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# Week 14: Lighting and Shadows

## Key Points

• Today's Lab: Lighting Lab
• [ ] Why is lighting important in a 3D scene?
• [ ] Why are shadows important in a 3D scene?
• [ ] What are some differences between point, spot, and directional lights?
• [ ] What is a normal and why is it important for lighting?
• [ ] What is the difference between diffuse, specular, and ambient lighting components?
• [ ] What is a material and how does it relate to shaders and textures?
• [ ] How do games implement reflections?
• [ ] How does forward rendering performance scale with the number of lights in a scene amd the number of models in a scene?
• [ ] What workarounds can you think of to make up for this?
• [ ] How are blob shadows implemented? What are some drawbacks?
• [ ] What is shadow-mapping and how does it fit into a rendering pipeline?
• This series of articles can be a nice overview for different aspects of rendering.

## Check-in: Progress on 3D Games

Pairs of teams should get together and discuss what you've been up to with your 3D games. What have you implemented so far? What are the roadblocks you're up against? Can you figure out together a way to solve them or work around them by tweaking the design?

What has teamwork been like? Can you think of ways to improve it or make it more equitable?

## Lighting

So far in our 3D scenes, every surface is evenly lit as if illuminated from within. This can make objects look flat, since our eyes rely on cues like differences in brightness to determine the shapes of objects (e.g., a solid color cube will look like a hexagon from almost all angles).

It's also difficult to perceive the relative positions of objects without cues like lighting and shadows—even though farther objects are drawn smaller, it is impossible to tell just by looking at two objects whether they are the same size (and they are equally close to the viewer) or whether one is larger (and further away). While texturing can help in this case, other similar situations are still problematic; it is difficult to tell where an object is relative to a ground plane if it does not cast a shadow, for example.

Lighting also gives us vital information about what objects are made of and what their physical properties might be. Smooth, soft objects tend not to take on harsh specular highlights (they scatter light fairly evenly), while reflective metal surfaces don't diffuse light (all the rays of light we see have come from a light or another object with no scattering). In our examples so far, the stone texture doesn't look very convincing because there is no self-shadowing or sense of depth in the material (as if it were just painted on plastic), and it can be hard to know whether the terrazzo-textured orbs are in tne air or on the ground.

We need lighting and shadows to know where things are and perceive what they are made of. How do we fake that in 3D games?

### Light Sources

Light comes from some type of light source. Photons are emitted and bounce off of objects, and some of these will reflect towards a viewer (in our case, the camera). Different lights emit photons in different ways, and we'll discuss four main types of lights.

First off is the directed light, mimicking something like the sun—infinitely far away, casting a uniform field of light oriented in a certain direction:

struct DirectedLight {
dir: Vec3, // just a normalized vector (technically a bivector) is fine
color: Vec3 // rgb intensities
}

Next is the point light, which is sort of the opposite: undirected, but located in space. It radiates light equally in all directions.

struct PointLight {
pos: Pos3,
color: Vec3
}

Finally we have the spot light, which has both position and orientation; it emits light in a cone shape defined by an angle θ and sometimes an attenuation exponent (so the center is brighter than the edges).

struct SpotLight {
pos: Pos3,
dir: Vec3,
theta: f32,
att: f32
}

In practice, we sometimes will make one data structure that captures all three using some tricks; Panda3D does this for example. This can be a sensible move since GPUs mainly work with four-element vectors.

struct Light {
pos: Pos4, // if w == 0.0, ignore; this is a directed light
dir: Pos4, // if w == 0.0, ignore; this is a point light
color: Vec4, // Pack theta, att into w coordinate
// For point lights, theta is 2PI and att is 1
// For directed lights, theta is PI and att is 1
// The shader could ignore those, but it might work out
}
impl Light {
pub fn point(pos:Pos3, color:Vec3) -> Self {
Self {
pos: Pos4::new(pos.x, pos.y, pos.z, 1.0),
dir: Vec4::new(0.0, 0.0, 0.0, 0.0),
color: Vec4::new(
color.x, color.y, color.z,
Self::pack_theta_att(std::f32::consts::TAU, 1.0)
)
}
}
pub fn directed(dir:Vec3, color:Vec3) -> Self {
Self {
pos: Pos4::new(0.0, 0.0, 0.0, 0.0),
dir: Vec4::new(dir.x, dir.y, dir.z, 1.0),
color: Vec4::new(
color.x, color.y, color.z,
Self::pack_theta_att(std::f32::consts::PI, 1.0)
)
}
}
pub fn spot(pos:Pos3, dir:Vec3, color:Vec3, theta:f32, att:f32) -> Self {
Self {
pos: Pos4::new(pos.x, pos.y, pos.z, 1.0),
dir: Vec4::new(dir.x, dir.y, dir.z, 1.0),
color: Vec4::new(
color.x, color.y, color.z,
Self::pack_theta_att(theta, att)
)
}
}
fn pack_theta_att(theta:f32, att:f32) -> f32 {
assert!(theta >= 0.0);
assert!(theta <= std::f32::consts::TAU);
assert!(att >= 0.0);
assert!(att <= 1.0);
// theta is in the 0..2PI range
let theta = theta / std::f32::consts::TAU;
// now theta is in 0..1
// att is in 0..1 already
f32::from_bits(
(((theta * (u16::MAX as f32)) as u32) << 16) |
((att * (u16::MAX as f32)) as u32 & 0x0000_FFFF)
)
}
// This would usually happen in a shader on the GPU,
// but this is the logic we need:
fn unpack_theta_att(theta_att:f32) -> (f32,f32) {
let bits = theta_att.to_bits();
let theta = (bits & 0xFFFF_0000) >> 16;
let att = bits & 0x0000_FFFF;
// normalize theta from 0..65535 to 0..1
let theta = theta as f32 / (u16::MAX as f32);
// back to the 0..2PI range
let theta = theta * std::f32::consts::TAU;
// now the same for att
let att = att as f32 / (u16::MAX as f32);
(theta, att)
}
}

### Lighting Models and Materials

We can't simulate every photon accurately (even ray tracing only samples from the emitted photons), so we need to fake it using statistical approximations. While we could explore the world of physically based rendering (where irradiance computations are combined with modeling of microfacets on materials to produce physically-sound, energy-conserving lighting computations), we'll start with a simpler model.

We want to come up with an equation to describe how much light hits a point on an object—i.e., how much of the object's color (from its texture, for example) is visible based on the light's intensity and its own color. It's enough to capture a ratio between 0 and 1, with 0 meaning unlit by this light and 1 meaning lit with maximum intensity.

First, we'll fake the phenomenon of ambient lighting, which is a result of photons continuing to bounce around many, many times before expending their energy. Even objects that aren't directly illuminated—or the rear faces of such objects—are usually not completely dark, because stray light has bounced around the room or other objects and reached its dark side. This portion of the equation is just a constant $$\alpha$$ which we add in. This constant could be set in a shader uniform or per-light.

The second phenomenon is the observation that all objects have microscopic bumps on them that absorb and disperse light of different frequencies differently, so when we look at a sheet or a curtain or a desk we mostly see the color of the object, modulated by the color of the light. We call this diffuse lighting. Statistically, we'll see the most rays from a light bouncing off of a surface (in all directions) when the light is directly pointed at the surface. We can imagine a linear model here where more "correct" angles use more of the light's intensity and less correct angles use less.

Mathematically, when lighting a fragment we need to find the incident angle of light onto the fragment. We can do this in one of two ways:

• For point and spot lights, by subtracting the fragment's position (obtained by interpolating between its component vertices) from the light's position (given in a uniform) and normalizing;
• For directional lights, the light is infinitely far, so there's an infinite number of these vectors. We just use the light's direction.

If we normalize that incident light vector, then we can find the diffuse intensity by taking the dot product of the incident light vector and the surface's normal. If the light is coming from directly "above" the fragment from the fragment's perspective, it will be at maximum intensity. The further away from the normal the light is, the less diffuse light it will spread off of the surface.

If it's a spot light, we also want to check the dot product $$d$$ of that incident light vector against the direction of the light, and if that does not exceed the cosine of the light's θ parameter (i.e. the fragment is not inside the cone) then force the intensity to 0; otherwise, we want to attenuate the light intensity by taking something like $$((1 - \mathrm{cos}(\theta))/(1 - d))^a$$, so that as the dot product is smaller (the lines are less parallel) the denominator gets larger and the term as a whole decreases. Any exponential function will work here.

