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Inequality (mathematics)
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In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality).
- The notation a ≠ b means that a is not equal to b.
- It does not say that one is greater than the other, or even that they can be compared in size.
If the values in question are elements of an ordered set, such as the integers or the real numbers, they can be compared in size.
- The notation a < b means that a is less than b.
- The notation a > b means that a is greater than b.
- In either case, a is not equal to b. These relations are known as strict inequalities. The notation a < b may also be read as "a is strictly less than b".
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
- The notation a ≤ b or a ⩽ b means that a is less than or equal to b (or, equivalently, not greater than b, or at most b); "not greater than" can also be represented by a ≯ b, the symbol for "greater than" bisected by a vertical line, "not". (The Unicode characters are: U+2264 ≤ LESS-THAN OR EQUAL TO, U+2A7D ⩽ LESS-THAN OR SLANTED EQUAL TO, and U+226F ≯ NOT GREATER-THAN.)
- The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, not less than b, or at least b); "not less than" can also be represented by a ≮ b, the symbol for "less than" bisected by a vertical line, "not". (The Unicode characters are: U+2265 ≥ GREATER-THAN OR EQUAL TO, U+2A7E ⩾ GREATER-THAN OR SLANTED EQUAL TO, and U+226E ≮ NOT LESS-THAN.)
In engineering sciences, a less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude. This implies that the lesser value can be neglected with little effect on the accuracy of an approximation. (For example, the ultrarelativistic limit in physics.)
- The notation a ≪ b means that a is much less than b. (In measure theory, however, this notation is used for absolute continuity, an unrelated concept.)
- The notation a ≫ b means that a is much greater than b.
Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and (in the case of applying a function) monotonic functions are limited to strictly monotonic functions.
The transitive property of inequality states:
- For any real numbers a, b, c:
- If a ≥ b and b ≥ c, then a ≥ c.
- If a ≤ b and b ≤ c, then a ≤ c.
- If either of the premises is a strict inequality, then the conclusion is a strict inequality:
- If a ≥ b and b > c, then a > c.
- If a > b and b ≥ c, then a > c.
- Since a = b implies a ≥ b these imply:
- If a = b and b > c, then a > c.
- If a > b and b = c, then a > c.
The relations ≤ and ≥ are each other's converse:
- For any real numbers a and b:
- If a ≤ b, then b ≥ a.
- If a ≥ b, then b ≤ a.
Addition and subtractionEdit
A common constant c may be added to or subtracted from both sides of an inequality:
- For any real numbers a, b, c
- If a ≤ b, then a + c ≤ b + c and a − c ≤ b − c.
- If a ≥ b, then a + c ≥ b + c and a − c ≥ b − c.
i.e., the real numbers are an ordered group under addition.
Multiplication and divisionEdit
The properties that deal with multiplication and division state:
- For any real numbers, a, b and non-zero c:
- If c is positive, then multiplying or dividing by c does not change the inequality:
- If a ≥ b and c > 0, then ac ≥ bc and a/c ≥ b/c.
- If a ≤ b and c > 0, then ac ≤ bc and a/c ≤ b/c.
- If c is negative, then multiplying or dividing by c inverts the inequality:
- If a ≥ b and c < 0, then ac ≤ bc and a/c ≤ b/c.
- If a ≤ b and c < 0, then ac ≥ bc and a/c ≥ b/c.
- If c is positive, then multiplying or dividing by c does not change the inequality:
More generally, this applies for an ordered field; see #Ordered fields.
Additive inverseEdit
The properties for the additive inverse state:
- For any real numbers a and b, negation inverts the inequality:
- If a ≤ b, then −a ≥ −b.
- If a ≥ b, then −a ≤ −b.
Multiplicative inverseEdit
The properties for the multiplicative inverse state:
- For any non-zero real numbers a and b that are both positive or both negative:
- If a ≤ b, then 1/a ≥ 1/b.
- If a ≥ b, then 1/a ≤ 1/b.
- If one of a and b is positive and the other is negative, then:
- If a < b, then 1/a < 1/b.
- If a > b, then 1/a > 1/b.
These can also be written in chained notation as:
- For any non-zero real numbers a and b:
- If 0 < a ≤ b, then 1/a ≥ 1/b > 0.
- If a ≤ b < 0, then 0 > 1/a ≥ 1/b.
- If a < 0 < b, then 1/a < 0 < 1/b.
- If 0 > a ≥ b, then 1/a ≤ 1/b < 0.
- If a ≥ b > 0, then 0 < 1/a ≤ 1/b.
- If a > 0 > b, then 1/a > 0 > 1/b.
Applying a function to both sidesEdit
Any monotonically increasing function may be applied to both sides of an inequality (provided they are in the domain of that function) and it will still hold. Applying a monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.
If the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. The rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.
A few examples of this rule are:
- Exponentiating both sides of an inequality by n > 0 when a and b are positive real numbers:
- a ≤ b ⇔ an ≤ bn.
