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TITLE: Divergence (Gauss-Ostrogradsky) theorem
QUESTION [1 upvotes]: Problem statement:
Calculate the integral of the vector field F (x, y, z) = ($ y^2 , z^2 , x^2 $) along
the following surface: r(u, v) = ($u \cdot cos(v), u \cdot sin(v), u^2$ ) , u ∈ [0, 1], v ∈
[0, 2π].
I am fully aware that the surface along ... | 6,960 |
\begin{document}
\title[Chen ranks and resonance varieties of the upper McCool groups]
{Chen ranks and resonance varieties of the \\ upper McCool groups}
\author[Alexander~I.~Suciu]{Alexander~I.~Suciu$^1$}
\address{Department of Mathematics,
Northeastern University,
Boston, MA 02115, USA}
\email{\href{mailto:a.suciu... | 132,549 |
\begin{document}
\pagestyle{myheadings}
\title{Rescalings of free products of II$_1$--factors}
\author{Ken Dykema}
\address{\hskip-\parindent
Department of Mathematics\\
Texas A\&M University\\
College Station TX 77843--3368, USA}
\email{Ken.Dykema@math.tamu.edu}
\author{Florin R\u adulescu}
\address{\hskip-\par... | 118,292 |
Have you ever been in this situation?
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Sweet Lily Natural Nail Spa located in New York City cares about the health of their customers. The products they use contain the most natural ingredients without sacrificing quality. Sweet Lily loves natural ingredients, so it comes as no surprise that they love natural nails and the best polish for natural nails is Z... | 124,313 |
Braids are a huge part of every little girl’s school days. Just when you decide you want to grow your hair longer than shoulder length, the hairstyle you are stuck with through most of your school days would be the two, neatly done, nicely oiled nightmarish braids.
I remember how mum would oil my hair every night to en... | 77,537 |
TITLE: The number of index $p$ subgroup and order $p$ subgroup for elementary abelian group
QUESTION [2 upvotes]: In my previous post, fact 2 says that
For elementary abelian group $A=(\Bbb Z/p)^n$, the number of subgroup of $A$ of order $p$ equals the number of subgroup of $A$ of index $p$.
Here's the proof (last pa... | 130,067 |
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Learn the best practices for developing a security awareness training program that is engaging. Engaging awareness programs have been shown to change more users’ behavior and are seen as an asset for your organization instead of annoyance.
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Introduction
Most likely you will agree that se... | 106,191 |
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TITLE: Do we have Hodge symmetry for char $p$?
QUESTION [2 upvotes]: Let $X$ be a smooth projective variety over a field $k$. Let $h^{p,q}=dim_k H^q(X,\Omega_{X/k}^p)$ be the Hodge numbers.
If $k$ is of char $0$, by Lefschetz principle, we always have Hodge symmetry, i.e. $h^{p,q}=h^{q,p}$, for the following reason: we... | 130,717 |
TITLE: Groups whose automorphism tower is finitely generated
QUESTION [3 upvotes]: Let $G$ be a finitely generated group. Then the automorphism group $\text{Aut}(G)$ of $G$ need not be finitely generated.
However, there are classes of f.g. groups for which the automorphism group will always be f.g., such as polycyclic ... | 188,395 |
The Upside of Regulatory Fines: Career Marketing!]
He's Belgian. More famous now than Van Damme.
Posted by: maccers | May 9, 2006 10:18 AM
Technically, he's Lebanese, no? (But as long as he's associated with running a French fund, we're going to consider him a French fund manager. The French are easier to mock.)
Posted... | 361,720 |
Seniors Halloween Pot Luck Luncheon
The Ancaster Senior Achievement Centre is hosting a Halloween Pot Luck Luncheon!
The Luncheon will be held on Friday October 25th, 2013 starting at 12:00 pm Wear your costume!
For those who cannot make or bring their share to the Pot Luck, the cost is $7.00. (If you do not feel likin... | 226,477 |
In heavily secured Cleveland, activists build Trump his wall
In a hot, high-ceilinged room just west of downtown Cleveland, a group of activists on Tuesday built a wall for Donald Trump.