Finally, we have the specular light component: harsh reflections that we see on metallic objects or other shiny surfaces, where light has come from a light source and bounced off of a surface towards our viewpoint. This glare can be reduced by changing your viewpoint or tilting the object, so it's clear it's a situation where that incident light angle will come in handy.

For today's discussion we'll look at an approach which is not physically motivated but gives a decent effect at low computational cost: the Phong and Blinn-Phong models.

Unlike diffuse lighting, specular lighting depends on the angle between the viewer and the incident light to the fragment. So in the Phong model, we need to compute the dot product of the incident with the normal, then the dot product of the reflected light vector with the vector from the fragment to the camera. This is used to scale the intensity of the specular highlight by exponentiating with some given "shininess" constant.

This approach is okay, but if the angle between the viewer and the reflected light is greater than 90 degrees, it breaks down. We need to provide some specularity in cases like this because no surface is perfectly smooth, and some of that reflected light still may have reached us. Blinn-Phong improves on Phong by using different vectors which are guaranteed to be less than 90 degrees apart from each other: first, it computes the so-called half-angle vector, which is halfway between the incident light and the viewpoint; second, it takes the dot product between the half-angle vector and the surface normal to determine the intensity of the specular reflection (if they're the same, the viewpoint is the angle of reflected light). Then that dot product is exponentiated by a material-dependent constant, as in Phong.

We haven't discussed it here, but lights also do tend to attenuate with distance, which can be nice visually and for efficiency reasons (e.g., culling lights that are too far to influence an object).

To obtain the final color of a fragment, we sum up the contributions of each light; each light contributes its ambient intensity multiplied by the diffuse color of the fragment, its diffuse intensity multiplied by the diffuse color, and its specular intensity multiplied by the fragment's specular color. These additions saturate at white (full intensity).

So far, this math has depended on three object-dependent properties: diffuse color, specular color (maybe the same as the diffuse color), shininess constants, and surface normals. Any of these could be given by a UV-wrapped texture and sampled, which can give some interesting illusions (look up normal mapping or bump mapping if you'd like to learn more) and effects (e.g., a metallic hairband with high specularity on a character's head).

Note that because all color contributions must be summed up, lighting any object depends on iterating through all of the lights in the scene! So each draw call requires iterating through every light for every fragment, leading to some very expensive computations. Note also that we need to make a new draw call for each new combination of textures, uniform parameters, or (sometimes) models we're rendering. If there are some things in your scene that don't need to be realistically lit, consider drawing them with a different, less expensive shader.

### Fancier Lighting

Most environments have more than a dozen or so lights, but lights are largely static. How do we render scenes like those?

For static geometry, we can fake it completely by baking the lights in advance: doing expensive lighting calculations when the level is exported, and exporting diffuse and specular information to be loaded by the game engine. We can also use a technique called environment mapping, where we put probes in the environment and render the scene from the viewpoint of each probe, facing six different directions, and creating a cube map which we can later sample like a texture to know how illuminated any point within that cube is. The more probes we have the more accurate this lighting can be, but the more expensive it is to sample them. These static lights can be added in to the lighting equations for dynamic objects as well.

A similar technique of rendering a scene into a cubemap can be used for reflections on metallic objects. If we have an environment map made from the perspective of the center of a reflective object, we can determine the color of a fragment on its surface by sampling the environment map and obtain reflections against static parts of the scene. If we want to have reflections of dynamic objects as well, or if the reflective object moves, we'll either need to re-render those environment maps often (expensive!) or do extra work to figure out what color from the dynamic object should be used for the reflection fragment.

In the special case where the reflective object is a plane, the job of rendering reflections is easier: just render the scene again from the perspective of the reflective plane, either clipping it to the region of screen space occupied by the reflective plane or rendering it to a texture and mapping that onto the reflective plane.

Other important lighting effects include Fresnel and rim lighting. This extended tutorial has some great examples.

Interestingly, a lot of these fancy tricks are just things you get for free with raytracing, so it's kind of silly how hard it is in rasterization-based approaches. But rasterization is just so fast!

The last topic for today is shadows. Shadows can help us figure out the relative positions of an object and the ground or other surfaces. The principle behind shadows is simple (and also trivial in raytracing): when light rays are blocked (occluded) by an object, they do not reach other objects which are behind the occluder! Unfortunately, our lighting equations would become horribly complex if we had to do collision checks along the incident light ray to figure out if a light should influence a fragment or not—so we have to do more hacks.

The simplest hack for shadows is to assume that light comes only from one direction and goes from somewhere mostly above the object towards the ground. We project a ray from the character's feet in the direction of the light source (usually, straight down) and, wherever it hits, place a small dark (sometimes translucent) circle or oval oriented along the normal to the surface. For greater realism in case the shadow should be cast on a couple of different surfaces, we might project rays from multiple points and deploy several shadow bits.

When combined with statically baked shadows, this can be fairly convincing. A more realistic way to achieve "I'm on the ground now" shadowing and more is screen space ambient occlusion, which I recommend looking up and reading about! We might discuss it in a couple of weeks.

If we want to get more realistic shadows that take the object's shape into account, we need to do some more sophisticated tricks. Shadow mapping is an old technique that's still used today for forward rendering pipelines, and it works on the observation that if we couldn't "see" a fragment from the perspective of a light source pointed at it, then that fragment should be in shadow—i.e., it should receive no light from this source.

Of course, we don't want to determine shaded-ness for each light for each fragment one at a time. It would be great if we could somehow calculate which positions in space are illuminated by each light, and then use that information during our fragment shader to determine how shadowed the fragment is. We're trying to view the scene from some perspective… sounds like a job for another render pass!

In this case, we don't care about the color of objects from the perspective of the light. It suffices to figure out the depth buffer we would obtain if we render the scene from the light's point of view. We can then sample from that depth buffer to determine how shadowed a fragment is: first by converting the fragment's position in world coordinates to its position from the light's perspective (multiplying by the inverse of the light's model-view-projection matrix), and then seeing if the depth of the depth buffer at that transformed $$x,y$$ position is greater or lower than the transformed $$z$$ value (which is how far forward from the light this fragment is). If the depth buffer sample is a lower value than the transformed fragment position's $$z$$, then the rays from this light were blocked from reaching this fragment.

We can figure out how shadow casting works for our three types of lights:

1. For spot lights, render the scene from the light's position, pointed in its direction, with a perspective projection based on θ. This is the simplest case.
2. For point lights, we render the scene six times from the light's position pointed in different directions to produce a cubemap; each time we use a perspective projection. Sampling the shadow cubemap is a little different.
3. For directed lights, render the scene with an orthographic projection since all light rays are parallel; but what position should we use and what width and height should our orthographic "box" be? It takes some care to be sure the box is large enough to hold the objects within (or even a bit outside of!) the camera's frustum which might either cast or receive shadows. To a first approximation, you could get nearby objects within the camera frustum; bound them with a box or other volume; and then add in objects that are in the inverse direction of the light from that bounding box.

Rendering the scene an additional time (or 6 times!) every time the lights move or objects in the scene move is pretty rough. Deferred rendering is an approach to do lighting and shadows in screen space once per actually rendered fragment, rather than once per possibly-rendered fragment. It's a really cool technique.

## Lighting Lab

You've been given a starter code package displaying a Blinn-Phong lit scene with randomly assorted spheres and boxes and a movable point light. Implement one of the following options into your renderer by modifying engine3d or light3d where necessary:

1. Additional lights (make them different colors) and types of lights (spotlight, directed light).
2. Physically based rendering shader model
3. Three different materials (different shaders + textures) that are visibly different; good ways to make them different include lighting properties, bump and normal mapping, vertex shader differences where appropriate, etc. They can be all visible at once or cycled through with a keyboard shortcut.
4. Reflective surface(s) and/or other uses of environment mapping


