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So you are a programmer. Why would you want to know about polynomials? One reason would be that this is the geometrical clay we can easily make different things of.

Mathematical analysis explained with Python, blood, and TNT shows how you can analyze and synthesize arbitrary functions as polynomials. But it doesn't even have to be functions. Sometimes you would want to model a spline out of some points or some properties like tangents of curvatures. Like when you want to do some animation; or nice video effect; or draw a curve that goes through the points you want; or build a surface that is flat in one place and curvy in another.

Polynomials or even polynomial splines might not always be the best tool for the job, but they possess some traits that programmers value. They are inherently simple, versatile, and, most noticeably, highly effective in terms of performance.

Let's say we have a polynomial like that:

P(x) = ax 3 + bx 2 + cx + d

It only takes 6 multiplications and 3 additions to calculate, and this is important because your model is meant to be calculated a lot. But even this can be reduced by using Horner's method. The same polynomial may be written as:

P(x) = ((ax+ b)x + c)x + d

And that's only 3 multiplications and 3 additions. See, we barely started, and you already learned how to throw away one-third of computation effort.

Polynomial interpolation

Fitting an n- degree polynomial in exactly n+1 points is called polynomial interpolation. There are several ways to do that. You can use Newton polynomials or Lagrange polynomials . But the very basic way you can get an interpolating polynomial is by solving a linear system.

If a polynomial P(x) goes through the point (x i , y i ) then, obviously enough, we can claim that P(x i ) = y i . Say we want to fit a polynomial into a set of three points. This means that:

P(x 1 ) = y 1 P(x 2 ) = y 2 P(x 3 ) = y 3

Generally, we can not fit a straight line into three arbitrary points. We would have to bend it forming a parabola. Or, put in other words, 2-nd degree polynomial also known as quadric.

P(x 1 ) = ax 1 2 + bx 1 + c = y 1 P(x 2 ) = ax 2 2 + bx 2 + c = y 2 P(x 3 ) = ax 3 2 + bx 3 + c = y 3

Since we know x s and y s, we only have to solve the system for (a, b, c) coefficients and since it's a three-equation system with three variables, it should generally give us one and only one solution.

Here, try it yourself. This is the canvas with three points on. Please drag them to see what happens.

This is also a good mental model for understanding linear systems. Generally, you can't fit a straight line into three points just like you can't find a solution for (n-1) variables for n-equations system. But sometimes you can. It's when some of the points coincide or they all deliberately lie on a straight line.

Validation Training Institute

Connecting with elders from the heart

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Validation is a method of communicating with and helping disoriented very old people. It is a practical way of working that helps reduce stress, enhance dignity and increase happiness. Validation is built on an empathetic attitude and a holistic view of individuals. When one can “step into the shoes” of another human being and “see through their eyes,” one can step into the world of disoriented very old people and understand the meaning of their sometimes bizarre behavior.

Validation theory explains that many very old disoriented people, who are often diagnosed as having Alzheimer type dementia, are in the final stage of life, trying to resolve unfinished issues in order to die in peace. Their final struggle is important and we, as caregivers, can help them. Using Validation techniques we offer disoriented elderly an opportunity to express what they wish to express whether it is verbal or non-verbal communication. Validation practitioners are caring, non-judgemental and open to the feelings that are expressed. When disoriented elderly can express the things that have often been suppressed for many years, the intensity of the feelings lessen, people communicate more and are less likely to withdraw into further stages of disorientation.

Validation has three basic components

1. Validation is a theory that very old people struggle to resolve unfinished life issues before death. Their behavior is age-specific. Their movements reflect human needs. Validation is a way of classifying their behaviors into four progressive stages:

Malorientation- Expressing past conflicts in disguised forms.

Time confusion- No longer holding onto reality; retreating inward.

Repetitive motion- Movements replace words and are used to work through unresolved conflicts.

Vegetation- Shuts out world completely and gives up trying to resolve living .

Each phase has specific physical and psycho-social characteristics.

2.Validation is based on a basic, empathetic attitude that respects and values very old people without judgment.

3. Validation includes specific techniques for individual as well as group work, based on the needs of the individual and his or her phase of resolution.

Principles were created by Naomi Feil, apply to maloriented and disoriented elderly; they help guide our actions and determine the Validating Attitude. Theoretical assumptions/bases: are created by other theorists, apply to the general population (not specific to disoriented elderly) and are useful in supporting Validation Principles when challenged by scientists or academics. Our Actions/behavior flow out of the Principles and support the Validation techniques.

