History and Applications of Matrices

History and Applications of Matrices Matrices find many applications at current time and very useful to us. Physics makes use of matrices in various domains, for example in geometrical optics and matrix mechanics; the latter led to studying in more detail matrices with an infinite number of rows and columns. Graph theory uses matrices to keep track of distances between pairs of vertices in a graph. Computer graphics uses matrices to project 3-dimensional space onto a 2-dimensional screen. Example of application A message is converted into numeric form according to some scheme. The easiest scheme is to let space=0, A=1, B=2, , Y=25, and Z=26. For example, the message Red Rum would become 18, 5, 4, 0, 18, 21, 13. This data was placed into matrix form. The size of the matrix depends on the size of the encryption key. Lets say that our encryption matrix (encoding matrix) is a 22 matrix. Since I have seven pieces of data, I would place that into a 42 matrix and fill the last spot with a space to make the matrix complete. Lets call the original, unencrypted data matrix A. There is an invertible matrix which is called the encryption matrix or the encoding matrix. Well call it matrix B. Since this matrix needs to be invertible, it must be square. This could really be anything, its up to the person encrypting the matrix. Ill use this matrix. The unencrypted data is then multiplied by our encoding matrix. The result of this multiplication is the matrix containing the encrypted data. Well call it matrix X. The message that you would pass on to the other person is the the stream of numbers 67, -21, 16, -8, 51, 27, 52, -26. Decryption Process Place the encrypted stream of numbers that represents an encrypted message into a matrix. Multiply by the decoding matrix. The decoding matrix is the inverse of the encoding matrix. Convert the matrix into a stream of numbers. Conver the numbers into the text of the original message. DETERMINANTS The determinant of a matrix A is denoted det(A), or without parentheses: det A. An alternative notation, used for compactness, especially in the case where the matrix entries are written out in full, is to denote the determinant of a matrix by surrounding the matrix entries by vertical bars instead of the usual brackets or parentheses. For a fixed nonnegative integer n, there is a unique determinant function for the nÃÆ'-n matrices over any commutative ring R. In particular, this unique function exists when R is the field of real or complex numbers. For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix Example. Evaluate Let us transform this matrix into a triangular one through elementary operations. We will keep the first row and add to the second one the first multiplied by . We get Using the Property 2, we get Therefore, we have which one may check easily. EIGEN VALUES AND EIGEN VECTORS In mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. The prefix eigen- is adopted from the German word eigen for innate, idiosyncratic, own. Linear algebra studies linear transformations, which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They are computed by a method described below, give important information about the matrix, and can be used in matrix factorization. They have applications in areas of applied mathematics as diverse as economics and quantum mechanics. In general, a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged (or possibly reversing it). These vectors are the eigenvectors of the matrix. A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with that eigenvector. An eigenspace is the set of all eigenvectors that have the same eigenvalue, together with the zero vector. These concepts are formally defined in the language of matrices and linear transformations. Formally, if A is a linear transformation, a non-null vector x is an eigenvector of A if there is a scalar ÃŽÂ » such that The scalar ÃŽÂ » is said to be an eigenvalue of A corresponding to the eigenvector x. Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wide-ranging application. For example, this problem is crucial in solving systems of differential equations, analyzing population growth models, and calculating powers of matrices (in order to define the exponential matrix). Other areas such as physics, sociology, biology, economics and statistics have focused considerable attention on eigenvalues and eigenvectors-their applications and their computations. Before we give the formal definition, let us introduce these concepts on an example. Example. Consider the matrix Consider the three column matrices We have In other words, we have Next consider the matrix P for which the columns are C1, C2, and C3, i.e., We have det(P) = 84. So this matrix is invertible. Easy calculations give Next we evaluate the matrix P-1AP. We leave the details to the reader to check that we have In other words, we have Using the matrix multiplication, we obtain which implies that A is similar to a diagonal matrix. In particular, we have for . Note that it is almost impossible to find A75 directly from the original form of A. This example is so rich of conclusions that many questions impose themselves in a natural way. For example, given a square matrix A, how do we find column matrices which have similar behaviors as the above ones? In other words, how do we find these column matrices which will help find the invertible matrix P such that P-1AP is a diagonal matrix? From now on, we will call column matrices vectors. So the above column matrices C1, C2, and C3 are now vectors. We have the following definition. Definition. Let A be a square matrix. A non-zero vector C is called an eigenvector of A if and only if there exists a number (real or complex) such that If such a number exists, it is called an eigenvalue of A. The vector C is called eigenvector associated to the eigenvalue . Remark. The eigenvector C must be non-zero since we have for any number . Example. Consider the matrix We have seen that where So C1 is an eigenvector of A associated to the eigenvalue 0. C2 is an eigenvector of A associated to the eigenvalue -4 while C3 is an eigenvector of A associated to the eigenvalue 3. It may be interesting to know whether we found all the eigenvalues of A in the above example. In the next page, we will discuss this question as well as how to find the eigenvalues of a square matrix. PROOFS OF PROPERTIES OF EIGEN VALUES::: PROPERTY 1 {Inverse of a matrix A exists if and only if zero is not an eigenvalue of A} Suppose A is a square matrix. Then A is singular if and only if ÃŽÂ »=0 is an eigenvalue of A. Proof We have the following equivalences: A is singular à ¢Ã¢â‚¬ ¡Ã¢â‚¬ there exists xà ¢Ã¢â‚¬ °Ã‚  0, Ax=0 à ¢Ã¢â‚¬ ¡Ã¢â‚¬ there exists xà ¢Ã¢â‚¬ °Ã‚  0, Ax=0x à ¢Ã¢â‚¬ ¡Ã¢â‚¬ ÃƒÅ½Ã‚ »=0 is an eigenvalue of A Since SINGULAR matrix A has eigenvalue and the inverse of a singular matrix does not exist this implies that for a matrix to be invertible its eigenvalues must be non-zero. PROPERTY-2 Eigenvalues of a matrix are real or complex conjugates in pairs Suppose A is a square