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## Introduction

Chemotherapy along with radiation and surgery is one of the three pillars of cancer treatment1. In clinical practice, it consists of systemic administration of cytotoxic chemicals, which due to a lack of selectivity, simultaneously attack many normal cells in the body, causing well-known side effects such as fatigue, hair loss, pain, constipation, nausea, and blood disorders. To minimize the side effects, researchers have developed targeted cancer therapies that aim to maximize drug interaction, specifically with cancer cells, while causing minimal damage to the healthy cells (immunotherapy being an example of latest developments in this area)2,3. One such technique is photodynamic therapy (PDT), which relies on selective optical activation of cytotoxic drug using a light source4,5. This provides spatial control through a synergistic combination of two otherwise non-toxic components: a photosensitizer (PS) and light. The photosensitizer generates cytotoxic agents only under light illumination with a particular wavelength. When absorbing photons, the photosensitizer becomes activated from the ground singlet state (S0) to an excited singlet state (S1). As it relaxes back to the ground state, the molecule can undergo two kinds of reactions, Type I and Type II6,7. Type I is the reaction with a molecule, transferring a hydrogen atom to form radicals that interact with oxygen to produce singlet oxygenated products (1O2). Type II refers to direct energy release to oxygen to form singlet oxygen (1O2) (Fig. 1a). Both reactions create reactive oxygen species (ROS, with a half-life of <40 ns and interaction radius of <20 nm) in a highly localized manner to attack the malignant cells8.

Despite its remarkable advantages as compared to standard chemotherapy, limited light penetration depth (<5 mm) at the wavelengths required by typical photosensitizers (500–600 nm) have restricted the applications of PDT to superficial dermatologic legions9. A possible approach to circumvent this limitation is deployment of wireless miniature light sources within the tumor10,11. Bansal. et al. recently presented radio-frequency powered miniature light sources, which could reach up to 5 cm12. Similarly, near-field communication LED chip in sandwiched by bio-adhesive polydopamine-modified poly(dimethylsiloxane) (PDMS) has introduced as metronomic PDT modality13. A few other relative easy to access organs (e.g., bladder, lung, esophagus, head and neck, glioblastoma, prostate cancer) could treated by an interstitial fiber optics14,15,16,17.

In this work, we outline an alternative approach of an ultrasonically powered implantable microlight source, μLight, to overcome the light delivery depth limitation and. The device converts the incoming ultrasonic wave into in-situ light. The use of ultrasonic powering enables for optimal size and form factor such that the device can be inserted using a biopsy needle. In addition, ultrasonic powering allows for deep tissue powering (>10 cm), not achievable with the traditional inductive methods18. The μLight can be used in primary or metastatic unresectable solid tumors. An important example of the former is pancreatic cancer in which 80% of tumors are unresectable and easily recurrent. The μLight can be used to ablate or shrink the tumor such that it can be removed as seen in clinical cases by Huggett et al.19. Metastatic liver tumors (commonly originating from colorectal, lung, or ovarian cancers) are examples of the latter. RF ablation is commonly used to treat such tumors; however, these procedures are associated with considerable morbidity and due to their invasiveness, they cannot be repeated. μLight allows for dose fractionation (i.e., the treatment can be administered in several sessions), thus both reducing the side effects and increasing the efficacy. Finally, recent advances in transmitting ultrasound across the skull to treat/ablate intracranial tumors has created the opportunity to place the μLights in the tumor through a burr hole and administer PDT without the need for a large and invasive craniotomy20.

## Results and Discussion

Figure 1b illustrates the microlight source, μLight, implanted in a deep-seated tumor (e.g., pancreas). An external ultrasonic transducer generates an acoustic wave which travels through tissue to the implant. Once energized, the onboard LEDs turn on and activate pre-delivered photosensitizers. The μLight consists of a small ultrasonic receiver (lead zirconate titanate, PZT-5A) with dimensions of 2 × 2 × 2 mm3 and 2 × 4 × 2 mm3 and super-bright surface mount LEDs (e.g., red, blue, or combinations). Such dimensions and form factors allow for easy insertion using a biopsy needle (gauge 8). The choice of LEDs is mainly dictated by its luminous intensity and excitation wavelengths of photosensitizers21,22,23. The μLight can carry up to four LEDs on the surfaces of the 2 × 2 × 2 mm3 embodiments or up to six on the 2 × 4 × 2 mm3 one; thus, capable of activating a large volume around the device.

### Ultrasonic Powering

Although inductive powering has been a clinically accepted wireless powering scheme for many decades24; at mm-scale it suffers from small efficiency, short transmission range, misalignment sensitivity, and manufacturing difficulty (e.g., it’s difficult to wound micro-coils at µLight dimensions). Inductive powering offers excellent performance only when the transmitter and receiver coils have a similar form factor, are aligned in parallel, and are in close proximity (e.g., cochlear implants). Ultrasonic powering is an attractive alternative wireless powering scheme since it offers unique advantages such as misalignment insensitivity, higher efficiency at mm-scale receiver size, and a larger penetration depth (>20 cm)18,25,26,27,28,29,30.

Figure 1c shows the ultrasonic powering schematic for in situ light delivery and its theoretical Krimholtz, Leedom, Mattaei (KLM) electrical equivalent circuit model31,32,33. The transmission system has three blocks; the ultrasonic transmitter, tissue, and the implanted receiver (i.e., μLight). The ultrasound transmitter is powered by an amplified and impedance matched sinusoidal signal from a function generator. At the input electrical port of the KLM model, amplified signal (Vin) passes through an electrode capacitance (CT) and frequency-dependent reactance (XT) before being converted to mechanical motion represented via the electromechanical coupling and modeled as a transformer (ϕ:1). The two faces of PZT transmitter and the acoustic matching layer are modeled as quarter wavelength transmission lines (ZTS, ZML). The mechanical input power to the receiver is denoted by αT where T is the transmitted mechanical power and α is the tissue attenuation factor (α = e−2μx, μ is the attenuation coefficient and x is the implantation depth). It is important to limit the transmitted ultrasonic power intensity to values approved by the FDA for ultrasonic imaging applications to avoid cavitation and heat induced damage to normal tissue (peak intensity of 720 mW/cm2)34. Similar to the transmitter, the receiver is also modeled as quarter-wave transmission lines (ZμLights) and mechanical-electrical conversion similarly by a transformer (ϕ:1). The output electrical voltage source (Vout) is again modelled with a parallel plate electrode capacitance (CR) and a frequency dependent acoustic reactance (XR). Note that the series reactance, XR becomes zero at resonant frequency operation; thus, the electrical part of receiver model could be modeled as an AC source with a source capacitance, CR.

