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statsmodels.graphics.functional.banddepth(data, method='MBD')[source]¶
Calculate the band depth for a set of functional curves.
Band depth is an order statistic for functional data (see fboxplot), with a higher band depth indicating larger “centrality”. In analog to scalar data, the functional curve with highest band
depth is called the median curve, and the band made up from the first N/2 of N curves is the 50% central region.
The vectors of functions to create a functional boxplot from. The first axis is the function index, the second axis the one along which the function is defined. So data[0, :] is the first
functional curve.
method{‘MBD’, ‘BD2’}, optional
Whether to use the original band depth (with J=2) of [1] or the modified band depth. See Notes for details.
Depth values for functional curves.
Functional band depth as an order statistic for functional data was proposed in [1] and applied to functional boxplots and bagplots in [2].
The method ‘BD2’ checks for each curve whether it lies completely inside bands constructed from two curves. All permutations of two curves in the set of curves are used, and the band depth is
normalized to one. Due to the complete curve having to fall within the band, this method yields a lot of ties.
The method ‘MBD’ is similar to ‘BD2’, but checks the fraction of the curve falling within the bands. It therefore generates very few ties.
The algorithm uses the efficient implementation proposed in [3].
[1] (1,2)
S. Lopez-Pintado and J. Romo, “On the Concept of Depth for Functional Data”, Journal of the American Statistical Association, vol. 104, pp. 718-734, 2009.
Y. Sun and M.G. Genton, “Functional Boxplots”, Journal of Computational and Graphical Statistics, vol. 20, pp. 1-19, 2011.
Y. Sun, M. G. Gentonb and D. W. Nychkac, “Exact fast computation of band depth for large functional datasets: How quickly can one million curves be ranked?”, Journal for the Rapid Dissemination
of Statistics Research, vol. 1, pp. 68-74, 2012.
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Words containing i and o
This list of words with i and o in them has 21170 entries. It may be helpful for people looking for a word that contains the letters O and I.
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bromide, bromides, bromidic, bromids, bromin, bromine, bromines, bromins, bromism, bromisms, bronchi, bronchia, bronchial, bronchitis, bronzier, bronziest, bronzing, bronzings, broodier, broodiest,
brooding, brooking, brookite, brookites, brookline, broomier, broomiest, brooming, broomstick, broomsticks, brothering, brotherliness.
brotherlinesses, browbeating, brownie, brownier, brownies, browniest, browning, brownish, browsing, bryologies, bryonies, bubonic, bucolic, bucolics, buffaloing, bulldogging, bulldozing, bullion,
bullions, buncoing, bunion, bunions, bunkoing, buoyancies, buoying, burgeoning, burrowing, bushido, bushidos, businesswoman, businesswomen, busybodies, buttoning, cabildo, cabildos, cabinetwork,
cabinetworks, cabriole.
cabrioles, cacomixl, cacomixls, cacophonies, cactoid, caisson, caissons, cajoleries, cajoling, calamitous, calamitously, calamitousness, calamitousnesses, calcification, calcifications, calculation,
calculations, calibration, calibrations, calibrator, calibrators, calico, calicoes, calicos, california, calliope, calliopes, callosities, callosity, callousing, caloric.
calorics, calorie, calories, calumniation, calumniations, calumnious, cambogia, cambogias, cameoing, camion, camions, camisado, camisadoes, camisados, camisole, camisoles, camomile, camomiles,
camouflaging, campion, campions, cancellation, cancellations, cancroid, cancroids, cannonading, cannoning, cannonries, canoeing, canoeist, canoeists, canonic.
canonical, canonically, canonise, canonised, canonises, canonising, canonist, canonists, canonization, canonizations, canonize, canonized, canonizes, canonizing, canonries, canopied, canopies,
canopying, cantoning, canzoni, capacious, capacitor, capacitors, capitalization, capitalizations.
capitol, capitols, capitulation, capitulations, caponier, caponiers, caponize, caponized, caponizes, caponizing, capricious, capriole, caprioled, caprioles, caprioling, caption, captioned,
captioning, captions, captious, captiously, captivation, captivations, captivator, captivators, caracoling, caracolling, carbinol, carbinols, carbonating.
carbonation, carbonations, carbonic, carcinogen, carcinogenic, carcinogenics, carcinogens, carcinoma, carcinomas, carcinomata, carcinomatous, cardiogram, cardiograms, cardiograph, cardiographic,
cardiographies, cardiographs, cardiography, cardioid, cardioids, cardiologies, cardiologist, cardiologists, cardiology, cardiotoxicities, cardiotoxicity, cardiovascular, caribou, caribous, carillon,
carillonning, carillons, carioca, cariocas, cariole, carioles, carious, carnation, carnations, carnivore, carnivores, carnivorous, carnivorously, carnivorousness, carnivorousnesses, caroli, caroling,
carolling, caroming, carotid, carotids, carotin, carotins, carousing, carriole, carrioles, carrion, carrions, carroming, carrotier, carrotiest, carrotin, carrotins, cartilaginous, cartographies,
cartoning, cartooning, cartoonist.
cartoonists, caryotin, caryotins, casino, casinos, cassino, cassinos, castigation, castigations, castigator, castigators, castration, castrations, cataloging, catastrophic, catastrophically,
categorical, categorically, categories, categorization, categorizations, categorize, categorized, categorizes, categorizing, catenoid.
catenoids, catheterization, cathodic, catholic, cation, cationic, cations, caudillo, caudillos, cauliflower, cauliflowers, causation, causations, cauterization, cauterizations, caution, cautionary,
cautioned, cautioning, cautions, cautious, cautiously, cautiousness, cautiousnesses, cavicorn, cavorting, ceboid, ceboids, celebration, celebrations, cementation.
cementations, cenobite, cenobites, censorial, censoring, censorious, censoriously, censoriousness, censoriousnesses, censorship, censorships, centimo, centimos, centralization, centralizations,
centroid, centroids, centurion, centurions, ceorlish, ceratoid, cerebration, cerebrations, cerebrospinal, ceremonial, ceremonies.
ceremonious, cerotic, certification, certifications, cessation, cessations, cession, cessions, cestoi, cestoid, cestoids, cetologies, chairwoman, chairwomen, chamiso, chamisos, chamois, chamoised,
chamoises, chamoising, chamoix, champion, championed, championing, champions, championship, championships, chancellories, chancellorship, chancellorships, chaotic, chaotically, chaperoning,
characterizations, charcoaling, chariot, charioted, charioting, chariots, checkpoint, checkpoints, checkrowing, cheerio, cheerios, cheloid, cheloids, chemotherapeutic, chemotherapeutical,
chemotherapies, cheviot, cheviots, chiao, chibouk, chibouks, chiccories, chiccory, chico, chicories, chicory, chicos, chiefdom, chiefdoms, chiffon, chiffons, chignon, chignons, chigoe, chigoes,
childhood, childhoods, chilopod, chilopods, chinbone.
chinbones, chino, chinone, chinones, chinook, chinooks, chinos, chiro, chiropodies, chiropodist, chiropodists, chiropody, chiropractic, chiropractics, chiropractor, chiropractors, chiros, chiton,
chitons, chivalrous, chivalrously, chivalrousness, chivalrousnesses, chlorambucil, chloric, chlorid, chloride, chlorides, chlorids, chlorin, chlorinate.
chlorinated, chlorinates, chlorinating, chlorination, chlorinations, chlorinator, chlorinators, chlorine, chlorines, chlorins, chlorite, chlorites, chloroforming, chocking, choice, choicely, choicer,
choices, choicest, choir, choirboy, choirboys, choired, choiring, choirmaster, choirmasters, choirs, chokier, chokiest.
choking, choleric, choline, cholines, chomping, choosier, choosiest, choosing, chopin, chopine, chopines, chopins, choppier, choppiest, choppily, choppiness, choppinesses, chopping, chopsticks,
choragi, choragic, chording, choregi, choreic, choreographic, choreographies, choreographing, choreoid, chorial, choriamb, choriambs, choric, chorine, chorines.
choring, chorioid, chorioids, chorion, chorions, chorister, choristers, chorizo, chorizos, choroid, choroids, chortling, chorusing, chorussing, chousing, chowdering, chowing, chowsing, chowtime,
chowtimes, chrismon, chrismons, chrisom, chrisoms, chromatic, chromic, chromide, chromides, chroming, chromite.
chromites, chromium, chromiums, chromize, chromized, chromizes, chromizing, chronaxies, chronic, chronicle, chronicled, chronicler, chroniclers, chronicles, chronicling, chronics, chronologic,
chronological, chronologically, chronologies, chthonic, churchgoing, churchgoings.
chymosin, chymosins, ciao, cibol, cibols, ciboria, ciborium, ciboule, ciboules, cicero, cicerone, cicerones, ciceroni, ciceros, cicisbeo, cicoree, cicorees, cilantro, cilantros, cinchona, cinchonas,
cineol, cineole, cineoles, cineols.
cinnamon, cinnamons, cion, cions, ciphonies, ciphony, cipolin, cipolins, circuitous, circulation, circulations, circulatory, circumcision, circumcisions, circumlocution, circumlocutions,
circumnavigation, circumnavigations, circumspection, circumspections, cirrhoses, cirrhosis, cirrhotic, cirrose, cirrous, cirsoid, cisco, ciscoes, ciscos, cissoid, cissoids.
cistron, cistrons, citation, citations, citatory, citola, citolas, citole, citoles, citreous, citron, citrons, citrous, civilization, civilizations, clairvoyance, clairvoyances, clairvoyant,
clairvoyants, clamoring, clamouring, clangoring, clangouring, clarification, clarifications, clarion, clarioned, clarioning, clarions, classification, classifications.
claustrophobia, claustrophobias, clavichord, clavichords, clipboard, clipboards, clitoral, clitoric, clitoris, clitorises, cloaking, clobbering, clocking, clockwise, cloddier, cloddiest, cloddish,
cloggier, cloggiest, clogging, cloister, cloistered, cloistering, cloisters, clomping, clonic, cloning, clonism, clonisms, clonking, clopping, closeting, closing.
closings, closuring, clothier, clothiers, clothing, clothings, clotting, cloturing, cloudier, cloudiest, cloudily, cloudiness, cloudinesses, clouding, clouring, clouting, clowneries, clowning,
clownish, clownishly, clownishness, clownishnesses, cloying, clupeoid, clupeoids, coaching, coacting, coaction, coactions, coactive, coadmire, coadmired, coadmires, coadmiring, coadmit.
coadmits, coadmitted, coadmitting, coagencies, coagulating, coagulation, coagulations, coalbin, coalbins, coalescing, coalfield, coalfields, coalfish, coalfishes, coalified, coalifies, coalify,
coalifying, coaling, coalition, coalitions.
coalpit, coalpits, coaming, coamings, coannexing, coappearing, coapting, coarsening, coassist, coassisted, coassisting, coassists, coassuming, coasting, coastings, coastline, coastlines, coati,
coating, coatings, coatis, coattail, coattails, coattending, coattesting, coauthoring, coauthorship, coauthorships, coaxial, coaxing, cobaltic, cobbier, cobbiest, cobbling, cobia, cobias.
cobwebbier, cobwebbiest, cobwebbing, cocain, cocaine, cocaines, cocains, cocaptain, cocaptains, cocci, coccic, coccid, coccidia, coccids, coccoid, coccoids, cochair, cochaired, cochairing,
cochairman, cochairmen, cochairs.
cochampion, cochampions, cochin, cochins, cocinera, cocineras, cockbill, cockbilled, cockbilling, cockbills, cockering, cockfight, cockfights, cockier, cockiest, cockily, cockiness, cockinesses,
cocking, cockish, cocklike, cockling, cockpit.
cockpits, cockshies, cocktail, cocktailed, cocktailing, cocktails, coconspirator, coconspirators, cocooning, cocreating, coddling, codeia, codeias, codein, codeina, codeinas, codeine, codeines,
codeins, coderive, coderived, coderives, coderiving, codesigner, codesigners, codeveloping, codfish, codfishes, codices, codicil, codicils.
codification, codifications, codified, codifier, codifiers, codifies, codify, codifying, coding, codirector, codirectors, codiscoverer, codiscoverers, codlin, codling, codlings, codlins, codpiece,
codpieces, coeditor, coeditors, coeducation, coeducational, coeducations, coefficient.
coefficients, coeliac, coelomic, coembodied, coembodies, coembodying, coemploying, coempting, coenacting, coenamoring, coenduring, coenuri, coequating, coercing, coercion, coercions, coercive,
coerecting, coexerting, coexist, coexisted, coexistence, coexistences.
coexistent, coexisting, coexists, coextending, coffering, coffin, coffined, coffing, coffining, coffins, coffling, cofinance, cofinanced, cofinances, cofinancing, cogencies, cogging, cogitate,
cogitated, cogitates, cogitating.
cogitation, cogitations, cogitative, cogito, cogitos, cognise, cognised, cognises, cognising, cognition, cognitions, cognitive, cognizable, cognizance, cognizances, cognizant, cognize, cognized,
cognizer, cognizers, cognizes, cognizing, cognomina, cognovit, cognovits, cohabit, cohabitation, cohabitations.
cohabited, cohabiting, cohabits, coheir, coheiress, coheiresses, coheirs, cohering, cohesion, cohesions, cohesive, cohobating, coif, coifed, coiffe, coiffed, coiffes, coiffeur, coiffeurs, coiffing,
coiffure, coiffured, coiffures, coiffuring, coifing, coifs, coign, coigne, coigned, coignes, coigning, coigns, coil, coiled, coiler, coilers, coiling, coils, coin, coinable.
coinage, coinages, coincide, coincided, coincidence, coincidences, coincident, coincidental, coincides, coinciding, coined, coiner, coiners, coinfer, coinferred, coinferring, coinfers, coinhere,
coinhered, coinheres, coinhering, coining, coinmate, coinmates, coins, coinsure, coinsured, coinsures, coinsuring, cointer.
cointerred, cointerring, cointers, coinventor, coinventors, coinvestigator, coinvestigators, coir, coirs, coistrel, coistrels, coistril, coistrils, coital, coitally, coition, coitions, coitus,
coituses, coking, coldish, colic, colicin, colicine.
colicines, colicins, colicky, colics, colies, coliform, coliforms, colin, colinear, colins, coliseum, coliseums, colistin, colistins, colitic, colitis, colitises, collaborating, collaboration,
collaborations, collaborative, collapsible, collapsing, collaring, collating, collectible, collecting.
collection, collections, collectivism, collegia, collegian, collegians, collegiate, colleting, collide, collided, collides, colliding, collie, collied, collier, collieries, colliers, colliery,
collies, collins, collinses, collision, collisions, colloguing, colloid, colloidal, colloids.
colloquial, colloquialism, colloquialisms, colloquies, colluding, collusion, collusions, colluvia, collying, collyria, colocating, coloni, colonial, colonials, colonic, colonies, colonise, colonised,
colonises, colonising, colonist, colonists, colonize, colonized, colonizes, colonizing, coloring, colorings, colorism, colorisms, colorist, colorists, colossi, colotomies, colouring, colpitis,
colpitises, coltish.
colubrid, colubrids, columbic, columnist, columnists, comatic, comatik, comatiks, combating, combative, combatting, combination, combinations, combine, combined, combiner, combiners, combines,
combing, combings, combining, comblike, combustibilities, combustibility, combustible.
combusting, combustion, combustions, combustive, comedian, comedians, comedic, comedienne, comediennes, comedies, comelier, comeliest, comelily, cometic, comfier, comfiest, comfit, comfits,
comforting, comic, comical, comics, coming, comings, comitia, comitial.
comities, comity, commandeering, commanding, commemorating, commemoration, commemorations, commemorative, commencing, commendation, commendations, commending, commentaries, commenting, commercial,
commercialize, commercialized, commercializes, commercializing, commercially, commercials, commercing, commie, commies, commiserate, commiserated, commiserates, commiserating, commiseration,
commiserations, commissaries, commissary, commission, commissioned, commissioner.
commissioners, commissioning, commissions, commit, commitment, commitments, commits, committal, committals, committed, committee, committees, committing, commix, commixed, commixes, commixing,
commixt, commodious, commodities, commodity, commotion, commotions, commoving, communicable, communicate, communicated, communicates.
communicating, communication, communications, communicative, communing, communion, communions, communique, communiques, communism, communist, communistic, communists, communities, community,
commutation, commutations, commuting, compacting, companied, companies, companion, companions, companionship, companionships, companying, comparative, comparatively, comparing, comparison,
comparisons, comparting.
compassing, compassion, compassionate, compassions, compatability, compatibilities, compatibility, compatible, compatriot, compatriots, compeering, compelling, compendia, compendium, compensating,
compensation, compensations, compering, competencies, competing, competition, competitions, competitive, competitor, competitors, compilation, compilations, compile, compiled, compiler, compilers,
compiles, compiling, comping, complacencies.
complain, complainant, complainants, complained, complainer, complainers, complaining, complains, complaint, complaints, complecting, complementing, completing, completion, completions, complexing,
complexion, complexioned, complexions, complexity, compliance, compliances, compliant, complicate, complicated, complicates, complicating, complication, complications, complice, complices,
complicity, complied, complier, compliers, complies, compliment, complimentary, compliments, complin, compline, complines, complins, complotting, complying, comporting, composing, composite,
composites, composition, compositions, composting, compounding, comprehending, comprehensible, comprehension, comprehensions, comprehensive, comprehensiveness, comprehensivenesses, compressing,
compression, compressions.
comprise, comprised, comprises, comprising, comprize, comprized, comprizes, comprizing, compromise, compromised, compromises, compromising, compting, compulsion, compulsions, compulsive, compunction,
compunctions, computation, computations, computerize, computerized, computerizes, computerizing, computing, comradeship, comradeships, comtemplating, conation, conations, conative, concatenating,
concaving, concavities, concavity, concealing, conceding, conceit, conceited, conceiting.
conceits, conceivable, conceivably, conceive, conceived, conceives, conceiving, concentrating, concentration, concentrations, concentric, conception, conceptions, conceptualize, conceptualized,
conceptualizes, conceptualizing, concerning, concerti, concertina, concerting, concession, concessions, conchies, conchoid, conchoids, conciliate, conciliated, conciliates, conciliating,
conciliation, conciliations, conciliatory.
concise, concisely, conciseness, concisenesses, conciser, concisest, concluding, conclusion, conclusions, conclusive, conclusively, concocting, concoction, concoctions, concomitant, concomitantly,
concomitants, concreting, concretion, concretions, concubine, concubines, concurring, concussing, concussion, concussions.
condemnation, condemnations, condemning, condensation, condensations, condensing, condescending, condescension, condescensions, condign, condiment, condiments, condition, conditional, conditionally,
conditioned, conditioner, conditioners, conditioning, conditions, condoling, condominium, condominiums, condoning, conducing, conducting, conduction, conductions, conductive, conduit, conduits,
confabbing, confecting, confederacies, conferring, confessing, confession, confessional.
confessionals, confessions, confetti, confidant, confidants, confide, confided, confidence, confidences, confident, confidential, confidentiality, confider, confiders, confides, confiding,
configuration, configurations, configure, configured, configures, configuring, confine, confined, confinement, confinements, confiner, confiners, confines, confining, confirm, confirmation,
confirmations, confirmed, confirming, confirms, confiscate.
confiscated, confiscates, confiscating, confiscation, confiscations, conflagration, conflagrations, conflating, conflict, conflicted, conflicting, conflicts, conforming, conformities, conformity,
confounding, confrontation, confrontations, confronting, confusing, confusion, confusions, confuting, congaing, congealing, congeeing, congenial.
congenialities, congeniality, congenital, congesting, congestion, congestions, congestive, congii, congius, conglobing, conglomerating, conglomeration, conglomerations, congratulating,
congratulation, congratulations, congregating, congregation, congregational, congregations, congresional, congressing, congruities, congruity, coni, conic, conical, conicities, conicity, conics,
conidia, conidial, conidian, conidium, conies, conifer, coniferous, conifers.
coniine, coniines, conin, conine, conines, coning, conins, conium, coniums, conjecturing, conjoin, conjoined, conjoining, conjoins, conjoint, conjugating, conjugation, conjugations, conjunction,
conjunctions, conjunctive, conjunctivitis, conjuring, conking, connecting, connection, connections, connective, conning, connivance, connivances, connive, connived.
conniver, connivers, connives, conniving, connoisseur, connoisseurs, connotation, connotations, connoting, connubial, conoid, conoidal, conoids, conquering, conquian, conquians, conscience,
consciences, conscientious, conscientiously, conscious, consciously, consciousness, consciousnesses, conscript, conscripted, conscripting, conscription, conscriptions, conscripts, consecrating,
consecration, consecrations, consecutive, consecutively.
consenting, consequential, conservation, conservationist, conservationists, conservations, conservatism, conservatisms, conservative, conservatives, conservatories, conserving, consider,
considerable, considerably, considerate, considerately, considerateness, consideratenesses, consideration, considerations, considered, considering.
considers, consign, consigned, consignee, consignees, consigning, consignment, consignments, consignor, consignors, consigns, consist, consisted, consistencies, consistency, consistent, consistently,
consisting, consists, consolation, consolidate, consolidated, consolidates, consolidating, consolidation, consolidations, consoling, consorting, consortium.
consortiums, conspicuous, conspicuously, conspiracies, conspiracy, conspirator, conspirators, conspire, conspired, conspires, conspiring, constabularies, constancies, constellation, constellations,
consternation, consternations, constipate, constipated, constipates, constipating, constipation, constipations, constituent, constituents, constitute.
constituted, constitutes, constituting, constitution, constitutional, constitutionality, constrain, constrained, constraining, constrains, constraint, constraints, constriction, constrictions,
constrictive, constructing, construction, constructions, constructive, construing, consultation, consultations, consulting, consuming, consummating, consummation, consummations, consumption,
consumptions, consumptive, contacting, contagia, contagion, contagions, contagious, contain.
contained, container, containers, containing, containment, containments, contains, contaminate, contaminated, contaminates, contaminating, contamination, contaminations, contemning, contemplating,
contemplation, contemplations, contemplative, contemporaries, contemptible, contending, contenting, contention, contentions, contentious, contesting.
contiguities, contiguity, contiguous, continence, continences, continent, continental, continents, contingencies, contingency, contingent, contingents, continua, continual, continually, continuance,
continuances, continuation, continuations, continue, continued, continues, continuing, continuities, continuity, continuo, continuos, continuous, continuousities, continuousity, contorting,
contortion, contortions, contouring.
contraception, contraceptions, contraceptive, contraceptives, contracting, contraction, contractions, contradict, contradicted, contradicting, contradiction, contradictions, contradictory,
contradicts, contrail, contrails, contraindicate, contraindicated, contraindicates, contraindicating, contraption, contraptions, contraries, contrarily, contrariwise, contrasting, contravening,
contribute, contributed, contributes.
contributing, contribution, contributions, contributor, contributors, contributory, contrite, contrition, contritions, contrivance, contrivances, contrive, contrived, contriver, contrivers,
contrives, contriving, controlling, controversial, controversies, controvertible, controverting, contumacies, contumelies, contusing.
contusion, contusions, convalescing, convecting, convection, convectional, convections, convective, convenience, conveniences, convenient, conveniently, convening, conventing, convention,
conventional, conventionally, conventions, convergencies, converging, conversation, conversational, conversations, conversing, conversion, conversions, convertible, convertibles, converting,
convexities, convexity, conveying, convict, convicted, convicting, conviction, convictions.
convicts, convince, convinced, convinces, convincing, convivial, convivialities, conviviality, convocation, convocations, convoking, convolution, convolutions, convolving, convoying, convulsing,
convulsion, convulsions, convulsive, cooeeing, cooeying, cooing, cooingly, cookeries, cookie, cookies, cooking, cookings, coolie, coolies, cooling, coolish, coonskin, coonskins, coontie.
coonties, cooperating, cooperation, cooperative, cooperatives, cooperies, coopering, cooping, coopting, cooption, cooptions, coordinate, coordinated, coordinates, coordinating, coordination,
coordinations, coordinator, coordinators, cootie, cooties, copaiba, copaibas, copartnership, copartnerships.
copied, copier, copiers, copies, copihue, copihues, copilot, copilots, coping, copings, copious, copiously, copiousness, copiousnesses, coplotting, coppering, coppice, coppiced, coppices, copping,
copremia, copremias, copremic, copresident, copresidents, coprincipal, coprincipals, coprisoner, coprisoners, coproducing, coproduction, coproductions, copromoting, coproprietor, coproprietors,
coproprietorship, coproprietorships.
copublish, copublished, copublisher, copublishers, copublishes, copublishing, copulating, copulation, copulations, copulative, copulatives, copycatting, copying, copyist, copyists, copyright,
copyrighted, copyrighting, copyrights, coquetries, coquetting.
coquille, coquilles, coquina, coquinas, coquito, coquitos, coracoid, coracoids, corbeil, corbeils, corbeling, corbelling, corbie, corbies, corbina, corbinas, cordial, cordialities, cordiality,
cordially, cordials, cording, cordite, cordites, cordlike, cordoning, corduroying, cordwain, cordwains, corecipient, corecipients.
coredeeming, coreign, coreigns, corelating, coremia, coremium, coresident, coresidents, corgi, corgis, coria, coring, corium, corkier, corkiest, corking, corklike, cormlike, cormoid, corncrib,
corncribs, cornering.
cornetcies, cornice, corniced, cornices, corniche, corniches, cornicing, cornicle, cornicles, cornier, corniest, cornily, corning, cornucopia, cornucopias, corodies, corollaries, coronaries,
coronation, coronations, corotating, corporation, corporations, corpulencies, corrading, corralling, correcting, correction, corrections, corrective, correlating, correlation, correlations,
correlative, correlatives, corresponding, corrida, corridas, corridor, corridors.
corrie, corries, corrival, corrivals, corroborating, corroboration, corroborations, corrodies, corroding, corrosion, corrosions, corrosive, corrugating, corrugation, corrugations, corruptible,
corrupting, corruption, corruptions, corsair, corsairs, corseting, cortical, cortices, cortin.
cortins, cortisol, cortisols, cortisone, cortisones, corvina, corvinas, corvine, coshering, coshing, cosie, cosier, cosies, cosiest, cosign, cosignatories, cosignatory, cosigned, cosigner, cosigners,
cosigning, cosigns, cosily, cosine, cosines, cosiness, cosinesses, cosmetic, cosmetics, cosmic, cosmical, cosmism, cosmisms, cosmist, cosmists.
cosmopolitan, cosmopolitans, cosseting, costarring, costing, costive, costlier, costliest, costliness, costlinesses, costmaries, costuming, coterie, coteries, cothurni, cotidal, cotillion,
cotillions, cotillon, cotillons, coting, cottier, cottiers, cottoning, cotyloid, couching, couchings, coughing, coulisse, coulisses, couloir, couloirs, coumaric, coumarin, coumarins, council,
councillor, councillors, councilman, councilmen.
councilor, councilors, councils, councilwoman, counseling, counselling, countenancing, counteraccusation, counteraccusations, counteracting, counteraggression, counteraggressions, counterarguing,
counterattacking, counterbalancing, counterbid, counterbids, countercampaign, countercampaigns, counterclaim, counterclaims, counterclockwise, countercomplaint, countercomplaints, countercriticism,
countercriticisms, counterdemonstration, counterdemonstrations, counterevidence, counterevidences, counterfeit, counterfeited, counterfeiter, counterfeiters.
counterfeiting, counterfeits, counterguerrila, counterinflationary, counterinfluence, counterinfluences, countering, counterintrigue, counterintrigues, countermanding, counterpetition,
counterpetitions, counterpoint, counterpoints, counterpropagation, counterpropagations, counterquestion, counterquestions, counterraid, counterraids, counterrallies, counterretaliation,
counterretaliations, counterrevolution, counterrevolutions, countersign, countersigns, counterstrategies, countersuggestion, countersuggestions, countersuing, countersuit, countersuits.
countertendencies, counterterrorism, counterterrorisms, counterterrorist, counterterrorists, countian, countians, counties, counting, countries, countryside, countrysides, couping, coupling,
couplings, courier, couriers, coursing, coursings, courtesied, courtesies, courtesying, courtier.
courtiers, courting, courtlier, courtliest, courtship, courtships, cousin, cousinly, cousinries, cousinry, cousins, couthie, couthier, couthiest, covenanting, covering, coverings, coverlid,
coverlids, coveting, coving, covings, cowardice, cowardices, cowberries, cowbind, cowbinds, cowbird, cowbirds, cowering, cowfish, cowfishes.
cowgirl, cowgirls, cowhide, cowhided, cowhides, cowhiding, cowier, cowiest, cowing, cowinner, cowinners, cowlick, cowlicks, cowling, cowlings, cowrie, cowries, cowskin, cowskins, cowslip, cowslips,
coxalgia, coxalgias, coxalgic, coxalgies.
coxing, coxswain, coxswained, coxswaining, coxswains, coying, coyish, cozening, cozie, cozier, cozies, coziest, cozily, coziness, cozinesses, cramoisies, cramoisy, cratonic, crayoning, creation,
creations, creditor, creditors, cremation, cremations, creosoting, cribrous, cribwork, cribworks, cricoid, cricoids, crimson, crimsoned, crimsoning.
crimsons, crinoid, crinoids, crinoline, crinolines, criollo, criollos, crisscross, crisscrossed, crisscrosses, crisscrossing, criterion, croakier, croakiest, croakily, croaking, crocein, croceine,
croceines, croceins, crocheting, croci, crocine, crockeries, crocking, crocodile, crocodiles, crocoite, crocoites, crojik, crojiks, cronies, cronyism, cronyisms, crooking, crooning, cropping.
croqueting, croquis, crosier, crosiers, crossbarring, crossbreeding, crosscutting, crossing, crossings, crosstie, crossties, crosswise, crouching, croupier, croupiers, croupiest, croupily, crowdie,
crowdies, crowding, crowing.
crowning, crozier, croziers, crucifixion, cruzeiro, cruzeiros, cryogenies, cryolite, cryolites, cryonic, cryonics, cryptographic, cryptographies, crystallization, crystallizations, ctenoid, cubiform,
cuboid, cuboidal, cuboids, cuckolding, cuckooing, cullion, cullions, culminatation, culminatations, cultivatation, cultivatations, cuniform, cuniforms, cupolaing, curculio.
curculios, curio, curios, curiosa, curiosities, curiosity, curious, curiouser, curiousest, cushion, cushioned, cushioning, cushions, cushiony, cuspidor, cuspidors, custodial, custodian, custodians,
custodies, customarily, customize.
customized, customizes, customizing, cyanosis, cyanotic, cyclitol, cyclitols, cycloid, cycloids, cyclonic, cyclopaedia, cyclopaedias, cyclopedia, cyclopedias, cyclophosphamide, cyclophosphamides,
cyclosis, cymoid, cystoid, cystoids, cytogenies, cytologies, cytopathological, cytosine, cytosines, dacoit, dacoities, dacoits, dacoity, dadoing, daemonic, daffodil, daffodils, daimio, daimios.
daimon, daimones, daimonic, daimons, daimyo, daimyos, dakoit, dakoities, dakoits, dakoity, daltonic, damnation, damnations, dandelion, dandelions, danio, danios, dariole, darioles, datapoint,
deaconing, deaconries, deadlocking, debarkation, debarkations, debonair, deboning, debouching, decapitatation.
decapitatations, decentralization, deception, deceptions, deciduous, decision, decisions, declamation, declamations, declaration, declarations, declension, declensions, declination, declinations,
decocting, decoding, decoloring, decolouring, decomposing, decomposition, decompositions, decorating, decoration, decorations, decorative, decoying, decrowning, decurion, decurions.
dedication, dedications, dedicatory, deduction, deductions, defamation, defamations, defecation, defecations, defection, defections, definition, definitions, deflation, deflations, deflection,
deflections, deflowering, defoaming, defogging, defoliant, defoliants, defoliate, defoliated, defoliates.
defoliating, defoliation, defoliations, deforcing, deforesting, deformation, deformations, deforming, deformities, deformity, defrocking, defrosting, degeneration, degenerations, degradation,
degradations, dehorning, dehorting, dehydration, dehydrations, deification, deifications, deiform, deionize, deionized, deionizes, deionizing.
dejection, dejections, delation, delations, delegation, delegations, deleterious, deletion, deletions, deliberation, deliberations, delicious, deliciously, delineation, delineations, delirious,
delousing, deltoid, deltoids, delusion, delusions, demagogies, demagogueries, demarcation, demarcations, demigod, demigods, demijohn, demijohns.
demivolt, demivolts, demobbing, demobilization, demobilizations, demobilize, demobilized, demobilizes, demobilizing, democracies, democratic, democratize, democratized, democratizes, democratizing,
demographic, demolish, demolished, demolishes, demolishing, demolition, demolitions, demoniac, demoniacs, demonian, demonic, demonise, demonised, demonises, demonising, demonism, demonisms, demonist,
demonists, demonize, demonized, demonizes, demonizing, demonstrating.
demonstration, demonstrations, demonstrative, demoralize, demoralized, demoralizes, demoralizing, demotic, demotics, demoting, demotion, demotions, demotist, demotists, demounting, dendroid,
denomination, denominational, denominations, denominator, denominators, denotation, denotations, denotative.
denoting, denotive, denouncing, dentition, dentitions, dentoid, denunciation, denunciations, deodorize, deodorized, deodorizes, deodorizing, depiction, depictions, depictor, depictors, depletion,
depletions, deploring, deploying, depolish, depolished, depolishes, depolishing, deponing, deportation, deportations.
deporting, deposing, deposit, deposited, depositing, deposition, depositions, depositor, depositories, depositors, depository, deposits, depravation, depravations, deprecation, deprecations,
depreciation, depreciations, depredation, depredations, depression, depressions, deputation, deputations, dereliction, derelictions, derision, derisions.
derisory, derivation, derivations, dermatologies, dermatologist, dermatologists, dermoid, derogating, description, descriptions, descriptor, descriptors, desecration, desecrations, desegregation,
desegregations, desiccation, desiccations, designation, designations, desirous, desmoid, desmoids, desolating, desolation, desolations, desorbing, desperation, desperations, despoil, despoiled,
despoiling, despoils, despondencies, desponding, despotic.
despotism, despotisms, desquamation, desquamations, destination, destinations, destitution, destitutions, destroying, destruction, destructions, detection, detections, detention, detentions,
deteriorate, deteriorated, deteriorates, deteriorating, deterioration, deteriorations, determination, determinations, detestation, detestations, dethroning, detonating, detonation, detonations,
detouring, detoxified, detoxifies, detoxify, detoxifying, detraction.
detractions, devaluation, devaluations, devastation, devastations, developing, deviation, deviations, deviator, deviators, devious, devisor, devisors, devoice, devoiced, devoices, devoicing, devoid,
devoir, devoirs, devolving, devoting, devotion, devotional.
devotions, devouring, dewooling, deworming, dhoolies, dhooti, dhootie, dhooties, dhootis, dhoti, dhotis, diabolic, diabolical, diabolo, diabolos, diaconal, diagnose, diagnosed, diagnoses, diagnosing,
diagnosis, diagnostic, diagnostics, diagonal, diagonally, diagonals, dialog, dialoger.
dialogers, dialogged, dialogging, dialogic, dialogs, dialogue, dialogued, dialogues, dialoguing, diamond, diamonded, diamonding, diamonds, diapason, diapasons, diaphone, diaphones, diaphonies,
diaphony, diarrhoea, diarrhoeas, diaspora, diasporas, diaspore, diaspores, diastole, diastoles, diatom, diatomic, diatoms, diatonic, diazo, diazole, diazoles, dichotic.
dichroic, dicot, dicots, dicotyl, dicotyls, dicrotal, dicrotic, dictation, dictations, dictator, dictatorial, dictators, dictatorship, dictatorships, diction, dictionaries, dictionary, dictions,
dido, didoes, didos, didymous.
diecious, diestock, diestocks, differentiation, diffusion, diffusions, diffusor, diffusors, digamous, digestion, digestions, digestor, digestors, diglot, diglots, digoxin, digoxins, digression,
digressions, dihedron, dihedrons, dilapidation, dilapidations, dilatation, dilatations, dilation, dilations, dilator, dilators, dilatory, dildo, dildoe, dildoes, dildos, dilution, dilutions, dilutor,
dilutors, diluvion, diluvions.
dimension, dimensional, dimensions, dimerous, dimorph, dimorphs, dimout, dimouts, dinero, dineros, dingdong, dingdonged, dingdonging, dingdongs, dingo, dingoes, dinosaur, dinosaurs, diobol, diobolon,
diobolons, diobols, diocesan, diocesans.
diocese, dioceses, diode, diodes, dioecism, dioecisms, dioicous, diol, diolefin, diolefins, diols, diopside, diopsides, dioptase, dioptases, diopter, diopters, dioptral, dioptre, dioptres, dioptric,
diorama, dioramas, dioramic, diorite, diorites, dioritic, dioxane, dioxanes.
dioxid, dioxide, dioxides, dioxids, diphthong, diphthongs, diploe, diploes, diploic, diploid, diploidies, diploids, diploidy, diploma, diplomacies, diplomacy, diplomaed, diplomaing, diplomas,
diplomat, diplomata, diplomatic, diplomats, diplont.
diplonts, diplopia, diplopias, diplopic, diplopod, diplopods, diploses, diplosis, dipnoan, dipnoans, dipodic, dipodies, dipody, dipolar, dipole, dipoles, dipteron, direction, directional, directions,
director, directories, directors, directory, disadvantageous, disaffection, disaffections, disallow, disallowed, disallowing, disallows.
disappoint, disappointed, disappointing, disappointment, disappointments, disappoints, disapproval, disapprovals, disapprove, disapproved, disapproves, disapproving, disastrous, disavow, disavowal,
disavowals, disavowed, disavowing, disavows, disbosom, disbosomed, disbosoming, disbosoms.
disbound, disbowel, disboweled, disboweling, disbowelled, disbowelling, disbowels, disclose, disclosed, discloses, disclosing, disclosure, disclosures, disco, discoid, discoids, discolor,
discoloration, discolorations, discolored, discoloring, discolors, discomfit, discomfited, discomfiting, discomfits, discomfiture, discomfitures, discomfort, discomforts, disconcert, disconcerted,
disconcerting, disconcerts, disconnect, disconnected, disconnecting, disconnects.
disconsolate, discontent, discontented, discontents, discontinuance, discontinuances, discontinuation, discontinue, discontinued, discontinues, discontinuing, discord, discordant, discorded,
discording, discords, discos, discount, discounted, discounting, discounts, discourage, discouraged, discouragement, discouragements, discourages, discouraging, discourteous, discourteously.
discourtesies, discourtesy, discover, discovered, discoverer, discoverers, discoveries, discovering, discovers, discovery, discretion, discretionary, discretions, discrimination, discriminations,
discriminatory, discrown, discrowned, discrowning, discrowns, discussion.
discussions, disembarkation, disembarkations, disembodied, disendow, disendowed, disendowing, disendows, disfavor, disfavored, disfavoring, disfavors, disfrock, disfrocked, disfrocking, disfrocks,
disgorge, disgorged, disgorges, disgorging, disharmonies, disharmonious, disharmony, dishcloth, dishcloths, dishonest, dishonesties, dishonestly, dishonesty, dishonor, dishonorable, dishonorably,
dishonored, dishonoring, dishonors, disillusion, disillusioned, disillusioning, disillusionment, disillusionments.
disillusions, disinclination, disinclinations, disinfection, disinfections, disintegration, disintegrations, disjoin, disjoined, disjoining, disjoins, disjoint, disjointed, disjointing, disjoints,
dislocate, dislocated, dislocates, dislocating, dislocation, dislocations, dislodge, dislodged, dislodges, dislodging, disloyal, disloyalties, disloyalty, dismount, dismounted, dismounting,
dismounts, disobedience.
disobediences, disobedient, disobey, disobeyed, disobeying, disobeys, disomic, disorder, disordered, disordering, disorderliness, disorderlinesses, disorderly, disorders, disorganization,
disorganizations, disorganize, disorganized, disorganizes, disorganizing, disown, disowned, disowning, disowns, dispassion, dispassionate, dispassions, dispensation, dispensations, dispersion,
dispersions, displode, disploded, displodes, disploding, disport, disported, disporting.
disports, disposable, disposal, disposals, dispose, disposed, disposer, disposers, disposes, disposing, disposition, dispositions, dispossess, dispossessed, dispossesses, dispossessing,
dispossession, dispossessions, disproof, disproofs, disproportion, disproportionate, disproportions, disprove, disproved, disproves, disproving, disputation, disputations, disqualification,
disqualifications, disrobe, disrobed, disrober, disrobers, disrobes.
disrobing, disroot, disrooted, disrooting, disroots, disruption, disruptions, dissatisfaction, dissatisfactions, dissection, dissections, dissemination, dissention, dissentions, dissertation,
dissertations, dissipation, dissipations, dissociate, dissociated, dissociates, dissociating, dissociation, dissociations, dissolute, dissolution, dissolutions, dissolve, dissolved, dissolves,
dissolving, dissonance.
dissonances, dissonant, dissuasion, dissuasions, distension, distensions, distention, distentions, distillation, distillations, distinction, distinctions, distome, distomes, distort, distorted,
distorting, distortion, distortions, distorts, distraction, distractions, distribution, distributions, distributor, distributors, disunion, disunions, disyoke, disyoked, disyokes, disyoking, dithiol,
ditto, dittoed, dittoing, dittos, diuron.
diurons, diversification, diversifications, diversion, diversions, divination, divinations, division, divisional, divisions, divisor, divisors, divorce, divorced, divorcee, divorcees, divorcer,
divorcers, divorces, divorcing, divot, divots, dizygous, doating, dobbies, dobbin, dobbins, dobie, dobies, docetic, docile, docilely, docilities, docility, docketing, docking.
dockside, docksides, doctoring, doctrinal, doctrine, doctrines, documentaries, documentation, documentations, documenting, doddering, dodgeries, dodgier, dodgiest, dodging, dodoism, dodoisms,
doeskin, doeskins, doffing, dogberries, dogeship, dogeships, dogfight, dogfighting, dogfights, dogfish, dogfishes.
doggeries, doggie, doggier, doggies, doggiest, dogging, doggish, doggoning, dogie, dogies, doglegging, doglike, dogmatic, dogmatism, dogmatisms, dognaping, dognapping, dogsbodies, dogtrotting,
doiled, doilies, doily, doing.
doings, doit, doited, doits, dolci, dolerite, dolerites, doling, dollied, dollies, dolling, dollish, dollying, dolomite, dolomites, dolphin, dolphins, doltish, domain, domains, domelike, domestic,
domestically, domesticate, domesticated.
domesticates, domesticating, domestication, domestications, domestics, domic, domical, domicil, domicile, domiciled, domiciles, domiciling, domicils, dominance, dominances, dominant, dominants,
dominate, dominated, dominates, dominating, domination, dominations, domine, domineer, domineered, domineering, domineers, domines, doming, dominick, dominicks, dominie.
dominies, dominion, dominions, dominium, dominiums, domino, dominoes, dominos, donating, donation, donations, donative, donatives, donning, donnish, donsie, doodling, doolie, doolies, dooming,
doornail, doornails, doorsill, doorsills, doozies, dopamine, dopamines, dopier, dopiest, dopiness, dopinesses, doping, dories, dormancies, dormice, dormie, dormient, dormin, dormins.
dormitories, dormitory, dornick, dornicks, dosimetry, dosing, dossier, dossiers, dossil, dossils, dossing, dotation, dotations, dotier, dotiest, doting, dotingly, dottier, dottiest, dottily, dotting,
doublecrossing, doubling, doubting, douching, doughier, doughiest, doughtier, doughtiest, dourine, dourines, dousing, dovekie, dovekies.
dovelike, dovening, dovetail, dovetailed, dovetailing, dovetails, dovish, dowdier, dowdies, dowdiest, dowdily, dowdyish, doweling, dowelling, doweries, dowering, dowie, dowing, downgrading, downhill,
downhills, downier, downiest, downing, downplaying, downright, downstairs, downtime, downtimes, downwind, dowries, dowsing, doxie, doxies, doxologies, doxorubicin, doylies, dozening, dozier.
doziest, dozily, doziness, dozinesses, dozing, draconic, dragooning, dramatization, dramatizations, driftwood, driftwoods, droit, droits, drolleries, drolling, dromedaries, droning, dronish,
drooling, droopier, droopiest, droopily, drooping, dropkick, dropkicks, dropping, droppings, dropsied, dropsies, droshkies, droskies, drossier, drossiest.
droughtier, droughtiest, drouking, drouthier, drouthiest, droving, drownding, drowning, drowsier, drowsiest, drowsily, drowsing, drypoint, drypoints, dubious, dubiously, dubiousness, dubiousnesses,
dumbfounding, dumfounding, duomi, duopolies, duopsonies.
duplication, duplications, duplicator, duplicators, duration, durations, durion, durions, dysautonomia, dysfunction, dysfunctions, dyspnoic, dystocia, dystocias, dystonia, dystonias, dystopia,
dystopias, dystrophies, easygoing, eavesdropping, ebonies, ebonise, ebonised, ebonises, ebonising, ebonite.
ebonites, ebonize, ebonized, ebonizes, ebonizing, ecbolic, ecbolics, echeloning, echinoid, echinoids, echoic, echoing, echoism, echoisms, eclogite, eclogites, eclosion, eclosions, ecologic,
ecological, ecologically, ecologies, ecologist, ecologists.
economic, economical, economically, economics, economies, economist, economists, economize, economized, economizes, economizing, ecotypic, ectopia, ectopias, ectopic, edacious, edification,
edifications, edition, editions, editor, editorial, editorialize, editorialized, editorializes, editorializing, editorially, editorials, editors, education, educational, educations, eduction,
eductions, efficacious, effusion, effusions, egestion.
egestions, egoism, egoisms, egoist, egoistic, egoists, egomania, egomanias, egotism, egotisms, egotist, egotistic, egotistical, egotistically, egotists, egregious, egregiously, eiderdown, eiderdowns,
eidola, eidolon, eidolons.
eidos, eightvo, eightvos, eikon, eikones, eikons, einkorn, einkorns, ejaculation, ejaculations, ejection, ejections, elaborating, elaboration, elaborations, elation, elations, elbowing, election,
elections, electrification, electrifications, electrocardiogram, electrocardiograms, electrocardiograph, electrocardiographs, electrocuting, electrocution, electrocutions, electroing, electrolysis,
electrolysises, electrolytic, electromagnetic, electronic, electronics, electroplating, elevation.
elevations, elicitor, elicitors, elimination, eliminations, elision, elisions, elocution, elocutions, eloign, eloigned, eloigner, eloigners, eloigning, eloigns, eloin, eloined, eloiner, eloiners,
eloining, eloins, elongating, elongation, elongations, eloping, elucidation, elucidations, elusion, elusions, elution, elutions, elytroid, emaciation, emaciations, emanation, emanations,
emancipatation, emancipatations, emancipation.
emancipations, emasculatation, emasculatations, embargoing, embarkation, embarkations, emblazoning, embodied, embodier, embodiers, embodies, embodiment, embodiments, embodying, emboldening, emboli,
embolic, embolies, embolism, embolisms, embordering, embosking, embosoming, embossing, emboweling, embowelling, embowering, embowing, embroider, embroidered.
embroidering, embroiders, embroil, embroiled, embroiling, embroils, embrowning, embryoid, embryonic, emendation, emendations, emeroid, emeroids, emersion, emersions, emigration, emigrations,
emission, emissions, emodin, emodins, emoting, emotion, emotional, emotionally, emotions, emotive, employing, empoison, empoisoned, empoisoning.
empoisons, emporia, emporium, emporiums, empowering, emulation, emulations, emulsification, emulsifications, emulsion, emulsions, emulsoid, emulsoids, enamoring, enamouring, enation, enations,
enchoric, enclosing, encoding, encomia, encomium, encomiums, encompassing, encoring, encountering, encouraging, encroaching, encyclopedia, encyclopedias.
encyclopedic, endeavoring, endocrine, endogamies, endogenies, endorsing, endowing, endozoic, enervation, enervations, enfeoffing, enfolding, enforcing, enginous, engorging, engrossing, enhaloing,
enjoin, enjoined, enjoiner, enjoiners, enjoining, enjoins, enjoying, ennobling, enolic, enologies, enormities, enormity, enosis, enosises, enouncing, enrobing, enrolling, enrooting.
ensconcing, enscrolling, enshrouding, ensiform, ensorceling, ensouling, enthroning, entoil, entoiled, entoiling, entoils, entombing, entomological, entomologies, entomologist, entomologists, entopic,
entozoic, entropies, enumeration, enumerations, enunciation, enunciations, enveloping, envenoming, envious, environ, environed, environing.
environment, environmental, environmentalist, environmentalists, environments, environs, envision, envisioned, envisioning, envisions, envoi, envois, enwombing, enzootic, enzootics, eobiont,
eobionts, eohippus, eohippuses, eolian, eolipile, eolipiles, eolith, eolithic, eoliths, eolopile, eolopiles, eonian, eonism.
eonisms, eosin, eosine, eosines, eosinic, eosins, epheboi, ephori, epibolic, epibolies, epiboly, epicotyl, epicotyls, epidemiology, epidote, epidotes, epidotic, epifocal, epigeous, epigon, epigone,
epigoni, epigonic, epigons, epigonus, epilog, epilogs, epilogue, epilogued, epilogues, epiloguing, epinaoi, epinaos, episcopal, episcope, episcopes, episode, episodes, episodic, episomal, episome,
episomes, epitome, epitomes, epitomic, epitomize, epitomized, epitomizes, epitomizing, epizoa, epizoic, epizoism, epizoisms, epizoite, epizoites.
epizoon, epizooties, epizooty, eponymic, eponymies, epopoeia, epopoeias, epoxide, epoxides, epoxied, epoxies, epoxying, epsilon, epsilons, equation, equations, equatorial, equinox, equinoxes,
equivocal, equivocate, equivocated, equivocates, equivocating, equivocation, equivocations, equivoke, equivokes, erasion, erasions, erection, erections, ergodic, ergotic, ergotism, ergotisms,
erigeron, erigerons, eringo, eringoes, eringos, erodible, eroding, erogenic, erosible, erosion, erosions, erosive, erotic, erotica, erotical, erotically, erotics, erotism, erotisms, erudition,
eruditions, eruption, eruptions, erythrocytosis, escalation, escalations, escalloping, escaloping, escorting.
escoting, escrowing, esophagi, esoteric, espionage, espionages, espousing, essoin, essoins, essonite, essonites, estimation, estimations, estimator, estimators, estopping, estriol, estriols, ethion,
ethions, ethmoid, ethmoids, ethnologic, ethnological, ethnologies, ethologies, etiolate, etiolated, etiolates, etiolating, etiologies, etiology, etoile.
etoiles, etymological, etymologist, etymologists, eulogia, eulogiae, eulogias, eulogies, eulogise, eulogised, eulogises, eulogising, eulogist, eulogistic, eulogists, eulogium, eulogiums, eulogize,
eulogized, eulogizes, eulogizing, euphonic, euphonies, euphonious, euphoria, euphorias, euphoric.
euphotic, euploid, euploidies, euploids, euploidy, eupnoeic, europium, europiums, eutrophies, evacuation, evacuations, evaluation, evaluations, evaporating, evaporation, evaporations, evaporative,
evasion, evasions, evection, evections, eversion, eversions, eviction.
evictions, evictor, evictors, evildoer, evildoers, evisceration, eviscerations, evocation, evocations, evocative, evoking, evolution, evolutionary, evolutions, evolving, evulsion, evulsions,
exaction, exactions, exaggeration, exaggerations, exaltation, exaltations, examination, examinations.
exasperation, exasperations, excavation, excavations, exception, exceptional, exceptionalally, exceptions, excision, excisions, excitation, excitations, exciton, excitons, excitor, excitors,
exclamation, exclamations, exclusion, exclusions, excommunicate, excommunicated, excommunicates, excommunicating, excommunication, excommunications, excretion, excretions, excursion, excursions,
execution, executioner, executioners, executions, exemplification, exemplifications.
exemption, exemptions, exertion, exertions, exhalation, exhalations, exhaustion, exhaustions, exhibition, exhibitions, exhibitor, exhibitors, exhilaration, exhilarations, exhortation, exhortations,
exhorting, exhumation, exhumations, exiguous, eximious, exocrine, exocrines, exodoi, exoergic, exogamic, exogamies, exonerating, exoneration, exonerations, exorbitant, exorcise, exorcised, exorcises,
exorcising, exorcism, exorcisms, exorcist.
exorcists, exorcize, exorcized, exorcizes, exorcizing, exordia, exordial, exordium, exordiums, exosmic, exoteric, exotic, exotica, exotically, exoticism, exoticisms, exotics, exotism, exotisms,
exotoxic, exotoxin, exotoxins, expansion, expansions, expectation, expectations, expedious, expedition, expeditions, expeditious, experimentation, experimentations.
expiation, expiations, expiator, expiators, expiration, expirations, explanation, explanations, exploding, exploit, exploitation, exploitations, exploited, exploiting, exploits, exploration,
explorations, exploring, explosion, explosions, explosive.
explosively, explosives, exponential, exponentially, exportation, exportations, exporting, exposing, exposit, exposited, expositing, exposition, expositions, exposits, expounding, expression,
expressionless, expressions, expulsion, expulsions, expurgation, expurgations, extension, extensions, extenuation, extenuations, exterior, exteriors, extermination, exterminations.
exterminator, exterminators, extinction, extinctions, extolling, extorting, extortion, extortioner, extortioners, extortionist, extortionists, extortions, extracommunity, extraconstitutional,
extracontinental, extraction, extractions, extradiocesan, extradition, extraditions, extranational, extraordinarily, extraordinary, extrascholastic, extravasation, extravasations, extraversion,
extraversions, extrication, extrications, exudation, exudations, eyepoint, eyepoints.
fabrication, fabrications, facetious, facetiously, facilitator, facilitators, faction, factional, factionalism, factionalisms, factions, factious, factitious, factories, factoring, faggoting,
fagoting, fagotings, fairground, fairgrounds, faitour, faitours, falchion, falchions, falconries, fallacious, fallowing, falsification, falsifications, fanion, fanions, farinose, farrowing,
fascination, fascinations, fashion, fashionable, fashionably, fashioned, fashioning.
fashions, fashious, fastiduous, fastiduously, fastiduousness, fastiduousnesses, fathoming, favonian, favoring, favorite, favorites, favoritism, favoritisms, favouring, federation, federations,
felicitation, felicitations, felicitous, felicitously, fellatio, fellatios, fellowing, fellowship, fellowships, felonies, felonious, felonries, feminization.
feminizations, feodaries, feoffing, feretories, fermentation, fermentations, fermion, fermions, ferocious, ferociously, ferociousness, ferociousnesses, ferocities, ferocity, fertilization,
fertilizations, festooning, fetation, fetations, fetologies, fiasco, fiascoes, fiascos, fiberboard, fiberboards, fibrillation, fibrillations, fibrocystic.
fibroid, fibroids, fibroin, fibroins, fibroma, fibromas, fibromata, fibroses, fibrosis, fibrotic, fibrous, fico, ficoes, fiction, fictional, fictions, fictitious, fido, fidos, fiefdom, fiefdoms,
figwort, figworts, filamentous, filemot, filiform, filmdom, filmdoms, filmgoer, filmgoers, filose.
filtration, filtrations, finfoot, finfoots, finochio, finochios, fiord, fiords, fireboat, fireboats, firebomb, firebombed, firebombing, firebombs, firebox, fireboxes, firedog, firedogs, firelock,
firelocks, fireproof, fireproofed, fireproofing, fireproofs, fireroom, firerooms, firewood, firewoods, firework, fireworks, fireworm, fireworms, fishboat, fishboats, fishbone, fishbones, fishbowl,
fishhook, fishhooks, fishpole, fishpoles, fishpond, fishponds, fission, fissionable, fissional, fissioned, fissioning, fissions, fistnote, fistnotes, fivefold, fixation, fixations, flagellation,
flagellations, flamingo, flamingoes, flamingos, flatfooting, flatiron, flatirons.
flavoring, flavorings, flavouring, flection, flections, flexion, flexions, flirtation, flirtations, flirtatious, floatier, floatiest, floating, flocci, floccing, flocculi, flockier, flockiest,
flocking, flockings, flogging, floggings, flooding, floodlit, flooring, floorings, floosies, floozie, floozies, floppier, floppiest, floppily, flopping, florid, floridly, florigen, florigens, florin,
florins, florist.
florists, floruit, floruits, flossie, flossier, flossies, flossiest, flotation, flotations, flotilla, flotillas, flouncier, flounciest, flouncing, floundering, flouring, flourish, flourished,
flourishes, flourishing, flouting, flowerier, floweriest, floweriness, flowerinesses, flowering, flowing, fluctuation, fluctuations, fluidounce, fluidounces, flummoxing, fluorescing, fluoric,
fluoridate, fluoridated, fluoridates, fluoridating, fluoridation, fluoridations, fluoride, fluorides, fluorids, fluorin, fluorine, fluorines, fluorins, fluorite, fluorites, fluoroscopic,
fluoroscopies, fluoroscopist, fluoroscopists, fluxion, fluxions, flyblowing, foaling, foamier, foamiest, foamily, foaming, foamlike, fobbing, focalise, focalised, focalises, focalising, focalize,
focalized, focalizes, focalizing, foci, focusing.
focussing, foddering, foetid, fogfruit, fogfruits, foggier, foggiest, foggily, fogging, fogie, fogies, fogyish, fogyism, fogyisms, foible, foibles, foil, foilable, foiled, foiling, foils, foilsman,
foilsmen, foin, foined, foining.
foins, foison, foisons, foist, foisted, foisting, foists, folacin, folacins, folding, folia, foliage, foliaged, foliages, foliar, foliate, foliated, foliates, foliating, folic, folio.
folioed, folioing, folios, foliose, folious, folium, foliums, folkish, folklike, folklorist, folklorists, folksier, folksiest, folksily, follicle, follicles, follies, follis, following, followings,
fomentation, fomentations.
fomenting, fonding, fondling, fondlings, fontina, fontinas, fooleries, foolfish, foolfishes, foolhardiness, foolhardinesses, fooling, foolish, foolisher, foolishest, foolishness, foolishnesses,
footbridge, footbridges, foothill, foothills, footier.
footiest, footing, footings, footlight, footlights, footlike, footling, footnoting, footprint, footprints, footsie, footsies, footslogging, foozling, fopperies, fopping, foppish, foraging, foramina,
foraying, forbearing, forbid, forbidal, forbidals, forbidden.
forbidding, forbids, forboding, forcible, forcibly, forcing, forcipes, fordid, fording, fordoing, forearming, forebodies, foreboding, forebodings, forecasting, foreclosing, foredating, foredid,
foredoing, foredooming, forefeeling, forefending, forefinger, forefingers, foregathering, foregoing, foreign, foreigner, foreigners, foreknowing, foreladies, forelimb, forelimbs, foremilk.
foremilks, forensic, forensics, foreordain, foreordained, foreordaining, foreordains, forerunning, foresaid, foresail, foresails, foreseeing, foreshadowing, foreshowing, foreside, foresides,
foresight, foresighted, foresightedness, foresightednesses, foresights, foreskin, foreskins, forestalling, foresting, forestries, foreswearing, foretasting, foretelling, foretime, foretimes,
forewarning, forewing, forewings, forfeit, forfeited, forfeiting.
forfeits, forfeiture, forfeitures, forfending, forgathering, forgeries, forgetting, forging, forgings, forgivable, forgive, forgiven, forgiveness, forgivenesses, forgiver, forgivers, forgives,
forgiving, forgoing, forint, forints, forjudging, forkier, forkiest, forking, forklift, forklifts, forklike.
formalin, formalins, formalities, formality, formalize, formalized, formalizes, formalizing, formation, formations, formative, formatting, formic, formidable, formidably, forming, formulating,
formulation, formulations, fornical, fornicate, fornicated, fornicates, fornicating, fornication, fornicator, fornicators, fornices, fornix, forrit.
forsaking, forswearing, forsythia, forsythias, forthcoming, forthright, forthrightness, forthrightnesses, forthwith, forties, fortieth, fortieths, fortification, fortifications, fortified, fortifies,
fortify, fortifying, fortis, fortitude, fortitudes, fortnight, fortnightly, fortnights, fortressing, fortuities, fortuitous, fortuity.
fortuning, forwarding, fossick, fossicked, fossicking, fossicks, fossil, fossilize, fossilized, fossilizes, fossilizing, fossils, fostering, fouling, foulings, foundation, foundational, foundations,
foundering, founding, foundling, foundlings, foundries, fountain, fountained, fountaining, fountains, fowling, fowlings, foxfire, foxfires, foxfish, foxfishes, foxier, foxiest, foxily, foxiness.
foxinesses, foxing, foxings, foxlike, foxskin, foxskins, foxtail, foxtails, fozier, foziest, foziness, fozinesses, fraction, fractional, fractionally, fractionated, fractioned, fractioning,
fractions, fragmentation, fragmentations, fraternization, fraternizations, freebooting, freeloading, frescoing, fricando, fricandoes, friction, frictional, frictions, frijol, frijole, frijoles,
frisson, frissons, frivol.
frivoled, frivoler, frivolers, frivoling, frivolities, frivolity, frivolled, frivolling, frivolous, frivolously, frivols, frocking, frogfish, frogfishes, froggier, froggiest, frogging, froglike,
frolic, frolicked, frolicking, frolicky, frolics, frolicsome, fromenties, frontier, frontiers, frontiersman, frontiersmen, fronting, frontispiece, frontispieces, frostbit, frostbite, frostbites,
frostbitten, frostier.
frostiest, frostily, frosting, frostings, frothier, frothiest, frothily, frothing, frouncing, frouzier, frouziest, frowning, frowsier, frowsiest, frowstier, frowstiest, frowzier, frowziest, frowzily,
fruition, fruitions, frustration, frustrations, fucoid, fucoidal, fucoids, fugio, fugios, fumatories.
fumigation, fumigations, fumitories, fumitory, function, functional, functionally, functionaries, functionary, functioned, functioning, functionless, functions, fungoid, fungoids, furbelowing,
furioso, furious, furiously, furloughing, furrowing, fusiform, fusion, fusions, gabbroic, gabbroid, gabion.
gabions, gadoid, gadoids, galiot, galiots, galipot, galipots, galliot, galliots, gallipot, gallipots, galloping, galvanization, galvanizations, gamboling, gambolling, gammoning, ganglion, ganglionic,
ganglions, ganoid, ganoids.
gaoling, gaposis, gaposises, garboil, garboils, garoting, garotting, garrison, garrisoned, garrisoning, garrisons, garroting, garrotting, gasiform, gasolier, gasoliers, gasoline, gasolines,
gastronomic, gastronomical, gastronomies, gavotting, gelatinous, gelation, gelations, gemologies, geneological, geneologically, geneologist, geneologists, generalization, generalizations, generation,
generosities, generosity, genitor, genitors, genitourinary, genocide, genocides, genomic, genuflection, genuflections, geodesic, geodesics, geodesies, geodetic, geodic, geognosies, geographic,
geographical, geographically, geographies, geoid, geoidal, geoids, geologic, geological, geologies, geologist, geologists, geomancies, geometric, geometrical, geometries.
geophagies, geophysical, geophysicist, geophysicists, geophysics, geoponic, georgic, georgics, geotaxis, geothermic, geraniol, geraniols, germination, germinations, gerontic, gheraoing, ghettoing,
ghostier, ghostiest, ghosting, ghostlier, ghostliest, ghostwrite, ghostwriter, ghostwriters, ghostwrites, ghostwritten, ghoulish.
