Inand , two types of cyclic codes, namely, BCH and RS codes, are constructed based on finite fields, and they not only have distinctive algebraic structures but are also powerful in correcting errors. In this chapter, we show that cyclic codes can be constructed based on finite geometries.

showed that cyclic codes can be constructed based on the lines of two classes of finite geometries, namely, Euclidean and projective geometries. It was shown that these cyclic finite-geometry (FG) codes can be decoded with the simple one-step majority-logic decoding (OSMLD) based on the orthogonal structure of their parity-check matrices.

In the previous five chapters, we have been primarily concerned with codes and coding techniques for channels on which transmission errors occur independently, i.e., random errors. However, there are communication and data-storage systems where errors tend to be localized in nature.

The objective of this chapter is to present some important elements of modern algebra and graphs that are pertinent to the understanding of the fundamental structural properties and constructions of some important classes of classical and modern error-correcting codes. The elements to be covered are groups, finite fields, polynomials, vector spaces, matrices, and graphs.

For many, the 2020 presidential election is most linked to the global COVID-19 pandemic, and indeed, this shaped every aspect of the election year, including the emphasis on health and health care policy issues, how campaigning occurred, and where, when and how voters cast their ballots. In reality, we navigated the 2020 election year amid twin pandemics in that the COVID-19 pandemic collided with the nation’s racial reckoning in response to the police killings of George Floyd by Minneapolis police officer Derek Chauvin and the police killing of Breonna Taylor during a raid of her Louisville, Kentucky home as she was in bed sleeping. Both the public health crisis and the protests that ensued from more police killings of unarmed African Americans created the unprecedented conditions of “pandemic politics” that centered race from multiple vantage points. Navigating pandemic politics prompted obvious changes to the methods many Americans used to cast their ballots, and unexpectedly created political opportunities that elevated African American women’s political leadership on the national stage as elected leaders and political activists.

Recall our discussion on internal fluid forces in . Here and in the next two sections we consider the explicit form of the shear stresses and in particular the deviatoric stress matrix. This is necessary if we want to consider/model any real fluid, i.e. non-ideal fluid. We explain shear stresses as follows – see , p. 31).

The well-posedness of smooth solutions to the three-dimensional incompressible Navier–Stokes equations globally in time is a major open mathematical problem. Our goal in this chapter is to provide a succinct though comprehensive introduction to the main well-posedness results that are known. In the three-dimensional case, we indicate how smooth solutions may develop singularities in finite time. At the same time we establish classical regularity assumptions/conditions that guarantee well-posedness globally in time, i.e. if we could prove three-dimensional incompressible Navier–Stokes solutions satisfied those assumptions/conditions, then we would establish global regularity.

Chapter 7 deals with fully-depleted SOI and double-gate MOSFETs. A general, asymmetric double-gate model is applied to long channel SOI MOSFETs. For symmetric double-gate MOSFETs – the generic form of FinFETs, an analytic potential model is described that covers all regions of operation continuously. The scale length model first introduced in Chapter 6 for bulk MOSFETs is modified for short-channel DG MOSFETs. Nanowire MOSFET models, both long and short channel, are also discussed. The last section examines the scaling limits of DG and nanowire MOSFETs based on quantum mechanical considerations.

Herein we focus on two-dimensional irrotational flows which have a rich structure in the sense that they are intimately connected to complex variable theory and analysis.

This chapter begins by reviewing MOSFET scaling – the guiding principle for achieving density, speed, and power improvements in VLSI evolution. The implications of the non-scaling factors, specifically, thermal voltage and silicon bandgap, on the path of CMOS evolution are discussed. The rest of the chapter deals with the key factors that govern the switching performance and power dissipation of basic digital CMOS circuits. After a brief description of static CMOS logic gates, their layout, and noise margin, Section 8.3 considers the parasitic resistances and capacitances that may adversely affect the delay of a CMOS circuit. These include source and drain series resistance, junction capacitance, overlap capacitance, gate resistance, and interconnect capacitance and resistance. In Section 8.4, a delay equation is formulated and applied to study the sensitivity of CMOS delay to a variety of device and circuit parameters such as wire loading, device width and length, gate oxide thickness, power-supply voltage, threshold voltage, parasitic components, and substrate sensitivity in stacked circuits. The last section addresses the performance factors of MOSFETs in RF circuits, in particular, the unity-current-gain frequency and unity-power-gain frequency.