- a ≤ b ⇔ a−n ≥ b−n.
- Taking the natural logarithm to both sides of an inequality when a and b are positive real numbers:
- a ≤ b ⇔ ln(a) ≤ ln(b).
- a < b ⇔ ln(a) < ln(b).
- This is true because the natural logarithm is a strictly increasing function.
Ordered fieldsEdit
If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:
- a ≤ b implies a + c ≤ b + c;
- 0 ≤ a and 0 ≤ b implies 0 ≤ a × b.
Both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field, because −1 is the square of i and would therefore be positive.
The non-strict inequalities ≤ and ≥ on real numbers are total orders. That is, given arbitrary a, b ∈ R, at least one of a ≤ b and b ≤ a holds; at the same time, at least one of a ≥ b and b ≥ a holds. The strict inequalities < and > on real numbers are strict total orders. That is, < on R has trichotomy property: given arbitrary a, b ∈ R, exactly one of a < b, b < a and a = b is true; likewise, > on R has the trichotomy property.
Chained notationEdit
The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, a < b + e < c is equivalent to a − e < b < c − e.
This notation can be generalized to any number of terms: for instance, a1 ≤ a2 ≤ ... ≤ an means that ai ≤ ai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to ai ≤ aj for any 1 ≤ i ≤ j ≤ n.
When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1/2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1/2.
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For instance, a < b = c ≤ d means that a < b, b = c, and c ≤ d. This notation exists in a few programming languages such as Python.
Sharp inequalitiesEdit
An inequality is said to be sharp, if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ, if ψ ⇒ φ holds, then ψ ⇔ φ also holds. For instance, the inequality ∀a ∈ ℝ. a2 ≥ 0 is sharp, whereas the inequality ∀a ∈ ℝ. a2 ≥ −1 is not sharp.[citation needed]
Inequalities between meansEdit
There are many inequalities between means. For example, for any positive numbers a1, a2, …, an we have H ≤ G ≤ A ≤ Q, where
(harmonic mean), (geometric mean), (arithmetic mean), (quadratic mean).
Power inequalitiesEdit
A "power inequality" is an inequality containing terms of the form ab, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.
- For any real x,
- If x > 0 and p > 0, then
- In the limit of p → 0, the upper and lower bounds converge to ln(x).
- If x > 0, then
- If x ≥ 1, then
- If x, y, z > 0, then
- For any real distinct numbers a and b,
- If x, y > 0 and 0 < p < 1, then
- If x, y, z > 0, then
- If a, b > 0, then
- This inequality was solved by I.Ilani in JSTOR,AMM,Vol.97,No.1,1990.
- If a, b > 0, then
- This inequality was solved by S.Manyama in AJMAA,Vol.7,Issue 2,No.1,2010 and by V.Cirtoaje in JNSA, Vol.4, Issue 2, 130–137, 2011.
- If a, b, c > 0, then
- If a, b > 0, then
Well-known inequalitiesEdit
Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:
- Azuma's inequality
- Bernoulli's inequality
- Bell's inequality
- Boole's inequality
- Cauchy–Schwarz inequality
- Chebyshev's inequality
- Chernoff's inequality
- Cramér–Rao inequality
- Hoeffding's inequality
- Hölder's inequality
- Inequality of arithmetic and geometric means
- Jensen's inequality
- Kolmogorov's inequality
- Markov's inequality
- Minkowski inequality
- Nesbitt's inequality
- Pedoe's inequality
- Poincaré inequality
- Samuelson's inequality
- Triangle inequality
Complex numbers and inequalitiesEdit
The set of complex numbers with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that becomes an ordered field. To make an ordered field, it would have to satisfy the following two properties:
- if a ≤ b, then a + c ≤ b + c;
- if 0 ≤ a and 0 ≤ b, then 0 ≤ a b.
Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ −a). In either case 0 ≤ a2; this means that and ; so and , which means ; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b, then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:
- , if or and
It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.
Vector inequalitiesEdit
Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors (meaning that and , where and are real numbers for ), we can define the following relationships:
- , if for .
- , if for .
- , if for and .
- , if for .
Similarly, we can define relationships for , , and . This notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).
The trichotomy property (as stated above) is not valid for vector relationships. For example, when and , there exists no valid inequality relationship between these two vectors. Also, a multiplicative inverse would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.
General existence theoremsEdit
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For a general system of polynomial inequalities, one can find a condition for a solution to exist. Firstly, any system of polynomial inequalities can be reduced to a system of quadratic inequalities by increasing the number of variables and equations (for example, by setting a square of a variable equal to a new variable). A single quadratic polynomial inequality in n − 1 variables can be written as
where X is a vector of the variables , and A is a matrix. This has a solution, for example, when there is at least one positive element on the main diagonal of A.
