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# comparing the reaction of magnesium and high melting point

Sodium Stearyl Fumarate (EP/USP-NF/JPE) CH (CH O US DMF

High melting point(220-240℃) THE PERFECT LUBRICANT Magnesium Stearate is incompatible with many APIs, mainly because of the electrophilic character of the Mg2+ ion and the basic reaction of the material itself. Magnesium Stearate shows higher voltage and retention times than SSF. Low electric charge and retention improve lubricant

Difference between Maleic and Fumaric Acid? | …

25.11.2013· Maleic has a 139 melting point, solubility of 79 and density of 1.590 g/mL; while fumaric has 287 C, solubility of 0.7 and density of 1.637 g/mL. fumaric having higher melting point, indies that it is a trans isomer and it is more stable compare to maleic acid (since it requires higher energy to break more stable compounds).

Melting point - Wikipedia

The melting point (or, rarely, liquefaction point) of a substance is the temperature at which it changes state from solid to liquid.At the melting point the solid and liquid phase exist in equilibrium.The melting point of a substance depends on pressure and is usually specified at a standard pressure such as 1 atmosphere or 100 kPa.. When considered as the temperature of the reverse change

What Are the Freezing, Melting, and Boiling …

The terms melting point or freezing point are often interchanged depending on whether a substance is being heated or cooled. For liquids it is known as the freezing point and for solids it is called the melting point. The melting point of a solid and the freezing point of the liquid are normally the same.

Which of the following is not a property of …

Which of the following is not a property of magnesium oxide? See below for the correct answer. 100% correct and accurate.

Alkali and Alkaline Earth Metals - TechnologyUK

Alkali and Alkaline Earth Metals. The elements in group one of the periodic table (with the exception of hydrogen - see below) are known as the alkali metals because they form alkaline solutions when they react with water. This group includes the elements lithium, sodium, potassium, rubidium, caesium and francium.Each of these elements has just one valence electron, which means that they form

Magnesium | Definition of Magnesium at …

Magnesium definition, a light, ductile, silver-white, metallic element that burns with a dazzling white light, used in lightweight alloys, flares, fireworks, in the manufacture of flashbulbs, optical mirrors, and precision instruments, and as a zinc substitute in batteries. Syol: Mg; atomic weight: 24.312; atomic nuer: 12; specific gravity: 1.74 at 20°C.

Why is the melting point of magnesium less …

Excellent question. I’ll try to answer it. You see Calcium and Magnesium are group two elements. The Group 2 elements are all metals with metallic bonding, so you expect their melting points to be high. In metallic bonding, metal ions in a meta

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Lakhmir Singh Chemistry Class 10 Solutions for Chapter 1 Chemical Reactions and Equations is provided here according to the latest CBSE syllabus. Download the solution in …

Melting Point of Chemical Elements - Periodic …

Hydrogen is a chemical element with atomic nuer 1 which means there are 1 protons and 1 electrons in the atomic structure.The chemical syol for Hydrogen is H. With a standard atomic weight of circa 1.008, hydrogen is the lightest element on the periodic table. Its monatomic form (H) is the most abundant chemical substance in the Universe, constituting roughly 75% of all baryonic mass.

The importance of melting temperature in …

11.04.2012· The melting temperature of an oligonucleotide duplex, or T m, is the temperature at which half of the oligonucleotide molecules are single-stranded and half are double-stranded, i.e., the oligonucleotide is 50% annealed to its exact complement.T m is a critical parameter to consider when designing and performing many molecular biology experiments, including PCR and qPCR.

pb metallic stearates pdf - Baerlocher

Contrary to most other lubricants and release agents, metallic stearates are characterised by their relatively high melting point. Therefore, metallic stearates in the form of a fine dry powder can act as a dry lubricant (like graphite) when used in this solid powder form. In most cases, the hydrophobic nature of metallic stearates is highly

Magnesium Aluminate Spinel Raw Materials for High

Magnesium aluminate spinel is a highly refractory material, with a melting point of 2135oC. Table 1 shows a comparison of the density, thermal expansion coefficient and thermal conductivity of the end-meers (MgO, Al2O3) compared to the spinel phase. The differences between these phases in thermal expansion are responsible, in part, for

Magnesium Reacts With Sodium Fluoride To …

The reaction of magnesium fluoride (MgF2) with sodium (Na) to form sodium fluoride (NaF) and magnesium (Mg) is shown in the equation below. Physical properties: Sodium oxide is a white crystalline solid. $2Na(s) + Cl_2(g) \rightarrow 2NaCl(s)$ Sodium Chloride is used as a preservative for meat and to melt the ice on the roads (via freezing point depression).

AQA A Level chemistry - A2 Unit 5: Section 3.5.2

Magnesium forms only one oxide, magnesium oxide MgO: Physical properties. It is a white powder with a high melting point. This is indiive of its giant ionic structure. Chemical properties. Magnesium oxide is a weakly basic oxide and is sparingly soluble in water, in which it reacts to form magnesium hydroxide solution: MgO(s) + H 2 O(l) Mg

Essay on The Melting Point Analysis of Crude …

The difference between the pure and impure samples was observed by comparison of melting points. It was found that impure sample had a lower and wider melting point range of 120.1-122.2 (C). The pure sample melting point range was 121.3-122.5 (C). These ranges helped determine purity by comparing the known melting point of pure benzoic

Why does manganese chloride (MgCl2) have a …

11.11.2012· But to compare manganese chloride or magnesium chloride (MnCl2 and MgCl2) to HCl is really comparing network solids to discrete molecules. Network solids whether the bonds have greater ionic character as in NaCl or greater covalent character as in SiO2 have high melting points because of the TYPE of structure, not the bond type. 1 0.

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Get an answer for ''Explain why Magnesium chloride has a high melting point. Please explain in details because this question has three marks.'' and find homework help for other Science questions at

Common IGCSE Chemistry Exam Questions With …

Why ionic compound does has high melting and boiling point? Explain in terms of bonding and structure. Sample Question: Explain, in terms of its structure and bonding, why magnesium oxide has a very high melting point.(4 marks) Ans: Magnesium Oxide is giant ionic structure, it contains Mg 2+ and O 2-ions, there is high attraction between the oppositely charged ions, takes lots of energy to

TRANSITION METALS chemical reactions …

The physical properties of Transition Metals like density, melting points, boiling points, strength are described and discussed along with a description of the important transition metal chemical properties of e.g. titanium, vanadium, manganese, iron, cobalt, nickel, copper and zinc. There are also sections on how metals can be improved to increase their usefulness e.g. alloys and they are

WebElements Periodic Table » Magnesium » …

Magnesium oxide is more normally made by heating magnesium carbonate. Calcium, immediately below magnesium in the periodic table is more reactive with air than magnesium. 2Mg(s) + O 2 (g) → 2MgO(s) 3Mg(s) + N 2 (g) → Mg 3 N 2 (s) Reaction of magnesium with water. Magnesium does not react with water to any significant extent.

Resistance of pure and mixed rare earth silies against

SiC, and a high melting point for appliions above 1200°C. Furthermore, Yb, have a tendency to form additional metastable reaction products.35 Comparing these studies, the factors dominating the re- silies against calcium-magnesium-aluminosilie

comparison of magnesium and calcium metal …

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Chapter 1 - The Alkaline Earths as Metals

magnesium metal. The metal is used in ﬂares, tracer bullets and incendiary bos as it burns with a brilliant white light. It has also been alloyed with aluminum to produce a low-density, strong material used in aircraft. Magnesium oxide has such a high melting point that it is used to line furnaces. The alkaline earth elements are found in

Physical Changes vs Chemical Changes Lab

Melting Points Wax melting points range from 121° F to approximately 160° F By adding Stearic Acid or other additives you can increase the melting point of wax so that the candle will burn more slowly.


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# Removing aluminum “corrosion?”

I have some aluminum alloy, possibly 6063 or 7050 (no clue really) which has corroded. The finish might be silver-anodized but I doubt it due to the high polish, could be urethane. The item is a fishing rod holder, used in a freshwater marine environment. Hardware is stainless and steel. Several screws and bolts need to be removed, and I was wondering if there was an acceptable reducing agent for the corrosion rather than busting driver bits and/or mangling threads.

Aluminum corrosion in most environment produces $\ce{Al2O3}$, corundum, which is extremely hard (9 on the Mohs scale, where 10 is diamond). Corundum is also quite resistant to most chemicals.
There are various products called "Aluminum Jelly", such as that from Loctite and from Henkel, which can remove some of the surface corrosion. They probably attack the aluminum at its point of contact with the oxide, rather than reacting with $\ce{Al2O3}$ itself. Regrettably, this probably means that though it might brighten the surface, the jelly will not substantially free the bolts.

