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User:Riveraelita

The Gabonese Education System

The Gabonese educational system stands as a beacon of hope and progress in Central Africa. With a strong commitment to intellectual growth and empowerment, Gabon's education system aims to equip its students with the necessary knowledge, skills, and values for a rapidly evolving world. This pages explains the core elements of the Gabonese educational system, highlighting its inclusive approach, curriculum structure, teacher development initiatives, and emphasis on research and higher education.

Introduction
Gabon is located on the west coast of Central Africa, bordered by Cameroon to the north, Equatorial Guinea to the northwest, the Atlantic Ocean to the west, and the Republic of the Congo to the east and south. The country's geographical coordinates range from approximately 2 degrees north latitude to 3 degrees south latitude and from 8 degrees east longitude to 14 degrees east longitude. Diverse geographical features, including dense tropical rainforests, savannas, and coastal plains, characterize Gabon. It is also home to the world's largest tropical rainforest region - the Congo Basin. Its coastline along the Atlantic Ocean provides access to ports and maritime resources.

Brief History of Gabon's Education System
The history of Gabon's education system is marked by significant developments and challenges that have shaped its current state. Before the arrival of European colonizers, education in Gabon was primarily informal and centered on traditional knowledge, skills, and cultural practices. Youth learned from elders, acquiring practical skills, and passing down oral traditions. During the late 19th and early 20th centuries, Gabon came under French colonial rule and was introduced to a Western-style education system to serve its administrative and economic interests. However, access to education was limited and geared towards the elite and children of colonial administrators. As Gabon moved towards independence, there was a growing demand for expanded educational opportunities. The country witnessed the establishment of additional primary and secondary schools, increasing access to education for a broader segment of the population. In the post-independence era, Gabon placed greater emphasis on educational development and aimed to improve literacy rates and provide educational infrastructure across the country. Economic challenges and budget constraints during this period affected the quality of education, and teacher strikes were prevalent. In recent years, Gabon has continued to focus on education as a priority sector for development. The government has invested in educational infrastructure, teacher training, and curriculum improvements and efforts have been made to promote gender equality in education and to expand access to education in rural and remote areas.

Early Childhood Education
Early childhood education at this level is not compulsory but serves as a foundation of formal education. It caters to children between the ages of 3 to 5 years and focuses on promoting social skills, basic language development and learning through play.

Primary and Secondary Schools

 * 1) Primary education is the first level of formal education and is compulsory for children. It generally covers students aged 6 to 11 years and lasts for six years and is divided into two cycles of three years each. The primary curriculum includes subjects such as mathematics, science, languages (French and local languages), social studies, and physical education.
 * 2) Secondary education covers students between the ages of 12-18 years and follows primary education and is also compulsory. The secondary education cycle lasts for seven years and is divided into two cycles:
 * 3) Lower secondary - Lasts for four years and offers a broader range of subjects, including vocational subjects. The curriculum includes subjects like mathematics, French, science, history, geography, technology, physical education, and arts. Additionally, there may be some vocational courses offered to provide practical skills to students.
 * 4) Upper secondary - Lasts for three years and allows students to specialize in academic or vocational tracks, preparing them for further education or the job market. The curriculum includes subjects like mathematics, sciences, humanities, foreign languages, and vocational subjects related to fields such as agriculture, commerce, industry, or technology.

Shared Responsibilities
Both governments collaborate to ensure that education is accessible and equitable for all citizens, regardless of their geographic location or socio-economic background. Funding for education is often shared, with contributions from both levels to ensure adequate financial resources for the education sector. State and Territory governments collaborate to align the curriculum with national educational goals and standards, promoting a consistent and coherent educational experience across the country.

The Gabonese Government's Key Education Forums, Organizations and Programs
The Gabonese government has established several key education forums, organizations, and programs to enhance the quality of education and promote lifelong learning opportunities for its citizens. These entities play a vital role in shaping the education landscape in Gabon.

 * 1) Ministry of National Education and Vocational Training
 * 2) National Education Council (Conseil National de l'Éducation, CNE)
 * 3) National Agency for Vocational Education and Training (Agence Nationale de la Formation Professionnelle, ANFPG)
 * 4) National Institute for Training and Educational Research (Institut National de Formation et de Recherche en Éducation, INFRE):
 * 5) National Center for Educational Documentation and Information (Centre National de Documentation et d'Information sur l'Éducation, CNDIE)
 * 6) Gabonese National Commission for UNESCO (Commission Nationale Gabonaise pour l'UNESCO)
 * 7) Education for All (EFA) Initiative

The Ministry of National Education
Public education in Gabon is governed by the Ministry of National Education, which is accountable for the education system's comprehensive administration, regulation, and growth. The ministry develops educational policies, establishes curriculum benchmarks, oversees teacher training, and ensures the quality of education in public schools. It works with regional education offices and local authorities to execute and survey educational programs nationwide.

Government and non-Government Schools
Gabon has private schools that provide alternative educational options. These private schools may follow different curricula or offer specialized programs. Some families may enroll their children in private schools for various reasons, such as specific educational philosophies or preferences.

Official Language
French is the official language of instruction in Gabonese schools. However, the curriculum may also incorporate the teaching of local languages to preserve cultural heritage and promote multilingualism.

Academic Year
Very similar to the United States, with the exception of the second term. This term is usually uninterrupted, with no major holidays or breaks. Schools in Gabon observe a break for Easter, which typically falls in March or April. The duration of the Easter break is typically around one to two weeks.

Higher Education

 * Higher education in Gabon includes universities and technical institutes.
 * Students who have completed upper secondary education can enroll in higher education institutions.
 * Universities offer academic programs, leading to bachelor's, master's, and doctoral degrees across various disciplines.
 * Technical institutes focus on vocational and technical training, preparing students for specific careers and industries.

Teacher Workforce
Gabon invests significantly in the professional development and support of its teachers. Well-trained and motivated teachers are fundamental to a thriving educational system. By providing continuous support and learning opportunities, Gabon ensures that its educators can inspire and guide the next generation effectively. Gabon provides formal education programs through universities and teacher training institutions, catering to aspiring teachers at various levels: primary, secondary, and vocational education. These specialized programs offer comprehensive training that encompasses pedagogy, subject-specific knowledge, classroom management, and an understanding of child development. The teacher education curriculum in Gabon covers a wide range of subjects, including pedagogy, curriculum development, psychology, assessment techniques, and educational technology. After completing their education programs, prospective teachers in Gabon must obtain teaching certification. This certification process ensures that educators meet the established standards of competence and professionalism, promoting quality assurance within the education system.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Riveraelita


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Cookbook:Akpi

Akpi, also called njangsa/djansang and other names, is a tropical African tree (Ricinodendron heudelotii) whose seeds are used in cooking.

Characteristics
Akpi seeds are brown to black in color and aromatic.

Use
The seeds are typically ground and used to flavor and thicken dishes. If dried, they should be prepared using a moist cooking method.

title: Cookbook:Akpi


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Radiation Oncology/Radiobiology/Radioresistance

 * Sven de Mey et al. Radioresistance of Human Cancers: Clinical Implications of Genetic Expression Signatures. Front Oncol. 2021; 11: 761901. ( 34778082)
 * Fabio Busato et al. Biological Mechanisms to Reduce Radioresistance and Increase the Efficacy of Radiotherapy: State of the Art. Int J Mol Sci. 2022 Sep; 23(18): 10211. ( 36142122)


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User talk:88.230.229.130

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If current server doesn't work please try other servers above Toggle light Watch List Comment 88.230.229.130 (discuss) 18:21, 30 July 2023 (UTC)

Blocked
You cannot edit for 2 weeks due to disruptive editing. You are welcome to contribute constructively when editing privileges are restored. You may appeal this action with a reasonable argument by adding the text below. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 18:00, 31 July 2023 (UTC)

title: User talk:88.230.229.130


text:
User talk:AshtonSchool1225

Blocked
You cannot edit indefinitely due to disruptive editing. You may appeal this action with a reasonable argument by adding the text below. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 14:03, 3 August 2023 (UTC)

 * Why did you block me? AshtonSchool1225 (discuss • contribs) 11:53, 4 August 2023 (UTC)
 * How can I edit back again? AshtonSchool1225 (discuss • contribs) 11:56, 4 August 2023 (UTC)
 * AshtonSchool1225 (discuss • contribs) 11:57, 4 August 2023 (UTC)
 * The reason is the warning you received above and ignored, which is adding hoax/fabricated information about Unicode blocks to Wikibooks. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 12:16, 4 August 2023 (UTC)

title: User talk:AshtonSchool1225


text:
Cookbook:Funje (Cassava Flour Pudding)

Funje is a traditional dish popularly eaten in Angola. It can be prepared using the modern method or the traditional method.

Ingredients

 * 3 cups water
 * 1½ cup cassava flour

Modern method

 * 1) Preheat the oven to about 350 °F.
 * 2) Put the water in a medium-size pot and let boil.
 * 3) Put the cassava flour in a medium-size bowl, and mix with a little cold water until it is just damp. Make sure to mix it very well so as to have an and even mixture.
 * 4) When the water is boiling, mix in the damp pre-mixed cassava flour.
 * 5) Whisk to combine, and continue to whisk until all lump-free.
 * 6) Cover with the lid and place it in the oven to cook for about 30 minutes.
 * 7) Whisk any further if you want, then remove and serve with any soup of your desire.

Traditional method

 * 1) Bring the water to a boil in a pot.
 * 2) When the water is boiling, remove from the stove and place on the floor. Sit on a small stool with the pot between your legs.
 * 3) Pour the cassava flour into the water and start to whisk as fast as you can using a wooden spoon.
 * 4) Continue to mix until the mixture is completely smooth.
 * 5) When you are done whisking it to your satisfaction, cover with a lid and leave on a reduced heat for about 3 minutes.
 * 6) Enjoy with any complimentary soup of your choice.

Notes, tips, and variations

 * Preparing this dish is not time time consuming, but if you are not careful enough it can get lumpy. Make sure to mix it very well as soon as you start pouring the cassava flour into the boiling water.

title: Cookbook:Funje (Cassava Flour Pudding)


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Movie Making Manual/Cinematography/Shot Sizes

The shot size refers to the amount of the setting or subject that is visible within a specific frame of a video, photograph, or animation. It determines the scope or size of the shot. In film or video production, various types of camera shots are utilized to convey different aspects of the narrative, and these shots are later combined in post-production to construct a compelling story. Filmmakers commonly use standardized names for shot sizes, often represented by abbreviated forms of 2 or 3 letters on a shot list or storyboard. For instance, a close-up shot may be shortened to "CU", while a wide shot is indicated as "WS".