Systems of inequalities can be written in terms of matrices A, B, C, etc., and the conditions for existence of solutions can be written as complicated expressions in terms of these matrices. The solution for two polynomial inequalities in two variables tells us whether two conic section regions overlap or are inside each other. The general solution is not known, but such a solution could be theoretically used to solve such unsolved problems as the kissing number problem. However, the conditions would be so complicated as to require a great deal of computing time or clever algorithms.
See alsoEdit
- Binary relation
- Bracket (mathematics), for the use of similar ‹ and › signs as brackets
- Fourier–Motzkin elimination
- Inclusion (set theory)
- Inequation
- Interval (mathematics)
- List of inequalities
- List of triangle inequalities
- Partially ordered set
- Relational operators, used in programming languages to denote inequality
- Hardy, G., Littlewood J. E., Pólya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8.CS1 maint: Multiple names: authors list (link)
- Beckenbach, E. F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2.CS1 maint: Multiple names: authors list (link)
- Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6.CS1 maint: Multiple names: authors list (link)
- Grinshpan, A. Z. (2005), "General inequalities, consequences, and applications", Advances in Applied Mathematics, 34 (1): 71–100, doi:10.1016/j.aam.2004.05.001
- Murray S. Klamkin. "'Quickie' inequalities" (PDF). Math Strategies.
- Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
- Harold Shapiro (2005, 1972–1985). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan. Check date values in:
- "3rd USAMO". Archived from the original on 2008-02-03.
- Pachpatte, B. G. (2005). Mathematical Inequalities. North-Holland Mathematical Library. 67 (first ed.). Amsterdam, The Netherlands: Elsevier. ISBN 0-444-51795-2. ISSN 0924-6509. MR 2147066. Zbl 1091.26008.
- Ehrgott, Matthias (2005). Multicriteria Optimization. Springer-Berlin. ISBN 3-540-21398-8.
- Steele, J. Michael (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press. ISBN 978-0-521-54677-5.
External linksEdit
|Wikimedia Commons has media related to Inequalities (mathematics).
- Hazewinkel, Michiel, ed. (2001) [1994], "Inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Graph of Inequalities by Ed Pegg, Jr., Wolfram Demonstrations Project.
- AoPS Wiki entry about Inequalities


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SoupOrSam wrote:Rick Tocchet @RealRocket22 30m
Players decide the game .John Steigerwald @Steigerworld 16m
@RealRocket22 Not calling the hook didn't let Malkin decide it. Helped Jagr decide it. Call 'em all.
This is a ginormous pet peeve of mine. "The refs shouldn't influence the game". Pfft. Not calling obvious penalties is a form of interference. Always hated that as a player.
Not saying the Malkin play in OT should've been a penalty, just chiming in on the concept.

subset: commoncrawl


text:
A couple weeks ago I was searching for something and came across a series of threads over on the GameDev.net forums that sought to address the prominent flaws in RPGs like Allacrost (that is, menu-based battles). I still haven't read them through all the way (in fact, I'm just starting), but I think there are some very good ideas and points brought up here that we may want to consider in our own designs. The links to these discussions are below. After I've read through them, I may suggest some ideas for us to consider here.
1) The "Fight" Command
http://www.gamedev.net/topic/626973-wee ... t-command/
2) Grinding
http://www.gamedev.net/topic/627204-wee ... -grinding/
3) Attrition
http://www.gamedev.net/topic/627645-wee ... attrition/
4) Exploration
http://www.gamedev.net/topic/627898-wee ... ploration/
5) Accessibility
http://www.gamedev.net/topic/628459-wee ... ssibility/
6) Safe Havens
http://www.gamedev.net/topic/629626-not ... fe-havens/
Ironically, a thread I posted on GameDev last year is what prompted these discussions to start.
Hello, we apologize but forum registrations are non-functional at this time. This issue should be fixed around mid-December. Until then, please stop by our Discord channel if you'd like to get in touch with the team. Thanks!
RPG Genre Flaws Discussion
A discussion area for general design issues that staff would like detailed feedback on.
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text: My wife has a 2010 F650GS with 1,200 miles on it and it is the standard suspension bike with a low seat. We need to still get it lowered a bit via the suspension and am curious if anyone has a F650GS with the factory low suspension rear shock that would like to trade for our standard height rear shock? If interested, let me know what mileage is on your shock and we can go from there. I figured if we make a swap, we each pay our own respective shipping / insurance costs. Thanks!


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Hey I know Facebook is taking over America. I'm not blind. While I refuse to jump on that train and get sucked into the vortex that is Facebook I get it. It's the coolest thing going. I've got my hands full trying to keep GMG the very best it could be. Signing up for Facebook would undoubtably mean that the blog would suffer and we can't have that now, can we?
Anyway, I'm asking all you Facebook freaks out there if you dig GMG could you pass along the link along to your pals on Facebook for them to check out GMG?