Not a real wall, of course. For the last five days, they've painted bricks and chain link upon muslin ponchos, including the slogan "... | 109,177 |
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\section{Ring graph}
\label{sec:ring_graph}
It is well-known that the metric dimension of a ring graph on $n$ nodes is two. A set of any two adjacent vertices is a resolving set. In this section we will see that metric dimension will remain two on adding an edge between any two vertices. Hence, this is an example of a... | 50,908 |
Or to give it it’s full name, “Neanderthals, bandits and farmers: How agriculture really began”, is an extremely short 53 page book from the Darwinism Today series, of which I think there are about five in total.
Really it’s a kind of academic paper, but not as dry and without any tedious referencing I picked this up s... | 131,796 |
\begin{document}
\title[An integral functional driven by fractional Brownian motion]{An integral functional driven by fractional Brownian motion$^{*}$}
\footnote[0]{$^{*}$The Project-sponsored by NSFC (11571071, 11426036) and Innovation Program of Shanghai Municipal Education Commission(12ZZ063).
${}^\S$litan-yan@ho... | 16,522 |
\begin{document}
\title[Witt kernels of quadratic forms]{Witt kernels of quadratic forms
for multiquadratic extensions in
characteristic $2$}
\author{Detlev W.\ Hoffmann}
\address{Fakult\"at f\"ur Mathematik, Technische Universit\"at Dortmund,
44221 Dortmund, Germany}
\email{detlev.hoffmann@math.tu-dortmund.de}
... | 2,160 |
TITLE: Is minimal polynomial of field of rational numbers and field of real numbers are same? Why?
QUESTION [0 upvotes]: Is minimal polynomial of field of rational numbers and field of real numbers are same? Why?
REPLY [0 votes]: Note that over the field $\mathbb{R}$ of the real numbers irreducible polynomials hav... | 37,412 |
TITLE: Generate a random binary full-rank rectangle matrix that is a basis of a subspace
QUESTION [0 upvotes]: Disclaimer: I think of vectors as row vectors.
I have a full-rank $m \times n$ ($m < n$) binary matrix $B$ which is a basis of $m$-dimensional subspace $V \subset\mathbb F_2^n$ (i.e. subspace $V$ is formed by ... | 89,671 |
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\begin{document}
\title[On the exponent of a finite group]{On the exponent of a finite group admitting a fixed-point-free four-group of automorphisms}
\keywords{automorphisms, Lie algebras, finite groups}
\subjclass{20D45,17B70}
\author[E. Romano]{Emanuela Romano}
\address[E. Romano]{Dipartimento di Matematica e Infor... | 96,243 |
Tarangire National Park — Africa’s little-known wonder!
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Getting its name from the Tarangire River which flows through it,Tarangire National Park is the sixth largest national park in Tanzania. It is located in Manyara Region covering an area of approximately ... | 230,458 |
TITLE: Automorphism group of $\mathbb{Q}/\mathbb{Z}$
QUESTION [4 upvotes]: Consider the group of all complex roots of unity, $\mathbb{Q}/\mathbb{Z}$ (where both groups are additive groups). I was wondering what its automorphism group is ?
REPLY [9 votes]: Instead of looking just at automorphisms of this abelian group,... | 18,271 |
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Ok..let's try something more basic..can you do a test page on the printer, without it being connected to the computer, at all? would be nice to know that the printer is known good, eh? | Facebook Social Network
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\section{Properties of Relation Compatible with Group Operation/CRG2}
Tags: Compatible Relations, Group Theory
\begin{theorem}
:$(1): \quad x \mathrel {\mathcal R} y \iff e \mathrel {\mathcal R} y \circ x^{-1}$
:$(2): \quad x \mathrel {\mathcal R} y \iff e \mathrel {\mathcal R} x^{-1} \circ y$
:$(3): \quad x \mathrel ... | 157,852 |
AT THE WHEEL OF HISTORY
Imagine a trip to the Amalfi Coast driving Alfa Romeo spider, enjoy the sunset in Portofino on the board of PGO Speedster. Open your Italy driving a retro car for many is an unforgettable experience.
Liveried chauffeur driving a retro car carries you to the castle, you have chosen for your weddi... | 297,109 |
NEEDHAM, Mass .– (COMMERCIAL THREAD) – As companies are tasked with collecting and reporting on environmental, social and governance (ESG) metrics, a market opportunity has arisen for software solutions that support the rapidly evolving requirements of asset management. ESG risks. According to a recent forecast from In... | 261,373 |
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4-H Newsletter
NEW MEMBERS: Any student who has turned 9 yrs. old is eligible to enroll in 4-H. Any new student also needs to fill out a new enrollment form, even if you have completed one at your previous school.
