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The four-year cycle cheap abana 60 pills free shipping cholesterol test variability, 2000-2004 abana 60pills line cholesterol levels during pregnancy, became known as the prophetic cycle of the Har rowing of Hell trusted 60pills abana cholesterol levels japan. Just as Jesus was crucified and then experienced the Harrowing of Hell before his resurrection buy 60pills abana cholesterol eyes, so too humanity has been crucified on the technospheric cross of false time, and is now in the Harrowing of Hell awaiting the resurrection in natural time. For having failed to take seriously all its messengers and all its prophets of all times, humanity was now to pass en masse through hell. Since the Galactic Storm year, 2000-2001, completed a Seed-Storm Year bundle begun in the fifth year of proph ecy (1997-1998), the 2000-2001 Blue Galactic Storm year was the year of evaluation. The Solar Seed year is the second year of the Harrowing of Hell, but the first year of the next four-year Seed-Storm Year bundle. The first half of the Yellow Solar Seed occurs during the second half of the first official year of the third millennium, 2001. The moment was now ripe for the Inevitable Event, for it was also the same Gregorian calendar year as 1945 and 1973. In the Telektonon Prophecy, verse 120, section 19, it is declared, "When the Lunar Moon has overflowed its banks, the G-7 will be no more. Only such a harmonic organizing prin ciple, universally accepted and applied, can ensure a time of harmony and the fulfill ment of the Harmonic Convergence. Is it possible that within the noosphere the Harmonic Convergence is awaiting its next climax on the Day Out of Time and the Thirteen Moon Synchronization date of the year 2004, that is, on Gregorian July 25 and 26, 2004? One reads through the Journal of Calendar Reform and the many texts and arguments for calendar re form that were put forth in the 1930s and feels the missed opportunity. Even Gandhi was in favor of a new universal timing standard as a principle means of unifying humanity in peace. The League of Nations had promoted calendar reform as one of its major agendas since the 1920s. In order to remain constant, any perpetual cal endar ordered by the seven-day week requires the principle of a null day, a Day Out of Time that is no day of the week at all. Reading through archives and literature, pro and con, it is clear that what defeated the calendar reform was the conservative, Vatican-inspired objection to the null day, the Day Out of Time. To observe such a day, which is no day of the week at all, it was argued, would break the weekly succession of days set in motion by God at the beginning of creation-such a break would plunge the world into barbarism, chaos, and war. The cult of Christianity is inextricably connected to the week whose origin reaches to the dawn of history. As a result of such propaganda, the League of Nations faltered, no calendar reform occurred at all, and the world was plunged into war-a war that ended with the beginning of the age of terror. This terror is now running its course, like a fever sweeping through the biosphere. The Advent of the Noosphere Who really lost out when the calendar reform issue was tabled in 1956 at the United Nations? It was not God who invented the week as we know it, but more likely the Babylonians. Behind the week is the mystery of the whole number seven, recalled in the Epic of Gilgamesh as the seven wise men ofU ruk. But the Babylonians clothed the seven-day cycle in the raiment of the sun, moon, and five principle plan ets-Mercury, Venus, Mars, Jupiter, and Saturn-to which the days of the week have no actual relation. Seen in this way, the seven-day week was intended to be a means on Earth for reenacting some remote mythic reality related to these seven heavenly bodies. What was that mythic reality, on what hidden knowledge is the count of the seven-day week presumed? Borrowing from Babylon, the ancient Hebrews incorpo rated the seven-day week into their purely lunar calendar. And it was thus from the Hebrews that the seven-day week entered the Julian and then the Gregorian calen dars. It was not, however, the Romans who adopted the week, but the Christian successor to the Roman empire, the Church of Rome. Colson, a Christian scholar writing as late as 1926, the seven-day cycle is described as "that intruder the week, consisting indeed of a fixed number of days, but paying no regard to months or years. To those whose minds are set on the synodic lunation cycle as the absolute standard, as does the Gregorian calendar, the week is what ensures the greatest disorder in the calendar and yet it is defended to the death. The week gains its power from the whole number 7, and not from the jumble of names in which the week is clothed. The seven-day cycle is the closest whole number that corresponds to the four phases of the moon, which, as whole numbers, are also seven days each. In fact, the fixed 7-day cycle only makes sense within the context of the Thirteen Moon calendar with its 28-day cycles divisible by four and seven, creating the annual cycle of fifty-two perfect weeks-plus the Day Out of Time, ensuring a perpetual harmony. Was the Babylonian naming of these days-with only one name a reference to the moon-meant to be a cover-up of some earlier knowledge, a knowledge both more purely lunar and harmonic, such as the Thirteen Moon calendar? What would really happen if there were a Day Out of Time, no day of the week or month at all? Since 1993, sixty years after the failure of the League of Nations to enact calendar reform, the Thirteen Moon Calendar Change Peace Movement has made certain that the the Adventof the Noosphere. All the while, the Gregorian world has arrogantly plunged blindly forward, locked into its mechanistically chaotic timing sensibility, only to receive the mortal wound of the Inevitable Event, causing it to fall through the seams of its own making, the technosphere. A Day Out of Time, it seems, might be a relief to a world now consumed by chaos, barbarism, and war. He sends down from the sky water for your drink and to grow trees for your benefit. With it He grows for you crops, olives, date palms, grapes, and all kinds of fruits.