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Video Transcript. Maths. Therefore the total negative state is -4. (a) CrO (b) \mathrm{CrO}_{3} (c) \mathrm{Cr}_{2} \mathrm{O}_{3} Enroll in one of our FREE online STEM summer camps. | EduRev NEET Question is disucussed on EduRev Study Group by 249 NEET Students. Options (a) 3 and +3 (b) 3 and 0 (c) 6 and +3 (d) 4 and +2. The coordination number and oxidation state number of Cr in K₃ Cr (C₂O₄)₃ are respectively . I understand that F is -1. Oxidation/Reduction Cycling of Iron and Sulfur is Driven by Microbial Respiration, and Can Lead to Cr(VI) Reduction and Attenuation as Insoluble Cr(III) Precipitates 138 Figure 5-1. Electronegativity of the ligand, i.e. Chemistry - oxidation numbers. CN – will cause pairing of electrons. The first is Cr+2, also known as Chromous. The oxidation states of Cr in [Cr(H 2 O) 6]Cl 3, [Cr(C 6 H 6) 2], and K 2 [Cr(CN) 2 (O) 2 (O 2) (NH 3)] respectively are +3, 0 and +6. Watch Queue Queue Favorite Answer. The EPMA measurement, however, cannot distinguish the oxidation states. The other one is Cr+3, also known as Chromic. Biology. I do not know the correct answer, however I can tell you that it is not +1 or +4 which were my two best guesses. K +1 Cr +6 O -2 What is the oxidation state of each element in SO32–? Laser scanning confocal microscopy (LSCM) was employed to identify Cr oct 3+ and Cr tetr 4+ ions, by excitation with a 532 nm frequency doubled Nd:YAG laser and a 1064 nm Yb-doped fiber laser, respectively. what is the oxidation number of Cr in Cr(OH)4^-? The oxidation state is a direct measure of the number of valence electrons in the outer shell of an atom. NCERT RD Sharma Cengage KC Sinha. Anonymous. Rep:? CrO3. 1 answer. Thus as the oxidation state of the central metal atom increases in the order . The oxidation state of Cr in[Cr(NH3)4 Cl2]^+ is (a) 0 (b) +1 (c) +2 (d) +3. Thus, it can either have a tetrahedral geometry or square planar geometry. 1 decade ago. #"Oxidation state/number"# is the charge left on the central atom, when all the bonding atoms are removed with the charge devolved to the more electronegative atom. Similarly in the MnO 4 -, Mn is present in the +7 oxidation state. Not what you're looking for? Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 … This video is unavailable. Correct Answer: 6 and +3. 0. reply. Therefore,X + (-2) + 4 ( -1) = 0. When we don't know both the oxidation state of central metal atom and charge on the coordination sphere, in this case how we find the oxidation state of central metal atom? Physics. The ions in which the central metal atom is present in the highest oxidation state will have the highest oxidising power. NCERT NCERT Exemplar NCERT Fingertips Errorless Vol-1 Errorless Vol-2. Relevance. An objective lens with a numerical aperture of 0.65 was employed to achieve a spatial resolution of about 1 μm. Chemistry. See the answer. > We can use the rules for oxidation numbers. Nov 28,2020 - What is the oxidation state of Cr in [Cr(NH3)4(Cl2)]?? 1 decade ago. Relevance? In compounds (which are brightly colored in the majority of cases), chromium displays several possible oxidation states - +2, +3, +4 (encountered quite rarely, chromium oxide CrO₂ is known), +6. Download PDF's . Space is limited so join now!View Summer Courses. NCERT RD Sharma Cengage KC Sinha. Explanation: No explanation available. The sum of the oxidation numbers of all atoms in a neutral compound is 0. Anonymous. Why is +6 the oxidation state of Cr in CrO2F2? Assign an oxidation state (or oxidation number) to an atom in a molecule or ion. Answer Save. What is the oxidation state of Cr in each compound? What is the oxidation state of Cr in each of the following compounds? Favorite Answer. In VO 2 +, vanadium is present in the +5 oxidation state, while in Cr 2 O 7 2- ion, Cr is present in the +6 oxidation state. Let oxidation number of Cr be X. 9 years ago. 1 Answer. It is diamagnetic in nature due to the unpaired electron. S +4 O -6 . This problem has been solved! We know that the sum of all oxidation numbers turns up to be 0. I thought it became +2 when in compound with fluorine? 1 Answer. Answer $+3$ Chapter 8 Introduction to Solutions and Aqueous Reactions Chemistry Structure and Properties Topics. 0 votes . CrO. The essential rules for this problem are: The oxidation number of a halogen in a compound is -1 unless it is with a more electronegative element. The oxidation state of Cr in K_(3)CrO_(8). Maths. To help remember the stability of higher oxidation states for transition metals it is important to know the trend: the stability of the higher oxidation states progressively increases down a group. What is the oxidation state of Cr in each compound? Oxygen is present with a preferred oxidation state of -2 each. The sum of the oxidation numbers of all atoms in an ion equals charge on the ion. Oxidation States Last updated; Save as PDF Page ID 31719; Contributors and Attributions; Skills to Develop. Transitions metals have more than one oxidation state. Cr, or Chromium, actually has two different oxidation numbers. Answer to 1) What is the oxidation state of the central metal in [Cr(acac)3]? Since … You must be signed in to discuss. How do you use a Lewis Structure to find the oxidation state of an element. Categories. NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. All oxidation reactions are accompanied by reduction reactions. Be the first to write the explanation for this question by commenting below. the oxidation state of cr in cro5 is 6, because it involves two peroxy bonds genaraly the valancy electron number(6) is an oxidation number. Books ; Test Prep; Summer Camps; Class; Earn Money; Log in ; Join for Free. What is the oxidation state of in each of the following ions? What is the name of K2Cr2O7 according to stock notation nomenclature method ? What is the oxidation number of Cr in Cr2O3? Relevance. I have this question using the oxidation rule i got +2, however how do i use it with Lewis structure. X =6. Half equations (Redox) AS Chemistry Redox question Reduction and oxidation half equations! Oxidation states of chromium. NCERT NCERT Exemplar NCERT Fingertips Errorless Vol-1 Errorless Vol-2. Since CN− ion is a strong field ligand, it causes the pairing of unpaired 3d electrons. a. CrO b. CrO3 c. Cr2O3. Answer / krishna prasad bobba. Answer Save. How many oxidation states does chromium have? The molecule has no net charge. chem 1411. what is the oxidation state of In in In2s3? Try… Oxidation States Help? Expert Answer 100% (2 ratings) Previous question Next question Get more help from Chegg. Chemistry. NCERT P Bahadur IIT-JEE Previous Year Narendra Awasthi MS Chauhan. Video from SHUBH RATHOD. Describe oxidation and reduction reactions in terms of oxidation state change. AIPMT 1995: The coordination number and oxidation state of Cr in K3 [Cr(C2 O4)3] are respectively (A) 3 and +3 (B) 3 and 0 (C) 6 and +3 (D) 4 and +2. HSO-3(not a negative sign) HSO-4(not a negative sign) Answer Save. The oxidation number of "Cr" is +3. Ni is in the +2 oxidation state i.e., in d 8 configuration. Favorite Answer. NCERT P Bahadur IIT-JEE Previous Year Narendra Awasthi MS Chauhan. If it has more electrons it tends to gain a few more electrons to complete its outer shell and acquires a negative oxidation state. You can view more similar questions or ask a new question. Biology. B. For example, in group 6, (chromium) Cr is most stable at a +3 oxidation state, meaning that you will not find many stable forms of Cr in the +4 and +5 oxidation states. in this case from Cl-to O2-, is an important factor for the edge energy difference between the CrCl 3 and Cr 2O 3 [4]. BobbJo Badges: 12. Chemistry. The sum of the oxidation numbers of each species is the charge on the ion.. We have the #Cr_2O_7^(2-)# anion, and, as is typical, each oxygen atom has a formal #-II# oxidation number: thus #7xx(-II)+2xxCr_"oxidation number"=-2#. O is alway -2. I have this same question on my mastering chemistry assignment due tomorrow. 1 decade ago. Every oxygen, .”O”, molecule has a -2 charge, or “oxidation state”. Which of the following is true about oxidation reactions? Question: What Is The Oxidation State Of Cr In Cr2O3. NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. Physics. But why is oxygen -2 in this case? Four oxygen atoms are to Cr through single bond. Cr2O3. Hence oxidation state of the oxygen atom, in this case, is -1. Class 12 Class 11 Class 10 Class 9 Class 8 … Oxidation State : The oxidation state, also called oxidation number, describes the degree of oxidation of an atom in a chemical compound. 1) what is the oxidation state of Cr in Cr(CO)6 in this case i only know that the oxidation state of oxygen is 2-. i need to know carbon's oxidation state to solve.. 2) oxidation state of Ag in AgNO3.. same problem as above 3)K2C2O4+KMnO4+H2SO4 >>>> K2SO4+MnSO4+H2O+CO2 from here, i know that carbon is oxidised C3+ >>> C4+ manganese is reduced +6. What is the oxidation state of Cr in Cr2O3. There are 4 CN− ions. Discussion. Contributors and Attributions . 3 Answers. asked Oct 23, 2018 in Redox reactions and electrochemistry by Sagarmatha (54.4k points) redox reaction; electrochemistry; jee; jee mains; 0 votes. QUESTION Use the Lewis structure of a thiosulfate . A. Oxidation reactions are the principal source of energy on Earth. So, Oxidation number of Cr in [Cr(NH3) ... 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## oxidation state of cr in cro6