To maximize the power transfer efficiency, the transmission link is operated at resonant frequency of the piezoelectric elements determined by the thickness of PZT (λ/2 = t, 1.09 MHz for 2 mm-thick PZT-5A). However, the simple aforementioned equation is true only if the aspect ratio (width/thickness) of transducer is greater than 10. In cuboidal form factors, such as the receivers used in the μLight, the resonant frequency deviates from the plate approximation due to Poisson’s ratio and other associated mode coupling effects of each axis35,36. The measured resonant frequency for the µLight with 2 × 2 × 2 mm3 and 2 × 4 × 2 mm3 receivers were 586 kHz and 650 kHz, respectively (Supplementary Fig. 1)37. Table 1 summarizes mechanical and electrical characteristics of the receiver components, i.e., PZT and LEDs. Figure 1d shows the fabrication process which consists of dicing a PZT wafer followed by soldering the LEDs and coating the device with a biocompatible polymeric thin film. (See the Methods sections for details).

The μLight was first characterized in terms of acoustic penetration depth and receiver directionality in water (a common model for soft tissue due to their similar acoustic properties (Zwater =1.48 MRayl, Ztissue = 1.63 MRayl, mean value for human tissue38). For each receiver size (2 × 2 × 2 mm3 and 2 × 4 × 2 mm3), devices were submerged in a water tank (dimension of 50 × 50 × 30 cm3) and excited by an ultrasonic transmitter (3.6 × 3.6 cm2) positioned within the near-field range, Fig. 2a. A signal generator (566 kHz sinusoidal wave for the μLight222 and 650 kHz for the μLight242) with a wideband power amplifier (ENI A300, Electronics & Innovation, Ltd. NY, USA) was used to drive the transmitter. The transmitted intensity was set to 185 mW/cm2, below the FDA limit for imaging applications. The acoustic power was measured with an ultrasound power meter (Precision Acoustic, Inc., UK). Figure 2b,c shows the normalized received electrical output power as a function of rotational and axial angular misalignment with respect to the ultrasonic input wave. As can be seen, the received output remains constant over 90-degree rotational and 50-degree axial angular misalignments (the axial angular was measured only up to 50 degrees due to experimental setup limitations), confirming the misalignment insensitivity of the receivers, as expected from the omni-directionality of ultrasound28.

The received electrical power for various load resistances at a constant transmitted ultrasonic intensity of 185 mW/cm2 was also measured, Fig. 2d. The 2 × 2 × 2 mm3 and the 2 × 4 × 2 mm3 PZT receivers were able to generate 108 μW (2 V and 54 μA) and 171 μW (3 and 57 μA), respectively, sufficient to power the LEDs. The μLight222 device can power red LEDs (VF = 1.8 V) and the μLight242 can power either red, blue, or green (VF = 2.8 V) LEDs. At the 185 mW/cm2 transmitted ultrasonic intensity, the maximum electrical power outputs were 2.2 mW for the μLight222 receiver and 3.3 mW for the μLight242 one (efficiencies of 1.7% and 2.7%, respectively, $$\eta =|{P}_{output}/{P}_{input}|$$)27.

### Light Delivery

The light generation performance of the μLight was evaluated by measuring the converted light intensity in a water tank setup placed inside of a black box to block the ambient light (Fig. 2a and Supplementary Fig. 2). The light irradiances of the μLight222 (1 LED facing the photodiode) and μLight242 (2 LEDs facing the photodiode) were measured using a PIN photodiode (VBP104S, Vishay Ltd.). In our case, we chose three different types of LEDs (AlGaAs for a red LED, InGaN for a blue and green LED) based on the output power measurements and forward turn-on voltages (1.8 V and 2.8 V). The measured current from the photodiode (2400 sourcemeter, Keithley Instruments) was converted back to light irradiance level. Figure 2e shows light irradiances for μLight242 (red, blue, and green LEDs) under increasing transmitted acoustic intensities. The difference in threshold voltage for μLight with different LEDs was firstly investigated. Irradiated lighting intensity in a non-reflection condition was firstly measured with μLight placed at 20 mm away from the transducer. When the device was turned on, the light intensity started increasing. The acoustic intensity generated near the transducer surface was measured for comparison. As shown in the Fig. 2e, the device with red LEDs can be turned on when the input acoustic intensity was around 60 mW/cm2, whereas the acoustic intensity needed for device with blue LED to be turned on is around 390 mW/cm2. It’s important to note that this acoustic input is not the transmitted intensity (which will be measured later), rather received intensity. As a result, the μLight with red LED requires relatively lower input to turn on as compared to blue or green LED.

Following the µLight activation intensity measurement, the transmitted acoustic intensity was also evaluated. It is important characterize since the light irradiance of μLight at various location within the body depends on the transmitted acoustic intensities. The non-reflective ultrasound propagation across the frontal region of the transmitter was investigated by measuring the acoustic intensity with a fiber-optics hydrophone (Precision Acoustic, UK), Fig. 2f–i. While the transmitter dimensions and wavelength define the near-field (~15 cm for the ultrasonic transmitter size of 3.6 × 3.6 cm2 at 650 kHz, N = D2/4λ, where N is near-field distance, D is the transducer diameter, λ is the ultrasonic wavelength), the acoustic intensities within the near-field region varies and maximize along the center axis; thus, it is much preferable region for the µLight.

To evaluate the performance of μLight within the proximity of transmitter (i.e., within the near-field region) in a reflective condition, the light intensity measurement was then repeated in a 5 cm diameter semispherical water tank with transducer placed at the bottom to create reflections in the water. The maximum light intensities for μLight222 were 48 μW/cm2 measured at the surface of the red LED and subsequently attenuated in the water according to the inverse square law (1/d2). The μLight242, meanwhile, could deliver light at much higher intensities, i.e., 1.1 mW/cm2 for the red LED and 6.5 mW/cm2 for the blue LED (both measured at the surface). Since the emitted light intensities are significantly lower than conventional external light sources (up to 1 W/cm2 for lasers)39, adequate therapeutic light energy dose (e.g. 20 J/cm2 for Foscan)6 requires longer operation time (e.g., a one hour operation with μLight at 5.5 mW/cm2 irradiance can achieve a light dosage of 20 J/cm2).

The light transmission was also measured ex vivo using soft bovine tissue (Supplementary Fig. 3). The light intensity emitted by the LEDs was measured at different tissue thicknesses. The results show blue light attenuates by 67%, when passing through 1.3 mm of soft tissue; whereas red light attenuates 50% less (33%) across 1.3 mm thick tissue. This was expected since red light interaction with soft tissue is known to be less than that of the shorter wavelength blue light.

### In vitro and In vivo Tests

The μLight devices were tested in vitro as to evaluate their functionality to initiate a cytotoxic response in a cell culture environment. Our choice of photosensitizer (PS) was verteporfin which has been studied for pancreatic cancer19. Verteporfin has two peak absorptions around wavelength of 400–450 nm and 665–685 nm40. Based on the device characterizations (e.g., light delivery and operational threshold), we selected μLight242 with two red LEDs (wavelength range of 637–671 nm with a peak at 655 nm) for in vivo experiment.