Officer Nominations: You may sign up to run for a 4-H office on the 4-H Bulletin board. Elections will be ... | 395,000 |
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An unexpected medical emergency, travel or wedding expenses can drain a government employee’s savings. Financial aid in the form of personal loans offers a cushion to incur such expenses at a short notice.
Whether it’s a travel-related expense, higher education loan or a medical emergency, financial needs can arise at ... | 1,399 |
Pallet Delivery UK
To request a pallet delivery UK, you’ll first need to measure your freight
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Previously, operators crossed the country to made point A to point B deliveries. This is wildly fuel-inefficient and costly in wear and tear. Now, the model has evolved. Specialists like Hall... | 46,380 |
TITLE: Why is "gravitational" red-shift neglected in galaxy and galaxy cluster scales?
QUESTION [8 upvotes]: The red-shift of the light of a star in a galaxy or that of a galaxy in a cluster of galaxies is generally interpreted as how fast the star or the galaxy is moving, i.e. it is interpreted in a purely special-rel... | 150,168 |
The People Who Own the Dark (1976)
Alternate Versions
The Spanish language print runs 94 minutes. When the film was released in the US it was dubbed in English and ran approximately 82 minutes and it was this shorter, re-edited version that was released onto home video in the states. The English version is not missing ... | 207,346 |
Dragons backs Will Harries and Ashley Smith return to action this weekend as Head Coach Darren Edwards names his squad for the region’s opening RaboDirect Pro 12 match against new Italian outfit, Zebre on Friday night (Rodney Parade, 7pm)
The Men of Gwent will be without captain Lewis Evans and fellow back row Darren W... | 232,374 |
By ERIN CALABRESE
June 10, 2007
LOS ANGELES - As if Paris Hilton didn't have enough to worry about, one of her fellow cons may have taken pictures of her at her most humiliated moment, The Post has learned.
Two inmates at the Twin Towers Correctional Facility said a woman who was brought in with Hilton on Friday night ... | 358,533 |
New £100,000 state-of-the-art CCTV cameras to be installed across the Dales
Derbyshire Dales District Council has started to roll out a state-of-the-art upgraded CCTV system in some of the area's busiest locations.
The purpose-designed digital CCTV system has already been installed at three sites in Matlock Bath - lau... | 268,070 |
TITLE: Check My Proof: if $a$ is even then $9(a +5)$ is odd
QUESTION [0 upvotes]: Just want to make sure I am doing this right.
If $a$ is an even integer then $9(a + 5)$ is odd
If $a$ is even, $a = 2k$, therefore, $9(a + 5) = 9(2k + 5)$ which is odd.
REPLY [2 votes]: You need to justify the assertion that $9(2k + 5)... | 149,523 |
TITLE: Calculating the integral curves of a vector field
QUESTION [16 upvotes]: How do I caluclate the integral curves of a vector field, i.e. how would I go about calculating the integral curves of:
Define the vector field in $\mathbb{R}^3$ by:
$ u = x_1\displaystyle\frac{\partial}{\partial x_2} +x_2\frac{\partial}{\p... | 201,893 |
My Virtual Girlfriend is the 1 dating simulation game available - no other games like that! Please note: The game text посмотреть еще in English language only. Go on activities like mini golf or dinner dates, offer gifts like new clothes and flirting moves that work through text phone app 2017 release will even wear th... | 235,807 |
Homosexuality is perceived as a new phenomenon in Africa and a taboo. It is outlawed in many African countries. Many African leaders have condemned homosexuality as being un-African. The Zimbabwean President, Robert Mugabe once described gays as worse than dogs and pigs. Former Namibia's President, Sam Nujoma, once sta... | 335,026 |
Saeed Ajmal has been a top class character ever since his late arrival in the national side. He was recommended by his domestic team mate Misbah Ul Haq initially. He came into the national side as an experienced seasoned campaigner with a handful of domestic matches behind him.