Here's the link for you to copy and paste-
Here is the email signup link-


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2012 Headlines
BSC names Larry D. Striplin Athletes of the Year at annual Athletic Awards BanquetMay 06, 2013BIRMINGHAM, Ala.–Birmingham-Southern held its annual Athletic Awards Banquet Monday evening in Bruno Great Hall and named the 2013 Newcomers, Athletes, Teams, and Man and Woman of the Year, as well as the Johnny Johnson Most Inspirational Senior Student Athletes.
Cross Country set to compete at NCAA Regionals this weekendNovember 10, 2012BIRMINGHAM, Ala.—Ten members of the Birmingham-Southern cross country teams are set to compete this Saturday, Nov. 10, at the NCAA Division III Regional Championships, hosted by Emory University at Nash Farm Battlefield in Hampton, Ga.
Willett finishes 17th at NCAA Cross Country Regional, earns All-Region honorNovember 10, 2012HAMPTON, Ga.—Senior runner Robert Willett finished 17th overall and earned All-Region honors at the NCAA South/Southeast Regional on Saturday, hosted by Emory at Nash Battlefield Farm in Hampton, Ga.
Cross Country set to compete at first SAA Championship this weekendOctober 27, 2012BIRMINGHAM, Ala.—The Birmingham-Southern cross country teams will travel to Choctaw Trails in Clinton, Miss., this weekend for the inaugural Southern Athletic Association Cross Country Championships. The race will take place on Saturday morning, Oct. 27.
Willett finishes second at SAA Cross Country Championship; men place fourth, women sixthOctober 27, 2012CLINTON, Miss.—Senior Robert Willett placed second overall to earn First-Team All-Conference honors at the inaugural Southern Athletic Association Cross Country Championship, held earlier today at Choctaw Trails in Clinton, Miss. The BSC men finished fourth overall, while the women were sixth.
Holdbrooks, Willett top Panther finishers at Coach O Cross Country InvitationalOctober 13, 2012TROY, Ala.—Senior Robert Willett finished third and freshman Kelsey Holdbrooks was eighth for the Birmingham-Southern cross country teams Saturday at the Coach O Invitational, hosted by Troy University in Troy, Ala.
Willett, BSC men win Tiger Twilight Cross Country InvitationalOctober 05, 2012SEWANEE, Tenn.—Senior Robert Willett won the men's individual title and helped the Birmingham-Southern men's cross country team win the Tiger Twilight Invitational, Friday evening in Sewanee, Tenn. On the women's side, freshman Kelsey Holdbrooks finished third in helping the Panthers to a second-place team finish.
Holdbrooks, Willett named SAA Runners of the WeekSeptember 25, 2012ATLANTA, Ga.—Senior Robert Willett and freshman Kelsey Holdbrooks have been named Southern Athletic Association Cross Country Runners of the Week, it was announced today.
Men's Cross Country places fifth at Montevallo Falcon ClassicSeptember 22, 2012MONTEVALLO, Ala.—The Birmingham-Southern cross country teams competed at the Montevallo Falcon Classic on Saturday.
Women seventh, men ninth at Commodore Cross Country ClassicSeptember 15, 2012NASHVILLE, Tenn.—The Birmingham-Southern cross country teams competed at the Commodore Classic, hosted by Vanderbilt University in Nashville, Tenn., Saturday. The Panther women placed seventh and the men were ninth.
Cross Country results from Auburn InvitationalSeptember 07, 2012AUBURN, Ala.—The Birmingham-Southern cross country teams competed at the Auburn Invitational today in Auburn, Ala.
Willett wins Montevallo Early BirdAugust 31, 2012MONTEVALLO, Ala.—The Birmingham-Southern men's and women's cross country teams opened the 2012 season on Friday at the Montevallo Early Bird, with senior Robert Willett finishing first in the men's 5K.
Cross Country opens 2012 season this FridayAugust 28, 2012BIRMINGHAM, Ala.—The Birmingham-Southern cross country teams begin their 2012 campaign this Friday evening with a short trip south to the University of Montevallo for the Early Bird Classic.


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Astros snap seven-game skid by toppling King Felix
SEATTLE -- The circumstances seemed about as dire as you could imagine. The Astros pulled into Safeco Field riding a seven-game losing streak Monday and with the unenviable task of facing Mariners ace Felix Hernandez, who has been dominant this year.
Baseball, as they say, is a funny game.View Full Game Coverage
SEATTLE -- The circumstances seemed about as dire as you could imagine. The Astros pulled into Safeco Field riding a seven-game losing streak Monday and with the unenviable task of facing Mariners ace Felix Hernandez, who has been dominant this year.
Baseball, as they say, is a funny game.View Full Game Coverage
The Astros enjoyed all the laughs in their first meeting of the season against the Mariners, getting a two-run homer from Matt Dominguez and a solid performance from starting pitcher Dallas Keuchel to send Seattle to its seventh consecutive loss, 7-2.
The seven runs tied a season high for the Astros. They had scored eight in their previous four games combined.
