Modulus Set Research Articles

Approximation on Slices in Balanced Strictly Convex Domains with C^{2} boundary

We show that a maximum modulus set K in a bounded strictly convex balanced domain Omega with C^{2} boundary can have a Lebesgue measure almost full on all slices. Additionally K has a so called slice continuity property.

폐타이어 재생 고무가 카본 블랙이 충전된 천연 고무/부타디엔 고무 복합재의 물리기계적, 열적 특성에 미치는 영향

Combinations of natural rubber (NR), butadiene rubber (BR), and reclaim rubber (RR) were prepared with a view to producing low-cost re-treading materials. It was found that cure time and scorch time decreased with the increase in the reclaim content. Only a small reduction in the properties was observed in almost all systems upon the addition of 30 phr of reclaim rubber. Systems with good abrasion resistance and tensile strength were obtained. High tensile values of 22.8 and 20.1 MPa were able to obtain for the NR-rich system. Tear strength can be enhanced by the addition of reclaim rubber. Elongation at break was higher for all the systems even after adding 30 phr of RR. Modulus and compression set values were increased with RR loading. The thermal behavior and activation energy of NR/BR/RR blends were improved by the addition of RR. The phase morphology of the blends was studied using a scanning electron microscope.

Effect of Modified Silica Fume Using MPTMS for the Enhanced EPDM Foam Insulation.

Silica fume (SF) is a by-product from the production of silicon metal, which has a relatively high silica concentration. The surface modified silica fume (mSF) is treated with (3-mercaptopropyl) trimethoxysilane (MPTMS) as filler in ethylene propylene diene monomer (EPDM) foam. The FTIR spectra of mSF clearly indicated that MPTMS can be successfully bonded to the SF surface. The reinforcing efficiency of mSF-filled EPDM foam insulation indicated that the mechanical properties such as hardness, tensile strength, modulus, and compression set enhanced higher than in case of SF and calcium carbonate. While the cure characteristics such as the maximum torque (MH), the minimum torque (ML) and the differential torque (MH-ML) are increasing in proportion to increasing filler contents, mainly with mSF. For the cure behavior, the mSF-filled EPDM foam insulation showed the fastest cure time (tc90) and scorch time (ts2) due to reduced accelerator adsorption. Whereas, the calcium carbonate-filled EPDM foam insulation increased the cure time (tc90) and scorch time (ts2), therefore, it also prevents compound scorching. The results indicated that the mSF with MPTMS can be used as an alternative filler for EPDM foam insulation.

On a Result of Hayman Concerning the Maximum Modulus Set

The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a “small” set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concerning the structure of this set near the origin, and make an interesting conjecture regarding the most general case. We prove this conjecture for polynomials of degree less than four.

The Maximum Modulus Set of a Polynomial

We study the maximum modulus set, {{mathcal {M}}}(p), of a polynomial p. We are interested in constructing p so that {{mathcal {M}}}(p) has certain exceptional features. Jassim and London gave a cubic polynomial p such that {{mathcal {M}}}(p) has one discontinuity, and Tyler found a quintic polynomial {tilde{p}} such that {{mathcal {M}}}({tilde{p}}) has one singleton component. These are the only results of this type, and we strengthen them considerably. In particular, given a finite sequence a_1, a_2, ldots , a_n of distinct positive real numbers, we construct polynomials p and {tilde{p}} such that {{mathcal {M}}}(p) has discontinuities of modulus a_1, a_2, ldots , a_n, and {{mathcal {M}}}({tilde{p}}) has singleton components at the points a_1, a_2, ldots , a_n. Finally we show that these results are strong, in the sense that it is not possible for a polynomial to have infinitely many discontinuities in its maximum modulus set.

The maximum modulus set of a quasiregular map

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are either the whole plane, or a countable union of analytic curves. We show that in the quasiregular case, by way of contrast, any closed set containing at least one point of each modulus can be attained as the maximum modulus set of a quasiregular map. These examples are all of polynomial type. We also show that, subject to an additional constraint, such sets can even be attained by quasiregular maps of transcendental type.