Dimethyl sulfoxide (DMSO) was used as the solvent for verteporfin and proper dosage of was determined prior to the test (see Methods section for details). HeLa cells (human cervical cancer cells) were selected for in vitro validation since it has been widely used for proof-of-concept evaluation. Cells were seeded onto culture dishes and were allowed to grow for 24 hours before the experiment. Five different cell groups were tested: (1) cells with only cell culture media (control); (2) cells with 1 mL cell culture media and DMSO (0.14% v/v); (3) cells with cell culture media and verteporfin without photodynamic treatment (1 mL cell culture media with 4 μM verteporfin and 0.14% v/v DMSO); (4) cells with cell culture media and verteporfin with photodynamic treatment (1 mL cell culture media with 4 μM verteporfin and 0.14% v/v DMSO); and finally (5) cells are killed by Triton X100. For group 4, during ultrasound treatment, all samples were kept in the dark to eliminate the effect of the ambient light that could activate verteporfin. Figure 3a shows the experimental setup in which two red μLight242s (655 nm) were placed under each well and powered through a 3 cm-thick 0.5% w/v agarose gel to mimic the power transfer through soft tissue. The ultrasonic power at 185 mW/cm2 was turned on for 30 minutes and generated light from the μLight could be visually detected/observed. The light dose delivered for each well was around 4 J/cm2. Figure 3b shows the relative cell viability for the five experimental groups. As can be seen, 30 minutes of photodynamic therapy using μLight could induce 70% decrease in average cell viability as compared to the cells that were just exposed to verteporfin (without the device being turned on, group 3) and 82% decrease as compared to the control (group 1) which shows a significant difference (one-way ANOVA, p < 0.05). In addition, through a separate control experiment, no significant difference (ANOVA) in cytotoxicity was introduced by pre-treated Verteporfin and we confirmed that the cytotoxic effect is only due to the in situ light generation. (See details on Supplementary Fig. 4).

We move forward with in vivo animal test after the positive in vitro results. We demonstrated the in vivo functionality of μLight in BALB/c mouse model. Figure 3c shows the in vivo experimental setup used to validate the capability of light delivered by μLight through the soft tissue. Tumor was induced in BALB/cJ mice by implanting cultured 4T1 cells in hind flank. We chose the 4T1 model in order to test the device in an immunocompetent setting where the immune system can participate in tumor eradication. Human pancreatic cancer cells would have to be tested in immunocompromised mice. After tumor had grown to average size of 200 mm3 in volume among mice, a μLight242 was implanted in the interstitial space around the tumor of sixteen mice (see Methods section for details). Allowing for the wound to recover for a week, near the implantation site, adjacent tumor cells migrated forming a second tumor and completely surrounding the device, resulting a deployment of the device inside tumor tissue. Upon the complete recovery, first PDT was initiated by intra-tumoral injection of Verteporfin on day 7. Out of eighteen mice, six mice were picked to a control group (no implants), eight mice were for a group with µLight, but no ultrasonic treatment, and four mice for a treatment group. We excluded the mice which died before the experiments and mice that reduced tumor naturally. Overall, each group had average of six animals.

For treatment group, μLight was activated wirelessly using ultrasound, after 15-minute of Verteporfin injection. The treatment time was set to 30 minutes (2 J/cm2 light dose). This protocol was designed to consider the drug deliver in the system as well as µLight performance. The μLight was generating light while the ultrasonic transducer supplied power from the outside, Fig. 3c. Tumor size was measured with a caliper and optical pictures were taken periodically. While the tumor size in control groups (non-operating µLight and no implants) shows faster growth, the test group with working µLight shows a decrease in average tumor size (~20% shrinkage, P = 0.009*) on day 8, Fig. 3e. Figure 3d is the picture taken 24 hours after the PDT treatment. The results and picture show necrotic tissue resulting from successful PDT (the small wound on the migrated adjacent tumor was observed before the PDT treatment and was probably a fungating legion). Necrosis was restricted to the tumor tissue where the device was implanted (device was surrounded by necrosis and thus not visible). The affected volume was limited by the light penetration depth into the tumor parenchyma, with larger tumors requiring multiple devices to achieve complete removal. Although additional studies are needed to evaluate the parameters required for optimal efficacy, this preliminary in vivo result indicates that μLight can be powered while being implanted and trigger the photosensitizer.

## Discussion

The expected treatment time estimated from the previously investigated clinical trials for PDT6,9,21 shows that μLight can have clinical relevance. Table 2 summarizes the current commercially available photosensitizers, their tested dosage (0.1–2 mg/kg delivered orally or intravenously), and light wavelength (lower wavelengths around 408 nm show more efficacy) and energy densities (between 3–200 J/cm2). After a post-infusion time (30 min to 24 h, determined by drug administration), various light energy densities have been used to treat lung, early-stage head/neck, and bladder cancers among others. Most of the studies utilized higher light energy densities using the external system (e.g., floor lamp, laser, etc.). However, a few studies suggested that the higher light intensities does not increases the PDT efficacy21. The 1.3 J/cm2 could be achieved from 20 minutes from μLight242 with 2 on-board LEDs. Twenty minutes of treatment time for PDT is not significantly longer than conventional PDT. Moreover, the actual treatment time could be even shorter with the μLight, since the light intensities reported from the clinical trials were not measured inside the tumors. For tumors larger than 0.5 cm3 in volume, multiple μLights should be implanted to cover the entire tumor volume (in a similar manner the brachytherapy seeds are placed inside a prostate cancer tumor). A recent study13 has presented the feasibility of applying low power light source to metronomic photodynamic therapy (low lighting dosage modality with extended treatment time).

## Methods

### Fabrication of µLight

The fabrication of μLight is illustrated in Fig. 1d. The lead zirconate titanate (PSI-5A4E, Piezo Systems Inc., MA, USA) was machined/diced into pieces of desired dimensions: 2 × 2 × 2 mm3 (henceforth known as μLight222) and 2 × 4 × 2 mm3 (henceforth known as μLight242), Fig. 1d(a). Super-bright red (λ = 640 nm, InGaN, APT2012SRCPRV, Kingbright Ltd.) or blue (λ = 470 nm, AlGaAs, LTST-C191TBKT, Lite-On Technology Corp.) LEDs were subsequently soldered on top of the nickel electrode surfaces of the PZT, Fig. 1d(b). Additional LEDs were soldered on other sides (multiple LEDs can be assembled to increase the angular coverage, up to 4 LEDs on μLight222 and up to 6 LEDs on μLight242), Fig. 1d(c). The red and blue LEDs have emission angles of 120° and 130°, respectively, allowing for a total angular coverage of nearly 360° by using multiple LEDs. Finally, the device was coated with 5 μm of parylene-C for passivation and biocompatibility, Fig. 1d(d). Several fabricated devices are shown in Fig. 1d(e).

### Preparation of cell culture media

Media solution for in vitro experiment was prepared using Gibco DMEM (Dulbecco Modified Eagle Media) supplemented with 10% fetal bovine serum, 1% penicillin-streptomycin and 1% L-glutamine.

0.5 mg verteporfin (Sigma Aldrich) was first dissolved in 250 μL DMSO (Sigma Aldrich). The solution was well stirred for complete dissolution. During device functionality evaluation, for treated group, 1.44 μL of verteporfin-DMSO solution was added to 998.56 μL of cell culture media to create a mixture of 1 mL with 4 μM concentrated verteporfin.