"That was arguably our best game that we played all year as far as complete baseball, from pitching to defense to timely hitting," Astros manager Bo Porter said. "Even a lot of the outs that were made were really quality at-bats across the board."
Keuchel (2-1) won by allowing six hits and two runs in six innings for the second start in a row, this time striking out eight batters. He has quality starts in each of his last three outings, lowering his ERA to 3.38. He said command of his off-speed stuff and improved confidence have been the key.
"I think I have some above-average off-speed stuff, and when I can throw my fastball for strikes with movement, it kind of sets up that off-speed," he said. "Any time I can get ahead and make those guys think about the fastball, it makes that off-speed so much better."
Hernandez, meanwhile, entered the game with a 2-1 career record in four starts against the Astros with a 2.00 ERA. The Astros' approach was to try to wait for him to elevate some changeups, and they were able to square up several balls.
"They put a couple swings on him," Seattle catcher Mike Zunino said. "They were aggressive all night and swung the bats quite a bit. It was one of those where the few mistakes you make, they're swinging and being aggressive and they ended up connecting on them."
Hernandez walked Jose Altuve to start the game before retiring the next nine batters he faced and 12 of 13. It appeared he was on his way to another big night when Alex Presley (3-for-4) singled with one out in the fifth and Dominguez launched a two-run homer to left to put Houston ahead, 2-0.
"We swung the bats well," Dominguez said. "We got a lot of good pitches to hit and put the barrel on it a good amount of times and got some guys in scoring positions and we were able to drive them in tonight."
The Mariners tied it moments later in the bottom of the fifth on a two-run double down the right-field line by Abraham Almonte, which caused Keuchel some grief.
"My whole thinking was go out there in the fourth and put a shutdown inning together, but I kind of messed around with a few guys and two runs later, it's tied up again," he said. "I've got to do a better job of shutting the door."
Altuve led off the sixth with a double to left and wound up safe at third when Kyle Seager dropped a throw from Zunino following a Dexter Fowler bunt. The error wound up leading to four unearned runs in the inning.
That's because Astros catcher Jason Castro hit a sac fly to make it 3-2, and Marc Krauss had an RBI single and Presley had a RBI triple with two outs. Presley then scored on a long Dominguez fly ball to left that hit off the wall for a double, pushing the lead to 6-2.
"We put a lot of barrels on the balls and were able to swing at a lot of pitches we wanted to swing at, as opposed to the stuff he wanted us to swing it," Presley said. "We hit the ball hard as a group and you see what happens when you do that."
Hernandez (3-1) worked seven innings and allowed six runs (two earned) and seven hits. Before giving up the homer to Dominguez, he had pitched 16 consecutive scoreless innings against the Astros since last year.
"The guys really accepted the challenge today," Porter said. "He's arguably one of the best pitchers in our league on the mound for those guys tonight, and I felt our guys really locked into the game plan tonight. [Hitting coaches] John Mallee and Ralph Dickenson did a really good job of preparing these guys and they were able to carry it out. They got the ball up and got some swings on the ball."
Krauss blasted his second homer of the season to right field in the eighth to stretch Houston's lead to five. Krauss and Presley, a pair of platoon players, went a combined 5-for-8 with four runs scored.
"When you start talking about a platoon, when it's your day you got to do your part," Porter said. "I've been saying that to our guys. Offensively, when a team struggles, one guy says, 'OK, I'm going to get us out of this.' It's not going to be just one guy. You just need to concentrate and do your part. Once we were able to get the big hit with Dominguez's two-run homer, I felt the guys relaxed even more and just continued to have good at-bats."
Chad Qualls, Matt Albers and Josh Fields combined for three scoreless innings in relief, allowing just one batter to reach. Fields struck out all three batters he faced in the ninth to end the game.
Brian McTaggart is reporter for MLB.com and writes an MLBlog, Tag's Lines. Follow @brianmctaggart on Twitter.


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Maysville Community and Technical College is one of the fastest-growing two-year colleges in America and a vital educational and arts resource for our northeastern Kentucky communities.
Our success would not be possible without the generous financial support of alumni, friends, and local businesses who value a strong, affordable college education right here in the surrounding counties of MCTC.
There are many convincing reasons to give to Maysville Community and Technical College. Your cash donation funds scholarships for students who may otherwise not have the opportunity to attend college and buys new computers and lab materials for technology and health care programs.
In addition to cash donations, the Resource Development Office can help you set up a tax-deferrable, planned gift in the form of a named endowment, a living will, and gifts of life insurance or retirement savings. Learn more about ways to support MCTC.
Our staff welcomes your questions about donating to MCTC.
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"Juntos" is Spanish for "together", and to collaborate is to work together for a common goal. The Juntos Collaborative is a group of adult education programs that share the goal of helping adults in Holyoke to get a free, high-quality education. The partners offer coordinated, comprehensive course sequences in Bilingual/Native Language Literacy, English ABE/ASE, and ESOL, along with counseling/referral services including college and vocational transition, to adult learners in Holyoke. We have been sharing an Adult Basic Education grant from the Massachusetts Department of Elementary and Secondary Education since 1995.
