This article connects field extensions with groups of automorphisms. It develops fixed fields and the Galois correspondence before applying them to cubic and quartic equations, roots of unity, Kummer extensions, and solvability by radicals.
This article turns group elements into symmetries of sets. Orbits and stabilizers lead to counting formulas, conjugacy and the class equation, permutation representations, and the classification of finite rotation groups.
This article studies how fields grow when elements and roots are adjoined, with particular attention to minimal polynomials, extension degrees, splitting fields, finite fields, Frobenius, and primitive elements.
This article replaces vector spaces by modules and follows the resulting path from presentations and Smith normal form to finitely generated abelian groups and rational canonical form.
This article introduces unital rings and their ideals, then develops polynomial rings, quotients, adjoining constructions, product decompositions, fraction fields, and the prime and maximal ideal tests.
This article develops the basic language of groups, from binary operations and subgroups through cosets, direct products, quotient groups, and the three isomorphism theorems.
Compared with combinational logic circuits, sequential logic circuits have memory and feedback loops, which means that the output not only depends on the current inputs but also on the past history of inputs. In this part, we will discuss the basic building blocks of sequential logic circuits, such as flip-flops and registers, and how to design and analyze more complex sequential circuits like counters and finite state machines.
Sequential logic circuits can be classified into three types based on their behavior: - Bistable device has two stable states and can store one bit of information. It can be implemented using flip-flops, which are the basic building blocks of sequential logic circuits. - Monostable device has one stable state and one unstable state. It can be implemented using a simple RC circuit or a 555 timer. - Astable device has no stable states and continuously oscillates between two states. It can be implemented using a 555 timer or a pair of cross-coupled NAND gates.
In this part, we will discuss the basic applications and combinations of logic gates, which are called combinational logic circuits. These circuits take multiple input signals and produce output signals based on the combination of the inputs, without any memory or feedback loops. How to design and simplify the circuit in convenience of combination and cascading will be the main focus of this part.
This chapter covers basic concepts of logic gates and the art of logic circuit design and simplication. In my own opinion, this part should be learned together with Discrete Mathematics.
Digital Electronics is a fundamental course for electronics and computer engineering. The course, Digital Electronics with course code EST2501, is offered by SAIS at SJTU to grade 1 SEIEE students, accounting for 2 credits. Although either a CS or IE undergraduate at SJTU should take this course, I think you can grasp a better understanding of this course if you have some background in discrete mathematics and computer architecture, especially for contents related to number systems and signed numbers. In the other hand, such contents like flip-flops serve as an important role in memory, which is a main focus in architecture course.
Textbook for this course is Digital Fundamentals by Thomas L. Floyd, which is a widely used textbook for these courses. It is really a detailed textbook with precise illustrations and clear explanations. However, it is also quite lengthy and too detailed - learning with reading every page exactly as it is written is a little inefficient. Therefore, I recommend to read the textbook in a more flexible way: just skimming through examples, too detailed descriptions and explanations, and focus on the main concepts and principles.
However, this course does not contain all chapters of this textbook, and only chapter 1 - 9 are included in the syllabus. These chapters are summarized into four parts, corresponding to the four articles in this series of notes. The first two parts, Digital Fundamentals and Logical Fundamentals, introduce the basic concepts and principles of digital electronics, while the last two parts, Combinational Logic Circuit and Sequential Logic Circuit, focus on the design and analysis of digital circuits.