Regular Multiplier Research Articles

Multiplier Hopf Monoids

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele's definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved --- in an appropriate, multiplier-valued sense --- which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved --- which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.

Heisenberg double as braided commutative Yetter–Drinfel’d module algebra over Drinfel’d double in multiplier Hopf algebra case

Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.

Integration on algebraic quantum groupoids

In this paper, we develop a theory of integration on algebraic quantum groupoids in the form of regular multiplier Hopf algebroids, and establish the main properties of integrals obtained by Van Daele for algebraic quantum groups before — faithfulness, uniqueness up to scaling, existence of a modular element and existence of a modular automorphism — for algebraic quantum groupoids under reasonable assumptions. The approach to integration developed in this paper forms the basis for the extension of Pontrjagin duality to algebraic quantum groupoids, and for the passage from algebraic quantum groupoids to operator-algebraic completions, which both will be studied in separate papers.

Novel design of array multiplier

In this paper a new array multiplier has been proposed, which has lower power consumption than the regular array multipliers. This technique has been applied on two conventional and leapfrog array multipliers. In the formation of 8×8 multiplier all designs proposed in this paper have been implemented using the HSPICE by the use of 180 nm TSMC technology at a supply voltage 1v. To verify the performance of the proposed structures, structures have been simulated in 130 nm & 65 nm PTM technologies. The simulation results show that applying the return technique in the array structures causes power consumption reduction and consequently PDP reduction. This improvement for 180 nm technology in the conventional array structure is 13.32 % and in the leapfrog array structure is 23.27 %. It should be noted that this technique substantially makes the number of transistors less and as a result area reduction.

Yetter-Drinfeld modules over weak multiplier bialgebras

We continue the study of the representation theory of a regular weak multiplier bialgebra with full comultiplication, started in [4, 2]. Yetter-Drinfeld modules are defined as modules and comodules, with compatibility conditions that are equivalent to a canonical object being (weakly) central in the category of modules, and equivalent also to another canonical object being (weakly) central in the category of comodules. Yetter-Drinfeld modules are shown to constitute a monoidal category via the (co)module tensor product over the base (co)algebra. Finite-dimensional Yetter-Drinfeld modules over a regular weak multiplier Hopf algebra with full comultiplication are shown to possess duals in this monoidal category.

Hardware architectures for the H.265/HEVC discrete cosine transform

This study presents a design methodology for the two-dimensional (2D) discrete cosine transform dedicated for H.265/HEVC hardware encoders. The methodology decomposes matrix multiplications for different transform sizes into some steps based on the division of transform units into fixed-size blocks. The modified order of processed blocks allows a significant reduction of the size of the transposition buffer. As a consequence, the resource consumption of the whole 2D-transform architecture is decreased. Separate transform cores assigned to two transform stages increase the throughput more than twice. The decomposition enables different hardware configurations of the architectures. Particularly, the architectures applying the proposed methodology are parametrically specified in VHDL, and configuration parameters enable the tradeoff between resources and the throughput. Furthermore, the interface adaptation to desired horizontal and vertical sizes is possible. The use of regular multipliers allows the support for transforms specified in other video standards. Computational elements embedded in architectures are well-suited to FPGA devices, which improves the area-speed efficiency. Synthesis results show that they can operate at 200 and 400 MHz when implemented in FPGA Arria II and TSMC 90 nm, respectively.

Cosmic ray test of mini-drift thick gas electron multiplier chamber for transition radiation detector

A thick gas electron multiplier (THGEM) chamber with an effective readout area of 10×10cm2 and a 11.3mm ionization gap has been tested along with two regular gas electron multiplier (GEM) chambers in a cosmic ray test system. The thick ionization gap makes the THGEM chamber a mini-drift chamber. This kind mini-drift THGEM chamber is proposed as part of a transition radiation detector (TRD) for identifying electrons at an Electron Ion Collider (EIC) experiment. Through this cosmic ray test, an efficiency larger than 94% and a spatial resolution ~220μm are achieved for the THGEM chamber at −3.65kV. Thanks to its outstanding spatial resolution and thick ionization gap, the THGEM chamber shows excellent track reconstruction capability. The gain uniformity and stability of the THGEM chamber are also presented.