However discount abana 60 pills without prescription cholesterol levels dogs, if there exists a pair with equal remainders when divided by 4 then no odd perfect number exists order 60pills abana otc cholesterol quick fix. Prove that if (m 60 pills abana overnight delivery lower cholesterol without medication quickly, n) is an amicable pair then 140 Perfect and amicable numbers Table 4 generic 60pills abana free shipping lowering good cholesterol foods list. In Whetstone, Recorde introduced the modern symbol of two horizontal line seqments for equals, `? Multiperfect numbers were studied extensively by French mathemati cians in the seventeenth century. In 1638, AndreA Jumeau, prior of Sainte Croix, Oloron-Ste-Marie, showed that 523 776 was a 3-perfect number, and issued a second challenge to Descartes to nd another 3-perfect number. Descartes responded that 1 476 304 896 is 3-perfect and listed six 4-perfect numbers, and two 5 perfect numbers. Descartes claimed that if n was 3-perfect and not divisible by 3 then 3n is 4-perfect; if 3 divides n and both 5 and 9 do not divide n then 45n is 4-perfect; if 3 divides n and 57, 9, and 13 do not divide n then 3. Multiperfect Number known type 2-perfect 37 3-perfect 6 4-perfect 36 5-perfect 65 6-perfect 245 7-perfect 516 8-perfect 1097 9-perfect 1086 10-perfect 25 Table 4. Eight years later, Fermat found the 3-perfect number 51 001 180 160, 2 4 perfect numbers, 2 5-perfect numbers, and the rst 2 6-perfect numbers. In 1647, Mersenne claimed that if n were 5-perfect and 5 did not divide n then 5n would be 6-perfect. In 1929, Poulet listed 36 4-perfect numbers, 55 5-perfect numbers, 166 6-perfect numbers, 69 7-perfect numbers and 2 8 perfect numbers, one of them being 262. In the 1950s, Benito Franqui and Mariano GarcoAa at the University of Puerto Rico and Alan Brown independently generated about 100 multiperfect numbers, albeit there were a few numbers common to both lists and some overlap with the multiperfect numbers generated by Poulet 25 years earlier. In 1974, Daniel Minoli and Robert Bear described a num ber of properties of hyperperfect numbers. They conjectured that for each posi tive integer k there exists a k-hyperperfect number. A positive integer n is called semiperfect or pseudoperfect if there exists a collection of distinct proper divisors of n such that their sum is n. Every multiple of a semiperfect number is semiper fect, hence, there are in?nitely many semiperfect numbers. A positive integer is called primitive semiperfect if it is semiperfect and is not divisible by any other semiperfect number. In 1680, Leibniz conjectured that if n was not prime then n did not divide 2n A 2. Even though all composite Fermat numbers are pseudoprime, pseudoprimes are much rarer than primes. A composite integer m is called a Carmichael 146 Perfect and amicable numbers Table 4. Chernick number m is a number that is a k-pseudoprime for all values of k, where gcd(k, m)? Korselt devised a criterion in 1899 for such numbers showing that a positive integer n is Carmichael if and only if n is squarefree and p A 1 divides n A 1 for all primes p which divide n. Pomerance showed that there are no more than n2a7 Carmichael numbers less than or equal to n. Richard Pinch of Cambridge University calculated all 105 212 Carmichael numbers less than 1015. Srinivasan de?ned a positive integer n to be practical if every positive integer less than n can be expressed as a sum of distinct divisors of n. The integer 10 is not practical since 4 cannot be expressed as a sum of distinct divisors of 10. A positive integer n is called unitary nonrepetitive, if, excluding the divisors 1 and n, it is possible to express n A 1 as a sum of some or all of the remaining divisors of n using each divisor once and only once. A positive integer is called harmonic if the harmonic mean of its divisors is an integer. Thabit ibn Qurra introduced two terms that describe the deviation of a number from being perfect. He de?ned the abundancy of an abundant number, denoted by a(n), as o(n) A 2n and the de?ciency of a de?cient number, denoted by a(n), as 2n A o(n). A positive integer n is called quasiperfect if it has an abundancy of 1 and almost perfect if it has a de?ciency of 1. Every quasiperfect number n is the square of an odd integer, is greater than 1020, and u(n) > 5, but so far none has been found. Suryanarayana showed that all even superperfect numbers are of the form 2 pA1, where 2 p A 1isa Mersenne prime. In 1975, Carl Pomerance showed that there are