He helped bowl Pakistan to the World T20 ... | 231,849 |
\begin{document}
\maketitle
\begin{abstract}
Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually e... | 123,728 |
TITLE: Using distributive property to prove that the product of two odd numbers is odd
QUESTION [0 upvotes]: I've been reading Algebra The Easy Way, and there's a problem at the end of the chapter to prove that the product of two odd numbers is an odd number.
In this problem, an even number is defined as two multiplied... | 51,355 |
I have been wanting to do something cookie dough and healthy for a long time. As I am sure you can imagine the two words are not often put together unless to say 'cookie dough is NOT healthy'.
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TITLE: Finding the domain of an equation and then solving.
QUESTION [1 upvotes]: Find domain and then solve:
$$\frac{x+2}{6x} +1 = \frac{x-7}{x}$$
Hello everyone, what does domain in this context mean? I'm not looking for the answer to this question. I'm just looking for a push in the right direction.
REPLY [1 votes]:... | 185,773 |
TITLE: How many different $7$-digit numbers can be formed from $0,1,2,2,3,3,3$ assuming no number can start with $0$? How many numbers will end with $0$?
QUESTION [0 upvotes]: I got this question and wanted to confirm my solution.
How many different $7$-digit numbers can be formed from $0,1,2,2,3,3,3$ assuming no numb... | 183,670 |
Home > Thank You, Bronson Heroes!
Since COVID-19 started to spread throughout our communities in early 2020, team members from every corner of the Bronson system have stepped up to help fight the virus. To our healthcare heroes – we thank you, we admire you and we appreciate you.
Since the start of the pandemic, commun... | 221,224 |
At a presented time, house and house homeowners will need to have plumbing companies to cater for water associated troubles. A plumber wants to be contacted to locate issues with the program, to unblock pipes, to unclog drains and to fix leakages. To conquer boiler engineer , it is a good idea for the home operator to ... | 112,456 |
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3 Fat Chicks on a Diet Weight Loss Community
(
)
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WW Clubs and Groups
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)
- -
Thursday Chatters Enjoying Summer
(
)
Tina
07-18-2013 08:07 AM
Thursday Chatters Enjoying Summer-18-2013 11:21 PM
Has anyone bought Twinkies this week? They're Baack !!!!!!
Unf... | 262,989 |
What's going on in arts, leisure and community May 28 to June 3
DAMASCUS, PA and ONLINE — Workshops from the Farm Arts Collective will be offered online and in-person at the Farm Arts Collective greenhouse on Willow Wisp Organic Farm. Its first workshop of the season, Organic Soil Management, takes place on Saturday, M... | 39,643 |
In this section we first formalize the properties of the online sub-sampling procedure. We then give complete proofs of these properties.
\subsubsection{Choice of the parameter $\beta$}
We elaborate the choice of $\beta$. In our algorithms, $\beta$ is used in the computation of the bonus function~\eqref{eqn:bonus} and ... | 4,322 |
TITLE: Is there a number $n$, such that there are $22$ groups of order $n$?
QUESTION [5 upvotes]: Denot : $N(n)$ : the number of groupfs of order $n$ ?
Is there a number $n$ with $N(n)=22$ ?
Checking the first about $2000$ numbers, I noticed that there is no $n\in [1,2000]$ with $N(n)=22$. Does such an $n$ exist ? An... | 16,438 |
TITLE: Are Sperner's Lemma and Brouwer's Fixed Point Theorem equivalent?
QUESTION [5 upvotes]: On this Wikipedia link I found the following statement:
In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which is equivalent to it.
At first, I found this statement very strange.... | 128,713 |
RSS Feeds Every Designer Should Follow Archive
07 Apr 2010
75 RSS Feeds Every Designer Should Follow
Tweet | 239,553 |
TITLE: Does strict convexity imply differentiability?
QUESTION [26 upvotes]: I know that convexity does not imply differentiability, for example f(x)=|x| is convex but not differentiable. However, |x| is not strictly convex. So I wonder whether strict convexity imply differentiability.
I did some search and found out t... | 51,307 |
Broward Country is situated in Florida and stretches from North to South. It is a big country which ranks in second as the highest populated country in the state of Florida. It has many populated cities and many attractions for its residents. A country which is densely populated will generally accumulate a lot of junk ... | 30,856 |
To see liquid phase reactions, reagents are mixed and reacted in a very small space of 1mm or less.
It’s possible to produce fine particles and capsules by using small flow channel and minute grooves.