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# Integrate - Area Under the Curve

• November 12th 2008, 11:26 PM
jhomie
Integrate - Area Under the Curve
This is the question in my book and I can't seem to get an answer:

14 (a) For the function: http://thestudentroom.co.uk/latexren...e9f3f6cc14.png the area under the curve from x = 0 to 1 and the x axis, can be found with the equation http://thestudentroom.co.uk/latexren...b2b1b5acc4.png. What is the equation for the area under the curve from y = 0 to 1 and the y axis? I got http://thestudentroom.co.uk/latexren...2014b70b3c.png. Is this right?

It then says

(b) With the same function (http://thestudentroom.co.uk/latexren...e9f3f6cc14.png), the equation for x = 0 to 2 and the x axis can be found with: http://thestudentroom.co.uk/latexren...83d3901726.png. What is the equation for the area unde the curve from y = 0 to 2 and the y axis? I don't know what to do here

And finally.. (however, if we can get the right answer to (b) i'm sure (c) will be done too)

(c) For the same function (http://thestudentroom.co.uk/latexren...e9f3f6cc14.png) the equation for x = 2 to 3 and the x axis is http://thestudentroom.co.uk/latexren...d89489fecd.png. What would the equation be for the area under the curve for y = 2 to 3 and the y axis?
• November 13th 2008, 12:21 AM
mr fantastic
Quote:

Originally Posted by jhomie
This is the question in my book and I can't seem to get an answer:

14 (a) For the function: http://thestudentroom.co.uk/latexren...e9f3f6cc14.png the area under the curve from x = 0 to 1 and the x axis, can be found with the equation http://thestudentroom.co.uk/latexren...b2b1b5acc4.png. What is the equation for the area under the curve from y = 0 to 1 and the y axis? I got http://thestudentroom.co.uk/latexren...2014b70b3c.png. Is this right? Mr F says: Yes. And how you got this answer is probably the way you need to get the answers to the next two questions.

It then says

(b) With the same function (http://thestudentroom.co.uk/latexren...e9f3f6cc14.png), the equation for x = 0 to 2 and the x axis can be found with: http://thestudentroom.co.uk/latexren...83d3901726.png. What is the equation for the area unde the curve from y = 0 to 2 and the y axis? I don't know what to do here

Mr F says: The area of the rectangle is ${\color{red}2 \cdot 2^n = 2^{n+1}}$. Then the required area is ${\color{red}2^{n+1} - \frac{2^{n+1}}{n+1} = \, ....}$

And finally.. (however, if we can get the right answer to (b) i'm sure (c) will be done too)

(c) For the same function (http://thestudentroom.co.uk/latexren...e9f3f6cc14.png) the equation for x = 2 to 3 and the x axis is http://thestudentroom.co.uk/latexren...d89489fecd.png. What would the equation be for the area under the curve for y = 2 to 3 and the y axis?

(c) is left for you to then.
• November 13th 2008, 02:16 AM
jhomie
Quote:

Originally Posted by mr fantastic
(c) is left for you to then.

Thanks, I appreciate your help

The thing is however, that for (a) I made that equation based on numerical trials - i.e substituting numbers and subtracting from the area. I didn't get it in an algebraic way which is why I'm lost for (b).

For (b) however, $2^n+^1 -\frac {2^n+^1}{n+1}$ can be simplified to $\frac {2n\times 2^n}{n+1}$ ??

(c) I thought solving (b) would solve (c), but now when i have this extra info, i see that in (c) from 2 to 3 it makes a rectangle, but the y axis is different from the x axis, no?
• November 13th 2008, 03:48 AM
mr fantastic
Quote:

Originally Posted by jhomie
Thanks, I appreciate your help

The thing is however, that for (a) I made that equation based on numerical trials - i.e substituting numbers and subtracting from the area. I didn't get it in an algebraic way which is why I'm lost for (b).

For (b) however, $2^{n+1} -\frac {2^{n+1}}{n+1}$ can be simplified to $\frac {2n\times 2^n}{n+1}$ ??

(c) I thought solving (b) would solve (c), but now when i have this extra info, i see that in (c) from 2 to 3 it makes a rectangle, but the y axis is different from the x axis, no?

$2^{n+1} -\frac {2^{n+1}}{n+1} = \frac{(n+1)2^{n+1}}{n+1} -\frac {2^{n+1}}{n+1} = \frac{n \cdot 2^{n+1}}{n+1}$.

(c) Get the length and width of the rectangle. Hence get the area of the rectangle. Subtract the area between the curve and the x-axis from the area of the rectangle.
• November 13th 2008, 04:08 AM
jhomie
Quote:

Originally Posted by mr fantastic
$2^{n+1} -\frac {2^{n+1}}{n+1} = \frac{(n+1)2^{n+1}}{n+1} -\frac {2^{n+1}}{n+1} = \frac{n \cdot 2^{n+1}}{n+1}$.

(c) Get the length and width of the rectangle. Hence get the area of the rectangle. Subtract the area between the curve and the x-axis from the area of the rectangle.

Thanks, for your help I have now done (a) (b) and (c) (Rofl)

2 more Qs though: The first is:

Doesn't $\frac{n \cdot 2^{n+1}}{n+1}$ the same/simplify to $\frac {2n\times 2^n}{n+1}$?

_____

The 2nd Q is somewhat related to the (a/b/c), but it is for integration 3 which i learnt last year and my paper 2 exam in 2 weeks is on this.

Area A is the area under the curve for x = a to b and the x axis
Area B is the area under the curve for y = a to b and the y axis

It says: " For a function $y=x^n$, the ratio of Area A: Area B is $n:1$. Given the function $y=x^n$ from x = a to x = b such that a < b, does this ratio hold true for the regions defined below? Prove and explain why or why not.

Area A: $y=x^n$, $y=a^n$, $y=b^n$ and the y axis
Area B: $y=x^n$, x = a, x = b, and the x axis

I'm not asking you to do it for me; you've already helped me a lot, but I don't know what to do or where to start, so just guide me a little?

• November 13th 2008, 04:12 AM
mr fantastic
Quote:

Originally Posted by jhomie and edited by Mr F (fixed the latex error)
Thanks, for your help I have now done (a) (b) and (c) (Rofl)

2 more Qs though: The first is:

Doesn't $\frac{n \cdot 2^{n+1}}{n+1}$ the same/simplify to $\frac{2n\times 2^n}{n+1}$?
[snip]

Yes it does. Sorry I didn't look carefully enough.
• November 13th 2008, 04:14 AM
mr fantastic
Quote:

Originally Posted by jhomie
Thanks, for your help I have now done (a) (b) and (c) (Rofl)

2 more Qs though: The first is:

Doesn't $\frac{n \cdot 2^{n+1}}{n+1}$ the same/simplify to $\frac {2n\times 2^n}{n+1}$?

_____

The 2nd Q is somewhat related to the (a/b/c), but it is for integration 3 which i learnt last year and my paper 2 exam in 2 weeks is on this.

Area A is the area under the curve for x = a to b and the x axis
Area B is the area under the curve for y = a to b and the y axis

It says: " For a function $y=x^n$, the ratio of Area A: Area B is $n:1$. Given the function $y=x^n$ from x = a to x = b such that a < b, does this ratio hold true for the regions defined below? Prove and explain why or why not.

Area A: $y=x^n$, $y=a^n$, $y=b^n$ and the y axis
Area B: $y=x^n$, x = a, x = b, and the x axis

I'm not asking you to do it for me; you've already helped me a lot, but I don't know what to do or where to start, so just guide me a little?

A ratio of n : 1 means the areas are a fraction $\frac{1}{n+1}$ and $\frac{n}{n+1}$ of the rectangular area.
• November 13th 2008, 01:05 PM
jhomie
Quote:

Originally Posted by mr fantastic
A ratio of n : 1 means the areas are a fraction $\frac{1}{n+1}$ and $\frac{n}{n+1}$ of the rectangular area.

Yup, I understand that, but I don't understand the rest of it. I don't know what is wants me and how to do it with the regions specified


text:
calculate $\prod_1^\infty k^{\frac1{k!}}$

Is it possible to find the point of convergence of $\prod_1^\infty k^{\frac1{k!}}$

$K!=k(k-1)!$.

My attempt:

If $S_n=\prod_1^\infty k^{\frac1{k!}}$ then $\ln S_n=\sum_1^\infty \frac{\ln k}{k!}< \sum\frac1{k!}=e$

So $S_n$ is converges but I don't know if it is possible to find the point of convergence or not.

