Nontrivial Multipliers Research Articles

Transchromatic extensions in motivic bordism

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum M G L MGL and of the Real bordism spectrum M U R MU_{\mathbb R} . These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height n n , the bracket itself will be of chromatic height ( n + 1 ) (n+1) . Using these, we deduce a family of exotic multiplications in the π ( ∗ , ∗ ) M G L \pi _{(\ast ,\ast )}MGL -module structure of the motivic Morava K K -theories, including non-trivial multiplications by 2 2 . These in turn imply the analogous family of exotic multiplications in the π ⋆ M U R \pi _{\star }MU_\mathbb R -module structure on the Real Morava K K -theories.

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

Abstract The Bogomolov multiplier of a group is the unramified Brauer group associated with the quotient variety of a faithful representation of the group. This object is an obstruction for the quotient variety to be stably rational. The purpose of this paper is to study these multipliers associated with nilpotent pro-$p$ groups by transporting them to their associated Lie algebras. Special focus is set on the case of $p$-adic Lie groups of nilpotency class $2$, where we analyse the moduli space. This is then applied to give information on asymptotic behaviour of multipliers of finite images of such groups of exponent $p$. We show that with fixed $n$ and increasing $p$, a positive proportion of these groups of order $p^n$ have trivial multipliers. On the other hand, we show that by fixing $p$ and increasing $n$, log-generic groups of order $p^n$ have non-trivial multipliers. Whence quotient varieties of faithful representations of log-generic $p$-groups are not stably rational, applications in non-commutative Iwasawa theory are developed.

High-Speed Continuous Wavelet Transform Processor for Vital Signal Measurement Using Frequency-Modulated Continuous Wave Radar

This paper proposes a high-speed continuous wavelet transform (CWT) processor to analyze vital signals extracted from a frequency-modulated continuous wave (FMCW) radar sensor. The proposed CWT processor consists of a fast Fourier transform (FFT) module, complex multiplier module, and inverse FFT (IFFT) module. For high-throughput processing, the FFT and IFFT modules are designed with the pipeline FFT architecture of radix-2 single-path delay feedback (R2SDF) and mixed-radix multipath delay commutator (MRMDC) architecture, respectively. In addition, the IFFT module and the complex multiplier module perform a four-channel operation to reduce the processing time from repeated operations. Simultaneously, the MRMDC IFFT module minimizes the circuit area by reducing the number of non-trivial multipliers by using a mixed-radix algorithm. In addition, the proposed CWT processor can support variable lengths of 8, 16, 32, 64, 128, 256, 512, and 1024 to analyze various vital signals. The proposed CWT processor was implemented in a field-programmable gate array (FPGA) device and verified through the measurement of heartbeat and respiration from an FMCW radar sensor. Experimental results showed that the proposed CWT processor can reduce the processing time by 48.4-fold and 40.7-fold compared to MATLAB software with Intel i7 CPU. Moreover, it can be confirmed that the proposed CWT processor can reduce the processing time by 73.3% compared to previous FPGA-based implementations.

Piecewise discretization of monodromy operators of delay equations on adapted meshes

<p style='text-indent:20px;'>Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized system and, in turn, the effectiveness of assessing local stability by approximating the Floquet multipliers. To overcome this problem when computing multipliers by collocation, the discretization grid should include the piecewise adapted mesh of the computed periodic solution. By introducing a piecewise version of existing pseudospectral techniques, we explain why and show experimentally that this choice is essential in presence of either strong mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is unrelated to that of the periodic solution.</p>

RevSCA-2.0: SCA-Based Formal Verification of Nontrivial Multipliers Using Reverse Engineering and Local Vanishing Removal

The formal verification of integer multipliers is one of the important but challenging problems in the verification community. Recently, the methods based on symbolic computer algebra (SCA) have shown very good results in comparison to all other existing proof techniques. However, when it comes to verification of huge and structurally complex multipliers, they completely fail as an explosion happens in the number of monomials. The reason for this explosion is the generation of redundant monomials known as vanishing monomials. This article introduces the SCA-based approach RevSCA-2.0 that combines reverse engineering and local vanishing removal to verify large and nontrivial multipliers. For our approach, we first come up with a theory for the origin of vanishing monomials, i.e., we prove that the gates/nodes where both outputs of half adders (HAs) converge are the origins of vanishing monomials. Then, we propose a dedicated reverse engineering technique to identify atomic blocks including HAs. The identified HAs are the basis for detecting converging cones and locally removing vanishing monomials, which finally results in a vanishing-free global backward rewriting. The efficiency of RevSCA-2.0 is demonstrated using an extensive set of multipliers with up to several million gates.