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23 Jan 2013 Now let's look at an example using this formula to calculate interest. You invest $5,000 in a bank for 2 years at a 3% interest rate. What is the Simple interest calculation formula. The simple interest amount is equal to the principal amount times the annual interest rate divided by the number of periods Calculating simple interest or the amount of principal, the rate, or the time of a loan can seem confusing, but it's really not that hard. Here are examples of how to use the simple interest formula to find one value as long as you know the others. Substituting all of these values into the simple-interest formula, I get: 150 = (500)(r)(3) 150 = 1500r 150 / 1500 = r = 0.10. Of course, I need to remember to convert this decimal to a percentage. I was getting 10% interest. For example, assume you have a car loan of$20,000 with simple interest at 4%. The loan is repayable over a five-year period in equal installments. Your payment would work out to be $368.33 per month over 60 months. In other words, interest is earned on top of interest and thus “compounds”. The compound interest formula can be used to calculate the value of such an investment after a given amount of time, or to calculate things like the doubling time of an investment. We will see examples of this below. Simple interest is money you can earn by initially investing some money (the principal). A percentage (the interest) of the principal is added to the principal, making your initial investment grow! ## Simple interest refers to the amount of money that is paid for a specific amount the problem asks about the total amount -- that is, the principal plus the interest. How to calculate the Simple Interest Formula, how to solve interest problems Example: Sarah deposits$4,000 at a bank at an interest rate of 4.5% per year. A review of the simple interest formula and examples of how to use it in the interest earned, the total amount, and other values depending on the problem. His bank offers him an interest rate of 6 % 6\% 6%6, percent per annum. How much money should he deposit in the bank? Let's solve problems involving principal, rate of interest, simple interest, and total amount.

### 26 Jan 2019 problem solving. GMAT simple interest explained with examples. GMAT Interest Rate Problems – Part 1: Simple Interest. By Dominate the

A review of the simple interest formula and examples of how to use it in the interest earned, the total amount, and other values depending on the problem. His bank offers him an interest rate of 6 % 6\% 6%6, percent per annum. How much money should he deposit in the bank? Let's solve problems involving principal, rate of interest, simple interest, and total amount. teacher for more information. The interest (I) is the dollar amount earned or owed. The interest rate (R) is per year (T)  What is the amount he gets after 1 year, 2 years and 3 years? Solution: In every $100, Robert gets$ 8. (Since rate is 8% → 8 for every 100) Find the interest earned and the amount at the end of those $3 \text{years}$ ? example 2: You deposit $\$12000$into a bank account paying$1.5\%$simple 11 Nov 2008 Try using the above calculator to solve the example problems listed below. Example 1: You take out a loan of$10,000 that charges a annual rate


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# Tag Info

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For small equipment mounting, let me introduce two simple methods: 1) Rigid Rotation Method (Classic) 2) Cantilever Method (Conservative approximation) Assume cantilever action, for which the applied load rotates about the point "A", and the rotation is restricted by the two bolts farthest from the point "A". Conclusion: I recommend ...

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As mentioned in kamran's answer its very important that the bar is rigid. Otherwise the following analysis is not relevant. Although you don't seem to consider it this way, -if the beam is rigid enough-, then the fulcrum at the far left edge of the beam (not on B1, so the force there is not zero). The equation I derived is based on that assumption, which is ...

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Assuming the lever is rigid enough not to bend to simlify our estimate: The lever will rotate about B1 bolt because it can not penetrate the concrete block. So B2 to and B3 will share the 288N proprtionaly to their distance from B1. Lets call their reactio RB2 and RB3 respectively. $$R_{B3} =(1250-742=508)/(1250-912=338)*R_{B2}= 1.5 R_{B2}$$ By substituting ...

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# Tag Info

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Firstly, you must know that there are many models for inflation which give different results to your a) and b) questions, and we still don't know which is the right one. I'll try to answer regarding the most accepted and simple models. a) During the period of inflation the distance between two separated points in the Universe increased at least ...

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Good question. Not clear to me it reaches infinity, or high enough for a singularity. It seems that if one is talking about non accelerating, i.e. Inertial, observers in a de Sitter spacetime, it is a constant temperature determined by the cosmological constant. The cosmological constant is constant in de Sitter, and per current observations. Please note ...

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Then we have alternatives, such as the Big Freeze transitioning into the new Big Bang in the Conformal Cyclic Cosmology Anyway, the short answer is nobody knows. There is too little known about the nature of Dark Energy especially to be able to say what the universe will be like in the far future.

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Heat Death and the Big Freeze are the same thing.

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The most straightforward theories of inflation assume there exists some scalar field $\phi$ that permeates the Universe and drives inflation. Over time this scalar field changes, and the rate of change is given by $\dot{\phi}$. There is also some "potential energy" associated with the scalar field, which is given by some function $V(\phi)$. The specific ...

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In a static bounded universe, what happens to particles that hit the edge? I venture to say nobody knows for sure. But it's an interesting question, because we have no evidence whatsoever that our universe is infinite or some kind of hypersphere. See this answer to a related subject. The story goes that in the old days, people could not conceive of a ...

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When we talk about universe inflation we should have in mind that the four cosmic forces did not appear yet, because at inflation no elements were formed yet. scientists say that at 0.03 second the inflation reached four light-years, also said that if it had continued at the same rate, it would have been disappeared during the second have of the first ...

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Conformal space is nice because in it, photons have straight world-lines, so we can easily see what we must do to achieve causal contact between two points in the CMB, after the physical time $t_i=0$ of the initial singularity, but before the physical time $t_{\text{CMB}}$ of decoupling. Since we have $$d\tau=\frac{dt}{a(t)},$$ ...

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# ankursinha.in/blog

neuroscience/fedora/musings

Thu 02 February 2012

Posted by ankur in Tech (339 words, approximately a 2 minute read)

I'm the one of the moderators of askfedora. Of late, I've seen a lot of new questions come in. I've observed that most of these do not make correct use of the forum and therefore am outlining the good practices of using it here:

1. Search before you post: Search the interweb and then askfedora itself before posting your question.
2. If you have more than one question,post them separately: Do not write a post that says "many fedora questions" and ask them all together. You're to ask each question separately.
3. Frame a question: This is most baffling. It's called ASKfedora, yet most of the queries are not in interrogative form. And yes, questions end in question marks.
4. Ask good questions: Here's a great post that tells you how to ask good questions
5. Use the tools: When you want to post information along with your question, please use the tools, such as quoting, code snippet etc. If folks can't read your questions, they cannot answer them.
6. Use correct tags: Make use of existing tags instead of adding new ones. This increases the visibility of your question. Remember that tags are without spaces, so "fedora16" is not the same as "fedora 16". The latter actually breaks up into two separate tabs "fedora" and "16" which is illogical.
7. Go through the sticky posts_!


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Gọi ngay để nhận ưu đãi 0988.900.116
Hotline 0988.470.212

# inductor energy

When a electric current is flowing in an inductor, there is energy stored in the magnetic field.Considering a pure inductor L, the instantaneous power which must be supplied to initiate the current in the inductor is . They also showed that the magnitude of voltage is dependent on the rate of change of flux.

The basic construction of an inductor contains an insulated (enameled) wire wound. Brief about Building Automation Systems.

An inductor is a passive circuit element which stores energy in the form of the magnetic field. In case of not supported by an internal core, is called air-core inductor. Energy in an Inductor. The current increases losses by producing heat.

Where for high frequencies, the core is made of soft ferrite because of not producing high eddy current losses.

Inductors resist change in current and they are commonly used in electronic circuits avoiding current surges. Inductors are passive electrical components which store energy in the form of a magnetic field. An inductor is a passive circuit element which stores energy in the form of the magnetic field. Where stored energy can be find by integrating both sides up to charging time, $\int_{t_{0}}^{t}{p (t)dt}=L\int_{t_{0}}^{t}{i(t) \frac{di}{dt’}} dt’$, $W_{Stored}=L\int_{i(t_{0})}^{i(t)}{i ‘(t)} di^{‘}$, $W_{Stored}=L\frac{1}{2}\lbrack i(t)^{2}-i(t_{0})^{2}\rbrack$. The induced voltage causes current inside the core, which is called Eddy current. If the inductor is initially not charged or left unconnected for a long time, so the initial current  $i(t_{0})$ will be zero.