matrix with real entries and x is an eigenvector of A for the eigenvalue ÃŽÂ ». Then x is an eigenvector of A for the eigenvalue ÃŽÂ ». à ¢- ¡ Proof Ax =Ax =Ax =ÃŽÂ »x =ÃŽÂ »x A has real entries x eigenvector of A Suppose A is an mÃÆ'-n matrix and B is an nÃÆ'-p matrix. Then AB=AB. à ¢- ¡ Proof To obtain this matrix equality, we will work entry-by-entry. For 1à ¢Ã¢â‚¬ °Ã‚ ¤ià ¢Ã¢â‚¬ °Ã‚ ¤m, 1à ¢Ã¢â‚¬ °Ã‚ ¤jà ¢Ã¢â‚¬ °Ã‚ ¤p, ABij =ABij =à ¢Ã‹â€ Ã¢â‚¬Ëœnk=1AikBkj =à ¢Ã‹â€ Ã¢â‚¬Ëœnk=1AikBkj =à ¢Ã‹â€ Ã¢â‚¬Ëœnk=1AikBkj =à ¢Ã‹â€ Ã¢â‚¬Ëœnk=1AikBkj =ABij APPLICATION OF EIGEN VALUES IN FACIAL RECOGNITION How does it work? The task of facial recogniton is discriminating input signals (image data) into several classes (persons). The input signals are highly noisy (e.g. the noise is caused by differing lighting conditions, pose etc.), yet the input images are not completely random and in spite of their differences there are patterns which occur in any input signal. Such patterns, which can be observed in all signals could be in the domain of facial recognition the presence of some objects (eyes, nose, mouth) in any face as well as relative distances between these objects. These characteristic features are called eigenfaces in the facial recognition domain (or principal components generally). They can be extracted out of original image data by means of a mathematical tool called Principal Component Analysis (PCA). By means of PCA one can transform each original image of the training set into a corresponding eigenface. An important feature of PCA is that one can reconstruct reconstruct any original image from the training set by combining the eigenfaces. Remember that eigenfaces are nothing less than characteristic features of the faces. Therefore one could say that the original face image can be reconstructed from eigenfaces if one adds up all the eigenfaces (features) in the right proportion. Each eigenface represents only certain features of the face, which may or may not be present in the original image. If the feature is present in the original image to a higher degree, the share of the corresponding eigenface in the sum of the eigenfaces should be greater. If, contrary, the particular feature is not (or almost not) present in the original image, then the corresponding eigenface should contribute a smaller (or not at all) part to the sum of eigenfaces. So, in order to reconstruct the origi nal image from the eigenfaces, one has to build a kind of weighted sum of all eigenfaces. That is, the reconstructed original image is equal to a sum of all eigenfaces, with each eigenface having a certain weight. This weight specifies, to what degree the specific feature (eigenface) is present in the original image. If one uses all the eigenfaces extracted from original images, one can reconstruct the original images from the eigenfaces exactly. But one can also use only a part of the eigenfaces. Then the reconstructed image is an approximation of the original image. However, one can ensure that losses due to omitting some of the eigenfaces can be minimized. This happens by choosing only the most important features (eigenfaces). Omission of eigenfaces is necessary due to scarcity of computational resources. How does this relate to facial recognition? The clue is that it is possible not only to extract the face from eigenfaces given a set of weights, but also to go the opposite way. This opposite way would be to extract the weights from eigenfaces and the face to be recognized. These weights tell nothing less, as the amount by which the face in question differs from typical faces represented by the eigenfaces. Therefore, using this weights one can determine two important things: Determine, if the image in question is a face at all. In the case the weights of the image differ too much from the weights of face images (i.e. images, from which we know for sure that they are faces), the image probably is not a face. Similar faces (images) possess similar features (eigenfaces) to similar degrees (weights). If one extracts weights from all the images available, the images could be grouped to clusters. That is, all images having similar weights are likely to be similar faces.

Cisco Systems: New Millenium †New acquisition strategy Essay

The case deals with the acquisition policy implemented by Cisco, by giving some real samples. The most interesting point concerns the way Cisco acquired companies during 90s with 4 main goals: a shared vision, shareholders’ satisfaction, motivating value added for employees, shareholders, customers and partners and a perfect â€Å"chemistry† (P.9). Contrary to the global trend of big companies’ acquisition, Cisco was involved in smaller companies, based on selection criteria (presented below) which fit the company’s needs and strategy: * Small companies: the advantage is that those companies are start-ups coming with new disruptive Ideas highly complementary with Cisco’s businesses. * Limited number of employees (but engineers as the large part of them): Cisco gives a huge importance to skilled people, because as they say, those people are the real asset which permits to increase future market shares, So people retention becomes one of acquisitions’ main goals (success driven by a very low turnover compared with the industry). * Very entrepreneurial systems focused on fast growing: the need to grow by forming part of Cisco is one of the motivating arguments to have win-win negotiations. On the other hand, this kind of companies is not risk adverse and promises future benefits to their owners. * Geographical proximity: for large acquisitions, it permits to shorten lost time and get directly to the real acquisition’s implementation in order to share the distribution channels (cost reduction) and obtain the new product earlier. * Comparable cultures and visions: since acquisition leads to the integration of the acquired company to the group, it is necessary to have the same vision of the future strategy and cultures which can be merged to obtain a coherent unit. Last, the company’s strength resides in its ability to decentralize decision making by creating a new business unit at each acquisition, because the acquired company is considered as a master in its domain. This system permits to make acquired company feel independent in the way it is managed (less change management to be processed). Cisco represents a benchmark as acquirer for the entrepreneurial firms which aim to be acquired because it proposes several advantages in terms of independency, people treatment, product valorization and so on, which form part of the most important criteria for acquisition. Moreover, in the last decade, it increased its attractiveness based on the personalized treatment provided to the new joiners and the restructuration by type of technology which makes economies of scale and allows people to share their knowledge with their previous â€Å"internal competitors† and increase their skills. So, in terms of win-win relationship, Cisco acquisition model can be a basis of comparison for any potential acquired company.

