A sphere is not a polyhedron. Unlike polyhedra, which are made up of flat faces, a sphere is a three-dimensional shape with a curved surface. It does not have any flat faces, edges, or vertices like polyhedra do. **

No, a cylinder is not a polyhedron. A polyhedron is a three-dimensional shape with flat faces and straight edges, while a cylinder has curved surfaces. A cylinder has two circular faces and one curved surface, making it a curved shape rather than a polyhedron. **

Yes, all faces of a polyhedron can have different numbers of vertices. In fact, this is a defining characteristic of a polyhedron - each face can have a different number of vertices, as long as the vertices are connected by edges to form a closed surface. This is what distinguishes a polyhedron from other geometric shapes, such as a prism or a pyramid, where all the faces have the same number of vertices. **

To receive dividends, you typically need to own the stocks before the ex-dividend date. This means you need to own the stocks at least one business day before the record date, which is the date set by the company to determine which shareholders are eligible to receive dividends. The exact timing can vary depending on the company and the specific dividend payment schedule. **

One long-term, PvP-based strategy game that is available is "Rise of Kingdoms." In this game, players can build and develop their own civilization, form alliances with other players, and engage in real-time battles against other players. The game offers a variety of strategic options, including managing resources, training troops, and conquering new territories. With its focus on player versus player combat and long-term progression, "Rise of Kingdoms" provides a challenging and immersive strategy gaming experience. **

The time it takes to start making a profit with dividends from stocks can vary depending on the company and the stock's performance. Some companies pay dividends quarterly, while others pay annually. Additionally, the amount of dividends can fluctuate based on the company's financial performance. Generally, it may take several years of consistent dividend payments and stock price appreciation to start making a significant profit from dividends. It's important to consider the long-term potential of the company and its ability to sustain and grow its dividend payments. **

The fundamentals of analysis typically cover the basic concepts and techniques used in mathematical analysis, such as limits, continuity, differentiation, and integration. These concepts are usually presented in a more introductory and accessible manner, aimed at students who are new to the subject. On the other hand, Analysis I is a more advanced and rigorous course that delves deeper into the theory and applications of analysis. It typically covers topics such as sequences and series, convergence, metric spaces, and the theory of differentiation and integration in multiple dimensions. Analysis I is usually taken by students majoring in mathematics or related fields, and it requires a solid foundation in calculus and mathematical reasoning. **

Stocks tend to grow exponentially in the long term. This is because as a company grows and becomes more profitable, its stock price tends to increase at an accelerating rate. This is often due to compounding effects, where the company reinvests its profits to generate even more growth. As a result, the stock price can experience exponential growth over time, rather than growing at a constant, linear rate. **