giaour, giaours, gibbon, gibbons, gibbose, gibbous, gigaton, gigatons, giglot, giglots, gigolo, gigolos, gigot, gigots, gingko, gingkoes, ginkgo, ginkgoes, gipon, gipons, girasol, girasole,
girasoles, girasols, girlhood, girlhoods, giro, giron, girons, giros, girosol, girosols, gismo, gismos, gitano, gitanos.
gizmo, gizmos, gladiator, gladiatorial, gladiators, gladiola, gladiolas, gladioli, gladiolus, glamorize, glamorized, glamorizes, glamorizing, glamouring, glassblowing, glassblowings, glenoid, glioma,
gliomas, gliomata, gloaming, gloamings, gloating, globin, globing, globins, globoid, globoids, globulin, globulins, glochid, glochids, glockenspiel, glockenspiels, glomming, gloomier, gloomiest,
gloominess, gloominesses, glooming, gloomings, gloria, glorias, gloried, glories, glorification, glorifications, glorified, glorifies, glorify, glorifying, gloriole, glorioles, glorious, gloriously,
glorying, glossarial, glossaries, glossier.
glossies, glossiest, glossily, glossina, glossinas, glossiness, glossinesses, glossing, glottic, glottides, glottis, glottises, glouting, gloving, glowering, glowflies, glowing, gloxinia, gloxinias,
glozing, glucosic, glutinous, gluttonies, glycolic, glyconic, glyconics, gnathion, gnathions, gnocchi, gnomic, gnomical, gnomish, gnomist, gnomists, gnomonic.
gnosis, gnostic, goading, goadlike, goalie, goalies, goaling, goatfish, goatfishes, goatish, goatlike, goatskin, goatskins, gobbing, gobbling, gobies, gobioid, gobioids, goblin, goblins, godchild,
godchildren, goddamming, goddamning, godding, godlier, godliest, godlike, godlily, godling, godlings, godship, godships, godwit.
godwits, goethite, goethites, goffering, gogglier, goggliest, goggling, going, goings, goiter, goiters, goitre, goitres, goitrous, goldbrick, goldbricked, goldbricking, goldbricks, goldfinch,
goldfinches, goldfish, goldfishes, goldsmith, goldstein, golfing, golfings, goliard.
goliards, golliwog, golliwogs, gomeril, gomerils, gomuti, gomutis, gonadial, gonadic, gondolier, gondoliers, gonging, gonglike, gonia, gonidia, gonidial, gonidic, gonidium, gonif, gonifs, gonion,
gonium, goodies, goodish, goodlier, goodliest.
goodwife, goodwill, goodwills, goodwives, goofier, goofiest, goofily, goofiness, goofinesses, goofing, gooier, gooiest, goonie, goonies, gooseberries, goosier, goosiest, goosing, gorbellies,
gorblimy, gorgerin, gorgerins, gorging, gorier, goriest, gorilla, gorillas, gorily, goriness, gorinesses, goring, gorsier, gorsiest, gosling, goslings, gossip, gossiped, gossiper, gossipers.
gossiping, gossipped, gossipping, gossipries, gossipry, gossips, gossipy, gothic, gothics, gothite, gothites, gouging, gourami, gouramis, goutier, goutiest, goutily, governing, governorship,
governorships, gowning.
goyim, goyish, gracioso, graciosos, gracious, graciously, graciousness, graciousnesses, graduation, graduations, graffito, grandiose, grandiosely, granulation, granulations, gratification,
gratifications, gratuitous, gravitation, gravitational, gravitationally, gravitations, graviton, gravitons, grazioso, gregarious, gregariously, gregariousness.
gregariousnesses, gridiron, gridirons, grievous, grievously, griffon, griffons, grillwork, grillworks, grindstone, grindstones, gringo, gringos, griseous, grison, grisons, groaning, groceries,
groggeries, groggier, groggiest, groggily, grogginess, grogginesses, groin, groined, groining, groins, grooming, groovier, grooviest.
grooving, groping, grossing, grouchier, grouchiest, grouching, grounding, groupie, groupies, grouping, groupings, groupoid, groupoids, grousing, groutier, groutiest, grouting, groveling, grovelling,
growing, growlier, growliest, growling, grunion, grunions, guaiacol, guaiacols, guaiocum.
guaiocums, gubernatorial, gudgeoning, guerdoning, guidebook, guidebooks, guidon, guidons, guillotine, guillotined, guillotines, guillotining, guiro, gulosities, gulosity, gumboil, gumboils, gumbotil,
gumbotils, gummosis, gumption, gumptions, gumshoeing, gunpoint, gunpoints, gynecoid, gynecologic, gynecological, gynecologies, gynecologist.
gynecologists, gynecomastia, gynoecia, gyration, gyrations, gyroidal, habitation, habitations, hadronic, haemoid, hailstone, hailstones, hailstorm, hailstorms, hairdo, hairdos, hairlock, hairlocks,
hairwork, hairworks, hairworm, hairworms, halation, halations, halidom, halidome, halidomes, halidoms.
halitosis, halitosises, halloaing, halloing, hallooing, hallowing, hallucination, hallucinations, hallucinatory, hallucinogen, hallucinogenic, hallucinogens, haloid, haloids, haloing, halolike,
handiwork, handiworks, haploid, haploidies, haploids, haploidy, haplopia, haplopias, haplosis, harboring, harbouring.
haricot, haricots, harlotries, harmonic, harmonica, harmonically, harmonicas, harmonics, harmonies, harmonious, harmoniously, harmoniousness, harmoniousnesses, harmonization, harmonizations,
harmonize, harmonized, harmonizes, harmonizing, harpooning, harpsichord, harpsichords, harrowing, hautbois, havior, haviors, haviour, haviours, havocking, hectoring, hedgehopping, hedonic, hedonics,
hedonism, hedonisms, hedonist.
hedonistic, hedonists, hegemonies, heinous, heinously, heinousness, heinousnesses, heirdom, heirdoms, heirloom, heirlooms, helicoid, helicoids, helicon, helicons, helicopt, helicopted, helicopter,
helicopters, helicopting, helicopts, helio, helios, heliotrope, heliport, heliports, helistop.
helistops, hellion, hellions, helloing, helotism, helotisms, helotries, hematocrit, hematoid, hematologic, hematological, hematologies, hematologist, hematologists, hematopenia, hemiola, hemiolas,
hemoglobin, hemoid, hemolyzing, hemophilia, hemophiliac, hemophiliacs, hemoptysis, hemorrhagic, hemorrhaging, hemorrhoids, herbivorous, herbivorously, hereinto, heriot, heriots, heritor, heritors,
hermaphrodite, hermaphrodites, hermaphroditic.
herniation, herniations, heroic, heroical, heroics, heroin, heroine, heroines, heroins, heroism, heroisms, heroize, heroized, heroizes, heroizing, heronries, herpetologic, herpetological,
herpetologies, herpetologist, herpetologists, hesitation, hesitations, hexapodies, hibernation, hibernations, hibernator, hibernators, hiccough, hiccoughed, hiccoughing.
hiccoughs, hickories, hickory, hidalgo, hidalgos, hideous, hideously, hideousness, hideousnesses, hideout, hideouts, hidroses, hidrosis, hidrotic, hieroglyphic, hieroglyphics, highborn, highboy,
highboys, highbrow, highbrows, highroad, highroads, hilarious, hilariously, hillo, hilloa, hilloaed, hilloaing, hilloas, hillock, hillocks, hillocky, hilloed, hilloing, hillos, hilltop, hilltops,
himations, hindmost, hipbone, hipbones, hippiedom, hippiedoms, hippiehood, hippiehoods, hippo, hippopotami, hippopotamus, hippopotamuses, hippos, hipshot, histogen, histogens, histogram, histograms,
histoid, histologic, histone, histones, histopathologic, histopathological, historian, historians, historic.
historical, historically, histories, history, hitherto, hoactzin, hoactzines, hoactzins, hoagie, hoagies, hoarding, hoardings, hoarier, hoariest, hoarily, hoariness, hoarinesses, hoarsening, hoatzin,
hoatzines, hoatzins, hoaxing, hobbies, hobbing, hobbling, hobbyist, hobbyists, hobgoblin, hobgoblins.
hoblike, hobnail, hobnailed, hobnails, hobnobbing, hoboing, hoboism, hoboisms, hocking, hocusing, hocussing, hodaddies, hoddin, hoddins, hoeing, hoelike, hogfish, hogfishes, hogging, hoggish,
hoglike, hogtie, hogtied.
hogtieing, hogties, hogtying, hoick, hoicked, hoicking, hoicks, hoiden, hoidened, hoidening, hoidens, hoise, hoised, hoises, hoising, hoist, hoisted, hoister, hoisters, hoisting, hoists, hoking,
hokypokies, holding, holdings, holibut, holibuts, holiday, holidayed, holidaying, holidays, holier, holies, holiest, holily, holiness, holinesses, holing, holism.
holisms, holist, holistic, holists, holking, hollaing, hollering, hollies, holloaing, holloing, hollooing, hollowing, holmic, holmium, holmiums, hologynies, holozoic, holstein, holsteins, holytide,
holytides, homaging, homebodies, homecoming, homecomings, homelier, homeliest.
homelike, homeliness, homelinesses, homemaking, homemakings, homering, homesick, homesickness, homesicknesses, homesite, homesites, homicidal, homicide, homicides, homier, homiest, homiletic,
homilies, homilist, homilists, homily, hominess, hominesses, homing, hominian, hominians, hominid, hominids, hominies, hominine, hominoid, hominoids, hominy, homogamies, homogeneities, homogeneity,
homogenies, homogenize, homogenized, homogenizer.
homogenizers, homogenizes, homogenizing, homogonies, homologies, homonymies, hondling, honesties, honeycombing, honeying, honeymooning, honied, honing, honkie, honkies, honking, honoraries,
honorarily, honoring, honouring, hoodie, hoodies, hooding, hoodlike, hoodooing, hoodwink, hoodwinked.
hoodwinking, hoodwinks, hoofing, hooflike, hookier, hookies, hookiest, hooking, hooklike, hoolie, hooligan, hooligans, hooping, hooplike, hoorahing, hooraying, hootier, hootiest, hooting, hoping,
hoplite, hoplites, hoplitic, hopping, hoppling, hordein, hordeins, hording, horizon, horizons.
horizontal, horizontally, hormonic, hornbill, hornbills, hornier, horniest, hornily, horning, hornito, hornitos, hornlike, hornpipe, hornpipes, horntail, horntails, horological, horologies,
horologist, horologists, horrible, horribleness, horriblenesses, horribles, horribly, horrid, horridly, horrific, horrified, horrifies, horrify, horrifying, horseflies, horsehair.
horsehairs, horsehide, horsehides, horsemanship, horsemanships, horseradish, horseradishes, horsier, horsiest, horsily, horsing, horticultural, horticulture, horticultures, horticulturist,
horticulturists, hosannaing, hosier, hosieries, hosiers, hosiery, hosing, hospice, hospices, hospitable, hospitably, hospital, hospitalities, hospitality, hospitalization.
hospitalizations, hospitalize, hospitalized, hospitalizes, hospitalizing, hospitals, hospitia, hosteling, hostelries, hostessing, hostile, hostilely, hostiles, hostilities, hostility, hosting,
hotching, hotdogging, hotelier, hoteliers, hotfooting, hotpressing, hotting, hottish, hounding, houri, houris.
housecleaning, housecleanings, houseflies, housekeeping, houseling, houselling, housemaid, housemaids, housewarming, housewarmings, housewife, housewifeliness, housewifelinesses, housewifely,
housewiferies, housewifery, housewives, housing, housings, hoveling, hovelling, hovering, howbeit, howdie, howdies, howitzer, howitzers, howking, howling, hoydening, hulloaing, hulloing,
humanization, humanizations, humanoid, humanoids, humidification, humidifications, humidor.
humidors, humiliation, humiliations, humoring, humorist, humorists, humouring, huntington, hyaloid, hyaloids, hybridization, hybridizations, hydrochloride, hydroelectric, hydroelectrically,
hydroelectricities, hydroelectricity, hydroid, hydroids, hydronic, hydrophobia, hydrophobias, hydropic, hydropsies, hyenoid, hygrometries, hylozoic, hymnodies, hyoid, hyoidal, hyoidean, hyoids,
hyoscine, hyoscines, hyperanxious, hypercautious, hyperconscientious, hyperemotional, hyperfastidious.
hypermoralistic, hypernationalistic, hyperromantic, hypersuspicious, hypertension, hypertensions, hyperthyroidism, hyphenation, hyphenations, hypnoid, hypnosis, hypnotic, hypnotically, hypnotics,
hypnotism, hypnotisms, hypnotizable, hypnotize, hypnotized, hypnotizes, hypnotizing, hypoacid, hypocalcemia, hypochondria, hypochondriac, hypochondriacs, hypochondrias, hypocrisies.
hypocrisy, hypocrite, hypocrites, hypocritical, hypocritically, hypodermic, hypodermics, hypogynies, hypoing, hypokalemia, hyponoia, hyponoias, hypotension, hypotensions, hypothesis, hypothetical,
hypothetically, hypothyroidism, hypoxia, hypoxias, hypoxic, hyracoid.
hyracoids, hysterectomies, hysterectomize, hysterectomized, hysterectomizes, hysterectomizing, iceboat, iceboats, icebound, icebox, iceboxes, icehouse, icehouses, ichor, ichorous, ichors,
ichthyologies, ichthyologist, ichthyologists, ichthyology, icon.
icones, iconic, iconical, iconoclasm, iconoclasms, iconoclast, iconoclasts, icons, idealization, idealizations, idealogies, idealogy, ideation, ideations, identification, identifications, ideogram,
ideograms, ideological, ideologies, ideology, idiocies, idiocy, idiolect, idiolects, idiom, idiomatic, idiomatically, idioms.
idiosyncrasies, idiosyncrasy, idiosyncratic, idiot, idiotic, idiotically, idiotism, idiotisms, idiots, idocrase, idocrases, idol, idolater, idolaters, idolatries, idolatrous, idolatry, idolise,
idolised, idoliser, idolisers, idolises, idolising, idolism, idolisms, idolize, idolized, idolizer, idolizers, idolizes, idolizing, idols, idoneities, idoneity, idoneous, igloo, igloos.
igneous, ignition, ignitions, ignitor, ignitors, ignitron, ignitrons, ignoble, ignobly, ignominies, ignominious, ignominiously, ignominy, ignoramus, ignoramuses, ignore, ignored, ignorer, ignorers,
ignores, ignoring, ikon, ikons, illation, illations, illogic, illogical.
illogically, illogics, illumination, illuminations, illusion, illusions, illusory, illustration, illustrations, illustrator, illustrators, illustrious, illustriousness, illustriousnesses,
imagination, imaginations, imago, imagoes, imbodied, imbodies, imbody, imbodying, imbolden, imboldened, imboldening.
imboldens, imbosom, imbosomed, imbosoming, imbosoms, imbower, imbowered, imbowering, imbowers, imbroglio, imbroglios, imbrown, imbrowned, imbrowning, imbrowns, imido, imino, imitation, imitations,
imitator, imitators, immemorial, immersion, immersions, immigration, immigrations, immobile, immobilities, immobility.
immobilize, immobilized, immobilizes, immobilizing, immoderacies, immoderacy, immoderate, immoderately, immodest, immodesties, immodestly, immodesty, immolate, immolated, immolates, immolating,
immolation, immolations, immoral, immoralities, immorality, immorally, immortal, immortalities, immortality, immortalize, immortalized, immortalizes, immortalizing.
immortals, immotile, immovabilities, immovability, immovable, immovably, immunization, immunizations, immunologic, immunological, immunologies, immunologist, immunologists, immunology, impactor,
impactors, impassioned, impasto, impastos, impecunious, impecuniousness, impecuniousnesses, impellor.
impellors, imperfection, imperfections, imperious, imperiously, impersonal, impersonally, impersonate, impersonated, impersonates, impersonating, impersonation, impersonations, impersonator,
impersonators, impervious, impetigo, impetigos, impetuous, impetuousities, impetuousity.
impetuously, impious, implementation, implementations, implication, implications, implode, imploded, implodes, imploding, implore, implored, implorer, implorers, implores, imploring, implosion,
implosions, implosive, impolicies, impolicy, impolite, impolitic, imponderable, imponderables, impone, imponed.
impones, imponing, imporous, import, importance, important, importantly, importation, importations, imported, importer, importers, importing, imports, importunate, importune, importuned, importunes,
importuning, importunities, importunity, impose, imposed, imposer, imposers, imposes, imposing.
imposingly, imposition, impositions, impossibilities, impossibility, impossible, impossibly, impost, imposted, imposter, imposters, imposting, impostor, impostors, imposts, imposture, impostures,
impotence, impotences, impotencies, impotency, impotent, impotently, impotents, impound, impounded, impounding, impoundment, impoundments.
impounds, impoverish, impoverished, impoverishes, impoverishing, impoverishment, impoverishments, impower, impowered, impowering, impowers, imprecision, impregnation, impregnations, impresario,
impresarios, impression, impressionable, impressions, imprison, imprisoned, imprisoning, imprisonment, imprisonments, imprisons, improbabilities, improbability, improbable, improbably, impromptu.
impromptus, improper, improperly, improprieties, impropriety, improvable, improve, improved, improvement, improvements, improver, improvers, improves, improvidence, improvidences, improvident,
improving, improvisation, improvisations, improviser, improvisers, improvisor, improvisors, impulsion, impulsions, imputation, imputations, inaction, inactions, inanition, inanitions, inapposite,
inappositely, inappositeness, inappositenesses.
inapproachable, inappropriate, inappropriately, inappropriateness, inappropriatenesses, inattention, inattentions, inauguration, inaugurations, inauspicious, inboard, inboards, inborn, inbound,
inbounds, incantation, incantations, incarceration, incarcerations, incarnation, incarnations, incautious, inception, inceptions, inceptor, inceptors, incestuous, inchoate.
inchworm, inchworms, incinerator, incinerators, incision, incisions, incisor, incisors, incisory, inclination, inclinations, inclose, inclosed, incloser, inclosers, incloses, inclosing, inclosure,
inclosures, inclusion, inclusions, incog, incognito, incogs.
incoherence, incoherences, incoherent, incoherently, incohesive, incombustible, income, incomer, incomers, incomes, incoming, incomings, incommensurate, incommodious, incommunicable, incommunicado,
incomparable, incompatibility, incompatible, incompetence, incompetences, incompetencies, incompetency, incompetent, incompetents, incomplete, incompletely, incompleteness, incompletenesses,
incomprehensible, inconceivable.
inconceivably, inconclusive, incongruent, incongruities, incongruity, incongruous, incongruously, inconnu, inconnus, inconsecutive, inconsequence, inconsequences, inconsequential, inconsequentially,
inconsiderable, inconsiderate, inconsiderately, inconsiderateness, inconsideratenesses, inconsistencies, inconsistency.
inconsistent, inconsistently, inconsolable, inconsolably, inconspicuous, inconspicuously, inconstancies, inconstancy, inconstant, inconstantly, inconsumable, incontestable, incontestably,
incontinence, incontinences, inconvenience, inconvenienced, inconveniences, inconveniencing, inconvenient, inconveniently, incony, incorporate, incorporated, incorporates, incorporating,
incorporation, incorporations.
incorporeal, incorporeally, incorpse, incorpsed, incorpses, incorpsing, incorrect, incorrectly, incorrectness, incorrectnesses, incorrigibilities, incorrigibility, incorrigible, incorrigibly,
incorruptible, incredulous, incredulously, incrimination, incriminations, incriminatory, incross, incrosses, incubation, incubations.
incubator, incubators, inculcation, inculcations, incurious, incursion, incursions, indecision, indecisions, indecorous, indecorously, indecorousness, indecorousnesses, indemnification,
indemnifications, indentation, indentations, indentor, indentors, indevout, indication.
indications, indicator, indicators, indictor, indictors, indigenous, indigestion, indigestions, indignation, indignations, indigo, indigoes, indigoid, indigoids, indigos, indirection, indirections,
indiscretion, indiscretions, indisposed, indisposition, indispositions, indissoluble, indocile, indoctrinate, indoctrinated, indoctrinates, indoctrinating, indoctrination, indoctrinations, indol,
indole, indolence, indolences, indolent, indoles, indols, indominitable.
indominitably, indoor, indoors, indorse, indorsed, indorsee, indorsees, indorser, indorsers, indorses, indorsing, indorsor, indorsors, indow, indowed, indowing, indows, indoxyl, indoxyls, induction,
inductions, inductor, inductors, industrialization, industrializations, industrious.
industriously, industriousness, industriousnesses, inebriation, inebriations, inexorable, inexorably, infamous, infamously, infatuation, infatuations, infection, infections, infectious, infector,
infectors, infelicitous, infeoff, infeoffed, infeoffing, infeoffs, inferior, inferiority, inferiors, inferno, infernos, infestation, infestations, infiltration, infiltrations, infixion, infixions,
inflammations, inflammatory, inflation, inflationary, inflator, inflators, inflection, inflectional, inflections, infliction, inflictions, inflow, inflows, info, infold, infolded, infolder,
infolders, infolding, infolds, inform, informal, informalities, informality, informally, informant, informants, information, informational, informations, informative, informed, informer, informers,
informs, infos, infraction, infractions, infusion, infusions, ingenious, ingeniously, ingeniousness, ingeniousnesses, ingenuous, ingenuously, ingenuousness, ingenuousnesses, inglenook, inglenooks,
inglorious, ingloriously, ingoing, ingot, ingoted, ingoting, ingots, ingroup, ingroups, ingrown, ingrowth, ingrowths.
inhalation, inhalations, inheritor, inheritors, inhesion, inhesions, inhibition, inhibitions, inion, iniquitous, initialization, initializations, initiation, initiations, initiatory, injection,
injections, injector, injectors, injudicious, injudiciously, injudiciousness, injudiciousnesses, injunction, injunctions, injurious, inkblot, inkblots, inkhorn, inkhorns, inkpot, inkpots, inkwood,
inkwoods, inmost, innermost, innersole, innersoles, innocence, innocences.
innocent, innocenter, innocentest, innocently, innocents, innocuous, innovate, innovated, innovates, innovating, innovation, innovations, innovative, innovator, innovators, innuendo, innuendoed,
innuendoes, innuendoing, innuendos, inocula, inoculate, inoculated, inoculates, inoculating, inoculation, inoculations.
inoculum, inoculums, inoffensive, inoperable, inoperative, inopportune, inopportunely, inordinate, inordinately, inorganic, inosite, inosites, inositol, inositols, inpour, inpoured, inpouring,
inpours, inquisition, inquisitions, inquisitor, inquisitorial.
inquisitors, inroad, inroads, insalubrious, inscription, inscriptions, inscroll, inscrolled, inscrolling, inscrolls, insecuration, insecurations, insertion, insertions, inshore, insidious,
insidiously, insidiousness, insidiousnesses, insinuation, insinuations, insofar, insolate, insolated, insolates, insolating, insole, insolence, insolences, insolent, insolents, insoles,
insolubilities, insolubility, insoluble, insolvencies, insolvency, insolvent, insomnia.
insomnias, insomuch, insouciance, insouciances, insouciant, insoul, insouled, insouling, insouls, inspection, inspections, inspector, inspectors, inspiration, inspirational, inspirations,
installation, installations, instantaneous, instantaneously, instigation.
instigations, instigator, instigators, institution, institutional, institutionalize, institutionally, institutions, instroke, instrokes, instruction, instructional, instructions, instructor,
instructors, instructorship, instructorships, instrumentation, instrumentations, insubordinate, insubordination, insubordinations, insulation, insulations, insulator, insulators, insupportable,
insurmounable, insurmounably, insurrection, insurrectionist, insurrectionists, insurrections, intaglio, intaglios.
integration, intensification, intensifications, intention, intentional, intentionally, intentions, interaction, interactions, interatomic, interborough, intercalation, intercalations, interception,
interceptions, interceptor, interceptors, intercession, intercessions, intercessor, intercessors, intercessory, intercoastal, intercollegiate, intercolonial, intercom, intercommunal, intercommunity.
intercompany, intercoms, intercontinental, interconversion, intercounty, intercourse, intercourses, interdenominational, interdiction, interdictions, interdivisional, interelectronic,
intergovernmental, intergroup, interinstitutional, interior, interiors, interjection, interjectionally, interjections, interlock, interlocked, interlocking, interlocks, interlope, interloped,
interloper, interlopers, interlopes, interloping, intermission, intermissions, intermolecular, intermountain, international.
internationalism, internationalisms, internationalize, internationalized, internationalizes, internationalizing, internationally, internationals, interoceanic, interoffice, interpersonal,
interpolate, interpolated, interpolates, interpolating, interpolation, interpolations, interpopulation, interpose, interposed, interposes.
interposing, interposition, interpositions, interpretation, interpretations, interprovincial, interregional, interrelation, interrelations, interrelationship, interreligious, interrogate,
interrogated, interrogates, interrogating, interrogation, interrogations, interrogative, interrogatives, interrogator, interrogators, interrogatory, interruption, interruptions, interscholastic,
intersection, intersectional, intersections, interspersion, interspersions, intertroop, intertropical, intervention, interventions, interwoven, interzonal, interzone, inthrone, inthroned.
inthrones, inthroning, intimation, intimations, intimidation, intimidations, into, intolerable, intolerably, intolerance, intolerances, intolerant, intomb, intombed, intombing, intombs, intonate,
intonated, intonates, intonating, intonation, intonations, intone, intoned, intoner, intoners, intones, intoning, intort, intorted, intorting, intorts, intown, intoxicant, intoxicants, intoxicate,
intoxicated, intoxicates, intoxicating, intoxication.
intoxications, intrados, intradoses, intravenous, intravenously, intro, introduce, introduced, introduces, introducing, introduction, introductions, introductory, introfied, introfies, introfy,
introfying, introit, introits, intromit, intromits, intromitted, intromitting, introrse, intros, introspect, introspected.
introspecting, introspection, introspections, introspective, introspectively, introspects, introversion, introversions, introvert, introverted, introverts, intrusion, intrusions, intuition,
intuitions, inundation, inundations, invasion, invasions, invention, inventions, inventor, inventoried, inventories, inventors, inventory, inventorying, inversion, inversions, invertor, invertors,
investigation, investigations, investigator, investigators, investor, investors, invidious.
invidiously, invigorate, invigorated, invigorates, invigorating, invigoration, invigorations, inviolabilities, inviolability, inviolable, inviolate, invitation, invitations, invocate, invocated,
invocates, invocating, invocation, invocations, invoice, invoiced, invoices, invoicing, invoke, invoked.
invoker, invokers, invokes, invoking, involuntarily, involuntary, involute, involuted, involutes, involuting, involve, involved, involvement, involvements, involver, involvers, involves, involving,
inwound, inwove, inwoven, iodate, iodated, iodates, iodating, iodation, iodations, iodic, iodid, iodide, iodides, iodids, iodin, iodinate, iodinated.
iodinates, iodinating, iodine, iodines, iodins, iodism, iodisms, iodize, iodized, iodizer, iodizers, iodizes, iodizing, iodoform, iodoforms, iodol, iodols, iodophor, iodophors, iodopsin, iodopsins,
iolite, iolites, ion, ionic, ionicities, ionicity, ionics, ionise, ionised, ionises, ionising, ionium, ioniums, ionizable, ionize, ionized, ionizer, ionizers, ionizes, ionizing, ionomer, ionomers,
ionone, ionones, ionosphere, ionospheres, ionospheric, ions, iota.
iotacism, iotacisms, iotas, ipomoea, ipomoeas, irksome, irksomely, iron, ironbark, ironbarks, ironclad, ironclads, irone, ironed, ironer, ironers, irones, ironic, ironical, ironically, ironies,
ironing, ironings, ironist.
ironists, ironlike, ironness, ironnesses, irons, ironside, ironsides, ironware, ironwares, ironweed, ironweeds, ironwood, ironwoods, ironwork, ironworker, ironworkers, ironworks, irony, irradiation,
irradiations, irrational, irrationalities, irrationality, irrationally, irrationals, irreconcilabilities, irreconcilability, irreconcilable, irrecoverable, irrecoverably, irreligious.
irreproachable, irresolute, irresolutely, irresolution, irresolutions, irresponsibilities, irresponsibility, irresponsible, irresponsibly, irrevocable, irrigation, irrigations, irritation,
irritations, isagoge, isagoges, isagogic, isagogics, isobar, isobare, isobares, isobaric, isobars, isobath, isobaths, isocheim, isocheims, isochime.
isochimes, isochor, isochore, isochores, isochors, isochron, isochrons, isocline, isoclines, isocracies, isocracy, isodose, isogamies, isogamy, isogenic, isogenies, isogeny, isogloss, isoglosses,
isogon, isogonal, isogonals, isogone, isogones, isogonic, isogonics, isogonies, isogons, isogony, isogram, isograms, isograph, isographs, isogriv, isogrivs, isohel, isohels, isohyet.
isohyets, isolable, isolate, isolated, isolates, isolating, isolation, isolations, isolator, isolators, isolead, isoleads, isoline, isolines, isolog, isologs, isologue, isologues, isomer, isomeric,
isomers, isometric, isometrics, isometries, isometry, isomorph, isomorphs, isonomic, isonomies, isonomy, isophote, isophotes, isopleth, isopleths, isopod, isopodan, isopodans.
isopods, isoprene, isoprenes, isospin, isospins, isospories, isospory, isostasies, isostasy, isotach, isotachs, isothere, isotheres, isotherm, isotherms, isotone, isotones, isotonic, isotope,
isotopes, isotopic, isotopically, isotopies, isotopy.
isotropies, isotropy, isotype, isotypes, isotypic, isozyme, isozymes, isozymic, isthmoid, italicization, italicizations, itemization, itemizations, iteration, iterations, ivories, ivory, ixodid,
ixodids, jacobin, jacobins, jailor, jailors, jalopies, jaloppies, jalousie, jalousies, janiform, janitor, janitorial, janitors, japonica.
japonicas, jargoning, jarosite, jarosites, jarovize, jarovized, jarovizes, jarovizing, jawboning, jealousies, jeapordize, jeapordized, jeapordizes, jeapordizing, jeopardies, jeoparding, jeopordize,
jeopordized, jeopordizes, jeopordizing, jettison, jettisoned, jettisoning, jettisons, jibboom, jibbooms, jigaboo, jigaboos, jillion, jillions, jimsonweed, jimsonweeds, jingko, jingkoes.
jingo, jingoes, jingoish, jingoism, jingoisms, jingoist, jingoistic, jingoists, jobberies, jobbing, jockeying, jocosities, jocosity, jogging, joggling, johnnies, join, joinable, joinder, joinders,
joined, joiner, joineries, joiners, joinery, joining, joinings, joins, joint, jointed, jointer, jointers, jointing.
jointly, joints, jointure, jointured, jointures, jointuring, joist, joisted, joisting, joists, joking, jokingly, jollied, jollier, jollies, jolliest, jollified, jollifies, jollify, jollifying,
jollily, jollities, jollity, jollying, joltier, joltiest, joltily, jolting, jonquil, jonquils, joshing, jostling, jotting, jottings, jouking, jouncier, jounciest, jouncing, journalism, journalisms.
journalist, journalistic, journalists, journeying, jousting, jovial, jovially, jowing, jowlier, jowliest, joying, joypopping, joyride, joyrider, joyriders, joyrides, joyriding, joyridings, joystick,
joysticks, jublilation, jublilations, judicious, judiciously, judiciousness, judiciousnesses, judoist, judoists, junction, junctions, junior, juniors, jurisdiction, jurisdictional, jurisdictions,
justification, justifications, juxtaposing, juxtaposition.
juxtapositions, kaleidoscope, kaleidoscopes, kaleidoscopic, kaleidoscopical, kaleidoscopically, kaoliang, kaoliangs, kaolin, kaoline, kaolines, kaolinic, kaolins, karyotin, karyotins, kathodic,
kation, kations, kayoing, keitloa, keitloas, keloid, keloidal, keloids, kenosis, kenosises.
kenotic, keratoid, kerosine, kerosines, ketonic, ketosis, ketotic, keyboarding, keynoting, kibosh, kiboshed, kiboshes, kiboshing, kickoff, kickoffs, kiddo, kiddoes, kiddos, killjoy, killjoys,
killock, killocks, kilo, kilobar, kilobars, kilobit, kilobits, kilocycle, kilocycles, kilogram, kilograms, kilohertz, kilometer, kilometers, kilomole.
kilomoles, kilorad, kilorads, kilos, kiloton, kilotons, kilovolt, kilovolts, kilowatt, kilowatts, kimono, kimonoed, kimonos, kinfolk, kinfolks, kingbolt, kingbolts, kingdom, kingdoms, kinghood,
kinghoods, kingpost, kingposts, kingwood, kingwoods, kinkajou, kinkajous, kino.
kinos, kinsfolk, kinswoman, kinswomen, kiosk, kiosks, kleptomania, kleptomaniac, kleptomaniacs, kleptomanias, knighthood, knighthoods, knobbier, knobbiest, knoblike, knocking, knolling, knotlike,
knottier, knottiest, knottily, knotting, knouting, knowing, knowinger, knowingest, knowings, kohlrabi, kohlrabies, koine, koines, kolinski, kolinskies, kolinsky, komatik, komatiks, kookie, kookier.
kookiest, koppie, koppies, koshering, kotowing, koumis, koumises, koumiss, koumisses, kowtowing, krikorian, krooni, kryolite, kryolites, kryolith, kryoliths, kurtosis, kurtosises, kyphosis, kyphotic,
laboratories, laboring, laborious, laboriously.
laborite, laborites, labouring, labroid, labroids, laceration, lacerations, laconic, laconically, laconism, laconisms, lactation, lactations, lactonic, ladino, ladinos, lambdoid, lamentation,
lamentations, lamination, laminations, laminose.
laminous, lampion, lampions, lampooning, landholding, landholdings, lanolin, lanoline, lanolines, lanolins, lanosities, lanosity, lascivious, lasciviousness, lasciviousnesses, lassoing, latigo,
latigoes, latigos, laughingstock, laughingstocks, lavation, lavations, lavatories, laxation, laxations, leapfrogging, lection, lections, legation, legations, legion, legionaries, legionary,
legionnaire, legionnaires, legions, legislation, legislations, legislator.
legislators, leguminous, lekythoi, lemonish, lemuroid, lemuroids, lentigo, lentoid, leonine, lepidote, leporid, leporids, leporine, leprosies, leprotic, leptonic, lesion, lesions, lessoning,
leukocytosis, leukopenia, leukophoresis, leukosis, leukotic, lewisson, lewissons, lexicographer, lexicographers.
lexicographic, lexicographical, lexicographies, lexicography, lexicon, lexicons, liaison, liaisons, lianoid, libation, libations, libeccio, libeccios, libellous, libelous, liberation, liberations,
liberator, liberators, libidinous, libido, libidos, libretto, librettos, licensor, licensors, licentious, licentiously, licentiousness.
licentiousnesses, lichenous, licorice, licorices, lictor, lictors, lido, lidos, lifeblood, lifebloods, lifeboat, lifeboats, lifelong, lifework, lifeworks, liftoff, liftoffs, ligation, ligations,
lighthouse, lighthouses, lightproof, ligneous, ligroin, ligroine, ligroines, ligroins, liguloid, likelihood, likelihoods.
limacon, limacons, limbo, limbos, limestone, limestones, limitation, limitations, limo, limonene, limonenes, limonite, limonites, limos, limousine, limousines, limuloid, limuloids, linalol, linalols,
linalool, linalools, lingcod, lingcods, lingo, lingoes, linkboy, linkboys, linkwork, linkworks, lino, linocut, linocuts, linoleum, linoleums, linos, linstock, linstocks, lintol.
lintols, lion, lioness, lionesses, lionfish, lionfishes, lionise, lionised, lioniser, lionisers, lionises, lionising, lionization, lionizations, lionize, lionized, lionizer, lionizers, lionizes,
lionizing, lionlike, lions, lipocyte, lipocytes, lipoid, lipoidal, lipoids, lipoma, lipomas, lipomata, liquefaction, liquefactions, liquidation, liquidations, liquor.
liquored, liquoring, liquors, lirot, liroth, lissom, lissome, lissomly, lithesome, litho, lithograph, lithographer, lithographers, lithographic, lithographies, lithographs, lithography, lithoid,
lithos, lithosol, lithosols, litigation.
litigations, litigious, litigiousness, litigiousnesses, litoral, litotes, littoral, littorals, livelihood, livelihoods, livelong, livestock, livestocks, loading, loadings, loafing, loamier, loamiest,
loaming, loaning, loanings, loathing, loathings, lobation, lobations, lobbied.
lobbies, lobbing, lobbying, lobbyism, lobbyisms, lobbyist, lobbyists, lobefin, lobefins, lobelia, lobelias, lobeline, lobelines, loblollies, lobotomies, lobstick, lobsticks, localise, localised,
localises, localising, localism, localisms, localist, localists, localite, localites, localities, locality, localization, localizations, localize, localized, localizes.
localizing, locating, location, locations, locative, locatives, lochia, lochial, loci, locking, locksmith, locksmiths, locoing, locoism, locoisms, locomoting, locomotion, locomotions, locomotive,
locomotives, loculi, locution.
locutions, locutories, lodging, lodgings, lodicule, lodicules, loessial, loftier, loftiest, loftily, loftiness, loftinesses, lofting, logarithm, logarithmic, logarithms, loggia, loggias, loggie,
loggier, loggiest, logging, loggings, logia.
logic, logical, logically, logician, logicians, logicise, logicised, logicises, logicising, logicize, logicized, logicizes, logicizing, logics, logier, logiest, logily, loginess, loginesses, logion,
logions, logistic, logistical, logistics, logoi.
logotypies, logrolling, loin, loins, loiter, loitered, loiterer, loiterers, loitering, loiters, lollies, lolling, lollipop, lollipops, lolloping, lollygagging, lonelier, loneliest, lonelily,
loneliness, lonelinesses, longeing, longevities, longevity, longhair, longhairs, longing, longingly, longings, longish, longitude.
longitudes, longitudinal, longitudinally, longline, longlines, longship, longships, longtime, longwise, loobies, looie, looies, looing, looking, looming, loonier, loonies, looniest, loopholing,
loopier, loopiest, looping, loosening, loosing, looting, loping, loppering, loppier, loppiest, lopping, lopsided, lopsidedly, lopsidedness, lopsidednesses, lopstick, lopsticks, loquacious.
loquacities, loquacity, lording, lordings, lordlier, lordliest, lordlike, lordling, lordlings, lordosis, lordotic, lordship, lordships, lorica, loricae, loricate, loricates, lories, lorikeet,
lorikeets, lorimer, lorimers, loriner, loriners, loris, lorises.
lorries, losing, losingly, losings, lothario, lotharios, lotic, lotion, lotions, lotteries, lotting, loudening, loudish, loudlier, loudliest, louie, louies, louis, lounging, louping, louring,
lousier, lousiest, lousily, lousiness, lousinesses, lousing, louting, loutish, loutishly, lovebird, lovebirds, lovelier, lovelies, loveliest, lovelily.
loveliness, lovelinesses, lovesick, lovevine, lovevines, loving, lovingly, lowering, lowing, lowings, lowish, lowlier, lowliest, lowlife, lowlifes, lowliness, lowlinesses, loxing, loyalism,
loyalisms, loyalist, loyalists, loyalties, lubrication, lubrications, lubricator, lubricators, ludicrous, ludicrously, ludicrousness.
ludicrousnesses, lugubrious, lugubriously, lugubriousness, lugubriousnesses, luminosities, luminosity, luminous, luminously, lunation, lunations, luscious, lusciously, lusciousness, lusciousnesses,
luteolin, luteolins, luxation, luxations, luxurious, luxuriously, lycanthropies, lymphocytopenia, lymphocytosis, lymphoid, lyophile, lyriform, lysogenies, macaroni, macaronies, macaronis,
machination, machinations, machismo, machismos, machzorim, mademoiselle.
mademoiselles, mafiosi, mafioso, magnanimous, magnanimously, magnanimousness, magnanimousnesses, magnetization, magnetizations, magnification, magnifications, magnolia, magnolias, mahogonies,
mahonia, mahonias, mahzorim, maidenhood, maidenhoods, maidhood, maidhoods, mailbox, mailboxes, maillot, maillots, mailperson, mailpersons, mailwoman, maintop, maintops, maiolica, maiolicas, majolica,
majolicas, majoring, majorities, majority, makimono, makimonos, maladroit.
malapropism, malapropisms, malediction, maledictions, malformation, malformations, malfunction, malfunctioned, malfunctioning, malfunctions, malicious, maliciously, malison, malisons, malleoli,
malnourished, malnutrition, malnutritions, mamboing, mammocking, manatoid, mandioca, mandiocas, mandolin, mandolins, manifestation, manifestations, manifesto, manifestos, manifold.
manifolded, manifolding, manifolds, manihot, manihots, manioc, manioca, maniocas, maniocs, manipulation, manipulations, manipulator, manipulators, manito, manitos, manitou, manitous, mannitol,
mannitols, manorial, manorialism, manorialisms, mansion, mansions, maraschino, maraschinos, marchioness, marchionesses, marigold, marigolds, marionette.
marionettes, mariposa, mariposas, marooning, marrowing, masculinization, masculinizations, masochism, masochisms, masochist, masochistic, masochists, masonic, masoning, masonries, massicot,
massicots, mastectomies, mastication, mastications, mastoid, mastoids, masturbation, masturbations, materialization, materializations, matriculation, matriculations, matrimonial, matrimonially,
matrimonies, matrimony, mattoid, mattoids, maturation, maturational, maturations, maxicoat.
maxicoats, mayonnaise, mayonnaises, mayoralties, mechanization, mechanizations, meconium, meconiums, medallion, medallions, mediation, mediations, mediator, mediators, medication, medications,
medico, medicos, mediocre, mediocrities, mediocrity, meditation.
meditations, medusoid, medusoids, meetinghouse, meetinghouses, meioses, meiosis, meiotic, melancholia, melancholic, melancholies, melanoid, melanoids, melatonin, melilot, melilots, meliorate,
meliorated, meliorates, meliorating, melioration, meliorations, meliorative.
mellifluous, mellifluously, mellifluousness, mellifluousnesses, mellowing, melodia, melodias, melodic, melodically, melodies, melodious, melodiously, melodiousness, melodiousnesses, melodise,
melodised, melodises, melodising, melodist, melodists, melodize, melodized, melodizes, melodizing, melodramatic, melodramatist, melodramatists, meloid, meloids, memoir, memoirs, memorabilia,
memorabilities, memorability, memorial, memorialize, memorialized, memorializes, memorializing, memorials.
memories, memorization, memorizations, memorize, memorized, memorizes, memorizing, mendacious, mendaciously, mendigo, mendigos, menologies, menstruation, menstruations, mention, mentioned,
mentioning, mentions, meowing, merino, merinos, meritorious, meritoriously, meritoriousness, meritoriousnesses, meropia.
meropias, meropic, mesdemoiselles, mesonic, mestino, mestinoes, mestinos, mestizo, mestizoes, mestizos, metabolic, metabolism, metabolisms, metabolize, metabolized, metabolizes, metabolizing,
metalworking, metalworkings, metamorphosing, metamorphosis, metaphorical, metazoic, meteoric, meteorically, meteorite, meteorites, meteoritic, meteorological, meteorologies, meteorologist,
meteorologists, methodic, methodical, methodically, methodicalness, methodicalnesses, methodological, methodologies.
meticulous, meticulously, meticulousness, meticulousnesses, metonymies, metopic, metrication, metrications, metropolis, metropolises, metropolitan, miaou, miaoued, miaouing, miaous, miaow, miaowed,
miaowing, miaows, micro, microbar, microbars, microbe, microbes, microbial, microbic, microbiological, microbiologies, microbiologist, microbiologists, microbiology, microbus, microbuses.
microbusses, microcomputer, microcomputers, microcosm, microfilm, microfilmed, microfilming, microfilms, microhm, microhms, microluces, microlux, microluxes, micrometer, micrometers, micromho,
micromhos, microminiature, microminiatures, microminiaturization, microminiaturizations, microminiaturized, micron, microns, microorganism, microorganisms, microphone, microphones, microscope,
microscopes, microscopic, microscopical, microscopically, microscopies, microscopy, microwave, microwaves, midiron, midirons, midmonth.
midmonths, midmost, midmosts, midnoon, midnoons, midpoint, midpoints, midstories, midstory, midtown, midtowns, mignon, mignonne, mignons, migratation, migratational, migratations, migrator,
migrators, migratory, mikado, mikados, mikron, mikrons, mikvoth, milepost, mileposts, milesimo, milesimos, milestone, milestones, milfoil, milfoils, milksop, milksops, milkwood, milkwoods, milkwort,
millimho, millimhos, milliohm, milliohms, million, millionaire, millionaires, millions, millionth, millionths, millpond, millponds, millstone, millstones, millwork, millworks, milo, milord, milords,
milos, mimeograph, mimeographed, mimeographing, mimeographs, mimosa, mimosas.
minatory, minerological, minerologies, minerologist, minerologists, minerology, minicalculator, minicalculators, miniclock, miniclocks, minicomponent, minicomponents, minicomputer, minicomputers,
miniconvention, miniconventions, minicourse, minicourses, minigroup, minigroups, minihospital, minihospitals, minimization, minination, mininations, mininetwork, mininetworks.
mininovel, mininovels, minion, minions, miniproblem, miniproblems, minirebellion, minirebellions, minirecession, minirecessions, minirobot, minirobots, minischool, minischools, minisocieties,
minisociety, ministration, ministrations, miniterritories, miniterritory, minivacation, minivacations, miniversion, miniversions, minnow, minnows.
minor, minorca, minorcas, minored, minoring, minorities, minority, minors, mioses, miosis, miotic, miotics, miraculous, miraculously, mirador, miradors, mirror, mirrored, mirroring, mirrors,
misanthrope, misanthropes, misanthropic, misanthropies, misanthropy, misapprehension.
misapprehensions, misappropriate, misappropriated, misappropriates, misappropriating, misappropriation, misappropriations, misatone, misatoned, misatones, misatoning, misbegot, misbehavior,
misbehaviors, misbound, miscalculation, miscalculations, miscegenation, miscegenations, miscellaneous, miscellaneously.
miscellaneousness, miscellaneousnesses, mischievous, mischievously, mischievousness, mischievousnesses, miscoin, miscoined, miscoining, miscoins, miscolor, miscolored, miscoloring, miscolors,
misconceive, misconceived, misconceives, misconceiving, misconception, misconceptions, misconduct, misconducts, misconstruction, misconstructions, misconstrue, misconstrued, misconstrues,
misconstruing, miscook.
miscooked, miscooking, miscooks, miscopied, miscopies, miscopy, miscopying, miscount, miscounted, miscounting, miscounts, misdemeanor, misdemeanors, misdo, misdoer, misdoers, misdoes, misdoing,
misdoings, misdone, misdoubt, misdoubted, misdoubting, misdoubts.
misdrove, misenrol, misenrolled, misenrolling, misenrols, misform, misformed, misforming, misforms, misfortune, misfortunes, misgrow, misgrowing, misgrown, misgrows, mishmosh, mishmoshes, misinform,
misinformation, misinformations, misinforms, misinterpretation, misinterpretations, misjoin.
misjoined, misjoining, misjoins, misknow, misknowing, misknown, misknows, mislabor, mislabored, mislaboring, mislabors, mislodge, mislodged, mislodges, mislodging, mismove, mismoved, mismoves,
mismoving, misnomer, misnomers, miso, misogamies.
misogamy, misogynies, misogynist, misogynists, misogyny, misologies, misology, misos, mispoint, mispointed, mispointing, mispoints, mispoise, mispoised, mispoises, mispoising, mispronounce,
mispronounced, mispronounces, mispronouncing, mispronunciation, mispronunciations, misquotation.
misquotations, misquote, misquoted, misquotes, misquoting, misrepresentation, misrepresentations, misshod, mission, missionaries, missionary, missioned, missioning, missions, missort, missorted,
missorting, missorts, missound, missounded, missounding, missounds, missout, missouts, misspoke, misspoken, misstop, misstopped, misstopping.
misstops, mistbow, mistbows, misthought, misthrow, misthrowing, misthrown, misthrows, mistletoe, mistook, mistouch, mistouched, mistouches, mistouching, mistutor, mistutored, mistutoring, mistutors,
misunion, misunions, misword, misworded, miswording, miswords, miswrote, misyoke, misyoked.
misyokes, misyoking, mitigation, mitigations, mitigator, mitigators, mitigatory, mitogen, mitogens, mitoses, mitosis, mitotic, mitsvoth, mitzvoth, mixologies, mixology, mnemonic, mnemonics, moaning,
moating, moatlike, mobbing, mobbish, mobile, mobiles, mobilise, mobilised, mobilises, mobilising, mobilities, mobility, mobilization.
mobilizations, mobilize, mobilized, mobilizer, mobilizers, mobilizes, mobilizing, moccasin, moccasins, mochila, mochilas, mockeries, mocking, mockingbird, mockingbirds, mockingly, modalities,
modality, modeling, modelings, modelling, moderating, moderation, moderations, modernities, modernity.
modernization, modernizations, modernize, modernized, modernizer, modernizers, modernizes, modernizing, modesties, modi, modica, modicum, modicums, modification, modifications, modified, modifier,
modifiers, modifies, modify, modifying, modioli, modiolus, modish, modishly, modiste, modistes, modularities.
modularity, modularized, modulating, modulation, modulations, moduli, mogging, mohair, mohairs, mohalim, moidore, moidores, moieties, moiety, moil, moiled, moiler, moilers, moiling, moils, moira,
moirai, moire, moires, moist, moisten, moistened, moistener, moisteners, moistening, moistens, moister, moistest, moistful, moistly, moistness.
moistnesses, moisture, moistures, molalities, molality, molarities, molarity, moldering, moldier, moldiest, moldiness, moldinesses, molding, moldings, molehill, molehills, moleskin, moleskins,
molestation, molestations, molesting, molies, moline, mollie, mollies, mollification, mollifications, mollified, mollifies, mollify, mollifying, mollycoddling, molting, molybdic, momentarily, momi,
momism, momisms, mommies, monacid.
monacids, monadic, monadism, monadisms, monandries, monarchic, monarchical, monarchies, monasterial, monasteries, monastic, monastically, monasticism, monasticisms, monastics, monaxial, monazite,
monazites, monecian, monetise, monetised, monetises, monetising, monetize, monetized, monetizes, monetizing, mongering, mongolism, mongolisms, monicker.
monickers, monie, monied, monies, moniker, monikers, monish, monished, monishes, monishing, monism, monisms, monist, monistic, monists, monition, monitions, monitive, monitor, monitored, monitories,
monitoring, monitors, monitory, monkeries, monkeying, monkeyshines, monkfish, monkfishes, monkish, monkishly, monkishness, monkishnesses, monoacid, monoacids, monodic, monodies, monodist.
monodists, monoecies, monofil, monofils, monogamic, monogamies, monogamist, monogamists, monogenies, monograming, monogramming, monogynies, monolingual, monolith, monolithic, monoliths, monologies,
monologist, monologists, monologuist, monologuists, monomial, monomials, mononucleosis, mononucleosises, monopodies, monopolies, monopolist, monopolistic, monopolists, monopolization,
monopolizations, monopolize, monopolized, monopolizes.
monopolizing, monorail, monorails, monosyllablic, monotheism, monotheisms, monotheist, monotheists, monotint, monotints, monotonies, monoxide, monoxides, monsieur, monsignor, monsignori, monsignors,
monstrosities, monstrosity, montaging, monteith, monteiths, monthlies, mooching, moodier, moodiest, moodily, moodiness, moodinesses, mooing, moonfish, moonfishes.
moonier, mooniest, moonily, mooning, moonish, moonlight, moonlighted, moonlighter, moonlighters, moonlighting, moonlights, moonlike, moonlit, moonrise, moonrises, moonsail, moonsails, moonshine,
moonshines, moorier, mooriest, mooring, moorings, moorish, mooting, moping, mopingly, mopish, mopishly, mopping, morainal, moraine, moraines, morainic, moralise.
moralised, moralises, moralising, moralism, moralisms, moralist, moralistic, moralists, moralities, morality, moralize, moralized, moralizes, moralizing, moratoria, moratorium, moratoriums, morbid,
morbidities, morbidity, morbidly, morbidness, morbidnesses, morbific, morbilli, mordancies.
mordanting, moribund, moribundities, moribundity, morion, morions, morning, mornings, moronic, moronically, moronism, moronisms, moronities, moronity, morosities, morosity, morphia, morphias,
morphic, morphin, morphine, morphines, morphins, morphologic, morphologically, morphologies, morrion, morrions, morris, morrises, morseling, morselling, mortalities, mortality, mortaring, mortgaging,
mortice, morticed, mortices, morticing.
mortification, mortifications, mortified, mortifies, mortify, mortifying, mortise, mortised, mortiser, mortisers, mortises, mortising, mortmain, mortmains, mortuaries, mosaic, mosaicked, mosaicking,
mosaics, moseying, moshavim, mosquito, mosquitoes, mosquitos, mossier, mossiest, mossing, mosslike, mothballing, mothering, mothier, mothiest, motif.
motifs, motile, motiles, motilities, motility, motion, motional, motioned, motioner, motioners, motioning, motionless, motionlessly, motionlessness, motionlessnesses, motions, motivate, motivated,
motivates, motivating, motivation, motivations, motive, motived, motiveless, motives, motivic, motiving, motivities, motivity, motlier, motliest.
motorbike, motorbikes, motorcyclist, motorcyclists, motoric, motoring, motorings, motorise, motorised, motorises, motorising, motorist, motorists, motorize, motorized, motorizes, motorizing,
mottling, mouching, mouchoir, mouchoirs, mouille, moujik, moujiks, mouldering, mouldier, mouldiest, moulding, mouldings, moulin, moulins.
moulting, mounding, mountain, mountaineer, mountaineered, mountaineering, mountaineers, mountainous, mountains, mountaintop, mountaintops, mounting, mountings, mourning, mournings, mousier, mousiest,
mousily, mousing, mousings, mouthier, mouthiest, mouthily, mouthing, mouthpiece, mouthpieces, movie, moviedom, moviedoms, movies, moving, movingly, mowing, moxie, moxies, mucilaginous.
mucinoid, mucinous, mucoid, mucoidal, mucoids, mucosities, mucositis, mucosity, mullion, mullioned, mullioning, mullions, multibillion, multicolored, multicounty, multidenominational,
multidimensional, multidirectional, multidivisional, multifarous, multifarously, multifunction, multifunctional, multihospital, multimillion, multimillionaire, multimodalities, multimodality,
multiplexor, multiplexors.
multiplication, multiplications, multipolar, multiproblem, multiproduct, multipurpose, multiroomed, multistory, multitudinous, multiunion, mummification, mummifications, munition, munitioned,
munitioning, munitions, munnion, munnions, muonic, mushrooming, mutation, mutational, mutations, muticous, mutilation, mutilations, mutilator, mutilators, mutinous, mutinously, myceloid, mycologies,
mycosis, mycotic, myeloid, myelosuppression, myelosuppressions, mylonite.
mylonites, myogenic, myoid, myologic, myologies, myopathies, myopia, myopias, myopic, myopically, myopies, myosin, myosins, myosis, myosotis, myosotises, myotic, myotics, myotonia, myotonias,
myotonic, myriapod, myriapods, myriopod, myriopods, myrmidon, myrmidons, mysterious, mysteriously, mysteriousness, mysteriousnesses, mystification, mystifications, mythoi, mythological.
mythologies, mythologist, mythologists, myxoid, naboberies, nabobism, nabobisms, naevoid, nailfold, nailfolds, nainsook, nainsooks, naoi, napiform, narcosis, narcotic, narcotics, narration,
narrations, narrowing, nasion, nasions, natation, natations, nation, national, nationalism, nationalisms, nationalist, nationalistic, nationalists, nationalities, nationality, nationalization,
nationalizations, nationalize, nationalized, nationalizes, nationalizing.
nationally, nationals, nationhood, nationhoods, nations, naturalization, naturalizations, navigation, navigations, navigator, navigators, necrologies, necromancies, necropsied, necropsies,
necropsying, necrosing, necrosis, necrotic, needlepoint, needlepoints, nefarious, nefariouses, nefariously, negation, negations, negotiable, negotiate, negotiated, negotiates, negotiating,
negotiation, negotiations, negotiator, negotiators, negroid, negroids, neighbor, neighbored, neighborhood.
neighborhoods, neighboring, neighborliness, neighborlinesses, neighborly, neighbors, nektonic, neolith, neoliths, neologic, neologies, neologism, neologisms, neomycin, neomycins, neotenic, neotenies,
neoteric, neoterics, nepotic, nepotism, nepotisms, nepotist, nepotists, neroli, nerolis, networking, neuroid, neurologic, neurological, neurologically, neurologies, neurologist, neurologists,
neuronic, neuropathies, neurosis, neurotic, neurotically, neurotics.
neurotoxicities, neurotoxicity, neutralization, neutralizations, neutrino, neutrinos, nevoid, nicol, nicols, nicotin, nicotine, nicotines, nicotins, niello, nielloed, nielloing, niellos,
nightclothes, nightgown, nightgowns, nigrosin, nigrosins, nimious, nimrod, nimrods, nincompoop, nincompoops, ninefold.
ninon, ninons, niobic, niobium, niobiums, niobous, niton, nitons, nitrator, nitrators, nitro, nitrogen, nitrogenous, nitrogens, nitroglycerin, nitroglycerine, nitroglycerines, nitroglycerins,
nitrolic, nitros, nitroso, nitrosurea, nitrosyl, nitrosyls, nitrous, niveous, nobbier, nobbiest, nobbily, nobbling, nobelium, nobeliums, nobilities, nobility, nobodies, nocking.
noctuid, noctuids, noctuoid, nodalities, nodality, noddies, nodding, noddling, nodi, nodical, nodosities, nodosity, noesis, noesises, noetic, noggin, nogging, noggings, noggins, noil, noils, noily,
noir, noise, noised.
noisemaker, noisemakers, noises, noisier, noisiest, noisily, noisiness, noisinesses, noising, noisome, noisy, nomadic, nomadism, nomadisms, nomarchies, nombril, nombrils, nomina, nominal, nominally,
nominals, nominate, nominated.
nominates, nominating, nomination, nominations, nominative, nominatives, nominee, nominees, nomism, nomisms, nomistic, nomoi, nomologies, nonabrasive, nonacademic, nonaccredited, nonacid, nonacids,
nonaddictive, nonadhesive, nonaffiliated, nonaggression, nonaggressions, nonalcoholic, nonaligned, nonautomatic, nonbasic, nonbeing, nonbeings, nonbeliever, nonbelievers, noncandidate, noncandidates,
noncitizen, noncitizens, nonclassical, nonclassified, noncombustible.
noncommercial, noncommittal, noncommunicable, noncompliance, noncompliances, nonconclusive, nonconflicting, nonconforming, nonconformist, nonconformists, nonconsecutive, nonconstructive,
noncontagious, noncontributing, noncontroversial, noncorrosive, noncriminal, noncritical, noncumulative, nondairy, nondeductible, nondeliveries, nondelivery, nondemocratic, nondenominational,
nondescript, nondestructive, nondiscrimination, nondiscriminations, nondiscriminatory, noneducational, nonelastic, nonelective, nonelectric, nonelectronic, nonemotional, nonentities, nonentity,
nonessential, nonexistence, nonexistences, nonexistent, nonexplosive, nonfattening, nonfictional, nonflowering, nonfluid, nonfluids, nonfunctional, nonguilt, nonguilts, nonhereditary, nonideal,
nonindustrial, nonindustrialized, noninfectious, noninflationary, nonintegrated, nonintellectual, noninterference, nonintoxicating, noninvolvement.
noninvolvements, nonionic, nonlife, nonliterary, nonlives, nonliving, nonmagnetic, nonmalignant, nonmedical, nonmetallic, nonmilitary, nonmusical, nonnarcotic, nonnative, nonnegotiable, nonobjective,
nonparametric, nonpareil, nonpareils, nonparticipant, nonparticipants, nonparticipating, nonpartisan, nonpartisans, nonpaying, nonperishable, nonphysical, nonplusing, nonplussing, nonpoisonous,
nonpolitical, nonpolluting, nonproductive, nonprofessional, nonprofit, nonproliferation, nonproliferations.
nonprossing, nonracial, nonradioactive, nonrealistic, nonrecurring, nonrefillable, nonregistered, nonreligious, nonrepresentative, nonresident, nonresidents, nonresponsive, nonrestricted,
nonreversible, nonrigid, nonrival, nonrivals, nonscientific, nonscientist, nonscientists, nonsensical, nonsensically, nonsignificant, nonskid, nonskier, nonskiers, nonslip, nonsmoking, nonsolid,
nonsolids, nonspeaking, nonspecialist, nonspecialists, nonspecific, nonstaining, nonstick, nonstrategic, nonstriker, nonstrikers.
nonstriking, nonsubscriber, nonsuit, nonsuited, nonsuiting, nonsuits, nonsurgical, nonswimmer, nonteaching, nontechnical, nontidal, nontitle, nontoxic, nontraditional, nontropical, nontypical,
nonunion, nonunions, nonusing, nonviolence, nonviolences, nonviolent, nonviral.
nonwhite, nonwhites, noodling, nookies, nooklike, nooning, noonings, noontide, noontides, noontime, noontimes, noosing, noria, norias, norite, norites, noritic, normalcies, normalities, normality,
normalization, normalizations, normalize, normalized, normalizes, normalizing, northing.
northings, noselike, noshing, nosier, nosiest, nosily, nosiness, nosinesses, nosing, nosings, nosologies, nostalgia, nostalgias, nostalgic, nostril, nostrils, notabilities, notability, notarial,
notaries, notarize, notarized, notarizes, notarizing, notating, notation, notations, notching, nothing, nothingness, nothingnesses, nothings, notice, noticeable, noticeably, noticed, notices,
notification, notifications, notified, notifier, notifiers, notifies, notify, notifying, noting, notion, notional, notions, notorieties, notoriety, notorious, notoriously, notornis, notturni,
notwithstanding, nourish, nourished, nourishes, nourishing, nourishment, nourishments, novalike, novation, novations, novelise, novelised, novelises, novelising, novelist, novelists, novelize.
novelized, novelizes, novelizing, novelties, novice, novices, nowise, noxious, nubilose, nubilous, nucleoli, nullification, nullifications, numerologies, numerologist, numerologists, numinous,
numinouses, nuncio, nuncios, nutation, nutations, nutrition, nutritional, nutritions, nutritious, nymphomania, nymphomaniac, nymphomanias, oafish, oafishly, oaklike, oarfish, oarfishes, oaring,
oarlike, oasis, oatlike, obduracies, obeahism.
obeahisms, obedience, obediences, obedient, obediently, obeisance, obeisant, obeli, obelia, obelias, obelise, obelised, obelises, obelising, obelisk, obelisks, obelism, obelisms, obelize, obelized,
obelizes, obelizing, obesities, obesity, obeying, obfuscating, obfuscation, obfuscations, obi, obia, obias, obiism, obiisms, obis, obit.
obits, obituaries, obituary, objecting, objection, objectionable, objections, objective, objectively, objectiveness, objectivenesses, objectives, objectivities, objectivity, oblasti, oblation,
oblations, obligate, obligated, obligates, obligati, obligating, obligation, obligations, obligato.