The Twilight Saga 2: New Moon Chapter 17 VISITOR

UNNATURALLY STILL AND WHITE, WITH LARGE BLACK EYES intent on my face, my visitor waited perfectly motionless in the center of the halt, beautiful beyond imagining. My knees trembled for a second, and I nearly fell. Then I hurled myself at her. â€Å"Alice, oh, Alice!† I cried, as I slammed into her. I'd forgotten how hard she was; it was like running headlong into a wall of cement. â€Å"Bella?† There was a strange mingling of relief and confusion in her voice. I locked my arms around her, gasping to inhale as much of the scent of her skin as possible. It wasn't like anything elsenot floral or spice, citrus or musk. No perfume in the world could compare. My memory hadn't done it justice. I didn't notice when the gasping turned into something elseI only realized I was sobbing when Alice dragged me to the living room couch and pulled me into her lap. It was like curling up into a cool stone, but a stone that was contoured comfortingly to the shape of my body. She rubbed my back in a gentle rhythm, waiting for me to get control of myself. â€Å"I'm sorry,† I blubbered. â€Å"I'm just so happy to see you!† â€Å"It's okay, Bella. Everything's okay.† â€Å"Yes,† I bawled. And, for once, it seemed that way. Alice sighed. â€Å"I'd forgotten how exuberant you are,† she said, and her tone was disapproving. I looked up at her through my streaming eyes. Alice's neck was tight, straining away from me, her lips pressed together firmly. Her eyes were black as pitch. â€Å"Oh,† I puffed, as I realized the problem. She was thirsty. And I smelled appetizing. It had been a while since I'd had to think about that kind of thing. â€Å"Sorry.† â€Å"It's my own fault. It's been too long since I hunted. I shouldn't let myself get so thirsty. But I was in a hurry today.† The look she directed at me then was a glare. â€Å"Speaking of which, would you like to explain to me how you're alive?† That brought me up short and stopped the sobs. I realized what must have happened immediately, and why Alice was here. I swallowed loudly. â€Å"You saw me fall.† â€Å"No,† she disagreed, her eyes narrowing. â€Å"I saw you jump.† I pursed my lips as I tried to think of an explanation that wouldn't sound nuts. Alice shook her head. â€Å"I told him this would happen, but he didn't believe me. ‘Bella promised,'† her voice imitated his so perfectly that I iroze in shock while the pain ripped through my torso. â€Å"‘Don't be looking for her future, either,'† she continued to quote him. ‘†We've done enough damage.† â€Å"But just because I'm not looking, doesn't mean I don't see† she went on. â€Å"I wasn't keeping tabs on you, I swear, Bella. It's just that I'm alreacy attuned to you when I saw you jumping, I didn't think, I just got on a plane. I knew I would be too late, but I couldn't do nothing. And then I get here, thinking maybe I could help Charlie somehow, and you drive up.† She shook her head, this time in confusion. Her voice was strained. â€Å"I saw you go into the water and I waited and waited for you to come up, but you didn't. What happened? And how could you do that to Charlie? Did you stop to think what this would do to him? And my brother? Do you have any idea what Edward† I cut her off then, as soon as she said his name. I'd let her go on, even after I realized the misunderstanding she was under, just to hear the perfect bell tone of her voice. But it was time to interrupt. â€Å"Alice, I wasn't committing suicide.† She eyed me dubiously. â€Å"Are you saying you didn't jump off a cliff?† â€Å"No, but† I grimaced. â€Å"It was for recreational purposes only.† Her expression hardened. â€Å"I'd seen some of Jacob's friends cliff diving,† I insisted. â€Å"It looked like fun, and I was bored† She waited. â€Å"I didn't think about how the storm would affect the currents. Actually, I didn't think about the water much at all.† Alice didn't buy it. I could see that she still thought I had been trying to kill myself. I decided to redirect. â€Å"So if you saw me go in, why didn't you see Jacob?† She cocked her head to the side, distracted. I continued. â€Å"It's true that I probably would have drowned if Jacob hadn't jumped in after me. Well, okay, there's no probably about it. But he did, and he pulled me out, and I guess he towed me back to shore, though I was kind of out for that part. It couldn't have been more than a minute that I was under before he grabbed me. How come you didn't see that?† She frowned in perplexity. â€Å"Someone pulled you out?† â€Å"Yes. Jacob saved me.† I watched curiously as an enigmatic range of emotions flitted across her face. Something was bothering herher imperfect vision? But I wasn't sure. Then she deliberately leaned in and sniffed my shoulder. I froze. â€Å"Don't be ridiculous,† she muttered, sniffing at me some more. â€Å"What are you doing?† She ignored my question. â€Å"Who was with you out there just now? It sounded like you were arguing.† â€Å"Jacob Black. He's sort of my best friend, I guess. At least, he was† I thought of Jacob's angry, betrayed face, and wondered what he was to me now. Alice nodded, seeming preoccupied. â€Å"What?† â€Å"I don't know,† she said. â€Å"I'm not sure what it means.† â€Å"Well, I'm not dead, at least.† She rolled her eyes. â€Å"He was a fool to think you could survive alone. I've never seen anyone so prone to life-threatening idiocy.† â€Å"I survived,† I pointed out. She was thinking of something else. â€Å"So, if the currents were too much for you, how did this Jacob manage?† â€Å"Jacob is strong.† She heard the reluctance in my voice, and her eyebrows rose. I gnawed on my lip for a second. Was this a secret, or not? And if it was, then who was my greatest allegiance to? Jacob, or Alice? It was too hard to keep secrets, I decided. Jacob knew everything, why not Alice, too? â€Å"See, well, he's sort of a werewolf,† I admitted in a rush. â€Å"The Quileutes turn into wolves when there are vampires around. They know Carlisle from a long time ago. Were you with Carlisle back then?