### Verteporfin dosage assay

The concentration range at which the drug itself is non-toxic to cells was determined by measuring lethal dose, 50% (LD50) value. HeLa cells were seeded at a density of 1.5 × 104 cells/well in a 96 well plate 24 hours before treatment. Cells were then treated with varying concentration of verteporfin (2.7 mM to 0.084 mM) (see Supplementary Fig. 5). The untreated wells had DMSO which corresponds to the amount of DMSO in the well with the highest concentration of the drug. Cells were incubated for 24 hours before analyzing the LD50 value using CellTiter 96 Aqueous One Solution Cell Proliferation Assay (MTS) according to the protocol. As the result, a concentration of 4 μM was selected which shows no cytotoxicity to the HeLa cells.

### Preparation of tissue phantom

TRIS borate- EDTA buffer solution (Fluka Analytical) was diluted by deionized water to 10% v/v. to prepare the base solution. Agarose powder (Sigma Aldrich) was dissolved at 100 °C while stirring in deionized water at a concentration of 0.5% w/v. Dissolved agarose solution was then stored in a vacuum chamber under 1 Torr for 10 min. The degassed pre-gel solution was poured into a glass beaker for cooling under the room temperature.

### Procedure for animal handling and in vivo experiments

Animal preparation and handling was in strict compliance with our approved Purdue University’s IACUC protocol (approval #: 1112000342 on 1/16/2016). Female BALB/cJ mice were obtained from Jackson Laboratory. Each group (control, dummy device, and test) had six to eight animals. 4T1 cell lines were obtained from American Type Culture Collection (ATCC) and cultured according to ATCC protocols. One million tumor cells were suspended in physiologic buffer and implanted into the hind flank area of mice. The device was sterilized with 70% ethanol for 30 minutes and surgically implanted when the size of the tumor was approximately 200 mm3.

Ultrasound excitation treatment was conducted after complete healing of the wound. To prepare the photosensitizer, 1 mg verteporfin was dissolved 0.25 mL DMSO following by 10 times dilution to 2.5 mL solution. For each mouse, 25 μL of verteporfin solution was injected resulting in a verteporfin dose of 0.5 mg/kg per mouse (weight of each mouse was around 20 g). The mouse was anesthetized using 3% isoflurane in a reclamation chamber and placed on a heating pad to maintain the body temperature during implantation as well as the µLight operation. For each experimental group, the tumor size for three mice was collected and measured.

Following the in vivo experiments, 21 days after tumor implantation, the mouse was euthanized per approved IACUC protocol. During the treatment, hair on the targeted site was shaved. Ultrasound gel (Medline) was applied to the skin as transmission media between mouse and ultrasound transmitter. Ultrasound transmitter was in touch with the gel during the treatment.

### Statistical analysis

For in vitro and in vivo studies, experimental data are presented as mean ± standard deviation, obtained from at least three independent measurements. For in vitro cell culture validation, the means comparison of cytotoxicity among groups were analyzed for statistical significance using one-way analysis of variance (ANOVA) followed by post-hoc Tukey test for comparing multiple conditions. For in vivo experiments, two-way ANOVA was used to compare control groups and treatment group. Difference was considered significant when *p < 0.05. Statistical analysis was performed using OriginPro software.


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# Simplification Problems

#### Chapter 4

5 Steps - 3 Clicks

# Simplification Problems

### Introduction

Simplification Problems is based on BODMASÂ rule, where

B â†’ Brackets,

O â†’ Of,

D â†’ Division,

M â†’ Multiplication,

S â†’ Subtraction.

### Methods

BODMAS Rule:

BODMAS is about simplifying an expression by firstly removing the brackets in the order i.e. (), {}, []. Removal ofÂ brackets isÂ followed byÂ addition, subtraction, multiplication, division,Â square roots, cube roots, powers, cancellation of numerator/ denominator and so on.

Example 1:
Simplify using BODMAS rule: 25 – 48 Ã· 6 + 12 Ã— 2

Solution:

25 – 48 Ã· 6 + 12 Ã— 2

= 25 – 8 + 12 Ã— 2, (Simplifying â€˜divisionâ€™ 48 Ã· 6 = 8)

= 25 – 8 + 24, (Simplifying â€˜multiplicationâ€™ 12 Ã— 2 = 24)

= 17 + 24, (Simplifying â€˜subtractionâ€™ 25 – 8 = 17)

= 41, (Simplifying â€˜additionâ€™ 17 + 24 = 41)

Example 2:
Simplify using BODMAS rule: 78 – [5 + 3 of (25 – 2 Ã— 10)]

Solution:

78 – [5 + 3 of (25 – 2 Ã— 10)]

= 78 – [5 + 3 of (25 – 20)], (Simplifying â€˜multiplicationâ€™ 2 Ã— 10 = 20)

= 78 – [5 + 3 of 5], (Simplifying â€˜subtractionâ€™ 25 – 20 = 5)

= 78 – [5 + 3 Ã— 5], (Simplifying â€˜ofâ€™)

= 78 – [5 + 15], (Simplifying â€˜multiplicationâ€™ 3 Ã— 5 = 15)

= 78 – 20, (Simplifying â€˜additionâ€™ 5 + 15 = 20)

= 58, (Simplifying â€˜subtractionâ€™ 78 – 20 = 58)

Example 3:
Simplify using BODMAS rule: 52 – 4 of (17 – 12) + 4 Ã— 7

Solution:

52 – 4 of (17 – 12) + 4 Ã— 7

= 52 – 4 of 5 + 4 Ã— 7, (Simplifying â€˜parenthesisâ€™ 17 – 12 = 5)

= 52 – 4 Ã— 5 + 4 Ã— 7, (Simplifying â€˜ofâ€™)

= 52 – 20 + 4 Ã— 7, (Simplifying â€˜multiplicationâ€™ 4 Ã— 5 = 20)

= 52 – 20 + 28, (Simplifying â€˜multiplicationâ€™ 4 Ã— 7 = 28)

= 32 + 28, (Simplifying â€˜subtractionâ€™ 52 – 20 = 32)

= 60, (Simplifying â€˜additionâ€™ 32 + 28 = 60)

Modulus of a real number:

In many engineering calculations you will come across the symbol “| |”. This is known as the modulus.

The modulus of a number is its absolute size. That is, we disregard any sign it might have.

Examples:

• The modulus of âˆ’8 is simply 8.

• The modulus of âˆ’$$\frac{1}{2}$$ is $$\frac{1}{2}$$.

• The modulus of 17 is simply 17.

• The modulus of 0 is 0.

So, the modulus of a positive number is simply the number.

The modulus of a negative number is found by ignoring the minus sign.

The modulus of a number is denoted by writing vertical lines around the number.

This observation allows us to define the modulus of a number quite concisely in the following way.

$| a | = \begin{cases} a & \quad \text{if } a > 0\\ -a & \quad \text{if } a < 0 \end{cases}$

Examples:

1. | 9 | = 9

2. | âˆ’ 11 | = 11

3. | 0.25 | = 0.25

4. | âˆ’ 3.7 | = 3.7

Virnaculum (or Bar):

When an expression contains Virnaculum, before applying the ‘BODMAS’rule, we simplify the expression under the Virnaculum.

