proof is of the form

$(L, R),\ (L_1, R_1), \ \ldots, \ (L_{\lceil \text{log}_2n \rceil - 1}, R_{\lceil \text{log}_2n \rceil - 1}), \ (\hat{a},\hat{b}).$

Note that the above argument is perfectly complete and sound, but not zero-knowledge. This is because we cannot construct a PPT simulator who could generate a valid transcript given statement and the crs.

### Weighted Inner Product Argument

The weighted inner product argument uses a weighted inner product operation $\odot_y: \mathbb{Z}^{n}_q \times \mathbb{Z}^{n}_q \mapsto \mathbb{Z_q}$ defined as $\textbf{a} \odot_{y} \textbf{b} = \langle \textbf{a}, \overrightarrow{y} \circ \textbf{b} \rangle$ where $\overrightarrow{y} = (y, y^2, \ldots, y^n) \in \mathbb{Z}_q^n$ and $n = |\textbf{a}| = |\textbf{b}|$. The weighted inner product is essentially a way to combine multiple equations into a single equation. For instance, $\textbf{a} \circ \textbf{b} = \textbf{c} \in \mathbb{Z}_q^n$ represent $n$ distinct equations. The weighted inner product combines these equations to give a single equation of the form $\sum_{i=1}^{n} y^{i} \cdot (a_ib_i) = \textbf{a} \odot_{y} \textbf{b} = \langle \textbf{c}, \overrightarrow{y} \rangle$

The weighted inner product protocol is also bilinear and satisfies the following property 3 $\textbf{a} \odot_{y} \textbf{b} = \textbf{a}_L \odot_{y} \textbf{b}_L + (y^{\frac{n}{2}} \cdot \textbf{a}_R) \odot_{y} \textbf{b}_R.$

We now wish to design a proof system based on the weighted inner product. The language of the weighted inner product protocol becomes $\mathcal{L}_{\textsf{WIP}}(\sigma) = \bigg\{ \underbrace{\textbf{g}, \textbf{h} \in \mathbb{G}^n, P \in \mathbb{G}, c \in \ \mathbb{Z}_q}_{\textsf{statement}}; \ \underbrace{\textbf{a}, \textbf{b} \in \mathbb{Z}_q^n}_{\textsf{witness}} \ \bigg| \ \underbrace{P = \textbf{g}^{\textbf{a}} \cdot \textbf{h}^{\textbf{b}} \cdot u^{c} \ \wedge \ c = \textbf{a} \odot_{y} \textbf{b}}_{\textsf{relation}} \bigg\}$

Similar to the inner product protocol, suppose the prover commits to vectors $(\textbf{a}_L, \textbf{b}_R)$ and $(\textbf{a}_R, \textbf{b}_L)$ and the scalars $c_L = \textbf{a}_L \odot_{y} \textbf{b}_R, \ c_R = (y^{\frac{n}{2}} \cdot \textbf{a}_R) \odot_{y} \textbf{b}_L$, the prover must convince the verifier of the relation $c = \textbf{a} \odot_y \textbf{b}$. For this, we blind the original witness vectors using a challenge scalar $x \in \mathbb{Z}_q$. $\hat{\textbf{a}} = x\textbf{a}_L + x^{-1}(y^{\frac{n}{2}} \cdot \textbf{a}_R), \quad \hat{\textbf{b}} = x^{-1}\textbf{b}_L + x\textbf{b}_R. \\[4pt] \implies \hat{\textbf{a}} \odot_y \hat{\textbf{b}} = x^2c_L + c + x^{-2}c_R.$

Now, similar to the inner product argument, the 4-tuple $(L,R,\hat{\textbf{a}}, \hat{\textbf{b}})$ becomes a valid proof of knowledge of the witness. We can shrink this argument too to logarithmic in the size of witness vectors. The WIP based argument presented in Bulletproofs+ paper is zero-knowledge as against the inner product argument. This is achieved by including randomly chosen blinding factors $d_L, d_R \in \mathbb{Z}_q$ in computing the Pedersen vector commitments $L, R$ as

$L = \textbf{g}_R^{\textbf{a}_L} \cdot \textbf{h}_L^{\textbf{b}_R} \cdot u^{\langle \textbf{a}_L, \textbf{b}_R\rangle} \cdot h^{d_L}, \quad R = \textbf{g}_L^{\textbf{a}_R} \cdot \textbf{h}_R^{\textbf{b}_L} \cdot u^{\langle \textbf{a}_R, \textbf{b}_L\rangle} \cdot h^{d_R}.$

where $h \in \mathbb{G}$. Further, in the last round of the protocol, instead of sending $\hat{a}, \hat{b} \in \mathbb{Z}$, we use a sigma-like protocol yeilding constant communication and computation. This is the main difference in the inner product and the weighted inner product arguments. We will see in the next blog how this helps building Bulletproofs+ - a range proof protocol with proof size shorter than that of Bulletproofs.

### References and Notes

1. Benedikt Bünz et al, “Bulletproofs: Short Proofs for Confidential Transactions and More”, in Cryptology ePrint Archive, Report 2017/1066, Available at https://eprint.iacr.org/2017/1066.pdf

2. Chung H. et al, “Bulletproofs+: Shorter Proofs for Privacy-Enhanced Distributed Ledger”, in Cryptology ePrint Archive, Report 2020/735, Available at https://eprint.iacr.org/2020/735.pdf

3. Note that $\textbf{a}_L \odot_{y} \textbf{b}_L = \langle \textbf{a}_L, (y,y^2, \ldots, y^{\frac{n}{2}}) \circ \textbf{b}_L \rangle$ for $\textbf{a}_L, \textbf{b}_L \in \mathbb{Z}_q^{\frac{n}{2}}$