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The Tools in Photoshop

Photoshop has many different tools, as shown in the image below. You can use any of these tools to manipulate images.

The most common tools include:

Rectangle Select – Spatial Selection

Rectangle Select is very useful for drawing objects. You can also use the Feather Selection tool, but Rectangle Select is not only faster, it is the best tool for drawing objects for cutting.

The tool enables you to draw an object (a rectangle, a line, a circle) and you can change its size, or you can select a rectangle and select the amount of feather (size) you want the selection to be.

The Rectangle Select tool is very easy to use: Draw a square, then select the rectangle button, as shown in Figure 2. You can change the size of your selection by changing the vertical and horizontal dimensions.

Figure 2: Rectangle Select tool. The rectangle is the selection area that you want to be in your picture.

Pen – Raster Drawing

The Pen tool is a powerful tool that is best used to create raster images (as opposed to vector images).

The Pen tool is used to draw objects or clean areas (erase unwanted objects or objects that are not needed) with a series of lines.

The Pen tool is very similar to the Rectangle Select tool, but it can be used to draw in any type of image, not only in the space inside a rectangle.

Figure 3: The Pen tool. A series of points are connected with lines. The points are selected and you can use the Draw Tool to convert the selected points to another type of shape.

Ellipse Select – Polygon Selection

The Ellipse Select tool works much like the Pen tool.

However, the Ellipse Select tool only draws a circle, rather than a line.

It’s a powerful tool for creating shapes and cleaning unwanted portions of the image.

Figure 4: The Ellipse Select tool. You can click and drag to create circular shapes.

Draw Tool – Raster Editing

The Draw tool is used to draw shapes in an image.

The drawing is converted into a shape after you click and drag in an image.

The drawing is saved as a new layer. You can resize the shape and use the paint bucket (or color) to change

The following article will help you get started with Photoshop Elements and become a productive user of this amazing program!

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Font Encoding and Versioning

For all of us who love fonts, then you’ll appreciate how great Photoshop is at handling fonts. I’ve never really had a problem with fonts, as I’ve never needed to create my own font from scratch. Photoshop does a great job of converting foreign fonts or changing the font you use to a different font.

When I first started using Photoshop Elements, I went to the File menu, under Format, and changed the font to a nice serif font (this was in 2007 when I first started using Photoshop Elements), but I noticed that when I go back and use that font in any text, it turns back into a regular old Times New Roman. I’m sure it’s just something simple I’m not aware of but I’m sure there’s a way to fix this so I have the text in this nice font in all my pages.

Another thing Photoshop does is it converts any fonts that are used in the current file to a special non-printing font. When you export a file, it looks for this special font in the font folder and if it doesn’t find it, Photoshop will go through and find any fonts and convert them all to this special non-printing font.

There’s nothing wrong with this. The problem is this is a hidden font. You can’t actually see it. When you open a photo and view it in Photoshop, you will see your regular type; but if you view the same photo in Photoshop Elements, the photo will use this special “non-printing” font in any text.

It seems a little silly that you can’t view this special font in the Photoshop editor, but it seems that’s the best way to use these fonts and share them.
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Brushes

Brushes are the most popular brushes, and generally speaking they are the most powerful, as well. They can be used to create various artistic effects. For example, a soft-edged brush can be used to draw smooth curves. A hard-edged brush can be used to draw subtle wrinkles in cloth. A feathered brush can be used to paint a smooth blend between two colors.

Brushes can be used on any open image or on a document. Brush size and feathering can be changed by clicking on the brush icon on the toolbox or selecting the brush from the toolbox and then pressing the up or down key. You can also select a brush from the Brush or Pen tool options from the Preferences dialog box (or press CTRL+K) to pick up an existing brush or pen tool. You can also change the size and opacity of a brush or pen tool with the Size and Opacity options in the Tool Options dialog box (or press U in the toolbox).

Foreground and Background

The Brush tool works on any open image, but sometimes it is helpful to select a non-painting area of the image to use as a “foreground” (the area that will appear through the effect) or background (the area behind the foreground). You can do this with the Brush or Pen tool’s Select tool. In either case, you can then paint on the selected image area with the Brush or Pen tool.

Type of Brush

The Type of brush determines the characteristics of the brush. For example, a Size brush is the most basic brush. It has no feathering or other effects, but it has a continuous, finite width. A Smooth brush has the same width as the pen. It has no “snow” or “hair” edge, but allows you to paint smooth curves and is useful for painting in fine detail.

Basic Brushes

Most Photoshop brushes are created with a brush size of 8 px and a normal size of 25 px. In addition, there are also Hard, Soft and Flat brushes. The most commonly used brushes are shown in the image below.

Normal Brush

You can create your own brushes by using the Brush tool, the New Brush tool or by using the Clone Stamp tool. The Brush tool allows you to draw a stroke that you use to paint an image, or it can be used to paint directly over an existing image. You can change the size of the brush with the brush

Q:

Convergence of a sequence on $2\mathbb{Z}$

$\{ a_n\}_{n\in \mathbb{N}}$ is a sequence on $2\mathbb{Z}$ such that $a_{2^k}=a_k$ for all $k\in \mathbb{N}$. Define the sequence $\{ b_n\}_{n\in \mathbb{N}}$ by $b_0=0$ and $b_{n}=a_{\lfloor \frac{n}{2}\rfloor }$ for $n>0$. Prove that
$\lim_{n\rightarrow \infty }\left( \frac{b_n}{2^n}\right) =1$.
I was able to prove by contradiction that, for all $n\in \mathbb{N}$, $b_n$ is not an integer. So it converges to $1$. Hence, the limit is $1$. I am stuck in showing that $b_n$ has at least one integer subsequence that converges to $1$.
Any hints on how to start would be appreciated.

A:

Consider all the subsequences of the form $\{a_{2^k},a_{2^k+1},a_{2^k+2},\ldots,a_{2^{k+1}-1},a_{2^{k+1}},a_{2^{k+1}+1},\ldots\}$. If one of these doesn’t converge, then there is a subsequence that doesn’t converge, and there is some point in it that is a limit point.
Since $a_{2^k}=a_k$ for all $k$, it follows that $\frac{b_k}{2^k}$ is a subsequence of the original sequence. So it is either divergent or a non-negative integer. Clearly, it cannot be a non-negative integer, so we are done.

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