Mathematics
http://repository.iiitd.edu.in/xmlui/handle/123456789/618
Fri, 23 Jul 2021 17:34:24 GMT2021-07-23T17:34:24ZSpatiotemporal linear stability of viscoelastic free shear flows
http://repository.iiitd.edu.in/xmlui/handle/123456789/871
Spatiotemporal linear stability of viscoelastic free shear flows
Bansal, Diksha; Sircar, Sarthok (advisor)
In this thesis, we have explored the temporal and spatiotemporal stability analyses offree shear, viscoelastic flows in the limit of low to moderate Reynolds number (Re) and Weissenberg number (We). The description of the six chapters are as follows:•The first chapter introduces the fundamental difference in the instability arising from Newtonian versus non Newtonian fluids: Newtonian fluid undergoes instabilities with increasing Re, although a direct transition to instability is also possible via bypass transition. Contrarily, non Newtonian fluids exhibit both inertial and purely elastic instabilities that arise even when the effect of inertia is too small to drive an instability in a Newtonian fluid at the same flow condition. Past research by several groups are highlighted.•The second chapter discusses the Compound Matrix Method (CMM) which is used to numerically integrate the eigenvalue problem using auxiliary variables emerging from the Orr Sommerfeld equation (OSE).•In the third chapter, a description of the different types of instabilities is included. The temporal modes refer to cases where the instability in the complex frequency is determined as a function of real wave number. The convectively unstable modes give rise to wave packets moving away from the source and ultimately leaving the medium in its undisturbed state. Absolutely unstable modes, by contrast, are gradually contaminated everywhere from a point-source disturbance. Evanescent modes (or the direct resonance mode) arise if the two coalescing modes originate from waves propagating in the same direction.• The fourth chapter highlights the stability analyses of antisymmetric, free shear, viscoelastic flows in the dilute regime, obeying the Oldroyd-B constitutive equation. The temporal stability analysis indicates that with increasing We, (a) the entire range of the most unstable mode is shifted toward longer waves, (b) the vorticity structure contours are dilated, and (c) the residual Reynolds stresses are diminished. The spatiotemporal analyses show that the free shear flow of dilute polymeric liquids is either (absolutely/convectively) unstable for all Re or the transition to instability occurs at comparatively low Re.
•In the fifth chapter, we provide a detailed comparison of the temporal and the spatiotemporal linearized analyses of free shear, viscoelastic flows in the limit of low to moderate Reynolds number and Elasticity number obeying four different types of stress-strain constitutive equations: Oldroyd-B(ε=0,a=1), Upper Convected Maxwell(ε=0,a=1,ν=0), Johnson-Segalman (ε=0,a=0.5) and Phan-Thien Tanner(ε=0.5,a=0.5). The temporal stability analysis indicates (a) elastic stabilization at higher values of elasticity number and (b) a non-monotonic instability pattern at low to intermediate values of elasticity number for the JS as well as the PTT model. The spatiotemporal phase diagram divulge the familiar regions of inertial and elastic turbulence, a recently verified region of elastoinertial turbulence and the unfamiliar temporally stable region for intermediate values of Reynolds and Elasticity number.• In the concluding chapter, we highlight the challenges that we have faced and intend to face in future numerical simulations as well as our future problems demandinga full spatiotemporal stability analyses: (a) Rayleigh-Plateau, describing the on set of the detachment of a droplet, (b) Saffman-Taylor, or the formation of patterns in a morphologically unstable interface between two fluids in a porous medium, (c)Faraday instability, or an unstable state of a flat hydrostatic surface due to a critical vibration frequency
Mon, 01 Mar 2021 00:00:00 GMThttp://repository.iiitd.edu.in/xmlui/handle/123456789/8712021-03-01T00:00:00ZMulti-twisted codes over finite fields and their generalizations
http://repository.iiitd.edu.in/xmlui/handle/123456789/870
Multi-twisted codes over finite fields and their generalizations
Chauhan, Varsha; Sharma, Anuradha (advisor)