An extension of compact operators by compact operators with no nontrivial multipliers

We construct a noncommutative, separably represented, type I and approximately finite dimensional C^* -algebra such that its multiplier algebra is equal to its unitization. This algebra is an essential extension of the algebra \mathcal K(\ell_2(\mathfrak{c})) of compact operators on a nonseparable Hilbert space by the algebra \mathcal K(\ell_2) of compact operators on a separable Hilbert space, where \mathfrak{c} denotes the cardinality of continuum. Although both \mathcal K(\ell_2(\mathfrak{c})) and \mathcal K(\ell_2) are stable, our algebra is not. This sheds light on the permanence properties of the stability in the nonseparable setting. Namely, unlike in the separable case, an extension of a nonseparable C^* -algebra by \mathcal K(\ell_2) does not have to be stable. Our construction can be considered as a noncommutative version of Mrówka’s \Psi -space; a space whose one point compactification is equal to its Cech–Stone compactification and is induced by a special uncountable family of almost disjoint subsets of \mathbb{N} .

An input-balanced realization-based structure for efficient implementation of digital IIR filters

Based on an input-balanced state-space realization, this paper derives a novel structure for efficient implementation of digital IIR filters. Such a structure is very similar to a normalized lattice and, for an N th order IIR filter, it possesses 5 N -1 non-trivial multipliers: two multipliers less than in a normalized lattice. Expressions for transfer function sensitivity measures and roundoff noise gain for the proposed structure are derived. Interestingly, it is shown that, when the filter is all-pass, the proposed structure requires 4 N multiplications in order to compute one output sample. Moreover, the corresponding roundoff noise gain is 4 N , much smaller than that of ( N +1) 2 given by the classical minimum roundoff state-space realizations. Results of simulations performed show that the proposed structure outperforms many classical ones in terms of implementation efficiency and robustness against finite-word-length effects.

Area-efficient FFT processor for MIMO-OFDM based SDR systems

In this letter, an area-efficient FFT processor is proposed for MIMO-OFDM based SDR systems. The proposed FFT processor can support variable lengths of 64, 128, 256, 512, 1024, 1536 and 2048. By reducing the required number of non-trivial multipliers with a mixed-radix algorithm, the complexity of the proposed FFT processor is dramatically decreased. The proposed FFT processor was designed in a hardware description language (HDL) and synthesized to gate-level circuits using a 0.13µm CMOS standard cell library. With the proposed architecture, the gate count for the proposed FFT processor is 78.8K and the size of memory is 393.22Kbits, which are reduced by 40.9% and 19.7%, respectively, compared with the 4-channel radix-2 single-path delay feedback (R2SDF) with the 4-channel radix-3 SDF (R3SDF) FFT processor. Also, compared with the 4-channel radix-2 multi-path delay commutator (R2MDC) with the 4-channel R3SDF FFT processor, it is shown that the gate count and memory size are reduced by 33.8% and 18.5%, respectively.

Low complexity pipeline FFT processor for MIMO-OFDM systems

In this paper, we propose a low complexity pipeline FFT processor for MIMO-OFDM systems with four transmitting and four receiving (4 × 4) antennas. The proposed FFT processor is based on multi-channel structure which enables to support multiple data streams efficiently. With mixed-radix algorithm, the number of non-trivial multiplications of the proposed FFT processor are decreased. Implementation results show that the proposed FFT processor reduces the required number of logic gates by 25% over the conventional 4-channel R4MDC FFT processor which has been considered to be the most area-efficient FFT processor for 4 × 4 MIMO-OFDM systems.

Fast Computation of Partial DFT for Comb Spectrum Evaluation

A comb spectrum evaluation problem arises in the channel estimation of pilot symbol-assisted modulation (PSAM) systems and also in the (de)modulation of multichannel communication system based on orthogonal frequency division multiplexing (OFDM). In this letter, we decompose an M-component comb spectrum evaluation with transform length N into 1) a fast Fourier transform evaluation of transform length M of a length M vector, 2) which in turn is obtained from the lengthN input vector (by folding and scaling). This decomposition allows for the application of split-radix and radix-4 algorithms for evaluation of M length transform and efficient computation of the length M vector from the length N input vector, which is classically termed as pruning. When N=2p, the comb spectrum computation requires only N/8+M/3log2 M nontrivial complex multiplications, if N/Mges8. In addition, conditions under which comb spectrum evaluation can be done in O(MlogM) complex multiplies are identified

New efficient FFT algorithm and pipeline implementation results for OFDM/DMT applications

In this paper, we propose a new efficient FFT algorithm for OFDM/DMT applications and present its pipeline implementation results. Since the proposed algorithm is based on the radix-4 butterfly unit, the processing rate can be twice as fast as that based on the radix-23 algorithm. Also, its implementation is more area-efficient than the implementation from conventional radix-4 algorithm due to reduced number of nontrivial multipliers like using the radix-2/sup 3/ algorithm. In order to compare the proposed algorithm with the conventional radix-4 algorithm, the 64-point MDC pipelined FFT processor based on the proposed algorithm was implemented. After the logic synthesis using 0.35 /spl mu/m CMOS technology, the logic gate count for the processor with the proposed algorithm is only about 70% of that for the processor with the conventional radix-4 algorithm. Since the proposed algorithm can achieve higher processing rate and better efficiency than the conventional algorithm, it is very suitable for the OFDM/DMT applications such as the WLAN, DAB/DVB, and ADSL/VDSL systems.