Thermal aging and chemical resistance evaluation of carbon black filled natural rubber blending: effect of the composition of acrylo nitrile and styrene butadiene rubber

Natural rubber (NR) has excellent tensile and abrasion properties, but its thermal and chemical resistance needs to be improved. The present study investigates the effect of acrylonitrile-butadiene rubber (NBR) and styrene-butadiene rubber (SBR) composition in NR compounding filled carbon black (CB). The effect of chemical and heat resistance was examined by varying the addition of SBR/NBR (phr/phr) as follows 40/0; 35/5; 30/0; 25/15; and 20/20 into 60 phr (part hundred rubber) NR filled by 50 phr CB respectively. Compared with NR composites, SBR/NBR addition improved the hardness and density slightly. While tensile strength, modulus, abrasion resistance, and compression set decreased significantly. The volume, hardness, and physical properties were measured before and after 28 days of immersion in acidic, alkalis, and salty solutions. It was found that the higher loading SBR/NBR enhances chemical resistance. The hydrochloric and sulphuric acid immersion influenced the physical changes more vigorously. The thermal aging evaluation was observed by measuring the hardness and tensile strength at hot air circulation of 70°C for 24; 48 and 72 hours. The higher SBR/NBR loading could minimize the changes in hardness and tensile strength. To sum up, SBR/NBR addition improves the chemical and heat resistance on ternary blends.

Thermal and mechanical behavior of SBR/devulcanized waste tire rubber blends using mechano–chemical and microwave methods

Abstract The engagement of waste tires rubber as source of raw materials for different applications can be a partial solution to the great environmental problems caused by these products. In this study, waste tire rubber was devulcanized using both mechano–chemical and microwave methods. This process was achieved using different concentrations of 2-mercapto benzothiazole disulfide (MBTS) and tetramethylthiuram disulfide (TMTD) as a devulcanizing agent and different microwave devulcanizing times. The optimum content of both MBTS, TMTD and suitable microwave treatment time to make continuous film were noted. The devulcanized waste rubber was then added, at different concentrations, to virgin styrene–butadiene rubber (SBR). The thermal properties and dynamic mechanical behaviors were investigated for all blends. The thermal analysis proved that natural and styrene butadiene rubber are the main two constituents of the waste tire rubber utilized in this study. The mechanical behavior of the SBR blends remarkably improved by using 20 phr waste rubber (WR) devulcanized by 2 phr MBTS and by exposure for 2.2 min to microwaves. Storage modulus, tearing strength and tension set behaviors notably improved for all SBR/WR blends by irradiating with gamma ionizing radiation with a dose of 100 kGy and further improvement could be attained by increasing the dose up to 200 kGy.

Effect of zinc oxide level as activator on the mechanical properties of natural rubber composite

The effect of Zinc Oxide level as an activator on the Mechanical properties of Natural Rubber Composite was studied. It has been discovered that technicians usually add additional Zinc oxide different from the recommended dosage used in the formulation receipt during the process of compounding and also recovery of scorched compounds. One hundred parts per hundred rubber (100phr) of Natural rubber was compounded with Zinc Oxide levels of, 0, 2, 3, 4, 5, 6, 7, 8, 10phr respectively. The effect of Zinc Oxide level on tensile strength, elongation at break, modulus, hardness, abrasion resistance and compression set were investigated. The result obtained showed that as the Zinc Oxide level increased, the tensile strength, elongation at break, modulus, hardness and compression set attained their maximum values at 5.0phr, after which there was a gradual decrease of same properties as a result of reversion. For the purpose of hardness and enhanced tensile properties, a known quantity of 5.0phr is recommended. Keywords : Activator, Composite, Mechanical properties, Natural Rubber, Zinc Oxide.