For R&D of drug delivery, food, cosmetics, etc
Highly efficient mixing by increasing the interface of 2 liquids
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11:11-6:2 ESV).
2 Corinthians 5:11-6:2
This is a famous passage, beloved by many Christians—and for good reason! Verse 17 beautifully describes the new creation, and verse 21 is perhaps the clearest description of the substitutionary atonement in the Bible.
But the message of the overall paragraph is not as well unders... | 7,367 |
TITLE: Contraction Mapping Theorem. Prove $\{ y_{1},f(y_{1}),f(f(y_{1})),\ \ldots) \} $ is Cauchy. (Abbott p 114 q4.3.9b)
QUESTION [0 upvotes]: Let $f$ be a function defined on all of $R$. Assume there is a constant $c$ such that $0< c <1$ and $ |f(x)\ -f(y)\leq c|x-y|$ for all $x,\ y\in R$.
(a) Show that $f$ is contin... | 140,090 |
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Fashion Designer Karl Lagerfeld's Death Certificate Cements French Legacy
Karl Lagerfeld Death Certificate Cements French Legacy
3/16/2019 12:30 AM PT3/16/2019 12:30 AM PT
Iconic fashion designer Karl Lagerfeld is being remembered in France as one of the nation's favorite adopted sons ... and it's all spelled out in hi... | 359,514 |
TITLE: Reconciling Donsker-Varadhan definition of KL divergence with the "usual" definition
QUESTION [8 upvotes]: Let $\mu$ and $\lambda$ be probability measures on a measurable space $(X, \Sigma)$.
In my experience, the usual definition of the Kullback-Liebler divergence of $\mu$ with respect to $\lambda$ is
$$
\tag{1... | 75,663 |
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TITLE: Let $f\in L^1(R)$. Suppose $\int_{a}^{b} f \,dx\ = 0$ whenever $a<b$. Prove that $f=0 $ a.e.
QUESTION [1 upvotes]: Let $f\in L^1(R)$. Suppose $\int_{a}^{b} f \,dx = 0$ whenever $a<b$.
Prove that $f=0 $ a.e.
My approach-
I thought of showing $m({x\in R : f\ne0})=0$
here there are two cases
$$m({x\in R : f>0})=0$$... | 182,695 |
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A slow website can cost your business lots of money and opportunities. There are simple steps you can take to make your site faster. This infographic from DesignMantic takes a look at a few things that can make your site slow:
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Arvada Subdivisions, Neighborhoods and Condominiums
COhomefinder allows you to browse or map all the subdivisions in Arvada. Some of the top Arvada subdivisions include Holland Hill Subdivision, Trail Village, Trail Village Flg 2, Arvada, Village Of Five Parks, Sunrise Ridge, Saddle Brook, and Village Of Five Parks Flg... | 36,386 |
TITLE: Remainder terms of congruence sums in sets of positive density
QUESTION [4 upvotes]: Let $\mathcal{A} \subset \mathbb{N}$ be an infinite sequence with positive density, in the sense that
$$
\tag{1}
\lim_{x\to\infty} \frac{|\mathcal{A} \cap x|}{x} = c > 0,
$$
and define the remainder term $r_p(x)$ by
$$
r_p(x) = ... | 30,346 |
74 members91 Comments 21 Likes
Rex Stone has not received any gifts yet
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Kingley Vale is located in West Sussex near the south central coast of England, a few miles from Chichester. It is said to contain some of the oldest living trees in Britain, with Yews thought be 2000 years old. During World War 2 it was used for target practice by the air force and if you look hard enough, apparently ... | 387,980 |
28 August, 2014
Date and place of birth:
June 25th, 1956, Charlotte, North Carolina
Place of residence:
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Website:
How did you end up in jazz photography, starting from jazz or from photography itself?
I grew up infused with the sounds of great black music. My parents record collection inclu... | 328,228 |
TITLE: How arithmetical is algebraic exponentiation?
QUESTION [7 upvotes]: Suppose $K$ is an exponential real closed field, i.e. there is an isomorphism, say exp, between the additive group of $K$ and the multiplicative group of its positive elements.
Assume further that $Z$ is an integer part of $K$ whose positive con... | 202,379 |
September 28, 2012
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This bangle of buffalo horn, silver, copper and resin was made by the Scottish artist Paul Godfrey in 1981.
