 of coordinates in formula_1 and formula_29, which takes formula_40 to formula_12 (and vice versa). One property that we can impose is that 
where "formula_47" denotes "transverse to". This property ensures that as we vary the unfolding parameters we can predict – by knowing how the orbit foliates formula_48 – how the resulting functions will vary.
Versal unfoldings.
There is an idea of a versal unfolding. Every versal unfolding has the property that 
formula_46, but the converse is false. Let formula_50 be local coordinates on formula_1, and let formula_52 denote the ring of smooth functions. We define the Jacobian ideal of formula_12, denoted by formula_54, as follows:
Then a basis for a versal unfolding of formula_12 is given by the quotient
This quotient is known as the local algebra of formula_12. The dimension of the local algebra is called the Milnor number of formula_12. The minimum number of unfolding parameters for a versal unfolding is equal to the Milnor number; that is not to say that every unfolding with that many parameters will be versal. Consider the function formula_60. A calculation shows that
This means that formula_62 give a basis for a versal unfolding, and that 
is a versal unfolding. A versal unfolding with the minimum possible number of unfolding parameters is called a miniversal unfolding.
Bifurcations sets of unfoldings.
An important object associated to an unfolding is its bifurcation set. This set lives in the parameter space of the unfolding, and gives all parameter values for which the resulting function has degenerate singularities.
Other terminology.
Sometimes unfoldings are called deformations, versal unfoldings are called versal deformations, etc.i didn't get to see the entire episode, but i caught these clips of rihanna singing "redemption song" bob marley & maxwell singing "fistful of tears" you can purchase both songs on itunes within 48 hours of the airing of the show & all proceeds go towards relief efforts in haiti. god bless.
pps. who made that dress Rihanna is wearing...tou'che girl!It's all about books, really, and where to put them. Writers need to be more interested in wall-space than square footage. Some of my shelves are made from books. The room isn't exactly a magical attic or atmospheric garret, but it's warm, quiet and connected to broadband. I have three desks. One's a sort of work-station: PC, printer etc, where I email, write prose, watch YouTube and gab on the phone. The second desk is notionally (laughably) for admin - letters, dealing with requests, ignoring threats and so on. The third is the writing desk, usually given over to a long-term project like a translation or dramatisation. Large, important-looking reference books are displayed here, especially when photographers visit. There's a lectern next to it, for my current favourite book. Not sure where I write the poems, but not in this room.
I call it an office, not a study. Studies have ink blotters, locked walnut bureaus and a splayed Bengal tiger on the floor. All my vinyl is in here, though in fact I transferred it all on to iTunes and listen to it through the computer, randomly. I'm not really fetishistic (don't need to be surrounded by lucky gonks), but I do have a display cabinet of photographs, souvenirs and mementoes. If I had my time again I'd collect small glass or carved animals from all over the world and line them up along the front of the bookcases, but there's no point starting that now. It's too late. There's a huge store-cupboard at one side, like a walk-in mortuary. Its contents are an unsolvable mystery.
My office is above the garage, a five-yard commute every morning, and I think the subliminal proximity of tools and gardening equipment is an aid to productivity. I'm tidy, except when immersed in a project, which is when the cups, pistachio shells and half-eaten meals pile up. I look out on to trees, until it gets dark and they look back at me. Then it's time to knock off.This page displays sold house prices for Carr House Gate in Bradford.
Carr House Gate in Wyke, Bradford consists predominantly of terraced houses. Properties on Carr House Gate typically have values around £100,000 - £100,000, but larger detached houses can command upwards of £200,000.
Map showing Carr House Gate in Bradford.I can't help but smile back at this picture. They look like they are having such fun!
Because I am a trad, i don't understand the whole Thousand word Thursday thing - but I do want to say 1. - I love your music selection and 2. I'm asking for your camera for my b-day next week!!!!!!!! Those photos look totally professional!
Awesome shot!!! I earn 2x what my husband is and no one would guess our income anyways because we drive basic cars, shop at kohls, etc... but I get so annoyed when people assume I stay at home or could stay at home with the automatic assumption that my husband is the breadwinner. And with all the COVID stuff it’s women no matter their circumstances that have had to take the hit in childcare and we just can’t. We just bought a new car to replace my old car and until we got to the financing part the dealer was pretty much just speaking to my husband until they saw the numbers and I started talking what we’re willing to do for numbers. My husband and I have joint accounts, we make all decisions together but I just get really frustrated at the assumptions people make and also I feel like an expectation of subservience because my husband must be making more.Welcome to day 14 of affirmation.
I am happy, whole and satisfied. God fills my life with unending joy.
Is your happiness tied to your partner?
Do you believe that you are whole and satisfied?
Today, I chose to be happy in this relationship. I am not defective. I have value that I am bringing into the relationship.
I choose to be satisfied with where I am at the moment. I see this current state of our relationship as a life long journey and not a destination. If it seems not satisfactory, I choose to work with my partner to move towards satisfaction.
There is a saying that a man and woman will come together in marriage to become one. Someone’s logical side will see this as half plus half make a whole. I hope you still remember the game of numbers.
You have to be whole(mature psychologically & spiritually) before you get into a serious relationship.
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