• But $\log k < 1$ only for $k=1,2$, no? – krvolok Aug 10 '15 at 16:53
• @krvolok Yeah sorry I should say $\sum\frac{\ln k}{k!}< \sum\frac k{k!}=\sum\frac1{(k-1)!}$ – mac Aug 10 '15 at 16:55
• The value of the product is $1.829024680$. – user64494 Aug 10 '15 at 16:57
• @user64494 Why!? – mac Aug 10 '15 at 18:27
• @ mac : That was found by the Maple code $$>evalf(product(k^{1/factorial(k)}, k = 1 .. infinity));$$ – user64494 Aug 10 '15 at 19:37

It's possible to rephrase your product as a nested radical, and sometimes these have closed-form solutions, like $\sqrt{2 \sqrt {2 \sqrt {2 \ldots}}} = 2$. However no one even knows a closed-form solution for the seemingly almost as innocent $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}}$. There all the nested radicals are square-roots. Your nested radicals are $k$th roots for increasing $k$. Thus it seems highly unlikely that anyone will be able to come up with a closed form for your expression, at least, not without some monumental effort.


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## Monday, May 24, 2010

### M Theory Lesson 331

To summarize the unitary $(a,b,c)$ parameterization: the CKM probabilities are obtained from the three values
$(a,b,c) = (-0.2314, 24, 0.0035)$,
which result in the angle $\beta_s = 0.0194 \simeq abc$. This value for $\beta_s$ is in perfect agreement with Standard Model fits.


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for Journals by Title or ISSN for Articles by Keywords help
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1 2 3 4 | Last

 Bulletin of the Malaysian Mathematical Sciences SocietyJournal Prestige (SJR): 0.626 Citation Impact (citeScore): 1Number of Followers: 0      Hybrid journal (It can contain Open Access articles) ISSN (Print) 0126-6705 - ISSN (Online) 2180-4206 Published by Springer-Verlag  [2351 journals]
• Capacity of a Condenser in the Heisenberg Group
• Authors: Najoua Gamara; Akram Makni
Abstract: We give new estimates on the capacity of a condenser in the first Heisenberg group. As an application, we establish new lower and upper bounds for the first eigenvalue of the Kohn–Laplace operator for a regular bounded open domain of the Heisenberg group.
PubDate: 2018-04-23
DOI: 10.1007/s40840-018-0628-7

• The Jacobian Conjecture and Injectivity Conditions
• Authors: Saminathan Ponnusamy; Victor V. Starkov
Abstract: One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.
PubDate: 2018-04-18
DOI: 10.1007/s40840-018-0626-9

• Fundamental Theorems of Doi–Hopf Modules in a Nonassociative Setting
• Authors: J. N. Alonso Álvarez; J. M. Fernández Vilaboa; R. González Rodríguez
Abstract: In this paper, we introduce the notion of weak nonassociative Doi–Hopf module and give the Fundamental Theorem of Hopf modules in this setting. Also, we prove that there exists a categorical equivalence that admits as particular instances the ones constructed in the literature for Hopf algebras, weak Hopf algebras, Hopf quasigroups and weak Hopf quasigroups.
PubDate: 2018-04-16
DOI: 10.1007/s40840-018-0624-y

• Multiplicity of Positive Solutions for Schrödinger–Poisson Systems with
a Critical Nonlinearity in $$\mathbb {R}^3$$ R 3
• Authors: Weihong Xie; Haibo Chen; Hongxia Shi
Abstract: This paper is dedicated to studying the multiplicity of positive solutions for the following Schrödinger–Poisson problem \begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\phi u=\lambda Q(x) u ^{q-2}u+ K(x) u ^4u, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ -\Delta \phi =u^2, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ \end{array}\right. \end{aligned} where $$4<q<6$$ or $$q=2$$ , $$\lambda >0$$ is a parameter, K(x) and Q(x) satisfy some mild assumptions. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where K(x) attains its global maximum for small $$\lambda$$ .
PubDate: 2018-04-11
DOI: 10.1007/s40840-018-0623-z

• Retraction Note to: Toeplitz Matrices Whose Elements are the Coefficients
of Starlike and Close-to-Convex Functions
• Authors: D. K. Thomas; S. Abdul Halim
Abstract: The authors have retracted this article because the article contains major flaws in the proof of the main results. The results of the paper are invalid, since the assumption that the functionals considered are rotationally invariant is not valid. All authors have agreed to this retraction.
PubDate: 2018-04-01
DOI: 10.1007/s40840-018-0620-2

• Edge Colorings of Planar Graphs Without 6-Cycles with Three Chords
• Authors: Wenwen Zhang; Jian-Liang Wu
Abstract: A graph G is of class 1 if its edges can be colored with k colors such that adjacent edges receive different colors, where k is the maximum degree of G. It is proved here that every planar graph is of class 1 if its maximum degree is at least 6 and any 6-cycle contains at most two chords.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0376-5

• Note on the Distribution Composition $$(x_+^\mu )^\lambda$$ ( x + μ ) λ

• Authors: Limonka Lazarova; Biljana Jolevska-Tuneska; İnci Aktürk; Emin Özc̣ağ
Abstract: The composition of the distributions $$x^\lambda$$ and $$x_+^\mu$$ is evaluated for $$\lambda = -1,-2,\ldots \quad \mu >0$$ and $$\lambda \mu \in {{\mathbb {Z}}}^- .$$ Further results are deduced.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0342-2

• Further Results on Distance Estrada Index of Random Graphs
• Authors: Yilun Shang
Abstract: Let G be a simple connected graph on n vertices. The distance Estrada index DEE(G) of G is defined as the sum of $$e^{\lambda _i(D)}$$ over $$1\le i\le n$$ , where $$\lambda _1(D),\lambda _2(D),$$ $$\ldots ,$$ $$\lambda _n(D)$$ are the eigenvalues of its distance matrix D. In this paper, we establish lower and upper bounds to DEE(G) for almost all bipartite graphs G.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0306-6

• A Class of Locally Conformal Almost Cosymplectic Manifolds
• Authors: Fortuné Massamba; Ange Maloko Mavambou
Abstract: In this paper, we study a class of almost contact manifolds, namely locally conformal almost manifolds. We investigate subclasses of such manifolds and prove that some of them contain the class of bundle-like metric structures. Under some conditions, we prove that the class of conformal changes of almost cosymplectic structures is a subclass of (almost)-cosymplectic structures. Examples are also obtained.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0309-3

• Some Reversed and Refined Callebaut Inequalities Via Kontorovich Constant
• Authors: Mojtaba Bakherad
Abstract: In this paper, we employ some operator techniques to establish some refinements and reverses of the Callebaut inequality involving the geometric mean and Hadamard product under some mild conditions. In particular, we show \begin{aligned}&K\left( \frac{M^{2t-1}}{m^{2t-1}},2\right) ^{r{^\prime }} \sum _{j=1}^n(A_j\sharp _{s}B_j)\circ \sum _{j=1}^n(A_j\sharp _{1-s}B_j)\\&\qquad +\left( \frac{t-s}{t-1/2}\right) \left( \sum _{j=1}^n(A_j\sharp _{t}B_j)\circ \sum _{j=1}^n(A_j\sharp _{1-t}B_j) \!-\!\sum _{j=1}^n(A_j\sharp B_j)\circ \sum _{j=1}^n(A_j\sharp B_j)\!\right) \\&\quad \le \sum _{j=1}^n(A_j\sharp _{t}B_j)\circ \sum _{j=1}^n(A_j\sharp _{1-t} B_j), \end{aligned} where $$A_j, B_j\in {\mathbb {B}}({\mathscr {H}})\,\,(1\le j\le n)$$ are positive operators such that $$0<m{^\prime } \le B_j\le m <M \le A_j\le M{^\prime }\,\,(1\le j\le n)$$ , either $$1\ge t\ge s>{\frac{1}{2}}$$ or $$0\le t\le s<\frac{1}{2}$$ , $$r{^\prime }=\min \left\{ \frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\}$$ and $$K(t,2)=\frac{(t+1)^2}{4t}\,\,(t>0)$$ .
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0364-9

• Wold-Type Decompositions for a Commutative Pair of Noncommutative
Semigroups of Isometries
• Authors: Tudor Bînzar; Cristian Lăzureanu
Abstract: In this paper, a pair of noncommutative isometric semigroups on a Hilbert space is considered. The existence and uniqueness of Wold-type decompositions for this pair are studied and connections with the noncommutative Wold decomposition for the product semigroup generated by this pair are given. Also, characterizations for the product semigroup to be a standard semigroup are presented.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0379-2