† Alice gawked at me for a moment, and then recovered herself, blinking rapidly. â€Å"Well, I guess that explains the smell,† she muttered. â€Å"But does it explain what I didn't see?† She frowned, her porcelain forehead creasing. â€Å"The smell?† I repeated. â€Å"You smell awful,† she said absently, still frowning. â€Å"A werewolf? Are you sure about that?† â€Å"Very sure,† I promised, wincing as I remembered Paul and Jacob fighting in the road. â€Å"I guess you weren't with Carlisle the last time there were werewolves here in Forks?† â€Å"No. I hadn't found him yet.† Alice was still lost in thought. Suddenly, her eyes widened, and she turned to stare at me with a shocked expression. â€Å"Your best friend is a werewolf?† I nodded sheepishly. â€Å"How long has this been going on?† â€Å"Not long,† I said, my voice sounding defensive. â€Å"He's only been a werewolf for just a few weeks.† She glowered at me. â€Å"A young werewolf? Even worse! Edward was rightyou're a magnet for danger. Weren't you supposed to be staying out of trouble?† â€Å"There's nothing wrong with werewolves,† I grumbled, stung by her critical tone. â€Å"Until they lose their tempers.† She shook her head sharply from side to side. â€Å"Leave it to you, Bella. Anyone else would be better off when the vampires left town. But you have to start hanging out with the first monsters you can find.† I didn't want to argue with AliceI was still trembling with joy that she was really, truly here, that I could touch her marble skin and hear her wind-chime voicebut she had it all wrong. â€Å"No, Alice, the vampires didn't really leavenot all of them, anyway. That's the whole trouble. If it weren't for the werewolves, Victoria would have gotten me by now. Well, if it weren't for Jake and his friends, Laurent would have gotten me before she could, I guess, so† â€Å"Victoria?† she hissed. â€Å"Laurent?† I nodded, a teensy bit alarmed by the expression in her black eyes. I pointed at my chest. â€Å"Danger magnet, remember?† She shook her head again. â€Å"Tell me everythingstart at the beginning.† I glossed over the beginning, skipping the motorcycles and the voices, but telling her everything else right up to today's misadventure. Alice didn't like my thin explanation about boredom and the cliffs, so I hurried on to the strange flame I'd seen on the water and what I thought it meant. Her eyes narrowed almost to slits at that part. It was strange to see her look so so dangerouslike a vampire. I swallowed hard and went on with the rest about Harry. She listened to my story without interrupting. Occasionally, she would shake her head, and the crease in her forehead deepened until it looked like it was carved permanently into the marble of her skin. She didn't speak and, finally, I fell quiet, struck again by the borrowed grief at Harry's passing. I thought of Charlie; he would be home soon. What condition would he be in? â€Å"Our leaving didn't do you any good at all, did it?† Alice murmured. I laughed onceit was a slightly hysterical sound. â€Å"That was never the point, though, was it? It's not like you left for my benefit.† Alice scowled at the floor for a moment. â€Å"Well I guess I acted impulsively today. I probably shouldn't have intruded.† I could feel the blood draining from my face. My stomach dropped. â€Å"Don't go, Alice,† I whispered. My fingers locked around the collar of her white shirt and I began to hyperventilate. â€Å"Please don't leave me.† Her eyes opened wider. â€Å"All right,† she said, enunciating each word with slow precision. â€Å"I'm not going anywhere tonight. Take a deep breath.† I tried to obey, though I couldn't quite locate my lungs. She watched my face while I concentrated on my breathing. She waited till I was calmer to comment. â€Å"You look like hell, Bella.† â€Å"I drowned today,† I reminded her. â€Å"It goes deeper than that. You're a mess.† I flinched. â€Å"Look, I'm doing my best.† â€Å"What do you mean?† â€Å"It hasn't been easy. I'm working on it.† She frowned. â€Å"I told him,† she said to herself. â€Å"Alice,† I sighed. â€Å"What did you think you were going to find? I mean, besides me dead? Did you expect to find me skipping around and whistling show tunes? You know me better than that.† â€Å"I do. But I hoped.† â€Å"Then I guess I don't have the corner on the idiocy market.† The phone rang. â€Å"That has to be Charlie,† I said, staggering to my feet. I grabbed Alice's stone hand and dragged her with me to the kitchen. I wasn't about to let her out of my sight. â€Å"Charlie?† I answered the phone. â€Å"No, it's me,† Jacob said. â€Å"Jake!† Alice scrutinized my expression. â€Å"Just making sure you were still alive,† Jacob said sourly. â€Å"I'm fine. I told you that it wasn't† â€Å"Yeah. I got it. ‘Bye.† Jacob hung up on me. I sighed and let my head hang back, staring at the ceiling. â€Å"That's going to be a problem.† Alice squeezed my hand. â€Å"They aren't excited I'm here.† â€Å"Not especially. But it's none of their business anyway.† Alice put her arm around me. â€Å"So what do we do now?† she mused. She seemed to talk to herself for a moment. â€Å"Things to do. Loose ends to tie.† â€Å"What things to do?† Her face was suddenly careful. â€Å"I don't know for sure I need to see Carlisle.† Would she leave so soon? My stomach dropped. â€Å"Could you stay?† I begged. â€Å"Please? For just a little while. I've missed you so much.† My voice broke. â€Å"If you think that's a good idea.† Her eyes were unhappy. â€Å"I do. You can stay hereCharlie would love that.† â€Å"I have a house, Bella.† I nodded, disappointed but resigned. She hesitated, studying me. â€Å"Well, I need to go get a suitcase of clothes, at the very least.