Mechanical properties of doubly crosslinked gels

We fabricated the doubly crosslinked gel, with internal stress by stretching the gel during the gelation reaction using tetra-PEG gels, which are formed by the AB type crosslink-coupling of two mutually reactive tetra-arm polyethylene glycol (PEG). In a doubly crosslinked gel, two networks having different reference states coexist with and balance each other, resulting in enhanced Young’s modulus and residual strain. We investigated these properties by tuning the time for the imposition of the stretching of the gels and the stretching ratio. The residual strain increased with increase in the time for the imposition of stretching and the stretching ratio. On the other hand, the Young’s modulus had a maximum value at a certain time for the imposition of the stretching of the gels and increased with increasing stretching ratio. These results were well explained by the balance between the strain energies of the first and second networks under the framework of the classical two-network theory, where the total strain energy is described as a simple sum of the energies of the first and second networks. We fabricated the doubly crosslinked gel by stretching the gel during the gelation using tetra-PEG gels. In this gel, two networks having different reference states coexist with and balance each other, influencing Young’s modulus (E) and permanent set (λs). We tuned the time for the stretching imposition on the gels, which influences the 1st network fraction (g1/g0). λs increased with increasing the 1st network fraction, while E had a maximum value against g1/g0, which were explained by the balance between the strain energies of the first and second networks.

Polybutadiene rubber/exfoliated graphite nanocomposites: Impact of filler on the thermal, mechanical and permeability characteristics

In this report, we demonstrate that both mechanical behaviour and thermal stability of polybutadiene rubber (PBR) nanocomposites could be improved by incorporating exfoliated graphite (EG) using melt mixing method. Incorporation of the filler EG in the polybutadiene (PBD) matrix is confirmed by infrared spectra and X-ray diffraction analysis. The structural characteristics, mechanical properties and thermal properties of these newly modified PBR nanocomposites were systematically analysed and studied. Thermal properties of the nanocomposites were studied using thermogravimetric analysis under nitrogen atmosphere. Thermogravimetric studies showed that PBR8 is having higher thermal stability than that of the PBR gum sample. Mechanical properties like tensile strength, Young’s modulus, tear strength, hardness and compression set of the nanocomposites were studied. Mechanical properties are also high for the modified PBR nanocomposites with maximum filler content. The permeability of organic vapours such as dichloromethane (CH2Cl2), chloroform (CHCl3) and carbon tetrachloride (CCl4) through PBR/EG nanocomposites was also studied. In vapour permeation studies, PBR4 exhibits the least permeability in CH2Cl2, CHCl3 and CCl4 solvents. Polybutadiene rubber–natural graphite (PBR/NG) nanocomposites were also prepared for comparison.

Transmit MIMO Radar Beampattern Design via Optimization on the Complex Circle Manifold

The ability of Multiple-Input Multiple-Output (MIMO) radar systems to adapt waveforms across antennas allows flexibility in the transmit beampattern design. In cognitive radar, a popular cost function is to minimize the deviation against an idealized beampattern (which is arrived at with knowledge of the environment). The optimization of the transmit beampattern becomes particularly challenging in the presence of practical constraints on the transmit waveform. One of the hardest of such constraints is the non-convex constant modulus constraint, which has been the subject of much recent work. In a departure from most existing approaches, we develop a solution that involves direct optimization over the non-convex complex circle manifold. That is, we derive a new projection, descent, and retraction (PDR) update strategy that allows for monotonic cost function improvement while maintaining feasibility over the complex circle manifold (constant modulus set). For quadratic cost functions (as is the case with beampattern deviation), we provide analytical guarantees of monotonic cost function improvement along with proof of convergence to a local minima. We evaluate the proposed PDR algorithm against other candidate MIMO beampattern design methods and show that PDR can outperform competing wideband beampattern design methods while being computationally less expensive. Finally, orthogonality across antennas is incorporated in the PDR framework by adding a penalty term to the beampattern cost function. Enabled by orthogonal waveforms, robustness to target direction mismatch is also demonstrated.