Extreme Long Shot (ELS)/Extreme Wide Shot (EWS)
An Extreme Long Shot (ELS) or Extreme Wide Shot (EWS) is a powerful camera shot that portrays the subject as a tiny figure in comparison to its surroundings. This shot captures an expansive view of the location, making the subject appear small, distant, and sometimes even insignificant in the grand scheme of things. The ELS or EWS can evoke a sense of vastness and scale, emphasizing the location and isolation of the subject.

Some key uses of a Extreme Long Shot or Extreme Wide Shot:

 * Emphasizing Location: The primary purpose of employing an ELS or EWS is to accentuate the setting or location where the scene takes place. By showcasing the subject in relation to the vast environment, the shot conveys the significance of the surroundings in shaping the narrative. Whether it's a majestic landscape, a bustling cityscape, or an eerie wilderness, this shot allows the audience to fully appreciate the context in which the events unfold.
 * Creating Distance and Unfamiliarity: An ELS or EWS is particularly useful when you want to establish emotional or physical distance between the subject and the audience. By visually separating the subject from the viewer, the shot can evoke a sense of detachment or unfamiliarity. This distancing effect can be used to enhance mystery, curiosity, or even a sense of isolation experienced by the characters within the story.
 * Conveying Overwhelm: In certain situations, an ELS or EWS can be employed to convey a feeling of overwhelm or powerlessness. Placing the subject amidst an expansive and dominating environment can make them appear small and insignificant, heightening the stakes and challenges they face. This technique is especially effective in portraying the struggle of characters against the forces of nature or overwhelming circumstances.

Long Shot (LS)/Wide Shot (WS)
A Long Shot (LS), also commonly referred to as a Wide Shot (WS), is a fundamental camera shot in filmmaking that captures a wide view of the scene, encompassing both the subject and a significant portion of the surrounding environment. This shot provides context, establishes the setting, and showcases the subject in relation to its surroundings. By revealing the subject's position within the larger frame, a Long Shot or Wide Shot offers the audience a comprehensive view of the location and the characters' interactions within it.

Some key uses of a Long Shot or Wide Shot:

 * Establishing the Scene: Long Shots or Wide Shots are often used to set the scene and provide the audience with a sense of the location's overall layout. By showing the subject in its environment, the shot helps viewers understand the physical space where the action takes place.
 * Emphasizing Setting and Context: Just like the Extreme Long Shot (ELS) or Extreme Wide Shot (EWS), Long Shots or Wide Shots are valuable for emphasizing the significance of the location. Whether it's a bustling city, a serene countryside, or an imposing castle, this shot allows the audience to immerse themselves in the world of the film.
 * Portraying Interaction: Long Shots or Wide Shots can effectively capture interactions between characters within the context of the environment. It showcases the characters' movements, spatial relationships, and actions within the larger frame.
 * Conveying Mood and Atmosphere: This shot can also contribute to the mood and atmosphere of a scene. For example, a vast and open landscape in a Wide Shot might evoke a sense of freedom and tranquility, while a crowded city street can convey a feeling of hustle and bustle.
 * Showing Isolation or Vulnerability: Similar to the Extreme Long Shot (ELS) or Extreme Wide Shot (EWS), a Long Shot or Wide Shot can be utilized to depict a character's isolation or vulnerability. Placing the subject within a vast and distant landscape can evoke a sense of solitude or insignificance.

Full Shot (FS)
A Full Shot (FS) portrays a character's entire body, reaching from the top of the frame to the bottom. In a Full Shot, the character is framed from head to toe, allowing the audience to observe not only their facial expressions and emotions but also their physicality, body language, and actions. This shot is versatile and can be used to highlight individual characters or showcase interactions between multiple characters within the same frame.

Some key uses of a Full Shot (FS) include:

 * Displaying Character Actions: The Full Shot effectively captures a character's movements and actions, allowing the audience to see how they interact with their environment and other characters.
 * Establishing Setting and Context: This shot can establish the setting and context of a scene by showing the character in relation to their surroundings. It gives the audience a better understanding of the physical space and location where the action takes place.
 * Revealing Character's Wardrobe: In addition to showcasing the character's physicality, the Full Shot displays their wardrobe and outfit, providing insight into their style and personality.
 * Showing Body Language and Interactions: The Full Shot is excellent for depicting body language and nonverbal communication between characters. It helps convey emotions and dynamics between individuals.
 * Presenting Multiple Characters: A Full Shot can accommodate one or more characters within the frame, showing their positions and interactions in relation to one another.

Medium Wide Shot (MWS)/Medium Long Shot (MLS)
A Medium Wide Shot (MWS), also known as a Medium Long Shot (MLS) or a ¾ shot, holds a place on the spectrum between a Full Shot and a Medium Shot. This versatile camera shot captures a subject from the knees upwards, offering a balanced view that includes essential visual elements from the surrounding while maintaining a focus on the subject's body language and expressions.

Some key uses of a Medium Wide Shot (MWS)/Medium Long Shot (MLS):

 * Conveying Context and Proximity: The Medium Wide Shot offers enough context to establish the setting while maintaining a visual closeness to the subject. It allows the audience to see the subject in relation to their surroundings without losing sight of their expressions and actions.
 * Balancing Character Focus: When a scene involves multiple characters or elements, the Medium Wide Shot can balance the focus among them. It ensures that individual characters are visible while still maintaining a sense of their collective presence within the scene.
 * Showcasing Body Language and Expressions: By framing the subject from the knees upwards, the Medium Wide Shot allows filmmakers to emphasize the character's body language and facial expressions. This shot is ideal for capturing emotional nuances and conveying a character's thoughts and feelings.
 * Facilitating Movement and Action: The Medium Wide Shot accommodates character movement and actions, making it suitable for dynamic scenes, such as walking, running, or interacting with the environment.
 * Versatility in Storytelling: The Medium Wide Shot can be creatively adapted to various narrative scenarios. It can highlight pivotal moments, dramatic reveals, or emphasize the significance of the subject in relation to their surroundings.

Cowboy Shot (CS)
A Cowboy Shot (CS), also known as an American shot, frames the actor from their mid-waist to just above their head. Frequently, the camera is positioned at hip level, providing a slightly low angle to the shot. The strategic framing of the Cowboy Shot enables filmmakers to depict characters in a heroic and confident fashion, while keeping the audience emotionally engaged through their facial expressions.

Some key uses of a Cowboy Shot (CS):

 * Signaling Heroism and Confidence: The Cowboy Shot's framing exudes a sense of heroism and confidence in the portrayed character. This shot is commonly used in Westerns and action films to portray protagonists as bold and courageous figures.
 * Emotional Engagement: By framing the actor from the mid-waist up, the Cowboy Shot allows the audience to connect emotionally with the character. The viewers can observe their facial expressions and register the emotions they convey during critical moments.
 * Showcasing Critical Action: The Cowboy Shot enables filmmakers to capture action occurring around the character's waistline, often showcasing the drawing of guns, swords, or other critical actions. This heightens the tension and intensity of the scene.

Medium Shot (MS)
A Medium Shot (MS), also known as a waist shot, is captured at a moderate distance from the subject, capturing waist up. It strikes a balance between the closeness of a close-up shot and the wider perspective of a long shot. The Medium Shot is frequently utilized in dialogue scenes, allowing filmmakers to capture body language, showcase costumes, and provide context to the setting.

Some key uses of a Medium Shot(MS):

 * Showing Important Action and Costumes: The Medium Shot is ideal for showcasing crucial actions, such as character movements or interactions. Additionally, it effectively captures costumes, allowing the audience to appreciate the visual details of the characters' attire.
 * Seamlessly Gluing Shots Together: In certain scenes, the Medium Shot can be used to create continuity by gluing separate shots together through constant actions. This technique creates a smooth flow between different camera angles and ensures seamless transitions.
 * Presenting Disarming, Comedic, and Informal Visuals: The Medium Shot is versatile in conveying a wide range of emotions and tones. It can create disarming or informal visuals, as well as add comedic elements to the scene.

Medium Close Up Shot (MCU)
A Medium Close-Up Shot (MCU) frames the subject from just above their head down to approximately midway on their torso. It strikes a delicate balance between the intimacy of a close-up shot and the inclusion of some background elements. The Medium Close-Up Shot allows filmmakers to capture the actor's emotions and facial expressions while retaining contextual information from the surroundings.

Some key uses of a Medium Close Up Shot (MCU):

 * Capturing the Actor's Performance: The Medium Close-Up Shot is an effective way to emphasize the actor's performance, allowing the audience to closely observe their emotions and expressions. It draws attention to the actor's face, making it a valuable shot for intense and emotional scenes.
 * Including Background and Context: Unlike a close-up shot that isolates the subject from the background, the Medium Close-Up Shot includes some contextual information, making it suitable for scenes where the environment is relevant to the narrative.
 * Neutral Shot Option for Standard Coverage: The Medium Close-Up Shot is often used as a versatile option for standard coverage. It provides a "neutral" shot that doesn't overwhelm or shock the viewer, making it suitable for a wide range of scenes.

Close Up (CU)
A Close-Up Shot (CU) typically focuses on a person's face, starting from the shoulders and ending at the top of their head. This shot establishes a strong emotional connection between the audience and the subject, intensifying the impact of the portrayed emotions and actions.

Some key uses of a Close Up (CU):

 * Generating Strong Emotions: By focusing on a person's face in a Close-Up Shot, filmmakers intensify the emotional impact of the scene. The audience can closely observe the character's emotions, thoughts, and reactions, creating a deep connection with their emotional journey.
 * Signaling Importance: The Close-Up Shot signifies the significance of the subject or object featured in the frame. It draws the audience's attention to the details and communicates that something crucial is unfolding in the story.
 * Portraying Facial Expressions: The Close-Up Shot is an invaluable tool for capturing actors' facial expressions. This is particularly effective during crucial moments, dialogue scenes, or emotionally charged sequences, allowing viewers to experience the characters' emotions intimately.

Extreme Close Up (ECU)
An Extreme Close-Up Shot (ECU) frames a subject very closely, often cutting off the outer portions of the subject by the edges of the frame. This shot is used to draw the audience's attention to specific portions of a subject, whether it's an actor's face, body parts, or inanimate objects. Extreme Close-Up Shots can go even closer, showing only an actor's mouth or a single eye, intensifying the focus on minute details for the desired effect.

Some key uses of Extreme Close Up (ECU):

 * Focusing on Specific Portions: The Extreme Close-Up Shot allows directors to isolate and emphasize particular portions of a subject, directing the audience's attention to crucial details.
 * Signaling Important Sensory Moments: In scenes with heightened sensory experiences, such as intense emotions, physical sensations, or impactful actions, the Extreme Close-Up Shot creates a more immersive and intimate connection with the audience.
 * Communicating Tiny Details: Extreme Close-Up Shots are employed to communicate subtle details that may be too small or unnoticed in wider shots. This technique draws the viewer's focus to the intricate elements that hold narrative significance.


