535 research outputs found

    The MacWilliams Identity for Krawtchouk Association Schemes

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    The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of its dual is the MacWilliams Identity, first developed for the Hamming association scheme. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via eigenvalues of the association scheme. The functional transformation form can, in particular, be used to derive important moment identities for the weight distribution of codes. In this thesis, we focus initially on extending the functional transformation to codes based on skew-symmetric and Hermitian matrices. A generalised b-algebra and new fundamental homogeneous polynomials are then identified and proven to generate the eigenvalues of a specific subclass of association schemes, Krawtchouk association schemes. Based on the new set of MacWilliams Identities as a functional transform, we derive several moments of the weight distribution for all of these codes

    The spectrum of the Neumann Poincaré operator on bow-tie curves

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    The study of the spectral properties of those operators which output potential functions has historically been of great value, particularly in the resolution of certain problems in potential theory and the mathematical study of gravitational and electromagnetic fields. The initial inspiration for this work was to build on the existing body of results that describe the spectrum of the Neumann Poincar´e operator on different surfaces, particularly those with corners and edges. The particular spectral properties of this integral operator align nicely with a number of scenarios discussed in physics, most notably in the study of plasmonics, where there is a noted coincidence between the elements of the spectrum of the Neumann Poincar´e operator when acting on specific function spaces and the phenomena of plasmon resonances. This work is a study of the spectral properties of the Neumann Poincar´e operator when considered over sets bounded by bow-tie curves, formed of two tear drop shapes or ’wings’, each with a corner that coincides, a scenario that distinguishes itself from prior studies in that the surface being acted on is neither completely smooth nor can it be characterized as a Lipschitz domain in the region of the curve’s singular point

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

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    Let XX be a smooth, separated, geometrically connected scheme defined over a number field KK and {ρλ}λ\{\rho_\lambda\}_\lambda a system of n-dimensional semisimple λ\lambda-adic representations of the \'etale fundamental group of XX such that for each closed point xx of XX, the specialization {ρλ,x}λ\{\rho_{\lambda,x}\}_\lambda is a compatible system of Galois representations under mild local conditions. For almost all λ\lambda, we prove that any type A irreducible subrepresentation of ρλQˉ\rho_\lambda\otimes \bar{\mathbb{Q}}_\ell is residually irreducible. When KK is totally real or CM, n6n\leq 6, and {ρλ}λ\{\rho_\lambda\}_\lambda is the compatible system of Galois representations of KK attached to a regular algebraic, polarized, cuspidal automorphic representation of GLn(AK)\mathrm{GL}_n(\mathbb{A}_K), for almost all λ\lambda we prove that ρλQˉ\rho_\lambda\otimes\bar{\mathbb{Q}}_\ell is (i) irreducible and (ii) residually irreducible if in addition K=QK=\mathbb{Q}.Comment: Revised version. To appear in JLM

    The Shimura lift and congruences for modular forms with the eta multiplier

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    The Shimura correspondence is a fundamental tool in the study of half-integral weight modular forms. In this paper, we prove a Shimura-type correspondence for spaces of half-integral weight cusp forms which transform with a power of the Dedekind eta multiplier twisted by a Dirichlet character. We prove that the lift of a cusp form of weight λ+1/2\lambda+1/2 and level NN has weight 2λ2\lambda and level 6N6N, and is new at the primes 22 and 33 with specified Atkin-Lehner eigenvalues. This precise information leads to arithmetic applications. For a wide family of spaces of half-integral weight modular forms we prove the existence of infinitely many primes \ell which give rise to quadratic congruences modulo arbitrary powers of \ell.Comment: 44 page