no odd superperfect numbers less than 7. In 1944, Paul ErdoEs and Alaoglu de?ned a positive integer n to be superabundant if o(n)an. Paul ErdoEs de?ned a positive integer n to be untouchable if there does not exist a positive integer x such that o(x)? The 148 Perfect and amicable numbers only unitary perfect numbers known are 6, 60, 90, 87 360, and 146 361 946 186 458 562 560 000 (218. In 1971, Peter Haggis de?ned a pair of positive integers (m, n)tobe unitary amicable if oA(m)? Nineteen unitary amicable pairs have been discovered including (114, 126), (1140, 1260), and (18 018, 22 302). It is an open question whether there are in?nitely many pairs of unitary amicable numbers. The arithmetic mean of the divisors of a positive integer is denoted by A(n) and given by A(n)? Determine the arithmetic mean of the divisors of pa, where p is prime and a is a positive integer.

The es fve most common incident sites of timated global annual numbers of cancers were the lung (13 generic 60 pills abana otc cholesterol ranges healthy. Among men buy generic abana 60pills online cholesterol in eggs pdf, the fve most com Estimated age-standardized in mon sites of cancer diagnosed in cidence and mortality rates for the Chapter 1 cheap abana 60 pills without a prescription definition de cholesterol ldl. Estimated world cancer incidence proportions by major sites generic abana 60pills with mastercard cholesterol levels statistics, in both sexes combined, in men, and in women, 2012. Estimated world cancer mortality proportions by major sites, in both sexes combined, in men, and in women, 2012. As in men, mon cancer sites worldwide are dence rates for all cancers com the stomach cancer mortality rate shown in Fig. Rates in the youngest rates for the other major cancers in vors diagnosed within the previous age group (0?14 years) are about women cancers of the colorectum 5 years, breast cancer is by far the 10 per 100 000, increasing to 150 (6. Prevalence in women and then become sub with cancer and are still alive at a refects the integration of incidence stantially higher from the age of specifc time [9]. The excess in women timates for 2012 show that there cancer, because of its very poor before the age of 50 years is due 20 Fig. Estimated world cancer 5-year prevalence proportions by major sites, in both sexes combined, in men, and in women, 2012. Estimated world cancer incidence rates per 100 000 by 5-year age group, of the upper rate category is higher for all sites combined (excluding non-melanoma skin cancer), in men and women, in men than in women. For example, Uruguay and Mongolia fall into the high-incidence category, Uruguay partly because of particularly high rates for lung cancer (and other smoking-related cancers), which are now being brought under con trol [10], and Mongolia because of the extraordinary high rate of liver cancer resulting from the particular ly high prevalence of infection with hepatitis B and C viruses [11]. These also show ex tensive international variation, but the contrasts are less marked than those for incidence. For example, there is less variation between North and South America and be tween western and eastern Europe. In both cases, this is a result of the impact of clinical care and generally improved cancer survival in North America and western Europe, lead ing to lower mortality rates (relative to the incidence rates) than is ob served in South America and east ern Europe [12]. In addition, and as described in the discussion of to the relatively earlier age at onset Global distribution regional patterns of cancer below, of cervical cancer and, especially, Incidence cancers associated with a lifestyle breast cancer compared with other World maps of estimated age typical of industrialized countries, major cancers. Above the age of standardized incidence rates for including cancers of the breast, 60 years, prostate cancer and lung all cancers combined (exclud colorectum, and prostate, have a cancer in men become more com ing non-melanoma skin cancer) relatively good prognosis, whereas mon. In general, and with mon in low-income countries, have 165 000 (95 000 in boys and 70 000 some exceptions, the highest in a signifcantly poorer prognosis. In in girls), and in teenagers and young cidence rates are associated with general, therefore, relatively more adults (15?39 years), the estimate the high-income countries of North cancer deaths are seen for a given is just more than 1 million (380 000 America and western Europe (to number of incident