The Juntos programs cooperate instead of competing with each other. Teachers from the different programs work together to improve our lesson plans and teaching skills. The program coordinators meet once a month to talk about how to make our programs as strong as we can. We share a common Advisory Board that helps us think about the best way to meet the needs of the adults who study with us and helps us work on long-term planning for adult education in Holyoke. Finally, every year we hold a large event called "Juntos Learner/Teacher Day" in which we bring together participants and staff from all of our programs to get to know each other and think together about a common theme.
The Holyoke Public Schools serves as the lead agency for Juntos.
Interested in tutoring? Learn about the Holyoke Tutor/Mentor Program.


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When to Plant Okra in Monterey,+california,+united+states+of+america
Planting vegetable seeds at the correct time is the most important step for getting the most out of your garden. Knowing exactly when to plant okra in your garden in Monterey,+california,+united+states+of+america will increase growing success and provide better yields.
Check optimal planting dates regularly for best results because the weather, and our results, change daily.
The optimal time to plant okra in Monterey,+california,+united+states+of+america is - . The best day is with a success rate of %.
Anywhere between 70-100% is great. If consistently below that you may want to start growing from seedling or young plant, or possibly grow in sheltered environment
Optimal time frame to start
Frost free days
Grow period
Tell us what you think about these results
You are very interesting!
Important okra growing information relevant to your exact garden location in Monterey,+california,+united+states+of+america:
The total number of frost free days for your garden in Monterey,+california,+united+states+of+america is days, but the total number of optimal days to plant okra is days.
For best results you should start planting okra in days starting .
Gardening Data Sources
University Research
Cooperative Extension
Gardening Websites
Gardening Blogs
Seed Suppliers
Seed Breeders
Home Gardeners
Other Sources
Gardening Data Sources
University Research
Cooperative Extension
Gardening Websites
Gardening Blogs
Seed Suppliers
Seed Breeders
Home Gardeners
Other Sources
How to Grow Okra in Monterey,+california,+united+states+of+america
Okra is referred to as a magical vegetable cultivated for its edible green fruits whose texture varies dramatically when cooked. Okra is one of those vegetables that divides people into two groups, those that love it and those that hate it. The exact origin of okra is not known but there are two theories. The most common is that it originated in what is present day Ethiopia and Eritrea. The second theory hypothesizes that ancestors of modern day okra are native to Northern India. Evidence suggests that it was cultivated by the Egyptians in 12th century BC and spread from there to North Africa, and from there to other countries in the Mediterranean. Okra arrived to the U.S. around the 1700's and was extremely popular in the South where European cuisine and African cooking produced the unique Creole cuisine commonly associated with New Orleans. India ranks first in the world for okra production, and its climate is highly suitable for growing okra which is a heat loving annual plants that requires a consistent temperature above 85 degrees Fahrenheit and its adaptability to various moisture conditions. Okra is rich in Vitamin C, Vitamin A, magnesium, calcium, and dietary fiber among others.
For more real-time hyper local detailed growing information download the GardenWeather App on IOS. It is now the leading gardening and weather app, more accurate than The USDA zone map for frost dates. GardenWeather™ is a technology and agri-tech startup providing real-time hyper local planting and gardening recommendations enabling home gardeners and homesteaders to make data-driven gardening decisions.
Use these images to know if your okra is heading in the right direction and growing properly. Exact number of growing days varies for each okra variety. Below is the average number of days for each stage gathered from many different varieties.
Okra Germination Guide
(Stage 1: Germination)
Germination is the process by which a plant grows from a seed. It is the first stage of the plant growing process and it begins when the seed is sown and ends when the radicale emerges out of the seed.
All seeds require proper soil temperature, moisture, air and light conditions to germinate successfully. However, during the germination stage the seeds require more soil warmth and moisture than light exposure.
Seeds respire just like any other living organism, requiring oxygen and producing carbon dioxide (CO2), so it is crucial to use a soil media that is sufficiently aerated so that the seeds do not "suffocate".
80% of our growing experts recommend to directly sow okra seeds into the ground. What is direct sowing and why do I care about it?
Required germination period
4-19 days
Optimal germination period
6-11 days
Required soil temp.
62.7º-101.6ºF / 17º-38.7ºC
Optimal soil temp.
84º-88.8ºF / 28.9º-31.6ºC
Required day temp.
59.6º-103ºF / 15.3º-39.4ºC
Optimal day temp.
87º-93.3ºF / 30.6º-34.1ºC
Required night temp.
57.6º-82.3ºF / 14.2º-27.9ºC
Optimal night temp.