Nowadays error-correcting codes are widely used in communication systems, returning pictures from deep space, designing registration numbers, and storage of data in memory systems. An important family of error-correcting codes is that of linear codes, which contain many well-known codes such as Hamming codes, Hadamard codes, cyclic codes and quasi-cyclic codes. Recently, Aydin and Halilovic [5] introduced and studied multi-twisted (MT) codes over the finite field Fq, whose block lengths are coprime to q: These codes are generalizations of well-known classes of linear codes, such as constacyclic codes and generalized quasi-cyclic codes, having rich algebraic structures and containing record-breaker codes. In the same work, they obtained subcodes of MT codes with best-known parameters [33, 12,12] over F3, [53, 18, 21] over F5, [23, 7, 13] over F7 and optimal parameters [54, 4, 44] over F7, Apart from this, they proved that the code parameters [53, 18, 21] over F5 and [33; 12; 12] over F3 can not be attained by constacyclic and quasi-cyclic codes, which suggests that this larger class of MT codes is more promising to find codes with better parameters than the current best known linear codes. In this thesis, we _rst investigate algebraic structures of MT codes over Fq, whose block lengths are coprime to q: We also study their dual codes with respect to Euclidean and Hermitian inner products, and derive necessary and sufficient conditions for a MT code to be (i) self-dual, (iii) self-orthogonal and (iii) linear with complementary-dual (LCD). Applying these results, we provide enumeration formulae for all Euclidean and Hermitian self-dual, self-orthogonal and LCD MT codes over Fq: We also derive some su_cient conditions under which a MT code is either Euclidean LCD or Hermitian LCD. We further develop generator theory for
these codes and determine their parity-check polynomials. We also obtain a BCH type bound on their minimum Hamming distances, and express generating sets of Euclidean and Hermitian dual codes of some MT codes in terms of their generating sets. Besides this, we provide a trace description for all MT codes by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure. Besides this, we explicitly determine all non-zero Hamming weights of codewords of several classes of MT codes over Fq: Using these results, we explicitly determine Hamming weight distributions
of several classes of MT codes with a few weights. Among these classes of MT codes with a few weights, we identify two classes of optimal equidistant MT codes that attain the Griesmer as well as Plotkin bounds, and several other classes of MT codes that are useful in constructing secret sharing schemes with nice access structures. We further extend the family of MT codes and study algebraic structures of MT codes over Fq; whose block lengths are arbitrary positive integers, not necessarily coprime to q, We study their dual codes with respect to the Galois inner product and derive necessary and sufficient conditions under which a MT code is (i) Galois
self-dual, (ii) Galois self-orthogonal and (iii) Galois LCD. We also provide a trace description for all MT codes over finite fields by using the generalized discrete Fourier transform (GDFT), which gives rise to a method to construct these codes. We further provide necessary and sufficient conditions under which a Euclidean selfdual MT code over a finite field of even characteristic is a Type II code. We also show that each MT code has a unique normalized generating set. With the help of a normalized generating set, we explicitly determine the dimension and the corresponding generating set of the Galois dual code of each MT code. Besides this, we identify several classes of MT codes over finite fields with a few weights and explicitly determine their Hamming weight distributions. We next study skew analogues of MT codes over finite fields, viz. skew multitwisted (MT) codes, which are linear codes and are generalizations of MT codes. We thoroughly investigate algebraic structures of skew MT codes over finite fields and
their Galois duals. Besides this, we view skew MT codes as direct sums of certain concatenated codes and provide a method to construct these codes. We also develop generator theory for these codes, and obtain two lower bounds on their minimum Hamming distances. Finally, we apply our results to obtain many linear codes with best known and optimal parameters from MT and skew MT codes over finite fields.