cases in the less in men and 670 000 in women). For gether with Japan, the Republic economically developed countries, more information on the descriptive of Korea, Australia, and New and cancer mortality rates are not epidemiology of cancer in these Zealand). This is the case for both very different from those in more age groups, see Chapter 1. A B 24 Regional patterns of cancer Europe incident sites in men and women, A regional breakdown of the global In Europe, the incidence pattern is respectively, and lung cancer is cancer incidence and mortality bur dominated by prostate and breast the most common cause of cancer den by continental region is provid cancers, which are the most com death in both sexes 1. However, unlike in Europe, cidence burden occurs in Asia, and spectively, together with lung and lung cancer is also relatively more almost a half of this, or 22% of the colorectal cancers (Fig. A quarter of the incidence because of its poor survival, is the alongside bladder and kidney can burden occurs in Europe, and the most important cause of cancer cers, malignant melanoma, non remainder is divided between the death in men (Fig. Bladder, Hodgkin lymphoma, and leukaemia Americas and Africa (with 1% in stomach, and kidney cancers also contribute signifcantly to the can Oceania). The mortality proportion contribute signifcantly to the bur cer burden in men, with incidence al distribution shows an increase den in men, with incidence rates rates of more than 10 per 100 000 in the proportion of cancer-related of more than 10 per 100 000, (Fig. This is also evident for deaths occurring in Asia and Africa, whereas in women, cancers of the malignant melanoma, non-Hodgkin together with a decrease in the pro corpus uteri and cervix also have lymphoma, corpus uteri, and thy portions occurring in the economi incidence rates of more than 10 per roid cancers in women. The prevalence pattern in deaths by sex, together with histo North America North America is very similar to that grams showing the age-standard In North America, patterns of the in Europe, with prostate, breast, ized incidence and mortality rates four major cancers are very similar and colorectal cancers together by cancer site and a chart showing to those in Europe. Prostate and accounting for half of all the 5-year the major 5-year prevalent cancers. Estimated world cancer incidence and mortality proportions by major world regions, in both sexes combined, 2012. Among most common cause of cancer these are much higher than those men, prostate and liver cancers are death in men 1. Colorectal prevalence in Oceania, as the third much higher but mortality is much cancer is the next most common most prevalent type. The importance of liver can cancer in women, but breast cancer pattern of prevalence is similar to cer in this region should be empha is the only type for which the inci that in Europe and North America sized; it is the second most com dence or mortality rates (43. Prostate, blad sub-Saharan Africa, especially in most common incident sites in der, colorectal, and liver cancers men, should also be noted. Kaposi men and women, respectively are the next most common, all with sarcoma is the third most common (Fig. Breast cancer is the incidence rates of more than 10 cancer in men and represents 9. In this region, breast of all cancer diagnoses, with an death in women, whereas prostate cancer represents a quarter of incidence rate of 7. However, and tively less prevalent compared with ated cancer sequelae before the unlike the situation in the more eco other world regions, bladder and advent of highly active antiretroviral nomically developed regions, cervi thyroid cancers make important therapy [13]. Cervical cancer and cal cancer makes a major contribu contributions to the overall pattern Kaposi sarcoma also make impor tion to the cancer burden in women, (Fig. Stomach is equivalent to breast cancer in taining 57% of the global popu cancer is also important, especially terms of incidence (each consti lation (19% in China and 18% in in men, where it is the fourth most tutes approximately a quarter of India). Among men, the most com common incident cancer (with lung the total burden) and is the most mon cancers and causes of cancer 26 death are cancers of the lung, corresponding rates for stomach a little different in the region, and stomach, liver, colorectum, and cancer (23. Due tant contributors to the pattern of stantively below those in Europe to the varying fatality rates of these prevalence (Fig.