If there is only one path for current the combination of inductors will be called series combination, as shown in the diagram below. It also confines the magnetic flux near the winding and increases the flux linkage. It can be represented mathematically.

so the energy input to build to a final current i is given by the integral Inductance is the measure of the property of an inductor that it opposes the change in current flow. In another case, the winding is supported by an iron core, which is called an iron core inductor.

The Lenz Law state that the direction of the induced voltage is always such that it opposes its cause which produces it.

Energy stored in the inductor is the multiplication of current through inductor and voltage across it.

In 1800’s Oersted, showed that current carrying conductor produces a magnetic field around it. As the core inside the inductor is subjected to the magnetic flux of the inductor.

As discussed earlier that the magnetic flux is directly related to the current flowing through it. The winding may be supported by a core or not.
Energy. Michal Faraday and Joseph Henry discovered that a changing magnetic field can produce a voltage across its neighboring circuits. To counter the eddy current losses, the core is made of laminations, the iron thin plates insulated from each other. It stores electric energy in the form of the magnetic field during the charging phase and releases the same energy to the circuit in the decay phase.

Practically the inductor shows some resistance, which absorbs part of the apparent power and reduces the efficiency of the inductor.
The energy stored is represented by the graph below. He is now working as a professional engineer for an internationally recognized organization as well as he is pursuing his master degree. Now applying KCL to the above diagram will give us the following equation, We know that voltages are same in a parallel combination, so putting that into the above equation, $i_{T}=\frac{1}{L_{1}}\int{vdt}+\frac{1}{L_{2}}\int{vdt}+\ldots +\frac{1}{L_{n}}\int{vdt} i=\frac{1}{L}\int{vdt}$, $i_{T}=(\frac{1}{L_{1}}+\frac{1}{L_{2}}+\ldots +\frac{1}{L_{n}}) \int{vdt}$, So equivalent inductance in case of the parallel combination will be, $\frac{1}{L_{eq}}=\frac{1}{L_{1}}+\frac{1}{L_{2}}+\ldots +\frac{1}{L_{n}}$. So inductor opposes a change in current either positive change or negative change explained by the Lenz Law.

If multiple inductors are connected such that there are multiple paths for current to flow, such combination is called parallel combination. So, when the current increases the flux also increases, this change in flux causes a voltage to induce in such direction that opposes the existing current.

The current flowing through the inductor generates the magnetic field where the energy is actually stored.

Just like capacitors and resistors, the inductor is also available in the market from few micro-Henry to tens of Henry, which may be fixed or variable. His keen interests include Electronics, Electrical, Power Engineering. Introduction to Microcontroller and its Benefits, What is Smart Building? An inductor is not anticipated to dissipate energy, it only stores energy and then delivers it to the circuit when required. After few years, Ampere showed by taking some careful experiments that magnitude of magnetic flux is directly related to the amount of current flowing through it. This is “choking” and that’s why inductors are called Choke. What is inductor, inductance and How it store energy, current through inductor and voltage across it, 10 LED Lighting electronic project ideas for fun, Resistor Color Coding System Explained: Calculator and Example, Explain: What is Resistor and Different Types of Resistor, What are Geometric dimensioning and tolerancing (GD&T) and its purpose, The process of Printed Circuit Board PCB Assembly: Introduction to PCBA, What is Automation?


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# Harmonics below oscillation frequency in the brain

Apologies if this question is trivial, I'm a neuroscientist and would hugely appreciate your help to correctly interpret my analysis.

I am recording EEG signals and am manipulating the brain using light flickering at 10Hz. I then compute the PSD of this signal using FFT of 4s long epochs. I am interested in what the manipulation does to frequencies around 5Hz.

My problem is that my 10Hz stimulation creates an artifact at 10Hz (light induces current flow). I see this clearly in the PSD and I also see higher harmonics at 20 and 30 Hz. My question is whether I should expect this artifact to affect the 5 Hz frequency as well? I don't see a clear peak but I don't fully understand the theory so I'm not sure.

• What is the sampling rate of the captured data you are working with? And have you low pass filtered your data to ensure harmonica above Nyquist don’t fold into your band and appear as other harmonics which may include 5 Hz? Feb 11, 2020 at 12:22
• The signal is digitized at 50kHz, low-pass filtered at 128 Hz and stored at 256Hz. Before analysis, I'm applying a regression-based detrending procedure to remove DC drift (effectively removing 0 -0.5 Hz bands). Could you explain in more detail what you mean by harmonica folding in and how this can be avoided?
– MCK
Feb 11, 2020 at 12:43
• What do you mean by "stored at 256 Hz"? I may need to update my answer with more details if this means that you down-sample to that lower rate. And if you do such down-sampling can you provide the full details of your 128 Hz low pass filter and the implementation of that down-sampler in your question? Feb 11, 2020 at 13:12
• Sorry for being unclear. The low pass filter at 128Hz is a second order biquad filter, the software does not tell the user how the downsampling is achieved.
– MCK
Feb 11, 2020 at 13:28
• What filtering is done in the analog before the 50KHz sampling? Feb 11, 2020 at 13:36

You explain in the comments that the signal is sampled at 50 KHz, then low pass filtered at 128 Hz with a 2nd order biquad filter and then resampled and stored at 256 Hz.

If you weren't prudent with choice of sampling rate and low pass filtering, then the harmonics of 10 Hz could create a component at 5 Hz through aliasing. For example this would occur if you used a sampling rate of 25 Hz: the third harmonic at 30 Hz would fold to the 5 Hz frequency. I explain aliasing in more detail at this post here:

Where should I set my anti-aliasing filter corner frequency for this signal?

That said, there are two prominent mechanisms that could create folding images in vicinity of 5 Hz due to aliasing during this sampling process.

The first would be due to interference components above the Nyquist frequency of 25 KHz prior to the sampling of the analog signal - specifically any frequency components at $$N f_s \pm 5$$ Hz would fold to 5 Hz in your digital signal where $$N$$ is any integer and $$f_s$$ here is 50 KHz. (So 50 KHz +/- 5 Hz, 100 KHz +/-5 Hz are example locations where interference can alias in). To avoid this be sure to use a good anti-aliasing filter and pay particular attention to its rejection around $$N f_s \pm 5$$ Hz with a rejection band around those frequencies that are within the frequency range that you want to maintain to be free of interference for your measurement. Given this is many of orders larger than your 10 Hz interference, it is less likely this is an issue however should not be discounted depending on your measurement sensitivity desired.

The second mechanism, and likely dominant one in this case, is your resampling approach. Resampling to 256 Hz is a fractional rate conversion from 50 KHz and there could be many opportunities to create harmonic folding effects. The actual implementation needs to be reviewed in detail including significantly the performance of the resampling filter prior to decimation. Resampling to 256 Hz is identical to sampling at 256 Hz directly, so any frequency content above the Nyquist frequency of 128 Hz would fold into band. Specifically in this case the frequencies at $$N 256 \pm 5$$ Hz would land at 5 Hz in the digital domain.

Without seeing the actual implementation or ability to change it this would be difficult to predict although certainly can be measured. Induce a 10 Hz test signal with high harmonic content (impulses) and measure the results without your actual device you are measuring connected to assess your measurement noise floor in that condition.

Regardless of aliasing, to be sure that the interference itself does not contain 5 Hz components, I would also recommend analyzing a measurement without your true data source connected (such that you are only evaluating background noise). This would also give you a fidelity of your measurement system in showing you what sensitivity can be achieved, and the related confidence of your results.

• Thank you! It makes sense that fractional resampling is tricky. I think the software may actually sample at slightly different rates to avoid that. If it were not fractional resampling then this second mechanism would be less relevant?
– MCK
Feb 11, 2020 at 15:14
• No still relevant - realize that resampling and sampling are the same thing: when you sample you are simply resampling from an infinite sampling rate so the same aliasing explanation applies. Feb 11, 2020 at 15:44
• OK great that makes sense, and - to summarise for dummies like me - you are saying that this second mechanism may still be relevant even if a lowpass filter was applied because it is not trivial to design an adequate lowpass filter for anti-aliasing.
– MCK
Feb 11, 2020 at 18:57
• It could be trivial and you certainly have enough margin for adequate rejection- the question is was it designed properly with consideration to the 5Hz aliasing frequencies. Since you do not have access to the internal design you need to test as I described to be certain it is not vulnerable to that. Feb 11, 2020 at 19:02


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# Overview

The whole nGraph Compiler stack

The nGraph Compiler stack consists of bridges, core, and backends. We’ll examine each of these briefly to get started.