• Weak Finite Intersection Characterizations of Existence in Optimization
• Authors: Phan Quoc Khanh; Vo Si Trong Long
Abstract: We propose a pure topological notion of a finite-intersection map (FI-map), which captures the idea of a typical generalized KKM-map, with no use of any generalized convexity structure and prove its full characterizations in terms of recent generalized KKM and connectedness structures. We establish an intersection point theorem in terms of FI-maps. Then, a half of the paper is devoted to apply this result to obtain characterizations of the existence of solutions in optimization. First, we discuss a general model of variational relations. Then, we move on to more particular, but typical and important cases: equilibrium problems, saddle points, constrained minimization, and Nash equilibria. We include discussions and examples to illustrate that our results improve or extend many recent contributions in the literature.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0370-y

• Inferences in Constant-Partially Accelerated Life Tests Based on
Progressive Type-II Censoring
• Authors: Rashad M. EL-Sagheer
Abstract: Accelerated life testing is widely used in product life testing experiments since it provides significant reduction in time and cost of testing. In this article, we assume that the lifetime of items under use condition follows the two-parameter distributions having power hazard function, and partially accelerated life tests based on progressive type-II censored samples are considered. The maximum likelihood, Bayes, and parametric bootstrap methods are used for estimating the unknown parameters. Based on normal approximation to the asymptotic distribution of MLEs, the approximate confidence intervals for the parameters are derived. Two bootstrap confidence intervals are also proposed. The classical Bayes estimates cannot be obtained in explicit form, so we propose to apply the Markov chain Monte Carlo (MCMC) technique. Gibbs within the Metropolis–Hasting algorithm has been applied to generate MCMC samples from the posterior density function. Based on the generated samples, the Bayes estimates and the highest posterior density credible intervals of the unknown parameters have been computed. Finally, analysis of a simulated data set has also been presented to illustrate the proposed estimation methods developed here.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0311-9

• One-Step Iterations for a Finite Family of Generalized Nonexpansive
Mappings in CAT(0) Spaces
• Authors: Hafiz Fukhar-ud-din; Khairul Saleh
Abstract: We study a one-step iterative scheme to establish strong convergence theorems and $$\triangle$$ -convergence theorems for a finite family of generalized nonexpansive mappings on a nonlinear domain. Our results generalize and extend several relevant results in the literature.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0310-x

• Existence Results for a Coupled System of Perturbed Functional
Differential Inclusions in Banach Algebras
• Authors: Aref Jeribi; Najib Kaddachi; Bilel Krichen
Abstract: In this paper, we establish some new fixed point theorems for a $$2\times 2$$ block operator matrix defined on nonempty, bounded, closed, and convex subsets of Banach algebras, where the inputs are multi-valued mappings. The obtained results are then applied to a coupled system of perturbed functional differential inclusions of initial and boundary value problems in order to prove the existence of solutions under a Carathéodory condition.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0374-7

• An Improvement to the Homotopy Perturbation Method for Solving Nonlinear
Duffing’s Equations
• Authors: A. R. Vahidi; E. Babolian; Z. Azimzadeh
Abstract: In this paper, a new form of the homotopy perturbation method has been adopted for solving nonlinear Duffing’s equations, which yields the Maclaurin series of the exact solution. The Laplace transformation is applied to the truncated Maclaurin series, and then the Padé approximation with fast convergence rate and high accuracy is used for the solution derived from the Laplace transformation. Illustrative examples are given to demonstrate the efficiency and the simplicity of the proposed method.
PubDate: 2018-04-01
DOI: 10.1007/s40840-015-0191-4

• On the Cayley Graphs of Completely Simple Semigroups
• Authors: Bahman Khosravi
Abstract: In this paper first we study and characterize the Cayley graph of Rees matrix semigroups (completely simple semigroups). Then we investigate Cayley D-saturated property of these graphs and we study the Cayley D-saturated property of Rees matrix semigroups.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0361-z

• Local Descriptions of Roots of Cubic Equations over P-adic Fields
• Authors: Mansoor Saburov; Mohd Ali Khameini Ahmad
Abstract: One of the most frequently asked question in the p-adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains $${\mathbb {Z}}_p^{*}, \ {\mathbb {Z}}_p{\setminus }{\mathbb {Z}}_p^{*}, \ {\mathbb {Z}}_p, \ {\mathbb {Q}}_p{\setminus }{\mathbb {Z}}_p^{*}, \ {\mathbb {Q}}_p{\setminus }({\mathbb {Z}}_p{\setminus }{\mathbb {Z}}_p^{*}), \ {\mathbb {Q}}_p{\setminus }{\mathbb {Z}}_p, \ {\mathbb {Q}}_p$$ or not. However, this question was open even for lower-degree polynomial equations. In this paper, we give local descriptions of roots of cubic equations over the p-adic fields for $$p>3$$ .
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0401-8

• Vector Exponential Penalty Function Method for Nondifferentiable
Multiobjective Programming Problems
• Authors: Tadeusz Antczak
Abstract: In this paper, a new vector exponential penalty function method for nondifferentiable multiobjective programming problems with inequality constraints is introduced. First, the case when a sequence of vector penalized optimization problems with vector exponential penalty function constructed for the original multiobjective programming problem is considered, and the convergence of this method is established. Further, the exactness property of a vector exact penalty function method is defined and analyzed in the context of the introduced vector exponential penalty function method. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto solutions of the considered nondifferentiable multiobjective programming problem and the associated vector penalized optimization problem with the vector exact exponential penalty function. This equivalence is established for nondifferentiable vector optimization problems with inequality constraints in which involving functions are r-invex.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0340-4

• Subscalarity, Invariant, and Hyperinvariant Subspaces for Upper Triangular
Operator Matrices
• Authors: Salah Mecheri
Abstract: In this paper, we introduce the class of quasi-nM-hyponormal operators and study various properties. We show that every quasi-nM-hyponormal operator is subscalar of order 2n; in particular, every M-hyponormal operator is subscalar of order two. Consequently, if the spectrum of T has a nonempty interior in $$\mathbb {C}$$ , then T has a nontrivial invariant subspace. We also examine the hyperinvariant subspace problem for quasi-nM-hyponormal operators. Finally, we consider some of the spectral properties of this class and show that they share many spectral properties with normal operators on Hilbert spaces, mainly concerning Fredholm theory and local spectral theory. We are primarily interested in isolated points of the spectra of quasi-nM-hyponormal operators as well as the isolated points of the approximate point spectra. These properties lead to the concept of a polaroid-type operator, which when combined with the single-valued extension property, an important property in local spectral theory, produces a general framework for several versions of Weyl-type theorems.
PubDate: 2018-04-01
DOI: 10.1007/s40840-016-0377-4

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# Abstracts

Michael Bennett (UBC)
Klein forms, Thue-Mahler equations and elliptic curves with bad reduction at a prescribed set of primes

Nicolas Billerey (Clermont-Ferrand)
Explicit large image results for modular Galois representations

Starting from the classical congruences of Ramanujan's tau function and after briefly recalling the history of the problem, I will give an explicit version of a large image theorem of Ribet for residual Galois representations attached to classical modular forms. In relation to this result I will also discuss the modularity of some specific reducible representations (joint works for Luis Dieulefait and Ricardo Menares).

Gebhard Boeckle (Heidelberg)
Computing multiplicative integrals over function fields and applications

Let $E$ be an elliptic curve over $\mathbb{Q}$. In the theoretical and experimental study of Heegner points on $E$, multiplicative integrals as introduced by Darmon play a central role. In the most basic case when $E$ has a uniformization by a Shimura curve, Heegner points as well as the Tate-period of $E$ can be expressed via multiplicative integrals. Work of Darmon-Greenberg-Pollack-Stevens, based on the theory of rigid analytic automorphic forms, show that such integrals can be computed in polynomial time and are thus well-suited for explicit computations.

In our talk we shall describe joint work in progress with the PhD student Yamidt Bermudez on multiplicative integrals for elliptic curves $E$ over a global function field. By a result of Drinfeld E corresponds to a function field automorphic form, and one disposes of multiplicative integrals as in the case over $\mathbb{Q}$. A main obstacle for algorithms in positive characteristic is the absence of a logarithm function, a minor obstacle that of a theory of overconvergent automorphic forms. We shall explain a polynomial time algorithm for the computation of such integrals in the absence of a logarithm and indicate possible applications.

Peter Bruin (Zurich)
Ranks of elliptic curves with prescribed level structure

I will explain joint work with J. Bosman, A. Dujella and F. Najman, and work in progress with F. Najman. We proved that every elliptic curve over a quadratic field with a point of order 13 or 18 has even rank, the same holds for every elliptic curve over a quartic field with a point of order 22, and similar (more complicated) results hold for elliptic curves over quadratic fields admitting a cyclic $n$-isogeny for certain values of $n$. The proofs rely on geometric and arithmetic properties of the modular curves $X_1(n)$ and $X_0(n)$.