obligatory, obligatos, oblige, obliged, obligee, obligees, obliger, obligers, obliges, obliging, obligingly, obligor, obligors, oblique, obliqued, obliquely, obliqueness, obliquenesses, obliques,
obliquing, obliquities.
obliquity, obliterate, obliterated, obliterates, obliterating, obliteration, obliterations, oblivion, oblivions, oblivious, obliviously, obliviousness, obliviousnesses, obloquies, obnoxious,
obnoxiously, obnoxiousness, obnoxiousnesses, oboist, oboists, oboli.
obovoid, obscenities, obscenity, obscuring, obscurities, obscurity, obsequies, obsequious, obsequiously, obsequiousness, obsequiousnesses, observation, observations, observatories, observing,
obsessing, obsession, obsessions, obsessive, obsessively, obsidian, obsidians, obsoleting, obstetrical, obstetrician, obstetricians, obstetrics, obstinacies, obstinacy, obstinate, obstinately.
obstructing, obstruction, obstructions, obstructive, obtain, obtainable, obtained, obtainer, obtainers, obtaining, obtains, obtesting, obtruding, obtrusion, obtrusions, obtrusive, obtrusively,
obtrusiveness, obtrusivenesses, obtunding, obturating.
obverting, obviable, obviate, obviated, obviates, obviating, obviation, obviations, obviator, obviators, obvious, obviously, obviousness, obviousnesses, ocarina, ocarinas, occasion, occasional,
occasionally, occasioned, occasioning, occasions, occident, occidental, occidents, occipita, occiput, occiputs, occluding, occulting, occupancies, occupation, occupational, occupationally,
occupations, occupied, occupier, occupiers.
occupies, occupying, occurring, oceangoing, oceanic, oceanographic, oceanographies, ocelli, oceloid, ochering, ochring, ochroid, ocotillo, ocotillos, octadic, octarchies, octonaries, octopi, octroi,
octrois, octupling, oculist, oculists, odalisk, odalisks, oddish, oddities, oddity, odic, odious, odiously, odiousness, odiousnesses, odium, odiums, odometries, odontoid, odontoids, odorize,
odorizes, odorizing, oecologies, oedipal, oedipean, oeillade, oeillades, oenologies, oestrin, oestrins, oestriol, oestriols, offending, offensive, offensively, offensiveness, offensivenesses,
offensives, offering, offerings, offertories, office.
officeholder, officeholders, officer, officered, officering, officers, offices, official, officialdom, officialdoms, officially, officials, officiate, officiated, officiates, officiating, officious,
officiously, officiousness, officiousnesses, offing, offings.
offish, offishly, offloading, offprint, offprinted, offprinting, offprints, offsetting, offside, offspring, oftentimes, ofttimes, oghamic, oghamist, oghamists, ogival, ogive, ogives, ogling, ogreish,
ogreism, ogreisms, ogrish, ogrishly, ogrism, ogrisms, ohia, ohias.
ohing, ohmic, oidia, oidium, oil, oilbird, oilbirds, oilcamp, oilcamps, oilcan, oilcans, oilcloth, oilcloths, oilcup, oilcups, oiled, oiler, oilers, oilhole, oilholes, oilier, oiliest, oilily,
oiliness, oilinesses, oiling.
oilman, oilmen, oilpaper, oilpapers, oilproof, oils, oilseed, oilseeds, oilskin, oilskins, oilstone, oilstones, oiltight, oilway, oilways, oily, oink, oinked, oinking, oinks, oinologies, oinology,
oinomel, oinomels, ointment, ointments, oiticica, oiticicas, okapi, okapis, okaying, oldie, oldies, oldish, oldwife, oldwives, olefin, olefine.
olefines, olefinic, olefins, oleic, olein, oleine, oleines, oleins, oleomargarine, oleomargarines, olibanum, olibanums, oligarch, oligarchic, oligarchical, oligarchies, oligarchs, oligarchy,
oligomer, oligomers, olio, olios, olivary, olive, olives, olivine, olivines, olivinic, ologies, ologist, ologists, olympiad, olympiads, omening, omicron, omicrons, omikron, omikrons.
ominous, ominously, ominousness, ominousnesses, omission, omissions, omissive, omit, omits, omitted, omitting, omniarch, omniarchs, omnibus, omnibuses, omnific, omniform, omnimode, omnipotence,
omnipotences, omnipotent, omnipotently, omnipresence, omnipresences, omnipresent, omniscience, omnisciences, omniscient, omnisciently, omnivora, omnivore, omnivores, omnivorous, omnivorously,
omnivorousness, omnivorousnesses.
omophagies, omphali, onagri, onanism, onanisms, onanist, onanists, oncidium, oncidiums, oncologic, oncologies, oncologist, oncologists, oncoming, oncomings, oneiric, onerier, oneriest, onetime,
ongoing, onion, onions.
onium, onomatopoeia, onside, ontic, ontogenies, ontologies, oodlins, oogamies, oogenies, oogonia, oogonial, oogonium, oogoniums, oohing, oolite, oolites, oolith, ooliths, oolitic, oologic, oologies,
oologist, oologists, oomiac, oomiack, oomiacks, oomiacs, oomiak.
oomiaks, oophytic, oorali, ooralis, oorie, oosporic, ootid, ootids, oozier, ooziest, oozily, ooziness, oozinesses, oozing, opacified, opacifies, opacify, opacifying, opacities, opacity, opalescing,
opaline, opalines, opaquing, opening, openings, operatic, operatics, operating, operation, operational, operationally, operations, operative.
ophidian, ophidians, ophite, ophites, ophitic, ophthalmologies, ophthalmologist, ophthalmologists, opiate, opiated, opiates, opiating, opine, opined, opines, oping, opining, opinion, opinionated,
opinions, opium, opiumism, opiumisms, opiums, oppidan, oppidans, oppilant, oppilate, oppilated, oppilates, oppilating, opportunism, opportunisms, opportunist, opportunistic, opportunists,
opportunities, opportunity, opposing.
opposite, oppositely, oppositeness, oppositenesses, opposites, opposition, oppositions, oppressing, oppression, oppressions, oppressive, oppressively, opprobrious, opprobriously, opprobrium,
opprobriums, oppugning, opsin, opsins, opsonic, opsonified, opsonifies, opsonify, opsonifying, opsonin, opsonins, opsonize, opsonized, opsonizes, opsonizing, optative, optatives, optic, optical,
optician, opticians, opticist, opticists, optics, optima.
optimal, optimally, optime, optimes, optimise, optimised, optimises, optimising, optimism, optimisms, optimist, optimistic, optimistically, optimists, optimize, optimized, optimizes, optimizing,
optimum, optimums, opting, option, optional, optionally, optionals, optioned, optionee, optionees.
optioning, options, optometries, optometrist, opulencies, opuntia, opuntias, oralities, orality, orating, oration, orations, oratorical, oratories, oratorio, oratorios, oratrices, oratrix, orbicular,
orbing, orbit, orbital, orbitals, orbited, orbiter, orbiters, orbiting, orbits, orcein, orceins, orchardist, orchardists.
orchestrating, orchestration, orchestrations, orchid, orchids, orchiectomy, orchil, orchils, orchis, orchises, orchitic, orchitis, orchitises, orcin, orcinol, orcinols, orcins, ordain, ordained,
ordainer, ordainers, ordaining, ordains, ordering, orderlies, orderliness, orderlinesses, ordinal, ordinals, ordinance, ordinances, ordinand, ordinands, ordinarier, ordinaries, ordinariest.
ordinarily, ordinary, ordinate, ordinates, ordination, ordinations, ordines, orectic, orective, oreide, oreides, organdie, organdies, organic, organically, organics, organise, organised, organises,
organising, organism, organisms, organist, organists, organization, organizational, organizationally, organizations, organize, organized, organizer, organizers, organizes, organizing, orgasmic,
orgastic, orgiac, orgic.
orgies, oribatid, oribatids, oribi, oribis, oriel, oriels, orient, oriental, orientals, orientation, orientations, oriented, orienting, orients, orifice, orifices, origami, origamis, origan, origans,
origanum, origanums, origin, original, originalities, originality, originally, originals, originate, originated, originates, originating, originator, originators, origins, orinasal, orinasals,
oriole, orioles.
orison, orisons, ornamentation, ornamentations, ornamenting, ornerier, orneriest, ornis, ornithes, ornithic, ornithological, ornithologist, ornithologists, ornithology, orogenic, orogenies, oroide,
oroides, orologies, orphaning, orphic, orphical, orpiment, orpiments, orpin, orpine, orpines, orpins, orreries.
orrice, orrices, orris, orrises, orthicon, orthicons, orthodontia, orthodontics, orthodontist, orthodontists, orthodoxies, orthoepies, orthographic, orthographies, orthopedic, orthopedics,
orthopedist, orthopedists, orthotic, oscillate, oscillated, oscillates, oscillating, oscillation, oscillations, oscine, oscines, oscinine, oscitant, osculating, osier, osiers, osmatic, osmic,
osmious, osmium.
osmiums, osmosing, osmosis, osmotic, ossein, osseins, ossia, ossicle, ossicles, ossific, ossified, ossifier, ossifiers, ossifies, ossify, ossifying, ossuaries, osteitic, osteitides, osteitis,
ostensible, ostensibly, ostentation, ostentations, ostentatious, ostentatiously, osteoid.
osteoids, osteopathic, osteopathies, osteopenia, ostia, ostiaries, ostiary, ostinato, ostinatos, ostiolar, ostiole, ostioles, ostium, ostomies, ostosis, ostosises, ostracism, ostracisms, ostracize,
ostracized, ostracizes, ostracizing, ostrich, ostriches, ostsis, ostsises, otalgia, otalgias, otalgic, otalgies, otherwise, otic.
otiose, otiosely, otiosities, otiosity, otitic, otitides, otitis, otolith, otoliths, otologies, otoscopies, ototoxicities, ototoxicity, ouabain, ouabains, oughting, ouistiti, ouistitis, ourari,
ouraris, ourebi, ourebis, ourie, ousting, outacting, outadding, outarguing, outasking, outbaking, outbarking, outbawling, outbeaming, outbegging, outbid, outbidden, outbidding, outbids, outblazing,
outbleating, outblessing.
outblooming, outbluffing, outblushing, outboasting, outboxing, outbragging, outbraving, outbreeding, outbribe, outbribed, outbribes, outbribing, outbuild, outbuilding, outbuildings, outbuilds,
outbuilt, outbullied, outbullies, outbullying, outburning, outcapering, outcatching, outcavil, outcaviled, outcaviling, outcavilled, outcavilling, outcavils, outcharming, outcheating, outchid,
outchidden, outchide, outchided.
outchides, outchiding, outclassing, outclimb, outclimbed, outclimbing, outclimbs, outcooking, outcrawling, outcried, outcries, outcropping, outcrossing, outcrowing, outcrying, outcursing, outdancing,
outdaring, outdating, outdid, outdistance, outdistanced, outdistances, outdistancing, outdodging, outdoing, outdrawing, outdreaming, outdressing, outdrink, outdrinking, outdrinks, outdrive,
outdriven, outdrives.
outdriving, outdropping, outeating, outechoing, outfabling, outfacing, outfasting, outfawning, outfeasting, outfeeling, outfield, outfielder, outfielders, outfields, outfight, outfighting, outfights,
outfind, outfinding, outfinds, outfire, outfired, outfires, outfiring, outfit.
outfits, outfitted, outfitter, outfitters, outfitting, outflanking, outflies, outflowing, outflying, outfooling, outfooting, outfoxing, outfrowning, outgain, outgained, outgaining, outgains,
outgassing, outgive, outgiven, outgives, outgiving.
outglaring, outglowing, outgnawing, outgoing, outgoings, outgrin, outgrinned, outgrinning, outgrins, outgrowing, outguessing, outguide, outguided, outguides, outguiding, outgunning, outhearing,
outhit, outhits, outhitting, outhowling, outhumoring, outing, outings, outjinx, outjinxed, outjinxes, outjinxing, outjumping, outjutting, outkeeping, outkick, outkicked, outkicking, outkicks,
outkiss, outkissed.
outkisses, outkissing, outlaid, outlain, outlandish, outlandishly, outlasting, outlaughing, outlawing, outlawries, outlaying, outleaping, outlearning, outlie, outlier, outliers, outlies, outline,
outlined, outlines, outlining, outlive, outlived, outliver, outlivers, outlives, outliving, outloving.
outlying, outmanning, outmarching, outmatching, outmoding, outmoving, outnumbering, outpacing, outpaint, outpainted, outpainting, outpaints, outpassing, outpatient, outpatients, outpitied, outpities,
outpity, outpitying, outplanning, outplaying, outplodding, outpoint.
outpointed, outpointing, outpoints, outpolling, outpouring, outpraying, outpreening, outpressing, outprice, outpriced, outprices, outpricing, outpulling, outpushing, outputting, outquoting,
outracing, outraging, outraise, outraised, outraises, outraising, outranging, outranking, outraving, outreaching, outreading, outridden, outride.
outrider, outriders, outrides, outriding, outright, outring, outringing, outrings, outrival, outrivaled, outrivaling, outrivalled, outrivalling, outrivals, outroaring, outrocking, outrolling,
outrooting, outrunning, outsail, outsailed, outsailing, outsails, outsavoring, outscolding, outscoring, outscorning, outseeing.
outselling, outserving, outshaming, outshine, outshined, outshines, outshining, outshooting, outshouting, outside, outsider, outsiders, outsides, outsight, outsights, outsin, outsing, outsinging,
outsings, outsinned, outsinning, outsins, outsit, outsits, outsitting, outsize, outsized, outsizes, outskirt, outskirts, outsleeping, outsmarting, outsmile, outsmiled, outsmiles, outsmiling,
outsmoking, outsnoring, outsoaring, outspanning.
outspeaking, outspelling, outspending, outstanding, outstandingly, outstaring, outstarting, outstating, outstaying, outsteering, outstrip, outstripped, outstripping, outstrips, outstudied,
outstudies, outstudying, outstunting, outsulking, outswearing, outswim, outswimming, outswims, outtalking, outtasking, outtelling, outthanking, outthink, outthinking, outthinks, outthrobbing,
outtowering, outtrading, outtrick, outtricked, outtricking, outtricks, outtrotting, outtrumping, outvaluing, outvaunting, outvoice, outvoiced, outvoices, outvoicing, outvoting, outwait, outwaited,
outwaiting, outwaits, outwalking, outwarring, outwasting.
outwatching, outwearied, outwearies, outwearing, outwearying, outweeping, outweigh, outweighed, outweighing, outweighs, outwhirl, outwhirled, outwhirling, outwhirls, outwile, outwiled, outwiles,
outwiling, outwill, outwilled, outwilling, outwills, outwind, outwinded, outwinding, outwinds, outwish, outwished, outwishes, outwishing, outwit, outwits, outwitted, outwitting, outworking, outwrit,
outwrites, outwriting, outwritten, outyelling, outyelping, outyield, outyielded, outyielding, outyields, ovalities, ovality, ovarial, ovarian, ovaries, ovariole, ovarioles, ovaritides, ovaritis,
ovation, ovations, ovenbird, ovenbirds, ovenlike, overachiever, overachievers, overacting, overactive.
overaggresive, overambitious, overamplified, overamplifies, overamplify, overamplifying, overanalyzing, overanxieties, overanxiety, overanxious, overapologetic, overarching, overarousing,
overassertive, overawing, overbaking, overbearing, overbetting, overbid, overbidden, overbidding, overbids, overbig, overbite, overbites.
overblowing, overbooking, overborrowing, overbright, overbuild, overbuilded, overbuilding, overbuilds, overburdening, overbuying, overcalling, overcapacities, overcapacity, overcapitalize,
overcapitalized, overcapitalizes, overcapitalizing, overcasting, overcautious, overcharging, overcivilized, overcoming, overcommit, overcommited, overcommiting, overcommits, overcompensating,
overcomplicate, overcomplicated, overcomplicates, overcomplicating, overconcerning, overconfidence, overconfidences, overconfident, overconscientious, overconsuming, overconsumption,
overconsumptions, overcontroling.
overcooking, overcooling, overcorrecting, overcramming, overcritical, overcropping, overcrowding, overdaring, overdecking, overdecorating, overdepending, overdeveloping, overdid, overdoing,
overdosing, overdramatic, overdramatize, overdramatized, overdramatizes, overdramatizing, overdrawing, overdressing, overdrink, overdrinks, overdyeing, overeating, overeducating, overemotional,
overemphasis, overemphasize, overemphasized, overemphasizes, overemphasizing, overenergetic, overenthusiastic, overestimate, overestimated, overestimates.
overestimating, overexaggerating, overexaggeration, overexaggerations, overexcite, overexcited, overexcitement, overexcitements, overexcites, overexciting, overexercise, overexertion, overexertions,
overexhausting, overexpanding, overexpansion, overexpansions, overexplain, overexplained, overexplaining, overexplains, overexploit, overexploited, overexploiting, overexploits, overexposing,
overextending, overextension, overextensions, overfamiliar, overfatigue, overfatigued, overfatigues, overfatiguing, overfearing, overfeeding, overfertilize, overfertilized.
overfertilizes, overfertilizing, overfill, overfilled, overfilling, overfills, overfish, overfished, overfishes, overfishing, overflies, overflowing, overflying, overgild, overgilded, overgilding,
overgilds, overgilt, overgird, overgirded, overgirding, overgirds, overgirt, overglamorize, overglamorized, overglamorizes, overglamorizing, overgoading, overgrazing, overgrowing, overhanding,
overhanging, overharvesting, overhating, overhauling, overheaping, overhearing, overheating.
overhigh, overholding, overhoping, overhunting, overidealize, overidealized, overidealizes, overidealizing, overidle, overimaginative, overimbibe, overimbibed, overimbibes, overimbibing,
overimpressed, overindebted, overindulge, overindulged, overindulgent, overindulges, overindulging, overinflate, overinflated, overinflates, overinflating.
overinfluence, overinfluenced, overinfluences, overinfluencing, overing, overinsistent, overintense, overintensities, overintensity, overinvest, overinvested, overinvesting, overinvests, overinvolve,
overinvolved, overinvolves, overinvolving, overjoying, overkill, overkilled, overkilling, overkills, overkind, overlading, overlaid, overlain, overlapping, overlaying, overleaping, overletting,
overliberal, overlie, overlies, overlive, overlived, overlives, overliving, overloading, overlooking, overlording.
overloving, overlying, overmanning, overmedicate, overmedicated, overmedicates, overmedicating, overmelting, overmild, overmix, overmixed, overmixes, overmixing, overnice, overnight, overobvious,
overoptimistic, overorganize, overorganized, overorganizes, overorganizing, overpaid, overparticular, overpassing, overpatriotic.
overpaying, overpermissive, overplaying, overplied, overplies, overplying, overpossessive, overpowering, overprasing, overprescribe, overprescribed, overprescribes, overprescribing, overprice,
overpriced, overprices, overpricing, overprint, overprinted, overprinting, overprints, overprivileged, overproducing, overproduction, overproductions, overpromise, overprotecting, overprotective.
overpublicize, overpublicized, overpublicizes, overpublicizing, overqualified, overrating, overreaching, overreacting, overreaction, overreactions, overrefine, overregulating, overregulation,
overregulations, overreliance, overreliances, overrepresenting, overresponding, overrich, overridden, override, overrides, overriding, overrife, overripe.
overruffing, overruling, overrunning, oversalting, oversaturating, oversaving, overseeding, overseeing, overselling, oversensitive, overserious, oversetting, oversewing, overshadowing, overshooting,
oversick, overside, oversides, oversight, oversights, oversimple, oversimplified, oversimplifies.
oversimplify, oversimplifying, oversize, oversizes, oversleeping, overslip, overslipped, overslipping, overslips, overslipt, oversoaking, oversolicitous, overspecialize, overspecialized,
overspecializes, overspecializing, overspending, overspin, overspins, overspreading, overstaffing, overstating, overstaying, overstepping, overstimulate, overstimulated.
overstimulates, overstimulating, overstir, overstirred, overstirring, overstirs, overstocking, overstrain, overstrained, overstraining, overstrains, overstressing, overstretching, overstrict,
oversupping, oversupplied, oversupplies, oversupplying, oversuspicious, oversweetening, overtaking, overtasking, overtaxing.
overthin, overtighten, overtightened, overtightening, overtightens, overtime, overtimed, overtimes, overtiming, overtire, overtired, overtires, overtiring, overtoil, overtoiled, overtoiling,
overtoils, overtopping, overtrain, overtrained, overtraining, overtrains.
overtreating, overtrim, overtrimmed, overtrimming, overtrims, overturing, overturning, overurging, overusing, overutilize, overutilized, overutilizes, overutilizing, overvaluing, overview, overviews,
overvoting, overwarming, overwearing, overweening, overweight, overwetting, overwhelming, overwhelmingly.
overwide, overwily, overwind, overwinding, overwinds, overwise, overworking, overwrite, overwrited, overwrites, overwriting, ovibos, ovicidal, ovicide, ovicides, oviducal, oviduct, oviducts, oviform,
ovine, ovines, ovipara, oviposit, oviposited, ovipositing, oviposits, ovisac, ovisacs, ovoid, ovoidal, ovoids.
ovoli, ovonic, ovulating, ovulation, ovulations, owing, owlish, owlishly, owllike, ownership, ownerships, owning, oxalating, oxalic, oxalis, oxalises, oxazine, oxazines, oxid, oxidable, oxidant,
oxidants, oxidase, oxidases, oxidasic, oxidate, oxidated, oxidates, oxidating.
oxidation, oxidations, oxide, oxides, oxidic, oxidise, oxidised, oxidiser, oxidisers, oxidises, oxidising, oxidizable, oxidize, oxidized, oxidizer, oxidizers, oxidizes, oxidizing, oxids, oxim, oxime,
oximes, oxims.
oxlip, oxlips, oxtail, oxtails, oxyacid, oxyacids, oxygenic, oxyphil, oxyphile, oxyphiles, oxyphils, oxytocic, oxytocics, oxytocin, oxytocins, oystering, oysterings, ozonic, ozonide, ozonides,
ozonise, ozonised, ozonises.
ozonising, ozonize, ozonized, ozonizer, ozonizers, ozonizes, ozonizing, pachouli, pachoulis, pacification, pacifications, paction, pactions, paddocking, padlocking, padroni, paisano, paisanos,
paleoclimatologic, palinode, palinodes, palliation, palliations, palomino, palominos, palpation, palpations, palpitation, palpitations, pandemonium, pandemoniums, pandowdies, pangolin, pangolins,
panoplies, panoptic, panoramic, pansophies.
pantomime, pantomimed, pantomimes, pantomiming, papillon, papillons, paradoxical, paradoxically, paradropping, paragoning, paranoia, paranoias, paranoid, paranoids, parashioth, parboil, parboiled,
parboiling, parboils, pardoning, paregoric, paregorics.
parishioner, parishioners, parochial, parochialism, parochialisms, parodic, parodied, parodies, parodist, parodists, parodoi, parodying, paroling, parotic, parotid, parotids, parotoid, parotoids,
parroting, parsimonies, parsimonious, parsimoniously, parsimony.
parsonic, participation, participations, participatory, partition, partitions, parturition, parturitions, parvolin, parvolins, passion, passionate, passionateless, passionately, passions,
pasteurization, pasteurizations, pastoring, pastromi, pastromis, pathologic, pathological, pathologies.
pathologist, pathologists, patio, patios, patois, patrimonial, patrimonies, patrimony, patriot, patriotic, patriotically, patriotism, patriotisms, patriots, patrolling, patronize, patronized,
patronizes, patronizing, pavilion, pavilioned, pavilioning, pavilions, pavior, paviors, paviour, paviours, pavonine, peacockier, peacockiest, peacocking, peccadillo, peccadilloes.
peccadillos, peculatation, peculatations, pedagogic, pedagogical, pedagogies, pediform, pedologies, peignoir, peignoirs, peloria, pelorian, pelorias, peloric, pemoline, pemolines, penetration,
penetrations, penologies, penpoint, penpoints, pension, pensione, pensioned, pensioner, pensioners, pensiones, pensioning, pensions, pentomic, penurious, peonies, peonism.
peonisms, peopling, peperoni, peperonis, peponida, peponidas, peponium, peponiums, peptonic, perambulation, perambulations, perception, perceptions, percoid, percoids, percolating, percussion,
percussions, peremptorily, perfection, perfectionist, perfectionists, perfections, perfidious, perfidiously, perforating, perforation, perforations, performing, pericopae, pericope.
pericopes, peridot, peridots, perigon, perigons, perilous, perilously, period, periodic, periodical, periodically, periodicals, periodid, periodids, periods, periotic, periscope, periscopes,
permeation, permeations, permission, permissions, pernicious, perniciously, perorating, peroxid, peroxide, peroxided, peroxides, peroxiding, peroxids.
perpetration, perpetrations, perpetuation, perpetuations, persecution, persecutions, personalities, personality, personalize, personalized, personalizes, personalizing, personification,
personifications, personifies, personify, perspicacious, perspiration, perspirations, persuasion, persuasions, pertinacious, perturbation, perturbations, perversion, perversions, pervious,
petalodies, petaloid, petiolar, petiole, petioled, petioles.
petition, petitioned, petitioner, petitioners, petitioning, petitions, petrifaction, petrifactions, petrolic, petticoat, petticoats, pettifog, pettifogged, pettifogging, pettifogs, pharmacologic,
pharmacological, pharmacologist, pharmacologists, phelonia, phenolic, phenolics, philanthropic, philanthropies, philanthropist, philanthropists, philanthropy, philharmonic, philodendron,
philodendrons, philomel, philomels, philosopher, philosophers.
philosophic, philosophical, philosophically, philosophies, philosophize, philosophized, philosophizes, philosophizing, philosophy, phimoses, phimosis, phimotic, phobia, phobias, phobic, phocine,
phoenix, phoenixes, phonating, phonemic, phonetic, phonetician.
phoneticians, phonetics, phonic, phonics, phonier, phonies, phoniest, phonily, phoning, phonographic, phosphatic, phosphid, phosphids, phosphin, phosphins, phosphoric, photic, photics, photoelectric,
photoelectrically, photogenic, photographic, photographies, photographing, photoing, photomapping, photonic.
photopia, photopias, photopic, photosetting, photosynthesis, photosynthesises, photosynthesize, photosynthesized, photosynthesizes, photosynthesizing, photosynthetic, phraseologies, phylloid,
phylloids, physiognomies, physiognomy, physiologic, physiological, physiologies, physiologist, physiologists, physiology, physiotherapies, physiotherapy.
phytoid, phytonic, piano, pianos, pibroch, pibrochs, picacho, picachos, picador, picadores, picadors, picaro, picaroon, picarooned, picarooning, picaroons, picaros, piccolo, piccolos, piceous,
picklocks, pickoff, pickoffs, pickpocket, pickpockets, picloram, piclorams, picogram, picograms, picolin, picoline, picolines, picolins, picot, picoted, picotee, picotees, picoting, picots,
pictorial, piddock, piddocks, piedfort, piedforts, piedmont, piedmonts, piefort, pieforts, pierrot, pierrots, pigboat, pigboats, pigeon, pigeonhole, pigeonholed, pigeonholes, pigeonholing.
pigeons, pigmentation, pigmentations, pignora, pileous, pilewort, pileworts, piliform, pillbox, pillboxes, pillion, pillions, pilloried, pillories, pillory, pillorying, pillow, pillowcase,
pillowcases, pillowed, pillowing, pillows, pillowy, pilose, pilosities, pilosity, pilot, pilotage, pilotages, piloted, piloting, pilotings, pilotless, pilots, pilous, pimento, pimentos, pimiento.
pimientos, pinafore, pinafores, pinbone, pinbones, pincushion, pincushions, pinecone, pinecones, pinewood, pinewoods, pinfold, pinfolded, pinfolding, pinfolds, pingo, pingos, pinhole, pinholes,
pinion, pinioned, pinioning, pinions, pinko, pinkoes, pinkos.
pinkroot, pinkroots, pinochle, pinochles, pinocle, pinocles, pinole, pinoles, pinon, pinones, pinons, pinpoint, pinpointed, pinpointing, pinpoints, pintado, pintadoes, pintados, pintano, pintanos,
pinto, pintoes, pintos, pinwork, pinworks, pinworm, pinworms, pinyon, pinyons, piolet, piolets, pion, pioneer, pioneered, pioneering, pioneers, pionic, pions, piosities.
piosity, pious, piously, pirog, pirogen, piroghi, pirogi, pirogue, pirogues, pirojki, piroque, piroques, piroshki, pirouette, pirouetted, pirouettes, pirouetting, pirozhki, pirozhok, piscator,
piscators, pisiform, pisiforms, pismo, pisolite, pisolites, pissoir, pissoirs, pistachio, pistol, pistole, pistoled, pistoles, pistoling, pistolled, pistolling, pistols, piston.
pistons, pitchfork, pitchforks, pitchout, pitchouts, piteous, piteously, piton, pitons, pivot, pivotal, pivoted, pivoting, pivots, placoid, placoids, planktonic, plantation, plantations, plasmoid,
plasmoids, platitudinous, platonic, platooning, plenipotentiaries, plenipotentiary, plimsol, plimsole, plimsoles, plimsoll, plimsolls, plimsols, plodding, ploddingly, ploidies, ploidy, plonking,
plosion, plosions, plosive, plosives, plottier, plotties, plottiest, plotting, ploughing, plowing, ploying, pluralization, pluralizations, plutocracies, plutocratic, plutonic, plutonium, plutoniums,
pluviose, pluvious, pneumonia, poachier, poachiest, poaching, pocketing, pocketknife.
pocketknives, pockier, pockiest, pockily, pocking, pockmarking, pocosin, pocosins, podagric, podding, podgier, podgiest, podgily, podia, podiatries, podiatrist, podiatry, podite, podites, poditic,
podium, podiums, podsolic, podzolic, poesies, poetic, poetical, poetics, poetise, poetised, poetiser, poetisers, poetises, poetising, poetize, poetized, poetizer.
poetizers, poetizes, poetizing, poetlike, poetries, pogies, pogonia, pogonias, pogonip, pogonips, pogroming, poi, poignancies, poignancy, poignant, poilu, poilus, poind, poinded, poinding, poinds,
poinsettia, point, pointe, pointed, pointer, pointers, pointes, pointier, pointiest, pointing, pointless.
pointman, pointmen, points, pointy, pois, poise, poised, poiser, poisers, poises, poising, poison, poisoned, poisoner, poisoners, poisoning, poisonous, poisons, poitrel, poitrels, pokier, pokies,
pokiest, pokily, pokiness, pokinesses, poking, polarise, polarised, polarises.
polarising, polarities, polarity, polarization, polarizations, polarize, polarized, polarizes, polarizing, poleaxing, poleis, polemic, polemical, polemicist, polemicists, polemics, polemist,
polemists, polemize, polemized, polemizes, polemizing, police, policed, policeman, policemen, polices, policewoman, policewomen, policies.
policing, policy, policyholder, poling, polio, poliomyelitis, poliomyelitises, polios, polis, polish, polished, polisher, polishers, polishes, polishing, polite, politely, politeness, politenesses,
politer, politest, politic, political, politician, politicians, politick, politicked, politicking, politicks, politico, politicoes, politicos, politics, polities, polity, polkaing, pollarding,
pollening, pollical.
pollices, pollinate, pollinated, pollinates, pollinating, pollination, pollinations, pollinator, pollinators, polling, pollinia, pollinic, pollist, pollists, polliwog, polliwogs, polluting,
pollution, pollutions, poloist, poloists, polonium, poloniums, polybrid, polybrids, polyenic, polygamies, polygamist, polygamists, polygonies, polygynies, polyparies, polypi, polypide.
polypides, polypodies, polypoid, polysemies, polysyllabic, polytechnic, polytenies, polytheism, polytheisms, polytheist, polytheists, polyuria, polyurias, polyuric, polyzoic, pomading, pommeling,
pommelling, pomologies, pomposities, pomposity, pondering, pondville, pongid, pongids, poniard, poniarded, poniarding, poniards, ponied, ponies, pontifex, pontiff, pontiffs, pontific, pontifical,
pontificate, pontificated, pontificates, pontificating.
pontifices, pontil, pontils, pontine, ponying, ponytail, ponytails, poohing, pooling, pooping, poori, pooris, poorish, poortith, poortiths, popelike, poperies, popinjay, popinjays, popish, popishly,
poplin, poplins, poplitic, poppied, poppies, popping, poppling, popularities, popularity, popularize.
popularized, popularizes, popularizing, populating, population, populations, populism, populisms, populist, populists, porcelain, porcelains, porcine, porcupine, porcupines, porgies, poring, porism,
porisms, porkier, porkies, porkiest, porkpie, porkpies, pornographic, porosities, porosity, porphyries, porpoise, porpoises.
porridge, porridges, porringer, porringers, portability, portaging, portending, portentious, portfolio, portfolios, portico, porticoes, porticos, portiere, portieres, porting, portion, portioned,
portioning, portions, portlier, portliest, portrait, portraitist, portraitists.
portraits, portraiture, portraitures, portraying, posies, posing, posingly, posit, posited, positing, position, positioned, positioning, positions, positive, positively, positiveness, positivenesses,
positiver, positives, positivest, positivity, positron, positrons, posits, posologies, possessing, possession, possessions, possessive, possessiveness, possessivenesses, possessives, possibilities,
possibility, possible, possibler, possiblest, possibly.
postbiblical, postcolonial, postdating, postelection, posterior, posteriors, posterities, posterity, postexercise, postfertilization, postfertilizations, postfix, postfixed, postfixes, postfixing,
postflight, postforming, postgraduation, posthospital, postiche, postiches, postimperial, postin, postinaugural.
postindustrial, posting, postings, postinjection, postinoculation, postins, postique, postiques, postmarital, postmarking, postnuptial, postoperative, postpaid, postponing, postproduction,
postradiation, postrecession, postretirement, postrevolutionary, postscript, postscripts, postsurgical, posttrial, postulating, posturing.
postvaccination, potamic, potassic, potassium, potassiums, potation, potations, potbellied, potbellies, potboil, potboiled, potboiling, potboils, potencies, potential, potentialities, potentiality,
potentially, potentials, pothering, potiche, potiches.
potion, potions, potlatching, potlike, potpie, potpies, potpourri, potpourris, potshotting, potsie, potsies, potteries, pottering, pottier, potties, pottiest, potting, pouchier, pouchiest, pouching,
poultice, poulticed, poultices, poulticing, poultries, pouncing, pounding, pouring, poussie.
poussies, poutier, poutiest, pouting, poverties, powdering, powering, powwowing, poxing, poxvirus, poxviruses, practitioner, practitioners, praiseworthy, preadmission, preadopting, preallocating,
preallotting, preauthorize, preauthorized, preauthorizes, preauthorizing, preboil, preboiled, preboiling, preboils, precancellation, precancellations, precarious, precariously, precariousness.
precariousnesses, precaution, precautionary, precautions, precious, preciouses, precipitation, precipitations, precipitous, precipitously, precision, precisions, precivilization, precocious,
precocities, precocity, precolonial, precombustion, precomputing, preconceive, preconceived, preconceives, preconceiving, preconception, preconceptions, precondition, preconditions, preconvention,
precooking, precooling, predesignation, predesignations, predication, predications, prediction, predictions, predilection, predilections, predispose.
predisposed, predisposes, predisposing, predisposition, predispositions, prednisone, prednisones, predominance, predominances, predominant, predominantly, predominate, predominated, predominates,
predominating, preelection, preelectronic, preemption, preemptions, prefabrication, prefabrications, prefocusing, prefocussing, preforming, prehistoric, prehistorical, preimmunization.
preimmunizations, preinoculate, preinoculated, preinoculates, preinoculating, preinoculation, premeditation, premeditations, premodified, premodifies, premodify, premodifying, premoisten,
premoistened, premoistening, premoistens, premonition, premonitions, premonitory, prenomina, prenotification, prenotifications, prenotified, prenotifies, prenotify, prenotifying, preoccupation.
preoccupations, preoccupied, preoccupies, preoccupying, preopening, preoperational, preordain, preordained, preordaining, preordains, preparation, preparations, preponderating, preposition,
prepositional, prepositions, prepossessing, preprocessing, preproduction, preprofessional, prepublication, prerecording, preregistration, preregistrations, prerevolutionary, prerogative,
prerogatives, prescoring, prescription, prescriptions, presentation, presentations, preservation, preservations, preshowing, presidio, presidios, presoaking.
pressurization, pressurizations, prestidigitation, prestidigitations, prestigious, presumption, presumptions, presupposing, presupposition, presuppositions, pretelevision, pretension, pretensions,
pretentious, pretentiously, pretentiousness, pretentiousnesses, preunion, preunions, prevarication, prevarications, prevaricator, prevaricators, prevention, preventions, previous, previously,
previsor, previsors, priesthood, priesthoods, primero, primeros, primo, primordial, primos, primrose, primroses, princock, princocks.
princox, princoxes, printout, printouts, prior, priorate, priorates, prioress, prioresses, priories, priorities, prioritize, prioritized, prioritizes, prioritizing, priority, priorly, priors, priory,
prismoid, prismoids, prison, prisoned, prisoner, prisoners, prisoning, prisons, privation, privations, probabilities, probability, probating, probation, probationary, probationer, probationers,
probing, probit, probities, probits, probity, problematic, problematical, proboscides, proboscis, procaine, procaines, proceeding, proceedings, processing, procession, processional, processionals,
processions, prochain, prochein, proclaim, proclaimed, proclaiming.
proclaims, proclamation, proclamations, proclivities, proclivity, procrastinate, procrastinated, procrastinates, procrastinating, procrastination, procrastinations, procrastinator, procrastinators,
procreating, procreation, procreations, procreative, proctorial, proctoring, procuring, prodding, prodigal, prodigalities, prodigality, prodigals, prodigies, prodigious, prodigiously.
prodigy, producing, production, productions, productive, productiveness, productivenesses, productivities, productivity, proemial, profaning, professing, profession, professional, professionalism,
professionalize, professionalized, professionalizes, professionalizing, professionally, professions, professorial, professorship, professorships, proffering, proficiencies, proficiency.
proficient, proficiently, profile, profiled, profiler, profilers, profiles, profiling, profit, profitability, profitable, profitably, profited, profiteer, profiteered, profiteering, profiteers,
profiter, profiters, profiting, profitless, profits, profligacies, profligacy, profligate.
profligately, profligates, profundities, profundity, profusion, profusions, progenies, progenitor, progenitors, progging, prognosing, prognosis, prognosticate, prognosticated, prognosticates,
prognosticating, prognostication, prognostications, prognosticator, prognosticators, programing, programmabilities.
programmability, programming, progressing, progression, progressions, progressive, progressively, prohibit, prohibited, prohibiting, prohibition, prohibitionist, prohibitionists, prohibitions,
prohibitive, prohibitively, prohibitory, prohibits, projectile, projectiles, projecting, projection, projections, prolamin, prolamins, prolapsing, proletarian, proletariat, proliferate, proliferated,
proliferates, proliferating, proliferation, prolific, prolifically, proline, prolines, prolix, prolixly.
prologing, prologuing, prolongation, prolongations, prolonging, promenading, prominence, prominences, prominent, prominently, promiscuities, promiscuity, promiscuous, promiscuously, promiscuousness,
promiscuousnesses, promise, promised, promisee, promisees, promiser, promisers, promises.
promising, promisingly, promisor, promisors, promissory, promontories, promoting, promotion, promotional, promotions, prompting, promulging, pronating, pronging, pronouncing, pronunciation,
pronunciations, proofing, proofreading, propagandist, propagandists, propagandize, propagandized.
propagandizes, propagandizing, propagating, propagation, propagations, propelling, propending, propensities, propensity, properties, prophecies, prophesied, prophesier, prophesiers, prophesies,
prophesying, prophetic, prophetical, prophetically, prophylactic, prophylactics, prophylaxis, propine, propined, propines, propining, propinquities.
propinquity, propitiate, propitiated, propitiates, propitiating, propitiation, propitiations, propitiatory, propitious, propolis, propolises, proponing, proportion, proportional, proportionally,
proportionate, proportionately, proportions, proposing, proposition, propositions, propounding, propping, proprietary, proprieties, proprietor, proprietors, proprietorship, proprietorships,
proprietress, proprietresses, propriety, propulsion, propulsions.
propulsive, propylic, prorating, proroguing, prosaic, prosaism, prosaisms, prosaist, prosaists, proscribe, proscribed, proscribes, proscribing, proscription, proscriptions, prosecting, prosecuting,
prosecution, prosecutions, proselytize, proselytized, proselytizes, proselytizing, prosier, prosiest, prosily, prosing, prosit, prosodic, prosodies, prospecting, prospective, prospectively,
prospering, prosperities, prosperity, prostatic, prosthesis.
prosthetic, prostitute, prostituted, prostitutes, prostituting, prostitution, prostitutions, prostrating, prostration, prostrations, protamin, protamins, protasis, protatic, protecting, protection,
protections, protective, protei, proteid, proteide, proteides, proteids, protein, proteins, proteinuria, protending, protestation, protestations, protesting, prothrombin, protist, protists, protium,
protiums, protocoling, protocolling, protonic, protoplasmic, protoxid.
protoxids, protracting, protruding, protrusion, protrusions, protrusive, prounion, proverbial, proverbing, provide, provided, providence, providences, provident, providential, providently, provider,
providers, provides, providing, province, provinces, provincial, provincialism, provincialisms, proving.
proviral, provirus, proviruses, provision, provisional, provisions, proviso, provisoes, provisos, provocation, provocations, provocative, provoking, prowling, proxemic, proxies, proximal, proximo,
pruinose, prurigo, prurigos, psalmodies, psiloses, psilosis, psilotic, psoai, psocid, psocids, psoriasis.
psoriasises, psychoanalysis, psychoanalytic, psychoanalyzing, psychological, psychologically, psychologies, psychologist, psychologists, psychopathic, psychosis, psychosocial, psychosomatic,
psychotherapies, psychotherapist, psychotherapists, psychotic, ptomain, ptomaine, ptomaines, ptomains, ptosis, ptotic, publication, publications, pugnacious, pulchritudinous, pulmonic, pulsation,
pulsion, pulsions, punctilious, punctuation, punition, punitions, punitory, pupation, pupations, purgatorial, purgatories, purification, purifications, purloin, purloined, purloining, purloins,
purporting, purposing, pusillanimous, putrefaction, putrefactions, pygmoid, pylori, pyloric, pyogenic, pyoid, pyosis, pyranoid, pyrenoid, pyrenoids, pyriform, pyritous, pyrologies.
pyrolyzing, pyromania, pyromaniac, pyromaniacs, pyromanias, pyronine, pyronines, pyrosis, pyrosises, pyrotechnic, pyrotechnics, pyrrolic, pythonic, qualification, qualifications, question,
questionable, questioned, questioner, questioners, questioning, questionnaire, questionnaires, questionniare.
questionniares, questions, quinoa, quinoas, quinoid, quinoids, quinol, quinolin, quinolins, quinols, quinone, quinones, quittor, quittors, quixote, quixotes, quixotic, quixotries, quixotry, quoin,
quoined, quoining, quoins, quoit, quoited, quoiting, quoits, quotation, quotations, quotient, quotients, quoting.
rabboni, rabbonis, racemoid, radiation, radiations, radiator, radiators, radio, radioactive, radioactivities, radioactivity, radioed, radioing, radiologies, radiologist, radiologists, radiology,
radioman, radiomen, radionuclide, radionuclides, radios, radiotherapies, radiotherapy, ragouting, railroad, railroaded, railroader, railroaders, railroading, railroadings, railroads, rainbow,
rainbows, raincoat.
raincoats, raindrop, raindrops, rainout, rainouts, rainstorm, rainstorms, raisonne, rambunctious, ramification, ramifications, ramiform, ramosities, ramosity, rampion, rampions, randomization,
randomizations, randomize, randomized, randomizes, randomizing, ransoming, rapacious, rapaciously, rapaciousness, rapaciousnesses, rarefaction, rarefactions, rasorial, ratification, ratifications,
ratio, ration, rational, rationale, rationales.
rationalization, rationalizations, rationalize, rationalized, rationalizes, rationalizing, rationally, rationals, rationed, rationing, rations, ratios, ratooning, rattooning, ravigote, ravigotes,
ravioli, raviolis, razoring, reabsorbing, reaction, reactionaries, reactionary, reactions, reactivation, reactivations, readopting, readorning, realization, realizations, reallocating, reallotting,
reanoint, reanointed.
reanointing, reanoints, reappoint, reappointed, reappointing, reappoints, reapportion, reapportioned, reapportioning, reapportions, reapproving, rearousing, reasoning, reasonings, reassociate,
reassociated, reassociates, reassociating, reassorting, reavowing, rebellion, rebellions, rebellious, rebelliously, rebelliousness, rebelliousnesses, reblooming, reboarding, reboil, reboiled,
reboiling, reboils, rebounding, rebroadcasting, rebuttoning.
recapitulation, recapitulations, reception, receptionist, receptionists, receptions, recertification, recertifications, recession, recessions, rechoosing, reciprocal, reciprocally, reciprocals,
reciprocate, reciprocated, reciprocates, reciprocating, reciprocation, reciprocations, reciprocities, reciprocity, recirculation, recirculations, recision, recisions, recitation, recitations,
reckoning, reckonings, reclamation, reclamations, reclassification, reclassifications, reclothing, recoaling, recocking.
recodified, recodifies, recodify, recodifying, recognition, recognitions, recognizable, recognizably, recognizance, recognizances, recognize, recognized, recognizes, recognizing, recoil, recoiled,
recoiler, recoilers, recoiling, recoils, recoin, recoined, recoining, recoins, recollection, recollections, recolonize, recolonized, recolonizes, recolonizing, recoloring, recombine, recombined,
recombines, recombing, recombining, recommendation, recommendations, recommending, recommit.
recommits, recommitted, recommitting, recompensing, recompile, recompiled, recompiles, recompiling, recomputing, reconceive, reconceived, reconceives, reconceiving, reconcilable, reconcile,
reconciled, reconcilement, reconcilements, reconciler, reconcilers, reconciles, reconciliation, reconciliations, reconciling, recondite, reconfigure, reconfigured, reconfigures, reconfiguring,
reconnaissance, reconnaissances, reconnecting.
reconnoiter, reconnoitered, reconnoitering, reconnoiters, reconquering, reconsider, reconsideration, reconsiderations, reconsidered, reconsidering, reconsiders, reconsolidate, reconsolidated,
reconsolidates, reconsolidating, reconstructing, recontaminate, recontaminated, recontaminates, recontaminating, reconvening, reconveying, reconvict, reconvicted, reconvicting.
reconvicts, recooking, recopied, recopies, recopying, recording, recounting, recouping, recoupling, recoveries, recovering, recreation, recreational, recreations, recrimination, recriminations,
recriminatory, recrossing, recrowning, rectification, rectifications, rectorial, rectories, recuperation, recuperations, redecorating, rededication, rededications, redemption, redemptions,
redeploying, redeposit, redeposited, redepositing, redeposits, redeveloping.
rediscover, rediscovered, rediscoveries, rediscovering, rediscovers, rediscovery, redissolve, redissolved, redissolves, redissolving, redocking, redoing, redoubling, redounding, reduction,
reductions, reechoing, reembodied, reembodies, reembodying, reemploying, reendowing, reenjoying, reevaluation, reevaluations, reevoking.
reexamination, reexaminations, reexporting, reflection, reflections, refloating, reflooding, reflowering, reflowing, refocusing, refocussing, refolding, reforesting, reforging, reformatories,
reformatting, reforming, reformulating, refounding, refraction, refractions, refrigeration, refrigerations, refrigerator, refrigerators, refronting, refutation, refutations, regeneration,
regenerations, regimentation, regimentations, region, regional, regionally, regionals, regions, registration, registrations.
reglossing, reglowing, regolith, regoliths, regorging, regression, regressions, regrooving, regrouping, regrowing, regulation, regulations, regurgitation, regurgitations, rehabilitation,
rehabilitations, rehospitalization, rehospitalizations, rehospitalize, rehospitalized, rehospitalizes, rehospitalizing, rehousing, reimport, reimported, reimporting, reimports, reimpose, reimposed,
reimposes, reimposing, reincarnation, reincarnations.
reincorporate, reincorporated, reincorporates, reincorporating, reinfection, reinfections, reinforcement, reinforcements, reinforcer, reinforcers, reinform, reinformed, reinforming, reinforms,
reinjection, reinjections, reinoculate, reinoculated, reinoculates, reinoculating, reinsertion, reinsertions, reintegration, reintegrations, reintroduce, reintroduced, reintroduces, reintroducing,
reinvestigation, reinvestigations, reinvigorate, reinvigorated, reinvigorates, reinvigorating.
reinvoke, reinvoked, reinvokes, reinvoking, reitbok, reitboks, reiteration, reiterations, rejection, rejections, rejoice, rejoiced, rejoicer, rejoicers, rejoices, rejoicing, rejoicings, rejoin,
rejoinder, rejoinders, rejoined, rejoining, rejoins, rejuvenation, rejuvenations, relation, relations, relationship, relationships, relaxation, relaxations, relegation, relegations, relievo,
relievos, religion, religionist, religionists, religions, religious.
religiously, reloading, reloaning, relocating, relocation, relocations, remission, remissions, remittor, remittors, remobilize, remobilized, remobilizes, remobilizing, remodeling, remodelling,
remodified, remodifies, remodify, remodifying, remoisten, remoistened, remoistening, remoistens.
remolding, remonstrating, remonstration, remonstrations, remorid, remotion, remotions, remotivate, remotivated, remotivates, remotivating, remounting, removing, remuneration, remunerations,
rendezvousing, rendition, renditions, renegotiate, renegotiated, renegotiates, renegotiating, reniform, renotified, renotifies.
renotify, renotifying, renouncing, renovating, renovation, renovations, renowning, renunciation, renunciations, renvoi, renvois, reobjecting, reobtain, reobtained, reobtaining, reobtains, reoccupied,
reoccupies, reoccupying, reoccurring, reoffering, reoil.
reoiled, reoiling, reoils, reopening, reoperating, reopposing, reorchestrating, reordain, reordained, reordaining, reordains, reordering, reorganization, reorganizations, reorganize, reorganized,
reorganizes, reorganizing, reorient, reoriented, reorienting, reorients, reovirus, reoviruses, reparation, reparations, repatriation, repatriations, repeopling, repercussion, repercussions,
repertoire, repertoires, repertories, repetition, repetitions, repetitious, repetitiously, repetitiousness.
repetitiousnesses, rephotographing, repletion, repletions, repolish, repolished, repolishes, repolishing, repopulating, reporting, reportorial, reposing, reposit, reposited, repositing, repository,
reposits, repossession, repossessions, repouring, repowering, reprehension, reprehensions, representation, representations.
repression, repressions, reproaching, reprobation, reprobations, reprobing, reprocessing, reproducible, reproducing, reproduction, reproductions, reproductive, reprograming, reproposing, reproving,
repudiation, repudiations, repudiator, repudiators, repulsion, repulsions, reputation, reputations, requisition.
requisitioned, requisitioning, requisitions, rerecording, rerolling, rerouting, rescission, rescissions, rescoring, reservation, reservations, reshoeing, reshooting, reshowing, resignation,
resignations, resinoid, resinoids, resinous, resistor, resistors, resmoothing.
resoldering, resolidified, resolidifies, resolidify, resolidifying, resoling, resolution, resolutions, resolving, resonating, resorbing, resorcin, resorcins, resorting, resounding, resoundingly,
resowing, respiration, respirations, respirator, respiratories, respirators, respiratory, responding, responsibilities, responsibility, responsible, responsibleness, responsiblenesses,
responsiblities, responsiblity, responsibly, responsive, responsiveness, responsivenesses, restitution, restitutions, restocking, restoration.
restorations, restorative, restoratives, restoring, restriction, restrictions, resummoning, resumption, resumptions, resurrection, resurrections, resuscitation, resuscitations, resuscitator,
resuscitators, retailor, retailored, retailoring, retailors, retaliation, retaliations, retaliatory, retardation, retardations.
retention, retentions, retiform, retinol, retinols, retooling, retorting, retouching, retraction, retractions, retribution, retributions, retributory, retroacting, retroactive, retroactively,
retrofit, retrofits, retrofitted, retrofitting, retrogressing, retrogression, retrogressions, retrospection, retrospections, retrospective, retrospectively, retrospectives, reunion, reunions,
reupholstering, revaccination, revaccinations, revelation, revelations, reverberation, reverberations, reversion, revision, revisions.
revisor, revisors, revisory, revocation, revocations, revoice, revoiced, revoices, revoicing, revoking, revolting, revolution, revolutionaries, revolutionary, revolutionize, revolutionized,
revolutionizer, revolutionizers, revolutionizes, revolutionizing, revolutions, revolving, revulsion, revulsions, rewording, reworking, rezoning, rhapsodic, rhapsodically, rhapsodies, rhapsodize,
rhapsodizes, rhapsodizing, rheologies, rheophil, rhetoric, rhetorical, rhetorician, rhetoricians, rhetorics, rhinestone, rhinestones, rhino, rhinoceri, rhinoceros, rhinoceroses, rhinos, rhizobia,
rhizoid, rhizoids, rhizoma, rhizomata.
rhizome, rhizomes, rhizomic, rhizopi, rhizopod, rhizopods, rhizopus, rhizopuses, rhodamin, rhodamins, rhodic, rhodium, rhodiums, rhombi, rhombic, rhomboid, rhomboids, rhonchi, rhyolite, rhyolites,
rialto, rialtos, ribbon, ribboned, ribboning, ribbons, ribbony, riboflavin, riboflavins, ribose.
riboses, ribosome, ribosomes, ribwort, ribworts, ricochet, ricocheted, ricocheting, ricochets, ricochetted, ricochetting, ricotta, ricottas, ridiculous, ridiculously, ridiculousness,
ridiculousnesses, ridotto, ridottos, rigadoon, rigadoons, rigatoni, rigatonis, rigaudon, rigaudons, righteous, righteously, righteousness.
righteousnesses, righto, rigmarole, rigmaroles, rigor, rigorism, rigorisms, rigorist, rigorists, rigorous, rigorously, rigors, rigour, rigours, rilievo, rimose, rimosely, rimosities, rimosity,
rimous, rimrock.
rimrocks, ringbolt, ringbolts, ringbone, ringbones, ringdove, ringdoves, ringtoss, ringtosses, ringworm, ringworms, riot, rioted, rioter, rioters, rioting, riotous, riots, ripcord, ripcords, ripieno.
ripienos, ripost, riposte, riposted, ripostes, riposting, riposts, risotto, risottos, rissole, rissoles, riverboat, riverboats, roaching, roadside, roadsides, roaming, roaring, roarings, roasting,
robberies, robbin, robbing, robbins, robin, robing, robins, robotics, robotism, robotisms, robotize, robotized.
robotizes, robotizing, robotries, rockabies, rockeries, rocketing, rocketries, rockfish, rockfishes, rockier, rockiest, rocking, rocklike, rockling, rocklings, rodding, rodlike, rogation, rogations,
rogueing, rogueries.
roguing, roguish, roguishly, roguishness, roguishnesses, roil, roiled, roilier, roiliest, roiling, roils, roily, roister, roistered, roistering, roisters, rolamite, rolamites, rollick, rollicked,
rollicking, rollicks, rollicky, rolling, rollings, romaine.
romaines, romancing, romanize, romanized, romanizes, romanizing, romantic, romantically, romantics, romping, rompish, ronion, ronions, roofing, roofings, rooflike, roofline, rooflines, rookeries,
rookie, rookier, rookies, rookiest, rooking, roomier.
roomiest, roomily, rooming, roosing, roosting, rootier, rootiest, rooting, rootlike, roperies, ropier, ropiest, ropily, ropiness, ropinesses, roping, roqueting, rosaria, rosarian, rosarians,
rosaries, rosarium, rosariums, rosefish, rosefishes, roselike, rosemaries, roseries.
rosier, rosiest, rosily, rosin, rosined, rosiness, rosinesses, rosing, rosining, rosinous, rosins, rosiny, roslindale, rosolio, rosolios, rotaries, rotating, rotation, rotations, rotative, rotifer,
rotifers, rotiform, rototill, rototilled, rototilling, rototills, rotting, roturier, roturiers, roughdried, roughdries, roughdrying.
roughening, roughhewing, roughing, roughish, rouging, rouletting, rounding, roundish, roupier, roupiest, roupily, rouping, rousing, rousting, routine, routinely, routines, routing, roving, rovingly,
rovings, rowdier, rowdies, rowdiest, rowdily, rowdiness, rowdinesses, rowdyish, rowdyism, rowdyisms, roweling, rowelling, rowing, rowings.
royalism, royalisms, royalist, royalists, royalties, roystering, rubigo, rubigos, ruction, ructions, ructious, rugosities, rugosity, ruinous, ruinously, rumoring, rumouring, sabotaging, sacrilegious,
sacrilegiously, sadiron, sadirons.
sagacious, sailboat, sailboats, sailor, sailorly, sailors, sainfoin, sainfoins, saintdom, saintdoms, sainthood, sainthoods, salacious, salivation, salivations, sallowing, salmonid, salmonids,
salubrious, salutation, salutations, salvation, salvations, salvoing, sanatoria, sanatorium, sanatoriums, sanctification, sanctifications, sanctimonious, sanction, sanctioned, sanctioning.
sanctions, sanious, sanitation, sanitations, santonin, santonins, saponified, saponifies, saponify, saponifying, saponin, saponine, saponines, saponins, saponite, saponites, saprobic, sarcoid,
sarcoids, sarcophagi, sardonic, sardonically, sarodist, sarodists, sartorii, satinpod, satinpods, satisfaction, satisfactions, satisfactorily, satisfactory.
satori, satoris, saturation, saturations, sautoir, sautoire, sautoires, sautoirs, savior, saviors, saviour, saviours, savorier, savories, savoriest, savorily, savoring, savourier, savouries,
savouriest, savouring, saxonies, scabiosa, scabiosas, scabious, scabiouses, scaffolding, scallion, scallions, scalloping, scammonies, scansion, scansions, scaphoid, scaphoids, scariose, scarious.
scenario, scenarios, schizo, schizoid, schizoids, schizont, schizonts, schizophrenia, schizophrenias, schizophrenic, schizophrenics, schizos, schmoosing, schmoozing, scholarship, scholarships,
scholastic, scholia, scholium, scholiums, schoolgirl, schoolgirls, schooling, scincoid.
scincoids, scintillation, scintillations, sciolism, sciolisms, sciolist, sciolists, scion, scions, scirocco, sciroccos, scission, scissions, scissor, scissored, scissoring, scissors, sciuroid,
scleroid, sclerosing, sclerosis.
sclerosises, sclerotic, scoffing, scolding, scoldings, scolices, scolioma, scoliomas, scolloping, sconcing, scooping, scooting, scorching, scoria, scoriae, scorified, scorifies, scorify, scorifying,
scoring, scorning, scorpion, scorpions, scotching, scotia, scotias, scotopia, scotopias, scotopic, scottie, scotties, scourging, scouring, scourings, scouthering, scouting, scoutings, scowdering,
scowing, scowling.
scroggier, scroggiest, scrooping, scrouging, scroungier, scroungiest, scrounging, scullion, scullions, scurrilous, seagoing, seasoning, seclusion, seclusions, secondi, seconding, secretion,
secretions, section, sectional, sectioned, sectioning, sections, sectoring, sedation, sedations, sedimentation, sedimentations, sedition, seditions, seditious, seduction, seductions, segregation,
segregations, seicento.
seicentos, seignior, seigniors, seignories, seignory, seismograph, seismographs, seisor, seisors, seizor, seizors, selection, selections, semicolon, semicolons, semicoma, semicomas, semiconductor,
semiconductors, semidome, semidomes, semiformal, semihobo, semihoboes, semihobos, semilog, semioses, semiosis, semiotic, semiotics, semipro, semipros, semisoft, semitone, semitones, semolina,
semolinas, senatorial, senecio.
senecios, senior, seniorities, seniority, seniors, senopia, senopias, senorita, senoritas, sensation, sensational, sensations, sensoria, sententious, sepaloid, separation, separations, sequoia,
sequoias, seraglio, seraglios, serendipitous, serious, seriously, seriousness, seriousnesses.
sermonic, serologies, serosities, serosity, serotine, serotines, serpigo, serpigoes, servitor, servitors, sesamoid, sesamoids, session, sessions, setiform, shadowier, shadowiest, shadowing,
shallowing, shammosim, shamois, shamosim, shamoying, shampooing, sharecroping, sharpshooting, sharpshootings, sheikdom, sheikdoms, sheikhdom, sheikhdoms, shinbone.
shinbones, shipboard, shipboards, shipload, shiploads, shippon, shippons, shipworm, shipworms, shkotzim, shoalier, shoaliest, shoaling, shocking, shoddier, shoddies, shoddiest, shoddily, shoddiness,
shoddinesses, shoebill, shoebills, shoehorning, shoeing, shogging, shoji, shojis, shooflies, shooing, shooling, shooting, shootings, shopgirl, shopgirls, shoplift, shoplifted.
shoplifter, shoplifters, shoplifting, shoplifts, shopping, shoppings, shorebird, shorebirds, shoring, shorings, shortchanging, shortcoming, shortcomings, shortening, shortia, shortias, shortie,
shorties, shorting, shortish, shortliffe, shortsighted, shotgunning, shotting, shouldering, shouting, shoveling, shovelling, shoving, showcasing, showering, showgirl, showgirls, showier, showiest,
showily, showiness, showinesses.
showing, showings, shroffing, shrouding, shylocking, sialoid, sickroom, sickrooms, sideboard, sideboards, sidelong, sideshow, sideshows, sierozem, sierozems, sigloi, siglos, sigmoid, sigmoids,
signatories, signatory, signification, significations, signior, signiori, signiories, signiors, signiory, signor, signora, signoras, signore, signori, signories, signors, signory.
signpost, signposted, signposting, signposts, silhouette, silhouetted, silhouettes, silhouetting, silicon, silicone, silicones, silicons, silkworm, silkworms, silo, siloed, siloing, silos, siloxane,
siloxanes, siluroid, siluroids, simioid, simious, simoleon, simoleons, simoniac, simoniacs, simonies, simonist, simonists, simonize, simonized, simonizes, simonizing, simony, simoom, simooms, simoon.
simoons, simpleton, simpletons, simplification, simplifications, simulation, simulations, simultaneous, simultaneously, simultaneousness, simultaneousnesses, sinfonia, sinfonie, singsong, singsongs,
sinkhole, sinkholes, sinologies, sinology, sinopia, sinopias.
sinopie, sinuous, sinuousities, sinuousity, sinuously, sinusoid, sinusoids, siphon, siphonal, siphoned, siphonic, siphoning, siphons, sirloin, sirloins, sirocco, siroccos, sisterhood, sisterhoods,
sistroid, sitologies, sitology, situation, situations, sixfold, sixmo, sixmos, skibob.
skibobs, skiddoo, skiddooed, skiddooing, skiddoos, skidoo, skidooed, skidooing, skidoos, skijorer, skijorers, skimo, skimos, skioring, skiorings, skoaling, skyphoi, skyrocketing, slaloming,
slingshot, slingshots.
slipform, slipformed, slipforming, slipforms, slipknot, slipknots, slipout, slipouts, slipover, slipovers, slipshod, slipslop, slipslops, slipsole, slipsoles, slivovic, slivovics, slobbering,
slobbish, slogging, sloid, sloids, sloping, sloppier, sloppiest, sloppily, slopping, sloshier, sloshiest, sloshing, slotting, slouchier, slouchiest, slouching, sloughier, sloughiest, sloughing,
slovenlier, slovenliest.