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Cookbook:Palaver Sauce

Palaver sauce is a delicious meal that has its origin from the West African countries, and particularly Liberia. It is often eaten with rice or other starches, such as yam. Apart from the normal way of preparation it can be prepared following the direction for Flair's Classical Palaver sauce method.

Ingredients

 * 1 lb meat (e.g. stew beef), cut into ½-inch cubes
 * ½ teaspoon salt or to taste
 * 3 large onions, chopped
 * 3 large garlic cloves, finely minced
 * 1 medium-size dried red chile pepper, ground
 * 2 large fresh tomatoes, diced
 * 1 tablespoon tomato paste
 * 1 lb frozen tilapia or cod fillet (or other firm white fish; optional)
 * 2-inch square piece of salted cod
 * 1 cup palm oil or vegetable oil
 * 3 lb mixed chopped greens (e.g. kale, spinach, or mustard leaves)
 * 5 oz smoked fish (e.g. mackerel)
 * 1½ cup agushi seeds
 * 3 fresh bay leaves (optional)
 * Ground dried shrimp (optional)

Method I

 * 1) Combine the beef, 1 cup water, and salt. Sprinkle some of the onions, garlic, and red pepper over the beef. Bring to a boil over medium heat, then cover and cook for about 15 minutes.
 * 2) Add the remaining onions and red pepper, along with the tomatoes, tomato paste, fish, salt cod, and palm oil. Cook for about 5–10 minutes.
 * 3) Add the chopped greens, then stir well and let it simmer over medium heat for another few minutes.
 * 4) While the sauce simmers, clean the smoked fish, removing the head, skin, and bones. If you notice the sauce is boiling you can reduce the heat.
 * 5) Gently put in the smoked fish into the saucepan, and let it simmer uncovered.
 * 6) Grind the agushi seed in a blender with an equal amount of water. Transfer the mixture to the saucepan along with the bay leaves.
 * 7) Try to avoid stirring the contents at this point, and leave the saucepan to simmer for about another 10 minutes.
 * 8) Uncover the saucepan and mix the sauce. After that, continue to simmer on low heat until majority of the liquid is gone.
 * 9) Taste and adjust the seasonings as needed to your desired taste. Enjoy with any complimentary dish of your choice.

Method II – Flair's Classical Method

 * 1) Cook the greens in ½–1 cup of boiling water for 5 minutes. Drain the water from the greens, setting both the greens and the water aside for later.
 * 2) Season and steam the meat as in the above method. Set aside.
 * 3) In a large pot, heat some oil along with some chopped onions and fresh bay leaves. Fry them over medium heat until they are browned in order to season the oil. Remove and discard the bay leaves.
 * 4) Stir in the ground red chili pepper, frozen fish, and salted cod. Cook for 10 minutes over medium heat.
 * 5) Stir in the tomato paste and the water from cooking the greens. Cook, stirring constantly, for about 4 minutes.
 * 6) Add the tomatoes and the cooked meat along with its juices.
 * 7) Add the ground dried shrimp and the smoked fish (cleaned as in the above method).
 * 8) Grind the agushi and stir it into the pot. Cover and cook for 10 minutes.
 * 9) Stir and adjust the seasonings to taste. Remove from the heat, and enjoy with any special dish of your choice.

Notes, tips, and variations

 * This sauce can be eaten with plain white rice, boiled yam, plaintain either ripe or unripe and potatoes.
 * You can also decide to freeze or reheat the sauce.
 * If you will prefer a thick sauce, try to limit the amount of water you will be adding.
 * Any type of oil you want to use for the sauce should not be rancid, as this can spoil the taste.
 * For a complimentary green you can use okra or spinach.

title: Cookbook:Palaver Sauce


text: Category:Juice recipes

title: Category:Juice recipes


text:
User:Mukesh devar

Devar means God in genesis first superbeing ex: bhiramha dev,maha dev Protecter of world. Etymology:this superbeing livied in towers of Himalayas (skylaayam) towerer... devar.. thevar..devs devathaas also called vaanavargal. Migration and foot prints :some thousand years ago a solar attack pushed to spread throughout four directions of Himalayas.majorty of them stteled in south western areas of Indian penninsular connected with western Asia and African continent. Surnames and foot prints:as a Indian we found high density in southern part of Tamil Nadu. Heritage similarity: Heritage matches very much only super natural Power missing.we found launguage roots matching,clans of devar surname communities connected to Mahabharata period. Note:I see somebody writtened even equal to some lower caste communities.in India there is no by born low caste people in India before arrivals of nowadays low caste communities.the identified low caste communities are migrated from African & east asian continent if we go through anthropology this low caste communities got birth after some thousand years after Devas & vayshiyas birth.just a constitutional machineries pushed up them to today status in just 40 yrs of Indian independence and sad to say they snatch the Devas comendapality, wealth, power and monopoly in social status.and tried to put as equal in all aspects.

title: User:Mukesh devar


text:
User talk:Radblox123

You cannot edit indefinitely due to disruptive editing. You may appeal this action with a reasonable argument by adding the text below.

--Xania   talk 06:10, 31 July 2023 (UTC)

title: User talk:Radblox123


text:
Wikibooks:Reading room/Administrative Assistance/Archives/2023/June

Wikistution
Could an admin please delete Wikistution as a broken redirect? (I can't tag it for speedy deletion because it's fully protected). Pppery (discuss • contribs) 05:12, 3 June 2023 (UTC)

 * ✅ Thank you! —Kittycataclysm (discuss • contribs) 16:17, 3 June 2023 (UTC)

2601:347:C301:8EF0:D03F:18FB:14F3:7229
The IP is creating non-sense pages. Esteban16 (discuss • contribs) 00:37, 27 June 2023 (UTC)

 * Taken care of, thank you! —Kittycataclysm (discuss • contribs) 00:43, 27 June 2023 (UTC)


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World War II/Fall of Berlin

By the beginning of 1945, Germany's troops were nearing collapse, with Nazi leader Adolf Hitler retreating to his Führerbunker in mid-January. After holding its Red Army for a time in Germany, the Soviet Union attacked Berlin on 16 April 1945 (four days after the death of American president Franklin D. Roosevelt). The Soviets began bombarding Berlin with its artillery on 20 April, Hitler's birthday. Around that time, Hitler began making preparations for his suicide. He rejected efforts to surrender from his officers.

By 28 April, the day of Italian dictator Benito Mussolini's death, the Red Army had closed in on Berlin. Hitler married his longtime lover, Eva Braun, shortly after midnight on 29 April and signed his last will and testament. The bodies of the couple were found the next afternoon. Separate Soviet and American inquiries produced details later used to confirm the deaths, as the bodies were reputedly almost completely burned to ashes. Hitler had died from a suicide gunshot and Braun from cyanide poisoning.

Berlin's garrison surrendered on 2 May. Some German units continuing to fight to the west in hopes of surrendering to the Western Allies instead of the Soviets. On 8 May, the war ended in Europe. The following month, Soviet leader Joseph Stalin began seeding a dubious narrative that Hitler had escaped. Ever since, conspiracy theorists have cited discrepancies seeming to imply Hitler's survival, being largely dismissed by mainstream historians.

title: World War II/Fall of Berlin


text: User:MihaiBooksWiki/CookBook

title: User:MihaiBooksWiki/CookBook


text: User talk:111.65.58.1

title: User talk:111.65.58.1


text: User:RuwikiAdm

title: User:RuwikiAdm


text:
User:Transe Ænd Danse

Books lover....

title: User:Transe Ænd Danse


text:
User talk:Jeth888

Welcome!
Welcome to Wikibooks, Jeth888!

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title: User talk:Jeth888


text:
Perspectives in Digital Literacy/Stop Scrolling and Listen to Me

Stop Scrolling and Listen to Me: The Psychology Behind Social Media
Introduction

Since the release of Facebook in 2004, the use of social media has increased enormously, and so has the threat to the emotional and social well-being of teenagers. As sociologist and psychologist, Dr. Sherry Turkle discusses in her TED Talk, “Connected but alone,” there are ways in which young people's reliance on technology is changing their relationships with each other. She argues that each of them is becoming increasingly disconnected from each other, both physically and emotionally, as they spend more time interacting with their devices; at one point she says that "No one is listening” anymore and “everyone is lonely but are afraid of intimacy." These statements highlight the serious impact that social media can have on young people's relationships with others. How so? Teenagers, for example, do not listen to each other anymore, either due to a lack of attention or because they do not care about each other, and so, they are encouraged to create a personal bubble or a safe space where they can be whoever they want without fear of judgment as they don’t trust others. With this in mind, to what extent can the overuse of social media affect teenagers' cognition and social relations? This paper argues that its frequent usage changes how their brain functions and modifies their social behavior so that they cannot live without the permanent interruption of social media. As a result, teenagers may experience several noticeable changes, such as a sense of loneliness despite being physically present with others, anxiety about others’ perceptions of them and a constant need for acceptance or fear of rejection, and sleep disorders, as they spend so much time scrolling through their social media feeds late into the night.

The Negative Effects of Social Media Overuse on Mental Health
Although social platforms can be a positive resource to connect with others, there are exceptions where they disconnect people by stealing their attention and making them irresponsive. In fact, a team from the Department of Life Science and the Zlotowski Centre for Neuroscience at Ben-Gurion University conducted a study that examined the relationship between cellphone usage and its potential negative effects on human and social cognition. They found that constant engagement with social media, text messages, and other platforms can lead to changes in how people think, remember, and interact with others. Some of the changes they refer to imply a decrease in mental health, wellbeing, and appearance of patterns from other disorders. Also, they cite other authors who explained the repercussions in mental health due to this relation such as experiencing low emotional stability and self-esteem, chronic stress, depression, and disruptions in sleep patterns. Teenagers are increasingly exposed to a variety of content, including bad news. For example, they may see news stories about mass shootings in schools. This exposure to bad news can lead to feelings of depression, anxiety and increased stress levels, as teenagers may start to worry that something similar could happen to them or their loved ones. The spread of bad news, or even fake news, can trigger these negative and overwhelming emotions. When they misuse social media, they can create significant negative effects on everyone in their surroundings, as they can amplify their insecurities and pressure them to conform or to fit in. As science writer for the American Psychological Association, Summer Allen analyzes in his article “Social media’s growing impact on our lives,” there are gradual changes of early exposure that social media is making to distinct aspects of a young mind, such as: cognitive development, face-to-face interaction, creating own identity and establishing relations. At one point he reveals:

 * [A] Common Sense Media survey found that 13 percent of teens reported being cyberbullied at least once. And social media can be a conduit for accessing inappropriate content like violent images or pornography. Nearly two-thirds of teens who use social media said they “'often' or ‘sometimes' come across racist, sexist, homophobic, or religious-based hate content in social media.