Sander Dahmen (Utrecht)

Some generalized Fermat equations of the form $x^p+y^p=z^q$

We discuss how the method of Chabauty-Coleman and the modular method can be combined to attack some new cases of the generalized Fermat equation $x^p+y^p=z^q$. In particular, we shall show how to solve this equation (in coprime integers $x$, $y$, $z$) for $(p,q) \in \{(5,7), (5,19), (7,5)\}$. We will also indicate how to obtain more results assuming GRH and some facts about Hilbert modular forms, and discuss the possibility of making these results unconditional. This is joint work with Samir Siksek.

Lassina Dembele (Warwick)
Examples of abelian surfaces with everywhere good reduction

In this talk, we will present explicit examples of abelian surfaces with everywhere good reduction. One class of examples has a particular connection with the Paramodularity Conjecture of Brumer-Kramer, which will be discussed in the process. This is joint work with Abhinav Kumar.

Luis Dieulefait (Barcelona)
Langlands functoriality for $\mathrm{GL}(2) \otimes \mathrm{GL}(n)$ and other related cases

We will explain how to combine recent Automorphy Lifting Theorems with the method of "Propagation of Automorphy" to deduce new cases of Langlands functoriality.

Neil Dummigan (Sheffield)
Testing congruences using a trace formula for algebraic modular forms

I will exhibit a simple formula for the trace of a Hecke operator on a space of algebraic modular forms, which can be used to find eigenvalues of Hecke operators in cases where the space has small dimension but the forms take values in a representation of dimension prohibitively high for making explicit the operators themselves. Such situations arise naturally when testing various analogues of Harder's conjectural congruences between Hecke eigenvalues of genus 2 and genus 1 cusp forms. We will look at two cases, involving compact forms of $U(2,2)$ and $\mathrm{GSp}_4$.

Ian Kiming (Copenhagen)
Higher congruences between modular forms

I will review recent work on modular forms and Galois representations modulo prime powers, focusing primarily on work by myself, I. Chen, G. Wiese, as well as P. Tsaknias, R. Adibhatla. I will also go into various open problems and report on some experimental/computational work related to those, primarily by my student N. Rustom.

Hartmut Monien (Bonn)
Modular forms for noncongruence subgroups of $\mathrm{PSL}_{2}\left(\mathbb{Z}\right)$

A powerful tool for investigating these non-congruence subgroups was introduced by Kulkarni in 1991 and is now known as Farey-Symbols. A complete and efficient implementation of it became available only recently. With the help of the Farey-Symbols it is possible to substantially extend methods from analytic number theory. We will discuss two methods. The first is a generalization of a numerical algorithm originally due to Hejhal. The second approach is based on series of papers by Rademacher and Zuckerman. We will present some noncongruence subgroups and and their modular forms.

On higher congruences between cusp forms and Eisenstein series

We will discuss certain new results of numerical investigation of congruences modulo prime powers between newforms and Eisenstein series at prime levels and with equal weights. In particular I would like to show the upper bound on the exponent of the congruence and formulate several observations based on the results of our computations, including a conjectural behavior of the congruence under the condition of ramification of prime ideals in coefficient fields of newforms.

Ariel Pacetti (Buenos Aires)
Half integral weight modular forms

During this talk we will recall the definition of half integral weight modular forms and some applications like the Shimura correspondence and its relation with special values of L-series. Then we will show how this theory generalizes to Hilbert modular forms over totally real fields.

Cris Poor (Fordham)
Paramodular cusp forms via Borcherds products

Evidence is given for the Paramodular Conjecture of Brumer and Kramer by using Borcherds products to construct examples of weight two paramodular cusp eigenforms that not are in the image of the Gritsenko lift.

Haluk Sengun (Warwick)
Cohomology of Bianchi Groups and Arithmetic

Given an imaginary quadratic field $K$ with ring of integers $R$, consider the Bianchi group $\mathrm{GL}(2,R)$. It is suspected since the late 1970's that there is a connection between the Hecke eigenclasses in the mod $p$ cohomology of (congruence subgroups of) Bianchi groups and the $2$-dimensional continuous mod $p$ representations of the absolute Galois group of $K$.
Most of the basic tools used for establishing this connection (and its surrounding problems) in the classical setting fail to work in the setting of Bianchi groups. The situation has an extra layer of complication by the fact that there are "genuinely mod $p$" Hecke eigenvalue systems, resulting from the existence of torsion in the integral cohomology. In this talk I will elaborate on the above paragraph, presenting numerical examples for illustration. Towards the end, I will also talk about how the "even" $2$-dimensional continuous mod $p$ representations of the absolute Galois group of $\mathbb{Q}$ come into the picture.

Nils Skoruppa (Siegen)
How to construct explicitly vector valued modular forms and Jacobi forms

In various theories one needs to construct explicitly spaces of vector valued elliptic modular forms. Examples for such theories are algebraic quantum field theory (where vector valued modular forms occur as traces of representations of infinite dimensional Lie algebras) or the geometry of moduli spaces in algebraic geometry (where vector valued modular forms occur in the construction of functions with distinguished divisors using Borcherds products or Gritsenko lifts). A recent theorem shows that vector valued modular forms can always be realized as Jacobi forms. For the latter there are various efficient constructions available. In this talk we explain the mentioned theorem, various constructions for Jacobi forms, and we show how to assemble everything to obtain useful explicit formulas for the objects mentioned in the title.

Fredrik Strömberg (Durham)
Dimension formulas for Hilbert modular forms

Dimension formulas for (scalar-valued) Hilbert modular forms have existed for a long time. However, due to the complex nature of the invariants which appear in the formulas, only a few cases have been worked out explicitly, in the sense of producing actual numbers. In a joint project together with N.-P. Skoruppa we have generalized the previous (theoretical) dimension formulas to the setting of vector-valued Hilbert modular forms of both integral and half-integral weights. Our goal is to also provide implementations of algorithms to compute the relevant invariants. The development of these algorithms is still in progress and I will give an overview of the necessary algorithms and present the current status of the project.

John Voight (Vermont)
Power series expansions for modular forms

We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real ﬁeld, including the image of CM points, and equations for Shimura curves. This is joint work with Michael Klug and John Willis.


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+0

# polynomial question

0
21
2

Find the remainder when the polynomial x^5 - 4x^3 + 7x^2 - 3 is divided by (x - 1)(x + 2).  I think there is a way to do this using Remainder Theorem, but I have no idea how.

Jan 4, 2021

#1
0

We can use long division

I got 4x+9 but I'm not sure

i suggest you search up polynomial division, there are some great tutorials like khan academy that are free

Jan 5, 2021
#2
0

Remainder theorem will get you there.

$$\displaystyle \frac{x^{5}-4x^{3}+7x^{2}-3}{(x-1)(x+2)}=q(x)+\frac{r(x)}{(x-1)(x+2)}.$$

q(x) will be a cubic and r(x) will be linear, (one degree lower than the quadratic on the bottom line), so let r(x) = ax + b.

Multiply throughout by (x - 1)(x + 2) to get

$$\displaystyle x^{5}-4x^{3}+7x^{2}-3=q(x)(x-1)(x+2)+ax+b$$,

and now substitute x = 1 and x = -2.

That gets you a pair of simultaneous equations in a and b which you can solve.

Jan 5, 2021


text:
## Greatest Common Divisor Calculator

The greatest common divisor (gcd) is a number that divides two or more integers without remainder. It is used often times in fractions to find the most reduced form of a fraction. The greatest common divisor is sometimes seen in the following notation: $$gcd(6, 9) = 3$$ It is also reffered to as the greatest common factor (gcf), the highest common factor (hcf), or the highest common divisor (hcd). Use the calculator below to solve your gcd problems.


text:
# Is there any intuitive way to think about the gamma function?

Is there a way to realize the gamma function intuitively? My first (and probably correct) guess is no, because, for example, $\Gamma(\frac 12)=\sqrt{\pi}$ doesn't make any intuitive sense at all. Also, to me it seems like negative integers would be more likely candidates to have defined values than fractions, but negative integers aren't defined with the gamma function. I saw this question which pointed out "Euler's reflection formula":

$$\Gamma(z)\Gamma(1-z)=\frac {\pi}{\sin(\pi z)}$$

This may offer some inspiration, but most likely not. Thank you for any responses to this question!

Edit: I saw the marked duplicate to this question. While it is good, I think it focuses more on the mathematical rigor than any intuition (when I say this I mean something that could be taught to a first or second year calculus student) that may exist.