Fri, 01 Jan 2021 00:00:00 GMThttp://repository.iiitd.edu.in/xmlui/handle/123456789/8702021-01-01T00:00:00ZConstacyclic codes over finite commutative chain rings
http://repository.iiitd.edu.in/xmlui/handle/123456789/838
Constacyclic codes over finite commutative chain rings
Sidana, Tania; Sharma, Anuradha (advisor)
Constructing codes that are easy to encode and decode, can detect and correct many errors and have a sufficiently large number of codewords is the primary aim of coding theory. Several metrics (e.g. Hamming metric, Lee metric, RT metric, etc.) have been introduced to study error-detecting and error-correcting properties of a code with respect to various communication channels. Among the prevalent metrics in coding theory, the Hamming metric is the most studied metric and it is suitable for orthogonal modulated channels. Singleton [74] derived the upper bound (called the Singleton bound) on the size of an arbitrary block code with respect to the Hamming metric. Linear codes that attain the Singleton bound are called maximum distance separable (MDS) Hamming codes. Later, motivated by the problem to transmit messages over several parallel communication channels with some channels not available for transmission, a non-Hamming metric, called the Rosenbloom-Tsfasman metric (or RT metric), was introduced by Rosenbloom and Tsfasman [70]; they also derived Singleton bound for codes with respect to the RT metric. Linear codes that attain the Singleton bound for the RT metric are called maximum distance separable (MDS) RT codes. Recently, Cassuto and Blaum [12, 13] established a new coding framework for channels whose outputs are overlapping pairs of symbols. Such channels are called symbol-pair read channels and the corresponding metric is called the symbol-pair metric. These channels are more suitable for high density data storage systems in which the spatial resolution of the reader is insufficient to isolate adjacent symbols. Chee et al. [15] derived a Singleton-type bound for codes with respect to the symbol-pair metric and constructed many
maximum distance separable (MDS) symbol-pair codes, i.e., the codes attaining the Singleton-type bound with respect to the symbol-pair metric. Recently, Yaakobi et al. [82] extended the framework of symbol-pair read channels to b-symbol read channels, whose outputs are consecutive sequences of b _ 3 symbols. The corresponding metric is called the b-symbol metric. In a recent work, Ding et al. [28] derived a Singleton-type bound for codes over finite fields with respect to the b-symbol metric. The codes that attain the Singleton-type bound with respect to the b-symbol metric are called MDS b-symbol codes. MDS codes have the highest possible error-detecting and error-correcting capabilities for given code length, code size and alphabet size, hence they are considered optimal codes in that sense. Thus it is of great interest to study and find MDS codes with respect to various metrics. The derivative is a well-known operator of sequences and is useful in investigating the linear complexity of sequences in game theory, communication theory and cryptography (see [6, 14, 33, 83]). Etzion [32] first applied the derivative operator on codewords of linear codes over finite fields, and defined the depth of a codeword in terms of the derivative operator. He showed that there are exactly k distinct non-zero depths attained by non-zero codewords of a k-dimensional linear code, and that any k non-zero codewords with distinct depths form a basis of the code. This shows that the depth distribution is an interesting parameter of linear codes. In this thesis, we study algebraic structures of repeated-root constacyclic codes over finite commutative chain rings and their dual codes. We also explicitly determine Hamming distances, symbol-pair distances, b-symbol distances, RT distances, and RT weight distributions of several classes of repeatedroot constacyclic codes over finite commutative chain rings. Using these results, we identify several
isodual, MDS Hamming, MDS RT, MDS symbol-pair and MDS b-symbol codes within the family of
repeated-root constacyclic codes over finite commutative chain rings. We also discuss a decoding
algorithm for repeated-root constacyclic codes of prime power lengths over finite commutative chain rings with respect to Hamming, symbol-pair and RT metrics. We also study depths of codewords of a class of repeated-root constacyclic codes over finite commutative chain rings. As a consequence, we explicitly determine depth distributions of this particular class of constacyclic codes over finite commutative chain rings. We also introduce two new turn-based roulette games and discuss their winning strategies by applying our results on depths of codewords of repeated-root constacyclic codes over finite commutative chain rings. These results are useful in encoding and decoding these codes and in studying their error-detecting and error-correcting capabilities with respect to various communication channels.
Wed, 01 Jul 2020 00:00:00 GMThttp://repository.iiitd.edu.in/xmlui/handle/123456789/8382020-07-01T00:00:00Z