We can interpret Allen’s findings by saying that teenagers are more vulnerable to harmful content without proper education, supervision, and communication. This can lead to them developing negative behaviors, such as racism, in order to fit in with their peers. As they are still developing their identity and sense of belonging, teens need to learn to identify harmful content and communicate with others about their online activity. Social media communities are known to encourage mental health problems and eating disorders by creating a lack of self-presentation/identity creation and body image disturbance. According to what scientific writers Pixie G. Turner and Carmen E. Lefevre analyze in their study “Eating and weight disorders,” there is a relation between the use of Instagram and the creation of Orthorexia Nervosa, an eating disorder that consists of eating obsessedly healthy which can lead to a restricted diet, nutritional deficiencies, or malnutrition. As they explain: “Social media use is ever increasing among young adults and has previously been shown to have negative effects on body image, depression, social comparison, and disordered eating". Therefore, social media platforms like Instagram can trigger social comparison, which can lead to negative body image and self-esteem. Turner and Lefevre found out factual evidence of the prevalence of Instagram as a beginning source for adolescents to experience loneliness, deception or frustration when they  do not look alike the people they follow, a diminished body image, altered diet by trying to eat like their influencers to achieve some desired image and the dangerous advisements they can  get about food-related content from people without the proper education to do so. Therefore, teens' perception of body image, nutrition, self-esteem, and identity are being disturbed by what is on trend and the content they are exposed to by the people they follow.

Upon the rise of more social media platforms, young people expect to socialize more with others, but the truth is they get lonelier and consequently, they isolate from society and limit their face-to-face interaction, all because the comfort it brings to be someone else. On the other hand, teenagers are so terrified to show their “imperfections” to the world that they hide behind a screen or a filter. As Dr. Turkle discusses in her TED Talk, they are being threatened by a loss in the capacity of choosing what they want and where they want to go. Also, diminished personal introspection is affected because they are afraid to look inside themselves, so they set a wall, a filter that can be changed based on what they want to show. As a result, they are unable to create real connections, do not trust themselves or others, and isolation and loneliness show up. When someone is alone, they choose to be alone to give themselves some time, so it is good emotionally; on the other hand, feeling lonely means they do not have company but also, they are empty, insecure, and misunderstood.

All the above can be expressed in researcher and professor Danah Boyd’s question, “does social media displace teens of socializing or we are displacing them when we restrict them to socialize with others?” (Allen). It would seem that they unintentionally isolate themselves from society due to fears, overprotection and other factors. At the same time, they are alone together when they decide to stop caring about their surroundings and get immersed in a personal virtual world while being physically with others.

Strategies for Regulating Social Media Usage
We have seen many negative consequences of social media overuse, but they vary depending on the person. For example, if teens check social media first thing in the morning and see something that upsets them, it can set their mood for the rest of the day. They may become angry, irritable or even depressed. This is because social media can be a breeding ground for negative emotions. Another example of this is feeling tired after scrolling through social media newsfeed all night. Therefore, it is truly important that everyone start to self-reflect about how their time management is and ask themselves why they are doing things and what keeps them motivated. By doing this, teenagers can recognize bad habits so later on they can reset goals and create a purpose (one that matters) for the things they do and use to make changes that guide them towards keeping that good workflow.

For these reasons, when young adults become conscious and recognize these red flags in their relationship with their phone and social media and also are aware of the many losses it creates, they can, therefore, develop strategies to overcome them. The Center for Humane Technology, a non-profit organization whose focus is on exposing the negative effects of social media and other attractive technologies that steal young adults' behavior and thoughts -among others-but also pursues a change in society and provides helpful resources have put together some ideas that can help teens reclaim control over themselves: Turning off notifications

So, they will not be interrupted at important moments.

Remove, manage, or log out of toxic apps

They promote misinformation, hate content and addiction (if they did not notice it). Experiencing a few days without media can be very healthy.

More critical less outrageous

Selectiveness with the content they choose to follow, more educational, inspiring, less outrage and polarization. Taking a break before arguing or criticizing someone, because whatever they do or say can be used against them and others.

 * Establishing limits

Taking control of the moments when they use their phone. Creating strategies that keep them the farthest from technology, exchange phone use with other activities.

And finally, reading, reading a lot: journals, magazines, books, comic books, whatever but read.

(“Take control of your social media use;” “Tips to take control of your tech use”)

In conclusion, overuse of social media can have a negative impact on teenagers' mental health and social behavior. It can diminish cognitive development, limit the ability for face-to-face interaction, and disturb their identity formation and self-esteem. It is important for them to be aware of the harmful effects of social media and to take steps to protect themselves and others from its negative effects. They can do this by identifying harmful content, educating themselves and their peers about its dangers, and using the strategies mentioned above to reduce its impact.

Time to reflect
It is important for young adults to reflect upon their daily activities and motivations in order for them to enhance their learning system and behavior.Therefore these questions are made for that purpose, to reflect about the power social media has over young adults life.

 * How many time do you spent daily on social media? if you know so, have you checked the screen time app in your phone?
 * How much do you know about social media usage and its effects?
 * How do you feel while being in social media vs doing other activities?
 * To what extent do you feel social media influence negatively your life and biased your decisions?
 * When was the last time you had a real conversation?

Further learning
If you are interested in this topic and would like to know more about it here are some recommendations on the topic:

Watch

 * Sherry,Turkle “Connected, But Alone?”, https://www.ted.com/talks/sherry_turkle_connected_but_alone.. In this TED TALK, American sociologist Sherry Turkle Examines our current relationship with technology and how it is affecting the deepness of our face to face interaction and how it is killing empathy.
 * The social dilemma https://vimeo.com/462049229 This documentary explores the impact that social platforms have over our lives, such as, how we think and what we choose to see. Also, it exposes the harmful content they have: Misinformation, polarization and mental health issues among others.

Listen

 * Speaking of psychology: How social media affects teens’ mental health, https://www.youtube.com/watch?v=ArMcYYZHZTM&list=PLxf85IzktYWK9aVxz9DhebQ5Y8hCIV27y&index=11 Psychologist Jacqueline Nesi from Brown University discusses the negative impact of social media among teenagers, if it should be banned for them and the political restrictions that are being developed to get social media out of their hands.
 * Center For Humane Technology: Are the Kids Alright? https://www.humanetech.com/podcast/26-are-the-kids-alright Social psychologist Jonathan Haidt and his coworker psychologist Jean Twenge reveal alarming statistics regarding self-harm and suicides rate among teens involving and increased social media use and a comparison between Generation Z and Millenials.

Read

 * Youth Toolkit https://www.humanetech.com/youth This guide created by Humantech proposes a safer way to navigate through social media and reduce its impact with self-awareness exercises that can be used by anyone who seeks to learn more.
 * Infographic: The Dark side of social Media https://www.humanetech.com/infographic-dark-side-social-media This infographic created by humantech exposes the harms of navigating through social media, how to identify its source and shows strategies to overcome this situation.


text:
User talk:AshtonTameirao

Hey MarcGarver, can I get back to editing? I am sorry for doing something wrong, I miss editing on Wikibooks and I should take another chance please.

Please unblock me! AshtonTameirao (discuss • contribs) 00:41, 30 August 2023 (UTC)

title: User talk:AshtonTameirao


text:
Cookbook:Panna Cotta (Basic)

Ingredients

 * ¾ tsp powdered gelatin
 * 2 cups (480 ml) heavy cream
 * ½ cup (60 g) white granulated sugar
 * 1½ tsp vanilla essence or 1 small vanilla bean
 * 1 pinch of salt

Procedure

 * 1) Allow the gelatin to bloom in 1 Tbsp of the cream for about 5 minutes.
 * 2) Combine the remaining ingredients in a medium saucepan over medium heat, and stir occasionally until the sugar dissolves and tiny bubbles appear around the edge of your pan.
 * 3) Remove from heat and add the gelatin, stirring until it is thoroughly incorporated.
 * 4) Pour the cream mixture into small ramekins and allow to set in the refrigerator for at least 4 hours, up to overnight.
 * 5) Serve with fresh fruit, balsamic vinegar, port, or a sweet sauce.

title: Cookbook:Panna Cotta (Basic)


text:
Cookbook:Hazelnut and Black Raspberry Panna Cotta

Ingredients

 * 4 cups whole milk
 * 2 cups white granulated sugar
 * 6 sheets gelatin
 * 2 tbsp vanilla extract
 * 1 vanilla bean, seeded
 * 1 oz (28 g) hazelnut liqueur
 * 1 oz (28 g) black raspberry liqueur

Procedure

 * 1) Heat all ingredients in a saucepan over medium-low heat until gelatin is melted.
 * 2) Pour into ramekins, and chill until set (usually 6 hours, but they are better left overnight).
 * 3) Serve with fresh berries or a berry sauce.


text: Category:Panna cotta recipes

title: Category:Panna cotta recipes


text: Category:Cream recipes

title: Category:Cream recipes


text: User talk:Healthfirstcenter12

title: User talk:Healthfirstcenter12


text:
Chess Opening Theory/1. c4/1...e5/2. Nc3/2...Nf6/3. Nf3/3...e4/4. Ng5/4...c6

Adhiban Gambit
The Adhiban Gambit rose to prominence in 2021, when Neural Network Leela Zero inspired strong GM Adhiban Baskaran to play this for the first time in the Meltwater Rapid Indian Qualifier, which is the qualifier for the prestigious Meltwater rapid. This has overtaken the popular Bellon Gambit (4..b5!?) which was refuted in the pre-computer era. As this opening is relatively new, theory is still developing.


text: Category:Book:Etiquette Guide

title: Category:Book:Etiquette Guide


text:
User:Ricks9001

This is a pseudonym I use to protect my privacy.

title: User:Ricks9001


text:
User talk:136.54.1.198

Xania   talk 03:09, 3 August 2023 (UTC)

title: User talk:136.54.1.198


text:
User talk:Ricks9001

Hello.

I have used the email address associated with this user for several years to protect my privacy.

title: User talk:Ricks9001


text:
Timeless Theorems of Mathematics/Intermediate Value Theorem

The Intermediate Value Theorem is a fundamental theorem in calculus. The theorem states that if a function, $$f(x)$$ is continuous on a closed interval $$[a,b],$$ then for any value, $$y$$ defined between $$(f(a)$$ and $$f(b),$$ there exists at least one value $$c \in (a,b)$$ such that $$f(c) = y$$.