A framework bridge interfaces with the “frontend” Core API. A framework bridge is a component that sits between a framework like TensorFlow or MXNet, and the nGraph Core frontend API. A framework bridge does two things: first, it translates a framework’s operations into graphs in nGraph’s in-memory Intermediary Representation. Second, it executes the nGraph IR graphs via the backend execution interface.

The details of bridge implementation vary from framework to framework, but there are some common patterns: a fairly typical example for a graph-based framework is illustrated here, and consists of basically two phases: a clustering phase and a translation phase.

Translation flow to an nGraph function

The clustering phase operates on the original framework’s graph. During this stage, we look for maximal subgraphs containing nodes that can be translated to data flow functions in nGraph. The ability to capture subgraphs of the original graph means that we maintain interoperability with the native framework runtime. Any node that is not placed in a cluster can still by handled by the native framework. On the other hand, identifying maximal subgraphs means that we can avoid unnecessary handoffs between the native framework runtime and nGraph; minimizing this is good for performance.

In the second phase, called translation, we cut out each cluster subgraph, translate it into an nGraph Function, and replace the cluster subgraph with a stand-in node called an “encapsulation node” that holds a pointer to the nGraph Function. Later, at runtime, those functions will be invoked when the framework asks us to execute the encapsulation node.

It’s worth noting that backends have total freedom to rewrite the nGraph Functions: they can do it for the sake of structural or algorithmic optimization of the graph, for easy integration with kernel libraries, or for any or no reason at all.

## Namespaces in nGraph¶

What follows here is a table of all documented namespaces with brief descriptions:

Namespace

Description

Location in Repo

Docs

ngraph

The Intel nGraph C++ API

ngraph

Implicit namespace omitted from most API documentation

builder

Convenience functions that create additional graph nodes to implement commonly-used recipes; for example, auto-broadcast

builder

Coming Soon

descriptor

Descriptors are compile-time representations of objects that will appear at run-time

descriptor

Coming Soon

op

Ops used in graph construction

op

List of Core ops

runtime

The objects and methods used for executing the graph

runtime

Backend APIs


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# Tag Info

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\documentclass{article} \usepackage{amsmath} \begin{document} \begin{align*}\renewcommand\arraystretch{2.1} \frac{2}{3} &= 0.66666\,66667 ~(-) \\ \frac{2}{3 \cdot 3 \cdot 9} &= 0.02469\,13580 ~(+) \\ \frac{2}{3 \cdot 5 \cdot 9^2} &= 0.00164\,60905 ~(+) \\ \frac{2}{3 \cdot 7 \cdot 9^3} &= ...

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As possible starting point: \documentclass{article} \usepackage{array,siunitx} \usepackage{mathtools} \begin{document} \begin{center} \begin{tabular}{>{$}c<{$}} \begin{aligned} \frac{2}{3} & = \num{0,6666666667}\ (-) \\ \frac{2}{3\cdot 3\cdot 9} & = \num{0,0246913580}\ (+) \\ \frac{2}{3\cdot 3\cdot ...

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I would construct this in a regular array, provided that there is no requirement for the construction to break across the page boundary. Here is such an implementation: \documentclass{article} \usepackage{array} \begin{document} \renewcommand{\arraystretch}{2.1}% http://tex.stackexchange.com/q/31672/5764 \begin{array}{ >{\displaystyle}r @{} ... 0 aligned equations are typically centered by default, so without more information about your preamble it's hard to know for certain what the problem is. That said, it's most likely that the problem arises from passing fleqn as an option to the document class. Removing this makes display math (including align environments) centered instead of left-aligned. ... 1 Made to work in all math styles. \documentclass{article} \usepackage{amsmath,scalerel,stackengine} \def\xyz#1{\ThisStyle{% \setbox0=\hbox{\SavedStyle#1}% \stackon[1\LMpt]{\SavedStyle#1}{\rule{\wd0}{.5\LMpt}}% \rule[-1\LMpt]{.5\LMpt}{\dimexpr\ht0+2.5\LMpt}}} \begin{document} P_{30:\xyz{20}}\quad \xyz{1234}\quad \scriptscriptstyle\xyz{567890} ... 1 You can use actuarialangle, see the “Comprehensive list of symbols”, Table 251; for the package documentation, texdoc actuarialangle \documentclass{article} \usepackage{actuarialangle} \begin{document} \[ P_{30{:}\actuarialangle{20}} \end{document}

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The command \showthe works exactly like \the (see The \the command), with the difference that TeX stops compilation for showing the resulting token list, instead of delivering it to the input stream. So \showthe\abovedisplayskip would interrupt the TeX run (if found in an interactive session) printing > 12.0pt plus 3.0pt minus 9.0pt. l.10 ...

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To have the numbering on the right, you have to replace \documentclass[english]{amsart} by \documentclass[english,reqno]{amsart} (See Placing equation numbers on the right) Edit : and like @Mico said, juste comment your numberwithin commands

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Yes, of course. \documentclass{article} \begingroup \catcode+=\active \gdef+{\mathbin{\mathrm{blurb}}} \endgroup \AtBeginDocument{\mathcode+="8000 } \begin{document} $x+y$ \end{document} Of course you can think to better definition for the “math active” +. ;-)

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Nest aligned properly; I defined a compute environment for convenience, also removing as much useless space as possible. \documentclass{article} \usepackage{mathptmx} \usepackage{amsmath} \newenvironment{compute} {\left.\kern-\nulldelimiterspace\!\aligned} {\endaligned\right\rbrace} \begin{document} \begin{equation*} \begin{compute} \begin{compute} ...

8

The following solution uses nested array environments, with the contents of all columns right-aligned. The innermost array, which is terminated by the first large curly brace, features a single column of type r. The next array features two such columns; note that its lower-left cell is empty. The outermost array also has two columns of type r, and its ...

4

Here's a tcolorbox solution (instead of mdframed) The \tcboxmath command can be used inside of an equation or as a standalone box, the configuration of colors etc. is quite simple with colframe and colback etc, either using tcbset or locally in the optional argument. \documentclass{article} \usepackage{mathtools} \usepackage[most]{tcolorbox} ...

2

Something like this? If you don't like the vertical bar, consider replacing \mid with \setminus. \documentclass{article} \usepackage{amsmath} % for 'align*' environment \begin{document} \begin{align*} x+2 &= 2 \qquad \mid -2\\ x\phantom{{}+2} &= 0 \end{align*} \end{document} Addendum: If your systems of equations tend to be more involved ...

2

Something like this: It is easy, just replace shape circle with ellipse: \documentclass{book} \usepackage{amsmath,amsfonts,amssymb,amsthm} \usepackage{mathptmx} \usepackage{calc} \usepackage{tikz} \usetikzlibrary{tikzmark,calc,,arrows,shapes,decorations.pathreplacing} \tikzset{every picture/.style={remember picture}} ...

3

Your equation will become shorter if you will move summation limits below and on the top of it symbol by use \limits after symbol \sum. Based on MWE from @Bernard answer: \documentclass{ieeetran} \usepackage{mathtools} \usepackage{lipsum} \begin{document} \lipsum[2] \begin{align}%\MoveEqLeft[12] TPR ...

2

To left align a moderately long equation, you can use the align environment, and the \MovEqLeft[number of ems](default is 1em), or, for the equation to begin at the left margin, the flalign environment. Needless to load amsmath since mathtools does it: \documentclass{ieeetran} \usepackage{mathtools} \usepackage{lipsum} \begin{document} \lipsum[2] ...

0

If you will use equation with {split} inside, \noindent will not help you, because equation will appear a little higher then top of page. Add showframe option to geometry package and put this code in your document: \clearpage \noindent $\begin{split} \left(a^{B^{B^{B}}}\right) \\ bb \end{split}$ Here is a solution which helped me How to place amsmath ...