slowing, slowish, smidgeon, smidgeons, smocking, smockings, smoggier, smoggiest, smokier, smokiest, smokily, smoking, smoldering, smooching, smoothening, smoothie, smoothies, smoothing, smothering,
smouldering, snapshotting, snobberies, snobbier, snobbiest, snobbily, snobbish, snobbishly, snobbishness, snobbishnesses, snobbism, snobbisms, snooding, snooking, snooling, snoopier, snoopiest,
snoopily, snooping.
snootier, snootiest, snootily, snooting, snoozier, snooziest, snoozing, snoozling, snoring, snorkeling, snorting, snottier, snottiest, snottily, snoutier, snoutiest, snouting, snoutish, snowballing,
snowbird, snowbirds, snowdrift, snowdrifts, snowier.
snowiest, snowily, snowing, snowlike, snowplowing, snowshoeing, snowsuit, snowsuits, soaking, soapier, soapiest, soapily, soaping, soaplike, soaring, soarings, sobbing, sobeit, sobering, soberize,
soberized, soberizes, soberizing, sobrieties, sobriety, sociabilities, sociability, sociable, sociables, sociably, social, socialism.
socialist, socialistic, socialists, socialization, socializations, socialize, socialized, socializes, socializing, socially, socials, societal, societies, society, sociological, sociologies,
sociologist, sociologists, sociology, socketing, socking.
sodalist, sodalists, sodalite, sodalites, sodalities, sodality, sodamide, sodamides, soddening, soddies, sodding, sodic, sodium, sodiums, sodomies, sodomite, sodomites, soffit, soffits, softening,
softie, softies, soggier, soggiest, soggily, sogginess, sogginesses, soigne, soignee, soil, soilage, soilages, soiled, soiling.
soilless, soils, soilure, soilures, soiree, soirees, sojourning, solacing, solanin, solanine, solanines, solanins, solaria, solarise, solarised, solarises, solarising, solarism, solarisms, solarium,
solariums, solarize, solarized, solarizes, solarizing, solatia, solating, solation, solations, solatium, soldering, soldi, soldier, soldiered, soldieries, soldiering, soldierly, soldiers.
soldiery, solecise, solecised, solecises, solecising, solecism, solecisms, solecist, solecists, solecize, solecized, solecizes, solecizing, solenoid, solenoids, solfeggi, soli, solicit, solicitation,
solicited, soliciting, solicitor, solicitors, solicitous, solicits, solicitude, solicitudes, solid, solidago, solidagos.
solidarities, solidarity, solidary, solider, solidest, solidi, solidification, solidifications, solidified, solidifies, solidify, solidifying, solidities, solidity, solidly, solidness, solidnesses,
solids, solidus, soliloquize, soliloquized, soliloquizes, soliloquizing, soliloquy.
soliloquys, soling, solion, solions, soliquid, soliquids, solitaire, solitaires, solitaries, solitary, solitude, solitudes, soloing, soloist, soloists, solstice, solstices, solubilities, solubility,
solution, solutions.
solvating, solvencies, solving, somatic, somebodies, somersaulting, somerseting, somersetting, somerville, something, sometime, sometimes, somewise, somital, somite, somites, somitic, somnambulism,
somnambulist, somnambulists, sonantic, sonatina, sonatinas, sonatine, songbird, songbirds, songlike, sonic, sonicate, sonicated, sonicates, sonicating, sonics, sonlike.
sonneting, sonnetting, sonnies, sonorities, sonority, sonship, sonships, sonsie, sonsier, sonsiest, soothing, soothsaid, soothsaying, soothsayings, sootier, sootiest, sootily, sooting, sophies,
sophism, sophisms, sophist, sophistic, sophistical, sophisticate, sophisticated, sophisticates, sophistication, sophistications.
sophistries, sophistry, sophists, sopite, sopited, sopites, sopiting, soporific, soppier, soppiest, sopping, soprani, sorbic, sorbing, sorbitol, sorbitols, sorceries, sordid, sordidly, sordidness,
sordidnesses, sordine, sordines, sordini, sordino.
sori, soricine, sorites, soritic, sorning, sororities, sorority, sorosis, sorosises, sorption, sorptions, sorptive, sorrier, sorriest, sorrily, sorrowing, sortie, sortied, sortieing, sorties,
sorting, sottish, souari, souaris.
soubise, soubises, soughing, soullike, sounding, soundings, soundproofing, soupier, soupiest, souping, sourdine, sourdines, souring, sourish, sousing, southing, southings, souvenir, souvenirs,
sovereign, sovereigns, sovereignties, sovereignty, soviet, soviets, sovranties, sowbellies, sowing, sozin, sozine, sozines, sozins, spacious, spaciously, spaciousness, spaciousnesses.
sparoid, sparoids, spasmodic, specialization, specializations, specification, specifications, specious, speculation, speculations, sphenoid, sphenoids, spheroid, spheroids, spiccato, spiccatos,
spigot, spigots, spinoff, spinoffs, spinor.
spinors, spinose, spinous, spinout, spinouts, spiroid, spittoon, spittoons, sploshing, splotchier, splotchiest, splotching, spoil, spoilage, spoilages, spoiled, spoiler, spoilers, spoiling, spoils,
spoilt, spoking, spoliate, spoliated, spoliates, spoliating, spondaic, spondaics, spongier, spongiest, spongily, spongin, sponging, spongins, sponsion, sponsions.
sponsoring, sponsorship, sponsorships, spontaneities, spontaneity, spoofing, spookier, spookiest, spookily, spooking, spookish, spooling, spoonier, spoonies, spooniest, spoonily, spooning, spooring,
sporadic, sporadically, sporing, sporoid.
sportier, sportiest, sportily, sporting, sportive, sportsmanship, sportsmanships, spotlight, spotlighted, spotlighting, spotlights, spottier, spottiest, spottily, spotting, spousing, spouting,
sprouting, spumoni, spumonis, spurious.
squadroning, squooshing, stabilization, stagnation, stagnations, stallion, stallions, stanchion, stanchions, standardization, standardizations, standpoint, standpoints, starvation, starvations,
stasimon, station, stationary, stationed, stationer, stationeries, stationers, stationery, stationing, stations, steinbok, steinboks, stenographic, stenosis, stenotic, stentorian, stereoing,
stereotyping, sterilization, sterilizations, steroid, steroids, stickout, stickouts, stiletto, stilettoed, stilettoes, stilettoing, stilettos, stillborn, stimulation, stimulations, stingo, stingos,
stinko, stinkpot, stinkpots, stipulation, stipulations, stoai, stobbing, stockading, stockier.
stockiest, stockily, stocking, stockings, stockish, stockist, stockists, stockpile, stockpiled, stockpiles, stockpiling, stodgier, stodgiest, stodgily, stodging, stogie, stogies, stoic, stoical,
stoically, stoicism, stoicisms, stoics, stokesia, stokesias, stoking.
stolid, stolider, stolidest, stolidities, stolidity, stolidly, stolonic, stomaching, stomatic, stomatitis, stomping, stoneflies, stonier, stoniest, stonily, stoning, stonish, stonished, stonishes,
stonishing, stooging, stooking, stoolie, stoolies, stooling, stooping, stoping, stoplight, stoplights, stoppering.
stopping, stoppling, storied, stories, storing, stormier, stormiest, stormily, storming, storying, storytelling, storytellings, stotinka, stotinki, stounding, stourie, stoutening, stoutish, stowing,
straightforward, straightforwarder, straightforwardest, stramonies, strangulation, strangulations, stratification, stratifications, stridor, stridors, strigose, strobic, strobil, strobila, strobilae,
strobiles, strobili, strobils, stroking, strolling, strontia, strontias, strontic, strontium, strontiums, strophic, stropping, strowing, stroying, stuccoing, studio, studios, studious, studiously,
stultification, stultifications, stupefaction, stupefactions, styloid, suasion, suasions, subatmospheric, subcategories, subclassification, subclassifications, subcommission, subcommissions,
subcommunities, subcommunity, subconscious, subconsciouses.
subconsciously, subconsciousness, subconsciousnesses, subcontracting, subcooling, subdivision, subdivisions, subequatorial, subito, subjection, subjections, subjoin, subjoined, subjoining, subjoins,
subjugation, subjugations, submersion, submersions, submission, submissions, suboceanic, suboptic, subordinate, subordinated, subordinates, subordinating, subordination, subordinations, suborning,
suboxide, suboxides, subpoenaing, subregion.
subregions, subroutine, subroutines, subscription, subscriptions, subsection, subsections, subsoil, subsoiled, subsoiling, subsoils, subsonic, substantiation, substantiations, substitution,
substitutions, subtonic, subtonics, subtopic, subtopics, subtotaling, subtotalling, subtraction, subtractions, succession, successions, succories, succoring, succouring, suction, suctions, sudation,
sudations, sudatories, suffixation, suffixations, suffocating, suffocatingly, suffocation.
suffocations, suggestion, suggestions, suitor, suitors, sulfonic, summarization, summarizations, summation, summations, summoning, summonsing, superambitious, supercilious, superconvenient,
superheroine, superheroines, superimpose, superimposed, superimposes, superimposing, superior.
superiorities, superiority, superiors, superpatriot, superpatriotic, superpatriotism, superpatriotisms, superpatriots, superpolite, supersonic, superstition, superstitions, superstitious,
supervision, supervisions, supervisor, supervisors, supervisory, supplication, supplications, supporting, supportive, supposing.
supposition, suppositions, suppositories, suppository, suppression, suppressions, suppuration, suppurations, surmounting, surreptitious, surreptitiously, surrounding, surroundings, survivor,
survivors, survivorship, survivorships, suspension, suspensions, suspicion, suspicions, suspicious.
suspiciously, swallowing, swinepox, swinepoxes, switchboard, switchboards, swobbing, swooning, swooping, swooshing, swopping, swordfish, swordfishes, swotting, swounding, swouning, syconia, syconium,
sycophantic, sycosis, symbion, symbions, symbiont, symbionts, symbiot, symbiote.
symbiotes, symbiots, symbolic, symbolical, symbolically, symboling, symbolism, symbolisms, symbolization, symbolizations, symbolize, symbolized, symbolizes, symbolizing, symbolling, symphonic,
symphonies, sympodia, symposia, symposium, symptomatically, synchronization, synchronizations, synchronize, synchronized.
synchronizes, synchronizing, syncopating, syncopation, syncopations, syncopic, syndication, synodic, synonymies, synopsis, synoptic, synovia, synovial, synovias, syntonic, syntonies, syphoning,
systolic, tabloid, tabloids, tabooing, taborin, taborine, taborines, taboring, taborins, tabouring, tabulation, tabulations, taconite, taconites, taction, tactions, tailbone, tailbones, tailcoat,
tailor, tailored, tailoring, tailors, talapoin, talapoins, talion, talions, talipot, talipots, tallitoth, tallowing, tallyhoing, tambourine, tambourines, tambouring, tampion, tampions, tamponing,
tangoing, tapioca, tapiocas, tattooing, taxation, taxations, taxonomies, technological, technologies, tectonic, tedious, tediously, tediousness, tediousnesses, teetotaling, teetotalling, tektronix,
telephoning, teleporting, telescopic, telescoping, television, televisions, teloi, telomic, telsonic, temporaries, temporarily, temptation, temptations, tenacious, tenaciously, tenderloin,
tenderloins, tenoning, tenorite, tenorites, tenotomies, tension, tensioned, tensioning, tensions, teocalli, teocallis.
teosinte, teosintes, teratoid, termination, terminologies, terminology, ternion, ternions, terpinol, terpinols, territorial, territories, territory, terrorism, terrorisms, terrorize, terrorized,
terrorizes, terrorizing, testimonial, testimonials, testimonies, testimony, tetanoid, tetroxid, tetroxids, thalloid, theocratic, theodicies, theodicy, theogonies, theologian, theologians,
theologies, theonomies, theoretical, theoretically, theories, theorise, theorised, theorises, theorising, theorist, theorists, theorize, theorized, theorizes, theorizing, thermion, thermions,
thermodynamics, thermometric, thermometrically, thermostatic, thermostatically, theroid, thiazol, thiazole, thiazoles, thiazols, thighbone, thighbones, thindown, thindowns, thio, thiol, thiolic,
thionate, thionates, thionic, thionin, thionine, thionines, thionins, thionyl, thionyls, thiophen, thiophens, thiotepa, thiotepas, thiourea, thioureas, tholepin, tholepins, tholing, tholoi, thoracic,
thoria, thorias, thoric, thorite, thorites, thorium, thoriums, thornier, thorniest, thornily, thorning, thouing, threnodies, throatier, throatiest.
throating, throbbing, thrombi, thrombin, thrombins, thrombocytopenia, thrombocytosis, thrombophlebitis, thromboplastin, thronging, throning, throttling, throwing, thyreoid, thyroid, thyroidal,
thyroids, thyroxin, thyroxins, thyrsoid, ticktock, ticktocked, ticktocking, ticktocks, tictoc, tictocked.
tictocking, tictocs, tiglon, tiglons, tigon, tigons, timeous, timeout, timeouts, timework, timeworks, timeworn, timorous, timorously, timorousness, timorousnesses, timothies, timothy, timpano,
tinamou, tinamous, tinfoil, tinfoils, tinhorn, tinhorns, tinstone, tinstones.
tinwork, tinworks, tipoff, tipoffs, tipstock, tipstocks, tiptoe, tiptoed, tiptoes, tiptoing, tiptop, tiptops, tiresome, tiresomely, tiresomeness, tiresomenesses, tiro, tiros, titanous, tithonia,
tithonias, titillation, titillations, titmouse, titrator, titrators, toadfish, toadfishes, toadied, toadies, toadish, toadlike, toadying.
toadyish, toadyism, toadyisms, toastier, toastiest, toasting, tobies, tobogganing, tochering, tocologies, tocsin, tocsins, toddies, toddling, todies, toeing, toelike, toenail, toenailed, toenailing,
toenails, toepiece, toepieces, toffies, toggeries, togging, toggling, toil, toile, toiled, toiler, toilers, toiles, toilet, toileted, toileting, toiletries.
toiletry, toilets, toilette, toilettes, toilful, toiling, toils, toilsome, toilworn, toit, toited, toiting, toits, tokening, tokenism, tokenisms, tokologies, tolerating, toleration, tolerations,
tolidin, tolidine, tolidines, tolidins, toling.
tolling, toluic, toluid, toluide, toluides, toluidin, toluidins, toluids, tomahawking, tombing, tomblike, tommies, tompion, tompions, tomtit, tomtits, tonalities, tonality, tondi, tonemic, tonetic,
tonetics, tonging, tonguing, tonguings, tonic, tonicities, tonicity, tonics, tonier, toniest, tonight, tonights, toning, tonish, tonishly, tonnish.
tonsil, tonsilar, tonsillectomies, tonsillectomy, tonsillitis, tonsillitises, tonsils, tonsuring, tontine, tontines, tooling, toolings, toothier, toothiest, toothily, toothing, toothpick, toothpicks,
tooting, tootling, tootsie, tootsies, topazine, tophi, topi, topiaries, topiary, topic, topical, topically.
topics, toping, topis, topkick, topkicks, toploftier, toploftiest, topographic, topographical, topographies, topoi, topologic, topologies, toponymies, topping, toppings, toppling, topsail, topsails,
topside, topsides, topsoil, topsoiled, topsoiling, topsoils, topworking, torchier, torchiers, torching, torchlight, torchlights.
toreutic, tori, toric, tories, torii, tormenting, tornadic, tornillo, tornillos, toroid, toroidal, toroids, torosities, torosity, torpedoing, torpid, torpidities, torpidity, torpidly, torpids,
torquing, torrefied, torrefies.
torrefying, torrential, torrid, torrider, torridest, torridly, torrified, torrifies, torrify, torrifying, torsi, torsion, torsional, torsionally, torsions, tortile, tortilla, tortillas, tortious,
tortoise, tortoises, tortoni, tortonis, tortrix, tortrixes, torturing, tossing, totaling, totalise, totalised, totalises, totalising, totalism, totalisms.
totalitarian, totalitarianism, totalitarianisms, totalitarians, totalities, totality, totalize, totalized, totalizes, totalizing, totalling, totemic, totemism, totemisms, totemist, totemists,
totemite, totemites, toting, tottering, totting, touchier, touchiest, touchily, touching, toughening, toughie, toughies, toughish, touring, tourings.
tourism, tourisms, tourist, tourists, touristy, tourneying, tourniquet, tourniquets, tousing, tousling, touting, touzling, tovarich, tovariches, tovarish, tovarishes, toweling, towelings, towelling,
towerier, toweriest, towering, towie, towies, towing, towline, towlines, townie, townies, townish, township, townships, toxaemia, toxaemias.
toxaemic, toxemia, toxemias, toxemic, toxic, toxical, toxicant, toxicants, toxicities, toxicity, toxin, toxine, toxines, toxins, toxoid, toxoids, toying, toyish, toylike, traction, tractional,
tractions, tradition, traditional, traditionally, traditions, traditor, traditores, trainload, trainloads, traitor, traitors, trampoline, trampoliner.
trampoliners, trampolines, trampolinist, trampolinists, transaction, transactions, transcription, transcriptions, transfiguration, transfigurations, transformation, transformations, transforming,
transfusion, transfusions, transgression, transgressions, transistor, transistorize, transistorized, transistorizes, transistorizing, transistors, transition, transitional, transitions, transitory,
translation, translations, transmission, transmissions, transpiration, transpirations, transplantation.
transplantations, transportation, transporting, transposing, transposition, transpositions, trapezoid, trapezoidal, trapezoids, travois, travoise, travoises, trefoil, trefoils, trepidation,
trepidations, triazole, triazoles, tribulation, tribulations, trichoid, trichome, trichomes, tricolor, tricolors, tricorn, tricorne, tricornes, tricorns, tricot.
tricots, trifocal, trifocals, trifold, triforia, triform, trigo, trigon, trigonal, trigonometric, trigonometrical, trigonometries, trigonometry, trigons, trigos, trillion, trillions, trillionth,
trillionths, trilobal, trilobed, trilogies, trilogy, trimorph, trimorphs, trimotor, trimotors, trinodal.
trio, triode, triodes, triol, triolet, triolets, triols, trios, triose, trioses, trioxid, trioxide, trioxides, trioxids, triploid, triploids, tripod, tripodal, tripodic, tripodies, tripody, tripoli,
tripolis, tripos, triposes, trisection, trisections, trisome, trisomes, trisomic, trisomics, trisomies, trisomy, tritoma.
tritomas, triton, tritone, tritones, tritons, troaking, trochaic, trochaics, trochil, trochili, trochils, trochoid, trochoids, trocking, troika, troikas, troilite, troilites, troilus, troiluses,
trois, troking, trolleying, trollied, trollies, trolling.
trollings, trollying, trombonist, trombonists, tromping, troopial, troopials, trooping, trophic, trophied, trophies, trophying, tropic, tropical, tropics, tropin, tropine, tropines, tropins, tropism,
tropisms, trothing.
trotline, trotlines, trotting, troubleshooting, troubling, trouncing, troupial, troupials, trouping, troutier, troutiest, troweling, trowelling, trowing, truncation, truncations, trunnion, trunnions,
trustworthiness, trustworthinesses, tuberculosis, tuberoid, tubiform, tuition, tuitions, turkois, turkoises, turmoil, turmoiled, turmoiling, turmoils, turquois, turquoise, turquoises, tutorial,
tutorials, tutoring, tutoyering, twinborn.
typhoid, typhoids, typhonic, typographic, typographical, typographically, typographies, typologies, tyronic, tyrosine, tyrosines, ubiquitous, ubiquitously, udometries, ulceration, ulcerations,
ulterior, ultimo, ultraviolet, umbonic, unaccompanied, unambiguous, unambiguously, unambitious, unanchoring, unanimous, unanimously, unauthorized, unavoidable, unbecoming, unbecomingly, unblocking,
unbodied, unbolting, unbonneting.
unbosoming, unboxing, unbuttoning, unceremonious, unceremoniously, unchoking, unciform, unciforms, uncloaking, unclogging, unclosing, unclothing, unclouding, uncocking, uncoffin, uncoffined,
uncoffining, uncoffins, uncoil, uncoiled, uncoiling, uncoils, uncoined, uncomic, uncommitted, uncomplimentary, uncompromising, unconcernedlies, unconditional, unconditionally, unconfirmed,
unconscionable, unconscionably, unconscious.
unconsciously, unconsciousness, unconsciousnesses, unconstitutional, unconventional, unconventionally, uncooperative, uncoordinated, uncorking, uncoupling, uncovering, uncrossing, uncrowning,
unction, unctions, undemocratic, underclothing, underclothings, underdoing, underexposing, undergoing, undernourished, undernourishment, undernourishments, underscoring, undocking, undoing, undoings,
undomesticated, undoubling, unemotional, unequivocal, unequivocally, unexotic, unfoiled.
unfolding, unforgivable, unforgiving, unfrocking, ungloving, ungodlier, ungodliest, ungodliness, ungodlinesses, unhallowing, unheroic, unholier, unholiest, unholily, unholiness, unholinesses,
unhooding, unhooking, unhorsing, unhousing, unicolor, unicorn.
unicorns, unidirectional, unification, unifications, uniform, uniformed, uniformer, uniformest, uniforming, uniformity, uniformly, uniforms, unilobed, unimportant, uninformed, unintentional,
unintentionally, union, unionise, unionised, unionises, unionising.
unionism, unionisms, unionist, unionists, unionization, unionizations, unionize, unionized, unionizes, unionizing, unions, unipod, unipods, unipolar, unironed, unison, unisonal, unisons, univocal,
univocals, unjoined.
unknotting, unknowing, unknowingly, unlikelihood, unloading, unlocking, unloosening, unloosing, unlovelier, unloveliest, unloving, unmodish, unmolding, unmooring, unmotivated, unmoving, unneighborly,
unnoisy, unnojectionable, unnoticeable, unnoticed, unobtainable, unobtrusive, unobtrusively, unoccupied, unofficial, unoiled, unorganized, unoriginal, unpatriotic, unpeopling, unpoetic, unpoised,
unpolite, unpopularities, unpopularity, unpretentious, unproductive, unprofessional.
unprofitable, unquestionable, unquestionably, unquestioning, unquoting, unreasoning, unresponsive, unrobing, unrolling, unroofing, unrooting, unrounding, unsatisfactory, unsocial, unsoiled,
unsoldering, unsolicited, unsolid, unsonsie, unsophisticated, unspoiled, unspoilt.
unstopping, unthroning, unvoice, unvoiced, unvoices, unvoicing, unwisdom, unwisdoms, unworthier, unworthies, unworthiest, unworthily, unworthiness, unworthinesses, unyoking, upboil, upboiled,
upboiling, upboils, upcoil, upcoiled, upcoiling, upcoils, upcoming, upflowing, upfolding, upgoing, upgrowing, uphoarding, upholding, upholsteries, upholstering, uppropping, uproarious.
uproariously, uprooting, uprousing, upshooting, upsidedown, upsilon, upsilons, upsoaring, upthrowing, uptossing, urination, urinations, urinose, urinous, urolagnia, urolagnias, urolith, uroliths,
urologic, urologies, uroscopies, ursiform, urushiol, urushiols, usurious, utilidor, utilidors, utilization, utopia, utopian, utopians, utopias, utopism, utopisms.
utopist, utopists, uxorial, uxorious, vacation, vacationed, vacationer, vacationers, vacationing, vacations, vaccination, vaccinations, vacillation, vacillations, vagabonding, vagotomies,
valedictorian, valedictorians, valedictories, valedictory, valgoid, validation, validations, valonia, valonias, valorise, valorised, valorises.
valorising, valorize, valorized, valorizes, valorizing, valuation, valuations, vamoosing, vamosing, vaporing, vaporings, vaporise, vaporised, vaporises, vaporish, vaporising, vaporization,
vaporizations, vaporize, vaporized, vaporizes, vaporizing, vapouring, variation, variations, varicose.
variegation, variegations, variform, variola, variolar, variolas, variole, varioles, variorum, variorums, various, variously, varistor, varistors, vasiform, vectoring, vegetation, vegetational,
vegetations, velocities, velocity, venation, venations, veneration, venerations, venison, venisons, venoming, venosities, venosity, ventilation, ventilations, ventilator, ventilators, ventriloquism,
ventriloquist, ventriloquists, ventriloquy, ventriloquys, veracious, verbalization, verbalizations, verbosities, verbosity, verification, verifications, verismo, verismos, veronica, veronicas,
version, versions, vertigo, vertigoes, vertigos, vetoing, vexation, vexations, vexatious, viator, viatores, viators, vibioid, vibration, vibrations, vibrato, vibrator, vibrators, vibratory, vibratos,
vibrio, vibrion, vibrions, vibrios.
vicarious, vicariously, vicariousness, vicariousnesses, viceroy, viceroys, vicious, viciously, viciousness, viciousnesses, vicomte, vicomtes, victimization, victimizations, victor, victoria,
victorias, victories, victorious, victoriously, victors, victory, video, videos, videotape, videotaped, videotapes, videotaping, vidicon, vidicons, viewpoint, viewpoints, vigor, vigorish, vigorishes,
vigoroso, vigorous, vigorously, vigorousness, vigorousnesses.
vigors, vigour, vigours, vilification, vilifications, villadom, villadoms, villianous, villianously, villianousness, villianousnesses, villose, villous, vindication, vindications, vindicator,
vindicators, vino, vinos, vinosities, vinosity, vinous, vinously, viol, viola, violable, violably, violas, violate, violated, violater, violaters, violates.
violating, violation, violations, violator, violators, violence, violences, violent, violently, violet, violets, violin, violinist, violinists, violins, violist, violists, violone, violones, viols,
viomycins, viperous, virago, viragoes, viragos, vireo, vireos, virion, virions, virologies, virology, viroses, virosis, virtuosa, virtuosas, virtuose, virtuosi, virtuosities, virtuosity, virtuoso,
virtuosos, virtuous, virtuously, viscoid, viscose, viscoses.
viscount, viscountess, viscountesses, viscounts, viscous, vision, visional, visionaries, visionary, visioned, visioning, visions, visitation, visitations, visitor, visitors, visor, visored, visoring,
visors, visualization, visualizations, vitiation, vitiations, vitiator, vitiators, vitiligo, vitiligos, vitreous, vitrification.
vitrifications, vitriol, vitrioled, vitriolic, vitrioling, vitriolled, vitriolling, vitriols, vituperation, vituperations, vivacious, vivaciously, vivaciousness, vivaciousnesses, vivisection,
vivisections, vizor, vizored, vizoring, vizors, vocabularies, vocalic, vocalics, vocalise, vocalised, vocalises.
vocalising, vocalism, vocalisms, vocalist, vocalists, vocalities, vocality, vocalize, vocalized, vocalizes, vocalizing, vocation, vocational, vocations, vocative, vocatives, vociferous, vociferously,
vogie, voguish, voice, voiced, voiceful, voicer, voicers, voices.
voicing, void, voidable, voidance, voidances, voided, voider, voiders, voiding, voidness, voidnesses, voids, voile, voiles, volatile, volatiles, volatilities, volatility, volatilize, volatilized,
volatilizes, volatilizing.
volcanic, volcanics, voleries, voling, volitant, volition, volitional, volitions, volitive, volleying, volplaning, voltaic, voltaism, voltaisms, volti, volubilities, volubility, voluming, voluminous,
voluntarily, volunteering, volutin, volutins, volution.
volutions, volvuli, vomerine, vomica, vomicae, vomit, vomited, vomiter, vomiters, vomiting, vomitive, vomitives, vomito, vomitories, vomitory, vomitos, vomitous, vomits, vomitus, vomituses,
voodooing, voodooism, voodooisms, voracious.
voraciously, voraciousness, voraciousnesses, voracities, voracity, vortical, vortices, votaries, votarist, votarists, voting, votive, votively, vouchering, vouching, vouchsafing, voussoir, voussoirs,
vowelize, vowelized, vowelizes, vowelizing, vowing, voyaging, vrooming, vulcanization, vulcanizations, waggoning, wagoning, wailsome, wainscot, wainscoted, wainscoting, wainscots, wainscotted,
walloping, wallowing, wanion, wanions, wantoning, warehousing, warison, warisons, warrior, warriors, washington, watchdogging, waterlogging, waterproofing, waterproofings, waygoing, waygoings,
weaponing, weaponries, wearisome, weatherproofing, weirdo.
weirdoes, weirdos, welcoming, wendigo, wendigos, whilom, whipcord, whipcords, whippoorwill, whippoorwills, whipworm, whipworms, whirlpool, whirlpools, whiteout, whiteouts, whitlow, whitlows,
whodunit, whodunits, wholesaling, wholism, wholisms, whomping, whooping, whooshing, whoosis, whoosises.
whopping, whoring, whorish, whosis, whosises, wickerwork, wickerworks, wicopies, wicopy, widgeon, widgeons, widow, widowed, widower, widowers, widowhood, widowhoods, widowing, widows, wifedom,
wifedoms, wifehood, wifehoods, wigeon, wigeons, wilco, wildfowl, wildfowls, wildwood, wildwoods, willow, willowed, willower, willowers, willowier, willowiest, willowing, willows.
willowy, willpower, willpowers, windigo, windigos, window, windowed, windowing, windowless, windows, windrow, windrowed, windrowing, windrows, windsock, windsocks, wineshop, wineshops, winesop,
winesops, wingbow, wingbows, wingover, wingovers, winnock, winnocks, winnow, winnowed, winnower, winnowers, winnowing, winnows.
wino, winoes, winos, winsome, winsomely, winsomeness, winsomenesses, winsomer, winsomest, wipeout, wipeouts, wirework, wireworks, wireworm, wireworms, wisdom, wisdoms, wishbone, wishbones, withhold,
withholding, withholds, without, withouts, withstood, witloof, witloofs.
wittol, wittols, wobblier, wobblies, wobbliest, wobbling, wolffish, wolffishes, wolfing, wolfish, wolflike, wolverine, wolverines, womaning, womanise, womanised, womanises, womanish, womanising,
womanize, womanized.
womanizes, womanizing, womankind, womankinds, womanlier, womanliest, womanliness, womanlinesses, wombier, wombiest, wondering, wonkier, wonkiest, wonning, wonting, woodbin, woodbind, woodbinds,
woodbine, woodbines, woodbins, woodier, woodiest, woodiness, woodinesses, wooding, woodpile, woodpiles, woodshedding, woodsia, woodsias, woodsier, woodsiest, woodwind, woodwinds, woofing, wooing,
wooingly, woolgathering.
woolgatherings, woolie, woolier, woolies, wooliest, woollier, woollies, woolliest, woollike, woolskin, woolskins, woorali, wooralis, woorari, wooraris, wooshing, woozier, wooziest, woozily,
wooziness, woozinesses.
wordier, wordiest, wordily, wordiness, wordinesses, wording, wordings, workability, working, workingman, workingmen, workings, workmanlike, workmanship, workmanships, worldlier, worldliest,
worldliness, worldlinesses, worldwide, wormier, wormiest, wormil, wormils.
worming, wormish, wormlike, worried, worrier, worriers, worries, worrisome, worrit, worrited, worriting, worrits, worrying, worsening, worship, worshiped, worshiper, worshipers, worshiping,
worshipped, worshipper, worshippers, worshipping, worships, worsting.
worthier, worthies, worthiest, worthily, worthiness, worthinesses, worthing, worthwhile, wotting, wounding, wowing, wrongdoing, wrongdoings, wronging, xenogamies, xenogenies, xenolith, xenoliths,
xenophobia, xerosis, xerotic, xiphoid, xiphoids, xyloid, xylophonist, xylophonists, xylotomies, xystoi, yahooism, yahooisms.
yellowing, yeomanries, yeshivoth, yodeling, yodelling, yodling, yogi, yogic, yogin, yogini, yoginis, yogins, yogis, yoicks, yokelish, yoking, yolkier, yolkiest, yomim, yoni, yonis, youngish,
youthening, yowie, yowies, yowing, yowling, zabaione, zabaiones, zealotries, zebroid, zecchino, zecchinos, zedoaries, zeolite.
zeolites, zeolitic, zeroing, zillion, zillions, zincoid, zincous, zingano, zingaro, zircon, zirconia, zirconias, zirconic, zirconium, zirconiums, zircons, zoaria, zoarial, zoarium, zodiac, zodiacal,
zodiacs, zoftig, zoic, zoisite, zoisites, zombi, zombie, zombies, zombiism, zombiisms, zombis.
zonation, zonations, zonetime, zonetimes, zoning, zoogenic, zooid, zooidal, zooids, zoolatries, zoologic, zoological, zoologies, zoologist, zoologists, zoomania, zoomanias, zoometries, zooming,
zoonosis, zoonotic, zoophile, zoophiles, zootomic, zootomies, zori, zoril, zorilla.
zorillas, zorille, zorilles, zorillo, zorillos, zorils, zowie, zoysia, zoysias, zygosis, zygosities, zygosity, zygotic, zymologies, zymosis, zymotic,
Glad you stopped by this reference page about words with i & o. To appear in the above I O word list these letters can appear in any order, each used at least once and perhaps multiple times,
adjacent or even with other letters between them.
Is this list missing any words? You can add them here. Thank you. | {"url":"http://wordscontaining.com/i-o/","timestamp":"2024-11-02T21:50:26Z","content_type":"application/xhtml+xml","content_length":"363857","record_id":"<urn:uuid:083717eb-f683-4b19-a2b9-e75fa9bbc8de>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00014.warc.gz"} |
Compass point between two locations
Given coordinates of two locations in decimal degrees, this calculator displays constant azimuth, distance and compass points for different compass roses.
This online calculator is created after the user's request, and it is just a convenient shortcut between Course angle and the distance between the two points on loxodrome (rhumb line). and Points of
the compass calculators. The first calculates a distance on loxodrome (rhumb line) and course angle (azimuth) between two points with given geographical coordinates. The second one outputs compass
point given course angle in degrees.
So, thanks to the ability to reuse calculator logic, this calculator accepts coordinates of two points, sends them over to the loxodrome calculator, receives a directional angle, sends it over to the
compass point calculator, and receives a compass point.
Finally, it outputs azimuth, distance in km and nm, compass point name, its degrees in DMS (degrees minutes seconds), and decimal format. You can also change a compass rose used.
PLANETCALC, Compass point between two locations | {"url":"https://embed.planetcalc.com/7042/","timestamp":"2024-11-14T21:13:24Z","content_type":"text/html","content_length":"41165","record_id":"<urn:uuid:0f793253-ea35-4aec-ab08-683e4c543bd8>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00522.warc.gz"} |
Core Decomposition on Uncertain Graphs Revisited
Core decomposition on uncertain graphs is a fundamental problem in graph analysis. Given an uncertain graph G, the core decomposition problem is to determine all (k,η)-cores in G, where a (k,η)-core
is a maximal subgraph of G such that each node has an η-degree no less than k within the subgraph. The η-degree of a node v is defined as the maximum integer r such that the probability that v has a
degree no less than r is larger than or equal to the threshold η ψ [0,1] The state-of-the-art algorithm for solving this problem is based on a peeling technique which iteratively removes the nodes
with the smallest η-degrees and also dynamically updates their neighbors' η-degrees. Unfortunately, we find that such a peeling algorithm with the dynamical h-degree updating technique is incorrect
due to the inaccuracy of the recursive floating-point number division operations involved in the dynamical updating procedure. To correctly compute the (k,η)-cores, we first propose a bottom-up
algorithm based on an on-demand η-degree computational strategy. To further improve the efficiency, we also develop a more efficient top-down algorithm with several nontrivial optimization
techniques. Both of our algorithms do not involve any floating-point number division operations, thus the correctness can be guaranteed. In addition, we also develop the parallel variants of all the
proposed algorithms. Finally, we conduct extensive experiments to evaluate the proposed algorithms using five large real-life datasets. The results show that our algorithms are at least three orders
of magnitude faster than the existing exact algorithms on large uncertain graphs. The results also demonstrate the high scalability and parallel performance of the proposed algorithms.
• Uncertain graphs
• cohesive subgraph mining
• uncertain core decomposition
Dive into the research topics of 'Core Decomposition on Uncertain Graphs Revisited'. Together they form a unique fingerprint. | {"url":"https://pure.bit.edu.cn/en/publications/core-decomposition-on-uncertain-graphs-revisited","timestamp":"2024-11-04T08:19:15Z","content_type":"text/html","content_length":"48478","record_id":"<urn:uuid:671ed7a9-0e7a-46f5-8341-949ecd466970>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00826.warc.gz"} |
The home appliance department of a
The home appliance department of a large department store is using inventory models to control the replenishment of a particular model of Microwave ovens. The daily demand follows a normal
distribution with a mean of 150 units and standard deviation of 16 units. The store pays $100 for each oven. Fixed costs of replenishment are $28. The accounting department recommends a 20% annual
holding cost rate. Assume that the average lead time is 5 days with a standard deviation of 1 day. Assume 365 days a year. Part A: What is the EOQ? Part B: What is the reorder point if the maximum
chance of 5% of stock out (i.e. 95% service level) is desired? Part C: What is the reorder point if a fill rate of 99% is required? Part D: Suppose management wants to simplify the process by setting
the reorder point to “1000” units. Based on this policy, what is the implied chance of stock ou
The home appliance department of a large department store is using inventory models to control the replenishment of a particular model of Microwave ovens. The daily demand follows a normal
distribution with a mean of 150 units and standard deviation of 16 units. The store pays $100 for each oven. Fixed costs of replenishment are $28. The accounting department recommends a 20% annual
holding cost rate. Assume that the average lead time is 5 days with a standard deviation of 1 day. Assume 365 days a year. Part A: What is the EOQ? Part B: What is the reorder point if the maximum
chance of 5% of stock out (i.e. 95% service level) is desired? Part C: What is the reorder point if a fill rate of 99% is required? Part D: Suppose management wants to simplify the process by setting
the reorder point to “1000” units. Based on this policy, what is the implied chance of stock ou
The home appliance department of a large department store is using inventory models to control
the replenishment of a particular model of Microwave ovens. The daily demand follows a
normal distribution with a mean of 150 units and standard deviation of 16 units. The store pays
$100 for each oven. Fixed costs of replenishment are $28. The accounting department
recommends a 20% annual holding cost rate. Assume that the average lead time is 5 days with a
standard deviation of 1 day. Assume 365 days a year.
Part A: What is the EOQ?
Part B: What is the reorder point if the maximum chance of 5% of stock out (i.e. 95% service
level) is desired?
Part C: What is the reorder point if a fill rate of 99% is required?
Part D: Suppose management wants to simplify the process by setting the reorder point to
“1000” units. Based on this policy, what is the implied chance of stock out?
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Philosophy 148 Home
Probability and Induction (Phil. 148 | Spring 2008 | UCB | Philosophy)
Professor GSI Lecture Sections
Branden Fitelson Raul Saucedo [S:240:S] [S:Mulford:S]
branden@fitelson.org rs339@cornell.edu 103 Moffitt
Office: 132 Moses Hall Office: 5323 Tolman See our sections page.
Hours: Tu/Th 2–3 Office Hours: W 10–12 Tu/Th 11–12:30
Tel: 642–0666 (or by appointment)
• 05/14/08: Raul's review session will be tomorrow (5/15) @ 6pm @ 100 wheeler.
• 05/12/08: I have posted solutions to HW #4 (except the extra-credit, since these are repeated on the final extra-credit assignment).
• 05/12/08: I have posted a set of extra-credit problems, as well as a "practice final exam". The extra-credit problems are due at the final exam, and we will discuss the "practice final" at the
review session on May 19 @ 4pm @ 122 Wheeler.
• 05/08/08: Branden will have office hours on Tuesday May 13 from 2–4pm. Raul will be holding review session for the final on Thursday May 15 @ 6pm (room TBA). I will be distributing extra credit
problems and a sample final exam by next Tuesday (5/13) morning (stay tuned).
• 05/06/08: I have posted a "hints and amendments" handout for HW #5, which I will also distribute at tonight's discussion section.
• 05/05/08: New plan for HW #5 — It will be due on the last class day (Thursday May 8). And, we'll have a discussion for it on Tuesday, May 6 @ 6pm @ 110 Wheeler. I'll also be announcing some
extra-credit problems in the last week of class (to be due ay the final exam on the 20th). We'll also have a review session for the final, which will take place on May 19 @ 4pm @ 122 Wheeler. The
final exam will take place at 20 Barrows (on 5/20 @ 8am). Finally, Branden will not have office hours on Tuesday May 6.
• 04/23/08: Branden has the flu. So, Thursday's (4/24) lecture is canceled.
• 04/21/08: The make up section for Raul's regular Tuesday 10-11 section will be on Thursday 10-11 @ 206 Wheeler. The make up section for Raul's regular Wednesday 9-10 section will be on Friday
12-1 (location TBA). Raul's make up OH will be Friday 11-12 and 1-2 at his office (5323 Tolman).
• 04/16/08: I have posted a handout on "some abstract properties of confirmation relations & four theories of confirmation". HW #5 has also been posted (it's due in two weeks).
• 04/14/08: Tuesday's (4/15) HW #4 discussion session will be @ 6pm @ 136 Barrows.
• 04/10/08: I've posted the notes for our two guest lectures this week (#19 & #20). Thanks, Mike and Kenny!
• 04/04/08: The grades for HW #2 are now posted on bspace. People did pretty well (mean = 88, standard deviation = 16). I'll post solutions soon.
• 04/02/08: Branden will be in Europe next week, so he will not be holding office hours. But, lectures will be given by guest lecturers. So, class will meet next week. HW #4 has been posted. We'll
have a discussion about HW #4 on Tuesday April 15 @ 6pm (stay tuned for the location).
• 03/30/08: I have posted solutions to HW #1. I'll be posting the HW #2 solutions soon. I will be running another "HW discussion" on Tuesday April 1 @ 6pm @ 251 Dwinelle.
• 03/20/08: I have updated my hints handout (also its LaTeX source) for HW #2 (to include Raul's hints handout). I will not have office hours tomorrow (after the mid-term). Good luck on the
mid-term, and have a great spring break! I have also posted HW #3. It will be due on April 3. I will be running another "HW discussion" on Tuesday April 1 @ 6pm. Stay tuned (here and by email)
for the location of that session.
• 03/11/08: By popular demand, here is the LaTeX source file for my HW #2 hints handout. Note: in all of my handouts and lecture notes for the course, I am using the Lucida Bright fonts, which are
not free. I have removed these from the above LaTeX source file (which just uses the standard/default/free Computer Modern fonts), since they would just cause the file to give errors on normal
LaTeX distributions.
• 03/05/08: Branden will be holding a "HW #2 / Mid-Term discussion / review session" on Monday March 17 @ 6pm @ 223 Dwinelle. Also, we will drop your lowest HW score.
• 03/04/08: HW #1 will be returned in sections. The median was 93 and the standard deviation was 22. There were "two curves" on this HW (this is not unusual for technical HW assignments). We will
allow you to drop your lowest HW score. We will also be scheduling "HW discussions" before each HW assignment is due. Stay tuned for time and location of the HW #2/Mid-Term discussion/review
session. HW #2 will be due on March 20 (the same day as the mid-term).
• 02/28/08: I've posted a handout on the "Package Principle" (and the conditions under which it is required to run a DBA).
• 02/27/08: I've made two changes to the assignment & exam schedule. Specifically, I have moved the mid-term back 2 weeks (from 3/6 to 3/20), and I have moved assignment #3 back one week (from 3/13
to 3/20). Finally, I have added a paper by van Fraassen on frequencies and probabilities (the first few pages explain why hypothetical infinite limiting frequencies do not satisfy the classical
probability axioms).
• 02/2708: I have posted HW #2 and a hints handout for HW #2. I strongly recommend studying the hints handout before you try to solve the HW #2 problems. I have also posted the Mathematica notebook
I used in lecture to simulate a fair coin being tossed n times.
• 02/19/08: I have posted the quiz solutions. People did well on the quiz (average: 93). I will be using straight grading scale for this course. Here it is: A+ >97, A (94,97], A- (90,94], B+
(87,90], B (84,87], B- (80,84], C+ (77,80], C (74,77], C- (70,74], D [50,70], F <50.
• 02/13/08: I have posted Homework Assignment #1. It is due on 2/28/08.
• 02/10/08: I have posted a handout explaining why Skyrms's "rules" for probability calculus are incomplete. Note: this handout was updated (on 2/10) since the first version I posted last week.
• 02/06/08: I have decided to move the quiz back from next Tuesday (2/12) to next Thursday (2/14).
• 02/06/08: Raul's office hours are being held in 5323 Tolman this semester (not 301 Moses, as I had previously listed on this website).
• 02/04/08: Branden's Tuesday office hours this week will be 3–4.
• 02/03/08: We have a permanent location for the Wednesday section: 2304 Tolman.
• 01/30/08: We have a permanent location for the Tuesday section: 206 Wheeler. Stay tuned for the Wed. section permanent location.
• 01/28/08: Branden's Thursday office hours will be 2:30-3:30 this week. I have posted a Mathematica notebook, which uses PrSAT to go through the algebraic probability calculus examples discussed
in Lecture #3.
• 01/27/08: The sections have been set. You should have received an email telling you which section you're in. For next week, sections will probably meet in 301 Moses Hall. Permanent locations will
be emailed to section members and posted on our sections page.
• 01/15/08: Note that our room has been changed from [S:240:S] [S:Mulford:S] to 103 Moffitt. | {"url":"http://fitelson.org/probability/","timestamp":"2024-11-08T00:39:31Z","content_type":"text/html","content_length":"16595","record_id":"<urn:uuid:5778aaa2-1afd-453e-aac0-d5cc6fb65205>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00136.warc.gz"} |
The production function for a firm is Q = −0.6L 3 + 18L 2K + 10L...
The production function for a firm is Q = −0.6L 3 + 18L 2K + 10L where Q is the amount of output, L is the number of labor hours per week, and K is the amount of capital.
(a)Use Excel to calculate the total short run output Q(L) for L = 0, 1, 2...20, given that capital is fixed in the short run at K = 1.
(b) Use Excel to calculate the total long run output Q(L) for L = 0, 1, 2...20 and K = 0, 1, 2.......20. (Note: You can use one formula to perform all calculations in this question).
(c) Suppose firm currently employs one unit of labor and one unit of capital. Use the matrix generated in part (b) to determine the returns to scale for this production function. Suppose that the
wage is $100, the rental rate is $800 per time period and the capital is fixed at K = 1
(d) For each quantity of labor in part (a), calculate the average product of labor (APL), marginal product of labor (MPL), total variable cost (T V C), total cost (T C), average variable cost (AV C),
average total cost (AT C) and marginal cost (MC).
(e) For each quantity of labor in part (a), calculate w/APL and w/MPL, and confirm that they equal AV C and MC, respectively. Explain why these relationships hold. | {"url":"https://justaaa.com/economics/162502-the-production-function-for-a-firm-is-q-06l-3-18l","timestamp":"2024-11-13T06:38:41Z","content_type":"text/html","content_length":"42079","record_id":"<urn:uuid:a9b6aa56-c114-442d-b7a6-8e3748a1548a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00776.warc.gz"} |
How to Make a Table in R: Master Data Visualization & Analysis
If you’re working with data in R, one of the essential tasks is to organize and present your data in a structured format. Tables are a powerful way to display data, and in this guide, we’ll explore
how to create tables in R. We’ll cover various aspects, including working with two-way tables, creating tables from data, and using different tools to manipulate and analyze the data within tables.
Two-Way Tables in R
Two-way tables are a fundamental tool for summarizing categorical data in R. They allow us to cross-tabulate two categorical variables and examine the relationship between them. In our example, we
will use the “smoker.csv” dataset, which contains information about individuals’ smoking status and socioeconomic status (SES).
To begin, let’s load the dataset and get a summary of its contents:
smokerData <- read.csv(file='smoker.csv', sep=',', header=T)
The output will display the distribution of smoking status and SES in the dataset, categorized as “current,” “former,” “never,” and “High,” “Low,” “Middle” respectively.
Creating a Table from Data
To create a two-way table from raw data, we can use the table() function in R. In our case, we’ll create a table that displays the number of individuals for each combination of smoking status and
smoke <- table(smokerData$Smoke, smokerData$SES)
This table will show the count of individuals in each category, making it easy to analyze the data and observe any patterns or trends.
Tools for Working with Tables
R provides several useful functions to work with tables and explore the data in various ways. Let’s delve into some of these tools:
The barplot() function is a handy tool to visualize two-way tables. It helps us understand the distribution of categories in each variable and how they interact. We can create a bar plot using the
following code:
barplot(smoke, legend=T, beside=T, main='Smoking Status by SES')
This will generate a bar plot showing the distribution of smoking status based on socioeconomic status.
The prop.table() function allows us to calculate proportions from the two-way table. We can use it to determine the proportion of individuals in each category, making it easier to compare the
distributions. Here’s how to use it:
This will give us a table of proportions, showing the percentage of individuals in each category for smoking status and SES.
Chi-Squared Test
The chi-squared test is a statistical test used to determine whether there is a significant association between two categorical variables. In R, we can perform a chi-squared test on our two-way table
using the chisq.test() function:
result <- chisq.test(smoke)
The output will provide information about the test, including the chi-squared statistic, degrees of freedom, and p-value.
Creating a Table Directly
Sometimes, instead of having raw data, we may already have a table and need to create a table directly from it. We can achieve this by creating an array of numbers and then converting it into a
Let’s consider an example where we want to create a table similar to our previous one:
data <- c(51, 43, 22, 92, 28, 21, 68, 22, 9)
rows <- c("current", "former", "never")
cols <- c("High", "Low", "Middle")
smoke_direct <- matrix(data, ncol=3, byrow=TRUE)
colnames(smoke_direct) <- cols
rownames(smoke_direct) <- rows
smoke_direct <- as.table(smoke_direct)
This will give us a two-way table created directly from the data specified in the arrays.
Graphical Views of Tables
In addition to numerical analysis, we can also create graphical views of tables to better understand the data.
Mosaic Plot
The mosaicplot() function is an excellent way to visualize the relationships between two categorical variables. It creates a mosaic plot that displays the proportion of individuals in each category.
mosaicplot(smoke, main="Smokers", xlab="Status", ylab="Economic Class")
This will generate a mosaic plot showing the distribution of smoking status based on socioeconomic status, helping us visualize the associations.
Sorting and Direction
We can customize the mosaic plot further by specifying the sort and direction options. These options allow us to change the orientation of the plot and the ordering of the categories.
mosaicplot(smoke, sort=c(2,1))
This will create a mosaic plot with the vertical axis determining the primary proportion.
mosaicplot(smoke, dir=c("v", "h"))
This will create a mosaic plot with the vertical and horizontal axes swapped.
Tables are essential tools in data analysis and are commonly used to present categorical data. In this tutorial, we explored how to create two-way tables from raw data, as well as how to work with
tables directly. We also learned about various tools to manipulate and analyze data within tables, including graphical views and statistical tests.
Understanding how to create and interpret tables in R will greatly enhance your data analysis skills, allowing you to gain valuable insights from your data. As you continue to work with R, you’ll
find that tables are a versatile and powerful way to organize and visualize data effectively.
Can I create a table directly from existing data in R?
Yes, you can create a table directly from existing data in R. R provides various functions to create tables from raw data. One common approach is to use the table() function, which allows you to
create a two-way table by cross-tabulating two categorical variables. Alternatively, you can create a table directly using the matrix() function and then convert it to a table using the as.table()
What tools are available for working with tables in R?
R offers several tools for working with tables, making it easier to manipulate and analyze data. Some of the essential tools include:
• table() function: To create two-way tables from raw data.
• prop.table() function: To calculate proportions from tables.
• margin.table() function: To get marginal distributions of the data.
• chisq.test() function: To perform the chi-squared test for table independence.
• mosaicplot() function: To visualize two-way tables using mosaic plots.
• barplot() function: To create bar plots for tables.
• summary() function: To get summary statistics of the table.
How can I visualize tables in R using graphical views?
You can visualize tables in R using graphical views like mosaic plots and bar plots. For mosaic plots, you can use the mosaicplot() function, which displays the proportion of individuals in each
category, making it easy to visualize the associations between two categorical variables. On the other hand, the barplot() function helps create bar plots that show the distribution of categories in
each variable.
How do I manage data in R for table creation?
Managing data in R for table creation involves various steps, including importing data, cleaning and preprocessing it if needed, and organizing it in a suitable format for table creation. You can
import data from various sources such as CSV files, Excel sheets, or databases using functions like read.csv(), read.table(), or specialized packages like readxl or readr. After importing, ensure
that your data is in the right format (e.g., factors for categorical variables) to create tables directly using the table() function or by converting it to a matrix.
Are there any time data types used in R tables?
Yes, R provides specific data types for handling time-related data. The most common ones are Date and POSIXct (or POSIXlt). The Date class is used to represent calendar dates without time, while
POSIXct represents dates and times with seconds precision. These data types are often used in tables when dealing with time-related data or when analyzing temporal patterns.
What are the steps to create a two-way table in R?
Creating a two-way table in R involves the following steps:
1. Import or generate the data: Load the data into R using functions like read.csv() or create data directly in R.
2. Organize data (if needed): Ensure that the variables you want to cross-tabulate are in the correct format (e.g., factors) for the table creation process.
3. Use the table() function: Create the two-way table using the table() function, passing the two categorical variables as arguments.
4. Optional: Use graphical views or statistical tests: Visualize the table using mosaicplot() or barplot() functions and perform a chi-squared test with chisq.test() to check for independence.
By following these steps, you can easily create and analyze two-way tables in R, helping you gain valuable insights from your data.
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compare_n_qualvars: Comparison for columns of factors for more than 2 groups with... in wrappedtools: Useful Wrappers Around Commonly Used Functions
Comparison for columns of factors for more than 2 groups with post-hoc
compare_n_qualvars( data, dep_vars, indep_var, round_p = 3, round_desc = 2, pretext = FALSE, mark = FALSE, singleline = FALSE, spacer = " ", linebreak = "\n", prettynum = FALSE )
data name of data set (tibble/data.frame) to analyze.
dep_vars vector of column names.
indep_var name of grouping variable.
round_p level for rounding p-value.
round_desc number of significant digits for rounding of descriptive stats
pretext for function formatP
mark for function formatP
singleline Put all group levels in a single line?
spacer Text element to indent levels, defaults to " ".
linebreak place holder for newline.
prettynum Apply prettyNum to results?
A tibble with variable names, descriptive statistics, and p-value of fisher.test and pairwise_fisher_test, number of rows is number of dep_vars.
# Separate lines for each factor level: compare_n_qualvars( data = mtcars, dep_vars = c("am", "cyl", "carb"), indep_var = "gear", spacer = " " ) # All levels in one row but with linebreaks:
compare_n_qualvars( data = mtcars, dep_vars = c("am", "cyl", "carb"), indep_var = "gear", singleline = TRUE ) # All levels in one row, separateted by ";": compare_n_qualvars( data = mtcars, dep_vars
= c("am", "cyl", "carb"), indep_var = "gear", singleline = TRUE, linebreak = "; " )
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STARKs, Part 3: Into the Weeds
2018 Jul 21
See all posts
Special thanks to Eli ben Sasson for his kind assistance, as usual. Special thanks to Chih-Cheng Liang and Justin Drake for review, and to Ben Fisch for suggesting the reverse MIMC technique for a
VDF (paper here)
Trigger warning: math and lots of python
As a followup to Part 1 and Part 2 of this series, this post will cover what it looks like to actually implement a STARK, complete with an implementation in python. STARKs ("Scalable Transparent
ARgument of Knowledge" are a technique for creating a proof that \(f(x)=y\) where \(f\) may potentially take a very long time to calculate, but where the proof can be verified very quickly. A STARK
is "doubly scalable": for a computation with \(t\) steps, it takes roughly \(O(t \cdot \log{t})\) steps to produce a proof, which is likely optimal, and it takes ~\(O(\log^2{t})\) steps to verify,
which for even moderately large values of \(t\) is much faster than the original computation. STARKs can also have a privacy-preserving "zero knowledge" property, though the use case we will apply
them to here, making verifiable delay functions, does not require this property, so we do not need to worry about it.
First, some disclaimers:
• This code has not been thoroughly audited; soundness in production use cases is not guaranteed
• This code is very suboptimal (it's written in Python, what did you expect)
• STARKs "in real life" (ie. as implemented in Eli and co's production implementations) tend to use binary fields and not prime fields for application-specific efficiency reasons; however, they do
stress in their writings the prime field-based approach to STARKs described here is legitimate and can be used
• There is no "one true way" to do a STARK. It's a broad category of cryptographic and mathematical constructs, with different setups optimal for different applications and constant ongoing
research to reduce prover and verifier complexity and improve soundness.
• This article absolutely expects you to know how modular arithmetic and prime fields work, and be comfortable with the concepts of polynomials, interpolation and evaluation. If you don't, go back
to Part 2, and also this earlier post on quadratic arithmetic programs
Now, let's get to it.
Here is the function we'll be doing a STARK of:
def mimc(inp, steps, round_constants):
start_time = time.time()
for i in range(steps-1):
inp = (inp**3 + round_constants[i % len(round_constants)]) % modulus
print("MIMC computed in %.4f sec" % (time.time() - start_time))
return inp
We choose MIMC (see paper) as the example because it is both (i) simple to understand and (ii) interesting enough to be useful in real life. The function can be viewed visually as follows:
Note: in many discussions of MIMC, you will typically see XOR used instead of +; this is because MIMC is typically done over binary fields, where addition is XOR; here we are doing it over prime
In our example, the round constants will be a relatively small list (eg. 64 items) that gets cycled through over and over again (that is, after k[64] it loops back to using k[1]).
MIMC with a very large number of rounds, as we're doing here, is useful as a verifiable delay function - a function which is difficult to compute, and particularly non-parallelizable to compute, but
relatively easy to verify. MIMC by itself achieves this property to some extent because MIMC can be computed "backward" (recovering the "input" from its corresponding "output"), but computing it
backward takes about 100 times longer to compute than the forward direction (and neither direction can be significantly sped up by parallelization). So you can think of computing the function in the
backward direction as being the act of "computing" the non-parallelizable proof of work, and computing the function in the forward direction as being the process of "verifying" it.
\(x \rightarrow x^{(2p-1)/3}\) gives the inverse of \(x \rightarrow x^3\); this is true because of Fermat's Little Theorem, a theorem that despite its supposed littleness is arguably much more
important to mathematics than Fermat's more famous "Last Theorem".
What we will try to achieve here is to make verification much more efficient by using a STARK - instead of the verifier having to run MIMC in the forward direction themselves, the prover, after
completing the computation in the "backward direction", would compute a STARK of the computation in the "forward direction", and the verifier would simply verify the STARK. The hope is that the
overhead of computing a STARK can be less than the difference in speed running MIMC forwards relative to backwards, so a prover's time would still be dominated by the initial "backward" computation,
and not the (highly parallelizable) STARK computation. Verification of a STARK can be relatively fast (in our python implementation, ~0.05-0.3 seconds), no matter how long the original computation
All calculations are done modulo \(2^{256} - 351 \cdot 2^{32} + 1\); we are using this prime field modulus because it is the largest prime below \(2^{256}\) whose multiplicative group contains an
order \(2^{32}\) subgroup (that is, there's a number \(g\) such that successive powers of \(g\) modulo this prime loop around back to \(1\) after exactly \(2^{32}\) cycles), and which is of the form
\(6k+5\). The first property is necessary to make sure that our efficient versions of the FFT and FRI algorithms can work, and the second ensures that MIMC actually can be computed "backwards" (see
the use of \(x \rightarrow x^{(2p-1)/3}\) above).
Prime field operations
We start off by building a convenience class that does prime field operations, as well as operations with polynomials over prime fields. The code is here. First some trivial bits:
class PrimeField():
def __init__(self, modulus):
# Quick primality test
assert pow(2, modulus, modulus) == 2
self.modulus = modulus
def add(self, x, y):
return (x+y) % self.modulus
def sub(self, x, y):
return (x-y) % self.modulus
def mul(self, x, y):
return (x*y) % self.modulus
And the Extended Euclidean Algorithm for computing modular inverses (the equivalent of computing \(\frac{1}{x}\) in a prime field):
# Modular inverse using the extended Euclidean algorithm
def inv(self, a):
if a == 0:
return 0
lm, hm = 1, 0
low, high = a % self.modulus, self.modulus
while low > 1:
r = high//low
nm, new = hm-lm*r, high-low*r
lm, low, hm, high = nm, new, lm, low
return lm % self.modulus
The above algorithm is relatively expensive; fortunately, for the special case where we need to do many modular inverses, there's a simple mathematical trick that allows us to compute many inverses,
called Montgomery batch inversion:
Using Montgomery batch inversion to compute modular inverses. Inputs purple, outputs green, multiplication gates black; the red square is the only modular inversion.
The code below implements this algorithm, with some slightly ugly special case logic so that if there are zeroes in the set of what we are inverting, it sets their inverse to 0 and moves along.
def multi_inv(self, values):
partials = [1]
for i in range(len(values)):
partials.append(self.mul(partials[-1], values[i] or 1))
inv = self.inv(partials[-1])
outputs = [0] * len(values)
for i in range(len(values), 0, -1):
outputs[i-1] = self.mul(partials[i-1], inv) if values[i-1] else 0
inv = self.mul(inv, values[i-1] or 1)
return outputs
This batch inverse algorithm will prove important later on, when we start dealing with dividing sets of evaluations of polynomials.
Now we move on to some polynomial operations. We treat a polynomial as an array, where element \(i\) is the \(i\)th degree term (eg. \(x^{3} + 2x + 1\) becomes [1, 2, 0, 1]). Here's the operation of
evaluating a polynomial at one point:
# Evaluate a polynomial at a point
def eval_poly_at(self, p, x):
y = 0
power_of_x = 1
for i, p_coeff in enumerate(p):
y += power_of_x * p_coeff
power_of_x = (power_of_x * x) % self.modulus
return y % self.modulus
What is the output of f.eval_poly_at([4, 5, 6], 2) if the modulus is 31?
Mouseover below for answer
\(6 \cdot 2^{2} + 5 \cdot 2 + 4 = 38, 38 \bmod 31 = 7\).
There is also code for adding, subtracting, multiplying and dividing polynomials; this is textbook long addition/subtraction/multiplication/division. The one non-trivial thing is Lagrange
interpolation, which takes as input a set of x and y coordinates, and returns the minimal polynomial that passes through all of those points (you can think of it as being the inverse of polynomial
# Build a polynomial that returns 0 at all specified xs
def zpoly(self, xs):
root = [1]
for x in xs:
root.insert(0, 0)
for j in range(len(root)-1):
root[j] -= root[j+1] * x
return [x % self.modulus for x in root]
def lagrange_interp(self, xs, ys):
# Generate master numerator polynomial, eg. (x - x1) * (x - x2) * ... * (x - xn)
root = self.zpoly(xs)
# Generate per-value numerator polynomials, eg. for x=x2,
# (x - x1) * (x - x3) * ... * (x - xn), by dividing the master
# polynomial back by each x coordinate
nums = [self.div_polys(root, [-x, 1]) for x in xs]
# Generate denominators by evaluating numerator polys at each x
denoms = [self.eval_poly_at(nums[i], xs[i]) for i in range(len(xs))]
invdenoms = self.multi_inv(denoms)
# Generate output polynomial, which is the sum of the per-value numerator
# polynomials rescaled to have the right y values
b = [0 for y in ys]
for i in range(len(xs)):
yslice = self.mul(ys[i], invdenoms[i])
for j in range(len(ys)):
if nums[i][j] and ys[i]:
b[j] += nums[i][j] * yslice
return [x % self.modulus for x in b]
See the "M of N" section of this article for a description of the math. Note that we also have special-case methods lagrange_interp_4 and lagrange_interp_2 to speed up the very frequent operations of
Lagrange interpolation of degree \(< 2\) and degree \(< 4\) polynomials.