Proof
Statement: If a function, $$f$$ is continuous on $$[a, b],$$ then for every $$y$$ between $$f(a)$$ and $$f(b),$$ there exists at least one value $$c \in (a, b)$$ such that $$f(c) = y$$

Proof: Assume that $$f(x)$$ is a continuous function on $$[a, b]$$ and $$f(a) 0$$

Since $$g(x)$$ is continuous and $$g(a)$$ is defined below the $$x$$-axis while $$g(b)$$ is defined above the $$x$$-axis, there must exist at least one point $$c$$ in the interval $$[a,b]$$ where $$g(c)=0$$.

Therefore, at the point $$c$$, $$g(c) = f(c) - y = 0 \implies f(c) = y$$

∴ There exists at least one point $$c$$ in the interval $$[a,b]$$ such that $$f(c)=y.$$ [Proved]


text:
Timeless Theorems of Mathematics/Product, Quotient, Composition Rules

The product rule, the quotient rule and the composition (or chain) rule are the most fundamental rules or formulas of differential calculus. For the functions $$u(x)$$ and $$v(x)$$, the rules are,
 * 1) Product Rule: $$D_x (uv) = v\cdot D_x u + u\cdot D_x v$$
 * 2) Quotient Rule: $$D_x (\frac{u}{v}) = \frac{1}{v^2} (v \cdot D_x u - u \cdot D_x v)$$
 * 3) Composition Rule: $$D_x v = D_y v\cdot D_x y$$

Product Rule
$$D_x (uv)$$ $$=\lim_ \frac {\Delta (uv)}{\Delta x}$$ $$=\lim_ \frac{u(x+\Delta x)\cdot v(x+\Delta x) - u(x)\cdot v(x)}{\Delta x}$$ $$=\lim_ \frac{u(x+\Delta x)\cdot v(x+\Delta x) - u(x)\cdot v(x+\Delta x) + u(x)\cdot v(x+\Delta x) - u(x)\cdot v(x)}{\Delta x}$$ $$= \lim_ \frac{[u(x+\Delta x)-u(x)]\cdot v(x+\Delta x) + u(x)\cdot [v(x+\Delta x)-v(x)]}{\Delta x}$$ $$=\lim_ \frac{(\Delta u) \cdot v(x+\Delta x) + u(x)\cdot (\Delta v)}{\Delta x}$$ $$= \lim_[\frac {\Delta u}{\Delta x}\cdot v(x + \Delta x) + \frac {\Delta v}{\Delta x}\cdot u(x)]$$ $$= \lim_[D_x u\cdot v(x + \Delta x) + u\cdot D_x v]$$ $$=v\cdot D_x u + u\cdot D_x v$$

$$\therefore D_x (uv) = v\cdot D_x u + u\cdot D_x v$$ [Proved]

Quotient Rule
$$D_x (\frac{u}{v})$$ $$=\lim_ \frac {\Delta (\frac {u}{v})}{\Delta x}$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)}{v(x + \Delta x)} - \frac {u(x)}{v(x)}]$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)\cdot v(x) - u(x)\cdot v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {u(x + \Delta x)\cdot v(x) -v(x)\cdot u(x) + v(x)\cdot u(x) - u(x)\cdot v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {v(x)\cdot (u(x + \Delta x)-u(x)) + u(x)\cdot (v(x) - v(x + \Delta x)}{v(x)\cdot v(x + \Delta x)}]$$ $$=\lim_ \frac {1}{\Delta x}\cdot [\frac {v(x)\cdot (\Delta u) + u(x)\cdot (\Delta v)}{v(x)\cdot v(x + \Delta x)}]$$ $$=\lim_ \frac {\frac {v(x)\cdot (\Delta u)}{\Delta x} + \frac {u(x)\cdot (\Delta v)}{\Delta x}}{v(x)\cdot v(x + \Delta x)}$$ $$=\lim_ \frac$$ $$=\lim_ \frac$$ $$=\frac$$ $$=\frac$$

$$\therefore D_x (\frac{u}{v}) = \frac{v \cdot D_x u - u \cdot D_x v}{v^2}$$

Composition Rule
$$D_xv$$ $$= \lim_{\Delta x\to 0}\frac{\Delta v}{\Delta x}$$ $$= \lim_{\Delta x\to 0}\frac{\Delta v\cdot\Delta y}{\Delta x\cdot\Delta y}$$ $$= \lim_{\Delta x\to 0}\frac{\Delta v}{\Delta y}\cdot\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}$$ $$= \lim_{\Delta y\to 0}\frac{\Delta v}{\Delta y}\cdot\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}$$ $$= D_yv\cdot D_xy$$

$$\therefore D_xv = D_yv\cdot D_xy$$


text:
Timeless Theorems of Mathematics/De Morgan's laws

De Morgan's Law is a fundamental principle in Logic and Set Theory. It establishes a useful relationship between the logical operators 'AND' and 'OR' when negations (NOT) are applied. There are two primary forms of De Morgan's Law, known as De Morgan's First Law and De Morgan's Second Law. These laws state that,
 * 1) The negation of a disjunction is the conjunction of the negations, $$(A \cup B)' = A' \cap B'$$
 * 2) The negation of a conjunction is the disjunction of the negations, $$(A \cap B)' = A' \cup B'$$

Proof
Let $$A$$ and $$B$$ are two sets. De Morgan's First Law states that the complement of the union of $$A$$ and $$B$$ sets is the same as the intersection of the complements of $$A$$ and $$B$$. That means, $$(A \cup B)' = A' \cap B'$$, or $$\overline {(A\cdot B)} = (\overline {A} + \overline {B})$$. And De Morgan's First Law states that the complement of the intersection of $$A$$ and $$B$$ is the same as the union of their complements, $$(A \cap B)' = A' \cup B'$$, or $$\overline {(A+B)} = (\overline {A} \cdot \overline {B})$$.

De Morgan's First Law
Assume $$x \in (A\cup B)'$$. Then $$x \not \in A \cup B$$.

$$\implies x \not \in A$$ AND $$x \not \in B$$

$$\implies x \in A'$$ AND $$x \in B'$$

$$\implies x \in (A'\cap B')$$

$$\therefore (A\cup B)' \subseteq (A'\cap B')$$

Again, Let, $$x \in (A'\cap B')$$. Then, $$x \in A'$$ AND $$x \in B'$$

$$\implies x \not \in A$$ AND $$x \not \in B$$

$$\implies x \not \in A \cup B$$.

$$\implies x \in (A\cup B)'$$

$$\therefore (A' \cap B') \subseteq (A\cup B)'$$

Therefore, $$(A \cap B)' = A' \cup B'$$ [Proved]

De Morgan's Second Law
Assume $$x \in (A\cap B)'$$. Then $$x \not \in A \cap B$$.

$$\implies x \not \in A$$ OR $$x \not \in B$$

$$\implies x \in A'$$ OR $$x \in B'$$

$$\implies x \in (A'\cup B')$$

$$\therefore (A\cap B)' \subseteq (A'\cup B')$$

Again, Let, $$x \in (A'\cup B')$$. Then, $$x \in A'$$ OR $$x \in B'$$

$$\implies x \not \in A$$ OR $$x \not \in B$$

$$\implies x \not \in A \cap B$$.

$$\implies x \in (A\cap B)'$$

$$\therefore (A' \cup B') \subseteq (A\cap B)'$$

Therefore, $$(A \cap B)' = A' \cup B'$$ [Proved]


text: Category:Dough recipes

title: Category:Dough recipes


text:
User:Ajayi Mololuwa

Every one needs to know the real concept of mathematics

title: User:Ajayi Mololuwa


text:
User talk:DenLebovsky

You cannot edit indefinitely due to disruptive editing. You may appeal this action with a reasonable argument by adding the text below.

--Xania   talk 07:46, 4 August 2023 (UTC)

title: User talk:DenLebovsky


text:
User:Germany paul whq2/sandbox

The invasion

Battle illustration

The beginning of the path

Muhammad and his companions went out with a high spirit of faith, loyal to God and certain of victory, rejoicing in the booty that God had promised them on their way back from Hudaybiyah, in the words of the Most High: God promised you many spoils that you would take, so He hastened these for you and sufficed the hand. people on your behalf and that it may be a sign for the believers and guide you to a straight path, and the Prophet did not allow For the hypocrites and the weak in faith who stayed behind in Al-Hudaybiyah to go out with him, and only the people of the tree went out with him, confirming the Almighty’s saying: And to God belong the hosts of the heavens and the earth, and God is Mighty, Wise. Good tidings and a warner to believe in God and His Messenger, to honor Him, to revere Him, and to glorify Him morning and evening. Imam Al-Qurtubi said in the interpretation of this verse: “It means the spoils of Khaybar, because God Almighty promised the people of Hudaybiyah to conquer Khaybar and that it is theirs exclusively, whoever is absent from them and whoever is present.”

A zeal erupted in the hearts of the Muslims to fight their enemy, and it did not weaken the strength of the Muslims by their numbers and strength. When the Jews of Khaybar did not think that the Messenger of God would attack them for their strength, fortifications, weapons and numbers, they used to come out every day ten thousand fighters in rows and then say: Muhammad is attacking us? Heh, heh, heh. Their resolve was not discouraged by the strength of their fortresses, but rather they moved to Khaybar filled with confidence in God and reassurance in His promise of victory, and their pure faith and sincerity motivated them to uphold the word of God, so they set out cheering, raising their voices with that, until the Messenger of God commanded them to be kind to themselves, and he said to them: “You do not pray.” Deaf or absent, you call upon the All-Hearing, near, and He is with you. The army marched on its way to Khaibar, and Amer bin Al-Akwa’, may God be pleased with him, came down to the people, saying:

Oh God, had it not been for You, we would not have been guided, nor believed in us, nor prayed, so forgive, in redemption for You, what has left us, and bestow tranquility upon us, and make our feet firm if we meet us. And with shouting they counted on us

And the army, led by the Prophet, descended in a valley called Al-Rajee’, and it was between the Jews and Bani Ghatafan, and that was to prevent them from supplying the people of Khaybar, and they had heard of the Prophet’s exit to Khaybar, so they gathered their crowds, and went out to support the Jews. The Muslims had disobeyed them, so they returned on their heels, and settled with their families and their wealth, and left between the Messenger and Khaybar, and God sufficed the believers with the burden of fighting them.

And when the Islamic army oversaw Khaybar, the Prophet stood supplicating his Lord and seeking his help, and he said: “Oh God, Lord of the seven heavens and what they shade, Lord of the seven earths and what they contain, Lord of the winds and what they leave, Lord of the devils and what they lead astray, we ask You for the good of this village and the best of its people, and we seek refuge in You from its evil And the evil of its people and the evil of what is in it.”