Nanocomposite materials based on zinc sulfide nanoparticles reinforced chlorinated styrene butadiene rubber

ABSTRACTThe effect of zinc sulfide (ZnS) nanoparticles in chlorinated styrene‐butadiene rubber (Cl‐SBR) was analyzed by X‐ray diffraction analysis (XRD), scanning electron microscopy (SEM), Differential Scanning Calorimetry, and impedance analyzer. The cure time, mechanical properties, and solvent transport of petroleum fuels through the Cl‐SBR/ZnS nanocomposites at different temperatures were evaluated. XRD and SEM studies showed that ZnS nanoparticles were well‐placed in the polymeric structure of Cl‐SBR. The increased glass transition temperature of the composite with the loading of nanoparticles indicated the increased molecular rigidity. Rheometric torque, alternating current conductivity, dielectric property, tensile strength, tear resistance, modulus, hardness, abrasion resistance, heat build‐up, and compression set were increased with the loading of nanoparticles, however, cure and scorch time, elongation at break, and resilience were reduced with the loading of nanoparticles. The diffusion results have been explained in terms of the size of liquid molecules and the diffusion mechanism was found to follow the anomalous trend. The activation energy for diffusion, sorption and permeation process was evaluated. These activation energy parameters were increased with the size of the penetrant molecules and also with the loading of nanoparticles. © 2018 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018, 135, 46538.

The effect of carbon black (CB) loading on curing characteristics and mechanical properties of virgin acrylonitrile butadiene rubber (Nbrv)/recycled acrylonitrile butadiene rubber (Nbrr) blends

Recently, the interest of polymer industry researchers have grown rapidly on the use of specific techniques which can reduce cost and utilize rubber waste into the processing form. The increasing of cognizance in environmental matters and the desire to sustain the resources had fortified the practice of recycling waste materials. In this work, the effect of carbon black loading on curing characteristics and mechanical properties of virgin acrylonitrile butadiene rubber/recycled acrylonitrile butadiene rubber (NBRv/NBRr) blends were studied. Cure time (t90), scorch time (tS2) and swelling percentage decreased but minimum torque (ML) and maximum torque (MH) increased with increasing carbon black (CB) loading in the blends. Increasing CB loading also increased tensile strength, tensile modulus (M100), hardness and compression set but decreased elongation at break (Eb) of NBRv/NBRr blends.

The k-resultant modulus set problem on algebraic varieties over finite fields

We study the k-resultant modulus set problem in the d-dimensional vector space Fqd over the finite field Fq with q elements. Given E⊂Fqd and an integer k≥2, the k-resultant modulus set, denoted by Δk(E), is defined asΔk(E)={‖x1±x2±⋯±xk‖∈Fq:xj∈E, j=1,2,…,k}, where ‖α‖=α12+⋯+αd2 for α=(α1,…,αd)∈Fqd. In this setting, the k-resultant modulus set problem is to determine the minimal cardinality of E⊂Fqd such that Δk(E)=Fq or Fq⁎. This problem is an extension of the Erdős–Falconer distance problem. In particular, we investigate the k-resultant modulus set problem with the restriction that the set E⊂Fqd is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof.

Design of Reverse Converters for a New Flexible RNS Five-Moduli Set { 2^k, 2^n-1, 2^n+1, 2^{n+1}-1, 2^{n-1}-1 } (n Even)

This paper presents the design methods of residue-to-binary (reverse) converters for the new flexible balanced five-moduli set { 2^k, 2^n-1, 2^n+1,2^{n+1}-1, 2^{n-1}-1 } for the pairs of positive integers {n ge 4} (even) and any {k>0}, which can provide the exact required dynamic range of the residue number system. This modulus set is the generalisation of the five-moduli set { 2^n, 2^n-1, 2^n+1, 2^{n+1}-1, 2^{n-1}-1 } (n even) with only a single parameter, n. The reverse converter for the new modulus set is the first ever proposed. Synthesis results obtained for the 65 nm technology for all dynamic ranges from 19 to 88 bits have shown that the state-of-the-art converters available for the five-modulus sets with a single parameter n{{ 2^n, 2^n-1, 2^n+1, 2^{n+1}-1, 2^{n-1}-1 }} (n even) and {{ 2^n-1, 2^n, 2^n+1, 2^{n+1}+1, 2^{n-1}+1 } } (n odd) not only introduce from 28 to 40% larger delay but also still consume more area and power than the converters proposed here.