Multiplier Hopf algebroids arising from weak multiplier Hopf algebras

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This relation has been considered already in the original paper on multiplier Hopf algebroids. In this note, we investigate the connection further. First we show that any regular weak multiplier Hopf algebra gives rise, in a natural way, to a regular multiplier Hopf algebroid. Secondly we give a characterization, mainly in terms of the base algebra, for a regular multiplier Hopf algebroid to have an underlying weak multiplier Hopf algebra. We illustrate this result with some examples. In particular, we give examples of multiplier Hopf algebroids that do not arise from a weak multiplier Hopf algebra.

A Principle for Critical Point Under Generalized Regular Constraint and Ill-Posed Lagrange Multipliers Under Nonregular Constraints

In this article, a kind of nonregular constraint and a principle for seeking critical point under the constraint are presented, where no Lagrange multiplier is involved. Let E, F be two Banach spaces, g: E → F a c 1 map defined on an open set U in E, and the constraint S = the preimage g −1(y 0), y 0 ∈ F. A main deference between the nonregular constraint and regular constraint is that g′(x) at any x ∈ S is not surjective. Recently, the critical point theory under the nonregular constraint is a concerned focus in optimization theory. The principle also suits the case of regular constraint. Coordinately, the generalized regular constraint is introduced and the critical point principle on generalized regular constraint is established. Let f: U → ℝ be a nonlinear functional. While the Lagrange multiplier L in classical critical point principle is considered, its expression is given by using generalized inverse g′+(x) of g′(x) as follows: if x ∈ S is a critical point of f| S , then L = f′(x) ○ g′+(x) ∈ F*. Moreover, it is proved that if S is a regular constraint, then the Lagrange multiplier L is unique; otherwise, L is ill-posed. Hence, in case of the nonregular constraint, it is very difficult to solve Euler equations; however, it is often the case in optimization theory. So the principle here seems to be new and applicable. By the way, the following theorem is proved: if A ∈ B(E, F) is double split, then the set of all generalized inverses of A, GI(A) is smooth diffeomorphic to certain Banach space. This is a new and interesting result in generalized inverse analysis.

Regularization in Sobolev Spaces with Fractional Order

We study the minimization of a quadratic functional where the Tichonov regularization term is an H s -norm with a fractional s > 0. Moreover, pointwise bounds for the unknown solution are given. A multilevel approach as an equivalent norm concept is introduced. We show higher regularity of the solution of the variational inequality. This regularity is used to show the existence of regular Lagrange multipliers in function space. The theory is illustrated by two applications: a Dirichlet boundary control problem and a parameter identification problem.

Design of a Novel Optimized MAC Unit using Modified Fault Tolerant Vedic Multiplier

In this study, the design of optimized Multiplication and Accumulation (MAC) unit with modified Vedic multiplier is presented. To design a MAC unit, efficient multiplier is used to increase speed and to reduce area and power. Conventional MAC is designed using without fault tolerant Vedic multiplier. But it consumes more area and power. And also less delay. So MAC unit is changed to design the efficient Vedic multiplier. Conventional MAC unit with regular Vedic multiplier is not working for some of the inputs condition. To overcome this fault, novel Vedic multiplier is proposed and designed using less half adder and Full Adder. Simulation is carried out using Modelsim 6.3c. Synthesis and Implementation is carried out using Xilinx and FPGA Spartan 3.

Comodules over weak multiplier bialgebras

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

Dual-Basis Superserial Multipliers for Secure Applications and Lightweight Cryptographic Architectures

Cryptographic algorithms utilize finite-field arithmetic operations in their computations. Due to the constraints of the nodes which benefit from the security and privacy advantages of these algorithms in sensitive applications, these algorithms need to be lightweight. One of the well-known bases used in sensitive computations is dual basis (DB). In this brief, we present low-complexity superserial architectures for the DB multiplication over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ). To the best of our knowledge, this is the first time that such a multiplier is proposed in the open literature. We have performed complexity analysis for the proposed lightweight architectures, and the results show that the hardware complexity of the proposed superserial multiplier is reduced compared with that of regular serial multipliers. This has been also confirmed through our application-specific integrated circuit hardware- and time-equivalent estimations. The proposed superserial architecture is a step forward toward efficient and lightweight cryptographic algorithms and is suitable for constrained implementations of cryptographic primitives in applications such as smart cards, handheld devices, life-critical wearable and implantable medical devices, and constrained nodes in the blooming notion of Internet of nano-Things.