And the Muslims presented Khaybar, and their morale as the best it could be, highness, sacrifice and sacrifice for the sake of God, and their arrival was at night, “And if the Prophet came to a people at night, he did not tempt them until morning, and when it became, the Jews went out with their scavenger hunts and battlefields, and when they saw him they said: Muhammad and God, Muhammad and Thursday The Prophet said: "Khaybar is ruined, and if we go down to the square of a people, it will be a bad morning for the warners."The Prophet thus achieved the element of surprise for the Jews, when they saw the legions of the Islamic army stationed in front of their city, which caused that disturbance between them, and they retreated to their fortresses, fleeing in terror.

March towards Khaibar

The Muslims besieged the fortresses of Khaybar, ready to fight the Jews, and they took their weapons and prepared their equipment for that, and the Messenger of God wanted to motivate his companions to confront their enemies, and to motivate in their souls sacrifice and love for God and His Messenger. And God and His Messenger love him.” And according to Ibn Ishaq: “He loves God and His Messenger. God opens his hands, not fleeing.” The Prophet explained to his companions two main conditions for obtaining victory, God willing, namely: love of God Almighty and His Messenger, and steadfastness in battle and not turning away in advancing. The Companions aspired to obtain this honor, and every one of them wished for it. Imam Al-Bukhari narrated in his Sahih: “When the morning came to the Messenger of God, all of them hoped that he would be given it, so he said: “Where is Ali bin Abi Talib?”. He said: “Then send for him.” So he was brought and the Messenger of God spat in his eyes and prayed for him, and he was cured.” Then he handed him the banner, and instructed him to call the Jews to Islam before raiding and fighting them, so the call to God is not only in a state of peace, but it may be while the Muslim is in the heart of the battle, so the Muslim calls to God Almighty before he is a fighter, and the call to God is the origin, and guidance is better than killing Muhammad said to him: "Go to your messengers until you descend in their yard, then invite them to Islam, and tell them what is obligatory for them regarding the rights of God in it. By God, if God guides one man through you, it is better for you than having red camels for you." The two groups met, and the Muslims showed courage and sacrifice in the way of God, which amazed the Jews, and made them desperate to defend themselves and their fortresses. And God opened the first forts, Na’em, to His Messenger, then he opened the fortress of Qamos, then made the Prophet open the fortresses of Khaybar one after the other, and whenever he opened a fortress, the Jews fled to the next fort, until he reached the last of their fortresses: the ramparts and the stairs, so he besieged them severely, until If they were certain of doom and saw that the circle was upon them, they asked the Prophet to spare their blood and to let them go, and he did. Then they asked him to keep them on cultivating the land of Khaybar in return for half of what he produces from its fruits, so he gave them that, on the condition that he would take them out whenever he wanted, and the Prophet had possessed all the money: the sheq, the bounty, the battalion and all their forts. Many duels took place during the fighting, in which the Muslims showed great courage and daring, and victory was the ally of the Muslims.

before the fight

The Companions arrived at Khaybar, and the Messenger began to study the situation. From which fortress does Khaybar enter? Khaybar is an area with many forts, and it has the right

title: User:Germany paul whq2/sandbox


text:
Timeless Theorems of Mathematics/Differentiability Implies Continuity

Statement: If a function $$f(x)$$ is differentiable at the point $$x_0,$$ then $$f$$ is continuous at $$x_0.$$

Proof: Assume $$f(x)$$ is differentiable at $$x_0$$. Then, $$D_{x}f(x_0)$$ exists.

The function $$f(x)$$ is continuous at $$x_0$$ when $$\lim_{x \to x_0} f(x) = f(x_0).$$ $$\Rightarrow \lim_{x \to x_0} f(x) - f(x_0) = 0$$ $$\Rightarrow \lim_{x \to x_0} f(x) - \lim_{x \to x_0} f(x_0) = 0$$ $$\Rightarrow \lim_{x \to x_0} [f(x) - f(x_0)] = 0.$$ So, to prove the theorem, it is enough to prove that $$\lim_{x \to x_0} [f(x) - f(x_0)] = 0.$$

$$\lim_{x \to x_0} [f(x) - f(x_0)]$$ $$=\lim_{x \to x_0} [\frac {f(x) - f(x_0)}{x - x_0} (x-x_0)]$$ $$=\lim_{x \to x_0} [\frac {f(x) - f(x_0)}{x - x_0}]\cdot\lim_{x \to x_0} (x-x_0)$$ $$=D_{x}f(x_0)\cdot 0 = 0$$

Therefore, When $$D_{x}f(x_0)$$ exists, $$\lim_{x \to x_0} [f(x) - f(x_0)] = 0$$ is true.

$$\therefore$$ If $$f(x)$$ is differentiable at the point $$x_0,$$ then $$f$$ is continuous at $$x_0.$$ [Proved]


text:
User talk:41.113.17.194

--SHB2000 (discuss • contribs) 10:32, 4 August 2023 (UTC)

title: User talk:41.113.17.194


text:
Timeless Theorems of Mathematics/Bézout's Identity

Bézout's Identity is a theorem of Number Theory and Algebra, which is named after the French mathematician, Étienne Bézout (31 March 1730 – 27 September 1783). The theorem states that the greatest common divisor, $$$$ of the integers, $$a$$ and $$b$$ can be written in the form, $$ax + by = d,$$ where $$x$$ and $$y$$ are integers. Here, $$x$$ and $$y$$ are called Bézout coefficients for $$(a, b)$$.

Computing the pairs, $$(x, y)$$
There are infinite number of pairs of $$(x, y)$$ which satisfies the equation $$ax + by = d,$$. A general formula can be developed to compute pairs as much as you want. To do that, first of all, it's required to calculate one pair of $$(x, y)$$. One simple way to calculate a pair is using the extended Euclidean algorithm.

General Formula for Computing
Once you have one pair $$(x_0, y_0),$$ you can apply the formula: $$(x_k, y_k) = (x_0 - k\frac{b}{d}, y_0 + k\frac{a}{d})$$where $$k\in \Z$$, that means $$k$$ is an integer.

Proof: As $$(x, y)=(x_0, y_0)$$ satisfies the equation $$ax + by = d,$$ then,

$$ax_0 + by_0 = d$$

Or, $$\frac {d(ax_0 + by_0)}{d} = d$$

Or, $$\frac {dax_0 + dby_0}{d} = d$$

Or, $$\frac {dax_0 -kba + kba + dby_0}{d} = d$$

Or, $$ax_0 - ak\frac{b}{d} + by_0 + bk\frac{a}{d} = d$$

Or, $$a(x_0 - k\frac{b}{d}) + b(y_0 + k\frac{a}{d}) = d$$

Or, $$a(x_0 - k\frac{b}{d}) + b(y_0 + k\frac{a}{d}) = ax_k + by_k$$

Therefore, the coefficients of $$a$$ are equal and the coefficients of $$b$$ are also equal, $$(x_k, y_k) = (x_0 - k\frac{b}{d}, y_0 + k\frac{a}{d})$$

''[Note: The formula only works when $$d \neq 0$$. Also, as $$x\in\Z$$, then $$k\in\Z$$.]''

Example: The greatest common divisor of $$a = 8$$ and $$b = 12$$ is $$\gcd(8, 12) = 4.$$ According to the identity, there exists integers $$x$$ and $$y$$, so that $$8x + 12y = 4$$ $$\Rightarrow 2x + 3y = 1$$. If you try to solve the equation, you may soon come up with a pair of solutions like $$(x_0, y_0) = (-1, 1)$$. So, $$(x_k, y_k) = (-(1 + k\frac{b}{d}), 1 + k\frac{a}{d})$$. By using this formula, you may find pairs as much as you want.

Proof
Assume $$S = \{ax+by:x,y\in\Z \text{ and } ax+by>0\}$$ where $$a$$ and $$b$$ are non zero integers. The set is not an empty set as it contains either $$a$$ or $$-a$$ when $$x = \pm 1$$ and $$y=0$$. Since $$S$$ is not an empty set, by the well-ordering principle, the set has a minimum element $$d = as + bt$$.

The Euclidean division for $$\frac {a}{d}$$ may be written as $$a = dq + r$$, where $$q=\left\lfloor {\frac{a}{d}} \right\rfloor$$, $$r = a - d \left\lfloor {\frac{a}{d}} \right\rfloor$$ and $$0 \leq r  0$$ as the remainder is zero and a is a non zero integer. Similarly, $$d$$ is also a divisor of $$b$$. Therefore, $$d$$ is a common divisor of $$a$$ and $$b$$.

Assume $$c$$ as any common divisor of $$a$$ and $$b$$; $$a = cu$$, $$b = cv$$. Again, $$d = as+bt$$ $$= cus + cvt$$ $$= c(us + vt)$$

Thus, $$c$$ is a divisor of d. Since $$d>0$$ $$\implies c\leq d$$. Therefore, any common divisor of $$a$$ and $$b$$ is less than or equals to $$d$$.

$$\therefore d$$ is the same as $$gcd(a, b)$$ and $$d$$ can be expressed as $$ax + by = d$$. [Proved]


text: User:KingCrocMan

title: User:KingCrocMan


text:
User talk:119.56.97.247

I am Faster328

title: User talk:119.56.97.247


text:
Sensory Systems/Owls

Introduction
Owls are nocturnal birds belonging to the order of Strigiformes. They possess extraordinary adaptations that enable them to thrive in low-light conditions. Their large tubular eyes, filled with light-sensitive cells, allow them to navigate almost effortlessly in near-total darkness. Furthermore, their remarkable auditory system, which is highly intricate and distinct, enables them to function effectively even in complete darkness. This exceptional auditory system incorporates specialized feathers that form a parabolic facial disc, adjustable operculum or flaps, and, in certain species, asymmetrical ear openings (where one ear is located higher than the other) which are facing forward and slightly to the side. Through this unique system, owls' brains can create an auditory representation of their surroundings, aiding them in locating their prey.

Visual System
Among the various features possessed by owls, their eyes stand out as particularly remarkable. They are large and positioned towards the front, and account for a significant proportion of the owl's body weight, ranging from one to five percent depending on the species. The forward-facing nature of their eyes, which contributes to their "wise" appearance, also grants them a broad range of binocular vision. This means that owls can perceive objects in three dimensions, encompassing height, width, and depth, much like humans. Owls have a field of view spanning approximately 110 degrees, with around 70 degrees comprising binocular vision. In contrast, humans have a field of view that encompasses 180 degrees, with approximately 140 degrees involving binocular vision. To compensate for its limited field of view, the owl possesses the remarkable ability to rotate its head up to 270 degrees both left and right from its forward-facing position. Furthermore, it can even rotate its head almost completely upside down. The size of owl eyes is optimized to enhance their effectiveness, particularly in situations with limited lighting. Interestingly, the eyes of an owl are not spherical, but elongated tubes. These tubular eyes are supported and fixed in position by bony structures known as sclerotic rings within the skull. Consequently, owls lack the ability to "roll" or rotate their eyes, limiting their gaze to a fixed forward direction.