1

The \left. and \right\} instructions have to be placed outside the array environment. \documentclass{article} \begin{document} $\left. \begin{array}{r} \frac{1}{2}( \pi_b+ \pi_a) \pi_b\\[1ex] {}+\frac{1}{2} \pi_b(1-\pi_a-\pi_b) \end{array} \right\} =\pi_b/2$ \end{document}

2

With array you can obtain the following: \documentclass{article} \usepackage{amsmath} \begin{document} \arraycolsep=1pt \begin{array}{cccc} \mathbf{A} & \mathbf{x} & = & \mathbf{b} \\ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} & \begin{bmatrix} x_1 \\ x_2 ...

2

as i said in a comment, don't do it. if you are submitting this document to a well-edited math journal, it will be reversed. however, if you must, this should work: \documentclass{article} \usepackage{amsmath} \begin{document} some text \vfill So that abutting triangles do not double-paint pixels, a bias of 1 is subtracted from edges that are top'' or ...

3

A Tikz solution \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{positioning} \begin{document} \begin{tikzpicture} [%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% node distance =.7cm ]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \node (A) {$\textbf{X}_{\textbf{n}}$}; \node (B) ...

4

Here are ways to do it, with the cases environment. In the second way, you can have the 1 centred easily with \hfill (thanks to @Mico's suggestion): \documentclass{article} \usepackage{amsmath} \begin{document} \label{xput_vc} \text{Throughput}_{VC} = \begin{cases} 1 & \scriptstyle N_{VC} \leq ...

6

A demo of what can be done with pstricks and with tikz-cd: \documentclass{article} \usepackage{amsmath} \usepackage{pst-node} \usepackage{tikz-cd} \usepackage{auto-pst-pdf} \begin{document} % $\psset{arrows=->, arrowinset=0.15, linewidth=0.5pt, nodesep=3pt, labelsep=2pt, rowsep=0.6cm, colsep = 1.1cm, shortput=nab} \everypsbox{\scriptstyle} ... 5 Another option is tikz-cd: \documentclass{article} \usepackage{amsmath} \usepackage{tikz-cd} \begin{document} \[ \begin{tikzcd} X_n \arrow[r,"\text{a.s.}"] \arrow[d,shift right=.15em,equal,"D"] & X \arrow[d,equal,"D"]\\ Y_n & Y \end{tikzcd}$ \end{document} The argument to \arrow (in brackets) has the direction, possible options like ...

1

This is only a sketch. Some tuning probably needed. (I am assuming, that it is about probability, not commutative diagrams). \documentclass{article} \usepackage{graphicx} \begin{document} \def\eqD{\stackrel{D}{=}} \def\roteqD{\rotatebox{-90}{$\eqD$}} $\begin{array}{c@{}c@{}c} X_n&\;\stackrel{\mathrm{a.s.}}{=}\;&X\\[-7pt] \roteqD ... 7 This is a commutative diagram, for which the Xy package is particularly useful. \documentclass{article} \usepackage{amsmath} \usepackage[all]{xy} \begin{document} \[ \xymatrix{ X_n\ar[r]^{\text{a.s.}} \ar@{=}[d]^D & X\ar@{=}[d]^D\\ Y_n & Y }$ \end{document} The package is really versatile for drawing ...

2

You can do it like this \text{\footnotesize $N_{VC} \leq N_{slave}$}

2

Here is a much simpler code with alignedat and cases environments. Use the bm package to have bold Greek letters.You don't need all those pairs of { }. Also note font commands such as \rm,\bf`, &c. shouldn't be used as they're plain TeX commands, and they don't work well w.r.t. New Font Selection Scheme (N.F.S.S.) which has been new for more than 20 ...

2

This answer is similar to the one by Enrico, but is using aligned instead of alignedat. \documentclass{article} \usepackage{mathtools} \begin{document} \begin{aligned} & \textrm{Dot product:} && \sigma_{ij} n_j \Leftrightarrow \mathbf{\sigma n} \\ & \textrm{Double dot product:} && \begin{cases} ...

5

MathType is not really doing a good work: it's much simpler to type in LaTeX directly. \documentclass{article} \usepackage{amsmath} \begin{document} \begin{alignedat}{2} &\text{Dot product:} &\quad& \sigma_{ij}{n_j} = \sigma\mathbf{n} \\ &\text{Double dot product:} &\quad& \begin{cases} ...

4

You should do two things: use \documentclass[fleqn]{article} (or whichever document class you're using), but also use gather instead of align. align and its cousins are for vertically aligning text around some specified character, usually =, but gather is for a general list of equations. (You can also use gather* to get an un-numbered version.) ...

3

Your coding style is, shall we way, not exactly tidy. Here's a minimalist fix-up that should get you going, at least for a little while. Good news: after the code clean-up, the equation numbers show up on the right-hand edge. \documentclass{article} \usepackage[utf8]{inputenc} \usepackage{natbib} \usepackage[section]{placeins} \usepackage{graphicx} ...

2

If you want to make your MWE work, then add \usepackage{graphicx} to the preamble. Otherwise \resizebox will not be defined. I suggest using multline though. I also adjusted the typography of the vectors, \vec{r_{ij}} → \vec{r}_{ij}. \documentclass[aps,pre,reprint]{revtex4-1} \usepackage{amsmath} \usepackage{lipsum} \begin{document} \lipsum[1] ...

3

Just remove the external pairs of braces. What are they here for? If you really need them, make several groups. Reduce the distance form the frame to the ledt margin of the text inside, and align the enumeration on the left margin with the enumitem package. Finally, if you reduce the font size, do it outside math mode: ...

1

This can be quite easily done using the tried and tested tikzmark idea, demonstrated throughout this site, for example How to draw arrows between circled coefficients of polynomials using tikz and How to draw arrows between parts of an equation to show the Math Distributive Property (Multiplication)? Here's the output: and the code: % arara: pdflatex % ...

7

If a binary relation such as = is used at the end or start of math list it does not get the additional space at the end of the list. align is designed to be used with &= and adds a {} at the start of the second column, so the effective list is ${}=def$ and you get space either side of the =. There are some rare occasions where it is convenient to move ...

2

Version 1: I don't know what precisely you want to change, but maybe the following can be a starting point for further improvements: \begin{equation*} \tikz[baseline]{\node(d13) {$\hat{\underline{\theta}}_{k+1}$}} = \tikz[baseline]{\node(d14){$\hat{\underline{\theta}}_{k}$}} + ...

5

It's easy to do with pstricks: just use the commands \circlenode and join them by an arc node connection, inside the psmatrix environment: \documentclass[x11names]{article} \usepackage[utf8]{inputenc} \usepackage{pst-node} \usepackage{auto-pst-pdf} \begin{document} \[ \begin{psmatrix} \psset{linewidth=0.5pt, linecolor=Coral1,arrows=<->, ...

1

Since Micro replaced his \phantom (more or less) based answer, here is my minipage solution: \documentclass{article} \usepackage{amsmath} \begin{document} \noindent\fbox{\parbox{\textwidth}{\refstepcounter{equation}% \begin{minipage}[t]{0.33\textwidth} \begin{align*} a_1 &= \frac{x}{h},\\ a_2 &= \frac{h \rho v}{\mu},\\ a_3 &= \frac{k}{2}, ...

4

Try: \documentclass{article} \usepackage{amsmath} \begin{document} \begin{aligned} a_1 & = \frac{x}{h}, &\qquad b_1 & = \frac{h}{h}, &\qquad c_1 & = \frac{\mu t}{\rho h}, \\ % a_2 & = \frac{h \rho v}{\mu}, &\qquad b_2 & = \frac{h_0^2 \rho p}{\mu^2} & c_2 & = \frac{h \rho ...

2

Here's a solution that uses an array environment to align the rows and columns. \documentclass{article} \usepackage{array} % for "\newcolumntype" and "\extrarowheight" macros \newcolumntype{L}{>{\displaystyle}l} \begin{document} \setlength{\extrarowheight}{2.5ex} % extra vertical space between rows \begin{array}{ r @{{}={}} L @{\qquad} ...

1

A single . can be used as the partner of a bracket, e.g.: \left| \frac{b}{c} \right. \left. \frac{b}{c} \right| There is also \middle in order to scale something in the middle of an expression, e.g.: \left. a * b = c \middle| :b \right. However, in your case, probably you should use some way to align your computation across several lines. This can be ...

1

Since the OP does not provide an MWE, it took a little bit to determine that the problem arises when trying to use the OP's syntax with the amsmath package loaded. The amsmath package redefines \dot, presumably for the best, but in this case not so much. If one needs this syntax (or an alternate-equivalent stackengine syntax) to co-exist with amsmath, then ...