Example 1:
Simplify: $$78 \,- [24 \,- {16 \,- (5 \,â€“ \overline{4 \,- 1})}]$$

Solution:

$$78 \,- [24 \,- {16 \,- (5 \,â€“ \overline{4 \,- 1})}]$$

= 78 – [24 – {16 – (5 – 3)}] (Removing vinculum)

= 78 -[24 – {16 – 2}] (Removing parentheses)

= 78 – [24 â€“ 14] (Removing braces)

= 78 – 10

= 68.

Example 2:
Simplify: $$197 \,- [1/9 \{42 \,+ (56 \,- \overline{8 \,+ 9})\} \,+ 108]$$

Solution:

$$197 \,- [1/9 \{42 \,+ (56 \,- \overline{8 \,+ 9})\} \,+ 108]$$

= 197 â€“ [1/9 {42 + (56 â€“ 17)} + 108] (Removing vinculum)

= 197 – [1/9 {42 + 39} + 108] (Removing parentheses)

= 197 – [(81/9) + 108] (Removing braces)

= 197 – [9 + 108]

= 197 – 117

= 80

Example 3:
Simplify: $$95 \,- [144 \,Ã· (12 \,\times 12) \,- (-4) \,- \{3 \,â€“ \overline{17 \,- 10}\}]$$

Solution:

$$95 \,- [144 \,Ã· (12 \,\times 12) \,- (-4) \,- \{3 \,â€“ \overline{17 \,- 10}\}]$$

= 95 – [144 Ã· (12 x 12) – (-4) – {3 -7}]

= 95 – [144 Ã· 144 – (-4) – {3-7)]

= 95 – [1 – (-4) – (-4)] [Performing division]

= 95 – [1 + 4 + 4]

= 95 – 9

= 86

### Samples

1. Simplify a-[a-(a+b)-{a-(a-b+a)}+2b]?

Solution:

Given that a-[a-(a+b)-{a-(a-b+a)}+2b]

â‡’a-[a-(a+b)-{a-a+b-a}+2b]

â‡’a-[a-(a+b)-{b-a}+2b]

â‡’a-[a-(a+b)-b+a+2b]

â‡’a-[a-a-b-b+a+2b]

â‡’a-a = 0

Therefore a-[a-(a+b)-{a-(a-b+a)}+2b] = 0

2. Find the value of $$x$$ if $$(\frac{12.24 Ã· x}{3.2 Ã— 0.2})$$ = 2

Solution:

Given expression is $$(\frac{12.24 Ã· x}{3.2 Ã— 0.2})$$ = 2

â‡’$$\frac{12.24}{x}$$ = 2 x 3.2 x 0.2

â‡’$$x$$ = $$\frac{12.24}{1.28}$$

â‡’$$x$$ = 0.011

Hence, the value of $$x$$ = 0.011

3. Find the value of $$\sqrt{30} Ã— \sqrt{10} = ?$$

Solution:

Given that $$\sqrt{30} Ã— \sqrt{10} = ?$$

Consider $$\sqrt{30} Ã— \sqrt{10}$$

â‡’$$\sqrt{{30} Ã— {10}}$$

â‡’$$\sqrt{6 Ã— 5 Ã— 5 Ã—2}$$

â‡’5$$\sqrt{6 Ã— 2}$$

â‡’5$$\sqrt{12}$$

â‡’5$$\sqrt{4 Ã— 3}$$

â‡’5 Ã— 2$$\sqrt{3}$$

â‡’10$$\sqrt{3}$$

Therefore the value of $$\sqrt{30} Ã— \sqrt{10}$$ = 10$$\sqrt{3}$$.

4. Simplify 6$$\frac{2}{9}$$ + 2$$\frac{7}{9}$$ – 4$$\frac{3}{11}$$ + 1$$\frac{3}{11}$$?

Solution:

Given that

6$$\frac{2}{9}$$ + 2$$\frac{7}{9}$$ – 4$$\frac{3}{11}$$ + 1$$\frac{3}{11}$$

â‡’$$\frac{56}{9}$$ + $$\frac{25}{9}$$ – $$\frac{47}{11}$$ + $$\frac{14}{11}$$

Now L.C.M. of 9, 9, 11, 11 is 99

â‡’$$\frac{616 + 275 – 423 + 126}{99}$$

â‡’$$\frac{1017 – 423}{99}$$

â‡’$$\frac{594}{99}$$

â‡’6

Therefore 6$$\frac{2}{9}$$ + 2$$\frac{7}{9}$$ – 4$$\frac{3}{11}$$ + 1$$\frac{3}{11}$$ = 6.

5. If $$\sqrt{a}$$ = 2b then find the value of $$\frac{b^2}{a}$$?

Solution:

Given

$$\sqrt{a}$$ = 2b

By taking Square root to other side,

â‡’ a = 4$$b^2$$

Now consider $$\frac{b^2}{a}$$

â‡’$$\frac{1}{4}$$ or 0.25

âˆ´ The value of $$\frac{b^2}{a}$$ = $$\frac{1}{4}$$ or 0.25

6. Find the unknown value from 25% of 180 = ? Ã· 0.25

Solution:

Assume the unknown value as $$x$$

Given 25% of 180 = ? Ã· 0.25

Substitute $$x$$ i.e.

$$\frac{25}{100}$$ Ã— 180 = $$x$$ Ã— $$\frac{1}{0.25}$$

â‡’$$x$$ = $$\frac{25 Ã— 180 Ã— 0.25}{100}$$

â‡’$$x$$ = $$\frac{1125}{100}$$

â‡’$$x$$ = 11.25

Therefore, the unknown value $$x$$ = 11.25

format: plain


text:
# Did this really happen?

### Help Support The Rocketry Forum:

#### mv1371

##### Active Member
I am at a trivia night and one question was What did NASA buy at Wal-Mart to protect the space shuttle from woodpeckers ?

6 plastic owls.

#### stantonjtroy

##### Well-Known Member
I can believe it. When I worked at the Marina, many of the boat owners would put plactic owls on their boats to keep other birds away. Crap on the deck and such.

#### n5wd

##### Well-Known Member
Looks like it probably did; at least, there are print stories from 1995:

Space Engineers Scheme To Outfox Birds
Didja notice the ad in the newspaper - an HP multifunction FAX/printer for only $800! :eyepop: Saw one in the local office supply store the other night, an HP no less, for right at$100.

Moore's Law in action!

#### sodmeister

##### Well-Known Member
Down a ways ,there`s a strip mall with a Pizza Hut that has a few owls on the roof edge to scare away pigeons.

The pigeons sit on its head ( no kidding)
I guess they get used to the percieved treat ,and after awhile use it as another place to roost.

P

#### AKPilot

##### Well-Known Member
I can see it. Plastic owls, rubber snakes, etc. are frequently used on DoD flight lines, along with propane cannons, falcons, etc. Even at Boeing we do the plastic owl thing. The trick is to move them.