Degenerate Optimal Control Problems with State Constraints

Standard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest, e.g., when the left endpoint of state trajectories is fixed at x0 and x0 lies in the boundary of the state constraint set; in these cases a nonzero, but nevertheless trivial, set of multipliers exists. We give conditions for the existence of nontrivial multipliers. A feature of these conditions is that they allow nonconvex velocity sets and measurably time-dependent data. The proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint, originating in the dynamic programming literature.

On polynomial reduction in the computation of LSP frequencies

This work describes a new version of the decimation-in-degree (DID) transformation used by Wu and Chen (see ibid., vol.5, p.105-15, 1997) as part of their procedure for computing line spectrum pair (LSP) coefficients. This new version eliminates all nontrivial multiplications, requires fewer total operations (additions and multiplications), and is performed in place-eliminating a memory buffer.

Computing partial DFT for comb spectrum evaluation

A comb spectrum evaluation problem arises in the (de)modulation for orthogonal frequency division multiplexing-based (OFDM-based) multichannel communication system. Efficient algorithms for this special type of partial discrete Fourier transform (DFT) computation are studied. For an M-component comb spectrum evaluation with transform length N, it is shown that only O(N+MlogM) multiplications are needed, compared with O(NlogM) multiplications necessary for a narrowband spectrum evaluation. Pruning radix-2 decimation-in-time fast Fourier transform (FFT) requires only (N/4+M/2log/sub 2/M-M) nontrivial complex multiplications. The frequency shift technique has also been applied to allow a modularized mixed-radix structure for the computation of comb spectrum with an initial component not starting from zero frequency point.

Nearring multiplications on groups

In this paper, the problem of nontrivial nearring multiplications on groups is closely investigated with a different approach. By using the notions of 5, 0-acts and the language of model theory, we give a detailed discussion on a large class of groups that possesses nontrivial nearring multiplications. In the last section, we apply our results to the theory of centralizer nearrings of a single endomorphism which is mainly due to the work of C. Maxson and K. Smith [28, 29]

Algorithms meeting the lower bounds on the multiplicative complexity of length-2/sup n/ DFTs and their connection with practical algorithms

In a previous work (see Electron. Lett., vol.20, no.17, p.690, 1984), the author described an algorithm that computes a length-2/sup n/ discrete Fourier transform using 2/sup 2+1/-2n/sup 2/+4n-8 nontrivial (i.e. not=+or-j=+or- square root -1) complex multiplications. In the present work, it is first shown that this algorithm actually provides the attainable lower bound on the number of complex multiplications. A slight modification of the last step of this algorithm is also shown to provide the attainable lower bound on the number of real multiplications. A connection with the split-radix FFT algorithm (SRFFT) is then explained, showing that SRFFT is another variation of these optimal algorithms, where the last step is computed recursively from shorter FFTs in a suboptimal manner. Finally, once the connection between the minimal complexity and SRFFT (which is the best known practical algorithm) is understood, it provides useful information on the possibility of further improvements of the SRFFT.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Recursive Relations for Complexities of Fast Hartley Transform Algorithms

Recursive relations for the number of non-trivial summations and multiplications required for the various fast Hartley transform algorithms are derived. These relations show that Bracewell's decimation-in-time [3] and Meckelburg and Lipka's decimation-in-frequency [5] algorithms are equally fast and slower than the split radix algorithm due to Pei and Wu[1] and Sorensen et al [2].

Existence of a 2n FFT algorithm with a number of multiplications lower than 2n+1

First we give a decomposition of an FFT of length 2n into a number of one-dimensional polynomial products. If these products are computed with minimum multiplication algorithms, we show that the 2n FFT can be computed with less than 2n+1 nontrivial complex multiplications. A variation of this algorithm is also shown to give the same multiplication count as the ‘split-radix’ FFT.

Compactness and Almost Periodicity of Multipliers

AbstractThe question as to the existence of nontrivial compact or weakly compact multipliers between spaces of functions on groups has been investigated for several years. Until now, however, no general method which is applicable to a large class of function spaces seems to be knownIn this paper we prove that the existence of nontrivial compact multipliers between Banach function spaces on which a group acts is related to the existence of nonzero almost periodic functions.

On the periods of abelian integrals and a formula of Chowla and Selberg

Given an imaginary quadratic field k of discriminant d , let E be an elliptic curve defined over Q, the algebraic closure of Q in C, which admits complex multiplication by some order in k. Let ~ be a non-zero differential of the first kind on E, defined over Q. We may consider ~o E as a holomorphic 1-form whose rational period lattice has dimension 1 over k; it is therefore reasonable to ask for the "transcendental factor" of this lattice. Since all curves with non-trivial multipliers in k are isogenous over Q, this factor will depend, up to an algebraic number, only on the field of multiplication. This problem was solved by Chowla and Selberg in 1949 [C-S]. Let