For an enlightening look at the subject of triskaidekaphobia 60 pills abana with visa cholesterol levels in pork chops, see Jonathon Cott purchase 60pills abana with visa cholesterol lowering foods drinks, Thirteen: A Journey into the Number (New York: Doubleday purchase 60 pills abana free shipping cholesterol levels targets, 1996) order abana 60 pills with visa cholesterol medications that don't affect the liver. Ben 4 and Ben 5, 13 Moon Natural Time Calendar,Ancient Sciencefor the Art of Now (Newport, Oreg. Nuestra Civilizacion Frente a la Reforma del Calendario(Our Civilization Confronted with Calendar Reform) (Buenos Aires: Casa Editora Sudamericana, 1944), 6. Colson, the Week: An Essay on the Origin and Development of the Seven-Day Cycle (Cambridge: Cambridge University Press, 1926), 2. Quoted in Jose Argiielles, CharlesHenry and the Formation of a PsychophysicalAesthetic (Chicago: University of Chicago Press, 1972), 128. Charles Muses, Destiny and Control in Human Systems: Studies in the Interactive Connect ednessof Time (Chronotopology)(Boston Dordrecht Lancaster: Kluwer-Nijhoff Pub lishing, 1984), 20-21. All further quotes from the Quran and from Rashad Khalifa are from this same work, which in addition to containing the complete Quran in translation also includes thirty-eight Appendices summarizing Dr. Though by no means ex haustive, it is hoped that the terminology used in the text is herein defined in a manner appropriate to cultivating a noospheric point of view. Chilam Balam: "Jaguar priest," name of Mayan prophet (ah bobat)who lived just before the time of conquest, which he accurately predicted; name given to lineage of Mayan prophets extending from the end of tenth baktun (circa A. Dreamspell: Any agreed-on consensus reality; a collective mind-set defining a noospheric epoch, such as "Dreamspell of history"; term given to proofs and tools demonstrating radial and fractal principles of fourth-dimensional mathematics; cosmo-mythic structure of Earth history and its future, otherwise known as Dreamspell: the Journey of Timeship Earth 2013. Hannonic Module: 260-unit mathematical matrix of 13:20 frequency, inclusive of a 52-unit binary triplet configuration or set of galactic activation portals; sometimes referred to as Tzolkin, 260-kin sacred count of the Maya; basis of fourth-dimensional vigesimal math ematics of time; coordinated with third-dimensional calendar of Thirteen Moons, estab lishes basis of 260 galactic signatures. Hunab Ku: "One giver of movement and measure"; in Mayan cosmology, can refer to God as the Supreme Creator. Inevitable Event: Law of Time definition of the point in synchronic order when the artifi cial time generated curves of machine, human population, and money converge exponen tially in one dramatic, prophecy-fulfilling moment, marking the conclusion oftechnospheric expansion; name given to the events of September 11, 2001 (9-11), epitomized by the col~ lapse of World Trade Center Twin Towers (Fall of Babylon). Julian Count: Established by Thomas Scaliger, 1583, the year after Gregorian calendar "reform," to establish a linear count of days beginning January 1, 4714 B. Law of Time: T(E) =Art, Energy factored by Time equals Art, where T (time) is the fourth dimensional universal synchronization frequency constant 13:20, E (energy) refers to any phenomenal manifestation, and Art to the fact that the synchronizing frequency of time harmonizes everything into a natural condition of beauty. Long Count: Mayan count of days (kin) measured by thirteen baktuns of 144,000 days each for a total of 1,872,000 kin, the measure of history from JulianiGregorian correlates August 13, 3113 B. Oxlahuntiku: the Thirteen Lords of Time that govern the power of the cosmic order of time that is based on circulatory power of prime number 13 (as opposed to 12, which is static and non-circulatory); coded as thirteen 28-day moons that govern terrestrial solar orbit; basis of prophecy cycle of Quetzalcoatl, Thirteen Heavens (Oxlahuntiku) and Nine Hells (Bolontiku); in Telektonon prophecy, the thirteen-year cycle-2 000-2 013-in which the first four years, 2000-2004, establish self-existing power of time, and last nine