Funny thing though, recently our local community college wanted to deter canadian geese in their ponds, so they put up cardboard coyotes near the pond. Funny thing is, the pond's full of ducks and geese and night - as the coyotes can only be seen during the daytime.

#### mv1371

I guess that since this was NASA I was expecting a million $high speed, low drag solution. But if low tech works, then good for them! #### dave carver ##### ....what hump? Naaa, it would only be a million$ owl if they let a Government contract but as they went to Walmart instead they got the low speed, high drag solution.

Now, if NASA would have written a contract to be filled they would have expected the owls to fly to new roosts on their own and have a ASB System(anti-small bird) Cost overruns could easily push it to a million per Owl.

#### Samuron

##### Well-Known Member
Didja notice the ad in the newspaper - an HP multifunction FAX/printer for only \$800! :eyepop:
Nah; I was too busy crying over the 6% interest rates. :bang:

#### sodmeister

##### Well-Known Member
I can see it. Plastic owls, rubber snakes, etc. are frequently used on DoD flight lines, along with propane cannons, falcons, etc. Even at Boeing we do the plastic owl thing. The trick is to move them.

Funny thing though, recently our local community college wanted to deter canadian geese in their ponds, so they put up cardboard coyotes near the pond. Funny thing is, the pond's full of ducks and geese and night - as the coyotes can only be seen during the daytime.
Again...down the way,there is a pond at a business complex that has a few plastic fox decoys to deter geese.

Results.....----------->>>>>:neener:

P

#### blackjack2564

##### Crazy Jim's Gone Banana's
TRF Supporter
Actually when I took the tour last summer the owls were one of the things the tour guide pointed out!

#### thomcat00

##### Well-Known Member
One of the local airports (SYR) uses them and they also have plastic flamingoes at the end of one runway for fun


text:
# How would one rescale the variance and standard deviation of a set of data?

I have a set of data with a certain mean, variance, and standard deviation. I centered the mean around the origin the standard way by subtracting it from the data.

Now how do I modify the data to make variance = 1?

-

Just divide everything by the standard deviation.

Suppose first that you have a finite data set $x_1,x_2,x_3,\ldots,x_n$.

Let $\overline{x} = (x_1+\cdots+x_n)/n$ be the mean and $s=\sqrt{(1/n)\sum_{i=1}^n (x_i - \overline{x})^2}$ be the standard deviation. Let $y_i=x_i/s$, i.e. we're dividing everything by the standard deviation. Then $$\overline{y} = \frac{y_1+\cdots+y_n}{n} = \frac{\frac{x_1}{s}+ \cdots+\frac{x_n}{s}}{n} = \frac 1 s \cdot \frac{x_1+\cdots+x_n}{n} = \frac{\overline{x}}{s}.$$ And the standard deviation of the $y$-values is $$\sqrt{\frac 1 n ((y_1-\overline{y})^2 + \cdots+(y_n-\overline{y})^2)} = \sqrt{\frac 1 n \left( \left(\frac{x_1}{s}-\frac{\overline{x}}{s}\right)^2+ \cdots + \left(\frac{x_n}{s}-\frac{\overline{x}}{s}\right)^2\right)}$$ $$= \frac{1}{\sqrt{s^2}}\sqrt{\frac 1 n ((x_1-\overline{x})^2 + \cdots+(x_n-\overline{x})^2)} = \frac 1 s \cdot s = 1.$$

For discrete data sets with non-uniform distributions, including infinite discrete data sets, one replaces $1/n$ with appropriate weights $p_i$. For continuous data sets, one used integrals, but the argument is essentially the same.

-
I knew that it would be something trivial like this. Can you show why this works? –  mathjacks May 21 '12 at 1:22
@flapjackery: Scaling the data by a value $a$ scales the standard deviation also by $a$, as you can see from the definition $\sigma = \sqrt{\frac1n\sum{x_i^2} - \big(\frac1n\sum x_i\big)^2}$. –  Rahul May 21 '12 at 1:48
@RahulNarain : That works if $a\ge 0$. In general multiplying everything by $a$ has the effect of multiplying the standard deviation by $|a|$. –  Michael Hardy May 21 '12 at 1:54
@MichaelHardy: You're right, of course. Thanks for the correction. –  Rahul May 21 '12 at 2:00


text:
# Structural induction proof

I need to find the relation between the number of comas in a term $p_c$ of language L = {f,g} and the number $p_f$ of f functional symbols and the number $p_g$ of g functional symbols in term $t$ .

$f$ is an unary func symbol, $g$ is ternary one.

My induction assumption is that this relation is $p_c(t) = 2p_g(t) +0p_f(t)$, where $p_g$ and $p_c$ show num of corresponding symbols in the term $t$ respectively.

Having this, base:

$p_c(x)=0$, where $x$ is some variable. Assumption holds

Step:

let $k_i \in \{k_a, k_b, k_c\}$ are some terms. $p_c(k_i) = 2p_g(k_i)$

Then $p_c(k_{i+1}) = p_c(f(k_i)) = 0 + 2 p_g(k_i)$ or

$p_c(k_{i+1}) = p_c(g(k_a, k_b, k_c)) = 2 + p_c(k_a)+ p_c(k_b)+ p_c(k_c) = 2 + 2p_g(k_a)+ 2p_g(k_b)+ 2p_g(k_c)$

what should I write next?

• Better late than never ... – Bram28 Nov 8 '16 at 18:34

Beforehand, I would state the claim that you are trying to prove simply as $p_c(t) = 2p_g(t)$
First say that you need to show $p_c(k_{i+1}) = 2p_g(k_{i+1})$
In the case where $k_{i+1} = f(k_i)$, the number of g functional symbols in $k_{i+1}$ equals the number of g functional symbols in $k_i$, and so $p_c(k_{i+1}) = p_c(k_i) = 2p_g(k_i) = 2p_g(k_{i+1})$
In the case where $k_{i+1} = g(k_a, k_b, k_c)$, the number of g functional symbols in $k_{i+1}$ is 1 + all the g functional symbols in the other terms, i.e. $p_g(k_{i+1}) = p_g(k_a) + p_g(k_b) + p_g(k_c) + 1$. So then: $p_c(k_{i+1}) = p_c(k_a) + p_c(k_b) + p_c(k_c) + 2 =$ (by inductive hypothesis) $= 2p_g(k_a) + 2p_g(k_b) + 2p_g(k_c) + 2 = 2(p_g(k_a) + p_g(k_b) + p_g(k_c) + 1) = 2p_g(k_{i+1})$
So yes, in both cases $p_c(k_{i+1}) = 2p_g(k_{i+1})$


text:
# Effective stress

Effective stress is a force that keeps a collection of particles rigid. Usually this applies to sand, soil, or gravel.

## History

Karl von Terzaghi first proposed the relationship for effective stress in 1925.[1][2][3] For him, the term "effective" meant the calculated stress that was effective in moving soil, or causing displacements. It represents the average stress carried by the soil skeleton.