71.7º-92.5ºF / 22º-33.6ºC
Optimal seed depth
0.7 inches / 1.8 cm
Optimal plant spacing
12 inches / 30.5 cm
Optimal row spacing
33.8 inches / 85.7 cm
Okra Growing Guide
(Stages 2-5: Growth)
Once seed has germinated and radicale has emerged the second stage of the plant growing process begins, developing from radicale to cotyledon -the embryonic first leaves of the seed forming a young seedling.
During the third stage of plant development cotyledon develops to 3-4 true leaves, stem begins to grow, becoming a mature seedling.
In the fourth stage the mature seedling develops to a young plant, ready for transplanting.
The fifth stage of plant development the plant is fully mature bearing fruit and/or flower.
The following guidelines apply to all of the okra plants' needs during stages 2-5 from radicale emergence to full maturity. During these stages sunlight becomes a more active participant in plant growth and development than in the germination phase.
Soil pH is an important factor during these stages as well. What is pH and why is it important?
Requires direct sunlight
Required soil temp.
60.9º-95ºF / 16.1º-35ºC
Optimal soil temp.
69.4º-86.6ºF / 20.8º-30.3ºC
Required day temp.
55º-NANºF / 12.8º-NANºC
Optimal day temp.
76.3º-83.3ºF / 24.6º-28.5ºC
Required night temp.
55º-NANºF / 12.8º-NANºC
Optimal night temp.
60º-68.8ºF / 15.6º-20.4ºC
Required sunlight hours
Optimal sunlight hours
Required soil pH
Optimal soil pH
Required watering amount
1-1 inches / 2.5-2.6 cm (per week)
Optimal watering amount
1-1 inches / 2.5-2.5 cm (per week)
Optimal depth
0.9 inches / 2.3 cm
Optimal plant spacing
15.6 inches / 39.5 cm
Optimal row spacing
25.4 inches / 64.6 cm
How difficult is it to grow okra?
Growing okra, or any other vegetable for that matter, is not hard, however some plants require more attention and care than others depending on the following factors:
Are you starting to grow your plant from seed? seedling? young plant? Depending on what stage plant you choose to plant will impact growing difficulty and patience.
The length of the growing season for your exact garden location.
If you have a short growing season and you want to start growing from seed then you may need to start the growing process in a sheltered environment (greenhouse or inside your home) which will require more involvement in the growing process.
Whether you direct sow in the garden or start in trays and then transplant also affects the level of difficulty
Growing difficulty level - 0
Growing patience level - 0
Companion Planting Guide
What is companion planting?
Companion planting is intentional planting of two or more vegetables and/or flowers next to each other in order to positively impact plant health and yield
Get More with Growmigo Pro
- No more estimated planting dates, only research based, accurate planting dates for your specific garden location
- Unlimited access to all research links
- Download Unlimited PDF's
- Access to growing information of new plant varieties that are added daily.
- Cancel anytime if you're not happy
Join the thousands of satisfied gardeners that have already signed up to Growmigo Pro


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Steven / Phil:
What size images? And, were they .jpg's?
I tested with a 230k image and it posted... and it converted it to about 30k.
Steven / Phil:
What size images? And, were they .jpg's?
I tested with a 230k image and it posted... and it converted it to about 30k.
Works great for iPhone and iPad.Tapatalk in my iPhone is working well for me. Very simple to post pics straight from the phone. The 3rd party hosting must all be done by magic. Here's my newly tidied workbench on board
BTW--great shot!
There are two kinds of boaters: those who have run aground, and those who lie about it.
That's a .png, started as an 1167 x 733, 1.5mb image, and posting converted it 800 x 500, 82k.
Doesn't work for me. I tried uploading from my desktop and got a message that the file exceeded the limit. I thought Scot just said that the forum would resize the image?
What size/type image? Want to send it to me? I'd be interested in testing it.
Hi Scot the image is a 2550X1702 jpeg. I tried to send it to you but can't without an email address to send it to.
Scot, thanks for getting to work on this. I hope you get there without too many headaches.
Robt's nice photo...
This is Phil's image, at 2.1mb prior to upload, automatically skinnier down to 53k.
Great, Scot - thanks! So if I post a big file stored on my hard drive, the WBF will automatically reduce the file size to something suitable?
Except that it wont do it for me
Yes, I tried to post a jpeg of 168 kb and was told I exceeded my quota by 24 kb.
RFNK... NOW try it.
IMG_3570.jpgHey! It works! This is fantastic!
I was born on a wooden boat that I built myself.
WooHoo! Thanks Scot!
Bruce entry.jpg
Yes, it works!
Thank you Scot - greatly appreciated!
It's great that photo posting has been simplified, but I don't understand why it is necessary to log in to see the images. I have no doubt there are many casual and even regular visitors to this forum who won't want to go to the trouble of registering just to see a photograph.