• possible duplicate of Intuition for the definition of the Gamma function? – Zev Chonoles Jun 7 '15 at 4:01
• – Zev Chonoles Jun 7 '15 at 4:02
• Also also related – Akiva Weinberger Jun 7 '15 at 4:58
• @columbus8myhw (and half of the world) please add some relevant text to your links (i.e., not "see this", "this page") Can you see why Zev Chonoles's links are much more useful than yours? – Rolazaro Azeveires Jun 7 '15 at 5:58
• Oh, OK… My link is an answer I made a while back trying to explain some of the intuition behind the gamma function. – Akiva Weinberger Jun 7 '15 at 6:00

$\Gamma\bigg(\dfrac12\bigg)=\sqrt\pi~$ doesn't make any intuitive sense at all.
Speak for yourself ! All factorials of argument $\dfrac1n$ are connected to geometric shapes of the form
$x^n+y^n=1$. We have $~\displaystyle\int_0^\infty e^{-x^n}~dx~=~\bigg(\dfrac1n\bigg){\large!}\quad$ and $\quad\displaystyle\int_0^1\sqrt[m]{1-x^n}~dx~=~\frac{\bigg(\dfrac1m\bigg){\large!}~\bigg(\dfrac1n\bigg){\large!}}{\bigg(\dfrac1m+\dfrac1n\bigg){\large!}}$
Connecting the latter integral to the superellipse $x^n+y^m=1$ is trivial. As for the former, just integrate along the two axes of coordinates. I wrote about all of this here, in my answer to this question.
$\Gamma(z)\Gamma(1-z)=\dfrac\pi{\sin(\pi z)}$
Read the article on the Basel problem, and then see my answer to this question. Similarly, we also have $\Gamma\bigg(\dfrac12+z\bigg)\Gamma\bigg(\dfrac12-z\bigg)=\dfrac\pi{\cos(\pi z)}~.$


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# Matrix norm inequality implying eigenvector norm inequality

For a matrix $A$ let $\|A\|$ be the norm given by $\|A\|=\sup_{v \neq 0}\frac{\|Av\|}{\|v\|}$ where $\|v\|$ is the Euclidian norm on the vector $v$.

Suppose we have matrices $M$ and $S$ with leading eigenvectors $u$ and $\hat{u}$ respectively. If we have $\|M-S\| \leq \theta\|M\|$ does this imply $\|\hat{u}-u\|\leq 2\theta$, and if so, how?

This comes from the paragraph containing equation (1) on page 2 of this paper http://www.stanford.edu/~montanar/RESEARCH/FILEPAP/GossipPCA.pdf but I don't see how the result is "immediate" as they claim.

In response to user1551, $M$ is an $n \times n$ symmetric matrix, $u$ is "the" eigenvector of $M$ corresponding to the eigenvalue of largest magnitude.

$S$ is some sparsification of $M$ obtained by setting each entry of $M$ to $0$ independently with probability $1-p$ then rescaling non-zero entries by $1/p$.

$\hat{u}$ is "the leading eigenvector" of $S$.

Let's assume the entries of $M$ are all positive, then by Perron-Frobenius say that $u$ and $\hat{u}$ are non-negative unit eigenvectors, if that makes it easier.

-
Ah, the old "it follows immediately" trick. – Alexander Gruber Dec 27 '12 at 5:09
Please elaborate more on the definitions of $M$ and $S$. For non-specialists like me, the descriptions of the two matrices in the paper are too vague. That how the eigenvectors are chosen is also unclear. For instance, if $u$ is an eigenvector, so is $ku$ for all $k\not=0$. So, does the cited paper claim that $\|k_1\hat{u}+k_2u\|\le2\theta$ for any $k_1,k_2\not=0$? Even if we take unit eigenvectors, the statement that $\|\hat{u}\pm u\|\le2\theta$ (if $u$ is an eigenvector, so is $-u$) is not true in general. – user1551 Dec 27 '12 at 7:27
@user1551 For matrices with positive entries the leading eigenvector has positive entries; this eliminates the sign ambiguity. – user53153 Dec 29 '12 at 8:09

I don't believe such a strong stability result. Consider the matrix $$M=\begin{pmatrix} 1.01 & 0.01 \\ 0.01 & 1 \end{pmatrix}$$ According to WolframAlpha, its leading eigenvector, normalized to unit length, is $u\approx (0.85, 0.525)^T$.

We don't need to be gurus of sparse computing to find $$S=\begin{pmatrix} 1.01 & 0 \\ 0 & 1 \end{pmatrix}$$ (up to irrelevant scaling). The leading eigenvector is obviously $\widehat u=(1,0)^T$.

In conclusion, $\|M-S\|\le 0.01\|M\|$ but $\|u-\widehat u\| > 0.5$.

I hear that the book The Algebraic Eigenvalue Problem by Wilkinson is where stability results of this kind are found. I never read the book myself, though.

-
This example doesn't work. By the OP's explanation on the construction of $S$, $S$ is either the zero matrix (when $p=1$) or it is scaled by a factor larger than $1$ (when $p<1$) after setting some entries of $M$ to zero. – user1551 Dec 29 '12 at 8:15
But apparently your example can be modified to work, though. So +1. – user1551 Dec 29 '12 at 8:18
@user1551 Scaling does not affect eigenvectors, as my parenthetical remark indicated. – user53153 Dec 29 '12 at 8:19
Yes, but scaling would affect $\|M-S\|$. So you need a continuity argument to fill the loophole. – user1551 Dec 29 '12 at 8:21
@user1551 Valid point. However, the purpose of scaling is to bring $S$ closer to $M$, not to make it more different. (The goal of sparsification is to produce a sparse matrix that is close to the original one). If keeping the size of errors much less than $1$ wasn't important, we could just say that $\|u-\widehat u\|\le 2$ and save ourselves a lot of time. – user53153 Dec 29 '12 at 8:38


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Links

# Burn - Reclaim collateral

Returning tokens from SmartPiggies
Whenever a piggy is created, it must be destroyed. This can happen just after creation or upon completion of the piggy life cycle. If a writer creates a piggy that has some mistake, like a mistyped parameter for instance, and the writer does not want to auction the piggy, she may burn the piggy, thereby deleting all the parameters and reclaiming the collateral that was sent to the piggy. In this scenario, the writer will still need to Claim the collateral from the SmartPiggies smart contract, but the piggy will cease to exist.
$^1$
After a piggy has been settled, the piggy can be burned as well. All of the parameters of the piggy will be zero'ed out, and the piggy will be burned, whereby no further action can be taken with regard to it.
To Burn a piggy, navigate to the Advanced Functions tab from the drop down panel on the row of the desired piggy.
The drop down panel can be accessed by clicking the switch located to the left of the piggy ID, and then navigate to the Advanced Functions tab to view the Burn button:
Then click the "Burn" button:
The piggy will be "burned" and deleted. If any collateral exists in the piggy, it will be returned to the writer, to be claimed.
$^2$

## Footnotes

1 The record of the piggy will exist in the contract as a zero'ed piggy, but all reference to ownership and associated collateral will be removed. The piggy is effectively deleted, it isn't active, can't be auctioned, cleared, settled, nor transferred.
2 There are a handful of circumstances when a piggy can be burned and when a piggy cannot. If the writer has never successfully auctioned the piggy, the writer can burn the piggy to reclaim the collateral. Consequently, a holder who requests a piggy, can burn it, if she decides the piggy is not worth auctioning. This will return the premium to the holder.
Last modified 1yr ago


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# Schedule for: 19w5044 - Ordered Groups and Rigidity in Dynamics and Topology

Beginning on Sunday, June 16 and ending Friday June 21, 2019

All times in Oaxaca, Mexico time, CDT (UTC-5).