Thermomechanical properties of zirconium tungstate/hydrogenated nitrile butadiene rubber (HNBR) composites for low-temperature applications

Rubber compounds for pressure sealing application typically have inferior dimensional stability with temperature fluctuations compared with their steel counterparts. This effect may result in seal leakage failures when subjected to decreases in temperature. Composites of hydrogenated nitrile butadiene rubber (HNBR) and zirconium tungstate as a negative thermal expansion filler were prepared in order to control the thermal expansivity of the material. The amount of zirconium tungstate (ZrW2O8) was varied in the range of 0 to about 40 vol%. The coefficient of thermal expansion (CTE), bulk modulus, uniaxial extension and compression set properties were measured. The CTE of the ZrW2O8-filled HNBR decreases with the filler content and it is reduced by a factor of 2 at the highest filler concentration used. The filler effect on CTE is found to be stronger when HNBR is below the glass transition temperature. The experimental thermal expansion data of the composites are compared with the theoretical estimates and predictions given by FEA. The effect of ZrW2O8 on the mechanical characteristics and compression set of these materials is also discussed.

Effects of silanization temperature and silica type on properties of silica‐filled solution styrene butadiene rubber (SSBR) for passenger car tire tread compounds

ABSTRACTInfluence of silanization temperature on properties of silica‐filled solution polymerized styrene butadiene rubber was investigated. Two types of silica, i.e., highly dispersible silica (HDSi) and conventional silica (CSi), were compared. Results show that the increased silanization temperature leads to the enhanced rubber–filler interaction, filler dispersion, and cross‐link density giving rise to the improvement in vulcanizate properties such as modulus, heat build‐up (HBU), and dynamic set, as well as tire performance, e.g., wet grip (WG), rolling resistance (RR), and abrasion resistance. Great care, however, must be taken to avoid the scorching phenomenon during the mixing process at too high temperature. Taken as a whole, the balanced properties are found at the silanization temperature of 140°C. Surprisingly, HDSi provides insignificant differences in degree of filler dispersion, WG, and RR, compared to CSi, despite its claimed greater dispersability. Probably, the relatively long mixing time used in this experiment may override the influence of silica type. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016, 133, 43342.

RDTP: reliable data transport protocol in wireless sensor networks

The paper deals with a new reliable data transport protocol for wireless sensor networks, referred as RDTP. One of the most prominent challenges in wireless sensor networks is reliable transport of data from sensor nodes to sink node. For designing protocols for such networks hardware, some constraining factors associated with energy and processing power must be taken into consideration. There are two generic methods, ARQ and FEC to achieve reliable data transport. Here, a reliable data transport protocol for wireless sensor networks is proposed, in which an efficient moduli set in redundant residue number system has been employed. The modulus set is exploited as a means for adding redundancy to transmitted data. Error controlling in proposed method is performed in a hop by hop manner. The simulation results indicate that the proposed method shows significant decreases in the energy consumption, compared to similar methods. The results also show that this leads to a rise in packet delivery ratio, with a simultaneous reduction in end to end delay.

Facile preparation of thermoplastic elastomer with high service temperature from dry selective curing of compatibilized EPDM/polyamide-12 blends

A facile preparation of thermoplastic elastomer (TPE) from simple polymer blends, elastically reinforced through a dry curing process, is introduced in this study. A TPE with a high service temperature of up to ∼180°C is prepared by melt mixing ethylene–propylene–diene rubber (EPDM) with polyamide-12 in the presence of maleated EPDM as a compatibilizer. A dry curing process employing electron beam (e-beam) irradiation is then applied for the selective cross-linking of the EPDM components, which is confirmed by the gel content and differential scanning calorimetry measurements. The mechanical and elastic properties of the TPE are improved by the e-beam irradiation, as evidenced by the measurements of tensile strength, elongation at break, Young’s modulus and tension set, affording a tough TPE. Repeated thermal recycling of the TPE demonstrates an excellent reprocessing capability, affording environmentally friendly TPEs using commercially available resources.