For some unfathomable reason, the photos that I've posted here (hosted on photobucket) are still visible. Perhaps I'm blessed. More likely it's because I have so few photos on photobucket that they don't care. Nevertheless this new manner of posting photos seems a lot more convenient so I am joining the chorus in giving thanks to the WoodenBoat folks, in particular to Scot!
My first photo under this new system:
First Solo & on the mooring 2.jpg
Test image of me on my new tractor.
If one advances confidently in the direction of his dreams, and endeavors to live the life which he has imagined, he will meet with a success unexpected in common hours.
-Henry David Thoreau-
Well I'll be....... Thanks Scot. This is awesome.
If one advances confidently in the direction of his dreams, and endeavors to live the life which he has imagined, he will meet with a success unexpected in common hours.
-Henry David Thoreau-
Test image
I used this new photo posting method and I think I'm doing right. But how do you rotate images Scot?
thanks for making the process a lot easier too. I don't understand it but that's pretty normal
The "quick" way to have vertical images show correctly is to crop to horizontal (or make the 'canvas' horizontal).
At this point the software won't keep the orientation... instead the longest side will be the width.
Testing......that is awesome!
2014-10-01 MASCF (14).jpg
Take Care,
Steve W
Honeoye Falls, New York
Building a B & B Core Sound 20 Mark III "Jazz Hands"
Spindrift 11N Suzy J Build Pictures
A little video of the Suzy J and my youngest son
thanks so much for working something out! I really appreciate it, having just restored some fairly big threads after losing all my photos to Photobucket.
You don't have to be prepared as long as you're willing to suffer the consequences.
I just checked back on this and it works...so easy! Here is a pic from the beginning of the 16 LYS build...two dreamers
Most ISPs furnish web space for personal web pages...
You guys have read the memo haven't you? You can post here direct from your device. No need for third party hosting these days.


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William B. Long, M.D.
Emeritus Associate Professor CE of Medicine
Department: Medicine
Contact information
Hospital of the University of Pennsylvania
3 Ravdin
3400 Spruce Street
Philadelphia, PA 19104
3 Ravdin
3400 Spruce Street
Philadelphia, PA 19104
Office: 215-615-4951
Fax: 215-662-6530
Fax: 215-662-6530
Education:Permanent link
Yale University, 1959.
University of Pennsylvania, 1963.
Yale University, 1959.
University of Pennsylvania, 1963.
Selected PublicationsChong A, Shah JN, Levine MS, Rubesin SE, Laufer I, Ginsberg GG, Long WL, Kochman ML.: Diagnostic Yield of Barium Enema Examination after Incomplete Colonoscopy. Radiology 223: 620-624, 2002.
Cope C, Tuite C, Burke DR and Long WB: Percutaneous management of Chronic Pancreatic Duct Stricture and External Fistulas with Long-term Results. J Vasc Interv Radiol 12: 104-110, 2001.
Pfau PR, Kochman ML, Lewis JD, Long WB, Lucey MR, Olthoff K, Shaked A, Ginsberg GG: Endoscopic Management of Postoperative Biliary Complications in orthotopic Liver Transplantation. Gastrointestinal Endoscopy 52: 55-63, 2000.
Scotiniotis I and Long WB: Common Bile Duct Stones. Current Treatment Options in Gastroenterology. 3: 133-140, 2000.
Lichtenstein GR, Park PD, Long WB, Ginsberg GG, and Kochman ML: Use of a push enteroscope improves ability to perform total colonoscopy in previously unsuccessful attempts at colonoscopy in adult patients. American Journal of Gastroenterology 94: 187-190, 1999.
Lipson DA, Berlin JA, Palevsky HI, Kotloff RM, Tino G, Bavaria J, Kaiser L, Long WB, Metz DC, Lichtenstein GR: Giant gastric ulcers and risk factors for gastroduodenal mucosal disease in orthotopic lung transplant patients. Digestive Diseases & Sciences 43: 1177-1185, 1998.
Long WB and Allen Pachence: Pathophysiology of bile, bile salts and bilirubin. 1998 Notes: Used for pathophysiology 205 in 1998.
Faigel DO, Stotland BR, Kochman ML, Hoops T, Judge T, KroserJ, Lewis J, Long WB, Metz DC and Ginsberg GG: Device choice and experience level in endoscopic foreign object retrieval: an in vivo study. Gastrointestinal Endoscopy 45: 490-494, 1997.
Kadish SL, Faigel DO; and Long WB: Safe duodenoscopic intubation in a patient with a large Zenker's diverticulum. Gastrointestinal Endoscopy 44: 101-102, 1996.
Amygdalos MA, Haskal ZJ, Cope C, Kadish SL and Long WB: Transjugular insertion of biliary stents (TIBS) in two patients with malignant obstruction, ascites, and coagulopathy. Cardiovascular & Interventional Radiology 19: 107-109, 1996.
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Last updated: 04/09/2008
The Trustees of the University of Pennsylvania
Last updated: 04/09/2008
The Trustees of the University of Pennsylvania