† I threw my arms around her. â€Å"Alice, you're the best!† â€Å"And I think I'll need to hunt. Immediately,† she added in a strained voice. â€Å"Oops.† I took a step back. â€Å"Can you stay out of trouble for one hour?† she asked skeptically. Then, before I could answer, she held up one finger and closed her eyes. Her face went smooth and blank for a few seconds. And then her eyes opened and she answered her own question. â€Å"Yes, you'll be fine. For tonight, anyway.† She grimaced. Even making faces, she looked like an angel. â€Å"You'll come back?† I asked in a small voice. â€Å"I promiseone hour.† I glanced at the clock over the kitchen table. She laughed and leaned in quickly to kiss me on the cheek. Then she was gone. I took a deep breath. Alice would be back. I suddenly felt so much better. I had plenty to do to keep myself busy while I waited. A shower was definitely first on the agenda. I sniffed my shoulders as I undressed, but I couldn't smell anything but the brine and seaweed scent of the ocean. I wondered what Alice had meant about me smelling bad. When I was cleaned up, I went back to the kitchen. I couldn't see any signs that Charlie ‘lad eaten recently, and he would probably be hungry when he got back. I hummed tunelessly to myself as I moved around the kitchen. While Thursday's casserole rotated in the microwave, I made up the couch with sheets and an old pillow. Alice wouldn't need it, but Charlie would need to see it. I was careful not to watch the clock. There was no reason to start myself panicking; Alice had promised. I hurried through my dinner, not tasting itjust feeling the ache as it slid down my raw throat. Mostly I was thirsty; I must have drunk a half gallon of water by the time I was finished. All the salt in my system had dehydrated me. I went to go try to watch TV while I waited. Alice was already there, sitting on her improvised bed. Her eyes were a liquid butterscotch. She smiled and patted the pillow. â€Å"Thanks.† â€Å"You're early,† I said, elated. I sat down next to her and leaned my head on her shoulder. She put her cold arms around me and sighed. â€Å"Bella. What are we going to do with you?† â€Å"I don't know,† I admitted. â€Å"I really have been trying my hardest.† â€Å"I believe you.† It was silent. â€Å"Doesdoes he† I took a deep breath. It was harder to say his name out loud, even though I was able to think it now. â€Å"Does Edward know you're here?† I couldn't help asking. It was my pain, after all. I'd deal with it when she was gone, I promised myself, and felt sick at the thought. â€Å"No.† There was only one way that could be true. â€Å"He's not with Carlisle and Esme?† â€Å"He checks in every few months.† â€Å"Oh.† He must still be out enjoying his distractions. I focused my curiosity on a safer topic. â€Å"You said you flew here Where did you come from?† â€Å"I was in Denali. Visiting Tanya's family.† â€Å"Is Jasper here? Did he come with your'† She shook her head. â€Å"He didn't approve of my interfering. We promised† she trailed off, and then her tone changed. â€Å"And you think Charlie won't mind my being here?† she asked, sounding worried. â€Å"Charlie thinks you're wonderful, Alice.† â€Å"Well, we're about to find out.† Sure enough, a few seconds later I heard the cruiser pull into the driveway. I jumped up and hurried to open the door. Charlie trudged slowly up the walk, his eyes on the ground and his shoulders slumped. I walked forward to meet him; he didn't even see me until I hugged him around the waist. He embraced me back fiercely. â€Å"I'm so sorry about Harry, Dad.† â€Å"I'm really going to miss him,† Charlie mumbled. â€Å"How's Sue doing?† â€Å"She seems dazed, like she hasn't grasped it yet. Sam's staying with her† The volume of his voice faded in and out. â€Å"Those poor kids. Leah's just a year older than you, and Seth is only fourteen† He shook his head. He kept his arms tight around me as he started toward the door again. â€Å"Um, Dad?† I figured I'd better warn him. â€Å"You'll never guess who's here.† He looked at me blankly. His head swiveled around, and he spied the Mercedes across the street, the porch light reflecting off the glossy black paint. Before he could react, Alice was in the doorway. â€Å"Hi, Charlie,† she said in a subdued voice. â€Å"I'm sorry I came at such a bad time.† â€Å"Alice Cullen?† he peered at the slight figure in front of him as if he doubted what his eyes were telling him. â€Å"Alice, is that you?† â€Å"It's me,† she confirmed. â€Å"I was in the neighborhood.† â€Å"Is Carlisle† â€Å"No, I'm alone.† Both Alice and I knew he wasn't really asking about Carlisle. His arm tightened over my shoulder. â€Å"She can stay here, can't she?† I pleaded. â€Å"I already asked her.† â€Å"Of course,† Charlie said mechanically. â€Å"We'd love to have you, Alice.† â€Å"Thank you, Charlie. I know it's horrid timing.† â€Å"No, it's fine, really. I'm going to be really busy doing what I can for Harry's family; it will be nice for Bella to have some company.† â€Å"There's dinner for you on the table, Dad,† I told him. â€Å"Thanks, Bell.† He gave me one more squeeze before he shuffled toward the kitchen. Alice went back to the couch, and I followed her. This time, she was the one to pull me against her shoulder. â€Å"You look tired.† â€Å"Yeah,† I agreed, and shrugged. â€Å"Near-death experiences do that to me So, what does Carlisle think of you being here?† â€Å"He doesn't know. He and Esme were on a hunting trip. I'll hear from him in a few days, when he gets back.† â€Å"You won't tell him, though when he checks in again?