The black oval bangle was carved from buffalo horn. It is decorated with open cubes of inlaid silver and copper on the narrow ends, and a design in silver and striped blue resin.. | 194,927 |
Turizmus Kft. edits an independent tourism magazine called Turizmus.com in partnership with Gazdasági Radio every Thursday in prime time, between 5.15-6 pm. The programme provides a behind-the-scenes glimpse of the tourism sector as seen through the eyes of experts. It supplies ideas, opportunities and useful tips for ... | 304,492 |
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A 12-foot Burmese python has been loose in a Louisiana mall store for days
Whither the US-German Relationship? by Joschka Fischer
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77-Year-Old Man Killed in Crash in Summit Townsh... | 336,935 |
\begin{document}
\title[Convergence to suitable weak solutions for a subgrid FEM model]{Convergence to suitable weak solutions for a finite element approximation of the Navier-Stokes equations with numerical subgrid scale modeling}
\author[S. Badia]{ Santiago Badia$^\dag$}
\thanks{$\dag$ Universitat Polit\`ecnica de... | 163,888 |
Why ‘Off-the-Rack’ Alavi is a Big Hit for
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See What Tiwa Savage Did To Her Son
And if you are not yet a mom, i’m sure this will make you want to have one asap, anyways new mother Tiwa savage is so proud of her cutie as she shares this cute message on her IG page.
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TITLE: Power of $ 2 $ congruent to 1 mod n
QUESTION [1 upvotes]: Fix some odd $ n $. For which values of $ m $, $ 1\leq m \leq n $, is $ 2^m $ congruent to $ 1 $ mod $ n $?
For $ n=3 $ the solutions are $ 2^2=4 \equiv 1 $
For $ n=5 $ the solutions are $ 2^4=16 \equiv 1 $
For $ n=7 $ the solutions are $ 2^3=8 \equiv 1 $... | 190,587 |
TITLE: How to solve this homogeneous recurrence relation
QUESTION [0 upvotes]: I have the homogeneous recurrence relation $x_n = x_{n-2}$ for $n \geq 2$ with $x_1 = 2$ and $x_0 = 1$. So for the characteristic polynomial I got $r^2 - r = 0$, then I factored out r: $r(r - 1)$ for which the roots then are: $r = 0, 1$. I t... | 130,721 |
Tuna Salad SushiPosted on February 8th, 2007 · Post a comment » · 17 Comments »Maki Recipes
As simple as it might sound, this roll is actually my favorite roll out of all the rolls I have made thus far! (It beat out the Crunchy Shrimp Roll, which is my most requested roll by those for whom I make sushi!) The tart tuna ... | 183,509 |
President Obama’s Univision appearance continues to divide commentators. Also, the legal fight over SB 1070 isn’t over, as Arizona again appeals a hold on one of SB 1070’s provisions, this time regarding violations for harboring and transporting illegal aliens. And Microsoft is still pushing for a STEM-based green card... | 329,727 |
TITLE: Folland Chapter 7, Exercise 13
QUESTION [0 upvotes]: Chapter 7 Exercise 13 of Folland's "Real Analysis" reads
Let $X=\mathbb R\times \mathbb R_d$ where $\mathbb R_d$ denotes the discrete topology. Let $f$ be a function on $X$ and define a positive linear functional on $C_c(X)$ by
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\begin{document}
\title {Homogeneous interpolation on ten points}
\author {Ciro Ciliberto and Rick Miranda}
\date{}
\maketitle
\begin{abstract}
In this paper we prove that for all pairs $(d,m)$ with $d/m\geq 174/55$,
the linear system of plane curves of degree $d$
with ten general base points of multiplicity $m$
h... | 186,723 |
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The boys are finally in bed and I am pooped. It's like someone flipped a switch sometime between yesterday and today and turned the babies into disobedient little animals. They were into EVERYTHING tonight. Oh to just be a fly on the wall, that would be nice.
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About writer’s retreat, it was a huge deal. Would everyone hate it? Would the manuscript be covered in red ink? Was it horrible?
At this particular retreat 6 people, including the mentor Dennis Hensley, reviewed the first 10 chapters of my book. It was a humbling, but at the same time, uplifting experience. I received ... | 60,340 |
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