## Description

Effective stress (σ') acting on a soil is calculated from two parameters, total stress (σ) and pore water pressure (u) according to:

${\displaystyle \sigma '=\sigma -u\,}$

Typically, for simple examples

{\displaystyle {\begin{aligned}\sigma &=H_{\mathrm {soil} }\,\gamma _{\mathrm {soil} }\\u&=H_{\mathrm {w} }\,\gamma _{\mathrm {w} }\end{aligned}}}

Much like the concept of stress itself, the formula is a construct, for the easier visualization of forces acting on a soil mass, especially simple analysis models for slope stability, involving a slip plane.[4] With these models, it is important to know the total weight of the soil above (including water), and the pore water pressure within the slip plane, assuming it is acting as a confined layer.

However, the formula becomes confusing when considering the true behaviour of the soil particles under different measurable conditions, since none of the parameters are actually independent actors on the particles.

Consider a grouping of round quartz sand grains, piled loosely, in a classic "cannonball" arrangement. As can be seen, there is a contact stress where the spheres actually touch. Pile on more spheres and the contact stresses increase, to the point of causing frictional instability (dynamic friction), and perhaps failure. The independent parameter affecting the contacts (both normal and shear) is the force of the spheres above. This can be calculated by using the overall average density of the spheres and the height of spheres above.

If we then have these spheres in a beaker and add some water, they will begin to float a little depending on their density (buoyancy). With natural soil materials, the effect can be significant, as anyone who has lifted a large rock out of a lake can attest. The contact stress on the spheres decreases as the beaker is filled to the top of the spheres, but then nothing changes if more water is added. Although the water pressure between the spheres (pore water pressure) is increasing, the effective stress remains the same, because the concept of "total stress" includes the weight of all the water above. This is where the equation can become confusing, and the effective stress can be calculated using the buoyant density of the spheres (soil), and the height of the soil above.

The concept of effective stress truly becomes interesting when dealing with non-hydrostatic pore water pressure. Under the conditions of a pore pressure gradient, the ground water flows, according to the permeability equation (Darcy's law). Using our spheres as a model, this is the same as injecting (or withdrawing) water between the spheres. If water is being injected, the seepage force acts to separate the spheres and reduces the effective stress. Thus, the soil mass becomes weaker. If water is being withdrawn, the spheres are forced together and the effective stress increases.[5]

Two extremes of this effect are quicksand, where the groundwater gradient and seepage force act against gravity; and the "sandcastle effect",[6] where the water drainage and capillary action act to strengthen the sand. As well, effective stress plays an important role in slope stability, and other geotechnical engineering and engineering geology problems, such as groundwater-related subsidence.

## References

### General references

• Terzaghi, K. (1925). Principles of Soil Mechanics. Engineering News-Record, 95(19-27).

### Inline citations

1. Terzaghi, Karl (1925). Erdbaumechanik auf Bodenphysikalischer Grundlage. F. Deuticke.
2. Terzaghi, Karl (1936). "Relation Between Soil Mechanics and Foundation Engineering: Presidential Address". Proceedings, First International Conference on Soil Mechanics and Foundation Engineering, Boston. 3, 13-18.
3. Archived June 18, 2006, at the Wayback Machine
4. Archived September 2, 2006, at the Wayback Machine
5. Archived May 30, 2008, at the Wayback Machine


text:
# Extract text file contents and put into string in xojo desktop app

As the title suggests, I’m having a hard time making my app take a look at a specific text file and then add its contents into an empty string. Any suggestions on how to make this happen would be appreciated.

Thank you

Take a look in the LR at TextInputStream http://documentation.xojo.com/index.php/TextInputStream

Look at the first example. Basically you could replace ‘TextArea1.Text’ in that example with the name of your variable.
What have you tried so far?

By using open Dialog I know that I am getting to the right location and that my program knows where the file is but I can’t get it to open it on its own. I have to actually click on the file and open to see the content(just one line of text).

dim myFile As FolderItem
myFile = GetFolderItem("\folder\text file I want to open")
Dim d As OpenDialog
Dim t As TextInputStream
d = New OpenDialog
myFile = d.ShowModal
If myFile <> Nil Then
t = TextInputStream.Open(myFile)
End If

What I want to be able to do is have the program open it without any user interaction and then copy the line of text in the text file into a string inside of the program.

What’s the actual path you’re using to the file?

Try something like this:

Dim f As FolderItem = GetFolderItem("absolute_path_to_file")
If f <> Nil Then
If f.Exists Then
// Be aware that TextInputStream.Open coud raise an exception
Dim t As TextInputStream
Try
t = TextInputStream.Open(f)
t.Encoding = Encodings.MacRoman
Catch e As IOException
t.Close
MsgBox("Error accessing file.")
End Try
End If
End If

If the file never changes, you could just drag it into your project and access its contents as if it were a constant.

For example, if you drag the file “MyTextFile.txt” into the project, within your code you can do stuff like:

dim s as string = MyTextFile // no extension

I also don’t care about displaying the text in the textarea I was just using it to see if the program would actually access the file.

Just tried the code you suggested above(Albin) and nothing happens. I don’t get an error(msgBox) but the text file isn’t displayed in the textarea. I tried using similar code earlier and it didn’t work. Absolute path = c:\\practice\test1.txt

Kem- the application won’t have any kind of text area that is why I’m trying to find a way to open and copy content from text doc without outside interaction.

Thank you guys for your suggestions, I’ve been stuck on this for a couple of days now.

My example, didn’t make use of a TextArea, but you certainly could do something like:

TextArea1.Text = MyTextFile

As for Albin’s code, it looks perfect, so make sure the file actually exists. Trace the code to make sure it’s actually going to all the steps.

// // Copy of the code from above post by "Albin K" // with extra "Else" added. // // The sample file in this example is located at // "C:\\MyData\\WorkArea\\MyFile.txt" // // Supply the correct file name and its absolute path. // Dim f As FolderItem = GetFolderItem( "C:\\MyData\\WorkArea\\MyFile.txt" ) If f <> Nil Then If f.Exists Then // Be aware that TextInputStream.Open coud raise an exception Dim t As TextInputStream Try t = TextInputStream.Open(f) t.Encoding = Encodings.MacRoman TextArea1.Text = t.ReadAll t.Close Catch e As IOException t.Close MsgBox("Error accessing file.") End Try Else MsgBox("f does not Exist.") End If Else MsgBox("f is Nil.") End If

The app opens the file now!!! It looks like I was missing “c:” from my absolute path.

Thank you guys!! Y’all are awesome

I was able to figure out the last part on my own. I needed to move the content into a string so i used

dim s as string = textare1.text
msgbox(s) //To make sure that the command worked as I wanted it to

The message box shows the same as the text area. Now do you guys have any suggestions on how to do this without using the textarea?
What I mean by this is copy the content of the file into a string without displaying it into the text area, or using the textarea as an intermediate place.

If this is not possible I could always just make the text are invisible in the app so it’s not an issue. I’m just curious.

Thanks Again!!

Simply substitute the variable name for the textarea name.

TextArea1.Text = t.ReadAll

becomes

s = t.ReadAll

Thank you everyone. You have been a lot of help.


text:
# Zech's logarithm

(Redirected from Jacobi logarithm)

Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator ${\displaystyle \alpha }$.