Fast Fourier Transforms
If you read the above algorithms carefully, you might notice that Lagrange interpolation and multi-point evaluation (that is, evaluating a degree \(< N\) polynomial at \(N\) points) both take
quadratic time to execute, so for example doing a Lagrange interpolation of one thousand points takes a few million steps to execute, and a Lagrange interpolation of one million points takes a few
trillion. This is an unacceptably high level of inefficiency, so we will use a more efficient algorithm, the Fast Fourier Transform.
The FFT only takes \(O(n \cdot log(n))\) time (ie. ~10,000 steps for 1,000 points, ~20 million steps for 1 million points), though it is more restricted in scope; the x coordinates must be a complete
set of roots of unity of some order \(N = 2^{k}\). That is, if there are \(N\) points, the x coordinates must be successive powers \(1, p, p^{2}, p^{3}\)... of some \(p\) where \(p^{N} = 1\). The
algorithm can, surprisingly enough, be used for multi-point evaluation or interpolation, with one small parameter tweak.
Challenge Find a 16th root of unity mod 337 that is not an 8th root of unity.
Mouseover below for answer
59, 146, 30, 297, 278, 191, 307, 40
You could have gotten this list by doing something like [print(x) for x in range(337) if pow(x, 16, 337) == 1 and pow(x, 8, 337) != 1], though there is a smarter way that works for much larger
moduluses: first, identify a single primitive root mod 337 (that is, not a perfect square), by looking for a value x such that pow(x, 336 // 2, 337) != 1 (these are easy to find; one answer is
5), and then taking the (336 / 16)'th power of it.
Here's the algorithm (in a slightly simplified form; see code here for something slightly more optimized):
def fft(vals, modulus, root_of_unity):
if len(vals) == 1:
return vals
L = fft(vals[::2], modulus, pow(root_of_unity, 2, modulus))
R = fft(vals[1::2], modulus, pow(root_of_unity, 2, modulus))
o = [0 for i in vals]
for i, (x, y) in enumerate(zip(L, R)):
y_times_root = y*pow(root_of_unity, i, modulus)
o[i] = (x+y_times_root) % modulus
o[i+len(L)] = (x-y_times_root) % modulus
return o
def inv_fft(vals, modulus, root_of_unity):
f = PrimeField(modulus)
# Inverse FFT
invlen = f.inv(len(vals))
return [(x*invlen) % modulus for x in
fft(vals, modulus, f.inv(root_of_unity))]
You can try running it on a few inputs yourself and check that it gives results that, when you use eval_poly_at on them, give you the answers you expect to get. For example:
>>> fft.fft([3,1,4,1,5,9,2,6], 337, 85, inv=True)
[46, 169, 29, 149, 126, 262, 140, 93]
>>> f = poly_utils.PrimeField(337)
>>> [f.eval_poly_at([46, 169, 29, 149, 126, 262, 140, 93], f.exp(85, i)) for i in range(8)]
[3, 1, 4, 1, 5, 9, 2, 6]
A Fourier transform takes as input [x[0] .... x[n-1]], and its goal is to output x[0] + x[1] + ... + x[n-1] as the first element, x[0] + x[1] * 2 + ... + x[n-1] * w**(n-1) as the second element, etc
etc; a fast Fourier transform accomplishes this by splitting the data in half, doing an FFT on both halves, and then gluing the result back together.
A diagram of how information flows through the FFT computation. Notice how the FFT consists of a "gluing" step followed by two copies of the FFT on two halves of the data, and so on recursively until
you're down to one element.
I recommend this for more intuition on how or why the FFT works and polynomial math in general, and this thread for some more specifics on DFT vs FFT, though be warned that most literature on Fourier
transforms talks about Fourier transforms over real and complex numbers, not prime fields. If you find this too hard and don't want to understand it, just treat it as weird spooky voodoo that just
works because you ran the code a few times and verified that it works, and you'll be fine too.
Thank Goodness It's FRI-day (that's "Fast Reed-Solomon Interactive Oracle Proofs of Proximity")
Reminder: now may be a good time to review and re-read Part 2
Now, we'll get into the code for making a low-degree proof. To review, a low-degree proof is a (probabilistic) proof that at least some high percentage (eg. 80%) of a given set of values represent
the evaluations of some specific polynomial whose degree is much lower than the number of values given. Intuitively, just think of it as a proof that "some Merkle root that we claim represents a
polynomial actually does represent a polynomial, possibly with a few errors". As input, we have:
• A set of values that we claim are the evaluation of a low-degree polynomial
• A root of unity; the x coordinates at which the polynomial is evaluated are successive powers of this root of unity
• A value \(N\) such that we are proving the degree of the polynomial is strictly less than \(N\)
• The modulus
Our approach is a recursive one, with two cases. First, if the degree is low enough, we just provide the entire list of values as a proof; this is the "base case". Verification of the base case is
trivial: do an FFT or Lagrange interpolation or whatever else to interpolate the polynomial representing those values, and verify that its degree is \(< N\). Otherwise, if the degree is higher than
some set minimum, we do the vertical-and-diagonal trick described at the bottom of Part 2.
We start off by putting the values into a Merkle tree and using the Merkle root to select a pseudo-random x coordinate (special_x). We then calculate the "column":
# Calculate the set of x coordinates
xs = get_power_cycle(root_of_unity, modulus)
column = []
for i in range(len(xs)//4):
x_poly = f.lagrange_interp_4(
[xs[i+len(xs)*j//4] for j in range(4)],
[values[i+len(values)*j//4] for j in range(4)],
column.append(f.eval_poly_at(x_poly, special_x))
This packs a lot into a few lines of code. The broad idea is to re-interpret the polynomial \(P(x)\) as a polynomial \(Q(x, y)\), where \(P(x) = Q(x, x^4)\). If \(P\) has degree \(< N\), then \(P'(y)
= Q(special\_x, y)\) will have degree \(< \frac{N}{4}\). Since we don't want to take the effort to actually compute \(Q\) in coefficient form (that would take a still-relatively-nasty-and-expensive
FFT!), we instead use another trick. For any given value of \(x^{4}\), there are 4 corresponding values of \(x\): \(x\), \(modulus - x\), and \(x\) multiplied by the two modular square roots of \(-1
\). So we already have four values of \(Q(?, x^4)\), which we can use to interpolate the polynomial \(R(x) = Q(x, x^4)\), and from there calculate \(R(special\_x) = Q(special\_x, x^4) = P'(x^4)\).
There are \(\frac{N}{4}\) possible values of \(x^{4}\), and this lets us easily calculate all of them.
A diagram from part 2; it helps to keep this in mind when understanding what's going on here
Our proof consists of some number (eg. 40) of random queries from the list of values of \(x^{4}\) (using the Merkle root of the column as a seed), and for each query we provide Merkle branches of the
five values of \(Q(?, x^4)\):
m2 = merkelize(column)
# Pseudo-randomly select y indices to sample
# (m2[1] is the Merkle root of the column)
ys = get_pseudorandom_indices(m2[1], len(column), 40)
# Compute the Merkle branches for the values in the polynomial and the column
branches = []
for y in ys:
branches.append([mk_branch(m2, y)] +
[mk_branch(m, y + (len(xs) // 4) * j) for j in range(4)])
The verifier's job will be to verify that these five values actually do lie on the same degree \(< 4\) polynomial. From there, we recurse and do an FRI on the column, verifying that the column
actually does have degree \(< \frac{N}{4}\). That really is all there is to FRI.
As a challenge exercise, you could try creating low-degree proofs of polynomial evaluations that have errors in them, and see how many errors you can get away passing the verifier with (hint, you'll
need to modify the prove_low_degree function; with the default prover, even one error will balloon up and cause verification to fail).
The STARK
Reminder: now may be a good time to review and re-read Part 1
Now, we get to the actual meat that puts all of these pieces together: def mk_mimc_proof(inp, steps, round_constants) (code here), which generates a proof of the execution result of running the MIMC
function with the given input for some number of steps. First, some asserts:
assert steps <= 2**32 // extension_factor
assert is_a_power_of_2(steps) and is_a_power_of_2(len(round_constants))
assert len(round_constants) < steps
The extension factor is the extent to which we will be "stretching" the computational trace (the set of "intermediate values" of executing the MIMC function). We need the step count multiplied by the
extension factor to be at most \(2^{32}\), because we don't have roots of unity of order \(2^{k}\) for \(k > 32\).
Our first computation will be to generate the computational trace; that is, all of the intermediate values of the computation, from the input going all the way to the output.
# Generate the computational trace
computational_trace = [inp]
for i in range(steps-1):
computational_trace.append((computational_trace[-1]**3 + round_constants[i % len(round_constants)]) % modulus)
output = computational_trace[-1]
We then convert the computation trace into a polynomial, "laying down" successive values in the trace on successive powers of a root of unity \(g\) where \(g^{steps}\) = 1, and we then evaluate the
polynomial in a larger set, of successive powers of a root of unity \(g_2\) where \((g_2)^{steps \cdot 8} = 1\) (note that \((g_2)^{8} = g\)).
computational_trace_polynomial = inv_fft(computational_trace, modulus, subroot)
p_evaluations = fft(computational_trace_polynomial, modulus, root_of_unity)
Black: powers of \(g_1\). Purple: powers of \(g_2\). Orange: 1. You can look at successive roots of unity as being arranged in a circle in this way. We are "laying" the computational trace along
powers of \(g_1\), and then extending it compute the values of the same polynomial at the intermediate values (ie. the powers of \(g_2\)).
We can convert the round constants of MIMC into a polynomial. Because these round constants loop around very frequently (in our tests, every 64 steps), it turns out that they form a degree-64
polynomial, and we can fairly easily compute its expression, and its extension:
skips2 = steps // len(round_constants)
constants_mini_polynomial = fft(round_constants, modulus, f.exp(subroot, skips2), inv=True)
constants_polynomial = [0 if i % skips2 else constants_mini_polynomial[i//skips2] for i in range(steps)]
constants_mini_extension = fft(constants_mini_polynomial, modulus, f.exp(root_of_unity, skips2))
Suppose there are 8192 steps of execution and 64 round constants. Here is what we are doing: we are doing an FFT to compute the round constants as a function of \((g_1)^{128}\). We then add zeroes in
between the constants to make it a function of \(g_1\) itself. Because \((g_1)^{128}\) loops around every 64 steps, we know this function of \(g_1\) will as well. We only compute 512 steps of the
extension, because we know that the extension repeats after 512 steps as well.
We now, as in the Fibonacci example in Part 1, calculate \(C(P(x))\), except this time it's \(C(P(x), P(g_1 \cdot x), K(x))\):
# Create the composed polynomial such that
# C(P(x), P(g1*x), K(x)) = P(g1*x) - P(x)**3 - K(x)
c_of_p_evaluations = [(p_evaluations[(i+extension_factor)%precision] -
f.exp(p_evaluations[i], 3) -
constants_mini_extension[i % len(constants_mini_extension)])
% modulus for i in range(precision)]
print('Computed C(P, K) polynomial')
Note that here we are no longer working with polynomials in coefficient form; we are working with the polynomials in terms of their evaluations at successive powers of the higher-order root of unity.
c_of_p is intended to be \(Q(x) = C(P(x), P(g_1 \cdot x), K(x)) = P(g_1 \cdot x) - P(x)^3 - K(x)\); the goal is that for every \(x\) that we are laying the computational trace along (except for the
last step, as there's no step "after" the last step), the next value in the trace is equal to the previous value in the trace cubed, plus the round constant. Unlike the Fibonacci example in Part 1,
where if one computational step was at coordinate \(k\), the next step is at coordinate \(k+1\), here we are laying down the computational trace along successive powers of the lower-order root of
unity \(g_1\), so if one computational step is located at \(x = (g_1)^i\), the "next" step is located at \((g_1)^{i+1}\) = \((g_1)^i \cdot g_1 = x \cdot g_1\). Hence, for every power of the
lower-order root of unity \(g_1\) (except the last), we want it to be the case that \(P(x\cdot g_1) = P(x)^3 + K(x)\), or \(P(x\cdot g_1) - P(x)^3 - K(x) = Q(x) = 0\). Thus, \(Q(x)\) will be equal to
zero at all successive powers of the lower-order root of unity \(g\) (except the last).
There is an algebraic theorem that proves that if \(Q(x)\) is equal to zero at all of these x coordinates, then it is a multiple of the minimal polynomial that is equal to zero at all of these x
coordinates: \(Z(x) = (x - x_1) \cdot (x - x_2) \cdot ... \cdot (x - x_n)\). Since proving that \(Q(x)\) is equal to zero at every single coordinate we want to check is too hard (as verifying such a
proof would take longer than just running the original computation!), instead we use an indirect approach to (probabilistically) prove that \(Q(x)\) is a multiple of \(Z(x)\). And how do we do that?
By providing the quotient \(D(x) = \frac{Q(x)}{Z(x)}\) and using FRI to prove that it's an actual polynomial and not a fraction, of course!
We chose the particular arrangement of lower and higher order roots of unity (rather than, say, laying the computational trace along the first few powers of the higher order root of unity) because it
turns out that computing \(Z(x)\) (the polynomial that evaluates to zero at all points along the computational trace except the last), and dividing by \(Z(x)\) is trivial there: the expression of \(Z
\) is a fraction of two terms.
# Compute D(x) = Q(x) / Z(x)
# Z(x) = (x^steps - 1) / (x - x_atlast_step)
z_num_evaluations = [xs[(i * steps) % precision] - 1 for i in range(precision)]
z_num_inv = f.multi_inv(z_num_evaluations)
z_den_evaluations = [xs[i] - last_step_position for i in range(precision)]
d_evaluations = [cp * zd * zni % modulus for cp, zd, zni in zip(c_of_p_evaluations, z_den_evaluations, z_num_inv)]
print('Computed D polynomial')
Notice that we compute the numerator and denominator of \(Z\) directly in "evaluation form", and then use the batch modular inversion to turn dividing by \(Z\) into a multiplication (\(\cdot z_d \
cdot z_ni\)), and then pointwise multiply the evaluations of \(Q(x)\) by these inverses of \(Z(x)\). Note that at the powers of the lower-order root of unity except the last (ie. along the portion of
the low-degree extension that is part of the original computational trace), \(Z(x) = 0\), so this computation involving its inverse will break. This is unfortunate, though we will plug the hole by
simply modifying the random checks and FRI algorithm to not sample at those points, so the fact that we calculated them wrong will never matter.
Because \(Z(x)\) can be expressed so compactly, we get another benefit: the verifier can compute \(Z(x)\) for any specific \(x\) extremely quickly, without needing any precomputation. It's okay for
the prover to have to deal with polynomials whose size equals the number of steps, but we don't want to ask the verifier to do the same, as we want verification to be succinct (ie. ultra-fast, with
proofs as small as possible).
Probabilistically checking \(D(x) \cdot Z(x) = Q(x)\) at a few randomly selected points allows us to verify the transition constraints - that each computational step is a valid consequence of the
previous step. But we also want to verify the boundary constraints - that the input and the output of the computation is what the prover says they are. Just asking the prover to provide evaluations
of \(P(1)\), \(D(1)\), \(P(last\_step)\) and \(D(last\_step)\) (where \(last\_step\) (or \(g^{steps-1}\)) is the coordinate corresponding to the last step in the computation) is too fragile; there's
no proof that those values are on the same polynomial as the rest of the data. So instead we use a similar kind of polynomial division trick:
# Compute interpolant of ((1, input), (x_atlast_step, output))
interpolant = f.lagrange_interp_2([1, last_step_position], [inp, output])
i_evaluations = [f.eval_poly_at(interpolant, x) for x in xs]
zeropoly2 = f.mul_polys([-1, 1], [-last_step_position, 1])
inv_z2_evaluations = f.multi_inv([f.eval_poly_at(quotient, x) for x in xs])
# B = (P - I) / Z2
b_evaluations = [((p - i) * invq) % modulus for p, i, invq in zip(p_evaluations, i_evaluations, inv_z2_evaluations)]
print('Computed B polynomial')
The argument is as follows. The prover wants to prove \(P(1) = input\) and \(P(last\_step) = output\). If we take \(I(x)\) as the interpolant - the line that crosses the two points \((1, input)\) and
\((last\_step, output)\), then \(P(x) - I(x)\) would be equal to zero at those two points. Thus, it suffices to prove that \(P(x) - I(x)\) is a multiple of \((x - 1) \cdot (x - last\_step)\), and we
do that by... providing the quotient!
Purple: computational trace polynomial (P). Green: interpolant (I) (notice how the interpolant is constructed to equal the input (which should be the first step of the computational trace) at x=1 and
the output (which should be the last step of the computational trace) at \(x=g^{steps-1}\). Red: \(P - I\). Yellow: the minimal polynomial that equals \(0\) at \(x=1\) and \(x=g^{steps-1}\) (that is,
\(Z_2\)). Pink: \(\frac{P - I}{Z_2}\).
Challenge Suppose you wanted to also prove that the value in the computational trace after the 703rd computational step is equal to 8018284612598740. How would you modify the above algorithm to
do that?
Mouseover below for answer
Set \(I(x)\) to be the interpolant of \((1, input), (g^{703}, 8018284612598740), (last\_step, output)\), and make a proof by providing the quotient \(B(x) = \frac{P(x) - I(x)}{(x - 1) \cdot (x -
g^{703}) \cdot (x - last\_step)}\)
Now, we commit to the Merkle root of \(P\), \(D\) and \(B\) combined together.
# Compute their Merkle roots
mtree = merkelize([pval.to_bytes(32, 'big') +
dval.to_bytes(32, 'big') +
bval.to_bytes(32, 'big') for
pval, dval, bval in zip(p_evaluations, d_evaluations, b_evaluations)])
print('Computed hash root')
Now, we need to prove that \(P\), \(D\) and \(B\) are all actually polynomials, and of the right max-degree. But FRI proofs are big and expensive, and we don't want to have three FRI proofs. So
instead, we compute a pseudorandom linear combination of \(P\), \(D\) and \(B\) (using the Merkle root of \(P\), \(D\) and \(B\) as a seed), and do an FRI proof on that:
k1 = int.from_bytes(blake(mtree[1] + b'\x01'), 'big')
k2 = int.from_bytes(blake(mtree[1] + b'\x02'), 'big')
k3 = int.from_bytes(blake(mtree[1] + b'\x03'), 'big')
k4 = int.from_bytes(blake(mtree[1] + b'\x04'), 'big')
# Compute the linear combination. We don't even bother calculating it
# in coefficient form; we just compute the evaluations
root_of_unity_to_the_steps = f.exp(root_of_unity, steps)
powers = [1]
for i in range(1, precision):
powers.append(powers[-1] * root_of_unity_to_the_steps % modulus)
l_evaluations = [(d_evaluations[i] +
p_evaluations[i] * k1 + p_evaluations[i] * k2 * powers[i] +
b_evaluations[i] * k3 + b_evaluations[i] * powers[i] * k4) % modulus
for i in range(precision)]
Unless all three of the polynomials have the right low degree, it's almost impossible that a randomly selected linear combination of them will (you have to get extremely lucky for the terms to
cancel), so this is sufficient.
We want to prove that the degree of D is less than \(2 \cdot steps\), and that of \(P\) and \(B\) are less than \(steps\), so we actually make a random linear combination of \(P\), \(P \cdot x^
{steps}\), \(B\), \(B^{steps}\) and \(D\), and check that the degree of this combination is less than \(2 \cdot steps\).
Now, we do some spot checks of all of the polynomials. We generate some random indices, and provide the Merkle branches of the polynomial evaluated at those indices:
# Do some spot checks of the Merkle tree at pseudo-random coordinates, excluding
# multiples of `extension_factor`
branches = []
samples = spot_check_security_factor
positions = get_pseudorandom_indices(l_mtree[1], precision, samples,
for pos in positions:
branches.append(mk_branch(mtree, pos))
branches.append(mk_branch(mtree, (pos + skips) % precision))
branches.append(mk_branch(l_mtree, pos))
print('Computed %d spot checks' % samples)
The get_pseudorandom_indices function returns some random indices in the range [0...precision-1], and the exclude_multiples_of parameter tells it to not give values that are multiples of the given
parameter (here, extension_factor). This ensures that we do not sample along the original computational trace, where we are likely to get wrong answers.
The proof (~250-500 kilobytes altogether) consists of a set of Merkle roots, the spot-checked branches, and a low-degree proof of the random linear combination:
o = [mtree[1],
prove_low_degree(l_evaluations, root_of_unity, steps * 2, modulus, exclude_multiples_of=extension_factor)]
The largest parts of the proof in practice are the Merkle branches, and the FRI proof, which consists of even more branches. And here's the "meat" of the verifier:
for i, pos in enumerate(positions):
x = f.exp(G2, pos)
x_to_the_steps = f.exp(x, steps)
mbranch1 = verify_branch(m_root, pos, branches[i*3])
mbranch2 = verify_branch(m_root, (pos+skips)%precision, branches[i*3+1])
l_of_x = verify_branch(l_root, pos, branches[i*3 + 2], output_as_int=True)
p_of_x = int.from_bytes(mbranch1[:32], 'big')
p_of_g1x = int.from_bytes(mbranch2[:32], 'big')
d_of_x = int.from_bytes(mbranch1[32:64], 'big')
b_of_x = int.from_bytes(mbranch1[64:], 'big')
zvalue = f.div(f.exp(x, steps) - 1,
x - last_step_position)
k_of_x = f.eval_poly_at(constants_mini_polynomial, f.exp(x, skips2))
# Check transition constraints Q(x) = Z(x) * D(x)
assert (p_of_g1x - p_of_x ** 3 - k_of_x - zvalue * d_of_x) % modulus == 0
# Check boundary constraints B(x) * Z2(x) + I(x) = P(x)
interpolant = f.lagrange_interp_2([1, last_step_position], [inp, output])
zeropoly2 = f.mul_polys([-1, 1], [-last_step_position, 1])
assert (p_of_x - b_of_x * f.eval_poly_at(zeropoly2, x) -
f.eval_poly_at(interpolant, x)) % modulus == 0
# Check correctness of the linear combination
assert (l_of_x - d_of_x -
k1 * p_of_x - k2 * p_of_x * x_to_the_steps -
k3 * b_of_x - k4 * b_of_x * x_to_the_steps) % modulus == 0
At every one of the positions that the prover provides a Merkle proof for, the verifier checks the Merkle proof, and checks that \(C(P(x), P(g_1 \cdot x), K(x)) = Z(x) \cdot D(x)\) and \(B(x) \cdot
Z_2(x) + I(x) = P(x)\) (reminder: for \(x\) that are not along the original computation trace, \(Z(x)\) will not be zero, and so \(C(P(x), P(g_1 \cdot x), K(x))\) likely will not evaluate to zero).
The verifier also checks that the linear combination is correct, and calls verify_low_degree_proof(l_root, root_of_unity, fri_proof, steps * 2, modulus, exclude_multiples_of=extension_factor) to
verify the FRI proof. And we're done!
Well, not really; soundness analysis to prove how many spot-checks for the cross-polynomial checking and for the FRI are necessary is really tricky. But that's all there is to the code, at least if
you don't care about making even crazier optimizations. When I run the code above, we get a STARK proving "overhead" of about 300-400x (eg. a MIMC computation that takes 0.2 seconds to calculate
takes 60 second to prove), suggesting that with a 4-core machine computing the STARK of the MIMC computation in the forward direction could actually be faster than computing MIMC in the backward
direction. That said, these are both relatively inefficient implementations in python, and the proving to running time ratio for properly optimized implementations may be different. Also, it's worth
pointing out that the STARK proving overhead for MIMC is remarkably low, because MIMC is almost perfectly "arithmetizable" - it's mathematical form is very simple. For "average" computations, which
contain less arithmetically clean operations (eg. checking if a number is greater or less than another number), the overhead is likely much higher, possibly around 10000-50000x. | {"url":"https://0xe4ba0e245436b737468c206ab5c8f4950597ab7f.arb-nova.w3link.io/general/2018/07/21/starks_part_3.html","timestamp":"2024-11-13T18:48:18Z","content_type":"text/html","content_length":"81071","record_id":"<urn:uuid:ad93a57c-f38d-4dc4-85a9-3a9a5c89edb0>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00800.warc.gz"} |
I don't know what I've been trying to prove.
Yeah, yeah. Why I feel like I do about generalization theory.
This is a live blog of Lecture 6 (part 3 of 3) of my graduate machine learning class “Patterns, Predictions, and Actions.” A Table of Contents for this series is here.
Since the generalization blogs this week have gotten little reaction on the interwebs, I was quite surprised by the excited conversations in class yesterday. Because I’m rather disappointed after
twenty years of thinking about generalization theory, it’s hard to write the uniform convergence bound on the board with a straight face. But the class pressed me to explain in detail why I was so
skeptical, and I think many made compelling arguments that I was being too harsh.
There is something wondrous about applying the Law of Large Numbers and concluding you can look at an infinite set of hypotheses and still have the empirical errors all be close to the errors on data
you are yet to see. This sort of inductive power looks incredible, and that it follows from undergraduate probability is kind of amazing. I’ll admit it: my mind is still regularly blown by the
concentration of measure phenomenon. But it is not clear that these probabilistic bounds help in machine learning because they assume an incorrect model of data generation.
After twenty years of being fascinated by the mathematics, I’m convinced generalization theory is almost entirely useless. I spent countless hours learning about covering numbers and normed
inequalities in Banach Spaces. But the guidance that comes out of this mathematics for machine learners just seems to be bad advice. And if your theory gives bad advice, perhaps we should drop the
Let me dive into the first fallacy that bothers me. Much of generalization theory, including bounds based on uniform convergence and algorithmic stability, begins with the notion that empirical error
(the error on the data I’ve seen) should be close to population error (the data I haven’t seen). The last decade of machine learning practice has emphatically demonstrated that this is not a
The best machine learning models have zero empirical error. Telling people that the empirical error is a good proxy for the population error is counterproductive advice. But if the empirical error is
always zero, what do these generalization theory bounds tell us to do? They say you should pick the “smallest” function space that gets the empirical error to zero.
But this is also terrible, counterproductive advice for practice. All of these models get zero empirical error on the CIFAR10 benchmark:
• A look-up table
• A quadratic polynomial
• A 2-layer fully connected neural net
• An Alexnet model
• A Wide ResNet
These function spaces are of increasing complexity. And all of them can fit any data pattern you’d like with 50,000 examples. But I have ranked them by their test error. The look-up table gets 90%
test error. The quadratic fit gets 64% test error. The wide ResNet get 2% test error. Why? Generalization theory has no answers.
OK, now you might point to the margin-esque generalization bounds like we proved using online-to-batch conversion. But even for linear problems, there is rarely a correlation between the post-hoc
margin measured after training and the performance on a benchmark. The margin bounds are just upper bounds, and they don’t help us other than to say that the error should go to zero if we can get
more data to train on.
This seems to be the only generally valid advice from generalization theory: “larger n should yield smaller population error.” I suppose this is more or less true. But how much complex geometric
functional analysis do we need to prove that more data is more better?
The main goal of applied math is to guide practice. We want theories that, while not perfect, give reasonable guidelines. But the advice from generalization theory just seems bad. I swear that all of
the following bullets were lessons from my first ML class 20 years ago and are still common in popular textbooks:
• If you perfectly interpolate your training data, you won’t generalize.
• High-capacity models don’t generalize.
• You have to regularize to get good test error.
• Some in-sample errors can reduce out-of-sample error.
• Good prediction balances bias and variance.
• You shouldn’t evaluate a holdout set too many times or you’ll overfit
This is all terrible advice!
If our theory gives bad advice for practice, we have to come clean and admit our theory has failed.
Why machine learning theory failed is a fascinating story for another day. The issue with a lot of statistical theory is that we take metaphysical beliefs (e.g., data is i.i.d.), turn these beliefs
into mathematical axioms, and then forget that these beliefs weren’t particularly grounded in the first place. What is the value of a bunch of theorems based on faulty axioms?
Rather than answering that question, let me end this blog with a proposed path out of this mess. What does “work” in practice? It’s hard to argue against this four-step procedure:
1. Collect as large a data set as you can
2. Split this data set into a training set and a test set
3. Find as many models as you can that interpolate the training set
4. Of all of these models, choose the model that minimizes the error on the test set
This method has been tried and true since 1962. You can say that step 4 is justified by the law of large numbers. Maybe that’s right. But there’s still a lot of magic happening in step 3.
I restructured the machine learning theory class this semester to front-load the theory for linear functions. I think it’s important to at least see the standard theory so we can discuss why it tells
us very little about practice. But the second quarter of this class will now try to figure out how ML practice works. We’ll turn to feature generation and try to understand how we decide on
representations. From there, we’ll discuss what happens when we try to fit models by iteratively tweaking these features. And then we’ll return to the resilience of the train-test paradigm. Let us
get swept into the depths.
On Twitter, Lily-Belle Sweet brought up a great question. How do we justify leave-one-out error or the holdout method or cross validation without an appeal to i.i.d.?
Here’s my take on this. When we collect a training data set, it serves as a population itself. If we subsample from this training data, then the i.i.d. assumption holds because we are being
intentional. Hence, bootstrap methods are telling us something about internal validity. They are telling us something about how the method performs when the superpopulation is our training set.
Then, to generalize beyond this to new data, we just have to convince ourselves that the training set is a representative sample of the data we’ll see.
Expand full comment
"If our theory gives bad advice for practice, we have to come clean and admit our theory has failed."
I agree with this, of course. It is important to take the next step though and state that we need new theory, consistent with empirical evidence.
Btw, I also find uniform-type bounds quite beautiful. That is a big part of their appeal, I suppose.
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Cite as
Vikrant Ashvinkumar, Aaron Bernstein, Nairen Cao, Christoph Grunau, Bernhard Haeupler, Yonggang Jiang, Danupon Nanongkai, and Hsin-Hao Su. Parallel, Distributed, and Quantum Exact Single-Source
Shortest Paths with Negative Edge Weights. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 13:1-13:15, Schloss
Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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author = {Ashvinkumar, Vikrant and Bernstein, Aaron and Cao, Nairen and Grunau, Christoph and Haeupler, Bernhard and Jiang, Yonggang and Nanongkai, Danupon and Su, Hsin-Hao},
title = {{Parallel, Distributed, and Quantum Exact Single-Source Shortest Paths with Negative Edge Weights}},
booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-338-6},
ISSN = {1868-8969},
year = {2024},
volume = {308},
editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.13},
URN = {urn:nbn:de:0030-drops-210849},
doi = {10.4230/LIPIcs.ESA.2024.13},
annote = {Keywords: Parallel algorithm, distributed algorithm, shortest paths} | {"url":"https://drops.dagstuhl.de/search/documents?author=Su,%20Hsin-Hao","timestamp":"2024-11-09T11:08:58Z","content_type":"text/html","content_length":"90186","record_id":"<urn:uuid:9dc88040-06f8-4084-a001-5e4875c58bdc>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00622.warc.gz"} |
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universe. A computer with 79 qubits has already been built. | {"url":"https://www.pgs.com/company/newsroom/news/quantum-computing-the-next-big-thing-for-oil-exploration/","timestamp":"2024-11-07T15:17:16Z","content_type":"text/html","content_length":"119749","record_id":"<urn:uuid:7feb512f-3b4f-4974-857a-d047aa474c06>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00190.warc.gz"} |
Perform polyphase FIR interpolation
The dsp.FIRInterpolator System object™ performs an efficient polyphase interpolation using an integer upsampling factor L along the first dimension.
Conceptually, the FIR interpolator (as shown in the schematic) consists of an upsampler followed by an FIR anti-imaging filter, which is usually an approximation of an ideal band-limited
interpolation filter. The coefficients of the anti-imaging filter can be specified through the Numerator property, or can be automatically designed by the object using the designMultirateFIR
The upsampler upsamples each channel of the input to a higher rate by inserting L–1 zeros between samples. The FIR filter that follows filters each channel of the upsampled data. The resulting
discrete-time signal has a sample rate that is L times the original sample rate.
Note that the actual object algorithm implements a direct-form FIR polyphase structure, an efficient equivalent of the combined system depicted in the diagram. For more details, see Algorithms.
To upsample an input:
1. Create the dsp.FIRInterpolator object and set its properties.
2. Call the object with arguments, as if it were a function.
To learn more about how System objects work, see What Are System Objects?
This object supports C/C++ code generation and SIMD code generation under certain conditions. For more information, see Code Generation.
firinterp = dsp.FIRInterpolator returns an FIR interpolator with an interpolation factor of 3. The object designs the FIR filter coefficients using the designMultirateFIR(3,1) function.
firinterp = dsp.FIRInterpolator(L) returns an FIR interpolator with the integer-valued InterpolationFactor property set to L. The object designs its filter coefficients based on the interpolation
factor L that you specify while creating the object using the designMultirateFIR(L,1) function. The designed filter corresponds to a lowpass with a cutoff at π/L in radial frequency units.
firinterp = dsp.FIRInterpolator(L,'Auto') returns an FIR interpolator with the NumeratorSource property set to 'Auto'. In this mode, every time there is an update in the interpolation factor, the
object redesigns the filter using the design method specified in DesignMethod.
firinterp = dsp.FIRInterpolator(L,num) returns an FIR interpolator with the InterpolationFactor property set to L and the Numerator property set to num.
firinterp = dsp.FIRInterpolator(L,method) returns an FIR interpolator with the InterpolationFactor property set to L and the DesignMethod property set to method. When you pass the design method as an
input, the NumeratorSource property is automatically set to 'Auto'.
firinterp = dsp.FIRInterpolator(___,Name,Value) returns an FIR interpolator object with each specified property set to the specified value. Enclose each property name in quotes. You can use this
syntax with any previous input argument combinations.
firinterp = dsp.FIRInterpolator(L,'legacy') returns an FIR interpolator where the filter coefficients are designed using fir1(15,0.25). The designed filter has a cutoff frequency of 0.25π radians/
Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.
If a property is tunable, you can change its value at any time.
For more information on changing property values, see System Design in MATLAB Using System Objects.
InterpolationFactor — Interpolation factor
3 (default) | positive integer
Specify the integer factor, L, by which to increase the sampling rate of the input signal. The polyphase implementation uses L polyphase subfilters to compute convolutions at the lower sample rate.
The FIR interpolator delays and interleaves these lower-rate convolutions to obtain the higher-rate output.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
NumeratorSource — FIR filter coefficient source
'Property' (default) | 'Input port' | 'Auto'
FIR filter coefficient source, specified as either:
• 'Property' –– The numerator coefficients are specified through the Numerator property.
• 'Input port' –– The numerator coefficients are specified as an input to the object algorithm.
• 'Auto' –– The numerator coefficients are designed automatically using the design method specified in DesignMethod.
Numerator — FIR filter coefficients
designMultirateFIR(L,1) (default) | row vector
Numerator coefficients of the anti-imaging FIR filter, specified as a row vector in powers of z^–1. The following equation defines the system function for a filter of length N+1:
$H\left(z\right)=\sum _{n=0}^{N}b\left(n\right){z}^{-n}$
The vector b = [b(0), b(1), …, b(N)] represents the vector of filter coefficients.
To act as an effective anti-imaging filter, the coefficients usually correspond to a lowpass filter with a normalized cutoff frequency no greater than the reciprocal of the InterpolationFactor. Use
designMultirateFIR to design such a filter. More generally, any complex bandpass filter can be used. For an example, see Double the Sample Rate Using FIR Interpolator.
The filter coefficients are scaled by the value of the InterpolationFactor property before filtering the signal. To form the L polyphase subfilters, Numerator is appended with zeros if necessary.
This property is visible only when you set NumeratorSource to 'Property'.
When NumeratorSource is set to 'Auto', the numerator coefficients are automatically redesigned using the design method specified in DesignMethod. To access the filter coefficients in the automatic
design mode, type objName.Numerator in the MATLAB^® command prompt.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Complex Number Support: Yes
DesignMethod — Auto design method
'Kaiser' (default) | 'ZOH' | 'Linear'
Design method of the FIR filter coefficients, specified as one of the following:
• 'Kaiser' –– Kaiser method. Approximate anti-imaging lowpass filter using the designMultirateFIR function.
• 'ZOH' –– Zero order hold method. Hold the input sequence values.
• 'Linear' –– Linear interpolation method.
This property is visible only when you set the NumeratorSource property to 'Auto', or if you pass the 'auto' keyword as an input while creating the object.
Fixed-Point Properties
FullPrecisionOverride — Full-precision override for fixed-point arithmetic
true (default) | false
Flag to use full-precision rules for fixed-point arithmetic, specified as one of the following:
• true –– The object computes all internal arithmetic and output data types using the full-precision rules. These rules provide the most accurate fixed-point numerics. In this mode, other
fixed-point properties do not apply. No quantization occurs within the object. Bits are added, as needed, to ensure that no roundoff or overflow occurs.
• false –– Fixed-point data types are controlled through individual fixed-point property settings.
For more information, see Full Precision for Fixed-Point System Objects and Set System Object Fixed-Point Properties.
RoundingMethod — Rounding method for fixed-point operations
'Floor' (default) | 'Ceiling' | 'Convergent' | 'Nearest' | 'Round' | 'Simplest' | 'Zero'
Rounding method for fixed-point operations. For more details, see rounding mode.
This property is not visible and has no effect on the numerical results when the following conditions are met:
• FullPrecisionOverride set to true.
• FullPrecisionOverride set to false, ProductDataType set to 'Full precision', AccumulatorDataType set to 'Full precision', and OutputDataType set to 'Same as accumulator'.
Under these conditions, the object operates in full precision mode.
OverflowAction — Overflow action for fixed-point operations
'Wrap' (default) | 'Saturate'
Overflow action for fixed-point operations, specified as one of the following:
• 'Wrap' –– The object wraps the result of its fixed-point operations.
• 'Saturate' –– The object saturates the result of its fixed-point operations.
For more details on overflow actions, see overflow mode for fixed-point operations.
This property is not visible and has no effect on the numerical results when the following conditions are met:
• FullPrecisionOverride set to true.
• FullPrecisionOverride set to false, OutputDataType set to 'Same as accumulator', ProductDataType set to 'Full precision', and AccumulatorDataType set to 'Full precision'
Under these conditions, the object operates in full precision mode.
CoefficientsDataType — Data type of FIR filter coefficients
Same word length as input (default) | Custom
Data type of the FIR filter coefficients, specified as:
• Same word length as input –– The word length of the coefficients is the same as that of the input. The fraction length is computed to give the best possible precision.
• Custom –– The coefficients data type is specified as a custom numeric type through the CustomCoefficientsDataType property.
CustomCoefficientsDataType — Word and fraction lengths of coefficients data type
numerictype([],16,15) (default) | custom numeric type
Word and fraction lengths of the coefficients data type, specified as an autosigned numerictype (Fixed-Point Designer) with a word length of 16 and a fraction length of 15.
This property applies when you set the CoefficientsDataType property to Custom.
ProductDataType — Data type of product output
'Full precision' (default) | 'Custom' | 'Same as input'
Data type of the product output in this object, specified as one of the following:
• 'Full precision' –– The product output data type has full precision.
• 'Same as input' –– The object specifies the product output data type to be the same as that of the input data type.
• 'Custom' –– The product output data type is specified as a custom numeric type through the CustomProductDataType property.
For more information on the product output data type, see Multiplication Data Types.
This property applies when you set FullPrecisionOverride to false.
CustomProductDataType — Word and fraction lengths of product data type
numerictype([],32,30) (default) | custom numeric type
Word and fraction lengths of the product data type, specified as an autosigned numeric type with a word length of 32 and a fraction length of 30.
This property applies only when you set FullPrecisionOverride to false and ProductDataType to 'Custom'.
AccumulatorDataType — Data type of accumulation operation
'Full precision' (default) | 'Same as input' | 'Same as product' | 'Custom'
Data type of an accumulation operation in this object, specified as one of the following:
• 'Full precision' –– The accumulation operation has full precision.
• 'Same as product' –– The object specifies the accumulator data type to be the same as that of the product output data type.
• 'Same as input' –– The object specifies the accumulator data type to be the same as that of the input data type.
• 'Custom' –– The accumulator data type is specified as a custom numeric type through the CustomAccumulatorDataType property.
This property applies when you set FullPrecisionOverride to false.
CustomAccumulatorDataType — Word and fraction lengths of accumulator data type
numerictype([],32,30) (default) | custom numeric type
Word and fraction lengths of the accumulator data type, specified as an autosigned numeric type with a word length of 32 and a fraction length of 30.
This property applies only when you set FullPrecisionOverride to false and AccumulatorDataType to 'Custom'.
OutputDataType — Data type of object output
'Same as accumulator' (default) | 'Same as input' | 'Same as product' | 'Custom'
Data type of the object output, specified as one of the following:
• 'Same as accumulator' –– The output data type is the same as that of the accumulator output data type.
• 'Same as input' –– The output data type is the same as that of the input data type.
• 'Same as product' –– The output data type is the same as that of the product output data type.
• 'Custom' –– The output data type is specified as a custom numeric type through the CustomOutputDataType property.
This property applies when you set FullPrecisionOverride to false.
CustomOutputDataType — Word and fraction lengths of output data type
numerictype([],16,15) (default) | custom numeric type
Word and fraction lengths of the output data type, specified as an autosigned numeric type with a word length of 16 and a fraction length of 15.
This property applies only when you set FullPrecisionOverride to false and OutputDataType to 'Custom'.
y = firinterp(x) interpolates the input signal x along the first dimension, and outputs the upsampled and filtered values, y.
y = firinterp(x,num) uses the FIR filter, num, to interpolate the input signal. This configuration is valid only when the 'NumeratorSource' property is set to 'Input port'.
Input Arguments
x — Data input
vector | matrix
Data input, specified as a vector or a matrix. A P-by-Q input matrix is treated as Q independent channels, and the System object interpolates each channel over the first dimension and generates a P*L
-by-Q output matrix, where L is the interpolation factor.
This object supports variable-size input and does not support complex unsigned fixed-point inputs.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi
Complex Number Support: Yes
num — FIR filter coefficients
row vector
FIR filter coefficients, specified as a row vector.
This input is accepted only when the 'NumeratorSource' property is set to 'Input port'.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi
Complex Number Support: Yes
Output Arguments
y — FIR interpolator output
vector | matrix
FIR interpolator output, returned as a vector or a matrix of size P*L-by-Q, where L is the interpolation factor.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi
Complex Number Support: Yes
Object Functions
To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named obj, use this syntax:
Specific to dsp.FIRInterpolator
freqz Frequency response of discrete-time filter System object
freqzmr Compute DTFT approximation of impulse response of multirate or single-rate filter
filterAnalyzer Analyze filters with Filter Analyzer app
info Information about filter System object
cost Estimate cost of implementing filter System object
polyphase Polyphase decomposition of multirate filter
generatehdl Generate HDL code for quantized DSP filter (requires Filter Design HDL Coder)
impz Impulse response of discrete-time filter System object
coeffs Returns the filter System object coefficients in a structure
outputDelay Determine output delay of single-rate or multirate filter
Common to All System Objects
step Run System object algorithm
release Release resources and allow changes to System object property values and input characteristics
reset Reset internal states of System object
Interpolate a Cosine Wave
Interpolate a cosine wave by a factor of 2. In the automatic filter design mode, change the underlying D/A signal interpolation model to 'linear' and interpolate the signal by a factor of 4, change
the underlying D/A signal interpolation model to 'ZOH' and interpolate the signal by a factor of 5.
The cosine wave has an angular frequency of $\frac{\pi }{4}$ radians/sample.
Design Default Filter
Create a dsp.FIRInterpolator object. The object uses an anti-imaging lowpass filter after upsampling. By default, the anti-imaging lowpass filter is designed using the designMultirateFIR function.
The function designs the filter based on the interpolation factor that you specify, and stores the coefficients in the Numerator property. For an interpolation factor of 2, the object designs the
coefficients using designMultirateFIR(2,1).
firinterp = dsp.FIRInterpolator(2)
firinterp =
dsp.FIRInterpolator with properties:
InterpolationFactor: 2
NumeratorSource: 'Property'
Numerator: [0 -2.0108e-04 0 7.7408e-04 0 -0.0020 0 0.0045 0 -0.0086 0 0.0153 0 -0.0257 0 0.0415 0 -0.0661 0 0.1084 0 -0.2003 0 0.6326 1 0.6326 0 -0.2003 0 0.1084 0 -0.0661 0 0.0415 0 -0.0257 0 0.0153 0 -0.0086 0 0.0045 0 ... ] (1x48 double)
Use get to show all properties
Visualize the filter response. The designed filter meets the ideal filter constraints that are marked in red. The cutoff frequency is approximately half the spectrum.
Interpolate by 2
Interpolate the cosine signal by a factor of 2.
Plot the original and interpolated signals. In order to plot the two signals on the same plot, you must account for the output delay of the FIR interpolator and the scaling introduced by the filter.
Use the outputDelay function to compute the delay value introduced by the interpolator. Shift the output by this delay value.
Visualize the input and the resampled signals. The input and output values coincide every other sample, due to the interpolation factor of 2.
[delay,FsOut] = outputDelay(firinterp)
nx = (0:length(x)-1);
ty = (0:length(y)-1)/FsOut-delay;
stem(ty,y,'filled',MarkerSize=4); hold on;
stem(nx,x); hold off;
ylim([-2.5 2.5])
legend('Interpolated by 2','Input signal','Location','best');
Interpolate by 4 in Automatic Filter Design Mode
Now interpolate by a factor of 4. In order for the filter design to be updated automatically based on the new interpolation factor, set the NumeratorSource property to 'Auto'. Alternately, you can
pass the keyword 'Auto' as an input while creating the object. The object then operates in the automatic filter design mode. Every time there is a change in the interpolation factor, the object
updates the filter design.
firinterp.NumeratorSource = 'Auto';
firinterp.InterpolationFactor = 4
firinterp =
dsp.FIRInterpolator with properties:
InterpolationFactor: 4
NumeratorSource: 'Auto'
DesignMethod: 'Kaiser'
Use get to show all properties
To access the filter coefficients in the automatic filter design mode, type firinterp.Numerator in the MATLAB command prompt.
The designed filter occupies a narrower passband that is approximately a quarter of the spectrum.
Interpolate the cosine signal by a factor of 4.
Plot the original and resampled signals. Recalculate the delay and the output sample rate values since the interpolation factor has changed. The input and output values coincide every 4 output
samples, owing to the interpolation factor of 4.
[delay,FsOut] = outputDelay(firinterp);
nx = (0:length(x)-1);
tyAuto = (0:length(yAuto)-1)/FsOut-delay;
stem(tyAuto,yAuto,'filled',MarkerSize=4); hold on;
stem(nx,x); hold off;
ylim([-2.5 2.5])
legend('Interpolated by 4','Input signal');
Specify Signal Interpolation Model
In the automatic design mode, you can also specify the underlying D/A signal interpolation model through the DesignMethod property.
Set DesignMethod to 'linear'
If you set the DesignMethod to 'linear', the object uses the linear interpolation model.
firinterp.DesignMethod = 'linear'
firinterp =
dsp.FIRInterpolator with properties:
InterpolationFactor: 4
NumeratorSource: 'Auto'
DesignMethod: 'Linear'
Use get to show all properties
Interpolate the signal using the linear interpolation model.
Plot the original and the linearly interpolated signal.
[delay,FsOut] = outputDelay(firinterp);
nx = (0:length(x)-1);
% Calculate output times for vector ylinear in input units
tylinear = (0:length(ylinear)-1)/FsOut-delay;
stem(tylinear,ylinear,'filled',MarkerSize=4); hold on;
hold off;
ylim([-2.5 2.5])
legend('Linear Interpolation by 4','Input signal');
Set DesignMethod to 'ZOH' and Change InterpolationFactor to 5
If you set the DesignMethod to 'ZOH', the object uses the zero order hold method. Change the interpolation factor to 5.
firinterp.DesignMethod = 'ZOH';
firinterp.InterpolationFactor = 5
firinterp =
dsp.FIRInterpolator with properties:
InterpolationFactor: 5
NumeratorSource: 'Auto'
DesignMethod: 'ZOH'
Use get to show all properties
Interpolate the signal using the zero order hold method.
Plot the original and ZOH interpolated signal.
[delay,FsOut] = outputDelay(firinterp);
nx = (0:length(x)-1);
% Calculate output times for vector yzoh in input units
tyzoh = (0:length(yzoh)-1)/FsOut-delay;
stem(tyzoh,yzoh,'filled',MarkerSize=4); hold on;
stem(nx,x); hold off;
ylim([-1.5 1.5])
legend('ZOH Interpolation by 4','Input signal');
Double the Sample Rate Using FIR Interpolator
Double the sample rate of an audio signal and play the interpolated signal using the audioDeviceWriter object.
Note: The audioDeviceWriter System object™ is not supported in MATLAB Online.
Create a dsp.AudioFileReader object. The default audio file ready by the object has a sample rate of 22050 Hz.
afr = dsp.AudioFileReader('OutputDataType',...
Create a dsp.FIRInterpolator object and specify the interpolation factor to be 2. The object designs the filter using the designMultirateFIR(2,1) function and stores the coefficients in the Numerator
property of the object.
firInterp = dsp.FIRInterpolator(2)
firInterp =
dsp.FIRInterpolator with properties:
InterpolationFactor: 2
NumeratorSource: 'Property'
Numerator: [0 -2.0108e-04 0 7.7408e-04 0 -0.0020 0 0.0045 0 -0.0086 0 0.0153 0 -0.0257 0 0.0415 0 -0.0661 0 0.1084 0 -0.2003 0 0.6326 1 0.6326 0 -0.2003 0 0.1084 0 -0.0661 0 0.0415 0 -0.0257 0 0.0153 0 -0.0086 0 0.0045 0 ... ] (1x48 double)
Use get to show all properties
Create an audioDeviceWriter object. Specify the sample rate to be 22050$×$2, which equals 44100 Hz.
adw = audioDeviceWriter(44100)
adw =
audioDeviceWriter with properties:
Device: 'Default'
SampleRate: 44100
Use get to show all properties
Read the audio signal using the file reader object, double the sample rate of the signal from 22050 Hz to 44100 Hz and play the interpolated signal.
while ~isDone(afr)
frame = afr();
y = firInterp(frame);
The FIR interpolation filter is implemented efficiently using a polyphase structure.
To derive the polyphase structure, start with the transfer function of the FIR filter:
N+1 is the length of the FIR filter.
You can rearrange this equation as follows:
$H\left(z\right)=\begin{array}{c}\left({b}_{0}+{b}_{L}{z}^{-L}+{b}_{2L}{z}^{-2L}+..+{b}_{N-L+1}{z}^{-\left(N-L+1\right)}\right)+\\ {z}^{-1}\left({b}_{1}+{b}_{L+1}{z}^{-L}+{b}_{2L+1}{z}^{-2L}+..+{b}_
{N-L+2}{z}^{-\left(N-L+1\right)}\right)+\\ \begin{array}{c}⋮\\ {z}^{-\left(L-1\right)}\left({b}_{L-1}+{b}_{2L-1}{z}^{-L}+{b}_{3L-1}{z}^{-2L}+..+{b}_{N}{z}^{-\left(N-L+1\right)}\right)\end{array}\end
L is the number of polyphase components, and its value equals the interpolation factor that you specify.
You can write this equation as:
E[0](z^L), E[1](z^L), ..., E[L-1](z^L) are polyphase components of the FIR filter H(z).
Conceptually, the FIR interpolation filter contains an upsampler followed by an FIR lowpass filter H(z).
Replace H(z) with its polyphase representation.
Here is the multirate noble identity for interpolation.
Applying the noble identity for interpolation moves the upsampling operation to after the filtering operation. This move enables you to filter the signal at a lower rate.
You can replace the upsampling operator, delay block, and adder with a commutator switch. The switch starts on the first branch 0 and moves in the counterclockwise direction, each time receiving one
sample from each branch. The interpolator effectively outputs L samples for every one input sample it receives. Hence the sample rate at the output of the FIR interpolation filter is Lfs.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
See System Objects in MATLAB Code Generation (MATLAB Coder).
The dsp.FIRInterpolator System object supports SIMD code generation using Intel^® AVX2 code replacement library under these conditions:
• Input signal is real-valued with real filter coefficients.
• Input signal is complex-valued with real or complex filter coefficients.
• Input signal has a data type of single or double.
The SIMD technology significantly improves the performance of the generated code. For more information, see SIMD Code Generation. To generate SIMD code from this object, see Use Intel AVX2 Code
Replacement Library to Generate SIMD Code from MATLAB Algorithms.
HDL Code Generation
Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.
This object supports HDL code generation with the Filter Design HDL Coder™ product. For workflows and limitations, see Generate HDL Code for Filter System Objects (Filter Design HDL Coder).
Version History
Introduced in R2012a
See Also | {"url":"https://it.mathworks.com/help/dsp/ref/dsp.firinterpolator-system-object.html","timestamp":"2024-11-14T04:44:37Z","content_type":"text/html","content_length":"173764","record_id":"<urn:uuid:f27334c0-e17f-4b6e-95b3-85e9ba263715>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00699.warc.gz"} |
Google Sheets: Conditional Formatting from Another Sheet | Online Tutorials Library List | Tutoraspire.com
Google Sheets: Conditional Formatting from Another Sheet
by Tutor Aspire
You can use the custom formula function in Google Sheets to apply conditional formatting based on a cell value from another sheet.
The following example shows how to use the custom formula function in practice.
Example: Conditional Formatting Based on Another Sheet
Suppose we have the following dataset in Sheet1 that shows the total points scored by various basketball teams:
And suppose we have the following dataset in Sheet2 that shows the total points allowed by the same list of teams:
Suppose we’d like to highlight each of the cells in the Team column of Sheet1 if the value in the Points Scored column is greater than the value in the Points Allowed column in Sheet2.
To do so, we can highlight the cells in the range A2:A11, then click the Format tab, then click Conditional formatting:
In the Conditional format rules panel that appears on the right side of the screen, click the Format cells if dropdown, then choose Custom formula is, then type in the following formula:
Note: It’s important that you include the equal sign (=) at the beginning of the formula, otherwise the conditional formatting won’t work.
Once you click Done, each of the cells in the Team column where the value in the Points Scored column is greater than the value in the Points Allowed column in Sheet2Â will be highlighted with a
green background:
The only Team values that have a green background are the ones where the value in the Points Scored column in Sheet1 is greater than the Points Allowed column in Sheet2.
Note that if the sheet name you’re referencing has spaces in the name, be sure to include single quotes around the sheet name.
For example, if your sheet is called Sheet 2 then you should use the following syntax when defining the formatting rule:
=B2>INDIRECT("'Sheet 2'!B2")
Additional Resources
The following tutorials explain how to perform other common tasks in Google Sheets:
Google Sheets: Conditional Formatting If Date is Before Today
Google Sheets: Conditional Formatting with Multiple Conditions
Google Sheets: Conditional Formatting if Another Cell Contains Text
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Mathematical Reasoning: Writing and Proof | Programming Valley
Mathematical Reasoning: Writing and Proof
Published January, 2023
Mathematical Reasoning: Writing and Proof by Ted Sundstrom is a comprehensive guide that delves into the art of mathematical reasoning and its connection to effective writing and proof construction.
With a clear and accessible approach, Sundstrom skillfully bridges the gap between theoretical mathematics and practical application, equipping readers with the tools needed to think critically,
analyze problems, and communicate mathematical ideas effectively.
In this illuminating book, Sundstrom emphasizes the importance of writing in the field of mathematics. He highlights how writing enhances the understanding of mathematical concepts and promotes
clarity of thought. By emphasizing the process of constructing coherent and rigorous proofs, the author encourages readers to think deeply about mathematical arguments and develop their own writing
The book is structured to provide a balanced blend of theory and practice. Sundstrom introduces fundamental concepts such as logic, set theory, and mathematical induction, while emphasizing the role
of clear and logical exposition. Through numerous examples, exercises, and thought-provoking problems, readers are given ample opportunities to apply the concepts they learn, strengthen their
mathematical reasoning skills, and enhance their ability to write mathematical proofs.
Sundstrom’s approachable writing style and engaging examples make this book suitable for a wide range of readers, from undergraduate mathematics students to self-study enthusiasts. Whether you are
new to proof writing or seeking to refine your skills, Mathematical Reasoning: Writing and Proof offers a solid foundation for honing your ability to think critically and present mathematical
arguments with precision.
As readers progress through the book, they will encounter a variety of proof techniques and strategies, including direct proofs, proof by contradiction, mathematical induction, and proof by cases.
Sundstrom demonstrates how these techniques can be effectively applied to different branches of mathematics, enabling readers to tackle a wide range of mathematical problems with confidence.
Moreover, the book addresses common pitfalls and misconceptions that students often encounter when learning to write mathematical proofs. Sundstrom provides insightful guidance on avoiding logical
fallacies, distinguishing between necessary and sufficient conditions, and effectively using mathematical notation to convey ideas accurately.
With Mathematical Reasoning: Writing and Proof, Ted Sundstrom invites readers into the fascinating world of mathematical reasoning and equips them with the necessary tools to become adept at writing
and constructing rigorous mathematical proofs. By emphasizing the importance of clear exposition and logical thinking, this book empowers readers to become proficient mathematical communicators and
critical thinkers, paving the way for success in advanced mathematics and related fields. | {"url":"https://programmingvalley.com/book/mathematical-reasoning-writing-and-proof/","timestamp":"2024-11-04T18:49:12Z","content_type":"text/html","content_length":"97697","record_id":"<urn:uuid:206f4d87-ba8e-4bd9-b8f5-e883bd28e16a>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00772.warc.gz"} |
Tom is playing at the opposite corner of the playing field to his friend . If the playing field is 180m wide and Tom is 300m away from his f - DocumenTVTom is playing at the opposite corner of the playing field to his friend . If the playing field is 180m wide and Tom is 300m away from his f
Tom is playing at the opposite corner of the playing field to his friend . If the playing field is 180m wide and Tom is 300m away from his f
Tom is playing at the opposite corner of the playing field to his friend . If the playing field is 180m wide and Tom is 300m away from his friend, how long is the playing field
in progress 0
Mathematics 3 years 2021-08-30T19:15:39+00:00 2021-08-30T19:15:39+00:00 1 Answers 21 views 0 | {"url":"https://documen.tv/question/tom-is-playing-at-the-opposite-corner-of-the-playing-field-to-his-friend-if-the-playing-field-is-21485117-92/","timestamp":"2024-11-06T15:15:04Z","content_type":"text/html","content_length":"79379","record_id":"<urn:uuid:9424c934-6ff8-4805-89e2-909f27eac08b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00399.warc.gz"} |
Natural transformation - (K-Theory) - Vocab, Definition, Explanations | Fiveable
Natural transformation
from class:
A natural transformation is a way of transforming one functor into another while preserving the structure of the categories involved. It provides a systematic way to relate different functors, making
it easier to study relationships between them, such as those found in K-Theory and cohomology. This concept is essential for understanding how transformations operate in the context of functorial
properties, basic constructions in KK-Theory, and reduced K-Theory.
congrats on reading the definition of natural transformation. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Natural transformations can be visualized as 'morphisms' between functors, allowing you to connect different mathematical structures.
2. In K-Theory, natural transformations help establish connections between different cohomological theories, showing how they can be transformed into one another.
3. The composition of natural transformations is associative, which means you can combine multiple transformations in a consistent manner.
4. Natural transformations are crucial for defining equivalences between categories and studying their properties within K-Theory.
5. When dealing with reduced K-Theory, natural transformations play a significant role in understanding suspension isomorphisms and their implications.
Review Questions
• How do natural transformations facilitate the relationship between functors in K-Theory?
□ Natural transformations allow us to systematically relate different functors within K-Theory by ensuring that the structure of categories is preserved. This means we can see how various
constructs like vector bundles and cohomology theories interact with each other. By providing a framework for connecting these functors, natural transformations enhance our understanding of
their underlying relationships and help us draw important conclusions about K-Theory.
• Discuss the implications of natural transformations in establishing connections between K-Theory and cohomology.
□ Natural transformations play a key role in linking K-Theory with cohomology by showing how one can transform between different cohomological theories. By using natural transformations,
mathematicians can demonstrate that various functors related to K-Theory behave similarly when applied to different topological spaces. This connection reveals deeper insights into how
algebraic structures correspond with topological properties, making it easier to study these complex relationships.
• Evaluate how natural transformations contribute to the basic constructions in KK-Theory and their significance in understanding reduced K-Theory.
□ Natural transformations are vital in KK-Theory as they help establish relationships between different constructions within this framework. They allow for the examination of how various
K-theoretical constructs can be transformed into one another while preserving their structural integrity. This becomes particularly significant when analyzing reduced K-Theory, where natural
transformations aid in understanding suspension isomorphisms and their applications, ultimately enhancing our grasp of K-Theory's overall landscape.
"Natural transformation" also found in:
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In the bag
Can you guess the colours of the 10 marbles in the bag? Can you develop an effective strategy for reaching 1000 points in the least number of rounds?
Ten marbles of varying colours are chosen at random and placed in a bag.
Can you guess the colours of the 10 marbles in the bag?
To help you make accurate predictions you can choose to see the results of 10 viewings - each viewing removes a marble out of the bag, records its colour and returns it to the bag before repeating
the process.
You can choose to perform as many 10 random viewings as you like before deciding to "Guess".
Each Round concludes when you have made a guess.
Can you reach a score of 1000 points?
You start with 500 points. Each run of 10 viewings "costs" you 10 points but you can "earn" points by predicting accurately the contents of the bag:
150 points for a perfect match
100 points for 9 matching marbles
0 points for 8 matching marbles
-150 points for 7 or fewer matching marbles.
Can you develop an effective strategy for reaching 1000 points?
Can you develop an effective strategy for reaching 1000 points in the least number of rounds?
Describe your strategies.
Getting Started
The sets of 10 marbles are chosen randomly, so could contain:
4 red, 3 green, 2 blue and 1 yellow or
4 red, 3 green and 3 yellow or
7 red and 3 blue or...
A run of 10 viewings may produce an outcome that is not representative of the actual contents of the bag.
Student Solutions
We received many solutions and ideas about this problem from schools and colleges including Robert Gordon College, Hawthorns School, Heckmondwike Grammar School and Wilson's School in the UK, the
Learning Enrichment Studio and the Frederick Irwin Anglican School in Australia, the Garden International School in Malaysia, PSBBMS in India and the British Vietnamese International School in Hanoi.
We were especially impressed by the answers we received which carefully explained the thinking behind your strategies. Well done, everyone!
Several responses commented on the fact that the distribution we obtain from taking random samples may not correspond to the actual distribution of the marbles. For example, if a sample of 10 marbles
did not contain a blue ball this did not necessarily mean that there was no blue in the bag.
Sydney and Maika from the Frederick Irwin Anglian School shared their strategies for overcoming this issue:
You have to draw at least 2 times to be confident enough.
We usually would draw 3 or 4 times.
One of the first things we checked was if only 3 colours showed we would have to check again.
Obviously, if it rarely appeared, that colour only has one or two marbles.
Briefly record the draws in your mind to see how frequently it shows up giving you a good idea of what to guess.
If there is many of the colour being pulled, then again, there must be about 3-5 of them in there.