† I asked. She knew I didn't mean Carlisle now. â€Å"No. He'd bite my head off,† Alice said grimly. I laughed once, and then sighed. I didn't want to sleep. I wanted to stay up all night talking to Alice. And it didn't make sense for me to be tired, what with crashing on Jacob's couch all day. But drowning really had taken a lot out of me, and my eyes wouldn't stay open. I rested my head on her stone shoulder, and drifted into a more peaceful oblivion than I had any hope of. I woke early, from a deep and dreamless sleep, feeling well-rested, but stiff. I was on the couch tucked under the blankets I'd laid out for Alice, and I could hear her and Charlie talking in the kitchen. It sounded like Charlie was fixing her breakfast. â€Å"How bad was it, Charlie?† Alice asked softly, and at first I thought they were talking about the Clearwaters. Charlie sighed. â€Å"Real bad.† â€Å"Tell me about it. I want to know exactly what happened when we left.† There was a pause while a cupboard door was closed and a dial on the stove was clicked off. I waited, cringing. â€Å"I've never felt so helpless,† Charlie began slowly. â€Å"I didn't know what to do. That first weekI thought I was going to have to hospitalize her. She wouldn't eat or drink, she wouldn't move. Dr. Gerandy was throwing around words like ‘catatonic,' but I didn't let him up to see her. I was afraid it would scare her.† â€Å"She snapped out of it though?† â€Å"I had Renee come to take her to Florida. I just didn't want to be the one if she had to go to a hospital or something. I hoped being with her mother would help. But when we started packing her clothes, she woke up with a vengeance. I've never seen Bella throw a fit like that. She was never one for the tantrums, but, boy, did she fly into a fury. She threw her clothes everywhere and screamed that we couldn't make her leaveand then she finally started crying. I thought that would be the turning point. I didn't argue when she insisted on staying here and she did seem to get better at first† Charlie trailed off. It was hard listening to this, knowing how much pain I'd caused him. â€Å"But?† Alice prompted. â€Å"She went back to school and work, she ate and slept and did her homework. She answered when someone asked her a direct question. But she was empty. Her eyes were blank. There were lots of little thingsshe wouldn't listen to music anymore; I found a bunch of CDs broken in the trash. She didn't read; she wouldn't be in the same room when the TV was on, not that she watched it so much before. I finally figured it outshe was avoiding everything that might remind her of him. â€Å"We could hardly talk; I was so worried about saying something that would upset herthe littlest things would make her flinchand she never volunteered anything. She would just answer if I asked her something. â€Å"She was alone all the time. She didn't call her friends back, and after a while, they stopped calling. â€Å"It was night of the living dead around here. I still hear her screaming in her sleep† I could almost see him shuddering. I shuddered, too, remembering. And then I sighed. I hadn't fooled him at all, not for one second. â€Å"I'm so sorry, Charlie,† Alice said, voice glum. â€Å"It's not your fault.† The way he said it made it perfectly clear that he was holding someone responsible. â€Å"You were always a good friend to her.† â€Å"She seems better now, though.† â€Å"Yeah. Ever since she started hanging out with Jacob Black, I've noticed a real improvement. She has some color in her cheeks when she comes home, some light in her eyes. She's happier.† He paused, and his voice was different when he spoke again. â€Å"He's a year or so younger than her, and I know she used to think of him as a friend, but I think maybe it's something more now, or headed that direction, anyway.† Charlie said this in a tone that was almost belligerent. It was a warning, not for Alice, but for her to pass along. â€Å"Jake's old for his years,† he continued, still sounding defensive. â€Å"He's taken care of his father physically the way Bella took care of her mother emotionally. It matured him. He's a good-looking kid, tootakes after his mom's side. He's good for Bella, you know,† Charlie insisted. â€Å"Then it's good she has him,† Alice agreed. Charlie sighed out a big gust of air, folding quickly to the lack of opposition. â€Å"Okay, so I guess that's overstating things. I don't know even with Jacob, now and then I see something in her eyes, and I wonder if I've ever grasped how much pain she's really in It's not normal, Alice, and it it frightens me. Not normal at all. Not like someone left her, but like someone died.† His voice cracked. It was like someone had diedlike I had died. Because it had been more than just losing the truest of true loves, as if that were not enough to kill anyone. It was also losing a whole future, a whole familythe whole life that I'd chosen Charlie went on in a hopeless tone. â€Å"I don't know if she's going to get over itI'm not sure if it's in her nature to heal from something like this. She's always been such a constant little thing. She doesn't get past things, change her mind.† â€Å"She's one of a kind,† Alice agreed in a dry voice. â€Å"And Alice† Charlie hesitated. â€Å"Now, you know how fond I am of you, and I can tell that she's happy to see you, but I'm a little worried about what your visit will do to her.† â€Å"So am I, Charlie, so am I. I wouldn't have come if I'd had any idea. I'm sorry.† â€Å"Don't apologize, honey. Who knows? Maybe it will be good for her.† â€Å"I hope you're right.† There was a long break while forks scraped plates and Charlie chewed. I wondered where Alice was hiding the food. â€Å"Alice, I have to ask you something,† Charlie said awkwardly. Alice was calm. â€Å"Go ahead.† â€Å"He's not coming back to visit, too, is he?† I could hear the suppressed anger in Charlie's voice. Alice answered in a soft, reassuring tone. â€Å"He doesn't even know I'm here. The last time I spoke with him, he was in South America.† I stiffened as I heard this new information, and listened harder. â€Å"That's something, at least.† Charlie snorted. â€Å"Well, I hope he's enjoying himself.