Vita. / Digitized by Kansas Correctional Industries

In 1967 Strocchi established that the quantisation of the electromagnetic field using a vector potential is impossible within the context of conventional field theory. Although this result is frequently referred to its significance is largely misunderstood. The fact that the electromagnetic field cannot be described in conventional field theory reflects more upon conventional field theory than theories of the electromagnetic field. A reappraisal of electromagnetic field theories should therefore be made. It could well be that features of these theories that have been previously regarded as deficiencies are not really deficiencies at all. This thesis is an account of the radiation gauge, Gupta - Bleuler and Fermi methods of quantising the electromagnetic field from that point of view. The radiation gauge and Gupta - Bleuler methods are well established schemes. Our discussion does not yield any results concerning these methods that cannot be found elsewhere. It does, however, serve to place them in a wider context. The Fermi method is little understood and hence most of this work is concerned with it. Even though the various formulations of field theory are by no means equivalent, they all eventually reproduce traditional field theory. Thus if we only require that the theory be rigorously formulated for such examples as the neutral scalar field it does not matter which formulation we choose. The differences are, however, important for applications to the quantisation methods of the electromagnetic field. The formulations have to be modified and the point at which such modifications must be made and their nature depends on both the general formulation and the quantisation method. The formalism that provides the most suitable framework for a rigorous formulation of the Fermi method turns out to be the C * algebra formulation of Segal. Following Segal, the Weyl algebra of the vector potential is constructed. The Fermi method is then related to a certain representation of the algebra. The representation is specified by a generating functional for a state on the algebra. Usually, dynamical and kinematical transformations are represented by unitarily implementable automorphisms of the algebra. We prove that this is not always true in the representation given by the Fermi method. The Weyl algebra of the physical field is then constructed as a factor algebra. Difficulties with both the Fermi and Gupta - Bleuler methods can he attributed to the need to use a factor algebra. The canonical commutation relations [ x [ subscript ] µ , p [ subscript ] ν ] = - i g [ subscript ] µ [ subscript ] ν are formulated as a Weyl algebra. We study the Schrödinger representation of the algebra and find that the Fermi method is just the generalisation of this representation to an infinite number of degrees of freedom. Further analogies are also possible. We can construct factor algebras from the Weyl algebra. The mechanics of such procedures can he studied without the additional complications of an infinite number of degrees of freedom. The Schrödinger representation of the Fermi method is then constructed. We conclude with a discussion of the results that have been obtained and an indication of ways in which the work might be extended. / Thesis (Ph.D.)--Department of Mathematical Physics, 1974.

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<p>This thesis consists of four papers about the Minkowski measure of asymmetry and the Minkowski (or Banach-Mazur) distance for convex bodies.We relate these two quantities by giving estimates for the Minkowski distance in terms of the Minkowski measure. We also investigate some properties of the Minkowski measure, in particular a stability estimate is given. More specifically, let <i>C</i> and <i>D</i> be n-dimensional convex bodies. Denote by As(<i>C</i>) and As(<i>D</i>) the Minkowski measures of asymmetry of <i>C</i> and <i>D </i>resp. and by <i>d</i>(<i>C,D</i>) the Minkowski distance between <i>C</i> and <i>D</i>.</p><p>In Paper I, by using a linearisation method for affine spaces and affine maps and using a generalisation of a lemma of D.R. Lewis, we proved that <i>d</i>(<i>C</i>,<i>D</i>) < <i>n</i>(As(<i>C</i>) + As(<i>D</i>))/2 for all convex bodies <i>C,D</i>.</p><p>In Paper II, by first proving some general existence theorems for a class of volume-increasing affine maps, we obtain the estimate that under the same conditions as in paper I, <i>d</i>(<i>C,D</i>) < (<i>n</i>-1) min(As(<i>C</i>),As(<i>D</i>)) + <i>n</i>.</p><p>In Paper III we consider the Minkowski measure itself. We determine the Minkowski measures for convex hulls of sets of the form <i>conv</i>(<i>C,p</i>) where <i>C</i> is a convex set with known measure of asymmetry and <i>p</i> is a point outside <i>C</i>.</p><p>In Paper IV, we focus on estimating the deviation of a convex body C from the simplex S if the Minkowski measure of C is close to the maximum value n (known to be attained only for the simplex). We prove that if As(C) > n - ε for 0 < ε < 1/δ where δ = 8(n+1), then d(C,S) < 1 + 8(n+1) ε .</p>