Leia and Vihaga, also from Frederick Irwin, drew this table to help them to keep track of their results:
They discussed how their table supported their mathematically thinking:
In the above picture you can see a table. The top row represents the colours (R, B, Y, and G). I have recorded the amount of each coloured marble for every round. After three rounds, I then used the
fourth row to average out the amounts of marbles for each colour. I then typed in my 'averaged amounts' as my answers. I kept doing this until I reached 1000.
Miraya at Heckmondwike School also used averages to work towards 1000 points:
The way I did it was that first of all I used the random picker once which selected a random choice of colours and I worked out that if you use that once then using probability you work out the
average number of times the colour appears by having one trial session then resetting once you have your average. Using this method I managed to get up to 1000! I know that the results are not always
the same but it does give you an approximate guide to work with and leave the rest to guess.
Oscar from Wilson's School thought about how many times to perform 10 random viewings, using an important idea that the more viewings we average over, the more likely we are to get the correct
distribution. This is a very important concept in probability - it's worth giving it some thought if it doesn't make sense yet. A good resource for seeing this phenomenon in action is Which Spinners?
This is Oscar's work:
To reach 1000 points, you would want to view the bag as many times to get an accurate answer but spending the least amount of points at the same time.
In just one viewing, each marble has $\frac 9{10}$ [probability] of NOT being picked out.
If all 10 views also only have a $\frac9{10}$ chance, it has roughly $\frac13$ a chance of not being picked in that round. (because $\left(\frac{9}{10}\right)^{10}\approx\frac13$)
This means, to maximise all marbles to be picked at least once, it would be best to view the bag at least three times before each guess.
If only 1 marble of a colour was in the bag, it might only be picked once out of all 30 viewings, this means it would be very low in percentage compared to other colours. However, if it was picked at
least once, we know there cannot be none of that colour and should round upwards to 10%. When guessing after 3 viewings, it is best to round up or down to the closest tenth (exception for less than
10% but greater than 0% as it would have to round upwards). If there are too many for 10 marbles, the lower percentages may need to lose a marble on the guess. This has a high chance of getting at
least 9 correct (100 points) which results in 70 points gained. There are chances in 10 correct (150 points) and that results in 120 points gained. However if only 8 were correct, only 30 points
would be deducted. In the rare chance of less than 7 correct, 180 points will be deducted.
Teachers' Resources
Why do this problem?
This problem introduces students to the limitations of making predictions based on small samples, and how predictions can become more reliable as the number of experiments increases. The problem
challenges students to make predictions based on the information they gather. It offers students a chance to test various strategies for reaching the target in the most efficient way.
The can be used as a starting point for developing the skills of making and testing hypotheses. Students can suggest various possible strategies, then make decisions about the information that is
required to compare their efficiency and how to analyse the data that is collected.
Possible approach
Place 10 marbles (or counters, multilink cubes...) in a bag/envelope/hat and ask students if they can predict the colours of the marbles.
To help them make accurate predictions allow them to pick one marble, record its colour and return it to the bag before repeating the process.
After this has been done 10 times:
"Do you now know the colours of the 10 marbles?"
"If not, why not?"
"Is there anything we can be sure of?"
Record the results of another 10 viewings.
"Are the results the same/different?" "Why?"
"Can you now predict the colours of the 10 marbles?"
Repeat a few more times.
"What can you do with the different sets of results?"
When the class become fairly confident that they can predict the contents of the bag, show them what the bag contains.
Demonstrate the interactivity and clarify the scoring system.
"Can you develop an effective strategy for reaching 1000 points?"
"Can you develop an effective strategy for reaching 1000 points in the least number of rounds?"
Allow students some time to test their strategies using the interactivity.
Bring students back together (possibly in a follow-up lesson).
Discuss, refine and list possible strategies. Ask students (possibly working in small groups) to select different strategies to test.
• how they could test the effectiveness of their strategy
• what data they will need to collect
• the amount of data needed to ensure meaningful results
Ask groups of students to write down a plan for what they will do to test their strategy before they carry out the investigation. Give them time to collect, analyse and interpret the data before
presenting their findings to other groups.
What methods of collection, analysis and representation were most appropriate and effective in communicating their findings? Discuss the merits and pitfalls of different approaches highlighting good
choices and appropriate use of the data.
Key questions
Is it better to have just a few viewings before making a guess or is it better to have more viewings and improve your chances of guessing correctly?
How do you decide on the most effective strategy?
Possible extension
Ask students to articulate two clear strategies, test them and then produce a clear justification of why one strategy is better than the other.
Possible support
Do the introductory activity several times before moving on to the interactivity.
Discuss the value of averaging the results. | {"url":"http://nrich.maths.org/problems/bag","timestamp":"2024-11-07T13:50:49Z","content_type":"text/html","content_length":"48378","record_id":"<urn:uuid:e9f56d89-b9f9-445c-92f5-a126031eb0f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00748.warc.gz"} |
Diving into Data Structures
Learn about data structures and algorithms.
We'll cover the following
Though a TPM is not held to the same level of proficiency as a software developer, we should understand basic programming concepts. We’ll cover the programming topics that come up the most in our
day-to-day activities.
A Data structures and algorithms class was likely in our first or second year in college if you took a traditional route to becoming a TPM. As with most programming fundamentals, we won’t use this
ourselves in our day-to-day work. However, we can think of them as a strong foundation for understanding the language our development team will use in most conversations we have with them.
We’ll briefly go over a few of the more common data structures we may encounter in design meetings, standups, and general work conversations. Even if you’ve taken the class and remember the concepts,
it’s always good to refresh your memory.
Space and time complexities
In a computer, random access memory (RAM) is where data is stored that is in active use, such as variables in an application. Because RAM is a limited resource, measuring the amount of space data
takes up in RAM is an important consideration. The other consideration is the amount of time it takes to perform an action such as searching, inserting, deleting, or accessing data. The amount of
time it takes to perform an action once is then compounded by the number of times the loop is run and can add up very quickly to a considerable time sink if the wrong data structure is utilized for
the task.
Both of these measurements use what is referred to as big O ($big$ $“Oh”$) notation. These measurements are essential categories used for reference to understand the performance of a data structure
or method. For this context, the big O notation is used assuming asymptotic growth and uses $n$ to denote the input that impacts the growth. Essentially, these mathematical functions represent the
curve, or behavior, that the space or time performance will start to match as $n$ gets large enough. As an example, if the amount of time it takes to access a specific element correlates linearly to
where it is in the data structure—for instance, the fifth element in a collection—the big O would be $O (n)$. As $n$ increases, so does the time it takes, which is called linear time. However, if the
amount of time it takes to access an element from a data structure is the same regardless of where in the data structure the element is, then the big O is $O (1)$, or in other words, constant time.
As a TPM, knowing where the complexity categories come from isn’t as important as knowing the relative costs associated with each big O. The figure below shows each big O category on a curve of
operations vs. elements:
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Jonathan Two parties, Alice and Bob, each have some input $x$ and $y$. Their goal is to communicate to compute a function $f(x,y)$ of their inputs. For which
February Tidor Communication functions $f$ (and which models of communication) is it possible for Alice and Bob to communicate few bits of information and still compute $f(x,y)$?
10th (MIT complexity Such problems are studied in communication complexity, a field of theoretical computer science. In this talk we will see several interesting
Mathematics) communication protocols and also many lower bounds using ideas from graph theory, linear algebra, and probabilistic and additive combinatorics.
Guanghao Ye Nested Dissection Meets We present a nearly-linear time algorithm for finding a minimum-cost flow in planar graphs with polynomially bounded integer costs and capacities. The
February (MIT IPMs: Planar Min-Cost previous fastest algorithm for this problem is based on interior-point methods (IPMs) and works for general sparse graphs in $O(n^{1.5}\text{poly}(\log
17th Mathematics) Flow in Nearly-Linear n))$ time [Daitch-Spielman, STOC'08]. Our results immediately extend to all families of separable graphs. This is joint work with Sally Dong, Yu Gao,
Time Gramoz Goranci, Yin Tat Lee, Richard Peng, and Sushant Sachdeva.
February Nitya Mani Nim is perhaps the most famous of impartial combinatorial games, with a simple solution initially given by Bouton. Many generalizations of Nim and other
24th (MIT Nim and variants impartial combinatorial games have been closely studied since the work of Bouton. We will survey a collection of interesting Nim variants and some
Mathematics) associated results, along the way encountering Fibonacci numbers, simple solutions, and surprisingly complicated dynamics.
Consider the task of communicating a message x to a receiver in an error resilient way. Classically, error correcting codes provide a non-interactive
March Rachel Zhang Interactive Error solution to this problem: the sender can simply encode x using an error correcting code, so that even if a constant fraction of the bits are
3rd (CSAIL) Correcting Codes adversarially corrupted, the receiver can still correctly learn x. In this talk, I will define the notion of an interactive error correcting code and
show that over a binary alphabet, they can tolerate more adversarial erasures than can (non-interactive) error correcting codes. This is joint work with
Meghal Gupta and Yael Tauman Kalai.
Advances in model inference and data-driven science have enabled the accurate discovery of governing equations from observations alone, accelerating our
George Learning and predicting understanding and control of dynamical systems. However, despite the ever-growing amount of experimental data collected, many physical and biological
March Stepaniants complex systems systems can only be partially observed. Here, building on recent progress in the inference and integration of nonlinear differential equations, we
10th (MIT dynamics from introduce an approach to learn a model using observations of just a single variable within a multi-variable dynamical system, and use this model to
Mathematics) single-variable accurately predict future dynamics. Furthermore, we validate our approach on a variety of physical, chemical and biological systems which exhibit
observations nonlinear dynamics such as relaxation oscillations and limit cycles. This is joint work with Alasdair Hastewell, Dominic Skinner, Jan Totz and Jörn
March No Seminar /
24th Spring Break
Alex Cohen A fractal uncertainty principle (FUP) states that a function `f' and its Fourier transform cannot both be large on a fractal set. These were recently
March (MIT A discrete 2D fractal introduced by Semyon Dyatlov and collaborators in order to prove new results in quantum chaos. So far FUPs are only understood for fractal sets in R, and
31st Mathematics) uncertainty principle fractal sets in $R^2$ remain elusive. In this talk, we prove a sharp fractal uncertainty principle for Cantor sets in Z/NZ x Z/NZ, a discrete model for
There is a surprising result in information theory where the quantum version of a conjecture is known to be true, and the classical one remains open. The
conjecture is that there exists a joint probability distribution for three parties, Alice, Bob, and Eve, that exhibits bound secrecy. Simply put, bound
Andrey Bound Secrecy and Bound secrecy is the idea that there can secret information present in the correlations of the random variables belonging to Alice and Bob but that is
April Khesin Entanglement: Where completely unknown to Eve, and yet despite this, no matter what Alice and Bob say to each other publicly, they will be unable to distill any bits of a
7th (MIT (Qu)Bits Go to Die secret key, random bits completely unknown to Eve. This is a very surprising fact, as it seems that there is a secret that Alice and Bob share and yet
Mathematics) cannot access despite their best efforts. The quantum version of this conjecture states that there exist joint states for Alice and Bob which are
entangled and therefore cannot be created without spending some amount of entanglement, but from which no pure entangled states, such as Bell pairs, can
be distilled. These joint states are called bound entangled, and not only are they known to exist, some small examples have been found.
April Byron Chin Reconstruction on the The problem of community detection is one of great relevance in machine learning and data science. The goal is to group vertices that behave similarly
14th (MIT stochastic block model within a graph or network. In the context of this problem, the Stochastic Block Model is one of the most well-studied random graph models. In this talk,
Mathematics) we discuss reconstruction results for this model, highlighting a provably optimal algorithm that is closely related to belief propagation.
We will describe recent progress on algorithmic hardness results for ``spin glass-like" random optimization problems. The landscapes of these problems
are highly nonconvex and often include many bad local maxima, impeding optimization by efficient algorithms. As a result, these problems exhibit gaps
between the largest objectives that exist and that can be found by efficient algorithms; characterizing the algorithmic limit requires developing sharp
hardness results. For mean field spin glasses, we describe a new hardness result that tightly characterizes the algorithmic limit for any algorithm with
$O(1)$-Lipschitz dependence on the disorder coefficients. This class includes the best algorithm known for this problem and natural formulations of
Tight Algorithmic gradient descent and approximate message passing.
April Brice Huang Thresholds from the
21th (MIT EECS) Branching Overlap Gap We will go over the Overlap Gap Property (OGP) framework of Gamarnik and Sudan, which shows hardness in random optimization problems against any
Property algorithm that is suitably stable in its inputs. We introduce the \emph{Branching OGP}, a generalization of the OGP to arbitrary ultrametric
constellations of solutions, which is the key tool in the proof of the aforementioned hardness result. By considering the Continuous Random Energy Model
(CREM), we will see by analogy why the Branching OGP yields tight algorithmic thresholds in mean field spin glasses and potentially in greater
Based on joint works with Guy Bresler and Mark Sellke.
We investigate various online packing problems in which convex polygons arrive one by one and have to be placed irrevocably into a container before the
next piece is revealed; the pieces must not be rotated, but only translated. The aim is to minimize the used space depending on the specific problem at
hand, e.g., the strip length in strip packing, the number of bins in bin packing, etc.
Anders Online Sorting and its We draw interesting connections to the following online sorting problem: We receive a stream of real numbers $s_1, \cdots ,s_n,$ each from the unit
April Aamand Connection to Geometric interval $[0,1],$ one by one. Each real must be placed in an array with n initially empty cells without knowing the subsequent arriving reals. The goal
28th (MIT CSAIL) Packing Problems is to minimize the sum of differences of consecutive reals in A. The offline optimum is to place the reals in sorted order, so the cost is at most 1. We
will see that no online algorithm can achieve a cost lower than $n^{(1/2)}$ in the worst case.
We use such lower bound to answer several fundamental questions about packing. Specifically, we prove the non-existence of competitive algorithms for
various online packing problems. A surprising corollary is that no online algorithm can pack translates of convex polygons into a unit square even if we
are guaranteed that the total area of the pieces is at most 0.000001 and the diameter of each piece is at most 0.000001.
Travis Is it always possible to cut a cake and distribute the pieces so that everyone at a party gets the piece they most desire? We'll show that when it comes
May 5th Dillon Fair partitioning? It's to cutting cakes, the old adage that you can't please everybody is dead wrong: Everyone can have their cake and eat it, too. Also, we'll prove Brouwer's
(MIT a piece of cake! fixed point theorem without using any topology. | {"url":"https://math.mit.edu/spams/2022sp.html","timestamp":"2024-11-04T08:25:32Z","content_type":"text/html","content_length":"19152","record_id":"<urn:uuid:2e4f83b9-6e8b-4ea2-87a6-20e11291b89e>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00560.warc.gz"} |
Miscellaneous Functions
If A is a square N-by-N matrix, poly (A) is the row vector of the coefficients of det (z * eye (N) - A), the characteristic polynomial of A.
For example, the following code finds the eigenvalues of A which are the roots of poly (A).
roots (poly (eye (3)))
⇒ 1.00001 + 0.00001i
1.00001 - 0.00001i
0.99999 + 0.00000i
In fact, all three eigenvalues are exactly 1 which emphasizes that for numerical performance the eig function should be used to compute eigenvalues.
If x is a vector, poly (x) is a vector of the coefficients of the polynomial whose roots are the elements of x. That is, if c is a polynomial, then the elements of d = roots (poly (c)) are
contained in c. The vectors c and d are not identical, however, due to sorting and numerical errors.
See also: roots, eig. | {"url":"https://docs.octave.org/v4.0.0/Miscellaneous-Functions.html","timestamp":"2024-11-12T12:29:31Z","content_type":"text/html","content_length":"6572","record_id":"<urn:uuid:09283d8f-e9ff-4e8d-83f1-2ddd7ffb4433>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00844.warc.gz"} |
HILAMA: High-dimensional multi-omic mediation analysis with latent confounding
Motivation The increasingly available multi-omic datasets have posed both new opportunities and challenges to the development of quantitative methods for discovering novel mechanisms in biomedical
research. One natural approach to analyzing such datasets is mediation analysis originated from the causal inference literature. Mediation analysis can help unravel the mechanisms through which
exposure(s) exert the effect on outcome(s). However, existing methods fail to consider the case where (1) both exposures and mediators are potentially high-dimensional and (2) it is very likely that
some important confounding variables are unmeasured or latent; both issues are quite common in practice. To the best of our knowledge, however, no methods have been developed to address these
challenges with statistical guarantees.
Results In this article, we propose a new method for HIgh-dimensional LAtent-confounding Mediation Analysis, abbreviated as “HILAMA”, that considers both high-dimensional exposures and mediators, and
more importantly, the possible existence of latent confounding variables. HILAMA achieves false discovery rate (FDR) control under finite sample size for multiple mediation effect testing. The
proposed method is evaluated through extensive simulation experiments, demonstrating its improved stability in FDR control and superior power in finite sample size compared to existing competitive
methods. Furthermore, our method is applied to the proteomics-radiomics data from ADNI, identifying some key proteins and brain regions relating to Alzheimer’s disease. The results show that HILAMA
can effectively control FDR and provide valid statistical inference for high dimensional mediation analysis with latent confounding variables.
Contact cinbo_w{at}sjtu.edu.cn
1 Introduction
The emergence of modern biotechnologies, such as high-throughput omics and multimodal neuroimaging, has led to the rapid accumulation of omic data at various levels, often including information on
genomics, epigenomics, transcriptomics, proteomics, radiomics, and clinical records. It then becomes possible to thoroughly study complex diseases such as cancer and Alzheimer’s disease by
integrating information from various scales [Subramanian et al., 2020, Kreitmaier et al., 2023]. For example, large collaborative consortia such as Alzheimer’s Disease Neuroimaging Initiative (ADNI)
have collected information across all the levels mentioned above to help unravel the causal mechanisms of Alzheimer’s disease [Bao et al., 2023]. It is thus urgently needed to develop rigorous
statistical methods for analyzing such datasets to reliably dissect the causal mechanisms [Tanay and Regev, 2017, Lv et al., 2021, Corander et al., 2022].
Such a problem falls into the category of causal mediation analysis [VanderWeele, 2015, Baron and Kenny, 1986], which can help disentangle the intermediate mechanisms between cause-effect pairs from
observational datasets [Tobi et al., 2018, Liu et al., 2022, Clark-Boucher et al., 2023]. The classical mediation analysis can be traced back to Wright [1934] in 1930s, and then reverberated by Baron
and Kenny [1986] in early 1980s based on regression techniques. Over the following decades, a vast literature has been engendered to put mediation analysis on a more rigorous ground both
mathematically and conceptually [Robins and Greenland, 1992, Pearl, 2001, VanderWeele and Vansteelandt, 2014, Lindquist, 2012]. We refer interested readers to VanderWeele [2015] for a textbook-level
However, methods for mediation analyses with a single or a few exposures and/or mediators often cannot be directly scaled to address high-dimensional omic data. For instance, traditional hypothesis
testing methods, such as Joint Significance Test (JST) [MacKinnon et al., 2002], Sobel’s method [Sobel, 2008], and bootstrap method [MacKinnon et al., 2007]], tend to be overly conservative
particularly in genome-wide epigenetic studies [Barfield et al., 2017, Huang, 2019].
In response to the above challenge, recent years have seen a surge in the development of new methods for high-dimensional mediation analysis. These methods aim to explore the biological mechanisms
derived from multi-omics data, as evidenced by studies such as Zeng et al. [2021] and Zhang et al. [2022a]. For instance, Zhang et al. [2016] and Gao et al. [2019] have focused on epigenetic studies
with high-dimensional mediators and a continuous outcome. They have primarily employed (debiased) penalized linear regression and multiple testing procedures to formulate their methods. Derkach et
al. [2019] have considered this similar problem by considering multiple latent variables as mediators that influence both the high dimensional biomarkers and the outcome. Moreover, Luo et al. [2020],
Zhang et al. [2021] and Tian et al. [2022] have extended the analysis to include a survival outcome, in addition to high-dimensional mediators. These studies have contributed to the growing
literature on exploring complex biological relationships. In a similar vein, Shao et al. [2021] has investigated high-dimensional exposures with a single mediator in an epigenetic study, utilizing a
linear mixed-effect model.
However, there have been limited works studying both multivariate exposures and mediators. Zhang [2022] consider high-dimensional exposures and mediators through two different procedures; however,
they require the mediators to be independent and mainly focus on mediator selection. Meanwhile, Zhao et al. [2022] develop a novel penalized principal component regression method that replaces the
exposures with their principal components in a lower dimension. This approach, however, lacks causal interpretation. More importantly, most high-dimensional mediation analyses approaches make the
untestable assumption of no latent confounding, which is highly problematic in multi-omic biological studies due to the prevalence of non-randomized study designs. If hidden confounding cannot be
eliminated, these methods may be influenced by spurious correlations and result in an inflated False Discover Rate (FDR). Recently, several works have addressed this issue by considering latent
confounding in high-dimensional linear models, which involve estimating overall causal effects under the Latent Structural Equation Modeling (LSEM) framework [Chernozhukov et al., 2017, Ćevid et al.,
2020, Guo et al., 2022, Bing et al., 2022a,b]. Specifically, Sun et al. [2022] are the first to address the large-scale hypothesis testing problem in the high-dimensional confounded linear model.
They achieve FDR control under finite sample size by introducing a decorrelating transformation before the debiasing step.
Inspired by the aforementioned works on high-dimensional linear regression under latent confounding [Chernozhukov et al., 2017, Ćevid et al., 2020, Guo et al., 2022, Bing et al., 2022a,b, Sun et al.,
2022], we propose a novel method called HILAMA, which stands for HIgh-dimensional LAtent-confounding Mediation Analysis. HILAMA addresses two critical challenges in applying mediation analysis (or
any causal inference method) to multi-omics studies: (1) accommodating both high-dimensional exposures and mediators, and (2) handling latent confounding. In contrast to competing methods [Baron and
Kenny, 1986, Zhang et al., 2016, Gao et al., 2019, Schaid et al., 2022, Zhao et al., 2022], our method maintains control over FDR at the nominal level for multiple mediation effect testing, even in
the presence of latent confounders. Now, we briefly sketch the essential components of our method. First, we employ the Decorrelate & Debias method in Sun et al. [2022] to obtain p-values for each
individual exposure and mediator’s effect on the outcome. Second, to estimate the effect matrix of exposures on mediators, we employ a column-wise regression strategy, again incorporating the
Decorrelate & Debias method [Sun et al., 2022]. To handle large and high-dimensional datasets, we utilize parallel computing in this step. Third, we apply the Min-Screen procedure in Djordjilovi et
al. [2019] to eliminate non-rejected hypotheses, retaining only the K most significant pairs for the final stage of multiple testing. Lastly, we compute p-values for all K pairs using the JST method
[MacKinnon et al., 2002], employing a data-dependent threshold determined by the Benjamini-Hochberg (BH) procedure [Benjamini and Hochberg, 1995] to maintain FDR at the nominal level α. We conduct
extensive simulations to evaluate our method’s performance, which demonstrates effective FDR control across various finite sample sizes, surpassing the capabilities of most other methods.
Furthermore, we apply HILAMA to a proteomics-radiomics dataset from the ADNI database (adni.loni.usc.edu) and identify key proteins and brain regions associated with learning, memory, and recognition
impairments in Alzheimer’s disease and cognitive impairment.
The rest of this article is organized as follows. In Section 2, we first describe our model and then introduce the HILAMA procedure under the Linear Structural Equation Models with highdimensional
exposures, high-dimensional mediators, continuous outcome, and latent confounders setting. In Section 3, we evaluate the FDR and Power performance of HILAMA across a wide range of simulations. In
Section 4, we apply the HILAMA to a proteomics-radiomics data of Alzheimer’s disease from ADNI. The paper is concluded with a discussion in Section 5. Technical details, some tables and figures are
collected in the online Supplementary Materials.
2 Methods
To describe the HILAMA methodology, we first need to briefly review mediation analyses. Mediation analyses are frequently utilized to disentangle the underlying causal mechanism between two sets of
variables, the exposures and outcomes, exerted by a third set of variables, the mediators. The overall causal effects can be decomposed into direct effects from exposures to outcomes, bypassing
mediators, and indirect effects via mediators. To be more precise, we ground our discussion under the Linear Structural Equation (LSE) framework, that models the causal mechanisms among p-dimensional
exposures X[i] = (X[i1], ⋯, X[ip])^⊤ ∈ ℝ^p, q-dimensional mediators M[i] = (M[i1], ⋯, M[iq])^⊤ ∈ ℝ^q, a scalar outcome Y[i] ∈ ℝ, and latent confounders H[i] = (H[i1], ⋯, H[is])^⊤ ∈ ℝ^s (e.g., batch
effects, disease subtypes, and lifestyle factors) as follows: where ϵ[i] and E[Mi] are the noise terms that are independent of X[i], M[i] and H[i]. In the outcome model (1), γ = (γ[1], ⋯, γ[p])^⊤ ∈ ℝ
^p is the direct effect vector of the exposures X[i] on the outcome Y[i], and β = (β[1], ⋯, β[q])^⊤ ∈ ℝ^q represents the effect vector of the mediators M[i] to the outcome Y[i] after adjusting for
the latent confounders H[i]. ϕ ∈ ℝ^s is the parameter vector that relates latent confounders H[i] to the outcome Y[i]. Here, we allow p and q to be larger than the sample size n, while s ≤ p + q. The
primary objective of our study is to identify the active direct/indirect effects (γ[k], θ[kl]β[l], k ∈ [p], l ∈ [q]) from p/pq possible paths, as shown in Figure 1.
In the mediator model (2), the matrix Θ = (θ[1], ⋯, θ[q]) = (θ[kl]) ∈ ℝ^p×q represents the regression coefficients of exposures on mediators and θ[kl] represents the effect of exposure X[ik] on
mediator M[il] after adjusting for the effect of latent confounders H[i]. Ψ[2] ∈ ℝ^s×q can be interpreted as the confounding effect of the latent confounders H[i] on mediators M[i]. The mediation
model (1) – (2) adopted here is similar to those proposed by Schaid et al. [2022] and Zhao et al. [2022], which were among the first works to consider both multivariate exposures and mediators.
However, we incorporate the latent confounders into our high-dimensional mediation analysis, which is a novel approach in the field. Furthermore, unlike the approach used in Zhao et al. [2022], our
mediation analysis is directly based on X[i], instead of a transformation of the original vector X[i].
As mentioned, the causal parameters of interest in mediation analyses are mainly the (average) natural direct and indirect effects. When the ignorability assumption (explained in Supplementary
Materials) holds, the natural direct effect of exposure k on outcome, denoted by NDE[k], and the nature indirect effect of exposure k on outcome, denoted by NIE[k], can be expressed as [Robins and
Greenland, 1992, Pearl, 2001] When x[k] and defer by one unit, NDE[k](1) = γ[k], the regression coefficient between X and Y in model (1), and NIE[kl](1) = θ[kl]β[l], the product of the regression
coefficient between X[k] and M[l] in model (2) and the regression coefficient between M[l] and Y in model (1). For their derivation, see Supplementary Materials.
However, when latent confounders exist (i.e. dim(H) > 0), neither NDEs nor NIEs are identifiable without making additional assumptions, which are generally based by domain-specific knowledge. To
identify the true parameter γ, β and θ[l] in the confounded linear model (1) and (2), it is necessary to make additional assumptions among the observed variables (X[i], M[i]) and the latent
confounders H[i]. Based on the works of Wang et al. [2017], Ćevid et al. [2020], Guo et al. [2022], Sun et al. [2022], a factor model is specified to characterize the relation between X[i] and H[i]:
where Cov(H[i], E[X,i]) = 0 and the random variable E[X,i] ∈ ℝ^p represents the unconfounded components of X[i]. Moreover, to accurately identify the true signals and effectively remove the
confounding effects, we impose a spiked singular value condition on the covariance between the exposures and mediators, as shown in Figure 6. Specifically, we require , where Ψ = (Ψ[1], Ψ[2]) ∈ ℝ^s×(
p+q). Our approach is particularly effective in scenarios where the confounding effect is dense, i.e., many observed variables in X ∈ ℝ^p and M ∈ ℝ^q are simultaneously influenced by the latent
confounders H ∈ ℝ^s.
Our goal is to identify the path-specific indirect effect θ[kl]β[l] (corresponding to the path X[k] → M[l] → Y) from the total p · q possible paths which corresponds to the following multiple
hypothesis testing problem: Here, we propose a novel framework called HILAMA to address the problem (4). The framework identifies the true paths with nonzero indirect effects and controls the
finite-sample FDR. It involves four major steps explained in detail below (as shown in Figure 2).
First, for the outcome model in equation (2), we utilize the Decorrelate & Debias approach presented in Sun et al. [2022] to carry out inference on the regression parameters γ and β. By applying this
method, we obtain the double debiased estimator and These estimators, after appropriate rescaling, asymptotically converge to centered Gaussian distributions: and individually under some mild
conditions. We denote the corresponding variance estimators as and Then p-values can be computed as follows: where Φ(·) denotes the cumulative distribution function of the standard normal
distribution N (0, 1).
Second, to estimate each column of the parameter matrix Θ in the multi-response mediation model defined in equation (2), we employ a column-wise regression strategy. For each sub-regression problem,
we utilize the Decorrelate & Debias approach [Sun et al., 2022]. To handle the computational challenges posed by large-scale datasets in multi-omics studies, we leverage parallel computing techniques
to accelerate the computation process. This allows us to efficiently calculate the point and variance estimators for the coefficients θ[kl](k ∈ [p] and l ∈ [q]), which are denoted as and ,
respectively. To assess the statistical significance of the coefficient estimates, we calculate the p-values using the formula: where Φ denotes the cumulative distribution function of the standard
normal distribution.
Third, we employ the MinScreen procedure [Djordjilovi et al., 2019] to screen the total p · q possible causal paths. The screened causal paths by MinScreen are defined as the top K significant paths:
where and α[0] is chosen such that This preliminary step eliminates the least promising causal paths before calculating the final p-value for H[0kl]. By doing so, it effectively reduces the
computational burden in the subsequent multiple testing phase.
Lastly, we apply the joint significance test (JST), also known as the MaxP test [MacKinnon et al., 2002], to obtain the p-value for the null hypothesis H[0kl] : θ[kl]β[l] = 0 which tests for no
indirect effect, for The p-values for JST are defined as We then sort the JST p-values and denote them as p[(i)], i = 1, ⋯, K, the notation for order statistics by convention. To protect the FDR at
the nominal level α, we follow the BH procedure [Benjamini and Hochberg, 1995] to find the data-driven p-value rejection threshold P* as Finally, we define the set containing statistically
significant non-zero path-specific effects as In this article, we evaluate HILAMA and other competitors by FDR and Power, which are defined as follows: where 𝒮 = {(k, l) : θ[kl]β[l] ≠ 0, k ∈ [p], l ∈
[q]} represents the true non-zero effect path-specific set and 𝒮^c = {(k, l) : θ[kl]β[l] = 0, k ∈ [p], l ∈ [q]} represents the zero effect path set.
3 Simulation Studies
In this section, we assess if HILAMA is capable of controlling the FDR with sufficient power across a wide range of simulation settings. The performance is compared against various other approaches.
As a baseline benchmark, we employ the univariate Baron & Kenny method [Baron and Kenny, 1986] (abbreviated as BK) for every possible individual exposure-mediator pair, using the R package mediation.
We also consider methods that only allow a univariate exposure and high-dimensional mediators, including HIMA [Zhang et al., 2016] and HDMA [Gao et al., 2019]. For these two methods, we directly use
their corresponding R packages HIMA and HDMA, analyze every individual exposure, and then aggregate the results. Finally, we compare two penalized methods developed for multiple exposures and
mediators. Specifically, for the method “mvregmed” [Schaid et al., 2022], we apply the R package regmed. While for the method developed by Zhao et al. [2022] (abbreviated as ZY), we implement their
penalized regression algorithm and omit the dimension reduction step for comparison. Here, we only compare the two penalized methods in the low dimensional setup in simulation 2 introduced below due
to their slow running time (See Figure 4d).
We first generate the exposure data X[i](i = 1, ⋯, n) according to model (3). The latent confounders H[i] ∈ ℝ^s and the elements of confounding matrix Ψ ∈ ℝ^s×p are independently drawn from the
standard normal distribution. The unconfounded components E[X,i] are drawn from N[p](0, Σ[E]), where Σ[E,kl] = κ^|k−l|(k, l ∈ [p]). The parameter κ controls the strength of correlation among
exposures, and it takes values in the range [0, 1).
Similarly, we generate the the mediator data M[i](i = 1, ⋯, n) according to model (2). The noise term E[M,i] are drawn from N[q](0, I) and the confounding effect matrix Ψ[2,kl](k [s], l∈ [q]) are
drawn from ξ · N (η, 1), where ξ is a Rademacher random variable, i.e. Then, for the signal coefficient matrix Θ ∈ ℝ^p×q, we randomly choose p · r[p] rows having non-zero elements, and choose δ
non-zero elements seperately in each of these rows, where δ follows uniform distribution on {5, 6, ⋯, 20}. The non-zero elements in Θ follow uniform distribution on [−1.5, −0.5] ∪ [0.5, 1.5].
Finally, we generate the outcome data Y[i](i = 1, ⋯, n) according to model (1). The coefficients γ are randomly selected with p·r[p] non-zero elements following a uniform distribution on [−1.5, −0.5]
∪ [0.5, 1.5]. As for the coefficients β, we choose q·r[q] non-zero elements following a uniform distribution on [−1.5, −0.5] ∪ [0.5, 1.5]. To determine the active location in β, we define 𝒜^c as the
set of columns in Θ with zero elements (∥Θ[.l]∥[1] = 0), and 𝒜 as the set of columns in Θ with non-zero elements (∥Θ[.l]∥[1] ≠ 0). From 𝒜^c, we randomly choose s[01] elements with equal probability,
where s[01] = min{0.2 · q · r[q], |𝒜^c|}. While from 𝒜, we randomly choose s[11] elements with unequal probability, where s[11] = q · r[q] − s[01]. The selection probability of l∈𝒜 is determined by
which represents the proportion of non-zero elements in column l relative to all non-zero elemen ts in Θ. The confounding effects ϕ are drawn from N[s](η, I) · ξ, and the noise terms ϵ[Y,i] are drawn
from N (0, 1).
For all the simulations below, we fix the sparsity proportions as r[p] = r[q] = 0.1 and the dimension of latent confounders s = 3. Additionally, we set the nominal FDR level at the α = 0.1 and all
the simulation results are averaged over 50 Monte Carlo replications.
Simulation 1
In the first simulation, we test the stability of our model under various scenarios. We evaluate the impact of changes in sample size (n ∈ {200, 400}), exposure dimension (p ∈ {100, 200, 400}),
mediator dimension (q ∈ {50, 100, 200}), correlation size among exposures (κ ∈ {0.4, 0.8}), and magnitude of latent effects (η ∈ {0.5, 1.5}). For the total 72 different settings, we present the
average value of empirical FDR and Power in Figure S1a and Figure S1b.
For simplicity, we only present scenarios for p = 400 in Figure 3. From Figure 3a, only HILAMA controls the FDR at the nominal level α = 0.1 in all scenarios, whereas the other three methods all fail
to do so. The reasons for their lack of control are due to their failure to correct for the effect of latent confounding, and their inability to accommodate high-dimensional exposure and mediator
settings. Turning to the power, by reading Figure 3b, we can easily see that HILAMA achieves the highest and most stable power close to 1 in large sample sizes (n = 400). However, in smaller sample
sizes (n = 200), the statistical power decreases as correlation coefficient κ or confounding effect η increases. The power of HILAMA is generally unaffected by the above parameters in larger samples.
The powers of the other three methods, on the other hand, are essentially meaningless since their FDRs are all close to 1. Moreover, the point estimates of mediation effects output by HILAMA has much
smaller bias than the other competing methods, represented as (see Figure S1c).
Simulation 2
In the second simulation, we assess how the denseness of latent confounding impacts the performance of HILAMA. We measure the denseness of latent confounding as the proportion (1− r[h]) of zero
entries in each row of Ψ[1] and Ψ[2]. If r[h] = 0, then Ψ[1] = Ψ[2] = 0, amounting to no latent confounding; whereas if r[h] = 1, all exposures and mediators are confounded by latent confounders, as
depicted in simulation 1. Here, we vary only r[h] ∈ {0, 0.1, 0.2, ⋯, ⋯ 0.9, 1} while holding n = 400, p = q = 100, κ = 0.6, η = 1 and compare HILAMA with the two aforementioned penalized methods. To
compare our p-value based method with the penalized methods mvregmed and ZY, we assume that the actual number of active pairs is known. We select the top K pairs that control the FDR at the level 0.1
and compare their power. If the FDR cannot be controlled at the 0.1 level, we choose the cut-off point associated with the lowest FDR and calculate the corresponding power.
Figure 4a indicates that HILAMA does not manage the FDR at the 0.1 level under the low-dimensional setting when only a few observed variables are confounded by latent confounders. Additionally, the
power of HILAMA is highly sensitive when a few observed variables are confounded by latent confounders from Figure 4b, although it tends to stabilize as the confounding density increases. However,
the mvregmed method does not control the FDR across all situations, even in the absence of latent confounding. Figure 4c demonstrates that HILAMA again has the minimum mean bias compared to the other
two competing methods, mvregmed and ZY. Specifically, Figure 4d shows that although the ZY method boasts the best FDR control and power performance, it takes hundreds of times longer to compute than
HILAMA, even in this low-dimensional setting.
Additionally, we consider n = 300, p = 200, q = 100, while all other components of the data generating distribution are held constant and compare HILAMA with the above three p-value based methods.
Figure 5a and Figure 5b display the FDR and power of the four p-value-based methods at the nominal FDR level of 0.1. HILAMA demonstrates effective FDR control in all cases, including under no latent
confounding, whereas the other three methods are unable to control the FDR in any setting. HILAMA sustains a stable power near 1 in most scenarios, except when r[h] = 0. Meanwhile, the powers of HIMA
and HDMA have a downward trend as r[h] increases. Figure 5c additionally explores the average bias of true mediation effect and it is obvious that HILAMA attains the minimum mean bias across all the
scenarios, while all the other other methods show an increasing trend in mean bias as r[h] increases.
Simulation 3
In the third simulation, our aim is to investigate the impact of signal strength on HILAMA. Specifically, the data generation process repeats that of the previous simulations, with the exception that
the non-zero components of Θ follow the distribution of ξ · N(ρ, 1) and the non-zero elements of β follow N(ρ, 0.1). We vary the signal strength ρ ∈ {0.1, 0.2, ⋯, 1.4, 1.5}while maintaining n = 400,
p = 300, q = 100, κ = 0.6, and η = 1.
According to Figure S2a, HILAMA can control the FDR to the nominal level of 0.1, whereas the other three methods fail to do so. Notably, from Figure S2b we can see that the power of HILAMA is
restricted when ρ - the signal strength is weak, and the power increases as ρ gets stronger, before stabilizing around 1 when ρ ≥1. Similar to these above findings, the mean bias of HILAMA is the
smallest among all the methods being compared, as shown in Figure S2c.
4 An application to the Proteomics-Radiomics Study of AD
In this final section, we apply HILAMA to a real multi-omic dataset collected by the ADNI. Before delving into the details, we emphasize that this data analysis should be viewed as at most
exploratory rather than confirmatory nature. It is highly likely that the linearity assumption imposed in the Structural Equation Model may not be a good approximation of the reality.
Alzheimer’s disease (AD) is an irreversible and complex neurological disease that affects millions of individuals worldwide. Currently, approximately 6.7 million Americans aged 65 years and older
live with AD, and this number is projected to dramatically increase to 13.8 million by the year 2060 [AD2, 2023]. AD is characterized by progressive memory loss and other cognitive impairments
resulting from the accumulation of amyloid-β(Aβ) and tau proteins in the brain, leading to neurodegenerative symptoms [Chen and Xia, 2020].
Unfortunately, there is currently no effective treatment for AD, underscoring the significance of early diagnosis and comprehending the disease’s pathogenesis. Therefore, it is crucial to develop
effective interventions to prevent, slow down, or even cure this disease through biomedical research. With this in mind, the Alzheimer’s Disease Neuroimaging Initiative (ADNI, adni.loni.usc.edu) was
established in 2003. Its primary goals are to develop biomarkers for AD, enhance the understanding of its pathophysiology, and improve early detection using various modalities such as magnetic
resonance imaging (MRI), positron emission tomography (PET), functional magnetic resonance imaging (fMRI), as well as clinical and neuropsychological assessments.
In this section, we utilize the HILAMA approach to examine the connection between proteins in the cerebrospinal fluid (CSF), whole-brain atrophy, and cognitive behavior. Our aim is to identify
critical biological pathways associated with AD by utilizing data from the ADNI database. The CSF proteomics data is acquired using a highly specific and sensitive technique called targeted liquid
chromatography multiple reaction monitoring mass spectrometry (LC/MS-MRM), resulting a list of 142 annotated proteins derived from 320 peptides. Additionally, the brain imaging data is obtained
through anatomical magnetic resonance imaging (MRI), and volumetric measurements are extracted from 145 brain regions-of-interest (ROI) [Doshi et al., 2016]. To assess the relationship between the
aforementioned variables and cognitive function, we consider the composite memory score as the response. This score is measured using the ADNI neuropsychological battery, with higher scores
indicating better cognitive function. In our model, we treat the 142 proteins as exposures (X), the 145 brain regions as mediators (M), and the memory score as the outcome (Y). For this study, we
focus on a total of 287 subjects who have both proteomics and imaging data available. These subjects consist of 86 cognitively normal individuals (CN), 135 patients with mild cognitive impairment
(MCI), and 66 AD patients. To account for potential confounding effects, we include covariates such as age, gender (Male = 1, Female = 2), years of education, and disease type (CN = 1, MCI = 2, AD =
3). For more detailed information on these baseline covariates, please refer to Table 1.
Prior to conducting the mediation analysis, we impute some volumetric measures recorded as zero with the corresponding median value, and then apply a log-transformation to achieve a more normal
distribution. Subsequently, we standardize both the protein data and MRI data to have a mean of zero and a standard deviation of one, while only centering the outcome cognitive score to have a mean
of zero. In Figure 6, we visualize the singular values of the protein and MRI data, allowing us to assess the potential presence of latent confounders. By examining Figure 6a and Figure 6b, we
observe the presence of two significantly larger singular values in the protein data and one in the MRI data. This finding suggests a distinct spiked structure, indicating the possible presence of
latent confounders as depicted in the models (2) and (3).
Following the preprocessing of data, we apply our method to the processed data. However, after implementing the BH procedure, no significant paths are obtained when controlling the FDR at a nominal
level of 0.1. In order to obtain meaningful results, we relax the criterion and set the significance threshold for p-values to 0.1 without applying multiple correction. Consequently, we identify 63
significant causal paths, corresponding to 45 proteins and 7 brain regions. The estimated path effects, including the θ[kl] and β[l], are presented in the Supplementary Table S1. In Figure 7, we
visualize the significant causal paths.
Our study has identified several brain regions associated with cognitive impairment and AD. Among them, R48 (left hippocampus) plays a crucial role in learning and memory, and is particularly
vulnerable to early-stage damage in AD [Nadel and Hardt, 2011]. Importantly, hippocampal atrophy has been universally recognized and validated as the most reliable biomarker for AD [Schröder and
Pantel, 2016]. Another crucial region in cognition is R106 (right angular gyrus), which is associated with language, spatial, and memory functions [Seghier, 2013, Humphreys et al., 2021]. The aging
process leads to structural atrophy in the angular gyrus, which is linked to subjective and mild cognitive impairments, as well as dementia [Karas et al., 2008, Jockwitz et al., 2023]. Additionally,
another significant region, R116 (right entorhinal area), often exhibits the earliest histological alterations in AD. Impaired neuronal activity in the area may cause memory impairments and spatial
navigation deficits at the initial stage of AD [Igarashi, 2023]. Furthermore, R205 (left triangular part of the inferior frontal gyrus), R105 (left anterior orbital gyrus), R148 (right postcentral
gyrus medial segment) and R207 (left transverse temporal gyrus) are also associated with AD and cognitive impairment. However, further investigation is necessary to comprehensively elucidate the
roles of these regions in AD pathology and cognitive function.
Several proteins have been identified as potentially critical biomarkers for AD. NPTX2 and NPTXR are proteins that bind to glutamate receptors, contributing to synaptic plasticity. Reductions in
NPTX2 have been linked to disruptions of the pyramidal neuron-PV interneuron circuit in an AD mouse model [Xiao et al., 2017]. PRDX1 and PRDX2 are peroxiredoxin proteins that provide protection
against neuronal cell death and oxidative stress [Kim et al., 2001]. PRDX3 plays a crucial role as a mitochondrial antioxidant defense enzyme, and its overexpression provides protection against
cognitive impairment while reducing the accumulation of Aβ in transgenic mice [Chen et al., 2012]. Furthermore, its overexpression reduces mitochondrial oxidative stress, attenuates memory impairment
induced by hydrogen peroxide and improves cognitive ability in transgenic mice [Chen et al., 2014]. Moreover, recent research has revealed that PRDX3 plays important roles in neurite outgrowth and
the development of AD [Xu et al., 2022a]. KNG1 is a protein involved in inflammatory responses, and leavage of KNG1 has been associated with the release of proinflammatory bradykininwhich may
contribute to AD-associated inflammation [Markaki et al., 2020]. SE6L1 is a potential neuronal substrate of the AD protease BACE1, which is a major drug target in AD [Pigoni et al., 2016]. Aberrant
function of SE6L1 may lead to movement disorders and neuropsychiatric diseases [Ong-Pålsson et al., 2022]. Overexpression of the neuropeptide precursor VGF has been found to partially rescue Aβ
mediated memory impairment and neuropathology in a mouse model, indicating a protective function against the development and progression of AD [Beckmann et al., 2020]. CUTA is a protein that has been
proposed to mediate acetylcholinesterase activity and copper homeostasis, which are important events in AD pathology. Overexpression of CUTA can reduce BACE1-mediated APP processing and Aβ
generation, while RNA interference increases it [Zhao et al., 2012]. PEDF is a unique neurotrophic and neuroprotective protein whose expression decays with aging. Experiments in a
senescence-accelerated mouse model show that PEDF negatively regulates Aβ and notably reduces cognitive impairment, suggesting that PEDF might play a crucial role in the development of AD [Huang et
al., 2018]. Knock-down of PIMT and treatment with AdOX significantly increase Aβ secretion, which serves as a negative regulator of Aβ peptide formation and a potential protective factor in the
pathogenesis of AD [Bae et al., 2011].
In summary, our study identified several critical brain regions, such as R48, R106 and R116, that are associated with learning, memory, and recognition. Moreover, we have identified several potential
biomarkers for AD, such as NPTX2, NPTXR, PRDX1, PRDX2, PRDX3, KNG1, SE6L1, VGF, CUTA, PEDF, etc., most of which are not selected by the method ZY [Zhao et al., 2022]. Nonetheless, it is crucial to
note that these findings are only suggestive and further experimental validation is warranted to fully understand their contributions to AD pathology and cognitive function.
5 Discussion
In this paper, we propose HILAMA, a new method for high-dimensional mediation analysis, an important statistical task in the analysis of multi-omics datasets increasingly available in biomedical
sciences. HILAMA effectively unravels the causal pathway between high-dimensional exposures and a continuous outcome, in the presence of possibly latent/unmeasured confounders. We validate the
practical performance of HILAMA through extensive simulations and by applying it to a real ADNI dataset, which allows for the identification of potential biomarkers for Alzheimer’s disease.
HILAMA features several key advantages over previous methods, designed towards better fitting into real-world multi-omics datasets. First, it is the first method to consider both high-dimensional
exposures and high-dimensional mediators in the presence of latent confounders without transforming exposures/mediators into principal components, rendering the analysis results more interpretable.
Second, it incorporates a novel Decorrelate & Debias method [Sun et al., 2022] to handle latent/unmeasured confounding and improve coefficient estimation, leading to better FDR control. Third, it
employs a MinScreen screening procedure [Djordjilovi et al., 2019] to reduce the number of hypotheses being tested, thereby enhancing the statistical power of the tests. Finally, the method is
computationally efficient and has implemented parallel computing techniques to handle the ever-increasing size and dimension of modern multi-omics datasets.
To conclude, we point out several venues for future research. First, HILAMA assumes linear models, which is standard practice in multi-omics studies. However, it will be interesting to generalize it
to nonlinear/nonparametric models via nonlinear factor analysis [Amemiya and Yalcin, 2001, Feng, 2020], autoencoders [Yang et al., 2021], kernel methods [Singh et al., 2021] or deep neural networks [
Xu et al., 2022b]. Second, HILAMA assumes that the effects of latent/unmeasured confounders on observables are dense. It may be possible to relax this assumption by extending the randomized
data-augmentation scheme proposed in Zhang et al. [2022b] for total effect to the mediation analysis setting. Third, considering the correlation structure among mediators could be beneficial in the
step of Mediator-Exposure deconfounded regression. Currently, this correlation structure is omitted due to challenges in controlling FDR in large-scale regression with multivariate response. However,
incorporating this information might be useful for removing the confounding effect and improving the power of the test [Zou et al., 2020, Kotekal and Gao, 2021]. Finally, other methods of dealing
with latent confounding can also be incorporated into HILAMA in its future version, such as the approaches [Miao et al., 2022, Tang et al., 2023] that directly leverage the majority rule [Kang et
al., 2016] or the plurality rule [Guo et al., 2018]. Overall, these future research directions have the potential to expand the capabilities of HILAMA, allowing for more accurate and robust causal
inference in multi-omics studies.
This work was partially supported by the National Science Foundations of China Grants No.12101397 (LL, XW) and No.12090024 (LL), Shanghai Municipal Science and Technology Commission Grants
No.21ZR1431000 (LL, XW) and 21JC1402900 (LL), Shanghai Municipal Science and Technology Major Project No.2021SHZDZX0102 (LL), the Neil Shen’s SJTU Medical Research Fund (XW, LL, HL) and SJTU Transmed
Awards Research (STAR) Grant No.20210106 (HL). LL is also affiliated with the Shanghai Artificial Intelligence Laboratory and the Smart Justice Lab of the Koguan Law School at Shanghai Jiao Tong
Conflict of Interest
none declared.
Supplementary Materials
Derivations of direct and indirect effects under LSEM (1) and (2)
We follow the potential outcomes framework [Rubin, 1972, Splawa-Neyman et al., 1990] to decompose the effect of exposures on the outcome. Specifically, we denote Y(x, m) as the potential outcome when
the exposure X is set to x = (x[1], ⋯, x[p])′, the mediator M is set to m = (m[1], ⋯, m[q])′. Formally, we define x[−k] = (x[1], ⋯, x[k−1], x[k+1], ⋯, x[p])′ and use as the potential value of the l
-th mediator where the k-th exposure is set to x[k] and the ot her exposures are set to . We denote the baseline adjusted covariates as Z = (Z[1], H), where Z[1] represents the observ ed confounders,
H represents the potential latent confounders, and as the reference level of exposure. Then, the average total effect (TE) of the k-th exposure on the outcome is defined as the average natural direct
effect (NDE) ofX[k] on Y is the average natural indirect effect (NIE) of X[k] on Y is Among the three effects, we have the following relationships: To identify the above NDE and NIE, we need the
following standard ignorability assumption [VanderWeele and Vansteelandt, 2014]:
(C1) Y(x, m) ⊥ X[k]| Z for ∀x, m, k ∈ [p], i.e. no unmeasured confounding between the exposures and the outcome;
(C2) Y(x, m) ⊥ M[l] | Z for ∀x, m, l ∈ [q], i.e. no unmeasured confounding between the mediators and the outcome;
(C3) M[l](x) ⊥ X[k] | Z for ∀ x, ∈ k [p], l ∈ [q], i.e. no unmeasured confounding between the exposures and the outcome;
(C4) Y(x, m) ⊥ M[l](x*) |Z for ∀x, x*, m, l ∈ [q], i.e. no unmeasured confounding between the mediators and the outcome that is itself affected by the exposures.
To simplify the representation, we replace H in LSEM (1) and (2) with Z to represent the above baseline covariates Z = (Z[1], H). From LSEM (2), we can express the potential mediator and potential
outcome as follows: Then, the (average) natural direct effect of exposure X[k] on outcome when the value of that exposure is manipulated from x[k] to denoted by can be derived directly based on
equation (6): Similarly, the natural indirect effect of exposure X[k] on outcome, denoted by can be derived as follows:
Supplementary tables and figures
The authors would like to thanks professor Yi Zhao for her valuable suggestions on accessing the ADNI data.
Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of
Defense award number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the
following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan
Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IX-ICO Ltd.; Janssen Alzheimer Immunotherapy
Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack
Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian Institutes of Health Research is
providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization
is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California. ADNI data are
disseminated by the Laboratory for Neuro Imaging at the University of Southern California. | {"url":"https://www.biorxiv.org/content/10.1101/2023.09.15.557839v1.full","timestamp":"2024-11-10T14:54:37Z","content_type":"application/xhtml+xml","content_length":"396716","record_id":"<urn:uuid:eef665e7-9267-49cc-832b-e81a83c42bde>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00105.warc.gz"} |
Sequence ratio calculations
summariseSequenceRatios {CohortSymmetry} R Documentation
Sequence ratio calculations
Using generateSequenceCohortSet to obtain sequence ratios for the desired outcomes.
cohortId = NULL,
confidenceInterval = 95,
movingAverageRestriction = 548,
minCellCount = 5
cohort A cohort table in the cdm.
cohortId The Ids in the cohort that are to be included in the analyses.
confidenceInterval Default is 95, indicating the central 95% confidence interval.
movingAverageRestriction The moving window when calculating nSR, default is 548.
minCellCount The minimum number of events to reported, below which results will be obscured. If 0, all results will be reported.
A local table with all the analyses.
cdm <- mockCohortSymmetry()
cdm <- generateSequenceCohortSet(cdm = cdm,
name = "joined_cohorts",
indexTable = "cohort_1",
markerTable = "cohort_2")
pssa_result <- summariseSequenceRatios(cohort = cdm$joined_cohorts)
version 0.1.2 | {"url":"https://search.r-project.org/CRAN/refmans/CohortSymmetry/html/summariseSequenceRatios.html","timestamp":"2024-11-07T22:31:51Z","content_type":"text/html","content_length":"3163","record_id":"<urn:uuid:d75358e9-dbef-4f69-a9d4-f4ad050598cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00779.warc.gz"} |
An Introduction to Game-Theoretic Modelling: Second Editionsearch
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• Student Mathematical Library
Volume: 11; 2001; 368 pp
MSC: Primary 91; Secondary 92
Now available in Third Edition: AMSTEXT/37
This book is about using game theory in mathematical modelling. It is an introductory text, covering the basic ideas and methods of game theory as well as the necessary ideas from the vast
spectrum of scientific study where the methods are applied.
It has by now become generally apparent that game theory is a fascinating branch of mathematics with both serious and recreational applications. Strategic behavior arises whenever the outcome of
an individual's action depends on actions to be taken by other individuals—whether human, as in the Prisoners' Dilemma, or otherwise, as in the “duels of damselflies”. As a result, game-theoretic
mathematical models are applicable in both the social and natural sciences. In reading this book, you can learn not just about game theory, but also about how to model real situations so that
they can be analyzed mathematically.
Mesterton-Gibbons includes the familiar game theory examples where they are needed for explaining the mathematics or when they provide a valuable application. There are also plenty of new
examples, in particular from biology, such as competitions for territory or mates, games among kin versus games between kin, and cooperative wildlife management.
Prerequisites are modest. Students should have some mathematical maturity and a familiarity with basic calculus, matrix algebra, probability, and some differential equations. As Mesterton-Gibbons
writes, “The recurring theme is that game theory is fun to learn, doesn't require a large amount of mathematical rigor, and has great potential for application.”
This new edition contains a significant amount of updates and new material, particularly on biological games. An important chapter on population games now has virtually all new material. The book
is absolutely up-to-date with numerous references to the literature. Each chapter ends with a commentary which surveys current developments.
Advanced undergraduates, graduate students, and research mathematicians interested in mathematical modelling; applied mathematicians; biologists, social scientists, and management scientists.
□ Chapters
□ Chapter 1. Noncooperative games
□ Chapter 2. Evolutionary stability and other selection criteria
□ Chapter 3. Cooperative games in strategic form
□ Chapter 4. Characteristic function games
□ Chapter 5. Cooperation and the prisoner’s dilemma
□ Chapter 6. More population games
□ Chapter 7. Appraisal
□ Chapter 8. The tracing procedure
□ Chapter 9. Solutions to selected exercises
□ This book helps not only to make game theory accessible, but also to convey both its power and scope in a variety of applications. The books deals in a unified manner with the central ideas
of both classical and evolutionary game theory. The key ideas are illustrated by a variety of well-chosen examples.
Zentralblatt MATH
□ Mesterton-Gibbons' book deals with mathematical modelling, not by an abstract discussion of how modelling should be done, but rather by presenting many concrete examples ... The mathematics
described [in the book] is fascinating and well worth studying ... The examples are great, and the author has clearly put enormous effort into building this collection ... a perfect source of
problems for a Moore method course ... a valuable contribution to the literature ... Everyone interested in game theory or mathematical modelling should take a look at it.
MAA Online
□ From reviews for the First Edition:
Readers will be hard-pressed to find a general introduction to game theory that blends biological and mathematical approaches more expertly. It is both a well-rounded survey and a reference
work of lasting value.
Behavioral Ecology
□ This book is an introduction to game theory with two specific features: it is written by a mathematician ... and it is written from the perspective of a mathematical modeller. This last
characteristic implies that all chapters start with examples and the general concepts are only presented once the specific examples have been carefully developed ... I find this book
excellent and ... worth considering when teaching an undergraduate course in game theory to students having some mathematical maturity (some calculus, some knowledge of matrix analysis and
Zentralblatt MATH
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 11; 2001; 368 pp
MSC: Primary 91; Secondary 92
Now available in Third Edition: AMSTEXT/37
This book is about using game theory in mathematical modelling. It is an introductory text, covering the basic ideas and methods of game theory as well as the necessary ideas from the vast spectrum
of scientific study where the methods are applied.
It has by now become generally apparent that game theory is a fascinating branch of mathematics with both serious and recreational applications. Strategic behavior arises whenever the outcome of an
individual's action depends on actions to be taken by other individuals—whether human, as in the Prisoners' Dilemma, or otherwise, as in the “duels of damselflies”. As a result, game-theoretic
mathematical models are applicable in both the social and natural sciences. In reading this book, you can learn not just about game theory, but also about how to model real situations so that they
can be analyzed mathematically.
Mesterton-Gibbons includes the familiar game theory examples where they are needed for explaining the mathematics or when they provide a valuable application. There are also plenty of new examples,
in particular from biology, such as competitions for territory or mates, games among kin versus games between kin, and cooperative wildlife management.
Prerequisites are modest. Students should have some mathematical maturity and a familiarity with basic calculus, matrix algebra, probability, and some differential equations. As Mesterton-Gibbons
writes, “The recurring theme is that game theory is fun to learn, doesn't require a large amount of mathematical rigor, and has great potential for application.”
This new edition contains a significant amount of updates and new material, particularly on biological games. An important chapter on population games now has virtually all new material. The book is
absolutely up-to-date with numerous references to the literature. Each chapter ends with a commentary which surveys current developments.
Advanced undergraduates, graduate students, and research mathematicians interested in mathematical modelling; applied mathematicians; biologists, social scientists, and management scientists.
• Chapters
• Chapter 1. Noncooperative games
• Chapter 2. Evolutionary stability and other selection criteria
• Chapter 3. Cooperative games in strategic form
• Chapter 4. Characteristic function games
• Chapter 5. Cooperation and the prisoner’s dilemma
• Chapter 6. More population games
• Chapter 7. Appraisal
• Chapter 8. The tracing procedure
• Chapter 9. Solutions to selected exercises
• This book helps not only to make game theory accessible, but also to convey both its power and scope in a variety of applications. The books deals in a unified manner with the central ideas of
both classical and evolutionary game theory. The key ideas are illustrated by a variety of well-chosen examples.
Zentralblatt MATH
• Mesterton-Gibbons' book deals with mathematical modelling, not by an abstract discussion of how modelling should be done, but rather by presenting many concrete examples ... The mathematics
described [in the book] is fascinating and well worth studying ... The examples are great, and the author has clearly put enormous effort into building this collection ... a perfect source of
problems for a Moore method course ... a valuable contribution to the literature ... Everyone interested in game theory or mathematical modelling should take a look at it.
MAA Online
• From reviews for the First Edition:
Readers will be hard-pressed to find a general introduction to game theory that blends biological and mathematical approaches more expertly. It is both a well-rounded survey and a reference work
of lasting value.
Behavioral Ecology
• This book is an introduction to game theory with two specific features: it is written by a mathematician ... and it is written from the perspective of a mathematical modeller. This last
characteristic implies that all chapters start with examples and the general concepts are only presented once the specific examples have been carefully developed ... I find this book excellent
and ... worth considering when teaching an undergraduate course in game theory to students having some mathematical maturity (some calculus, some knowledge of matrix analysis and probability).
Zentralblatt MATH
Permission – for use of book, eBook, or Journal content
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Discrete Random Variables - The Culture SGDiscrete Random Variables
As usual, definitions first.
The cumulative distribution function (CDF),
A discrete random variable, X, has probability mass function (PMF),
The expected value of a discrete random variable, X is given by
The variance of an random variable, X, is defined as
Now we can look at some examples of discrete random variables.
We say the X has a binomial distribution, that is,
For example, X can represent the number of heads in n independent coin tosses, where
A simpler case of binomial where there is only one event is called the Bernoulli distribution.
I’ll give a simple illustration of binomial model used in finance. This was also used in finance engineering in the beginning.
Suppose a fund manager outperforms the market in a given year with probability p and under performs the market with probability 1 – p. She has a track record of 10 years and has outperformed the
market in 8 out of 10 years. We also note that performance in any one year is independent of performance in other years.
From this illustration, we note that there are only two outcomes, she outperforms or underperforms. We can let X be the number of outperforming years. Assuming the fund manager has no skill, at least
8 out of 10 years.
An extension here will be to consider there are M fund managers instead of 1 now.
Poisson Distribution
We say that X has a
Next, we look at Bayes Theorem, also known as “conditional probability” in H2 Mathematics.
Let A and B be two events for which
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Optimal Design of Fractional-Order Electrical Network for Vehicle Mechatronic ISD Suspension Using the Structure-Immittance Approach
Automotive Engineering Research Institute, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China
School of Automotive and Traffic Engineering, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China
School of Electromechanical Engineering, Guangdong University of Technology, Guangzhou 510006, China
Author to whom correspondence should be addressed.
Submission received: 30 November 2022 / Revised: 30 December 2022 / Accepted: 3 January 2023 / Published: 4 January 2023
In order to more effectively design the structure of vehicle ISD (Inerter Spring Damper) suspension system using the inerter, this paper proposed a design method using a fractional-order electrical
network structure of a mechatronic inerter for fractional-order electrical network components, according to the characteristics that the external electrical network of a mechatronic inerter can
simulate the corresponding mechanical network structure equivalently. First, the 1/4 dynamic model of the suspension is constructed. The improved Oustaloup filtering algorithm is used to simulate
fractional calculus, and the fractional order components are simulated. Then, the simulation model of the vehicle mechatronic ISD suspension is established. In order to simplify the electrical
network, one resistance, one fractional inductance and one fractional capacitance are limited in the design of the fractional electrical network at the outer end of the mechatronic inerter. The
structure-immittance approach is used to obtain two general layouts of all possible structures of three elements. At the same time, the optimal fractional electrical network structure and parameters
are obtained by combining the optimization algorithm. The simulation results verify the performance of the fractional ISD suspension with the optimized structure, which can provide a new idea for the
structural design of a fractional-order electrical network applied in vehicle mechatronic ISD suspension.