1. All very old people are unique and worthwhile.

A 90 year old woman lives in a nursing home. The caregiver calls her “sweetie” or mama or grandma.” Validation: Caregivers address her as “Mrs. Smith.” Validation Worker (V/W) action: we address older people in a respectful way and treat each person as an individual. from Humanistic psychology (Maslow, Rogers, etc.): know your client as a unique individual.

2. Maloriented and disoriented old people should be accepted as they are: we should not try to change them.

a 90 year old woman demands her breakfast after she has just eaten. The nurse does not say: “Honey, its 8 o’clock. You just finished all your breakfast. You can’t be hungry”. we know that this woman is perhaps psychologically hungry for her family, for love. We ask, `what fills you up?’ we do not try to change the person’s behavior; we accept it and try to help the individual fulfill the needs that are being expressed. from Humanistic psychology (Rogers, Maslow, etc.): accept your client without judgment. Additionally, Freud said that the therapist cannot give insight or change behavior if the client is not ready to change or does not have the cognitive capacity for insight.

3. Listening with empathy builds trust, reduces anxiety and restores dignity.

A woman angrily declares that the caregiver threw a basin of water over her clothes and now she’s all wet. The trusted, familiar Validation caregiver rephrases, asks, “Does that happen every morning? “Yes, every morning.’ “Is there a morning when she doesn’t throw water over you?’ “Well, only when the very, nice, young lady comes to look in on me in the night. She asks if I need to go to the toilet. You see, I’m getting older now and I have a problem sometimes with pee-pee.’ we understand that this woman is ashamed of being incontinent and empathize with the feelings of the older person without concentrating on the “truth’ of the facts. from Humanistic psychology, most especially Rogers who focused his client-centered approach on using empathy.

4. Painful feelings that are expressed, acknowledged and validated by a trusted listener will diminish. Painful feelings that are ignored or suppressed will gain in strength

An old woman gets up at 3.30 every afternoon to go home to her children. When staff says, “Sit down. Everything’s OK.’ She gets more and more nervous and upset. When staff asks, “What is the worst thing that can happen to your children?” The old woman expresses her vivid memory of having left her children alone. Her fears are expressed to a validating caregiver and the old woman feels relieved. we are open to all feelings that are expressed by older people. Through empathy w share these feelings and encourage expression. We acknowledge that disoriented older people freely express emotions in order to heal themselves. from Psychoanalytical psychology (Freud, Jung, etc.): “the cat ignored becomes a tiger”

5. There is a reason behind the behavior of very old maloriented and disoriented people

A woman accuses the cook of poisoning the food. The nurse does not say, “We have the best cook from Paris.” The nurse knows the old woman may have been overfed by her mother (instead of loved), and now she expresses anger against the cook. The validating caregiver asks, “What does she put into the soup?” An old woman refuses to eat soup and vomits each time she is forced to do so. As a Jewish woman during WWII, she hid her identity papers in the soup tureen during a raid at her home. although we do not always know why the person behaves in a certain way, we help him or her express emotions to resolve unfinished business. The brain is not the exclusive regulator of behavior in very old age. Behavior is a combination of physical, social and intrapsychic changes that happen during the life span. (Adrian Verwoerdt is the original source for this, but this idea has been generally accepted by most geriatricians.)

6. The reasons that underlie the behavoir of maloriented or disoriented very old people can be one or more of the following basic human needs:

7. Early learned behaviors return when verbal ability and recent memory fails:

movement of tongue, teeth and lips create new word combinations – it is often an expression of basic human needs; an old woman sucks on her fingers to feel safe and pleasure (like she felt as a baby) and it is self-stimulating – she is alive. Early learned movement can replace speech when verbal ability fails. A woman moans: HEALVEN. She tells the worker she needs help from heaven. Her mother is in heaven. She combines images and sounds to form, HEALVEN. The worker asks, “You really miss your mother. Was she always with you when you needed help?” We calibrate the breathing, movements, gestures, body tension, mirror movements and sounds. That allows us to get onto the same wavelength as the old person and meet them where they are in that moment, even if we can’t explain their behavior logically. Piaget’s theory: movement comes before speech in human cognitive development.

8. Personal symbols used by maloriented or disoriented elderly are people or things (in present time) that represent people, things or concepts from the past that are laden with emotion.

An old man who has been oppressed by his father, accuses the administrator of tying him up at night; a doll is treated like a baby. A wedding ring can represent love, a handbag, the identity or self. An institutional hall can become a street, a wheelchair can become an auto, an old man who used to be a car mechanic gets under his bed everyday to repair a car. we accept that symbols are used to express needs and feelings; we try to explore and react with empathy. Freud and Jung wrote extensively about symbols, describing them as representations.