<p>This thesis consists of three papers on topics related to optimization and estimation of solutions of Riccati equations. We are concerned with the initial value problem</p><p><i>f</i>'+<i>f</i>² =<i>r</i>², <i>f</i>(0)=0, (*)</p><p>and we want to optimise</p><p><i>F</i>(<i>T</i>)= ∫₀^T <i>f</i>(<i>t</i>) <i>dt</i></p><p>when <i>r</i> is allowed to vary over the set <i>R</i>(φ ) of all <i>equimeasurable</i> rearrangements of a decreasing function φ and its convex hull <i>CR</i>(φ). </p><p>In the second paper we give a new proof of a lemma of Essén giving lower and upper bounds for the solution to the above equation, when <i>r</i> is increasing. We also generalize the lemma to a more general equation.</p><p>It was proved by Essén that the infimum of <i>F</i>(<i>T</i>) over <i>R</i>(φ) and <i>RC</i>(φ) is attained by the solution <i>f</i> of (*) associated to the increasing rearrangement of an element in <i>R</i>(φ). The supremum of <i>F</i>(<i>T</i>) over <i>RC</i>(φ) is obtained for the solution associated to a decreasing function <i>p</i>, though not necessarily the decreasing rearrangement φ, of an element in <i>R</i>(φ). By changing the perspective we determine the function <i>p </i>that solves the supremum problem.</p>

Modular Multiplication is a time-consuming arithmetic operation because it involves multiplication as well as division. Modular exponentiation can be performed as a sequence of modular multiplications. Speeding the modular multiplication increases the speed of modular exponentiation. Modular exponentiation and modular multiplication are heavily used in current cryptographic systems. Well-known cryptographic algorithms, such as RSA and Diffie-Hellman key exchange, require modular exponentiation operations. Elliptic curve cryptography (ECC) needs modular multiplication. Information security is increasingly becoming very important. Encryption and Decryption are very likely to be in many systems that exchange information to secure, verify, or authenticate data. Many systems, like the Internet, cellular phones, hand-held devices, and E-commerce, involve private and important information exchange and they need cryptography to make it secure. There are three possible solutions to accomplish the cryptographic computation: software, hardware using application-specific integrated circuits (ASICs), and hardware using field-programmable gate arrays (FPGAs). The software solution is the cheapest and most flexible one. But, it is the slowest. The ASIC solution is the fastest. But, it is inflexible, very expensive, and needs long development time. The FPGA solution is flexible, reasonably fast, and needs shorter development time. Montgomery multiplication algorithm is a very smart and efficient algorithm for calculating the modular multiplication. It replaces the division by a shift and modulus-addition (if needed) operations, which are much faster than regular division. The algorithm is also very suitable for a hardware implementation. Many designs have been proposed for fixed precision operands. A word-based algorithm and the scalable Montgomery multiplier based on this algorithm have been proposed later. The scalable multiplier can be configured to meet the design area-time tradeoff. Also, it can work for any operand precision up to the memory capacity. In this thesis, we develop a prototyping environment that can be used to verify the functionality of the scalable Montgomery multiplier on the circuit level. All the software, hardware, and firmware components of this environment will be described. Also, we will discuss how this environment can be used to develop cryptographic applications or test procedures on top of it. We also present two FPGA designs of the processing unit of the scalable Montgomery multiplier. The FPGA design techniques that have been used to optimize these designs are described. The implementation results are analyzed and the designs are compared against each other. The FPGA implementation of the first design is also compared against its ASIC implementation. / Graduation date: 2003