The owl's eyes possess an exceptionally large cornea, which is the transparent outer layer of the eye, as well as a sizable pupil located at the center. The larger cornea facilitates a larger pupil size, leading to an increased number of photons reaching the retina, which is the light-sensitive tissue responsible for image formation. As a result, visual sensitivity is enhanced.

The iris, a colored membrane situated between the cornea and lens, regulates the size of the pupil. When the pupil expands, it allows a greater amount of light to pass through the lens and reach the extensive retina. Within the retina, light-sensitive cells serve as receptors and contribute to the formation of images. Similar to humans, owls possess two distinct types of light-sensitive cells in their retinas: rods, responsible for detecting light and movement, and cones, which enable color differentiation. However, there is a significant difference in the ratio of rods to cones between owls and humans. While humans typically have approximately 20 rods for every cone, owls exhibit a ratio closer to 30 to one. This discrepancy grants owls exceptional capabilities in detecting movement, even in low-light conditions.

Beyond the rod-rich retina, there exists an additional layer known as the tapetum lucidum behind an owl's eye. This specialized layer captures any light that penetrates through the retina and reflects it back to the highly sensitive rods. When combined, all of these adaptations result in remarkable low-light sensitivity, with some owl eyes being potentially up to 100 times more adept than ours.

Although it is commonly believed that owls are blind in bright light due to their exceptional night vision, this assumption is false. Owls possess pupils with a wide range of adjustment, allowing them to regulate the amount of light that reaches their retinas effectively. In fact, certain owl species can even perceive bright light better than humans.

To safeguard their eyes, owls possess three eyelids. They have the typical upper and lower eyelids, with the upper eyelid closing during blinking and the lower eyelid rising when the owl is asleep. Additionally, owls possess a third eyelid known as a nictitating membrane. This thin layer of tissue diagonally crosses the eye from the inside to the outside, serving to clean and protect the eye's surface.

Due to the significant size of their "tubular" eyes and being firmly secured by a bony sclerotic ring, owls possess very limited mobility of their eyes. However, in order to overcome this constraint and compensate for their relatively narrow field of view, owls have developed the remarkable ability to smoothly and rapidly swivel their heads laterally by 270 degrees and vertically by 90 degrees.

Auditory system
They possess a well-developed auditory system due to their nocturnal nature, with their ears situated on the sides of their head, behind their eyes, and concealed by the feathers of the facial disc. The apparent "Ear Tufts" found in certain species are actually decorative feathers and not functional ears.

An owl's range of audible sounds is comparable to that of humans. However, their hearing becomes remarkably acute, especially at frequencies of 5 kHz and above. This heightened sensitivity allows them to perceive the slightest movements of their prey within leaves or undergrowth, maximizing their hunting accuracy within the frequency range of 4 to 8 kHz. The sound localization in Owl’s involves utilizing two binaural cues: the interaural time difference (ITD) and the interaural intensity difference (IID). The utilization of continuous time differences imposes limitations on the range of sounds that can be accurately located, and it also places exceptional demands on the auditory systems of owls. One issue is the presence of phase ambiguity, as ongoing time differences are determined by the phase delay between the two ears. When owls rely solely on low frequencies for sound localization, the maximum time delay, occurring when the sound source is directly opposite one ear, always results in a phase delay of less than 180°. Consequently, each phase delay corresponds to a distinct interaural time difference (ITD) and a specific spatial position. However, with higher frequencies, phase delays exceeding 180° can occur, making it impossible to determine from the ongoing waveform whether the sound is leading in the left or right ear, or leading in one ear by more than one period of the sound frequency. Therefore, for high-frequency tones, the ITDs generated at multiple spatial positions may yield the same relative phase at both ears.

Additionally, the process of sound localization is based on ongoing time differences, as auditory neurons are unable to encode the phase of extremely high-frequency sounds.

Certain owl species possess ear openings that are placed asymmetrically, meaning one ear is positioned higher than the other. This asymmetry creates distinct auditory directional sensitivity patterns for high frequencies, resulting in differences in sound elevation perception between the two ears. As a result, owls are able to localize sounds in the vertical plane by comparing the intensity and spectral composition of sound captured by each ear. In simpler terms, when an owl hears a noise, it can determine the source of the sound by analyzing the minuscule time difference in which it reaches the left and right ears. This time difference, known as the interaural time difference, can be as short as 10 millionths of a second.

The strategic arrangement of feathers on the barn owl's facial periphery forms a specialized disc that effectively captures and concentrates sound, akin to the external ears of humans. As sound waves traverse the owl's ear canal, they ultimately reach the eardrum, pass through the ossicles, and enter the inner ear, enabling the owl to precisely determine the whereabouts of its prey. Given that owls rely on their remarkable auditory abilities to track and capture prey, comprehending the intricate structures within the barn owl's ear that facilitate sound transmission is of utmost significance.

The basilar membrane of the barn owl consists of two primary components: the vestibular part and the tympanic part. In the vestibular part, the basilar membrane is composed of supporting cells, along with a limited number of border cells located at the lower edge of the membrane. Towards the outermost section of the papilla, the basilar membrane is comparatively thin, while it gradually thickens into a fibrous mass as it approaches the innermost section.

In the owl's brain, the signals indicating left, right, up, and down are rapidly integrated to form a cohesive mental image of the spatial location of the sound source. Research conducted on owl brains has unveiled a remarkable complexity in the medulla, the region associated with hearing, surpassing that of other avian species. For instance, it is estimated that a Barn Owl's medulla contains a minimum of 95,000 neurons, three times the number found in a Crow.

Auditory processing pathways in the Owl’s brain
The auditory nerve fibers provide innervation to two main cochlear nuclei within the brainstem: the cochlear nucleus magnocellularis the cochlear nucleus angularis. The neurons in the nucleus magnocellularis exhibit phase-locking but are relatively insensitive to variations in sound pressure. On the other hand, the neurons in the nucleus angularis show poor or no phase-locking but are sensitive to sound pressure fluctuations. These two nuclei serve as the initial points for separate yet parallel pathways to the inferior colliculus. The pathway from the nucleus magnocellularis processes interaural time differences (ITDs), while the pathway from the nucleus angularis processes interaural intensity differences (IIDs).

The first location of binaural convergence in the temporal pathway is the nucleus laminaris, which is comparable to the mammalian medial superior olive. It uses coincidence detection and neuronal delay lines to detect and encode ITDs. Laminaris neurons fire most intensely when phase-locked impulses from both ears coincide at the neuron. As a result, the nucleus laminaris functions as a delay-line coincidence detector, converting the sound's travel time into a map of interaural time delays. The anterior lateral lemniscal nucleus and the core of the inferior colliculus' central nucleus receive projections from the nucleus laminaris' neurons.

The posterior lateral lemniscal nucleus, which is similar to the lateral superior olive in mammals, functions as the site of binaural convergence and processes IIDs in the sound level pathway. The neurons are inhibited when the contralateral ear is stimulated, whereas they are excited when the ipsilateral ear is stimulated. The difference between the strength of the inhibitory and excitatory inputs determines the level of excitation and inhibition, which in turn affects how quickly lemniscal nucleus neurons fire. Thus, these neurons' response reflects how ‘loudly each ear is hearing the sound’.

At the lateral shell of the inferior colliculus's central nucleus, the temporal and sound pressure pathways come together. The lateral shell then sends a signal to the external nucleus, where each neuron has a specific receptive field that only responds to sounds coming from that area of space. Only binaural signals with ITD and IID properties similar to those produced by a sound source in the neuron's receptive field will elicit a response from these neurons. Therefore, these space-specific neurons' receptive fields result from their tuning to particular ITD and IID combinations within a constrained range. The positions of receptive fields in space are thus projected in an isomorphic manner onto the anatomical locations of the neurons by these space-specific neurons, resulting in a map of auditory space.

title: Sensory Systems/Owls


text: User talk:Bestwesternslidell

title: User talk:Bestwesternslidell


text:
User:Allahu Jehovah

Allahu Jehovah also known as the Sarahu Nagarazan.

title: User:Allahu Jehovah


text:
Timeless Theorems of Mathematics/Rolle's Theorem

The Rolle's theorem says that, If a real-valued function $$f$$ is continuous on a closed interval $$[a, b]$$, differentiable on an open interval $$(a, b)$$ and $$f(a) = f(b)$$, then there exists at least a number $$c$$ such that $$D_xf(c) = 0$$. It means that if a function satisfies the three conditions mentioned in the previous sentence, then there is at least a point in the graph of the function, where the slope of the tangent line at the point is $$0$$, or the tangent line is parallel to the $$x$$-axis.


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User talk:ShikiEdit1

This is the first entry of a daily series of hypothetitical communiques to simulate modern neocaptilist 1980 style messaging

title: User talk:ShikiEdit1


text:
User:ShikiEdit1

This is a source template

title: User:ShikiEdit1


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Timeless Theorems of Mathematics/Napoleon's theorem

The Napoleon's theorem states that if equilateral triangles are constructed on the sides of a triangle, either all outward or all inward, the lines connecting the centers of those equilateral triangles themselves form an equilateral triangle. That means, for a triangle $$\Delta ABC$$, if three equilateral triangles are constructed on the sides of the triangle, such as $$\Delta ACM$$, $$\Delta BCX$$ and $$\Delta ABZ$$ either all outward or all inward, the three lines connecting the centers of the three triangles, $$ML$$, $$LN$$ and $$MN$$ construct an equilateral triangle $$LMN$$.

Proof
Let, $$\Delta ABC$$ a triangle. Here, three equilateral triangles are constructed, $$\Delta BCE$$, $$\Delta ABD$$ and $$\Delta ACF$$ and the centroids of the triangles are $$P$$, $$Q$$ and $$R$$ respectively. Here, $$AC=b$$, $$AB=c$$, $$BC=a$$, $$PQ=r$$, $$PR=q$$ and $$QR=p$$. Therefore, the area of the triangle $$\Delta ABC$$, $$T=\frac{1}{2}bc\cdot\sin A$$ $$\Rightarrow bc\cdot\sin A=2T$$

For our proof, we will be working with one equilateral triangle, as three of the triangles are similar (equilateral). A median of $$\Delta ACF$$ is $$AG = (m+k)$$, where $$AR = m$$ and $$RG = k$$. $$ AQ=n $$ and, as $$\Delta ACF$$ is a equilateral triangle, $$\angle AGC=90^\circ$$.