years, 2004-2013 are coded by the Bolontiku, the Nine Lords of Time. Pax Cultura, Pax Biospherica: Principle of new human social order following establishment of Thirteen Moon/28-Day calendar and inaugurating the advent of the noosphere (2004) in which nation-state is replaced by biospheric regionalism governed by a Planetary Biospheric Assembly; due to perfect harmony of new world standard calendar of thirteen moons/28 days, harmony, art, and aesthetic values elevate culture as the new operating norm of the human reintegrated within the biosphere, hence Pax Cultura, Pax Biospherica. Vernadsky defining the next geological era; meaning the era of the spiritualization of life, it is coincident with the noosphere in its full emergence after the omega point has been reached, 2012-2013. Quran, Holy Quran: Literally "recitation;" last revelation and moral-spiritual criterion for historic humanity; received over a twenty-three-year period by the Prophet Muhammad (A. Rashad Khalifa (1935-1990) to be a radically nonlinear text rigorously constructed of a nineteen-based mathematical code. Glossary names of the seven plasmas replace the names of the week in the Thirteen Moon/2 8-Day calendar; activated in the 7:7::7:7 Telektonon Revelation practice. Rinri Project: Law of time practice for coordinating the opening of the psi bank, psi chrono unit per psi chrono unit, on a daily basis over a four-year cycle by means of the Thirteen MoontTelektonon. Telektonon: "Earth Spirit Speaking Tube," the name of the psychoduct leading from the tomb of Pacal Votan beneath the Temple of the Inscriptions to the temple floor atop the pyramid, the discovery of which, in 1949, led to the excavation and revelation of the long concealed tomb in 1952; name given to the prophecy ofPacal Votan, decoded in 1993-1994, prophesying the Thirteen Moon/2 8-Day calendar as the only means to keep the human race from destroying the biosphere and itself; definition of the fourth-dimensional structure of the heliosphere in which consciousness is a function of the planetary orbits; basis of a board "game" coded to synchronic order via Thirteen Moon calendar for the awakening of telepa thy; basis of the Cube of the Law, days 7-22, containing numerous principles of the Law of Time as a daily practice; basis of other practices for establishing synchronic order as prin ciples of a new consciousness, including 20 Tabletsof the Law of Time and the 7:7::7:7. Tzolkin: "Sacred Count," 260-day calendar of the Maya, basis of Mayan time knowledge and the Long Count; temporal manifestation of Harmonic Module or universal 13:20 timing matrix. Wave Harmonic of history: Fourth-dimensional description of thirteen baktun cycle (3113 B. Living Energies:An Exposition of ConceptsRelated to the Theoriesof Viktor Schauberger. Vednie V Systemy "Ekoprognos":MetodologiyaPrognosirovaniyaAnomalii Ekosperi (Ecoprognosis: Methodology for Making Prognosis of Anomolous Phenom ena in the Ecosphere). Cycles of the Sun, Mysteries of the Moon: the Calendar in Mesoamerican Civilization. Destiny and Control in Human Systems: Studies in the Interactive Connected ness of Time (Chronotopology). Nuestra Civilizacion Frente a la Reforma del Calendario (Our Civilization Confronted with Calendar Reform). The Theory and Use of the Church Calendar in the Measurement and Distribution of Time. The Study of Time: Proceedings of the First Conference of the International Society for the Study of Time, Oberwolfach, Black Forest, West Germany. Principia Mathematica of the Fourth Dimension: Introduction and Annotated Bibliography. About the Inevitable Event: Prophecy, History, and Politics Ahmad, Mirza Ghulam of Qadian. From the Beginnings of Lift to the Day of Purification: Teachings, History & Prophecies of the Hopi People. Beneath the Moon and Under the Sun: A Reappraisal of the Sacred Calendar and the Prophecies of Ancient Mexico. Tulku Thondup Rinpoche, Hidden Teachings of Tibet: An Explanation of the Terma Tradition of Tibetan Buddhism. Professor Department of Biological Sciences Drug Discovery & Development: Bench, Bedside, & Beyond I. Ethical Issues: Money, Data, & Politics Disclaimer Those who have knowledge, don?t predict.