9. Maloriented and disoriented old people live on several levels of awareness, often at the same time.

An old woman runs out of the nursing home calling, “mama’; she needs to find her mother. When asked, “Where is your mother?’ the old woman says, “My mother is with the dear Lord.” V/W action: we never lie to older people because we know that on some level they know what is the truth. Freud, preconscious, conscious and unconscious

10. When the 5 senses fail, maloriented and disoriented elderly stimulate and use their “inner senses’. They see with their “mind’s eye’ and hear sounds from the past.

A mother hears her children calling her – she wants to be a good mother to her children. The validating caregiver asks, “How many children do you have?” An old mother hears her daughter crying next door. This happens every night. Her 17 year old daughter died and the mother never had enough time to grieve. She wants to express her guilt. The caregiver asks, “What do you miss the most?” when older people see or hear things that we do not, we accept those as being part of their personal reality and understand that they are trying to meet their human needs. Wilder Penfield: Human beings can stimulate their brains to recreate vivid visual, auditory and kinesthetic memories.

11. Events, emotions, colors, sounds, smells, tastes and images create emotions, which in turn trigger similar emotions experienced in the past. Old people react in present time, the same way they did in the past.

A woman hides behind the sofa scared to death each time the meal cart clatters onto the department. She feels the same fear she felt when tanks clattered through the streets of her hometown and bombed her home when she was 4 years old. The worker asks, “Did something terrible happen?” we accept and acknowledge that experiences from the past can be triggered and explore the feelings without judgment and with empathy. Early, well-established emotional memories survive to very old age. Schettler Boyd. Proust Was a Neuroscientist, by Jonah Lehrer, 2007, published by Houghton Mifflin Co., NY

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Before Your Breast Augmentation Surgery

Before undergoing any Breast enhancement surgery, we will need to meet to go over your particular procedure. Because each woman is unique, both physically and in terms of her desired results, it is important that we go over your options. The four questions that you and I will discuss your surgery are noted below.

Based on your anatomy and aesthetic desires, we will develop a customized plan for you. Having realistic expectations for your breast augmentation surgery is important. Women often think of breast size in terms of bra cup size – but most women are actually wearing the wrong bra size!

In our office we have a wide range of sizes that you can put in your bra and test out (the best way to pick your new size). I’ll have you feel the difference between saline and silicone breast implants and we will talk about the differences. Generally speaking, silicone breast implants tend to have a more natural feel and retain their shape better than saline. However, saline breast implants can be filled after being placed, allowing for a higher degree of precision and control over your size.

The location of the incision and placement of the implants will also play in important part in your final results. The incision can be made in a concealed area, such as the armpit, although this may not be ideal for the type of implant you want. Determining the placement is similar; placing them over the muscle may be easier, but the benefits of placing the breast implants under the muscle include a more natural appearance. This is why we will have to weigh all your options during your consultation.

It is important to me that your consultation is a relaxed, fun experience. I always think back to when I went in for my breast augmentation consultation. I remember feeling confused and overwhelmed, and had trouble picking a size. In my office, there is no rush; I will explain to you in a clear way what your surgical procedure will entail, and answer any of your questions or concerns. I want you to feel excited about your breast implants!

You must have someone drive you home and stay with you for 24 hours after your augmentation. You will experience some pain the first day or two, although this is easily controlled with pain pills. Some tightness in the breast area is also normal as your body adjusts to the implants.

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To get started simplysign up for a free plan and log into your account. You will be presented with the following options:

If you already have an email list elsewhere, you can import it to SendPulse.

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Once the subscription form is embedded into your website you can start collecting subscribers’ personal information and building your email list.

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Unless you’re proficient with HTML, I’d advise you to use any of SendPulse’s pre-designed email templates to save time and effort. Below is a sample of email templates that you’d be using with SendPulse:

Email automation is one of the most effective ways to engage and nurture your email list. With SendPulse, you can easily setup email automation for your website using 3 types of autoresponders:

To set that up you will need to click “Create an autoresponder”, fill in the email subject, select (or create) an email template, and fill it with the text. You can also add triggered messages to this chain. The triggers include: the first (previous) email being opened / not being opened, a click on the link, etc.

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As mentioned earlier, SendPulse is one of the few ESPs in the market to offer both bulk SMS messages sending and web push notifications in addition to bulk email.

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