Zech logarithms are named after Julius Zech,[1][2][3][4] and are also called Jacobi logarithms,[5] after Carl G. J. Jacobi who used them for number theoretic investigations.[6]

## Definition

Given a primitive element ${\displaystyle \alpha }$  of a finite field, the Zech logarithm relative to the base ${\displaystyle \alpha }$  is defined by the equation

${\displaystyle Z_{\alpha }(n)=\log _{\alpha }(1+\alpha ^{n}),}$

or equivalently by

${\displaystyle \alpha ^{Z_{\alpha }(n)}=1+\alpha ^{n}.}$

The choice of base ${\displaystyle \alpha }$  is usually dropped from the notation when it is clear from the context.

To be more precise, ${\displaystyle Z_{\alpha }}$  is a function on the integers modulo the multiplicative order of ${\displaystyle \alpha }$ , and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol ${\displaystyle -\infty }$ , along with the definitions

${\displaystyle \alpha ^{-\infty }=0}$
${\displaystyle n+(-\infty )=-\infty }$
${\displaystyle Z_{\alpha }(-\infty )=0}$
${\displaystyle Z_{\alpha }(e)=-\infty }$

where ${\displaystyle e}$  is an integer satisfying ${\displaystyle \alpha ^{e}=-1}$ , that is ${\displaystyle e=0}$  for a field of characteristic 2, and ${\displaystyle e={\frac {q-1}{2}}}$  for a field of odd characteristic with ${\displaystyle q}$  elements.

Using the Zech logarithm, finite field arithmetic can be done in the exponential representation:

${\displaystyle \alpha ^{m}+\alpha ^{n}=\alpha ^{m}\cdot (1+\alpha ^{n-m})=\alpha ^{m}\cdot \alpha ^{Z(n-m)}=\alpha ^{m+Z(n-m)}}$
${\displaystyle -\alpha ^{n}=(-1)\cdot \alpha ^{n}=\alpha ^{e}\cdot \alpha ^{n}=\alpha ^{e+n}}$
${\displaystyle \alpha ^{m}-\alpha ^{n}=\alpha ^{m}+(-\alpha ^{n})=\alpha ^{m+Z(e+n-m)}}$
${\displaystyle \alpha ^{m}\cdot \alpha ^{n}=\alpha ^{m+n}}$
${\displaystyle \left(\alpha ^{m}\right)^{-1}=\alpha ^{-m}}$
${\displaystyle \alpha ^{m}/\alpha ^{n}=\alpha ^{m}\cdot \left(\alpha ^{n}\right)^{-1}=\alpha ^{m-n}}$

These formulas remain true with our conventions with the symbol ${\displaystyle -\infty }$ , with the caveat that subtraction of ${\displaystyle -\infty }$  is undefined. In particular, the addition and subtraction formulas need to treat ${\displaystyle m=-\infty }$  as a special case.

This can be extended to arithmetic of the projective line by introducing another symbol ${\displaystyle +\infty }$  satisfying ${\displaystyle \alpha ^{+\infty }=\infty }$  and other rules as appropriate.

Notice that for fields of characteristic two,

${\displaystyle Z_{\alpha }(n)=m}$ ${\displaystyle Z_{\alpha }(m)=n}$ .

## Uses

For sufficiently small finite fields, a table of Zech logarithms allows an especially efficient implementation of all finite field arithmetic in terms of a small number of integer addition/subtractions and table look-ups.

The utility of this method diminishes for large fields where one cannot efficiently store the table. This method is also inefficient when doing very few operations in the finite field, because one spends more time computing the table than one does in actual calculation.

## Examples

Let α ∈ GF(23) be a root of the primitive polynomial x3 + x2 + 1. The traditional representation of elements of this field is as polynomials in α of degree 2 or less.

A table of Zech logarithms for this field are Z(−∞) = 0, Z(0) = −∞, Z(1) = 5, Z(2) = 3, Z(3) = 2, Z(4) = 6, Z(5) = 1, and Z(6) = 4. The multiplicative order of α is 7, so the exponential representation works with integers modulo 7.

Since α is a root of x3 + x2 + 1 then that means α3 + α2 + 1 = 0, or if we recall that since all coefficients are in GF(2), subtraction is the same as addition, we obtain α3 = α2 + 1.

The conversion from exponential to polynomial representations is given by

${\displaystyle \alpha ^{3}=\alpha ^{2}+1}$  (as shown above)
${\displaystyle \alpha ^{4}=\alpha ^{3}\alpha =(\alpha ^{2}+1)\alpha =\alpha ^{3}+\alpha =\alpha ^{2}+\alpha +1}$
${\displaystyle \alpha ^{5}=\alpha ^{4}\alpha =(\alpha ^{2}+\alpha +1)\alpha =\alpha ^{3}+\alpha ^{2}+\alpha =\alpha ^{2}+1+\alpha ^{2}+\alpha =\alpha +1}$
${\displaystyle \alpha ^{6}=\alpha ^{5}\alpha =(\alpha +1)\alpha =\alpha ^{2}+\alpha }$

Using Zech logarithms to compute α6 + α3:

${\displaystyle \alpha ^{6}+\alpha ^{3}=\alpha ^{6+Z(-3)}=\alpha ^{6+Z(4)}=\alpha ^{6+6}=\alpha ^{12}=\alpha ^{5}}$ ,

or, more efficiently,

${\displaystyle \alpha ^{6}+\alpha ^{3}=\alpha ^{3+Z(3)}=\alpha ^{3+2}=\alpha ^{5}}$ ,

and verifying it in the polynomial representation:

${\displaystyle \alpha ^{6}+\alpha ^{3}=(\alpha ^{2}+\alpha )+(\alpha ^{2}+1)=\alpha +1=\alpha ^{5}}$ .


text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Congruences of $p$-adic integer order Bernoulli numbers. (English) Zbl 0866.11013

The $p$-adic integer order Bernoulli numbers ${B}_{n}^{\left(p\right)}$ of the title are defined by

${\left(\frac{t}{{e}^{t}-1}\right)}^{p}·{e}^{xt}=\sum _{n=0}^{\infty }{B}_{n}^{\left(p\right)}\left(x\right)·\frac{{t}^{n}}{n!},$

with ${B}_{n}^{\left(p\right)}={B}_{n}^{\left(p\right)}\left(0\right)$, where $p$ is an arbitrary $p$-adic integer. In this very well-written paper the author proves some new congruences for ${B}_{n}^{\left(p\right)}$. These generalize e.g. the classical Kummer congruences for ordinary Bernoulli numbers. Several results by F. T. Howard or L. Carlitz are also extended.

The congruences (which are too complicated to be stated here) have ramifications for the Stirling numbers of the first and second kind. These are consequences of irreducibility theorems on certain Bernoulli polynomials of order divisible by $p$, as an application of the very general congruence properties obtained by the author for ${B}_{n}^{\left(p\right)}$ and related numbers.

In the Erratum a correct statement of Theorem 1 (iii) is given as well as two minor misprint corrections.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11S80 Other analytic theory of local fields