Modular arithmetic operations (i.e., inversion, multiplication and exponentiation) are used in several cryptography applications, such as decipherment operation of RSA algorithm, Diffie-Hellman key exchange algorithm, elliptic curve cryptography, and the Digital Signature Standard including the Elliptic Curve Digital Signature Algorithm. The most important of these arithmetic operations is the modular multiplication operation since it is the core operation in many cryptographic functions. Given the increasing demands on secure communications, cryptographic algorithms will be embedded in almost every application involving exchange of information. Some of theses applications such as smart cards and hand-helds require hardware restricted in area and power resources. Cryptographic applications use a large number of bits in order to be considered secure. While some of these applications use 256-bit precision operands, others use precision values up to 2048 or 4096 such as in some exponentiation-based cryptographic applications. Based on this characteristics, a scalable multiplier that operates on any bit-size of the input values (variable precision) was recently proposed. It is replicated in order to generate long-precision results independently of the data path precision for which it was originally designed. The multiplier presented in this work is based on the Montgomery multiplication algorithm. This thesis work contributes by presenting a modified radix-4 Montgomery multiplication algorithm with new encoding technique for the multiples of the modulus. This work also describes the scalable hardware design and analyzes the synthesis results for a 0.5 ��m CMOS technology. The results are compared with two other proposed scalable Montgomery multiplier designs, namely, the radix-2 design, and the radix-8 design. The comparison is done in terms of area, total computational time and complexity. Since modular exponentiation can be generated by successive multiplication, we include in this thesis an analysis of the boundaries for inputs and outputs. Conditions are identified to allow the use of one multiplication output as the input of another one without adjustments (or reduction). High-radix multipliers exhibit higher complexity of the design. This thesis shows that radix-4 hardware architectures does not add significant complexity to radix-2 design and has a significant performance gain. / Graduation date: 2003

Given a holomorphic self-map of complex projective space of de-gree larger than one, we prove that there exists a finite collection oftotally invariant algebraic sets with the following property: given anypositive closed (1,1)-current of mass 1 with no mass on any element of this family, the sequence of normalized pull-backs of the current converges to the Green current. Under suitable geometric conditions on the collection of totally invariant algebraic sets, we prove a sharper equidistribution result. / <p>QC 20110530</p>

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This thesis consists of four papers about the Minkowski measure of asymmetry and the Minkowski (or Banach-Mazur) distance for convex bodies.We relate these two quantities by giving estimates for the Minkowski distance in terms of the Minkowski measure. We also investigate some properties of the Minkowski measure, in particular a stability estimate is given. More specifically, let C and D be n-dimensional convex bodies. Denote by As(C) and As(D) the Minkowski measures of asymmetry of C and D resp. and by d(C,D) the Minkowski distance between C and D. In Paper I, by using a linearisation method for affine spaces and affine maps and using a generalisation of a lemma of D.R. Lewis, we proved that d(C,D) &lt; n(As(C) + As(D))/2 for all convex bodies C,D. In Paper II, by first proving some general existence theorems for a class of volume-increasing affine maps, we obtain the estimate that under the same conditions as in paper I, d(C,D) &lt; (n-1) min(As(C),As(D)) + n. In Paper III we consider the Minkowski measure itself. We determine the Minkowski measures for convex hulls of sets of the form conv(C,p) where C is a convex set with known measure of asymmetry and p is a point outside C. In Paper IV, we focus on estimating the deviation of a convex body C from the simplex S if the Minkowski measure of C is close to the maximum value n (known to be attained only for the simplex). We prove that if As(C) &gt; n - ε for 0 &lt; ε &lt; 1/δ where δ = 8(n+1), then d(C,S) &lt; 1 + 8(n+1) ε .