1. Introduction
The proposition of the inerter [
] breaks through the inherent structure of the existing suspension system “spring damper” parallel connection, and forms a new suspension structure system. This suspension, consisting of spring,
damper and inerter elements, is called ISD suspension. Scholars all over the world have adopted many methods to realize the inerter [
], and after the application of the inerter, the performance potential of the vibration isolation system has also been expanded to include aircraft [
], trains [
], buildings [
], bridges [
], etc. The structural design of ISD suspension plays an important role in meeting various performance indicators of vehicles. The question of how to design the structure of ISD suspension has
attracted the attention of scholars at home and abroad.
Common ISD suspension structure design methods include the structure approach, the immitance approach and the structure-immitance approach. The structure approach [
] limits the number of components in the suspension, and integrates them into parameter optimization according to the feasible range of component parameters. The disadvantage is that the arrangement
and combination method has a huge workload, which makes it difficult to cover a wide range of mechanical networks, and it is easy to omit structures with excellent performance. The immitance approach
] replaces the suspension structure with a fixed form of impedance or admittance expression, uses the parameter optimization method to optimize the solution and finally realizes it passively through
network synthesis. The suspension component parameters obtained by the immitance approach often do not conform to the routine, which is not conducive to engineering realization. The
structure-immitance approach [
] can be used to express the structure with a predetermined number of elements with a general impedance expression. However, the structural complexity of a pure mechanical network is high, which has
affected the engineering design of ISD suspension. The mechatronic inerter [
] is a device coupled by a mechanical inerter and a rotating motor. The external circuit impedance of the mechatronic inerter can be used to simulate the target mechanical impedance to achieve the
passive structure design of complex mechanical network, overcome the space limitation of pure mechanical network suspension structure and expand the design idea of suspension system structure.
However, for the electrical network at the outer end of the mechatronic inerter, the increase of the order of its impedance transfer function will bring higher performance improvement [
], and at the same time, the difficulty of network integration will also increase greatly. For example, the bicubic impedance transfer function requires no more than 13 elements to realize passively
], and its structure is complex.
In the structural design of the suspension system, fractional calculus theory has also been commonly used [
], and its feasibility has also been verified [
], indicating that the fractional-order function can more accurately describe the dynamic characteristics of complex systems than the integral-order function. Using fractional-order electrical
network elements to replace the original integer-order electrical network at the outer end of the mechatronic inerter can effectively avoid the high complexity of pure integer order network
structures. However, the structural design of fractional-order electrical networks in vehicle mechatronic ISD suspension has not been reported yet, and in the design of a fractional-order vehicle
mechatronic ISD suspension, a simple and clear fractional-order electrical network structure is essential. Therefore, this paper will use the structure-immitance approach to study the optimal design
of the fractional-order electrical network structure for a vehicle mechatronic ISD suspension. The content layout of the rest of this paper is as follows.
First, in
Section 2
, the definition and algorithm realization of fractional calculus are introduced, and the equivalent realization relationship between fractional electrical components and fractional mechanical
components are analyzed. In
Section 3
, a quarter suspension dynamic simulation model is established, and fractional-order electrical components are used in the design of the electrical network, and the structure-immitance approach is
used to design the electrical network structure. Then, in
Section 4
, the electrical network structure and parameters of the suspension system are obtained through optimization. Finally, in
Section 5
, the dynamic performance of the optimized fractional-order ISD suspension is evaluated by comparison, and some conclusions are made in
Section 6
2. Equivalent Realization of Fractional Passive Network Elements
Fractional calculus has the basic operator
$t 0 D t α$
, among which, α is limited to real numbers,
are the upper and lower bounds of the operator. The unified definition of fractional calculus operator [
] is:
$t 0 D t α f ( t ) = d α d t α f ( t ) , α > 0 f ( t ) , α = 0 ∫ t 0 t f ( τ ) d τ − α , α < 0$
There are many definitions of fractional calculus. This paper adopts the Grünwald–Letnikow [
] fractional calculus definition. The Grünwald–Letnikow of the α derivative of a given function
) is defined as:
$D t 0 G L t α f ( t ) = lim h → 0 1 h α ∑ j = 0 [ ( t − t 0 ) / h ] ( − 1 ) j α j f ( t − j h )$
where [·] means taking the nearest integer. At the same time, in order to ensure the approximation effect at the frequency band boundary and ensure that the transfer function is regular, the improved
Oustaloup filtering algorithm [
] is considered to approximate fractional calculus. The mathematical model of the improved Oustaloup filter is:
$s γ ≈ d ω h b γ d s 2 + b ω h s d ( 1 − γ ) s 2 + b ω h s + d γ ∏ k = 1 N s + ω ′ k s + ω k$
$ω ′ k = ω b ω u ( 2 k − 1 − γ ) / N , ω k = ω b ω u ( 2 k − 1 + γ ) / N , ω u = ω h / ω b$
is the filter order,
is the fractional order,
are zero point and the pole, respectively.
are the upper and lower limits of frequency bands, respectively. In general, the weighting parameters
= 10,
= 9. In this paper, the filter frequency band is (10
, 10
) rad/s. The larger the filter order, the higher the approximation accuracy. In this frequency band, the fifth order Oustaloup filtering effect has met the accuracy requirements, so the selection
order is five.
In the new mechanical and electrical analogy, the spring and the inductance, the damper and the resistance, and the inerter and capacitance are similar, respectively [
]. According to the above fractional definition and approximation method, the impedance expression of fractional network elements (including mechanical network and electrical network) is obtained by
using the form of pull transform, with excitation force as the input and corresponding speed as the output, as shown in
Table 1
, where s is the Laplace variable,
are fractional orders.
3. Model Construction of Vehicle Mechatronic ISD Suspension System
3.1. The Ball-Screw Mechatronic Inerter
A mechatronic inerter is considered in this paper, which is formed by coupling a ball-screw inerter with a rotary motor. Its structural diagram is shown in
Figure 1
. The relative linear motion of the two ends of the mechanical inerter can be converted into the rotary motion of the motor. The inductor, resistor and capacitor in the electrical network at the
outer end of the rotary motor can equivalently simulate the spring, damper, and inerter in the mechanical network structure.
3.2. Mechatronic ISD Suspension Structure Layout
The quarter suspension model is a typical vibration model of vehicle suspension system, which is a basic dynamic model for studying its vertical performance. In this paper, a quarter vehicle
mechatronic ISD suspension dynamics model is established, as shown in
Figure 2
. Based on a mature vehicle model,
Table 2
illustrates the parameters for the model.
The dynamic Laplace equation of the suspension model is shown in Equation (5):
$m s s 2 Z s + [ k + c s + s B ( s ) ] ( Z s − Z u ) = 0 m u s 2 Z u − [ k + c s + s B ( s ) ] ( Z s − Z u ) + k t ( Z u − Z r ) = 0$
, and
are spring stiffness, tire stiffness, and the damping coefficient, respectively,
are the sprung mass and the unsprung mass, respectively.
are the vertical displacements of the sprung mass, the unsprung mass, and road roughness, respectively, and
are their Laplace transforms, respectively.
) is the impedance expression of the mechatronic inerter, which is shown as follows [
$B ( s ) = b s + K m Z e ( s ) K m = 2 π P 2 k t k e$
is the inertance of the ball-screw mechatronic inerter,
is the pitch of the ball-screw mechanism,
is the induced electromotive force coefficient of the rotary motor,
is the thrust coefficient of the rotary motor.
is the electromechanical parameter conversion coefficient of the ball-screw mechatronic inerter, which is taken as 7056 HN/m in this paper.
) is the impedance expression of the external electrical network of the rotary motor. The fractional-order external electrical network of the mechatronic inerter includes resistor(s),
fractional-order capacitor(s) and fractional-order inductor(s). In order to simplify the electrical network, the number of resistors, fractional capacitors and fractional inductors is limited to one
in the optimal design. Eight structures of the three element arrangement are summarized using the structure-immittance approach, and two general structures are used for general expression, as shown
Figure 3
Figure 4
The impedance transfer function expressions of the two general structures in
Figure 3
Figure 4
are, respectively, as follows:
$Y 1 ( s ) = C 1 R s α + β + C ( 1 L 4 + 1 L 6 ) s β + 1 R ( 1 L 2 + 1 L 6 ) C 1 R L 3 s 2 α + β + C s α + β + 1 R s α + 1 L 2 + 1 L 4$
$Y 2 ( s ) = C 1 R ( L 1 + L 2 ) s 2 α + β + C s α + β + 1 R s α + 1 L 3 C ( L 1 + L 5 ) s 2 α + β + 1 R ( L 2 + L 5 ) s α + β + s α$
, and
are fractional-order inductors,
are the resistor and the fractional-order capacitor, respectively.
are the fractional-order inductance order and the fractional-order capacitance order, respectively. In the
) structure, at least three of
, 1/
are zero, and in the
) structure, at least three of 1/
, 1/
and 1/
are zero. For example, in
Figure 3
, when
are zero, it is a structure in which a fractional-order inductor is connected in parallel with a fractional-order capacitor, and then connected in series with a resistor. In
Figure 4
, when 1/
, 1/
and 1/
are zero, it is a fractional-order capacitor in series with a fractional-order inductor, and then in parallel with a resistor.
4. Parameter Optimization Design
4.1. Pattern Search Optimization Algorithm
In this paper, the pattern search method [
] is used for the optimization design of the suspension system. As a general algorithm for solving the optimal value of a function, the greatest advantage of pattern search method is that it does not
need to use the derivative of the objective optimization function in the algorithm program of pattern search method. Therefore, pattern search method can effectively solve the optimization problems
of non-derivative functions and complex derivative functions. The specific steps of pattern search method are shown in
Figure 5
4.2. Optimization Results
To ensure vehicle ride comfort, the RMS (root-mean-square) values of the vehicle body acceleration, the suspension working space and the dynamic tire load are selected as evaluation indicators, and
the traditional passive suspension is chosen as evaluation benchmark to establish the optimization objective function, as shown below:
$f = B A P B A pas + S W S P S W S pas + D T L P D T L pas$
$P = b c L e C e R e α β$
are the RMS values of the vehicle body acceleration of the suspension to be optimized and the traditional passive suspension, respectively,
are the RMS values of the suspension working space of the suspension to be optimized and the traditional passive suspension, respectively, and
are the RMS values of the dynamic tire load of the suspension to be optimized and the traditional passive suspension, respectively.
are calculated by a mature traditional passive suspension [
], and their performances have reached a high level, which are 1.3096 m·s
, 0.0130 m and 900.4704 N, respectively.
represents the set of parameters to be optimized for the suspension system, including inertance
, damping coefficient
, fractional-order inductance coefficient
, fractional-order capacitance coefficient
, resistance coefficient
, fractional-order inductance order
, and fractional-order capacitance order
. Their constraints are as follows:
$b , c ≥ 0 L e , R e , C e ≥ 0 1 ≥ α , β ≥ 0$
The optimized fractional-order electrical network structure is shown in
Figure 6
. This structure is the case when
, 1/
are zero in
) structure. Set the fractional-order inductance order
and the fractional-order capacitance order
to 1 for optimization, and get the integer-order ISD suspension system parameters. The optimization parameters of fractional-order ISD suspension and integral-order ISD suspension are shown in
Table 3
5. Simulation Analysis
5.1. The Characteristics of Bode Diagram
Compared with the Bode diagram of the traditional passive suspension,
Figure 7
shows the Bode diagram of vehicle mechatronic ISD suspension applying the optimized fractional-order electrical network structure.
It can be seen that for the fractional-order ISD suspension, in the low frequency range [10
, 2] Hz, the optimized structure shape is similar to the spring. In the range [
] Hz, the structure shape is similar to the damper, and above 4 Hz, the optimized structure is similar to the inerter, which is the difference between the traditional passive suspension system and
the optimized structure. The traditional suspension system composed of “spring damper” mechanical components cannot show inertia characteristics, which is the main factor limiting the performance
improvement of the traditional suspension structure, and also the reason why the ISD suspension of vehicles containing the inerter has better vibration isolation performance.
5.2. Random Road Input
The random road input is selected as the road input model to study the advantages of the optimized fractional-order ISD suspension compared with the integral-order ISD suspension and the traditional
passive suspension. The random road input model is as follows [
$z ˙ r ( t ) = − 0.111 [ u z r ( t ) + 40 G q ( n 0 ) u w ( t ) ]$
) and
) are the vehicle speed, the vertical input displacement, the white noise with the mean value of 0, and the road roughness coefficient, respectively. Class C pavement is selected in this paper, and
the pavement roughness coefficient is 2.56 × 10
Figure 8
Figure 9
Figure 10
Table 4
show the comparison of the RMS values of the vehicle body acceleration, the suspension working space and the dynamic tire load of the three suspension systems at a speed of 20 m/s.
The optimization adopts multi-objective optimization, and the final optimization result is the best case of comprehensive improvement. From the data point of view, the RMS value of suspension working
space has the best effect, and the other two indexes have also been improved. It can be seen that, compared with the traditional passive suspension, the RMS values of the vehicle body acceleration of
the integral-order ISD suspension and the fractional-order ISD suspension are reduced by 3.44% and 4.12%, respectively. The RMS values of the suspension working space of the two suspensions are
reduced by 22.31% and 23.08%, respectively. Furthermore, the RMS values of the dynamic tire load of the two suspensions are reduced by 2.73% and 5.31%, respectively, showing the advantages of the
designed fractional-order electric network structure. It shows that the vehicle mechatronic ISD suspension with optimized fractional-order electrical network structure can further improve the
vibration isolation performance of the suspension system.
6. Conclusions
In this paper, the optimal design of fractional-order electrical network for vehicle mechatronic ISD suspension is studied. An optimization design method of fractional-order electrical network for
vehicle mechatronic ISD suspension is proposed by using the structure-immittance approach. The structural parameters of the fractional-order vehicle mechatronic ISD suspension are optimized by
establishing a 1/4 dynamic model of the suspension. Through simulation comparison, the results show that the performance of the vehicle mechatronic ISD suspension system applying the fractional-order
electrical network structure obtained by optimization design is further improved, which provides a reference for the structural design of fractional-order electrical network components based vehicle
mechatronic ISD suspension.
Author Contributions
Conceptualization, J.H. and Y.S.; methodology, Y.Z.; software, J.H.; validation, X.Y.; formal analysis, Y.L.; investigation, Y.S.; writing—original draft preparation, J.H.; writing—review and
editing, X.Y.; supervision, Y.Z. All authors have read and agreed to the published version of the manuscript.
This research was funded by the National Natural Science Foundation of China under Grant 52002156, 52072157 and 52008259, the Natural Science Foundation of Jiangsu Province under Grant BK20200911.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
Mechanical Network Elements Impedance Electrical Network Elements Impedance
Spring k/s^α Inductor 1/Ls^α
Damper c Resistor 1/R
Inerter bs^β Capacitor Cs^β
Parameters Values
Sprung Mass m[s]/kg 320
Unsprung Mass m[u]/kg 45
Spring Stiffness k/N m^−1 22,000
Tire Stiffness k[t]/N m^−1 190,000
Fractional-Order ISD Suspension Integer-Order ISD Suspension
Parameters Values Parameters Values
Inertance b/kg 5 Inertance b/kg 13
Damping coefficient c/N·s·m^−1 1074 Damping coefficient c/N·s·m^−1 232
Fractional-order inductance coefficient L[e]/H 1.05 Inductance coefficient L[e]/H 1.34
Fractional-order capacitance coefficient C[e]/F 0.06 Capacitance coefficient C[e]/F 0.03
Resistance coefficient R[e]/Ω 320.73 Resistance coefficient R[e]/Ω 5.56
Fractional-order inductance order α 0.28 - -
Fractional-order capacitance order β 0.81 - -
Performance Index Traditional Passive Suspension Integer-Order Isd Suspension Improvement Fractional-Order Isd Suspension Improvement
RMS of vehicle body acceleration/(m·s^−2) 1.3096 1.3051 3.44% 1.3042 4.12%
RMS of suspension working space/(m) 0.0130 0.0101 22.31% 0.0100 23.08%
RMS of dynamic tire load/(N) 900.4704 875.8558 2.73% 852.6704 5.31%
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MDPI and ACS Style
Hua, J.; Shen, Y.; Yang, X.; Zhang, Y.; Liu, Y. Optimal Design of Fractional-Order Electrical Network for Vehicle Mechatronic ISD Suspension Using the Structure-Immittance Approach. World Electr.
Veh. J. 2023, 14, 12. https://doi.org/10.3390/wevj14010012
AMA Style
Hua J, Shen Y, Yang X, Zhang Y, Liu Y. Optimal Design of Fractional-Order Electrical Network for Vehicle Mechatronic ISD Suspension Using the Structure-Immittance Approach. World Electric Vehicle
Journal. 2023; 14(1):12. https://doi.org/10.3390/wevj14010012
Chicago/Turabian Style
Hua, Jie, Yujie Shen, Xiaofeng Yang, Ying Zhang, and Yanling Liu. 2023. "Optimal Design of Fractional-Order Electrical Network for Vehicle Mechatronic ISD Suspension Using the Structure-Immittance
Approach" World Electric Vehicle Journal 14, no. 1: 12. https://doi.org/10.3390/wevj14010012
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Show that the expression \
Hint: In the question, LHS has inverse sine functions and RHS has inverse cosine function. We have inverse cosine functions in the RHS of the given expression and inverse sine function in the LHS of
the given expression. So first we will try to transform them. We can do so by taking the term \[x={{\sin }^{-1}}\dfrac{3}{5}\] and \[y={{\sin }^{-1}}\dfrac{8}{17}\]. Then, we can apply the formula \
[\cos (x-y)=cosx.cosy+sinx.siny\] to simplify and solve further.
Complete step by step answer:
Now, we have to remove the inverse functions from the terms present in LHS.
Let us assume,
\[x=si{{n}^{-1}}\dfrac{3}{5}\] ………….. (1)
The RHS part in the question is given in inverse cosine function. So, we need to convert this inverse cosine into a cosine function.
Now, applying sine in LHS as well as RHS in the equation,
\[ x=si{{n}^{-1}}\dfrac{3}{5} \]
\[ \Rightarrow \sin x=\dfrac{3}{5} \]
Also, we have to convert the sine function into cosine.
We know the identity \[{{\sin }^{2}}x+{{\cos }^{2}}x=1\] ,using this identity we get
\[\cos x=\sqrt{1-{{\sin }^{2}}x}\] ………….(2)
Putting the value of “sin x” in the equation (2), we get
\[\cos x=\sqrt{1-{{\sin }^{2}}x}\]
\[ \cos x=\sqrt{1-{{\left( \dfrac{3}{5} \right)}^{2}}} \]
\[ \Rightarrow \cos x=\sqrt{1-\dfrac{9}{25}} \]
\[ \Rightarrow \cos x=\sqrt{\dfrac{25-9}{25}} \]
\[ \Rightarrow \cos x=\sqrt{\dfrac{16}{25}} \]
\[ \Rightarrow \cos x=\dfrac{4}{5} \]
Similarly, let us assume,
\[y=si{{n}^{-1}}\dfrac{8}{17}\]………….. (3)
Now, applying sine in LHS as well as RHS in the equation,
\[ y=si{{n}^{-1}}\dfrac{8}{17} \]
\[ \Rightarrow \sin y=\dfrac{8}{17} \]
Also, we have to convert the sine function into cosine.
We know that, \[\operatorname{cosy}=\sqrt{1-{{\sin }^{2}}y}\] .
Putting the value of “sin y” in the equation, , we get
\[\operatorname{cosy}=\sqrt{1-{{\sin }^{2}}y}\]
\[ \operatorname{cosy}=\sqrt{1-{{\left( \dfrac{8}{17} \right)}^{2}}} \]
\[ \Rightarrow \operatorname{cosy}=\sqrt{1-\dfrac{64}{289}} \]
\[ \Rightarrow \operatorname{cosy}=\sqrt{\dfrac{289-64}{289}} \]
\[ \Rightarrow \operatorname{cosy}=\sqrt{\dfrac{225}{289}} \]
\[ \Rightarrow \operatorname{cosy}=\dfrac{15}{17} \]
Putting the equation(1) and equation (3) in the given expression, we get,
\[ {{\sin }^{-1}}\dfrac{3}{5}-{{\sin }^{-1}}\dfrac{8}{17} \]
\[ =x-y \]
This means we have to find (x-y).
As in RHS, we have cosine terms, so using the formula expand cos(x-y).
\[ \cos (x-y)=cosx.cosy-sinx.siny \]
\[ \cos (x-y)=\dfrac{4}{5}.\dfrac{15}{17}+\dfrac{3}{5}.\dfrac{8}{17} \]
\[ \cos (x-y)=\dfrac{84}{85} \]
\[ (x-y)=co{{s}^{-1}}\dfrac{84}{85}. \]
Therefore, LHS=RHS.
Hence, proved.
Note: We can also solve this question, after the conversion of the inverse of cosine present in LHS as well as RHS into the inverse of tan. We can do so by taking \[{{\sin }^{-1}}(3/5)=x\] and \[{{\
sin }^{-1}}(8/17)=y\].
\[ {{\sin }^{-1}}\left( \dfrac{3}{5} \right)=x \]
\[ \Rightarrow sinx=\dfrac{3}{5} \]
Using the Pythagoras theorem, we can find the perpendicular.
\[Perpendicular=\sqrt{{{\left( hypotenuse \right)}^{2}}-{{\left( base \right)}^{2}}}\]
\[ =\sqrt{{{5}^{2}}-{{3}^{2}}} \]
\[ =\sqrt{25-9} \]
\[ =\sqrt{16} \]
\[ =4 \]
\[\tan x=\dfrac{Perpendicular}{Base}\]
\[\tan x=\dfrac{3}{4}\]
\[ si{{n}^{-1}}\left( \dfrac{8}{17} \right)=y \]
\[ \Rightarrow siny=\dfrac{8}{17} \]
Using the Pythagoras theorem, we can find the base.
\[Base=\sqrt{{{\left( Hypotenuse \right)}^{2}}-{{\left( Perpendicular \right)}^{2}}}\]
\[ =\sqrt{{{17}^{2}}-{{8}^{2}}} \]
\[ =\sqrt{289-64} \]
\[ =\sqrt{225} \]
\[ =15 \]
\[\tan x=\dfrac{Perpendicular}{Base}\]
\[\tan x=\dfrac{8}{15}\]
And then using the formula \[{{\tan }^{-1}}A+{{\tan }^{-1}}B={{\tan }^{-1}}\left( \dfrac{A+B}{1-AB} \right)\] , we can get the required result. | {"url":"https://www.vedantu.com/question-answer/show-that-the-expression-sin-1dfrac35sin-class-12-maths-cbse-5fb2caffb7fb205f4fd9a884","timestamp":"2024-11-07T15:22:08Z","content_type":"text/html","content_length":"217723","record_id":"<urn:uuid:fa0528fd-e53c-4866-a96e-b6fb3c542924>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00506.warc.gz"} |
The Positioner Coordinate System (O[p], X[p], Y[p], Z[p]) is defined as follows:
The quantities SID and ISO are specified by the Attributes Distance Source to Detector (0018,1110) and Distance Source to Isocenter (0018,9402) respectively.
The Positioner Coordinate System (O[p], X[p], Y[p], Z[p]) is characterized, with respect to the Isocenter Coordinate System (O, X, Y, Z), by two angles describing the X-Ray center beam, and a third
angle describing the rotation of the X-Ray detector plane. These angles are relative to the Isocenter reference system, and independent from the patient position on the equipment.
Positioner Isocenter Primary Angle (0018,9463) (so-called Ap[1] in Figure C.8.19.6-6) is defined in the plane XY, as the angle between the plane YZ and the plane Y[p]Z. The axis of rotation of this
angle is the Z axis. Angle from -Y to +X is positive. The valid range of this angle is -180 to +180 degrees.
Positioner Isocenter Secondary Angle (0018,9464) (so-called Ap[2] in Figure C.8.19.6-6) is defined in the plane Y[p]Z, as the angle of the X-Ray Center Beam (i.e., Y[p]) relative to the XY plane. The
axis of rotation of this angle is perpendicular to the plane Y[p]Z. Angle from the plane XY to +Z is positive. The valid range of this angle is -180 to +180 degrees.
Positioner Isocenter Detector Rotation Angle (0018,9465) (so-called Ap[3] in Figure C.8.19.6-6 and in Figure C.8.19.6-7) is defined in the detector plane, as the angle of the vertical scan-lines of
the detector (i.e., Z[p]) relative to the intersection between the detector plane and the plane Y[p]Z. The sign of this angle is positive clockwise when facing on to the detector plane (see
Figure C.8.19.6-7). The valid range of this angle is -180 to +180 degrees.
The table coordinate system (O[t], X[t], Y[t], Z[t]) is defined as follows:
- Origin O[t], so-called Table Reference Point, is on the Table Top plane
- +X[t] direction to the TABLE LEFT
- +Z[t] direction to the TABLE HEAD
- +Y[t] direction to the TABLE DOWN
The table coordinate system (O[t], X[t], Y[t], Z[t]) is characterized, with respect to the Isocenter coordinate system (O, X, Y, Z), by a 3D translation and 3 angles describing the tilting and
Table X Position to Isocenter (0018,9466) (so-called T[X] in Figure C.8.19.6-8) is defined as the translation of the Table Reference Point O[t] with respect to the Isocenter Coordinate System in the
X direction. Table motion toward +X is positive.
Table Y Position to Isocenter (0018,9467) (so-called T[Y] in Figure C.8.19.6-8) is defined as the translation of the Table Reference Point O[t] with respect to the Isocenter Coordinate System in the
Y direction. Table motion toward +Y is positive.
Table Z Position to Isocenter (0018,9468) (so-called T[Z] in Figure C.8.19.6-8) is defined as the translation of the Table Reference Point O[t] with respect to the Isocenter Coordinate System in the
Z direction. Table motion toward +Z is positive.
A translation of ( T[X],T[Y],T[Z] ) = (0, 0, 0) means that the Table Reference Point O[t] is at the System Isocenter.
Table Horizontal Rotation Angle (so-called At[1] in Figure C.8.19.6-9) is defined in the horizontal plane XZ, as the angle of the projection of the +Zt axis in the XZ plane relative to the +Z axis.
The axis of rotation of this angle is the vertical axis crossing the Table Reference Point Ot. Zero value is defined when the projection of +Zt in the XZ plane is equal to +Z. Angle from +Z to +X is
positive. The valid range of this angle is -180 to +180 degrees.
Table Head Tilt Angle (so-called At[2] in Figure C.8.19.6-9) is defined in the vertical plane containing Z[t] (i.e., YZ[t]), as the angle of the +Z[t] axis relative to the horizontal plane XZ. The
axis of rotation of this angle is defined as the intersection between the horizontal plane XZ and the plane X[t]Y[t]. Zero value is defined when +Z[t] is contained in the horizontal plane XZ. Angle
from horizontal (plane XZ) to -Y direction (upwards) is positive, indicating that the head of the table is above the horizontal plane. The valid range of this angle is -45 to +45 degrees.
Table Cradle Tilt Angle (so-called At[3] in Figure C.8.19.6-9) is defined in the X[t]Y[t] plane, as the angle of the +X[t] axis relative to the intersection between the X[t]Y[t] plane and the
horizontal plane XZ. The axis of rotation of this angle is the axis Z[t]. Zero value is defined when +X[t] is contained in the horizontal plane XZ. Angle from horizontal (plane XZ) to -Y direction
(upwards) is positive, indicating that the left of the table is above the horizontal plane. The valid range of this angle is -45 to +45 degrees.
The angles At[1] , At[2] and At[3] are independent from any specific mechanical design of the table rotation axis defined by a manufacturer. In particular, they don't require the three rotation axis
to cross on a single point. If a mechanical rotation axis does not cross the Table Reference Point O[t], a mechanical rotation around this axis will generate a change in one or more table angles as
well as a translation of the Table Reference Point. | {"url":"https://dicom.nema.org/medical/dicom/current/output/chtml/part03/sect_C.8.19.6.13.html","timestamp":"2024-11-12T13:33:50Z","content_type":"application/xhtml+xml","content_length":"52668","record_id":"<urn:uuid:f25f58c2-b62e-4752-8966-011695532fe2>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00117.warc.gz"} |
GRE Quant - Counting Methods Theory
This post is a part of [
Frequency of the concepts tested
Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.
Enumeration is a method of counting all possible ways to arrange elements. Although it is the simplest method, it is often the fastest method to solve hard GRE problems and is a pivotal principle for
any other combinatorial method. In fact, combination and permutation is shortcuts for enumeration. The main idea of enumeration is writing down all possible ways and then count them. Let's consider a
few examples:
Example #1
Q:. There are three marbles: 1 blue, 1 gray and 1 green. In how many ways is it possible to arrange marbles in a row?
Solution: Let's write out all possible ways:
Answer is 6.
In general, the number of ways to arrange n different objects in a row
Example #2
Q:. There are three marbles: 1 blue, 1 gray and 1 green. In how many ways is it possible to arrange marbles in a row if blue and green marbles have to be next to each other?
Solution: Let's write out all possible ways to arrange marbles in a row and then find only arrangements that satisfy question's condition:
Answer is 4.
Example #3
Q:. There are three marbles: 1 blue, 1 gray and 1 green. In how many ways is it possible to arrange marbles in a row if gray marble have to be left to blue marble?
Solution: Let's write out all possible ways to arrange marbles in a row and then find only arrangements that satisfy question's condition:
Answer is 3.
Arrangements of n different objects
Enumeration is a great way to count a small number of arrangements. But when the total number of arrangements is large, enumeration can't be very useful, especially taking into account GRE time
restriction. Fortunately, there are some methods that can speed up counting of all arrangements.
The number of arrangements of n different objects in a row is a typical problem that can be solve this way:
1. How many objects we can put at 1st place? n.
2. How many objects we can put at 2nd place? n - 1. We can't put the object that already placed at 1st place.
n. How nany objects we can put at n-th place? 1. Only one object remains.
Therefore, the total number of arrangements of n different objects in a row is
\(N = n*(n-1)*(n-2)....2*1 = n!\)
A combination is an unordered collection of k objects taken from a set of n distinct objects. The number of ways how we can choose k objects out of n distinct objects is denoted as:
knowing how to find the number of arrangements of n distinct objects we can easily find formula for combination:
1. The total number of arrangements of n distinct objects is n!
2. Now we have to exclude all arrangements of k objects (k!) and remaining (n-k) objects ((n-k)!) as the order of chosen k objects and remained (n-k) objects doesn't matter.
\(C^n_k = \frac{n!}{k!(n-k)!}\)
A permutation is an ordered collection of k objects taken from a set of n distinct objects. The number of ways how we can choose k objects out of n distinct objects is denoted as:
knowing how to find the number of arrangements of n distinct objects we can easily find formula for combination:
1. The total number of arrangements of n distinct objects is n!
2. Now we have to exclude all arrangements of remaining (n-k) objects ((n-k)!) as the order of remained (n-k) objects doesn't matter.
\(P^n_k = \frac{n!}{(n-k)!}\)
If we exclude order of chosen objects from permutation formula, we will get combination formula:
\(\frac{P^n_k}{k!} = C^n_k\)
Circular arrangements
Let's say we have 6 distinct objects, how many relatively different arrangements do we have if those objects should be placed in a circle.
The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a
circle. So, for the number of circular arrangements of n objects we have:
\(R = \frac{n!}{n} = (n-1)!\)
Tips and Tricks
Any problem in Combinatorics is a counting problem. Therefore, a key to solution is a way how to count the number of arrangements. It sounds obvious but a lot of people begin approaching to a problem
with thoughts like "Should I apply C- or P- formula here?". Don't fall in this trap: define how you are going to count arrangements first, realize that your way is right and you don't miss something
important, and only then use C- or P- formula if you need them. | {"url":"https://gre.myprepclub.com/forum/gre-quant-counting-methods-theory-18819.html","timestamp":"2024-11-13T04:18:11Z","content_type":"application/xhtml+xml","content_length":"205880","record_id":"<urn:uuid:0105c837-7cc7-43df-b7ae-03d1b831c9fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00587.warc.gz"} |
Method of surface tensiometer and interfacial tensiometry for measurement of surface tension and interface tension
1, Background
Surface tension or interfacial tension is the macroscopic manifestation of a host of molecular phenomena at the interface between liquid-fluid (or liquid-gas) system. Surfactants (or surface active
agents) may change a liquid's wetting characteristics, and alter surface tension and mass transfer at liquid interfaces. The usual interpretation of surface tension as the force per unit length
exerted across any line lying in the plane of the liquid surface has led to the development of a variety of force-balance surface or interfacial tensiometers. The distinction between surface tension
and surface free-energy per unit area (Ip and Toguri 1994) is unessential here. These devices typically rely on placing a solid object into the liquid of interest, determining the length of the
macroscopic solid-liquid contact line (hereafter referred to as the wetted perimeter), and measuring the added force on the object resulting from its contact with the liquid. The interfacial surface
tension is then recovered by dividing the measured force by the wetted perimeter. Implicit in these techniques are the assumptions that: i) surface tension does not depend on liquid-surface
curvature, ii) the liquid does not apply any force to the submerged portion of the solid beyond hydrostatic pressure, and iii) the angle that the interface makes with the vertical at the contact line
is known (usually assumed to be zero for the receding contact angle in the experimental determination of surface tension).
Surface tensiometer is frequently associated with extraordinary experimental hygiene. Yet the surface tension of ordinary tap-water/air interfaces subject to air-borne particulate contamination is
still of interest in many hydrodynamic studies involving large wave tanks or towing basins where high-purity water cannot be used and even daily water changes are not practical. Here, vertical-pull
film balances are superior to horizontal or Langmuir film balances (Harkins and Anderson 1937) and surface properties must be monitored in situ because any type of sampling will disturb any
intentional or unintentional surfactants. Unlike some surface or interfacial tensiometer (tensiometry) methods, the technique described here is robust in ordinary laboratory environments and yields
consistent results across a variety of wetted-object geometries even when high-purity liquids and clean-room conditions are unavailable.
Accurate surface tension measurements with force balances have proved difficult because the results depend on the contact condition between the object ( such as plate or ring, rod) and the liquid
interface, the shape of the meniscus (contact angle? And it is actually differ from that of sessile drop method. Visit www.uskino.com and find surface tensiometer model 80 series for more information
about contact angle correction method.), the object's buoyancy (refer surface tension meter model A80 series), and other possible molecular attraction or repulsion forces between the object and the
liquid. These problems have been ignored through simplifying assumptions, treated by ad hoc corrections or special ways, mitigated by constraining the measurement technique, or partially corrected by
additional measurements. These remedial actions typically prevent in situ measurements or complicate the overall technique reducing its utility and flexibility. More elaborate surface tension
measurement techniques have been pursued, but these are application specific and require more resources and process time than is typically available for basic surface tensiometer.
There are two main techniques for force-balance surface / interfacial tensiometry ( tension meter): the Wilhelmy plate, and DuNouy ring methods (Adamson 1990; Gaines 1966; Davies and Rideal 1963).
These are discussed in the next two subsections. The final subsection covers other force-balance techniques. Additional information is available in Rusanov and Prokhorov (1996).
2, Wilhelmy plate method
There are several variations of the Wilhelmy plate methods. All are based on balancing the static forces of surface tension, gravity, and buoyancy acting on a thin plate (usually made of glass or
platinum) suspended vertically in the air-liquid interface. Figure 1 shows a cross sectional free body diagram of the active part of the balance while for a known wetted plate perimeter, the
experimentalist measures the pull on the balance, and, in some cases, the vertical position of the bottom plate edge relative to the undisturbed free surface. The surface tension σ is then determined
from (Allan 1958; Jordan and Lane 1964)
Fig. 1. Free body diagram of the Wilhelmy plate balance. The total pull on the plate of thickness t and length L (into the page) is balanced by its own weight, the force from surface tension at the
contact line, and the negative buoyancy resulting from raising the plate above the mean free surface a distance h. The angle from the vertical h of the meniscus in contact with the plate is a true
contact angle when capillary rise up the plate occurs (h’≠0). For the case of no capillary rise, h’ is zero
Here, W[total] is the weight registered by a hook balance, W[app] is the dry weight of the apparatus (plate and the harness), P is the wetted perimeter of the plate, σ is surface tension, h is the
angle the liquid meniscus makes with the vertical at its point of contact with the plate, Do is the density difference between the liquid and the air, g is the gravitational acceleration, t and L are
the plate thickness and length (for a rectangular plate P=2(L + t) ), and h is the height of the bottom of the plate above the undisturbed mean surface. The angle with the vertical θ is a
generalization of the macroscopic or apparent contact angle that remains well-defined at a solid surface discontinuity. The contact angle is the angle defined by Young's equation (Adamson 1990). When
a meniscus contacts an object at a corner and no capillary rise up the vertical sides of the object occurs (h’= 0), h is well defined but the contact angle is not. For the frequently encountered case
where some capillary rise occurs (h≠’0), h is both the contact angle and the angle with the vertical. All the parameters in (1), exceptσ, θ, and h, are easy to determine accurately. The final term in
(1) is called the buoyancy correction. While not large, typically 1 to 10% of W[total] , the buoyancy correction is typically not negligible (Gaonkar and Neuman 1984). Note that surface curvature
effects indirectly enter (1) through the contact angle θ.
Four variations of the Wilhelmy plate measurement are commonly used: the zero-buoyancy method, the detachment method, the immersion method (Gaines 1966), and the maximum-pull method (La Mer and
Robbins 1958; Loglio et al. 1976; Gaonkar and Neuman 1984). In principle, the four variations are similar except for procedural adjustments that simplify the final two terms in (1) so only the first
two are considered here. The immersion method gave inconsistent results and the maximum-pull method failed since the meniscus always ruptured before the pull reached a maximum for the thin plates we
2.1 The zero-buoyancy method for measurement of surface tension or interfacial tension
In this method, the plate is quasi-statically lowered while keeping the lower plate edge parallel to the plane of the undisturbed liquid surface until the slide first contacts the liquid surface
(Padday 1957; Zotova and Trapeznikov 1960; Padday and Russell 1960; Slowinski and Masterton 1961; Pallas and Pethica 1983; Gaonkar and Neuman 1987). The surface tension induced force is then measured
under the assumption that h is zero. However, the advancing contact line leads to variability in the contact angle θ. This method is susceptible to significant error if the measured pull at first
contact is used. Instead, the plate should be further lowered into the fluid and then withdrawn to the first-touch height to promote better plate wetting (Kawanishi et al. 1970; Lane and Jordan
2.2 The detachment method (Furlong and Hartland 1979) for measurement of surface tension or interfacial tension
In this method, the plate is quasi-statically pulled from the liquid until the meniscus ruptures. Withdrawal of the plate ensures wetting through a receding contact line that drives h toward zero.
The hope is that θ approaches zero on the entire plate perimeter as the contact line comes toward the corner near rupture. A thin plate ensures that the buoyancy term is small and that there cannot
be much under-cutting of the meniscus before rupture occurs. The detachment method is subject to uncertainty arising from non-repeatability of dynamic meniscus rupture (Padday and Russell 1960;
Padday 1969, Loglio et al. 1976).
The most common implementation of either Wilhelmy plate method assumes that the free surface is vertical at the point of attachment (or close enough to justify cos θ=1) and that the buoyancy term can
be neglected for a sufficiently thin plate. Hence, Eq. (1) reduces to
The two neglected effects leading to Eq. (2) partially cancel, and this has led to a lack of consistency between sources about the terms in Eq. (1) that are prudently modified or neglected.
The use of smooth or roughened plates to enhance plate wetting is controversial (Kawanishi et al. 1969). Some investigations generally support roughening plates (Princen 1970; Furlong and Hartland
1979; Gaonkar and Neuman 1987), while others generally oppose it (Jordan and Lane 1964; Lane and Jordan 1970, 1971; Pallas and Pethica 1983). Some investigators motivated by practicality (Pike and
Bonnet 1970), like ourselves, merely use the finish obtained on commercially available glassware.
Additional controversy surrounds the necessity of an empirical correction. In some studies (Jordan and Lane 1964; Pike and Bonnet 1970; Lane and Jordan 1971; Furlong and Hartland 1979), as well as
the present, a “film deficit” is observed near the plate ends. The consequence of this variation in meniscus shape is a perimeter-location dependent value of θ which causes measured surface tension
values to have an unexpected dependence on plate thickness. This θ variation is commonly ignored or dismissed (Taylor and Mingins 1975; Orr et al. 1977; Furlong and Hartland 1979; Pallas and Pethica
1983; Sauer and Carney 1990; Palas and Harrison 1990; Mennella and Morrow 1995), or treated with
an empirical end or peripheral correction (Padday 1957, 1969; Padday and Russell 1960; Kawanishi et al. 1970; Pike and Bonnet 1970; Gaonkar and Neuman 1984, 1987). Apparently, this end-correction
controversy will not soon be settled either by experiments (Pallas and Pethica 1989, 1991; Gaonkar and Neuman 1991; Pallas and Pethica 1991) or three-dimensional theory (Orr et al. 1975, Orr et al.
3, surface tensiometer base on Du Nouy ring method for measurement of surface tension or interfacial tension
The Du Nouy ring method may be the most common force balance method. Here, a platinum, wire ring lying in a plane parallel to the liquid surface is submerged in the liquid and then slowly withdrawn
while the net fluid force on the ring is measured. In general, the ring with attached liquid meniscus is raised above the mean undisturbed surface until the pull on the ring reaches a maximum.
Further raising of the ring causes a reduction of pull. The maximum force, Wmax obtained during ring withdrawal can be directly related to the surface tension if the ring is perfectly wetted by the
fluid (Harkins and Jordan 1930; Freud and Freud 1930; Cini et al. 1972; Huh and Mason 1975). However, a non-zero contact angle has been found to be important for ring surface tensiometer ( or
tensiometry ) (Princen and Mason 1965; Cram and Haynes 1971; Gifford 1978). Hence, one needs a priori knowledge of the contact angle to properly implement the Du Nouy ring method, so the Wilhelmy
plate method is often recommend for work where liquid wetting characteristics are not known (Padday and Russell 1960; Gaines 1966; Gaonkar and Neuman 1984; Adamson 1990).
Flaming the du NouK y ring (or platinum Wilhelmy plates) is a controversial cleaning procedure. Many investigations support flaming (La Mer and Robbins 1958; Lane and Jordan 1970; Cini et al. 1972;
Huh and Mason 1977; Furlong et al. 1983), while others (Gaines 1960; Kawanishi et al. 1970; Gaonkar and Neuman 1984) claim that the procedure affects the wetting and thereby affects the surface
tension measurement.
Another problem common to the ring (and the plate) is the interpretation of the actual force measurement. Equating the force on a Du Nouy ring just before meniscus rupture to any of the points on the
force versus height curve produced from theory to determine the surface tension is not recommended (Padday and Russell 1960; Padday 1969; Huh and Mason 1975). Perhaps the largest single source of
misunderstanding in surface tensiometer (tensiometry) is confusing Wmax with the experimentally determined point of meniscus detachment. The experimental point of meniscus detachment may occur
anywhere along the force versus height curve as determined by ®lm stability and the detailed procedures of the particular force balance method. A typical torsion balance either controls h or W, or
both, and usually uses the force immediately prior to detachment in the measurement. Some methods are designed to cause meniscus detachment as close as possible to Wmax, however, the actual proximity
is always in doubt once rupture occurs. Certain geometries, like
the thin Wilhelmy plates used in this study, cause meniscus rupture before Wmax is reached.
4, Diagram of different measuring methods of surface tensiometer
4.1 Modified Wilhelmy plate method
4.2 Classical Wilhelmy plate method (Max pull)
4.3 Classical Wilhelmy plate method (Zero buoyance method)
Note：This method is used in surface tensiometer made by Kruss, KSV, KYOWA. And it cannot used to measure surface tension of cationic surface active agent and sample with viscosity due to its
immersing and drawing out process.
4.4 DuNouy ring method
Note: Sometimes, we will repeat process 2, 3 after process 3 by rising stage and lowering stage several times, and calculate surface tension by average measured data. | {"url":"http://www.uskino.com/newsshow_55.html","timestamp":"2024-11-11T18:01:49Z","content_type":"text/html","content_length":"41895","record_id":"<urn:uuid:50e94c72-be88-496d-a5e3-800e2dc34cb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00205.warc.gz"} |
Custom Performance Metrics
In this section we describe how to develop custom performance metric functions.
We’ll go through the P50 function in detail to see how it works:
## function (MSEobj = NULL, Ref = 0.5, Yrs = NULL)
## {
## Yrs <- ChkYrs(Yrs, MSEobj)
## PMobj <- new("PMobj")
## PMobj@Name <- "Spawning Biomass relative to SBMSY"
## if (Ref != 1) {
## PMobj@Caption <- paste0("Prob. SB > ", Ref, " SBMSY (Years ",
## Yrs[1], " - ", Yrs[2], ")")
## }
## else {
## PMobj@Caption <- paste0("Prob. SB > SBMSY (Years ", Yrs[1],
## " - ", Yrs[2], ")")
## }
## PMobj@Ref <- Ref
## PMobj@Stat <- MSEobj@SB_SBMSY[, , Yrs[1]:Yrs[2]]
## PMobj@Prob <- calcProb(PMobj@Stat > PMobj@Ref, MSEobj)
## PMobj@Mean <- calcMean(PMobj@Prob)
## PMobj@MPs <- MSEobj@MPs
## PMobj
## }
## <bytecode: 0x000000001a531a20>
## <environment: namespace:MSEtool>
## attr(,"class")
## [1] "PM"
Functions of class PM must have three arguments: MSEobj, Ref, and Yrs:
## function (MSEobj = NULL, Ref = 0.5, Yrs = NULL)
## NULL
1. The first argument MSEobj is obvious, an object of class MSE to calculate the performance statistic.
2. The second argument Ref must have a default value. This is used as reference for the performance statistic, and will be demonstrated shortly.
3. The third argument Yrs can have a default value of NULL or specify a numeric vector of length 2 with the first and last years to calculate the performance statistic, or a numeric vector of length
1 in which case if it is positive it is the first Yrs and if negative the last Yrs of the projection period.
The first line of a PM function must be Yrs <- ChkYrs(Yrs, MSEobj). This line updates the Yrs variable and makes sure that the specified year indices are valid. For example:
MSEobj <- runMSE(silent=TRUE) # example MSE object
ChkYrs(NULL, MSEobj) # returns all projection years
## [1] 1 50
ChkYrs(c(1,10), MSEobj) # returns first 10 years
## [1] 1 10
ChkYrs(c(60,80), MSEobj) # returns message and last 20 years
## Yrs[2] is greater than MSEobj@proyears. Setting Yrs[2] =
## MSEobj@proyears
## Yrs[1] is greater than MSEobj@proyears. Setting Yrs[1] = Yrs[2] -
## Yrs[1]
## [1] 30 50
ChkYrs(5, MSEobj) # first 5 years
## [1] 1 5
ChkYrs(-5, MSEobj) # last 5 years
## [1] 46 50
ChkYrs(c(50,10), MSEobj) # returns an error
## Error: Yrs[1] is > Yrs[2]
When the default value for Yrs is NULL, the Yrs variable is updated to include all projection years:
Yrs <- ChkYrs(NULL, MSEobj)
## [1] 1 50
Creating a PM function
We’ll go through the steps of creating the P50 function.
First, we create a new object of class PMobj, and populate the Name slot with a short but descriptive name:
PMobj <- new("PMobj")
PMobj@Name <- "Spawning Biomass relative to SBMSY"
The next line populates the Caption slot with a brief caption including the years over which the performance statistic is calculated. The if statement is not crucial, but avoids the redundant SB > 1
SBMSY in cases where Ref=1.
Next we store the value of the Ref argument in the PMobj@Ref slot so that information is contained in the function output.
PMobj@Ref <- Ref
The Stat slot is an array that stores the variable which we wish to calculate the performance statistic; an output from the runMSE function with dimensions MSE@nsim, MSE@nMPs, and MSE@proyears (or
fewer if the argument Yrs != NULL).
In this case we want to calculate a performance statistic related to the biomass relative to BMSY, and so we assign the Stat slot as follows:
PMobj@Stat <- MSEobj@SB_SBMSY[, , Yrs[1]:Yrs[2]]
Note that we are including all simulations and MPs and indexing the years specified in Yrs.
Next we use the calcProb function to calculate the mean of PMobj@Stat > PMobj@Ref over the years dimension. This results in a matrix with dimensions MSE@nsim, MSE@nMPs:
PMobj@Prob <- calcProb(PMobj@Stat > PMobj@Ref, MSEobj)
Note that in order to calculate a probability the argument to the calcProb function must be a logical array, which is achieved using the Ref slot.
Also note that in this case PMobj@Stat > PMobj@Ref is equivalent to MSEobj@SB_SBMSY[, , Yrs[1]:Yrs[2]] > 0.5.
The PM functions have been designed this way so that in most cases the PMobj@Prob <- calcProb(PMobj@Stat > PMobj@Ref) line is identical in all PM functions and does not need to be modified. The
exception to this is if we don’t want to calculate a probability but want the actual mean values of PMobj@Stat, demonstrated in the example below.
In the next line we calculate the mean of PMobj@Prob over simulations using the calcMean function:
PMobj@Mean <- calcMean(PMobj@Prob)
Similar to the previous line, this line is identical in all PM functions and can be simply copy/pasted from other PM functions without being modified. The Mean slot is a numeric vector of length
MSEobj@nMPs with the overall performance statistic, in this case the probability of \(\text{SB} > 0.5\text{SB}_\text{MSY}\) across all simulations and years.
Finally, we store the names of the MPs and return the PMobj.
Creating Example PMs and Plot
As an example, we will create another version of DFO_plot using some custom PM functions and a customized version of TradePlot.
First we create the plot using DFO_plot:
From the help documentation (?DFO_plot) we can see that this function plots mean spawning biomass relative to \(\text{SB}_\text{MSY}\) and fishing mortality rate relative to \(F_\text{MSY}\) over the
final 5 years of the projection.
First we’ll develop a PM function to calculate the mean \(\text{SB}_\text{MSY}\) for the last 5 years of the projection period. Notice that this is very similar to P50 described above, with the
modification of the Caption and the Prob slots, and the Yrs argument. We are calculating a mean here instead of a probability and are not using the Ref argument:
MeanB <- function(MSEobj = NULL, Ref = 1, Yrs = -5) {
Yrs <- ChkYrs(Yrs, MSEobj)
PMobj <- new("PMobj")
PMobj@Name <- "Spawning Biomass relative to SBMSY"
PMobj@Caption <- paste0("Mean SB/SBMSY (Years ", Yrs[1], " - ", Yrs[2], ")")
PMobj@Ref <- Ref
PMobj@Stat <- MSEobj@SB_SBMSY[, , Yrs[1]:Yrs[2]]
PMobj@Prob <- calcProb(PMobj@Stat, MSEobj)
PMobj@Mean <- calcMean(PMobj@Prob)
PMobj@MPs <- MSEobj@MPs
We develop a PM function to calculate average F/FMSY in a similar way:
MeanF <- function(MSEobj = NULL, Ref = 1, Yrs = -5) {
Yrs <- ChkYrs(Yrs, MSEobj)
PMobj <- new("PMobj")
PMobj@Name <- "Fishing Mortality relative to FMSY"
PMobj@Caption <- paste0("Mean F/FMSY (Years ", Yrs[1], " - ", Yrs[2], ")")
PMobj@Ref <- Ref
PMobj@Stat <- MSEobj@F_FMSY[, , Yrs[1]:Yrs[2]]
PMobj@Prob <- calcProb(PMobj@Stat, MSEobj)
PMobj@Mean <- calcMean(PMobj@Prob)
PMobj@MPs <- MSEobj@MPs
Similar to developing custom MPs we need to tell R that these new functions are PM methods:
class(MeanB) <- "PM"
class(MeanF) <- "PM"
Now we can test our performance metric functions:
## MP SB_SBMSY F_FMSY
## 1 curEref 0.3258801 2.186392e+00
## 2 FMSYref 0.8950520 1.008874e+00
## 3 FMSYref50 1.9335515 5.044369e-01
## 4 FMSYref75 1.3170624 7.566554e-01
## 5 NFref 4.2925589 4.017549e-15
How do these results compare to what is shown in DFO_plot?
We could also use the summary function with our new PM functions, but note that these results are not probabilities:
summary(MSEobj, 'MeanB', 'MeanF')
## Calculating Performance Metrics
## Performance.Metrics
## 1 Spawning Biomass relative to SBMSY Mean SB/SBMSY (Years 46 - 50)
## 2 Fishing Mortality relative to FMSY Mean F/FMSY (Years 46 - 50)
## Performance Statistics:
## MP MeanB MeanF
## 1 curEref 0.33 2.2e+00
## 2 FMSYref 0.90 1.0e+00
## 3 FMSYref50 1.90 5.0e-01
## 4 FMSYref75 1.30 7.6e-01
## 5 NFref 4.30 4.0e-15
Finally, we will develop a customized plotting function to reproduce the image produced by DFO_plot.
We can produced something fairly similar quite quickly using the TradePlot function:
TradePlot(MSEobj, 'MeanB', 'MeanF', Lims=c(0,0))
## MP MeanB MeanF Satisificed
## 1 curEref 0.33 2.2e+00 TRUE
## 2 FMSYref 0.90 1.0e+00 TRUE
## 3 FMSYref50 1.90 5.0e-01 TRUE
## 4 FMSYref75 1.30 7.6e-01 TRUE
## 5 NFref 4.30 4.0e-15 TRUE
Adding the shaded polygons and text requires a little more tweaking and some knowledge of the ggplot2 package. We will wrap up our code in a function:
NewPlot <- function(MSEobj) {
# create but don't show the plot
P <- TradePlot(MSEobj, 'MeanB', 'MeanF', Lims=c(0,0), Show='none')
P1 <- P$Plots[[1]] # the ggplot objects are returned as a list
# add the shaded regions and the text
P1 <- P1 + ggplot2::geom_rect(ggplot2::aes(xmin=c(0,0,0), xmax=c(0.4, 0.8, Inf),
ymin=c(0,0,1), ymax=rep(Inf,3)),
alpha=c(0.8, 0.6, 0.4), fill="grey86") +
ggplot2::annotate(geom = "text", x = c(0.25, 0.6, 1.25), y = Inf,
label = c("Critical", "Cautious", "Healthy") ,
color = c('white', 'darkgray', 'darkgray'),
size=5, angle = 270, hjust=-0.25)
# Re-order layers so text and points are not covered
P1$layers <- c(P1$layers, P1$layers[[3]], P1$layers[[4]])
P1$layers[[3]] <- P1$layers[[4]] <- NULL | {"url":"https://openmse.com/features-calculating-performance/custom-performance-metrics/","timestamp":"2024-11-13T15:47:29Z","content_type":"text/html","content_length":"20629","record_id":"<urn:uuid:8f8342e1-8a93-4a4d-833f-548d94efc70f>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00435.warc.gz"} |
The compression ratio of an engine is 10 and the temperature and
pressure at the start...
The compression ratio of an engine is 10 and the temperature and pressure at the start...
The compression ratio of an engine is 10 and the temperature and pressure at the start of compression is 370 Celcius degree and 1 bar. The compression and expansion processes are both isentropic and
the heat rejected at exhaust at constant volume. The amount of heat added during the cycle is 3000 kJ/kg.
Determine the mean effective pressure and thermal efficiency of the cycle if the maximum pressure is limited to 70 bar and heat added at both constant volume and constant pressure.
Assume that the charge has the same physical properties as that of air.
Note: k=1.4, Cv=0.717 kJ/kgK, R=0.287 kJ/kgK
75 minutes,
To find Q(out) T5 is found out
From Q(out) thermal efficiency is found out
From thermal efficiency Work is found out and then P(mean) can be computed. | {"url":"https://justaaa.com/mechanical-engineering/1306488-the-compression-ratio-of-an-engine-is-10-and-the","timestamp":"2024-11-04T12:08:04Z","content_type":"text/html","content_length":"41906","record_id":"<urn:uuid:60659d6a-f9b9-4ac8-b53f-bad1a8076ad7>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00680.warc.gz"} |
Please help me to solve this problem .
Aloysius had some money at first. He spent $360 on a suit. He then spent 1/7 of his remaining money on shoes and had 2/5 of his money left. How much money did he have at first?
sugabreaddoughman May 6, 2024
For this problem, you would have to set the amount of money as a variable.
They key is to essentially write a solvable equation to figure it out!
In fact, my parents used to give me a lot of problems similar to this one :)
So let's let that be x.
From the problem, we can get the equation \(\frac{2}{5}x+\frac{1}{7}(x-360)+360=x\).
We have \(\frac{2}{5}x + \frac{1}{7}x - \frac{360}{7}+360 = x\). Combining like terms and isolating x, we have \(\frac{2160}{7}=\frac{16}{35}x\).
We multiply both sides by 35/16, and we have \(\frac{35}{16} \cdot \frac{2160}{7} = x\).
So we have \(x = 675\)
NotThatSmart May 6, 2024 | {"url":"https://web2.0calc.com/questions/please-help-me-to-solve-this-problem","timestamp":"2024-11-11T19:25:17Z","content_type":"text/html","content_length":"21893","record_id":"<urn:uuid:829a982a-2046-4025-b43b-66bbd6da7964>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00345.warc.gz"} |
Kinodynamic Conflict-Based Search (K-CBS) is a complete and scalable MRMP algorithm.
Multi-robot motion planning (MRMP) is the fundamental problem of finding non-colliding trajectories for multiple robots acting in an environment, under kinodynamic constraints. Due to its complexity,
existing algorithms utilize simplifying assumptions, are incomplete, or both. To this end, this work introduces a decoupled, scalable, and (proabilistically) complete MRMP algorithm that treats the
kinodynamic constraints of the robots natively. Unlike related works that directly adopt MAPF algorithms on a discretized version of the problem, we incorporate the ideas that make CBS successful
into the continuous domain using a sampling-based method. Our algorithm, dubbed Kinodynamic CBS (K-CBS), like CBS, uses a low-level search and maintains a constraint tree. The low-level search can be
any (kinodynamic) sampling-based tree planner (e.g., RRT or KPIECE).
In each iteration of K-CBS, a trajectory is computed for an individual robot given a set of constraints (time-dependent obstacles). Then, collisions between the robot trajectories are checked. If one
exists, constraints are defined in the constraint tree, and a new planning query is specified accordingly. To ensure probabilistic completeness, we introduce a \emph{merge and restart} method, by
which we merge robots whose plans often conflict, into a single meta-robot.
The main contribution of this work is a decoupled, probabilistically-complete MRMP algorithm that is capable of generating kinodynamically feasible plans efficiently. Our algorithm, K-CBS, naturally
adapts CBS from MAPF to MRMP. This lifts every off-the-shelf (kinodynamic) random tree-based planner to the multi-robot setting and removes all the limitations (assumptions) of state-of-the-art MRMP
planners. Specifically, K-CBS operates completely in the continuous state space of the agents, hence, it does not require discretization nor a feedback-control design. It easily works with arbitrary,
possibly heterogeneous, nonlinear dynamical models, and is capable of solving very complex MRMP instances efficiently. Some example solutions are shown below.
I invite you to investigate any of these great resources to learn more about K-CBS:
1. The original K-CBS IROS 2022 Conference Paper
2. The most-current implementation of K-CBS (to my knowledge) located inside The Multi-Robot OMPL Repository
3. The extended K-CBS Demos Repository that pairs with the MR-OMPL implementation | {"url":"https://justinkottinger.com/projects/K-CBS/","timestamp":"2024-11-02T08:39:58Z","content_type":"text/html","content_length":"16118","record_id":"<urn:uuid:ea90e10a-a5ae-4bf5-8370-7ec119eebfac>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00412.warc.gz"} |
Bounds relating the weakly connected domination number to the total domination number and the matching number
Let G = (V, E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V, E ∩ (S × V)) of G with vertex set V that consists of all edges of G incident
with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted γ[wc] (G). A set S of vertices in G is
a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ[t] (G) of G. In this paper,
we show that frac(1, 2) (γ[t] (G) + 1) ≤ γ[wc] (G) ≤ frac(3, 2) γ[t] (G) - 1. Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as
well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of
G is called the matching number of G, denoted α^′ (G). We also establish that γ[wc] (G) ≤ α^′ (G), and show that γ[wc] (T) = α^′ (T) for every tree T.
• Bounds
• Matching number
• Total domination
• Weakly connected domination
ASJC Scopus subject areas
• Discrete Mathematics and Combinatorics
• Applied Mathematics
Dive into the research topics of 'Bounds relating the weakly connected domination number to the total domination number and the matching number'. Together they form a unique fingerprint. | {"url":"https://pure.uj.ac.za/en/publications/bounds-relating-the-weakly-connected-domination-number-to-the-tot","timestamp":"2024-11-07T00:31:54Z","content_type":"text/html","content_length":"55673","record_id":"<urn:uuid:9f416827-9524-469e-a02e-967e856a191d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00664.warc.gz"} |
Pll refout vs carriertracking
All - Do I understand correctly that gr.pll_carriertracking_cc() is
supposed to downconvert to DC? I don’t see it doing that, and can’t
see in the work functions where that magic would be accomplished.
Just want to make sure I’m building the most efficient graph possible.
I tried both in an existing AM detector with a 7.5KHz IF and both work
just the same with an external mixer.
gr_pll_refout_cc::work (int noutput_items,
gr_vector_const_void_star &input_items,
gr_vector_void_star &output_items)
const gr_complex *iptr = (gr_complex *) input_items[0];
gr_complex *optr = (gr_complex *) output_items[0];
float error;
float t_imag, t_real;
int size = noutput_items;
while (size-- > 0) {
error = phase_detector(*iptr++,d_phase);
d_freq = d_freq + d_beta * error;
d_phase = mod_2pi(d_phase + d_freq + d_alpha * error);
if (d_freq > d_max_freq)
d_freq = d_max_freq;
else if (d_freq < d_min_freq)
d_freq = d_min_freq;
*optr++ = gr_complex(t_real,t_imag);
return noutput_items;
gr_pll_carriertracking_cc::work (int noutput_items,
gr_vector_const_void_star &input_items,
gr_vector_void_star &output_items)
const gr_complex *iptr = (gr_complex *) input_items[0];
gr_complex *optr = (gr_complex *) output_items[0];
float error;
float t_imag, t_real;
for (int i = 0; i < noutput_items; i++){
error = phase_detector(iptr[i],d_phase);
d_freq = d_freq + d_beta * error;
d_phase = mod_2pi(d_phase + d_freq + d_alpha * error);
if (d_freq > d_max_freq)
d_freq = d_max_freq;
else if (d_freq < d_min_freq)
d_freq = d_min_freq;
optr[i] = gr_complex(t_real,t_imag);
d_locksig = d_locksig * (1.0 - d_alpha) + d_alpha*(iptr[i].real() *
t_real + iptr[i].imag() * t_imag);
if ((d_squelch_enable) && !lock_detector())
optr[i] = 0;
return noutput_items;
You are correct. pll_carriertracking_cc returns the recovered carrier.
Sending to baseband is then done by a complex multiply block, don’t
forget the conjugation.