Here, $$m+k=\frac{b\sqrt{3}}{2}$$. As the centroid of a triangle divides a median of the triangle as $$1:2$$ ratio, then $$m=\frac{2}{3}\cdot\frac{b\sqrt{3}}{2}$$ $$=\frac{b}{\sqrt{3}}$$. Similarly, $$n=\frac{c}{\sqrt{3}}$$.

According to the Law of Cosines, $$a^2 = b^2 + c^2 - 2bc\cdot \cos A$$ (for $$\Delta ABC$$) and for $$\Delta AQR$$, $$p^2 = m^2 + n^2 - 2mn\cdot\cos(A+60^\circ)$$

$$= (\frac{b}{\sqrt{3}})^2 + (\frac{c}{\sqrt{3}})^2 - 2\frac{b}{\sqrt{3}}\cdot\frac{c}{\sqrt{3}}\cdot\cos(A+60^\circ)$$

$$= \frac{b^2 + c^2 - 2bc\cdot\cos(A+60^\circ)}{3}$$

$$= \frac{b^2 + c^2 - 2bc(\cos A\cdot cos60^\circ - \sin A\cdot\sin 60^\circ)}{3}$$

$$= \frac{b^2 + c^2 - 2bc(\cos A\cdot \frac{1}{2} - \sin A\cdot\frac{\sqrt{3}}{2})}{3}$$

$$= \frac{b^2 + c^2 - bc(\cos A\cdot - \sin A\cdot\sqrt{3})}{3}$$

$$= \frac{b^2 + c^2 - bc\cos A\cdot - 2T\sqrt{3})}{3}$$

$$= \frac{a^2 + 2bc\cos A - bc\cos A - 2T\sqrt{3})}{3}$$ [According to the law of cosines for $$\Delta ABC$$]

$$= \frac{a^2 + bc\cos A - 2T\sqrt{3})}{3}$$

$$= \frac{a^2 + \frac {b^2 + c^2 - a^2}{2} - 2T\sqrt{3})}{3}$$

$$= \frac{\frac {b^2 + c^2 + a^2 - 4T\sqrt{3}}{2}}{3}$$

Therefore, $$p = \sqrt{\frac{1}{6} (a^2 + b^2 + c^2 - 4T\sqrt{3})}$$

In the same way, we can prove, $$q = \sqrt{\frac{1}{6} (a^2 + b^2 + c^2 - 4T\sqrt{3})}$$ and $$r = \sqrt{\frac{1}{6} (a^2 + b^2 + c^2 - 4T\sqrt{3})}$$. Thus, $$p = q = r$$.

$$\therefore\Delta PQR$$ is an equilateral triangle. [Proved]


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Short guide to printing objects using 3D printers

Three-dimensional printing (3D printing) is the process of printing an object as it was designed with a 3D software, usually requires a previous slicing with a 3D printer software, this facilitate the printing of the object step by step. Many 3D printers use a solid resin filament is used as "ink" a header heat the filament and then printing use it to print the designed objecets.

The 3D printer machines that this book works with are made by Ultimaker company, the models worked with are:

 * Ultimaker Original and Ultimaker Original+,
 * Ultimaker 2, Ultimaker 2 to go, Ultimaker 2 extended, Ultimaker 2 extended +, Ultimaker 2+ Connect,
 * Ultimaker 3 series, Ultimaker 3 extended,
 * Ultimaker S5,
 * Ultimaker S3.

This guide may be useful when printing objects from other manufactures such the models produced by Aleph Objects, Airwold 3D, AIO Robotics, 3D Maker, Sindoh, Voxeljet, Mcor, Materialise NV, Fusion3, FormLabs, EnvisionTEC, Creality, Carbon 3D, among others.

3D printers
3D printers follow instructions on a file that gives it the steps to print segment by segment an object until its completion. They use a solid material, usually a solid resin at room temperature, the filament is loaded into the printer header which heat the resin, liquifying it, the segments are printed and in a semiliquid segments added one at a time to the plate, the segments are the building blocks of the object, they are fused due effects of the gravity force while at the time they solidify when reach room temperature, at the time that the complete object is printed. The company Made In Space is developing 3D printers to be used in microgravity for 3D printing in planets with lower gravity or during interplanetary travel.

3D printing files can be stored in USB drives and then inserted in the printers USB slots. Newer printer versions can also work with files stored in computers or also use Bluetooth, wife and cloud services to access these files.

User interface
3D printers can be used and controlled with a touch screen frontal panel and also with a traditional frontal LCD panel and a dial to select the printer options and the files stored in a USB drive. Other 3D printers can be controlled with a computer connected to them.

They use an internal software to operate, most of the time the user interaction with the printers is to select the file to print and press start to begin the printer.

Newer 3D printer versions recognize what was the last file added to the usb drive, the older version allow to select the file with a dial located in the front of the printer while reading the files also in the screen located in the front of the printer.

The material used to create objects may vary, the current most use material is Polylactic acid (PLA), but other materials are also used. 3D printers liquify and then print the object while is solidified when returned to room temperature.

3D printers most used format is .stl, but they also work with the following formats: OBJ, X3D, 3MF, 3DS, OBJ, AMF, STEP, COLLADA, PLY, FBX, 3MF, IGES, among others.

Materials
One of the most common material used with 3D printers are resins of Polylactic acid (PLA) they come in a spool that is placed in one of the spoon holders of 3D printers to be used as a printer material. 3D printers may also work with the following materials:

 * Polypropylene (PP)
 * Polyvinyl acetate (PVA)
 * Acrylonitrile butadiene styrene (ABS)
 * Polylactic acid (PLA)
 * Tough PLA,[29]
 * Copolyester (CPE)
 * Nylon
 * Polycarbonate (PC)
 * Thermoplastic polyurethane (TPU 95A)
 * Breakaway
 * Thermoplastic elastomers (TPE)
 * Metals,
 * Ceramics.

Luminous PLA also can be used in some 3D printers, these are materials that absorb light energy and glow in the dark while release the absorbed energy. Many Ultimaker printers are not capable to print Luminous PLA.

Other options
Many 3D printer machines allow the use of more than one filament and colors, these options can be selected in Cura, the software will instruct the printer how to use the filaments.

3D printer use
The use of a 3D machine can be summarized to 3 steps, loading a filament, selecting the file to print and wait until the object is completed.

The steps to use a 3D printer are the followed:

Turn on the printer: The on off button is located in the back of the printer.

Choose and load a filament
Select a spool of filament, it can be a resin pla filament of any choose color, or another material. Check that is not the filament is not bended or melted, if it is, cut the bended or melted part.

Put the filament spool in the spool holder in the back of the printer, and insert it into the 3D printer following the printer instructions (sometimes this is pressing a button to allow the filament be introduced into the printer). Put the spool in the holder in a way that the filament is oriented towards the machine filament input feeder.

Load the filament using the frontal menu in the printer, select the chosen resin type. If the filament does not reach the header, select load material again, when it reaches the header will extrude the material.

Wait until the material is printed in an extension of the long of 2 arms extended, and select the option accept. confirm if this is the case or in case of failure, reload the material and confirm.

If the filament is not entirely loaded and does not reach the header and stays in the middle of the hose, it could be due the material is bended, if this is the case, unload it, cut it and restart the process. In some printers the header blinks in blue when is not heating and in red when is heating

Select the file to print
Input the USB drive into the 3D printer, then select the object to print. Wait the minutes or hours that are indicated in the printers frontal screen.

Choose the material used and load it, in case that the material is not detected, select the material and confirm. If necessary take the previous material out and reinsert, then if the material is being extruded, confirm when material is being extruded. If the file is not detected, use a .stl format instead.

Print the file
In the LCD screen choose the Print option, the building plate will be heated for a few minutes, then it will print the object, this time can be a few minutes or a few hours the time will be displayed in the LCD screen.

When the object is completely printed the message will be shown in the LCD screen.

Final steps
Wait two minutes after the object is printed in order to allow the base and object cool down, then Unload the material using an spatula.

Remove the USB flash drive.

Clic the return option, material, unload material, let the filament exit the printer.

Push the button to take the filament out of the printer, ant put the resin in its box.

Turn off the printer.

Cura (free software)
Since 1970 many 3D printer manufacturers designed their own software, but their software was not designed to operate in other 3D printer machines. In 2012 David Braam created the first 3D printing free software, Cura. This software allow the slicing of 3D objects and add the instructions that the 3D printers require in order to print a complete object using small slices or segments that are fused into a complete object. It also allows make modifications in the object and also allows changes of the 3d printing process such are speed, temperature, among others.

The company Ultimaker continued the developing of Cura since 2015, renaming it as Ultimaker Cura.

The main function of 3d printing software is slicing the 3D object, this will allow to create each section of the object at once in an stablished order, the format used is .slt, the files generated can range from 10 MB to 1 GB. .SLT files can be saved into the USB drive. Once the .slt file is generated it can be printed in a 3D printer with just a file saved into a USB drive. The software also indicates the time required to printer a file with the model of printer selected.

Other adjustments
Other adjustments that can be done with 3D printing software are speed, size of the object, temperature, etc. The speed selected can affect the quality and the shape of the object printed.

Objects that do not have a heavy base may require to add a flat base at the base to avoid that the head of the printer move the object while printing it. Users can modify this selecting printing settings option (it is located below the monitor option and above the model window), there choose the "build plate adhesion" and add an adhesion type, these could be: Skirt, Brim and Raft.

When one of the pieces does not touch the surface of the printer place, support can be added, select the option generate support, support placement everywhere, this will increase the printing time a few more minutes, but it will allow the accurate and complete printing of all the segments of the object.

Requirements
Cura software use less than 1 GB of space and requires less than 500 MB RAM, it can be installed in many operative systems, including Linux, Windows and Mac OS.

Creation of 3D objects
Cura can be used to create 3D objects, but other software can also be used, among them can be named Blender (also free software), Autodesk 3Ds Max, Siemens NX, SolidWorks among others. They can generate 3D objects and also 3D texts.

Use of free 3D object designs
Other option for 3D printer users is to print objects designed for other users, they are available in many websites that have a list of 3D objects and the option to download the objects, some designers are benefited by donations given by the user of their objects.


text:
TeX/definition/control word

It is a set of letters that are preceded by an escape character, usually the backslash character. This occurs when macros (commands) are defined. The letters that give the command its name constitute the control word.

title: TeX/definition/control word