† For the first time, Alice's voice had a bit of steel in it. â€Å"I wouldn't make assumptions, Charlie.† I knew how her eyes would flash when she used that tone. A chair scooted from the table, scraping loudly across the floor. I pictured Charlie getting up; there was no way Alice would make that kind of noise. The faucet ran, splashing against a dish. It didn't sound like they were going to say anything more about Edward, so I decided it was time to wake up. I turned over, bouncing against the springs to make them squeak. Then I yawned loudly. All was quiet in the kitchen. I stretched and groaned. â€Å"Alice?† I asked innocently; the soreness rasping in my throat added nicely to the charade. â€Å"I'm in the kitchen, Bella,† Alice called, no hint in her voice that she suspected my eavesdropping. But she was good at hiding things like that. Charlie had to leave thenhe was helping Sue Clearwater with the funeral arrangements. It would have been a very long day without Alice. She never spoke about leaving, and I didn't ask her. I knew it was inevitable, but I put it out of my mind. Instead, we talked about her familyall but one. Carlisle was working nights in Ithaca and teaching part time at Cornell. Esme was restoring a seventeenth century house, a historical monument, in the forest north of the city. Emmett and Rosalie had gone to Europe for a few months on another honeymoon, but they were back now. Jasper was at Cornell, too, studying philosophy this time. And Alice had been doing some personal research, concerning the information I'd accidentally uncovered for her last spring. She'd successfully tracked down the asylum where she'd spent the last years of her human life. The life she had no memory of. â€Å"My name was Mary Alice Brandon,† she told me quietly. â€Å"I had a little sister named Cynthia. Her daughtermy nieceis still alive in Biloxi.† â€Å"Did you find out why they put you in that place?† What would drive parents to that extreme? Even if their daughter saw visions of the future She just shook her head, her topaz eyes thoughtful. â€Å"I couldn't find much about them. I went through all the old newspapers on microfiche. My family wasn't mentioned often; they weren't part of the social circle that made the papers. My parents' engagement was there, and Cynthia's.† The name fell uncertainly from her tongue. â€Å"My birth was announced and my death. I found my grave. I also filched my admissions sheet from the old asylum archives. The date on the admission and the date on my tombstone are the same.† I didn't know what to say, and, after a short pause, Alice moved on to lighter topics. The Cullens were reassembled now, with the one exception, spending Cornell's spring break in Denali with Tanya and her family. I listened too eagerly to even the most trivial news. She never mentioned the one I was most interested in, and for that I was grateful. It was enough to listen to the stories of the family I'd once dreamed of belonging to. Charlie didn't get back until after dark, and he looked more worn than he had the night before. He would be headed back to the reservation first thing in the morning for Harry's funeral, so he turned in early. I stayed on the couch with Alice again. Charlie was almost a stranger when he came down the stairs before the sun was up, wearing an old suit I'd never seen him in before. The jacket hung open; I guessed it was too tight to fasten the buttons. His tie was a bit wide for the current style. He tiptoed to the door, trying not to wake us up. I let him go, pretending to sleep, as Alice did on the recliner. As soon as he was out the door, Alice sat up. Under the quilt, she was fully dressed. â€Å"So, what are we doing today?† she asked. â€Å"I don't knowdo you see anything interesting happening?† She smiled and shook her head. â€Å"But it's still early.† All the time I'd been spending in La Push meant a pile of things I'd been neglecting at home, and I decided to catch up on my chores. I wanted to do something, anything that might make life easier for Charliemaybe it would make him feel just a little better to come home to a clean, organized house. I started with the bathroomit showed the most signs of neglect. While I worked, Alice leaned against the doorjamb and asked nonchalant questions about my, well, our high school friends and what they been up to since she'd left. Her face stayed casual and emotionless, but I sensed her disapproval when she realized how little I could tell her. Or maybe I just had a guilty conscience after eavesdropping on her conversation with Charlie yesterday morning. I was literally up to my elbows in Comet, scrubbing the floor of the bathtub, when the doorbell rang. I looked to Alice at once, and her expression was perplexed, almost worried, which was strange; Alice was never taken by surprise. â€Å"Hold on!† I shouted in the general direction of the front door, getting up and hurrying to the sink to rinse my arms off. â€Å"Bella,† Alice said with a trace of frustration in her voice, â€Å"I have a fairly good guess who that might be, and I think I'd better step out.† â€Å"Guess?† I echoed. Since when did Alice have to guess anything? â€Å"If this is a repeat of my egregious lapse in foresight yesterday, then it's most likely Jacob Black or one of his friends.† I stared at her, putting it together. â€Å"You can't see werewolves?† She grimaced. â€Å"So it would seem.† She was obviously annoyed by this factvery annoyed. The doorbell rang againbuzzing twice quickly and impatiently. â€Å"You don't have go anywhere, Alice. You were here first.† She laughed her silvery little laughit had a dark edge. â€Å"Trust meit wouldn't be a good idea to have me and Jacob Black in a room together.† She kissed my cheek swiftly before she vanished through Charlie's doorand out his back window, no doubt. The doorbell rang again.