    Easily decoded error correcting codes

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    This thesis is concerned with the decoding aspect of linear block error-correcting codes. When, as in most practical situations, the decoder cost is limited an optimum code may be inferior in performance to a longer sub-optimum code' of the same rate. This consideration is a central theme of the thesis. The best methods available for decoding short optimum codes and long B.C.H. codes are discussed, in some cases new decoding algorithms for the codes are introduced. Hashim's "Nested" codes are then analysed. The method of nesting codes which was given by Hashim is shown to be optimum - but it is seen that the codes are less easily decoded than was previously thought. "Conjoined" codes are introduced. It is shown how two codes with identical numbers of information bits may be "conjoined" to give a code with length and minimum distance equal to the sum of the respective parameters of the constituent codes but with the same number of information bits. A very simple decoding algorithm is given for the codes whereby each constituent codeword is decoded and then a decision is made as to the correct decoding. A technique is given for adding more codewords to conjoined codes without unduly increasing the decoder complexity. Lastly, "Array" codes are described. They are formed by making parity checks over carefully chosen patterns of information bits arranged in a two-dimensional array. Various methods are given for choosing suitable patterns. Some of the resulting codes are self-orthogonal and certain of these have parameters close to the optimum for such codes. A method is given for adding more codewords to array codes, derived from a process of augmentation known for product codes

    Quantum modularity for a closed hyperbolic 3-manifold

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    This paper proves quantum modularity of both functions from Q\mathbb{Q} and qq-series associated to the closed manifold obtained by 1/2-1/2 surgery on the figure-eight knot, 41(1,2)4_1(-1,2). In a sense, this is a companion to work of Garoufalidis-Zagier where similar statements were studied in detail for some simple knots. It is shown that quantum modularity for closed manifolds provides a unification of Chen-Yang's volume conjecture with Witten's asymptotic expansion conjecture. Additionally we show that 41(1,2)4_1(-1,2) is a counter-example to previous conjectures relating the Witten-Reshetikhin-Turaev invariant and the Z^(q)\widehat{Z}(q) series. This could be reformulated in terms of a "strange identity", which gives a volume conjecture for the Z^\widehat{Z} invariant. Using factorisation of state integrals, we give conjectural but precise qq-hypergeometric formulae for generating series of Stokes constants of this manifold. We find that the generating series of Stokes constants is related to the 3d index of Dimofte-Gaiotto-Gukov of 41(1,2)4_1(-1,2) proposed by Gang-Yonekura. This extends the equivalent conjecture of Garoufalidis-Gu-Mari\~no for knots to closed manifolds. This work appeared in a similar form in the author's thesis.Comment: 73 pages, 6 figures, (v2 added a conjectural link between the 3d index and generating series of Stokes constants

    50 Years of quantum chromodynamics – Introduction and Review

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    Faster Complete Formulas for the GLS254 Binary Curve

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    GLS254 is an elliptic curve defined over a finite field of characteristic 2; it contains a 253-bit prime order subgroup, and supports an endomorphism that can be efficiently computed and helps speed up some typical operations such as multiplication of a curve element by a scalar. That curve offers on x86 and ARMv8 platforms the best known performance for elliptic curves at the 128-bit security level. In this paper we present a number of new results related to GLS254: - We describe new efficient and complete point doubling formulas (2M+4S) applicable to all ordinary binary curves. - We apply the previously described (x,s) coordinates to GLS254, enhanced with the new doubling formulas. We obtain formulas that are not only fast, but also complete, and thus allow generic constant-time usage in arbitrary cryptographic protocols. - Our strictly constant-time implementation multiplies a point by a scalar in 31615 cycles on an x86 Coffee Lake, and 77435 cycles on an ARM Cortex-A55, improving previous records by 13% and 11.7% on these two platforms, respectively. - We take advantage of the completeness of the formulas to define some extra operations, such as canonical encoding with (x, s) compression, constant-time hash-to-curve, and signatures. Our Schnorr signatures have size only 48 bytes, and offer good performance: signature generation in 18374 cycles, and verification in 27376 cycles, on x86; this is about four times faster than the best reported Ed25519 implementations on the same platform. - The very fast implementations leverage the carryless multiplication opcodes offered by the target platforms. We also investigate performance on CPUs that do not offer such an operation, namely a 64-bit RISC-V CPU (SiFive-U74 core) and a 32-bit ARM Cortex-M4 microcontroller. While the achieved performance is substantially poorer, it is not catastrophic; on both platforms, GLS254 signatures are only about 2x to 2.5x slower than Ed25519

    The Fifteenth Marcel Grossmann Meeting

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    The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity
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