Sunday, June 16
14:00 - 23:59 Check-in begins (Front desk at your assigned hotel)
19:30 - 22:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
20:30 - 21:30 Informal gathering (Hotel Hacienda Los Laureles)
Monday, June 17
07:30 - 09:00 Breakfast (Restaurant at your assigned hotel)
09:15 - 09:30 Introduction and Welcome (Conference Room San Felipe)
09:30 - 10:30 Dale Rolfsen: Ordered groups and topology — a personal journey
This talk will be an introduction to ordered groups and their connections with topology. Beginning with learning of the left-orderability of braid groups some decades ago, I will describe how I came to fall in love with the subject.
(Conference Room San Felipe)
10:30 - 11:00 Coffee Break (Conference Room San Felipe)
11:00 - 12:00 Nathan Dunfield: Ordering 3-manifold groups
I will discuss practical computational techniques for proving the fundamental group of a closed hyperbolic 3-manifold is or is not orderable and describe the result of applying them to some 300,000 manifolds. I will include discussion of a new practical algorithm for solving the word problem for the fundamental group of a hyperbolic 3-manifold based on interval arithmetic and effective versions of the inverse function theorem.
(Conference Room San Felipe)
12:10 - 13:20 Cameron Gordon: ADE links and cyclic branched covers
The Dynkin diagrams of types A,D and E arise in many classification problems in mathematics. We conjecture a modest addition to this list: the fibered links that induce the standard tight contact structure on $S^3$ and have some cyclic branched cover whose fundamental group is not left-orderable. We will discuss progress towards a proof of this conjecture. This is joint work with Michel Boileau and Steve Boyer.
(Conference Room San Felipe)
13:20 - 13:30 Group Photo (Hotel Hacienda Los Laureles)
13:30 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 - 16:00 Michele Triestino: Ping-pong partitions for virtually free groups
We formulate a ping-pong lemma for actions of fundamental groups of graph of groups, listing sufficient conditions for the action to be faithful. This allows to generalize the notion of ping-pong partitions for actions on the circle (appearing in recent works by Matsumoto and Mann-Rivas), to the case of groups with torsion. In concrete terms, a ping-pong partition is a collection of finitely many combinatorial data that determines the semi-conjugacy class of the action. We apply this to give a partial positive solution to an old conjecture by Dippolito on codimension-one foliations with exceptional minimal set. This is part of a long-term project with many collaborators.
(Conference Room San Felipe)
16:00 - 16:30 Coffee Break (Conference Room San Felipe)
16:30 - 17:15 Yago Antolin: Complexity of positive cones of limit groups
A left-order on a group G is totally determined by its positive cone P, that is the elements in G that are greater than the identity. The set P is a subsemigroup and we can ask ourselves when this semigroup can befinitely generated, described by a regular language,etc. A recent resultof S.Hermiller and Z.Sunic shows that non-abelian free groups do not have regular positivecones. I will discuss this and how to generalize this result to limit groups. The talk will be based on joint work with C.Rivas and I will report some results of my PhD student H.L.Su. (Yago Antolin is from the Unversidad Autónoma de Madrid)
(Conference Room San Felipe)
17:15 - 18:00 Cristóbal Rivas: Some examples of groups (not)supporting regular positive cones
A positive cone in a finitely generated group is called regular if it can be represented by a regular language over the group-generators. In this talk, I will discuss examples and non-examples of groups supporting regular positive cones. For instance we will see that every left-orderable polycyclic group admits a regular positive cone, but this is no longer true for the class solvable groups. I will also try to relate to presence of regular positive cone with the presence/absence of isolated orders.
(Conference Room San Felipe)
18:00 - 18:15 Yago Antolin: Complexity of positive cones of limit groups, part 2 (Conference Room San Felipe)
19:00 - 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Tuesday, June 18
07:30 - 09:00 Breakfast (Restaurant at your assigned hotel)
09:30 - 10:30 Hyungryul Baik: Topology, Geometry, and Dynamics of Laminar groups
A notion of laminar groups was introduced by D. Calegari. A group acting on the circle by orientation-preserving homeomorphisms is called a laminar group if it admits an invariant lamination. Abundant examples arise naturally in the study of low-dimensional topology and geometric group theory. We will discuss how topology, geometry, and dynamics interplay when we study laminar groups. Some old and new results will be discussed as examples.
(Conference Room San Felipe)
10:30 - 11:00 Coffee Break (Conference Room San Felipe)
11:00 - 12:00 Thomas Koberda (Conference Room San Felipe)
12:10 - 13:30 Sang-Hyun Kim: RAAGs in Diff(S^1)
Let $Diff^r(S^1)$ denote the group of orientation-preserving $C^r$ diffeomorphisms on the circle. We prove that a right-angled Artin group $G$ embeds into $Diff^2(S^1)$ if and only if $G$ does not contain $(F_2 x Z) * Z$. This extends a previous joint work with Baik and Koberda, and answers a question of Kharlamov. (joint work with Koberda)
(Conference Room San Felipe)
13:30 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 - 16:00 Tyrone Ghaswala: Promoting circular-orderability to left-orderability
I will present a new characterisation for a circularly-orderable group to be left-orderable, and introduce the obstruction spectrum of a circularly-orderable group. This raises a plethora of intriguing questions. This is joint work with Jason Bell and Adam Clay.
(Conference Room San Felipe)
16:00 - 16:30 Coffee Break (Conference Room San Felipe)
16:30 - 17:00 Tetsuya Ito: Generalized torsion in 3-manifold groups and normal closures of slope elements
If a 3-manifold group does not admit a bi-ordering, then we may expect that it has a generalized torsion element. As a particular case, the fundamental group of any 3-manifold obtained by non zero surgery on a knot in the 3-sphere may have such an element. Then there are two situations: (1) a generalized torsion element in a knot group becomes a generalized torsion element in the surgered 3-manifold, or (2) a generalized torsion element arises via the Dehn filling. The first situation leads us to study of normal closures of slope elements in a knot group. In the first talk we investigate relationships among such normal subgroups. In particular, we establish the peripheral Magnus property. In the second talk we focus on generalized torsion elements in Dehn surgered manifolds arisen from the first or the second situations. We also take a closer look at some explicit examples. In the third talk we prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the former result, we give an upper bound for the stable commutator length of generalized torsion elements.
(Conference Room San Felipe)
17:00 - 17:30 Kimihiko Motegi: Generalized torsion in 3-manifold groups and normal closures of slope elements
If a 3-manifold group does not admit a bi-ordering, then we may expect that it has a generalized torsion element. As a particular case, the fundamental group of any 3-manifold obtained by non zero surgery on a knot in the 3-sphere may have such an element. Then there are two situations: (1) a generalized torsion element in a knot group becomes a generalized torsion element in the surgered 3-manifold, or (2) a generalized torsion element arises via the Dehn filling. The first situation leads us to study of normal closures of slope elements in a knot group. In the first talk we investigate relationships among such normal subgroups. In particular, we establish the peripheral Magnus property. In the second talk we focus on generalized torsion elements in Dehn surgered manifolds arisen from the first or the second situations. We also take a closer look at some explicit examples. In the third talk we prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the former result, we give an upper bound for the stable commutator length of generalized torsion elements.
(Conference Room San Felipe)
17:30 - 18:00 Masakazu Teragaito: Generalized torsion and Dehn filling
If a 3-manifold group does not admit a bi-ordering, then we may expect that it has a generalized torsion element. As a particular case, the fundamental group of any 3-manifold obtained by non zero surgery on a knot in the 3-sphere may have such an element. Then there are two situations: (1) a generalized torsion element in a knot group becomes a generalized torsion element in the surgered 3-manifold, or (2) a generalized torsion element arises via the Dehn filling. The first situation leads us to study of normal closures of slope elements in a knot group. In the first talk we investigate relationships among such normal subgroups. In particular, we establish the peripheral Magnus property. In the second talk we focus on generalized torsion elements in Dehn surgered manifolds arisen from the first or the second situations. We also take a closer look at some explicit examples. In the third talk we prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the former result, we give an upper bound for the stable commutator length of generalized torsion elements.
(Conference Room San Felipe)
19:00 - 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Wednesday, June 19
07:30 - 09:00 Breakfast (Restaurant at your assigned hotel)
09:30 - 10:30 Liam Watson: Khovanov homology and the L-space conjecture
The L-space conjecture relates non-left-orderability of 3-manifold groups to Heegaard Floer homology lens-spaces, or, L-spaces. In this talk I will give the definition of an L-space , and attempt to give a feeling for this class of manifolds by focusing on some examples. One class of examples is due to Ozsváth and Szabó: branched double covers of the 3-sphere with branch set a non-split alternating link. This leads to a surprising conjecture (implied by the L-space conjecture) relating simplicity in Khovanov homology to non-left-orderability of the fundamental group of the branched double cover.
(Conference Room San Felipe)
10:30 - 11:00 Coffee Break (Conference Room San Felipe)
11:00 - 11:45 Siddhi Krishna: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture
The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll build taut foliations for manifolds obtained by surgery on positive 3-braid closures. Our theorem provides the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. As an example, we'll construct taut foliations in every non-L-space obtained by surgery along the P(-2,3,7) pretzel knot.
(Conference Room San Felipe)
11:45 - 12:15 Rachel Roberts: Modifying branched surfaces
A branched surface that meets the torus boundary of a compact 3-manifold transversely (in a train track \tau) can sometimes be upgraded'' to a branched surface that fully carries CTFs that strongly realize all boundary slopes except one. We give a condition on \tau that guarantees that such an upgrade is possible. This approach succeeds for all alternating and Montesinos knots without L-space surgeries, for certain Murasugi sums, and for all nontrivial connected sums of alternating knots, Montesinos knots, or fibered knots. This is joint work with Charles Delman.
(Conference Room San Felipe)
12:15 - 13:30 Lunch (Restaurant Hotel Hacienda Los Laureles)
13:30 - 19:00 Free Afternoon (Oaxaca)
19:00 - 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Thursday, June 20
07:30 - 09:00 Breakfast (Restaurant at your assigned hotel)
09:30 - 10:30 Nicolás Matte Bon: Groups of piecewise linear homeomorphisms of flows
Given a compact space endowed with a flow, every group of orbit-preserving homeomorphisms of the space naturally acts on the real line (identified with an orbit of the flow). This simple observation can be used to define interesting examples of left-orderable groups. In a joint work with Michele Triestino, we explore this idea by defining and studying a class of groups acting on suspension flows of homeomorphisms of the Cantor set. I will explain how this can be used to give a short and conceptual construction of finitely generated simple left-orderable groups, whose existence was recently obtained by Hyde and Lodha. I will also discuss several additional properties of these groups, such as the inability to act on the circle without fixed points, and the lack of subgroups with property (T).
(Conference Room San Felipe)
10:30 - 11:00 Coffee Break (Conference Room San Felipe)
11:00 - 12:00 Ying Hu: Euler class and taut foliations on surgered 3-manifolds
This talk is motivated by the conjecture: the fundamental group of a QHS is left-orderable if and only if it admits a co-orientable taut foliation. It is known that if the Euler class of the taut foliation vanishes, then the fundamental group is left-orderable. In this talk, we will investigate the Euler class of co-orientable taut foliations on 3-manifolds which are obtained by Dehn surgery along a null-homologous knot.
(Conference Room San Felipe)
12:10 - 13:30 Hélène Eynard-Bontemps: Smooth times of a flow in dimension one
Consider a one-parameter group of homeomorphisms of some interval $I$, i.e. a $C^0$ flow on I, and assume its time-1 and $\alpha$ maps are smooth, for some irrational number $\alpha$. Does this imply that the flow itself is smooth? We will explain that the answer depends on the arithmetic nature of $\alpha$, namely: the answer is yes if alpha is diophantine, and no otherwise. For flows without fixed points, this follows from famous linearization results for circle diffeomorphisms, and we will see how some of the techniques from this context adapt to our general situation.
(Conference Room San Felipe)
13:30 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 - 16:00 Zoran Sunic: Left relatively convex subgroups
Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order. When G is left orderable, this holds if and only if H is convex in G under some left ordering of G. We give a criterion for H to be left relatively convex in G that generalizes a well known criterion of Burns and Hale. We then use this criterion to show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a result of Duncan and Howie. We show that if G is left orderable, then each free factor of G is left relatively convex in G. More generally, for any graph of groups, if each edge group is left relatively convex in each of its vertex groups, then each vertex group is left relatively convex in the fundamental group; this generalizes a result of Chiswell. Finally, we show that all maximal cyclic subgroups in locally residually torsion-free nilpotent groups are left relatively convex. This is a joint work with Yago Antolin and Warren Dicks.
(Conference Room San Felipe)
16:00 - 16:30 Coffee Break (Conference Room San Felipe)
16:30 - 18:00 Problem session
This will be a session for free discussion of open problems.
(Conference Room San Felipe)
19:00 - 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Friday, June 21
07:30 - 09:00 Breakfast (Restaurant at your assigned hotel)
10:30 - 11:00 Coffee Break (Conference Room San Felipe)
13:30 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)