2

I'm not sure this is the best way to present the data. However, a dirty trick seems to give what you're looking for. \documentclass{article} \usepackage{amsmath} \newcommand{\rb}[1]{\raisebox{3.5ex}{$\displaystyle#1$}} \begin{document} \begin{align} & \rb{\text{Replenishment Indicator:}} & \rb{O_t} & \rb{{}=\{1,\text{if }t=\sum_{\kappa=1}^k ...

2

align can be found in the amsmath guide. Note that the ampersands define what to align. In this situation, put & before the "=" of each equation. \begin{align} m_{1,1}^i &= \sum (x - \bar{x}^i) (y - \bar{y}^i)\text{,} \\ m_{2,0}^i &= \sum (x - \bar{x}^i)^2 \text{ and} \\ m_{0,2}^i &= \sum (y - \bar{y}^i)^2 \text{ } \forall \text{ } ...

5

The gathered environment, used within math mode, is what you're looking for. \documentclass[12pt]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage[utf8]{inputenc} \begin{document} $$\begin{gathered} 3(a-x) = 3.5x + a - 1 \\ 3a - 3x = 3.5x + a - 1 \\ a = \frac{13}{4}x - \frac{1}{2} \end{gathered}$$ \end{document} ...

1

If you want to write text in math mode use \text{æ,ø,å}. See also Umlauts in math mode and utf-8 characters in latex math mode

2

Your input has several errors: the environment is called multline $$and$$ should not appear inside it several braces are not matched up correctly Here is a possibly correct version \documentclass{article} \usepackage{amsmath} \begin{document} \begin{multline} ...

2

Some comments: In the input, I would not split parts of one and the same cell across different matrix environments, as doing so messes up the spacing around plus and minus symbols. For the line break in the third equation, do consider inserting the line break earlier, i.e., right after the 2x2 (inverted) matrix. This point is also made in egreg's answer. ...

4

I would not split the wide matrix across two lines, but break at the multiplication, which is a more natural break point. You can use the same idea for the different break style. \documentclass[11pt]{book} \usepackage[top=3cm,bottom=3cm,left=3.2cm,right=3.2cm,headsep=10pt,a4paper]{geometry} \usepackage{amsmath,amsthm} \begin{document} ...

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# How to add a picture in TeX (a jpeg)? [duplicate]

I saved the jpeg file in the same place where all my TeX files are placed at; which commands should I be using to make the pic appear?

I am using Bakoma TeX 11.85, but I believe it's the same in every software of TeX, but I may be wrong.

You may use the package `graphicx`: https://ctan.org/pkg/graphicx.
• `\includegraphics[scale=0.5]{IMAGE.jpg}` it halves the size of the image. You can choose any percentage, even greater than 1.
• `\includegraphics[width=0.5\textwidth]{IMAGE.jpg}` it rescales the image by setting its width as the 50% of the text width and automatically adjust the height to maintain the proportion. You can choose any percentage: for example, `\includegraphics[width=0.1\textwidth]{IMAGE.jpg}` set the width as 1/10 of the textwidth.
• `\includegraphics[height=0.5\textheight]{IMAGE.jpg}` it rescales the image by setting its height as the 50% of the text width and automatically adjust the width to maintain the proportion. You can choose any percentage, also greater than 1.
• You can use some options for the `\includegraphics` command, such as `\includegraphics[scale=0.5]{(b).jpg}`, which halves the image; you can choose the percentage of the scaling as you desire. Nonetheless, the option that I prefer is to use `\includegraphics[width=0.5\textwidth]{(b).jpg}` which rescales the image's dimension as a percentage of the width of the text (in this case, it sets as 50% of the text width). You may play a little bit with this commands and see what better suits you. – Eddymage Apr 24 at 19:43


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# $f=\infty$ on a set of measure 0, then $\int_E f = 0$

Let $E$ be a set of measure zero and define $f = \infty$ on $E$. Show that $\int_E f = 0$.

This is out of Royden 4E, p 84.

I know how to prove this if $f=0$ on $E$. But I'm curious, as stated, won't this result in a situation in which $\infty \cdot 0$.

Royden's definition (at least in the Second Edition - I doubt that it has changed) of $\int_E f$, where $f$ is a nonnegative measurable function on measurable set $E$, is $$\int_E f = \sup_{h \le f} \int_E h$$ where $h$ is a bounded measurable function such that $m\{x: h(x) \ne 0\}$ is finite. So, if $h$ is such a function and $\mu(E) = 0$, what is $\int_E h$?

• Are you using the idea of finite support? – emka Nov 11 '12 at 10:47
• Your functions are defined on $E$. So $\{x: h(x) \ne 0\} \subseteq E$, which has measure $0$. – Robert Israel Nov 11 '12 at 20:08
• Should it be $\sup\limits_{h \leq f}\int_{E_0} h$ where $E_0=\{x: h(x) \neq 0\}$? – emka Nov 11 '12 at 22:27

The Lebesgue integral doesn't "see" sets of measure zero. That is, if $X$ is any set and $A$ a set of measure zero, then $$\int_X f = \int_{X \setminus A} f.$$ For your question, we have $$\int_E f = \int_{E \setminus E} f = \int_\emptyset f = 0.$$

The convention in measure theory is that $\infty \cdot0=0$. That convention is taken specifically so that the $\int_E f d\mu = 0$ whenever $\mu(E)=0$ regardless of $f$.

• Does this mean the proof is the same? – emka Nov 11 '12 at 7:09
• @emka Yes, it's the same as if $f$ were bounded – Logan Stokols Nov 11 '12 at 21:33

Another way would be to define $f_n=n$ on E and then apply the monotone convergence theorem. So $\int_{E}f=\lim_{n}\int_{E}f_n=\lim_{n}\int_{E}n=0$ , all the limits are as n goes to infinity.

With some simple theorems about integration over subsets we have that

$\qquad\displaystyle\int_E f \,d\mu = \int_X \chi_E\cdot f\,d\mu = \int_X 0 \,d\mu = 0 \cdot \mu(X) = 0$,

where we've used that $f \cdot \chi_E = 0$ a.e. and that the integral of two a.e. equal functions agree.

Of course this all depends on the order that one develops measure theory in and thus what definitions one is working with. One can also define integrals over subsets by restriction of measure in a suitable way and then prove that

$\qquad\displaystyle\int_E f\,d\mu = \int_X \chi_E\cdot f\,d\mu$.


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# What is the steric number of NH_3?

Jul 28, 2016

$4$

#### Explanation:

For a given molecule, the steric number is calculated by counting how many regions of electron density surround the central atom.

A region of electron density can be

• a single, double, or triple bond -- all three count as one region of electron density
• a lone pair of electrons

In order to calculate the steric number of nitrogen, which is the central atom in ammonia, ${\text{NH}}_{3}$, you must first draw the molecule's Lewis structure.

The molecule will have a total of $8$ valence electrons, $5$ coming from nitrogen and $1$ coming from each of the three hydrogen atoms.

The nitrogen atom will form single bonds with the three hydrogen atoms, with each of these bonds accounting for $2$ valence electrons. The remaining $2$ valence electrons will be placed on the nitrogen atom as a lone pair of electrons.

The Lewis structure for ammonia will thus looks like this

Notice that the central atom is surrounded by a total of $4$ regions of electron density

• three single bonds
• one lone pair of electrons

You can thus say that the nitrogen atom has a steric number equal to $4$. The steric number tells you the hybridization of the central atom.

In this case, a steric number equal to $4$ implies that the nitrogen atom is ${\text{sp}}^{3}$ hybridized.


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# Data Visuals

## dc.js - Oil and Ruble dashboard

This is a simple dashboard that shows the futures price of oil as well as the price of a dollar in rubles. The reason I included the ruble is to show visually how it is dependent on the price of oil. There are other factors that affect the USD/RUB exchange rate like economic sanctions, political circumstaces, supply and demand, etc. However, by far the oil and gas industry has nearly direct influence on the ruble economy.

I used the javascript charting library dc.js for data visualization, crossfilter for data manipulation, and quandl.com for collecting data. Following resources were helpful for this exercise:

The data is updated as soon as the Quandl database is updated, usually on a daily basis.

TODO:

- Update Exchange rate api request.

Daily Futures Price of Brent and WTI grade oil

Source: Quandl: Brent , WTI

Daily Price of a Dollar in

Source: Quandl: USD/RUB

Composite Chart

selected out of records
Date WTI Price, $WTI Price Change, % Brent Price,$ Brent Price Change, % RUB per $RUB per$ Change, %