Making better Maxent models of species distributions: complexity, overfitting and evaluation

AbstractAimModels of species niches and distributions have become invaluable to biogeographers over the past decade, yet several outstanding methodological issues remain. Here we address three critical ones: selecting appropriate evaluation data, detecting overfitting, and tuning program settings to approximate optimal model complexity. We integrate solutions to these issues for Maxent models, using the Caribbean spiny pocket mouse, Heteromys anomalus, as an example.LocationNorth‐western South America.MethodsWe partitioned data into calibration and evaluation datasets via three variations of k‐fold cross‐validation: randomly partitioned, geographically structured and masked geographically structured (which restricts background data to regions corresponding to calibration localities). Then, we carried out tuning experiments by varying the level of regularization, which controls model complexity. Finally, we gauged performance by quantifying discriminatory ability and overfitting, as well as via visual inspections of maps of the predictions in geography.ResultsPerformance varied among data‐partitioning approaches and among regularization multipliers. The randomly partitioned approach inflated estimates of model performance and the geographically structured approach showed high overfitting. In contrast, the masked geographically structured approach allowed selection of high‐performing models based on all criteria. Discriminatory ability showed a slight peak in performance around the default regularization multiplier. However, regularization levels two to four times higher than the default yielded substantially lower overfitting. Visual inspection of maps of model predictions coincided with the quantitative evaluations.Main conclusionsSpecies‐specific tuning of model parameters can improve the performance of Maxent models. Further, accurate estimates of model performance and overfitting depend on using independent evaluation data. These strategies for model evaluation may be useful for other modelling methods as well.

Estimating optimal complexity for ecological niche models: A jackknife approach for species with small sample sizes

Algorithms for producing ecological niche models and species distribution models are widely applied in biogeography and conservation biology. However, in some cases models produced by these algorithms may not represent optimal levels of complexity and, hence, likely either overestimate or underestimate the species’ ecological tolerances. Here, we evaluate a delete-one jackknife approach for tuning model settings to approximate optimal model complexity and enhance predictions for datasets with few (here, <10) occurrence records. We apply this approach to tune two settings that regulate model complexity (feature class and regularization multiplier) in the presence-background modeling program Maxent for two species of spiny pocket mice in Ecuador and southwestern Colombia. For these datasets, we identified an optimal feature class parameter that is more complex than the default. Highly complex features are not typically recommended for use with small sample sizes in Maxent. However, when coupled with higher regularization, complex features (that allow more flexible responses to environmental variables) can obtain models that out-perform those built using default settings (employing less complex feature classes). Although small sample sizes remain a serious limitation to model building, this jackknife optimization approach can be used for species with few localities (<approximately 20–25) to produce models that maximize the utility of the little information available.

On Braided T-Categories over Multiplier Hopf Algebras

Let A be a regular multiplier Hopf algebra. , the subcategory of Aℳ𝒴𝒟A consisting of finite dimensional objects, is a braided T-category with the left and right dualities. We also show that to a multiplier T-coalgebra we can also associate another braided T-category.

A New Conical Regularization for Some Optimization and Optimal Control Problems: Convergence Analysis and Finite Element Discretization

In this article, we study an abstract constrained optimization problem that appears commonly in the optimal control of linear partial differential equations. The main emphasis of the present study is on the case when the ordering cone for the optimization problem has an empty interior. To circumvent this major difficulty, we propose a new conical regularization approach in which the main idea is to replace the ordering cone by a family of dilating cones. We devise a general regularization approach and use it to give a detailed convergence analysis for the conical regularization as well as a related regularization approach. We showed that the conical regularization approach leads to a family of optimization problems that admit regular multipliers. The approach remains valid in the setting of general Hilbert spaces and it does not require any sort of compactness or positivity condition on the operators involved. One of the main advantages of the approach is that it is amenable for numerical computations. We consider four different examples, two of them elliptic control problems with state constraints, and present numerical results that completely support our theoretical results and confirm the numerical feasibility of our approach. The motivation for the conical regularization is to overcome the difficulties associated with the lack of Slater's type constraint qualification, which is a common hurdle in numerous branches of applied mathematics including optimal control, inverse problems, vector optimization, set-valued optimization, sensitivity analysis, variational inequalities, among others.