If p is an odd prime and d divides p A 1 discount 60 pills abana cholesterol levels garlic, show that xd A 1 0 (mod p) has exactly d incongruent solutions modulo p cheap 60pills abana free shipping cholesterol medication vytorin side effects. If p is an odd prime and d divides p A 1 buy abana 60 pills without prescription worst high cholesterol foods, determine the d incongruent solutions to buy discount abana 60pills on-line how do cholesterol lowering foods work xd A 1 0 (mod p). A group G is called cyclic if it contains an element a, called a generator, such that for every element g in G there is an integer k such that g? Cryptology is the study of secrecy systems, cryptography, the design and implementation of secrecy systems, and cryptanalysis, the study of systems or methods of breaking ciphers. The message to be altered into secret form, the message we want to send, is called the plaintext. The device used to transform the plaintext into the ciphertext is called a cipher. Plaintext and ciphertext may be composed of letters, numbers, punctuation marks, or other symbols. Decryption or deciphering is the process of changing ciphertext back into plaintext. In order to make decryption more dif?cult, plaintext and ciphertext are often broken up into message units of a xed number of characters. The enciphering transformation can be thought of as a one-to-one function that takes plaintext message units into corresponding ciphertext message units. The process or method used in going from the plaintext to ciphertext and back to the plaintext is called a cryptosystem. Encryption or decryption is often mistaken for encoding or decoding, respectively. A code, however, is a system used for brevity or secrecy of communication, in which arbitrarily chosen words, letters, or phrases are assigned de?nite symbols. The demand for and use of cryptography are directly proportional to the literacy and paranoia of the peoples involved. One of the earliest references to Greek writing is found in Book 6 of the Iliad when King Proetus sends Bellerophon to Lycia with a document containing secret writing. In Book 5 of the History, Herodotus remarked that Histiaeus, the despot of Miletus who was being held by Darius, shaved and tattooed a message to revolt against the Persians on the head of a trusted slave. After waiting for the hair to grow in again, Histiaeus sent the slave to his son-in-law Aristagoras in Miletus who shaved the head and found the message. Thrasybulus, despot of Miletus, gives no written or verbal message to a messenger from Periander, tyrant of Corinth and one of the seven sages of the ancient world, but while walking through a eld of corn with him, cuts down any corn that was growing above the rest. This act of removing the fairest and strongest is related to Periander by the messenger and he interprets it as having to murder the most eminent citizens of Corinth. They enciphered some messages by wrapping a strip of papyrus or parchment helically around a long cylindrical rod called a skytale. Given a rod of the same radius and length, the strip could be wound around it helically and the message deciphered. Valerius Probus, a grammarian, wrote a treatise on the ciphers used by Julius Caesar. Suetonius, the Roman historian, wrote that Caesar used a cipher which simply replaced each letter in the alphabet by the letter three letters to the right, with the stipulation that X, Y, and Z were replaced by A, B, and C respectively, as shown in Table 7. One can hardly fail to get a feeling for the dearth of literacy during this period of Roman history. We can generalize character ciphers mathematically by translating the letters of the alphabet of any plaintext into numerical equivalents, for example, using Table 7. Let the letter P denote the numerical equivalent of a letter in the plaintext and the letter C denote the numerical equivalent 7. Accordingly, the corresponding deciphering transfor mation is given by P C A k (mod 26). In a relatively long sample of English text, the most frequently occurring letter will normally be e, followed by t, n, i, r, o and a, respectively. However, we cannot always assume that the natural frequency prevails in the plaintext, for it is not impossible to circumvent the natural frequencies of a language as well. The encipherer has divided the ciphertext into a uniform set of letters, quintuplets in this case, to disguise any natural lengths that may be apparent in the plaintext. The plaintext message expressed in quintuplets would read ifiha vesee nalit tlefa rther thano thers itisb ecaus eihav estoo donth eshou lders ofgia ntsxx, or with natural word length if i have seen a little farther than others it is because i have stood on the shoulders of giants, a quote attributed to Isaac Newton. The deciphering transformation for an af?ne cipher is given by P aA1(C A b) (mod 26), where 0 < P < 25 and aaA1 1 (mod 26). Nevertheless, a deciphering technique using the relative frequency of letters can be used to decipher most af?ne transformations as illustrated in the next example. Multiplying both sides of the congruence by 7, the inverse of 15 modulo 26, we get a 19 (mod 26). The barbarians were at the gates, culture and literacy went seriously into decline, and with them went cryptography. Almost singlehandedly, Bene dictine monasteries continued to serve as effective educational institutions throughout the Dark Ages. According to conservative estimates over 90 percent of the literate men between 600 and 1100 received their instruction in a monastic order. Very few scienti?c commentaries appeared and many of those that did were woefully primitive. As with mathematics and science, cryptology developed in India and Islamic countries during the European Dark Ages. The Kamasutra, written sometime between the third and fth centuries and attributed to Vatsyaya na, lists secret writing as one of the arts a woman should understand and practice.

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