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# Timeseries of odds in race - how to pick a model

Being new to AI/ML I'd like some pointers to where to begin. I got data from horse races. Specifically, I got the odds for each runner during the race - ca 5 times per second.

t1  r1   r2   r3 ...
1   5.25 2.04 3.25
2   5.10 2.50 2.75
...

I also know if the runner won/placed/lost My goal is to be able to say that runnerx X will win/place after ca 50-75% of the expected racetime with say 80% accuracy.

My problem is that I don't know how to model this situation. I've seem tournament strategies - ie who out of two runner will win - but here's more data - both in time and in participants

What model should I pay attention to?

/Björn

• Start simple. How well does predicting that "the favourite at time T wins the race" perform? As T increases I'd expect its accuracy to increase, but it might meet your threshold. At the very least you'll get an idea of how much better you'll need an algorithm to perform. And of course it might not even be possible. – Spacedman Oct 30 '18 at 16:08
• Hmm - that is a good idea. I'll think about for a while, and test it. Thanks. – bnl Oct 31 '18 at 8:19

Odds are directly connected to prediction accuracy. 4 means that you loose your money in 4 cases and win in one case. This is 20% probability $$\left(\frac1{4+1}\right)$$. If the bet goes down to 0.25 then the probability is 80% $$\left(\frac1{0.25+1}\right)$$. This logic is true if the spread between betting for the horse and against the horse is low. Betting is a financial market where people with the best betting algorithms bet the most and actually decide what the bet should be. If there is a situation on the market where someone could earn money it will be closed in milliseconds by trading robots. This is called an Efficient Market Hypothesis. In your data you can actually check if the hypothesis is true for horse betting.

If you would like to use ML algorithms to try to predict the outcomes better:

There are several ways to interpret the data.

1. You predict the probability that each horse is a winner.
2. You predict a rank of each horse.
3. You predict time difference between horses and the winner
4. You predict time

Some algorithms could be better at predicting one problem or the other.

Possible algorithms:

1. Linear regression. I always start with this because it is easy to calculate and easy to interpret. You build separate models for each horse number. Can be used for all problem definition.
2. Logistic regression. Similar to regression, but can only be used for the first problem definition.
3. LGBoost and XGBoost. Most likely the best models for such data, because the data is structured (not an image, sound). Can be used for all problem definitions.
4. Neural networks. For example MLP (multilayer perceptron).

You can also calculate some additional features. For example, multiply some odds, or calculate inverse normal distribution from the odds.

I would also recommend to collect more data. For example weather data. Then it is good to know the horse. Some horses could be more popular than others and receive better odds. Also time of the year because the horses could have a training plan according to the time of the year.

• This is data from Betfair - a large bet-exchange. While what you say is true, I'm looking for 'the holy graal' here. I'd like to find out if there are patterns in odds movement early in the race that might indicate a winner early - while the odds are high. But the odds are not truely inversed to the probability. They are inversed to what - in this case the bookmaker - thinks the probability is. So the odds curve is not smooth, there are plenty of hops in odds level and therefore sometimes a large spread. But I was looking of a way to feed the series to a model and ask it to point to a winner – bnl Oct 31 '18 at 8:30


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# zbMATH — the first resource for mathematics

Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions. (English) Zbl 1190.11005
This paper under review studies tilings and representation spaces related to the $$\beta$$-transformation when $$\beta$$ is a Pisot number, that is not a unit. The obtained results are applied to study the set of rational numbers having a purely periodic $$\beta$$-expansion. The authors make use of the connection between pure periodicity and a compact self-similar representation of number having no fractional part in their $$\beta$$-expansion, called center tile. For elements $$x$$ of the ring $$\mathbb Z[1/\beta]$$, so-called $$x$$-tiles are introduced, so that the central tile is a finite union of $$x$$-tiles up to translation. These $$x$$-tiles provide a covering (and even in some cases a tiling) of the space the authors working in. This space, called complete representation space, is based on archimedean as well as on the non-archimedean completions of the number field $$\mathbb Q(b)$$ corresponding to the prime divisors of the norm of $$\beta$$. This representation space has numerous potential implications. The authors focus on the gamma function $$\gamma(\beta)$$ defined as the supremum of the set of elements $$v$$ in $$[0,1]$$ such that every positive rational number $$p/q$$, with $$p/q\leq v$$ and $$q$$ coprime with the norm of $$\beta$$, has a purely periodic $$\beta$$-expansion. The key point relies on the description of the boundary of the tiles in terms of paths on a graph called “boundary graph”. The paper ends with explicit quadratic examples, showing that the general behaviour of $$\gamma(\beta)$$ is slightly more complicated than in the unit case.

##### MSC:
 11A63 Radix representation; digital problems 03D45 Theory of numerations, effectively presented structures 11S99 Algebraic number theory: local fields 28A75 Length, area, volume, other geometric measure theory 52C23 Quasicrystals and aperiodic tilings in discrete geometry
##### Keywords:
beta-numeration; tilings; periodic expansions
Full Text:
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