Lee Harvey Oswald Was Not the Lone Assassin - 1985 Words

Lee Harvey Oswald Was No Loner On November 22nd, 1963, President John F. Kennedy (JFK) was shot and killed in a motorcade running through Dealy Plaza, in Dallas, Texas. Shortly after, a man by the name of Lee Harvey Oswald was arrested and charged with the murder of President Kennedy. Over the years there has been much controversy over if Lee Harvey Oswald was in fact, the only man involved in the assassination of JFK. The assassination is still a topic of debate to this day and has spawned many conspiracy theories. At the time, there was little persuasive evidence to prove that Oswald was involved in any sort of conspiracy to assassinate the president, but as time went on people began to grow suspicious of certain things. In 1966,†¦show more content†¦Although Ruby claims to have murdered Oswald as revenge for killing President Kennedy, most people believe it was part of a cover up for a much larger conspiracy. After Ruby’s arrest, he was sentenced to death, only to have his trials postponed and die of lung cancer shortly after postponing the trial. While Oswald was in interrogation, he changed his story many times when asked routine questions such as â€Å"where were you at the time of the shooting†, or â€Å"explain to us what you did the day of the assassination†. He also denied any involvement in the incident even though there was evidence against him. Oswald was bound to face further, more intense interrogation in the future, and this is the reason most people believe that assassination of Oswald was to prevent him from talking about what actually happened in the Kennedy assassination. There were also many strange reports of President Kennedy’s brain being switched from when it was seen in evidence, to when the autopsy was performed. There were pictures of Kennedy’s brain for evidence showing immense damage to the rear, consistent with an exit wound and therefore evidence of a shot from the front. Meanwhile, the autopsy brain did not nearly show the same amount of damage in the back of the brain, and an exit wound in the front. Douglas Horne, the Record Review Boards chief analyst for military records said that he was â€Å"90-95%† certain that these brains were notShow MoreRelatedThe Assassination Of President Kennedy1491 Words   |  6 Pagesis no absolute, 100 percent clear evidence that proves Lee Harvey Oswald’s guilt or innocence of the assassination of President Kennedy (JFK,) but there is much evidence that points towards his innocence, or the fact that he was not the lone gunman to kill Kennedy. Being established a week after the assassination, on the 29th of November 1963, the Warren Commission, after a year long investigation and a singular report, concluded that Oswald had acted alone in assassinating President Kennedy, butRead More Kennedy Assassination Essay example1846 Words   |  8 Pages â€Å"President Kennedy was dead. I stopped by a giant live oak tree on the vast front lawn of Parkland Hospital and cried.quot; As this reporter said, people everywhere were distraught when they heard that President Kennedy had been assassinated. News reporters from other countries flew in, called up, and sent letters back to tell the people in their country what had happened and to give advice to the Dallas Police Station. Everyone everywhere was upset by this tragic loss. People were sent home fromRead MoreNoah Cooper. Mrs. Sites. English 10A. 15 April 2017. Lee1749 Words   |  7 PagesEnglish 10A 15 April 2017 Lee Harvey Oswald: The Sole Assassin? The public must be satisfied that Oswald was the assassin; that he did not have confederates who are still at large; and that evidence was such that he would have been convicted at trial. Unfortunately the facts on Oswald seem about too pat- too obvious (Marxist, Cuba, Russian wife, etc.) The Dallas police have put out statements on the Communist conspiracy theory and it was they who were in charge when he was shot and thus silenced (â€Å"TheRead MoreEssay on The J.F.K. Conspiracy941 Words   |  4 PagesThe J.F.K. Conspiracy Time after time again we are told in our schools, our homes, and by our media that Lee Harvey Oswald acted alone in the assassination of former President John F. Kennedy. How can this conclusion by made about a case that was never completely solved? If we as Americans accept the government given truth, such as in this case, without questioning how they came to their conclusion, then what would stop the government from altering other truths givenRead MoreAnalysis Of The Book New Orleans 1971 Words   |  8 Pagesreturned to the United States Oswald moves around quite frequently, eventually ending up in New Orleans. This is where the most mysterious and perplexing chapter of his life took place. In 1963, Oswald is employed at the Reily Coffee Company, where he is fired from only months later. He also starts a one-man chapter of the Fair Play for Cuba Committee and began pro-Castro propaganda in New Orleans. Oddly enough, the return address on Oswald’s pro-Castro leaflets was 544 Camp Street, New Orleans.Read MoreConspiracy Theories Related to the Assassination of John F. Kennedy1200 Words   |  5 PagesAmerica, John Fitzgerald Kennedy, was assassinated in Dallas, Texas. Lee Harvey Oswald was charged with the crime, but was shot and killed two days later by Jack Ruby before he could be taken to trial. The Warren Commission official ly determined that Oswald was the lone assassin, however, this conclusion has not been accepted by many. In fact, a 2003 poll reported that 75% of Americans do not believe that Lee Harvey Oswald acted alone. Most believe that the assassination was the result of a conspiracy,Read MoreThe Assassination Of John Fitzgerald Kennedy Essay1556 Words   |  7 Pagesassassination of John Fitzgerald Kennedy. Friday on November 22, 1963, the 35th President of the United States, JFK, was shot at 12:30 PM while traveling in Dallas, Texas to appear in the next presidential campaign. Thousands of Americans whom was present at Dallas, Texas just witness the murder of the United States president. Many Americans believed that the United States government was behind the JFK assassination. Thousands of important evidence about the JFK assassination has been locked and sealedRead MoreThe Death Of John F. Kennedy1183 Words   |  5 PagesOn November 22, 1963, our president, John F. Kennedy, was shot an d killed while riding through Dealey Plaza in Dallas, Texas. Lee Harvey Oswald was identified as the lone assassin hours after thousands of Texans witnessed the horrific last shot shattering their beloved president’s right temple. Six months later, a report of the investigation was released by the Warren Commission. Unreported findings caused the majority of America to question the accuracy of the results to what happened that day.Read MoreThe Hidden History Of The Jfk Assassination2336 Words   |  10 PagesKennedy assassination was from a lone gunman or as part of a bigger conspiracy cover-up. On November 22, 1963, President John F. Kennedy was shot as he paraded through Dallas, Texas. That same afternoon, Dallas police had arrested their suspect, Lee Harvey Oswald. Oswald, an itinerant ex-U.S. marine and self-described Marxist-Leninist, previously lived in the Soviet Union prior to his move back to the United States. However, within days of his arrest, Oswald was de ad. Oswald was murdered by JackRead More Book Report On Cover Up Essay2031 Words   |  9 PagesKennedy Jr. was a very emotional time in our nations history. This horrifying incident occurred on November 22, 1963, in a motorcade procession in Dallas, Texas. At 12:30 in the afternoon the procession was going down Elm Street in Dealy Plaza, when shots were fired. One struck President Kennedy in the throat and moments later a bullet tore apart his head. At 1:00 p.m., President JFK was pronounced dead. That same afternoon, Lee Harvey Oswald was arrested as a suspect of murder. Oswald however, pleaded