Faster and Energy-Efficient Signed Multipliers

We demonstrate faster and energy-efficient column compression multiplication with very small area overheads by using a combination of two techniques: partition of the partial products into two parts for independent parallel column compression and acceleration of the final addition using new hybrid adder structures proposed here. Based on the proposed techniques, 8-b, 16-b, 32-b, and 64-b Wallace (W), Dadda (D), and HPM (H) reduction tree based Baugh-Wooley multipliers are developed and compared with the regular W, D, H based Baugh-Wooley multipliers. The performances of the proposed multipliers are analyzed by evaluating the delay, area, and power, with 65 nm process technologies on interconnect and layout using industry standard design and layout tools. The result analysis shows that the 64-bit proposed multipliers are as much as 29%, 27%, and 21% faster than the regular W, D, H based Baugh-Wooley multipliers, respectively, with a maximum of only 2.4% power overhead. Also, the power-delay products (energy consumption) of the proposed 16-b, 32-b, and 64-b multipliers are significantly lower than those of the regular Baugh-Wooley multiplier. Applicability of the proposed techniques to the Booth-Encoded multipliers is also discussed.

Optimal statistical tolerance allocation for reciprocal exponential cost–tolerance function

Statistical tolerancing has been widely employed in industry as it is more practical compared with the worst-case tolerancing in achieving lower manufacturing cost while satisfying design specification. As reciprocal exponential function is one of the commonly employed cost–tolerance models in practice and current approach is difficult to allocate statistical tolerances for such a function, this article investigates a method for optimal statistical tolerance allocation with such a cost–tolerance function. The method is to minimize manufacturing cost subject to constraints on tolerance target and machining capabilities. The optimization problem is solved by applying the algorithmic approach. Particularly, a closed-form expression of the tolerance optimization problem is further derived based on the Lagrange multiplier method with integrating the Lambert W function, a multivalue complex function. In addition, for constrained minimization problems with only equality constraints, the optimal tolerance allocation can be obtained by solving simultaneous equations without the time-consuming computing on differentiating while keeping the solution accurate. An example is illustrated to demonstrate the application of this approach. Through comparisons with the regular Lagrange multiplier method applied to reciprocal exponential cost–tolerance type, the result reveals that tolerances can be allocated much faster using this proposed method.

A Unified Forward/Inverse Transform Architecture for Multi-Standard Video Codec Design

SUMMARY This paper describes a unified VLSI architecture which can be applied to various types of transforms used in MPEG-2/4, H.264, VC-1, AVS and the emerging new video coding standard named HEVC (High Efficiency Video Coding). A novel design named configurable butterfly array (CBA) is also proposed to support both the forward transform and the inverse transform in this unified architecture. Hadamard transform or 4/8-point DCT/IDCT are used in traditional video coding standards while 16/32-point DCT/IDCT are newly introduced in HEVC. The proposed architecture can support all these transform types in a unified architecture. Two levels (architecture level and block level) of hardware sharing are adopted in this design. In the architecture level, the forward transform can share the hardware resource with the inverse transform. In the block level, the hardware for smaller size transform can be recursively reused by larger size transform. The multiplications of 4 or 8-point transform are implemented with Multiplierless MCM (Multiple Constant Multiplication). In order to reduce the hardware overhead, the multiplications of 16/32 point DCT are implemented with ICM (input-muxed constant multipliers) instead of MCM or regular multipliers. The proposed design is 51% more area efficient than previous work. To the author’s knowledge, this is the first published work to support both forward and inverse 4/8/16/32-point