Charles S. wrote:
float t_imag, t_real;
else if (d_freq < d_min_freq)
Discuss-gnuradio mailing list
[email protected]
Discuss-gnuradio Info Page
AMSAT VP Engineering. Member: ARRL, AMSAT-DL, TAPR, Packrats,
NJQRP/AMQRP, QRP ARCI, QCWA, FRC. ARRL SDR Wrk Grp Chairman
“An expert is a man who has made all the mistakes which can be
made in a very narrow field.” Niels Bohr
Bob –
I think we have a problem with carriertracking – it was supposed to mix
the input signal down, and output that, but it looks like it just
outputs the reference, like refout.
Charles –
To detect AM, you can:
1 - take output of pll_refout_cc, take its complex conjugate and complex
multiply that by the input signal. The audio you want is in the real
output. The imaginary output should only have noise. If the RMS value
of both is about the same you don’t have lock, or the signal is very
2 - Fix pll_carriertracking_cc to do what I said in 1, above, since that
is what it was supposed to do.
Robert McGwier wrote:
I did that on purpose. There are myriad instances where you want the
recovered carrier or tone but not have the complex input mixed to the
“new zero”. If you want a recovery and baseband, I suggest that
rather than do an if test, we make a new module that is track and mix.
Yes, but that is what refout is for. Right now refout and
carriertracking do the exact same thing.
Sorry. I missed what you referred to in the early email. I misread
it. I agree that refout and carriertracking should not be doing the
same thing. I will look at it.
Matt E. wrote:
AMSAT VP Engineering. Member: ARRL, AMSAT-DL, TAPR, Packrats,
NJQRP/AMQRP, QRP ARCI, QCWA, FRC. ARRL SDR Wrk Grp Chairman
“An expert is a man who has made all the mistakes which can be
made in a very narrow field.” Niels Bohr
I did that on purpose. There are myriad instances where you want the
recovered carrier or tone but not have the complex input mixed to the
“new zero”. If you want a recovery and baseband, I suggest that rather
than do an if test, we make a new module that is track and mix.
Discuss-gnuradio mailing list
[email protected] | {"url":"https://www.ruby-forum.com/t/pll-refout-vs-carriertracking/97831","timestamp":"2024-11-03T18:21:50Z","content_type":"text/html","content_length":"30853","record_id":"<urn:uuid:e389e931-2954-47a1-80d8-2e4438bd0ef7>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00223.warc.gz"} |
ECONOMICS 252
INTERMEDIATE MACROECONOMIC THEORY
For a Word version, click here.
Professor Steven Horwitz Fall 2000
108 Hepburn Hall T Th 2:30 - 3:50
229-5731 (office) 113 Hepburn Hall
379-9737 (home, before 9pm)
Office Hourse: M,W 1-2; F 9 -10:30 and by appointment
Email: sghorwitz@stlawu.edu AIM: sghorwitz Web: http://myslu.stlawu.edu/~shorwitz
This course is one of the two required courses in intermediate economic theory. Along with Econ 251, this course forms the core of the theoretical courses in economics. Like 251, this course has a
fair deal of mathematical and geometrical analysis. Also like 251, we will try to apply this theoretical work to real world problems of economic growth and economic policy. However, there are two big
differences between this course and 251. The first one is that this is a macro course and our focus will be on issues of GDP, inflation, unemployment, and monetary and fiscal policy. The second
difference requires some additional explanation.
Unlike 251 where you were generally presented with one theoretical structure that was then deployed to explain economic phenomena, there is far less agreement on what counts as good macroeconomic
theory. As a result, one aspect of this course will be the exploration of the differences between various macroeconomic theories. We will explore these theoretical differences in terms of how they
relate to one general model. Your textbook constructs a loanable funds-based circular flow model that I think nicely captures many of the major elements of good macroeconomics. The first two-thirds
of the semester will be devoted to understanding that model and exploring how it can be used to shed light on alternative macroeconomic approaches. These sections of the course move from the
immediate effects of macroeconomic disturbances and policy changes to the short run, intermediate run, and long run effects. This is an effective way of developing your "macroeconomic way of
thinking." One of the important aspects of this course is to acquire and refine the tools of macroeconomic theory so that you can make use of them in later courses.
After we work through all of those runs of time, we will look at the conduct of monetary and fiscal policy. We will focus on the issues of inflation and the federal government's budget to illustrate
both the way policy is made and how it relates to our theoretical discussions. There is only one text for this course, Meir Kohn's Macroeconomics. The bulk of the work is homework questions/problems
and exams. Be prepared for a constant level of work and exams that ask you to interpret, apply and assess course material.
I will be archiving this syllabus and all other course material possible on my web site. Go there and click on the cell for Economics 252 - http://myslu.stlawu.edu/~shorwitz/Fall00/Econ252/index.htm.
Your grade will be based four exams, one quiz, a number of homework assignments, and a short policy analysis paper. The first three hour exams will count 15% each and will be based on the material
discussed in class. The fourth exam will take place at the scheduled time for the final, and will be slightly longer and worth more (20%) than the other two. It will ask you to draw a little bit on
material from earlier in the semester. My exams are short-answer oriented, along with some mathematical or geometrical problems. They will ask you both to demonstrate your understanding of the
material and critically compare and assess it. The quiz will count 5% and will be in-class and around five questions long.
The homework assignments involve answering questions and solving problems from the textbook. The exact questions corresponding to each reading assignment are listed later in the syllabus. You are
responsible for doing 9 of the 11 assignments. If you do more, you can receive extra credit. Each assignment is worth 9 points and I will take your total score on however many you do, and that total
is worth 15% of your final grade. There will also be a writing assignment in the policy section that I will discuss as it nears. That will be worth 15% of your grade. Although the general format of
the class will be lectures, there will be plenty of room for questions, discussions, and applications to current events. I will make every effort to avoid lecturing at you too much.
Below is the grading breakdown along with the dates and times of all the major assignments. THE DATES FOR THE EXAMS ARE NOT WRITTEN IN STONE. I haven't taught this course for a few semesters, and
have never taught it on Tuesday and Thursday, so things might wind up being off by a day or two. I will give you plenty of warning about any changes, HOWEVER, IT IS YOUR RESPONSIBILITY TO BE AWARE OF
Grading breakdown and due dates: Grading scale:
Quiz Thur Sept 14 5% 4.0 (93-100)
First hour exam Thur Sept 28 (evening) 15% 3.5 (88-92)
Second hour exam Thur Oct 19 15% 3.0 (83-87)
Third hour exam Thur Nov 16 15% 2.5 (78-82)
Fourth hour exam Wed Dec 20 6pm 20% 2.0 (73-77)
Homework (various) 15% 1.5 (68-72)
"Start to finish" paper Tues Dec 12 15% 1.0 (60-67)
The grading scale reflects the baseline from which any curve will be constructed. The curve will be no harder than this, but it may be easier. Although this course will ask a lot of you, it is one
that both has great importance and it is one I enjoy teaching. If you have questions or problems, please call or stop by my office.
SCHEDULE OF TOPICS AND READING ASSIGNMENTS
No class on Thursday November 9^th
Topic (approximate # of class days) Readings
Introduction Kohn (1)
The market for time Kohn (2)
The supply and demand for money Kohn (3)
Quiz THURSDAY SEPTEMBER 14^TH
THE SHORT RUN: SHOCKS AND POLICY IN THE CIRCULAR FLOW (4)
The circular flow in a closed economy Kohn (4)
Loanable funds and foreign exchange Kohn (5)
EXAM #1 THURSDAY SEPTEMBER 28^TH - EVENING
IS-LM AND THE QUANTITY THEORY (5)
The Keynesian cross and ISLM models Kohn (7)
Money prices and output in the long run Kohn (8)
EXAM #2 THURSDAY OCTOBER 19^TH
AGGREGATE SUPPLY AND DEMAND IN THE LONG RUN (6)
Aggregate supply and demand Kohn (9)
Inflation and its consequences Kohn (10)
Goals, institutions, and expectations Kohn (11)
EXAM #3 THURSDAY NOVEMBER 16^th
MACROECONOMIC POLICY (6)
The development of macroeconomic policy Kohn (12)
Monetary policy Kohn (13)
Fiscal policy Kohn (16
EXAM #4
HOMEWORK QUESTIONS
│ CHAPTER │ PAGES │ QUESTIONS TO DO │ DUE DATE │
│ Chapter 2 │ p. 52 │ #4, 7, 8, 10 (do the last three graphically) │ Friday Sept. 9 │
│ Chapter 3 │ p. 85 │ #1, 3 │ Tuesday Sept. 12 │
│ Chapter 4 │ p. 112 │ #1-4 (do the loanable funds part graphically) │ Friday Sept. 22 │
│ Chapter 5 │ p. 140 │ #4-6 │ Tuesday Sept. 26 │
│ Chapter 7 │ p. 205 │ #1, 2, 5, 7-10 (do 1, 2, 5, 7 and 8 graphically) │ Friday Oct. 6 │
│ Chapter 8 │ p. 235 │ #2, 3, 5-7 │ Wednesday Oct. 11 │
│ Chapter 9 │ p. 271-2 │ #1, 4, 8, 10, 11 │ Friday Nov. 3 │
│ Chapter 10 │ p. 305 │ #1, 4-6 │ Thursday Nov. 9 (noon) │
│ Chapter 12 │ p. 362 │ #1, 2, 5, 7-9 │ Friday Dec. 1 │
│ Chapter 13 │ p. 389 │ #1, 2, 5-6 │ Friday Dec. 8 │
│ Chapter 16 │ p. 480 │ #1, 3 │ Friday Dec. 15 │
If a chapter reading is not listed here, there is no homework assignment that goes with it.
Homeworks are due at the start of class on Tues/Thurs due dates, and at 3pm on other days (with the exception of Thursday Nov. 9^th, which is due at noon).
There are 11 different assignments here. You are expected to do 9 of them. Each one will be worth 11 points for a total of 99 (everyone gets a bonus point to make it 100). You may do more than 9, and
any additional points will be considered extra credit. It is therefore possible to get a homework grade of more than 100 if you do more than 9 assignments.
For the questions that require written, not numerical, answers, typing them would be preferred. I will return them corrected as soon as possible so that you can use them as study aids.
Kohn, Meir. 1997. Macroeconomics. Cincinnati: South-Western College Publishing | {"url":"http://myslu.stlawu.edu/~shorwitz/Teaching/f00252.htm","timestamp":"2024-11-09T20:32:20Z","content_type":"text/html","content_length":"16069","record_id":"<urn:uuid:7a4fdfab-32e8-46e3-8200-385ff06d8a7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00001.warc.gz"} |
Late Again
Moira is late for school. What is the shortest route she can take from the school gates to the entrance?
Here is a picture of Moira's school which has a paved playground at the front:
If Moira was sitting on the bench nearest the school gate, to get to the climbing frame she could:
Go forward $2$ squares to the pond.
Turn to the right and go $1$ square forward.
Turn to the left and go $2$ squares forward.
This morning, Moira is late for school.
What is the shortest route she can take from the school gate to the school door?
Is there more than one way she could go?
Getting Started
Which direction might you go in first?
Try and describe any route. Can you make it shorter?
How will you know which routes you have tried?
Student Solutions
David from Thomas Reade School sent in the following solution:
The shortest route is forward $1$, right $1$, forward $6$ and right $3$ which is $11$ spaces.
Another way is forward $1$, right $4$ and forward $6$ which is $11$ spaces.
Another way is forward $3$, right $4$ and forward $4$ which is $11$ spaces.
(This assumes that Moira is standing just outside the school gate and finished standing outside the school doors.) Thank you David.
Teachers' Resources
Why do this problem?
This problem
introduces children to the language involved in describing position and direction. It could be done in a practical context and adapted to suit your playground.
It would be useful to project the plan onto a screen for the children to see and you could give pairs a copy of it on paper.
This sheet contains two plans.
Key questions
Which direction might you go in first?
Try and describe any route.
Can you make it shorter?
How will you know which routes you have tried?
Possible extension
Children could be encouraged to make a plan of their own playground and set each other challenges.
Possible support
Having a
paper copy
of the plan will help children get started. Encourage them to work with a partner. | {"url":"https://nrich.maths.org/problems/late-again","timestamp":"2024-11-13T17:55:50Z","content_type":"text/html","content_length":"41059","record_id":"<urn:uuid:daa2f9d5-b961-4036-8339-2a97eddfcab3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00770.warc.gz"} |
Interpolation - Wikipedia Republished // WIKI 2
In the mathematical field of numericalanalysis, interpolation is a type of estimation, a method of constructing (finding) new datapoints based on the range of a discreteset of known data points.^[
In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the
independentvariable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.
A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few
data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from
interpolation error and give better performance in calculation process.
An interpolation of a finite set of points on an epitrochoid. The points in red are connected by blue interpolated splinecurves deduced only from the red points. The interpolated curves have
polynomial formulas much simpler than that of the original epitrochoid curve.
YouTube Encyclopedic
• 1/5
• Linear Interpolation. Quick & Easy!
• Interpolation | Lecture 43 | Numerical Methods for Engineers
• What is Interpolation and Extrapolation?
• 1. Mathematics of linear interpolation | Animation | Computer animation | Khan Academy
This table gives some values of an unknown function ${\displaystyle f(x)}$.
Plot of the data points as given in the table
${\displaystyle x}$ ${\displaystyle f(x)}$
1 0 . 8415
2 0 . 9093
3 0 . 1411
4 −0 . 7568
5 −0 . 9589
6 −0 . 2794
Interpolation provides a means of estimating the function at intermediate points, such as ${\displaystyle x=2.5.}$
We describe some methods of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and smoothness of the resulting interpolant function.
Piecewise constant interpolation
Piecewise constant interpolation, or nearest-neighborinterpolation
The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost
as easy, but in higher-dimensional multivariateinterpolation, this could be a favourable choice for its speed and simplicity.
Linear interpolation
Plot of the data with linear interpolation superimposed
One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating f(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f(2.5)
midway between f(2) = 0.9093 and f(3) = 0.1411, which yields 0.5252.
Generally, linear interpolation takes two data points, say (x[a],y[a]) and (x[b],y[b]), and the interpolant is given by:
${\displaystyle y=y_{a}+\left(y_{b}-y_{a}\right){\frac {x-x_{a}}{x_{b}-x_{a}}}{\text{ at the point }}\left(x,y\right)}$
${\displaystyle {\frac {y-y_{a}}{y_{b}-y_{a}}}={\frac {x-x_{a}}{x_{b}-x_{a}}}}$
${\displaystyle {\frac {y-y_{a}}{x-x_{a}}}={\frac {y_{b}-y_{a}}{x_{b}-x_{a}}}}$
This previous equation states that the slope of the new line between ${\displaystyle (x_{a},y_{a})}$ and ${\displaystyle (x,y)}$ is the same as the slope of the line between ${\displaystyle (x_{a},y_
{a})}$ and ${\displaystyle (x_{b},y_{b})}$
Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point x[k].
The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by g, and suppose that x lies between x[a] and x[b] and that g is
twice continuously differentiable. Then the linear interpolation error is
${\displaystyle |f(x)-g(x)|\leq C(x_{b}-x_{a})^{2}\quad {\text{where}}\quad C={\frac {1}{8}}\max _{r\in [x_{a},x_{b}]}|g''(r)|.}$
In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including polynomialinterpolation and spline interpolation (described
below), is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants.
Polynomial interpolation
Plot of the data with polynomial interpolation applied
Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a linearfunction. We now replace this interpolant with a polynomial of higher degree.
Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:
${\displaystyle f(x)=-0.0001521x^{6}-0.003130x^{5}+0.07321x^{4}-0.3577x^{3}+0.2255x^{2}+0.9038x.}$
Substituting x = 2.5, we find that f(2.5) = ~0.59678.
Generally, if we have n data points, there is exactly one polynomial of degree at most n−1 going through all the data points. The interpolation error is proportional to the distance between the data
points to the power n. Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of the problems of linear
However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see computationalcomplexity) compared to linear interpolation.
Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see Runge'sphenomenon).
Polynomial interpolation can estimate local maxima and minima that are outside the range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at x ≈
1.566, f(x) ≈ 1.003 and a local minimum at x ≈ 4.708, f(x) ≈ −1.003. However, these maxima and minima may exceed the theoretical range of the function; for example, a function that is always positive
may have an interpolant with negative values, and whose inverse therefore contains false verticalasymptotes.
More generally, the shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental
system which has generated the data points. These disadvantages can be reduced by using spline interpolation or restricting attention to Chebyshevpolynomials.
Spline interpolation
Plot of the data with spline interpolation applied
Linear interpolation uses a linear function for each of intervals [x[k],x[k+1]]. Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that
they fit smoothly together. The resulting function is called a spline.
For instance, the naturalcubicspline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating
the points in the table above is given by
${\displaystyle f(x)={\begin{cases}-0.1522x^{3}+0.9937x,&{\text{if }}x\in [0,1],\\-0.01258x^{3}-0.4189x^{2}+1.4126x-0.1396,&{\text{if }}x\in [1,2],\\0.1403x^{3}-1.3359x^{2}+3.2467x-1.3623,&{\text
{if }}x\in [2,3],\\0.1579x^{3}-1.4945x^{2}+3.7225x-1.8381,&{\text{if }}x\in [3,4],\\0.05375x^{3}-0.2450x^{2}-1.2756x+4.8259,&{\text{if }}x\in [4,5],\\-0.1871x^{3}+3.3673x^{2}-19.3370x+34.9282,&{\
text{if }}x\in [5,6].\end{cases}}}$
In this case we get f(2.5) = 0.5972.
Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in
polynomial interpolation. However, the global nature of the basis functions leads to ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in
Boost.Math and discussed in Kress.^[3]
Mimetic interpolation
Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic
interpolation evaluates the integral of fields on target lines, areas or volumes, depending on the type of field (scalar, vector, pseudo-vector or pseudo-scalar).
A key feature of mimetic interpolation is that vectorcalculusidentities are satisfied, including Stokes'theorem and the divergencetheorem. As a result, mimetic interpolation conserves line, area
and volume integrals.^[4] Conservation of line integrals might be desirable when interpolating the electricfield, for instance, since the line integral gives the electricpotential difference at the
endpoints of the integration path.^[5] Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the
potential at the end points of the integration path, regardless of the length of the integration path.
Linear, bilinear and trilinearinterpolation are also considered mimetic, even if it is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area
weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed.^[6]
Function approximation
Interpolation is a common way to approximate functions. Given a function ${\displaystyle f:[a,b]\to \mathbb {R} }$ with a set of points ${\displaystyle x_{1},x_{2},\dots ,x_{n}\in [a,b]}$ one can
form a function ${\displaystyle s:[a,b]\to \mathbb {R} }$ such that ${\displaystyle f(x_{i})=s(x_{i})}$ for ${\displaystyle i=1,2,\dots ,n}$ (that is, that ${\displaystyle s}$ interpolates ${\
displaystyle f}$ at these points). In general, an interpolant need not be a good approximation, but there are well known and often reasonable conditions where it will. For example, if ${\displaystyle
f\in C^{4}([a,b])}$ (four times continuously differentiable) then cubicsplineinterpolation has an error bound given by ${\displaystyle \|f-s\|_{\infty }\leq C\|f^{(4)}\|_{\infty }h^{4}}$ where ${\
displaystyle h\max _{i=1,2,\dots ,n-1}|x_{i+1}-x_{i}|}$ and ${\displaystyle C}$ is a constant.^[7]
Via Gaussian processes
Gaussianprocess is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for
fitting an interpolant that passes exactly through the given data points but also for regression; that is, for fitting a curve through noisy data. In the geostatistics community Gaussian process
regression is also known as Kriging.
Other forms
Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rationalfunctions using Padéapproximant, and
trigonometricinterpolation is interpolation by trigonometricpolynomials using Fourierseries. Another possibility is to use wavelets.
The Whittaker–Shannoninterpolationformula can be used if the number of data points is infinite or if the function to be interpolated has compact support.
Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to Hermiteinterpolation problems.
When each data point is itself a function, it can be useful to see the interpolation problem as a partial advection problem between each data point. This idea leads to the displacement interpolation
problem used in transportationtheory.
In higher dimensions
Comparison of some 1- and 2-dimensional interpolations.
Black and red/yellow/green/blue dots correspond to the interpolated point and neighbouring samples, respectively.
Their heights above the ground correspond to their values.
Multivariate interpolation is the interpolation of functions of more than one variable. Methods include bilinearinterpolation and bicubicinterpolation in two dimensions, and trilinearinterpolation
in three dimensions. They can be applied to gridded or scattered data. Mimetic interpolation generalizes to ${\displaystyle n}$ dimensional spaces where ${\displaystyle n>3}$.^[8]^[9]
• Nearest neighbor
• Bilinear
• Bicubic
In digital signal processing
In the domain of digital signal processing, the term interpolation refers to the process of converting a sampled digital signal (such as a sampled audio signal) to that of a higher sampling rate (
Upsampling) using various digital filtering techniques (for example, convolution with a frequency-limited impulse signal). In this application there is a specific requirement that the harmonic
content of the original signal be preserved without creating aliased harmonic content of the original signal above the original Nyquistlimit of the signal (that is, above fs/2 of the original signal
sample rate). An early and fairly elementary discussion on this subject can be found in Rabiner and Crochiere's book Multirate Digital Signal Processing.^[10]
Related concepts
The term extrapolation is used to find data points outside the range of known data points.
In curvefitting problems, the constraint that the interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within
some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the error. In the simplest case this leads to leastsquares approximation.
Approximationtheory studies how to find the best approximation to a given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound
on how well the interpolant can approximate the unknown function.
If we consider ${\displaystyle x}$ as a variable in a topologicalspace, and the function ${\displaystyle f(x)}$ mapping to a Banachspace, then the problem is treated as "interpolation of
operators".^[11] The classical results about interpolation of operators are the Riesz–Thorintheorem and the Marcinkiewicztheorem. There are also many other subsequent results.
See also
External links
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At relatively high reference 2 frequencies (on the Inhibitors,Modulators,Libraries order of kHz) C0 starts leading, while C1 and C2 are short circuits, so the equivalent circuit becomes as in Figure
Inhibitors,Modulators,Libraries 4.Figure 4.Equivalent circuit for sixth frequency domain (on the order of kHz).The impedance of such circuit is:Z=SR0R123C0+R0+R123SR123C0+1and corner frequencies
are:f5=12��R123C0f6=R0+R1232��R0R123C0At the end, the lowest horizontal part of Bode plot is obtained at highest frequencies (on the order of tens of kHz) when C0 is in short circuit, too, so:Z4=
R0The theoretical Bode plot for the whole equivalent circuit given in Figure 1 is presented in Figure 5.3.?ExperimentalTesting of the system and developed method was done on a physical model of the
electrochemical system, constructed of known elements in a defined arrangement as in Figure 1.
The elements that the physical model was made of were: R0 = 3 ��; R1 = 39 ��, R2 =90 ��; C0 = 0,12 ��F; C1 = 30 mF; C2 = 1,6 F and R3 = 1 k�� (alternatively R3 = 150 ��). Experiments were performed
using the following parameters: DC level 10 mV, AC amplitude 5 mV, frequency range 30 Inhibitors,Modulators,Libraries ��Hz up to 1 Hz. The obtained curves are presented in Figures 6 and and77.Figure
6.Experimentaly obtained Bode plot for the physical model (R3 = 1 k��).Figure 7.Experimentaly obtained Bode plot for the physical model (R3 = 150 ��).From the experimentally obtained Bode curve, all
parameters of the system have been determined by following the next steps:From the plateau 4, R0 is obtained immediately from R0 = Z4;Horizontal region 1 is equal to Z1, and then R3 can be calculated
from:R3=Z1?R0Plateau 2 gives Z2, and then applying:R23=Z2?R0?and?R2=R23R3R3+R23From horizontal part 3, we get Z3 and calculate R123 = Z3 �C R0.
Then R1 can be estimated from:R1=R123R23R23+R123From the corner frequency f1, capacitance C2 is calculated from:C2=12��f1?(R2+R3)From Inhibitors,Modulators,Libraries the corner frequency f3, C1 can
be calculated as:C1=12��f3?(R1+R23);Finally, from the corner frequency f5, C0 is estimated as:C0=12��f5?R123.Using the method described above, values of the circuit parameters have been calculated
from the plot given in Figure 6. The results are compared with those obtained using the commercial software EqCwin applied to the data from Figure 6 (Table 1).Table 1.Parameters of the investigated
equivalent circuit.The plot in Figure 7 gives similar results, except R3, that is, in this case, 150 ��.
Plots in Figures 6 and and77 do not have the fourth plateau for highest Drug_discovery frequencies, so R0 could not be determined from such a curve.4.?ConclusionsTable 1 shows a very good agreement
between the actual values of the electrical components forming the investigated physical model, the values obtained by the method described different in this work and the values obtained using a
commercial software product. In that way the method, hardware and software are fully confirmed.
While some near PLs saturate the user��s AGC during their own dut
While some near PLs saturate the user��s AGC during their own duty selleck screening library cycles, the other PL signals can be received during their own duty cycles. Pulsing is effective but cannot
be a complete solution, because its performance decreases as the number of signal sources increases. In addition, fine scheduling on the pulsing timing according to the PL constellation and AGC
characteristics is required.While pulsing is a solution for the near�Cfar problem in PL systems, we propose a vector tracking loop (VTL) algorithm with PL systems as a new solution, to be implemented
Inhibitors,Modulators,Libraries in a receiver. Combining a pulsing scheme in PLs and a VTL in a receiver could make a more robust PL navigation system and improve navigation availability.The main
feature of the VTL is that it has one big loop that combines tracking and navigation.
Conventional receivers use an independent tracking loop (ITL) and navigation functions. Inhibitors,Modulators,Libraries VTL tracking control input is generated from pseudorange and range-rate
estimates that are estimated from navigation results, while the navigation results are calculated from the tracking results. Tracking results, i.e., discriminator output, are not directly connected
to the tracking control inputs, but are used in estimating pseudorange and range rate from receiver position, velocity and satellite position. These in turn are used to generate tracking control
inputs. In this case, even though some satellite signals are attenuated or blocked at times, the receiver can track them using the navigation results obtained from undisturbed visible satellites.
It can make use of the redundancy Inhibitors,Modulators,Libraries of visible satellites. This is a well-known technique for improving tracking robustness. Spilker [1] commented that in the nonlinear
conditions in PL navigation systems, a VTL should improve tracking performance. However, the application of VTL to PL systems has not yet been properly studied. We propose a VTL algorithm applicable
to asynchronous PL navigation systems, and assess its ability to mitigate the near�Cfar problem and improve PL navigation availability.This paper starts with a brief review of the VTL concept,
comparing it with a conventional ITL in Section 2. Section 3 describes the construction of a vector delay/frequency lock loop (VDFLL) based on the extended Kalman filter (EKF). In Section 4, a
measurement model and a navigation algorithm for asynchronous PL navigation systems are reviewed.
Section 5 proposes a VTL algorithm for an asynchronous Inhibitors,Modulators,Libraries PL navigation system Dacomitinib and its receiver implementation. In Sections 6 and 7, a simulation and test
results will be described. The test was performed using the Seoul National University GNSS Laboratory (SNUGL) indoor navigation system. The results show that VTL could be a good solution for the
near�Cfar problem and improve PL navigation availability.2.?Brief Review of VTLIn 1980, the basic concept selleck chemical of VTL was described in Copps�� paper [5].
The temperature calibrations
The temperature calibrations http://www.selleckchem.com/products/DAPT-GSI-IX.html showed the decrease in I with increase temperature.Figure 2.Temperature calibrations of AA-PSPs with varying the
dipping duration.The Inhibitors,Modulators,Libraries value of �� was determined from Equation (6) (Table 2). With increase the dipping duration, we can see that �� decreased until 100 s and increased
over this dipping duration. The difference of �� was roughly a factor of 2. The variation of �� was greater than that of the error bar. Compared to the effect on the pressure sensitivity, the dipping
duration showed a greater effect on the temperature dependency.3.3. Signal LevelThe value of �� was determined from Equation
Microcantilever-based sensors have emerged as a powerful, universal and highly sensitive tool to study various physical, chemical, and biological phenomena.
They are found to be especially attractive in biochemical and biological sensor applications because of Inhibitors,Modulators,Libraries their rapid, label-free and real-time detection abilities
[1�C7]. The application of microcantilevers in modern sensors was greatly enhanced by the invention of atomic force microscopy (AFM) and the advancements in associated micro-fabrication technologies.
The widespread availability of inexpensive micro-fabricated cantilevers has resulted in renewed interest in using surface stress-based cantilever sensors as a means of detecting biomolecule
absorption [8].Microcantilever biosensors exploit surface stress-induced deflections to assay the analyte.
The surface stresses, in general, are generated either by the redistribution of the electronic charge at the surface, due to the change in the equilibrium positions of the atoms near the surface, or
by the adsorbtion of foreign atoms onto its surface to saturate the dangling Inhibitors,Modulators,Libraries bonds [9]. When the target molecules attach onto the functionalized top surface of the
cantilever, the surface stress distribution on the surface is changed, resulting in a differential stress across the top and bottom surfaces of Inhibitors,Modulators,Libraries the cantilever. The
differential stress ultimately generates deflection in the cantilever, whose measurement give information on type and concentration of the analyte. The deflections are usually measure by optical
read-out technique. The optical detection technique of deflection measurement in microcantilever sensors has several disadvantages. First, it requires external devices Anacetrapib for deflection
measurement, i.e.
, a laser beam and position sensitive detector (PSD), which makes the sensor system bulky and restricts its out-of-lab usage. Second, perfect alignment between laser source, selleck chem Wortmannin
cantilever and PSD is required that necessitates frequent calibration. In addition, the optical properties of the analyte are also critical. If the analyte is translucent or opaque to laser, the
electrical signal from the PSD can be diminished significantly. It reduces the resolution of the sensor. These disadvantages can be avoided by integrating the detection elements or devices into the
This effect is utilized in semiconductor based gas sensors fabric
This effect is utilized in semiconductor based gas sensors fabricated inhibitor Belinostat on various semiconductor materials such as Si [8], SiC [4,5], and GaN [2,3]. The interaction of hydrogen
with semiconductor devices has long been studied, and intensive research led to a model which attributes the reaction mechanism of the devices to hydrogen to the formation of a hydrogen-induced
dipole layer at the metal/dielectric/semiconductor interface [8�C12]. Lundstr?m and co-workers investigated the influence of hydrogen on Pd or Pt�CSiO2�CSi structure using various methods, including
internal photoemission, polarization currents, C�CV measurements, and Kelvin probe. As a result, they concluded Inhibitors,Modulators,Libraries the interaction mechanism as follows: molecular
hydrogen adsorbs on Pd or Pt surface and dissociates.
Hydrogen atoms diffuse through Pd or Pt and adsorb at the metal�Coxide interface, forming a dipole layer. The dipole layer is responsible for the work function change, for example, showing up as a
voltage shift in the C�CV characteristics of the device. Despite the existence of a considerable Inhibitors,Modulators,Libraries quantity of experimental data, however, there are still some debates
as to the origin of the hydrogen sensitivity. For example, a work function Inhibitors,Modulators,Libraries decrease in the Schottky metals, such as Pd and Pt, on exposure to hydrogen is reported to
be the origin of the changes in the characteristics of devices [13,14]. The role of the interface state density in the interaction of hydrogen with semiconductor devices is also discussed in previous
reports [13]. Even now, the interaction mechanism of hydrogen with semiconductor devices still remains to be mysterious.
In order to fabricate hydrogen sensors with higher performances, for example, those with selectivity for hydrogen, the interaction mechanism of hydrogen with semiconductor
Inhibitors,Modulators,Libraries devices should be elucidated. Especially, the metal/semiconductor interfaces play a key role in Cilengitide the interaction mechanism in the devices. Here, I
investigate the interaction mechanism of hydrogen with the nitride-based semiconductor diodes, focusing on the metal/semiconductor interfaces.2.?ExperimentalMetal organic chemical vapor deposition
(MOCVD) grown undoped GaN, Si-doped GaN (n-type 5 �� 1017 cm?3) epilayers, and AlGaN/GaN heterostructures on (0001) Al2O3 substrates were used for this study, respectively.
For Pt�CGaN Schottky barrier diodes (SBDs), Ti(20 nm)/Al(100 nm)/Pt(40 nm)/Au(100 selleck chemical Axitinib nm) multi-layers were formed on either 2 ��m undoped GaN films or 2 ��m Si-doped GaN films
grown on undoped 1 ��m GaN layers by lift off of electron beam evaporation as ohmic contacts. The contacts were subsequently annealed at 750 ��C for 30 s under a flowing N2 ambient in a rapid thermal
annealing (RTA) system. Then, Schottky contacts were formed by lift-off of electron beam deposited Pt(25 nm).
However, baseline separations of
However, baseline separations of thereby NH4+ from alkali and alkaline earth metal ions in water samples were non achievable. For potentiometric detection of NH4+ ion, nonactin has been widely used
as sensing material. Even though nonactin-based ion-selective electrodes show good sensitivity toward NH4+ ion, they suffer interference from other ions such as K+ [13,14]. Flow injection systems
combined with spectrophotometric methods, e.g., the Berthelot reaction involving a colour change in the presence of NH4+ ion, have very slow reaction kinetics [15], whereas fluorimetric flow
injection analysis requires pretreatment of the samples with long diffusion times to avoid background interferences [2,16].Today, there is a well-recognised trend towards the
Inhibitors,Modulators,Libraries simplification and miniaturisation of analytical processes [2].
An amperometry approach employing a miniaturised SPE with immobilized enzyme as tranducer Inhibitors,Modulators,Libraries considerably improves the operation cost, providing for a simple, reliable,
rapid and reproducible analytical procedure. A few biosensors for the amperometric determination of NH4+ ion employing glutamate dehydrogenase (GLDH) have been reported where the enzyme was
immobilized onto the working electrode in several ways [17�C19]. However, to effect the enzymic GLDH reaction, a substrate and co-factor normally needed to be introduced and this leads to an extra
Inhibitors,Modulators,Libraries step during the assay of NH4+ ion. In order to obviate the needs for external reagent treatment during measurement, which may also cause contamination of the reference
electrode, we describe in this work an approach employing a stacked membranes system for the immobilization of enzyme, co-factor and also substrate that eventually leads to a reagentless biosensor
for NH4+ ion determination.
In this work, we have used alanine dehydrogenase (AlaDH) to construct a biosensor for the determination of NH4+ ion. To our knowledge, the use of AlaDH in an NH4+ ion biosensor has not been reported.
The concept of the biosensor based on AlaDH is the reversible amination of pyruvate to L-alanine by AlaDH in the presence of NADH co-factor and NH4+ ion (Equation (1)) [20�C22]. The current
Inhibitors,Modulators,Libraries generated from the electrochemical AV-951 process was measured based on the oxidation of NADH (Equation (2)) whilst the enzyme redox reaction consumed NH4+ ion in the
process. Thus, the redox current is proportional to the NH4+ ion concentration changes under optimal conditions at an applied potential of +0.
55 V:AlaDHPyruvate+NADH+NH4+��L-alanine+NAD++H2O(1)NADH��NAD++H++2e?(2)To construct the stacked membrane biosensor, AlaDH enzyme was first entrapped in the photoHEMA membrane, whereby the membrane
with the entrapped enzyme was formed via UV photopolymerisation of 2-hydroxylethyl methacrylate monomer. Past studies have shown that the use of photoHEMA since is compatible with many enzymes
without leaching problems.
1 3 Related WorkThe research into AUVs is progressing towards so
1.3. Related WorkThe research into AUVs is progressing towards solutions in the commercial, military and research fields. Some examples are Slocum [21,22], based on a buoyancy engine and Ictineu
[23], that uses propellers as a propulsion system. In [21] the proposal was the use of the Slocum as a thermal glider www.selleckchem.com/products/CP-690550.html using the heat flow between the
vehicle engine and the thermal gradient of the ocean temperature in order to propel itself. In this case the control of the pitch and roll was performed by moving an internal mass and the control of
the yaw and heading by the hydrodynamic yawing moment due to the roll. In [22] the proposal Inhibitors,Modulators,Libraries was the use of an electric glider based on the use of an electromechanical
displacement actuator to change their weight.
In this case the roll was set by the position of the glider��s static center of gravity (CG) and pitch was controlled by moving its internal mass. Yaw and heading were controlled using the rudder
mounted on the vertical tail of Inhibitors,Modulators,Libraries the glider. In the case of Ictineu [23], developed by the University of Girona, the AUV prototype was developed to fulfill several
aims: moving the robot from a launch/release point and submerging, passing through a 3 �� 4 meter validation gate, locating a cross situated on the bottom of the pool and dropping a marker over it,
and locating a mid-water target.Other AUVs, such as Finnegan [14], Madeleine [24], AQUA [25], and NTU turtle robot [26] are examples of robots that alternatively use hydrofoils as a propulsion system
to improve maneuverability [20].
Finnegan is a prototype developed by MIT Department of Ocean Engineering Towing Tank, which uses four fins located symmetrically on each side of the robot to generate thrust force. Each fin is
started by a pair Inhibitors,Modulators,Libraries of actuators allowing an unlimited motion in pitch. The main target of this research was to improve the maneuvering performance of AUVs, while
providing the agility to control six degrees of freedom. Madeleine is a prototype developed in 2005 as a result of the cooperation between three institutions: Nekton Research, Monterey Bay Aquarium
Research Institute and Vassar College. Like Finnegan, Madeline uses four fins, but in this case, each fin is started by a single actuator. The motivations of this Inhibitors,Modulators,Libraries
project were to predict efficient fin pitching operation, and build a platform for testing the fin��s locomotion.
AQUA is the result of collaboration between McGill and York Universities. Batimastat This robot is able to swim or walk using six legs, which can be changed depending on the robot��s function. The
selleckchem DAPT secretase vehicle uses a variety of sensors to fulfill a range of real tasks in applications that require large autonomy. The NTU turtle robot was developed by Nanyang Technological
University. This robot can swim using two fore limbs, which are started by two actuators, while its two hind limbs are used for steering.
Consequently, there has been
Consequently, there has been www.selleckchem.com/products/XL184.html little research on the urban and regional impacts of utility restructuring and the changing environment for urban and regional
governance [20,21] Inhibitors,Modulators,Libraries with a large-scale introduction of PV. To take advantage of PV technology’s continued price declines, an understanding of the urban local potential
(roof space and solar exposure among others) is critical for utility planning, accommodating grid capacity, deploying financing schemes and formulating future adaptive policies [22].The paper
describes a methodology that is part of the complex process of assessing solar PV potential for a region using the Renewable Energy Region (RER) of Southeastern Ontario as an example [22,23].
Specifically the methodology provides an application of Light Detection and Ranging (LiDAR) of urban terrain to automated solar PV deployments on a municipal unit, which can be scaled up first to the
level of a city and then the cities within the RER Inhibitors,Modulators,Libraries region. The primary stakeholders for this research are local and regional utilities companies (e.g., Utilities
Kingston), municipal government (e.g., the City Council of Kingston) and academic research on regional energy modeling (e.g., Queen’s University and GEOIDE). Challenges in urban information
extraction and management for solar PV deployment Inhibitors,Modulators,Libraries assessment are determined and quantified. This study provides the following contributions: (i) a methodology that
integrated the cross-disciplinary competences in remote sensing (RS), GIS, computer vision and urban environmental studies; (ii) a robust methodology that can work with low-resolution, spatially and
temporally inconsistent and incomprehensive data and reconstruct vegetation Inhibitors,Modulators,Libraries and buildings separately and concurrently; (iii) recommendations for future generations of
It then presents a case study as an example of the methodology applied to realistic, complex data for Kingston, Ontario. Experience from the case study such as trade-offs between time consumption and
data quality is discussed. This discussion highlights a need for connectivity between demographic information, electrical engineering schemes and geographical Brefeldin_A information selleck chemical
systems (GIS) and a typical factor of solar PV suitable roof area that can be extracted per method. Finally conclusions are developed to provide guidelines for a final methodology with the most
useful information in situations of incomprehensive GIS data to facilitate the processing of LiDAR, low budgets for both time and finance, and personnel with diverse expert in computer vision. The
methodology can be adapted for use anywhere that LiDAR and urban GIS data is available.2.?Background2.1.
In fact very
In fact very Volasertib mechanism few studies target the handheld sensor case and in general only Inhibitors,Modulators,Libraries the case of sensors held in the user’s phoning or texting hand is
considered [13]. Indeed in this context, the sensor is mainly experiencing the inertial force produced by the global motion of the user, which is similar to the body fixed case. Conversely, the cases
of the sensor held in the swinging hand and when the sensor’s placement changes while the user is moving are omitted. In [14], different sensor carrying modes are examined, including carrying the
sensor in the swinging hand, but only traditional techniques, designed for body fixed sensors, are adopted. When the above techniques are applied to handheld smart phones, they produce lower
performance than the ones obtained with body Inhibitors,Modulators,Libraries fixed sensors.
Facing the identified Inhibitors,Modulators,Libraries limitations of existing techniques in the context of autonomous indoor navigation based on smart phones, Inhibitors,Modulators,Libraries a
dedicated and extensive analysis of the hand case has been performed herein. Its results are presented in this paper and lead to the development of a handheld based step length model. Algorithms are
proposed for estimating the step length of pedestrians walking on a flat ground using handheld devices without constraining the sensor’s carrying mode. The proposed step length model combines the
user’s step frequency and height. Step frequency evaluation is performed directly in the frequency domain and independently from the step detection process. In order to adapt the model to the
handheld case, the relationship between step frequency and hand frequency is deeply investigated.
Performance of the proposed model is assessed in the position domain by combining the step length model with a step detection algorithm presented in [15]. The assessment part, performed with 10 test
subjects, shows that the handheld step length model achieves comparable performances as the ones obtained in the literature but with body fixed sensors only.The structure Brefeldin_A of the paper is
the following. In Section 2, the signal model is introduced and the signal preprocessing phase is sellckchem illustrated. In Section 3, the analysis of human gait using handheld devices is described.
Then, in Section 4, the proposed step length model is presented with a description of a novel technique used to extract the user’s step frequency from the user’s hand frequency. Section 5 deals with
the assessment of the proposed algorithm with 10 test subjects. Finally, Section 6 draws conclusions.2.?Signal Model and Pre-ProcessingIn this paper step length estimation is performed using a
six-degree of freedom (6DoF) IMU. It comprises a tri-axis gyroscope and accelerometer that sense angular rates and accelerations of the body frame.
44 else mg) together with EDC (19.14 mg) and NHS (11.49 mg) were dissolved in 0.01 M phosphate buffer saline (PBS, pH 7.4, 0.1 mL) and stirred for 15 min at room temperature (RT). This solution
Inhibitors,Modulators,Libraries was then added dropwise to polylysine (PL) solution (polylysine: 450 mg in 1 mL of 0.01 Inhibitors,Modulators,Libraries M PBS at pH 7.2) so the PL: biotin molar ratio
was 1:1. The mixture was stirred at RT for 1 h and excessive unreacted biotin and ions in aqueous solution were removed by size exclusion (5 KD) column chromatography against PBS buffer (0.01 M, pH
7.4). Then the biotin-labeled PL was labeled with Atto 647N according to the Atto 647N protein labeling kit protocol from Sigma. Firstly, the solution of biotin-labeled PL was added with sodium
bicarbonate buffer solution (pH 9.5), adjusted to pH 8.5�C9.
0 and Inhibitors,Modulators,Libraries transferred to Atto 647N (1 mg Atto 647N dissolved in 10 ��L DMSO), incubated at room temperature under gentle shaking for 2 h, followed by being separated
conjugates from free dye by a size-exclusion column. Finally, the Atto 647N labeled polylysine (FLPL conjugate) fraction was collected and stored at 4 ��C. The rabbit polyclonal antibody-Atto 647N
conjugate (RIgG-FL) was obtained and purified with the same process.2.3.2. Preparation of Biotinylated-mAb1Biotinylation of Mouse monoclonal antibody (mAb1) was performed as described in Section
2.3.1. The molar ratio of biotin to mAb1 Inhibitors,Modulators,Libraries was 2:1. After labeling, excessive unreacted biotin and ions in aqueous solution were removed by dialysis against PBS buffer
(0.01 M, pH 7.4) for 2 days. Then, it was stored in ?20 ��C until use.2.3.3.
Preparation of FLPL-BSAS-mAb1 ConjugateThe biotinylated-mAb1, streptavidin and FLPL conjugate were mixed in a molar ratio equal to 1:1:3 or 1:2:6 (Table 1). The conjugating mechanism of
FLPL-BSAS-mAb1 conjugates was shown in Figure 1. Finally, FLPL-BSAS-mAb1 conjugate and RIgG-FL (1:1, v/v) were dispersed in stock solution (1% BSA and 0.005% sodium azide in 0.01 M PBS pH 7.4),
respectively, Dacomitinib and kept at 4 ��C until use.Figure 1.Preparation process of FLPL-BSAS-mAb1 conjugate.Table 1.Properties of FLPL-BSAS-mAb1 conjugate prepared with different conjugate
types.2.4. Preparation of Lateral Flow Immunoassay SystemThe main body of the test strip consisted of five parts, including plastic backing, sample pad, conjugate pad, absorbent pad and NC membrane
(Figure 2).
Every component of the strip should be given a pretreatment described as follows: the NC membrane was attached to a plastic backing layer for cutting and handling. The pAb2 and GAR were immobilized
at test line (T line) and control line (C line), respectively. The glass fiber following website was cut into two sizes 0.5 cm �� 0.4 cm and 2.2 cm �� 0.4 cm for the conjugate pad and sample pad.
Conjugate pad contained FLPL-BSAS-mAb1 conjugate and RIgG-FL diluted by 0.01 M PBS buffer (pH 7.4) containing BSA (0.5%, w/v) and sucrose (3%, w/v). Sample pad was pretreated with BSA (3%, w/v) and
Tween-20 (0.5%, w/v). Absorbent pad was 2.
mutation indeed might lead to loss of SUMO 1 binding as described
mutation indeed might lead to loss of SUMO 1 binding as described in, our data raise the possibility that loss of interaction could also be the result of a more general, unspecific effect of TDG
misfolding in this part of the molecule and subsequent aggregation of TDG D133A into high molecular weight precipitates. therefore In contrast, the TDG E310Q mutant behaves as the TDG wild type
protein and few discrepancies were detectable in far UV spectra obtained by circular dichro ism as well as on the HSQC resonances between both spectra. This is, given our previous analysis of TDG CAT
NMR behavior, Inhibitors,Modulators,Libraries explained by the fact that the mutated residue is part of the very rigid region not detected in the HSQC spectra.
Moreover, since few differences between mutant and wild type proteins are observed when comparing the HSQC spectra, we can Inhibitors,Modulators,Libraries reasonably assume that the E310Q mutation
does not, unlike the D133A mutation, strongly affect the structure of Inhibitors,Modulators,Libraries TDG. We have further investigated the SUMO 1 binding to TDG E310Q. Under the same conditions used
as for wild type TDG, no modification of neither C terminal nor RD resonances of TDG E310Q were detected in the presence of a 10 fold molar excess of SUMO 1 indicating that SUMO 1 binding to TDG is
abolished by the E310Q mutation and SUMO 1 binding to the TDG C terminal SBM is solely responsible for both the C and N terminal conforma tional changes. Moreover, in contrast to wild type TDG, the
overall signal intensity of 15N SUMO 1 does not decrease in presence of a 3 fold excess of TDG E310Q, confirming that SUMO 1 does not interact with TDG E310Q.
Furthermore, the CD spectra of TDG or TDG E310Q in presence of SUMO 1 point to a slight modification of protein structures for the Inhibitors,Modulators,Libraries wild type TDG only confirming the
TDG SUMO 1 inter molecular interaction and subsequent structural rearran gement. No competition between cis and trans SUMO 1 for TDG CAT binding Interestingly, SUMO 1 was also able to bind SBM2 in
the context of sumoylated TDG. We have detected modifications of the C terminal resonances of 15N labeled sumoylated TDG when adding a 10 fold molar excess of unlabeled SUMO 1 as well as appearance
of TDG RD resonances AV-951 similarly to unmodified TDG. However, except of SUMO 1 resonances observable at natural abundance, no additional 15N labeled SUMO 1 signals coming from sumoylated TDG were
detected indicating that SBM2 bound SUMO 1 does not displace intramolecular SUMO 1.
These data show that intermolecular SUMO 1 choose size binding does not fully compete with cis SUMO 1 and that SBM2 remains accessible to SUMO 1 interactions. Based on these observations, we can
speculate for a lar ger C terminal SBM than the one that has been described. Additionally, the 15N 1H HSQC spec trum of the sumoylated TDG E310Q mutant shows no significant modification of TDG E310Q
resonances and no SUMO signals except the amino terminal residues also detectable for the SUMO modified wild type TDG. These data confirm the existence of distinct SUMO interfaces for e | {"url":"https://her2signaling.com/2015/10","timestamp":"2024-11-14T13:56:44Z","content_type":"text/html","content_length":"79526","record_id":"<urn:uuid:9f72619d-8cb7-43d8-a1cb-1b1385cc72cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00253.warc.gz"} |
American Mathematical Society
Variation of conformal spheres by simultaneous sewing along several arcs
HTML articles powered by AMS MathViewer
by T. L. McCoy
Trans. Amer. Math. Soc. 231 (1977), 65-82
DOI: https://doi.org/10.1090/S0002-9947-1977-0444922-6
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Let M be a closed Riemann surface of genus zero, $\Gamma$ a tree on M with branches ${\Gamma _j}$, and ${p_0}$ a point of $M - \Gamma$. A family of neighboring topological surfaces $M(\varepsilon )$
is formed by regarding each ${\Gamma _j}$ as a slit with edges $\Gamma _j^ -$ and $\Gamma _j^ +$, and re-identifying p on ${\Gamma ^{{ - _j}}}$ with $p + \varepsilon {\chi _j}(p,\varepsilon )$ on $\
Gamma _j^ +$, with ${\chi _j}$ vanishing at the endpoints of ${\Gamma _j}$. We assume the ${\Gamma _j}$ and ${\chi _j}$ are such that, under a certain natural choice of uniformizers, the $M(\
varepsilon )$ are closed Riemann surfaces of genus zero. Then there exists a unique function $f(p,\varepsilon ;{p_0})$ mapping $M(\varepsilon )$ conformally onto the complex number sphere, with
normalization $f({p_0},\varepsilon ;{p_0}) = 0,f’({p_0},\varepsilon ;{p_0}) = 1$. Under appropriate smoothness hypotheses on $\Gamma$ and the ${\chi _j}$, we find the first variation of f as a
function of $\varepsilon$. Further, we obtain smoothness results for f as a function of $\varepsilon$. The problem is connected with the study of the extremal schlicht functions; that is, the
schlicht mappings of the unit disc corresponding to boundary points of the coefficient bodies. References
A. E. Taylor, Advanced calculus, Blaisdell, Waltham, Mass., 1955.
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Bibliographic Information
• © Copyright 1977 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 231 (1977), 65-82
• MSC: Primary 30A30
• DOI: https://doi.org/10.1090/S0002-9947-1977-0444922-6
• MathSciNet review: 0444922 | {"url":"https://www.ams.org/journals/tran/1977-231-01/S0002-9947-1977-0444922-6/?active=current","timestamp":"2024-11-12T09:49:30Z","content_type":"text/html","content_length":"58627","record_id":"<urn:uuid:9c719ac0-8793-456f-bee7-ccce6132ac2c>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00319.warc.gz"} |
Transactions Online
Shoya OOHARA, Mitsuji MUNEYASU, Soh YOSHIDA, Makoto NAKASHIZUKA, "Image Regularization with Total Variation and Optimized Morphological Gradient Priors" in IEICE TRANSACTIONS on Fundamentals, vol.
E102-A, no. 12, pp. 1920-1924, December 2019, doi: 10.1587/transfun.E102.A.1920.
Abstract: For image restoration, an image prior that is obtained from the morphological gradient has been proposed. In the field of mathematical morphology, the optimization of the structuring
element (SE) used for this morphological gradient using a genetic algorithm (GA) has also been proposed. In this paper, we introduce a new image prior that is the sum of the morphological gradients
and total variation for an image restoration problem to improve the restoration accuracy. The proposed image prior makes it possible to almost match the fitness to a quantitative evaluation such as
the mean square error. It also solves the problem of the artifact due to the unsuitability of the SE for the image. An experiment shows the effectiveness of the proposed image restoration method.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.1920/_p
author={Shoya OOHARA, Mitsuji MUNEYASU, Soh YOSHIDA, Makoto NAKASHIZUKA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Image Regularization with Total Variation and Optimized Morphological Gradient Priors},
abstract={For image restoration, an image prior that is obtained from the morphological gradient has been proposed. In the field of mathematical morphology, the optimization of the structuring
element (SE) used for this morphological gradient using a genetic algorithm (GA) has also been proposed. In this paper, we introduce a new image prior that is the sum of the morphological gradients
and total variation for an image restoration problem to improve the restoration accuracy. The proposed image prior makes it possible to almost match the fitness to a quantitative evaluation such as
the mean square error. It also solves the problem of the artifact due to the unsuitability of the SE for the image. An experiment shows the effectiveness of the proposed image restoration method.},
TY - JOUR
TI - Image Regularization with Total Variation and Optimized Morphological Gradient Priors
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1920
EP - 1924
AU - Shoya OOHARA
AU - Mitsuji MUNEYASU
AU - Soh YOSHIDA
AU - Makoto NAKASHIZUKA
PY - 2019
DO - 10.1587/transfun.E102.A.1920
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E102-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 2019
AB - For image restoration, an image prior that is obtained from the morphological gradient has been proposed. In the field of mathematical morphology, the optimization of the structuring element
(SE) used for this morphological gradient using a genetic algorithm (GA) has also been proposed. In this paper, we introduce a new image prior that is the sum of the morphological gradients and total
variation for an image restoration problem to improve the restoration accuracy. The proposed image prior makes it possible to almost match the fitness to a quantitative evaluation such as the mean
square error. It also solves the problem of the artifact due to the unsuitability of the SE for the image. An experiment shows the effectiveness of the proposed image restoration method.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.1920/_p","timestamp":"2024-11-03T10:51:49Z","content_type":"text/html","content_length":"61413","record_id":"<urn:uuid:9ccacd3b-9d26-4962-945a-18fe116ef0ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00575.warc.gz"} |
when perfect intervals are inverted, they
The final lesson step explains how to invert each interval. Compound intervals are intervals bigger than an octave e.g. For example, to calculate Compound intervals are larger than the octave and are
heard as expanded variants of their simple counterparts: a tenth (octave plus a third, such as C–C′–E′) is associated by the ear with a third (an interval encompassing three scale steps, such as
C–E).. Quality: Perfect. ) Similarly, all 7ths when inverted become 2nds. Intervals that use the same keys on the piano but are spelled differently, such as the augmented third, C-E, and the perfect
fourth, C-F, are known as enharmonic equivalents. ( Perfect intervals also include fourths and fifths. Augmented ↔️ Diminished. The diatonic intervals as they normally occur up from the tonic of the
major scale are called either major or perfect. Adapted from Measures 14–16, Parry H (1897) "Rustington". Texts that follow this restriction may use the term position instead, to refer to all of the
possibilities as a category. Augmented changes to diminished. These four permutations (labeled prime, retrograde, inversion, and retrograde inversion) for the tone row used in Arnold Schoenberg's
Variations for Orchestra, Op. Then we apply the transposition operation … 5 A major interval when inverted becomes a minor interval while a … 3. A compound … If no letter is added, the chord is
assumed to be in root inversion, as though a had been inserted. An interval of an octave (8th) or less can be inverted. C-G (perfect 5th) becomes G-C which is a perfect 4th, a 3rd would become a 6th.
Fourth and fifth intervals are used interchangeably most of the time. The rearrangement of the notes above the bass into different octaves (here, the note E) and the doubling of notes (here, G), is
known as voicing – the first voicing is close voicing, while the second is open. This is the opposite way previously explained to determine intervals. 4th ↔️ … In an inverted chord, the root is not
the lowest note. T I However, theorists before Rameau spoke of different intervals in different ways, such as the regola delle terze e seste ("rule of sixths and thirds"), which requires the
resolution of imperfect consonances to perfect ones and would not propose a similarity between 64 and 53 sonorities, for instance. One note is obviously being counted twice). {\displaystyle T_{5}I
(3)} = n C to D is a major 2nd, whereas D to C is a minor 7th. Perfect intervals are labeled with a capital "P." The Major prefix is only used for seconds, thirds, sixths and sevenths. To invert an
interval, either make the top note the new bottom note or the bottom note the new top note. 1) Perfect intervals include adding a note above the first note of a major scale that represents the
distance of a unison (prime), 4th, 5th or 8th (octave) interval. When an interval is inverted the size and quality change: The size of the original and the inverted interval will always adds to …
(Listen to pieces suggested below for perfect intervals.) All perfect intervals, when inverted, are still perfect (this is why they are called “perfect”). The second part of an interval name is based
on the quality of the interval. means "transpose by some interval For example, the root of a C-major triad is C, so a C-major triad will be in root position if C is the lowest note and its third and
fifth (E and G, respectively) are above it – or, on occasion, don't sound at all. You can practice the concept of inversion with intervals by flipping the position of the two notes by either moving
the lower note up an octave or the upper note down an octave. 1. {\displaystyle T_{5}I(3)=2} Examples of interval naming: The interval from C (1) to D (2) is a "Second" because it includes two tones,
the interval from C (1) to E (3) and the interval from E (3) to G (5) are both a "Third" because they include three diatonic tones. The notation of octave position may determine how many lines and
spaces appear to share the axis. When perfect intervals are inverted they remain perfect; major intervals become minor (and vice versa); augmented intervals become diminished (and vice versa). About
interval qualities. Figured-bass numerals express distinct intervals in a chord only as they relate to the bass note. Perfect intervals stay perfect when inverted. Take a look at the note circle
again. In the case of the other interval qualities, they change their qualities when inverted: Maj>min (and the converse) dim>Aug (and its converse) INVERTED to that of a major sixth (M6). The
interval from 1 to itself is a perfect unison. This is the basis for the terms given above such as "64 chord" for a second inversion triad. Inverted Intervals. Sets are said to be inversionally
symmetrical if they map onto themselves under inversion. But the simplest explanation I've seen so far, and my favorite, I found on another website: "Perfect intervals are the ones that don't have
two forms: major and minor." So, the first interval (m3) has now been. Inverting perfect intervals. Intervals that span three half steps are minor thirds; those that comprise four half steps are
major thirds. In contrapuntal inversion, two melodies, having previously accompanied each other once, accompany each other again but with the melody that had been in the high voice now in the low,
and vice versa. This will determine the distance of the inverted interval. Simple intervals encompass one octave or less. If you didn’t know the Cipher’s half-step values of intervals, where … For
example, if you were to invert a perfect 4th it would become a perfect 5th and vice versa, when you invert a perfect 5th it becomes a perfect 4th. (Doubly diminished intervals become doubly augmented
intervals, and vice versa.). If inverted, or flipped upside down, these intervals will always equal another interval from the list. 1. 2) A perfect interval does not have to include the first note of
the major scale. answer the question about why 1, 4, 5, and 8 are called the perfect intervals. , first subtract 3 from 12 (giving 9) and then add 5 (giving 14, which is equivalent to 2). Quality:
Reversing pairs To determine the quality you must remember the following pairs. Play the following example of all 5ths and notice the ... Any interval larger than an octave is called a compound
interval. That specificity comes in ... way of counting off diatonic intervals, where the number includes the starting and ending pitches, and when combining inverted intervals, there is always one
note that gets counted twice—in this case, E4.) Inverted intervals are simply intervals which have been turned upside down. Perfect — Perfect; Study these examples that illustrate the change of both
number size and quality under inversion. T e.g. C-G (perfect 5th) becomes G-C which is a perfect 4th, a 3rd would become a 6th. The inversions are numbered in the order their lowest notes appear in a
close root-position chord (from bottom to top). As you can see below by taking the C at the bottom of the interval and moving it above the G, the initial interval of a 5th turns into a 4th when
turned upside down. A major third interval, inverted, becomes a minor sixth interval. " measured in number of semitones. chromatic. Inverted intervals identifying note C. This table inverts the above
intervals, so that each link in the last column leads to note C. C 1st inverted intervals; Short Medium Long Note name Link to inverted interval; P1: Cperf1: C perfect Unison: C <-(!? C to D an
octave and one more note above it is a major 9th. ... An interval may be inverted by placing the lower note an Inverted Intervals (With Interval Exercise) Beyond the interval quality (major, minor,
perfect) and its name, there is one more property of intervals which is important to understand. The size of an interval between two notes may be measured by the ratio of their frequencies.When a
musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios, such as 1:1 (), 2:1 (), 5:3 (major sixth), 3:2 (perfect
fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third).Intervals with small-integer ratios are often called just … The interval from 1 to 4 is known as a perfect fourth, from 1 to 5 is a
perfect fifth, and from 1 to 8 is a perfect octave. I The following categories will be essential for your work in strict voice-leading, and they will be a helpful guide for free … 2) A perfect
interval does not have to include the first note of the major scale. Big intervals are called “wide” intervals. Transformation of an interval that results from displacing one pitch by an octave such
that the interval size and quality change. That specificity comes in the form of an interval’s quality. These intervals include: 3-7, 6-3, 2-6, 5-2, 1-5, 4-1 . Intervals are categorized as consonant
or dissonant based on their sound (how stable, sweet, or harsh they sound), how easy they are to sing, and how they best function in a passage (beginning, middle, end; between certain other
intervals; etc.). They may be thought of as their smaller counterparts by subtracting seven from whatever the number is. Simple intervals mean that they are an octave or smaller in size, while
compound intervals means that intervals are larger than an octave. Here, no less than five themes are heard together: The whole passage brings the symphony to a conclusion in a blaze of brilliant
orchestral writing. For instance, a C-major triad contains the tones C, E and G; its inversion is determined by which of these tones is the lowest note (or bass note) in the chord. the higher note
becomes the lower note and vice versa). Combining quality with a generic interval name produces a specific interval. Once you understand the results of interval inversion, you can apply the technique
to help write and identify intervals. A perfect interval usually has 2 other intervals grouped around it - one higher and one lower: ... but it also describes the number of either lines or spaces on
the staff between the tonic note and all intervals sharing that number - 1st, be they called diminished, minor, major, perfect or augmented. Thus, All perfect intervals, when inverted, are still
perfect (this is why they are called “perfect”). Inverted Intervals. If you subtract any of these from 9, you still get a 1st, 4th, 5th or 8th, which are all perfect intervals. {\displaystyle T_{n}}
n When intervals are inverted they reverse the relative position of the notes. 2 Bach's The Well-Tempered Clavier, Book 1, the following passage, from bars 9–18, involves two lines, one in each hand:
When this passage returns in bars 26–35 these lines are exchanged: J.S. An inverted 6th is a 3rd.) {\displaystyle n} The term inversion often categorically refers to the different possibilities,
though it may also be restricted to only those chords where the lowest note is not also the root of the chord. An interval from C to F is called a perfect fourth. the lower of the two notes is raised
an octave, or the higher one is dropped an octave, the interval becomes minor e.g. C to D an octave and one more note above it is a major 9th. Lastly, the major interval inverts into a minor, and
vice versa. Perfect Interval - raised by one semitone becomes an Augmented Interval. The lower note of a music interval is always classed as the keynote or root of the interval in question, even when
inverted. T Traditional interval numbers add up to nine: seconds become sevenths and vice versa, thirds become sixths and vice versa, and so on. This is sometimes known as the parent chord of its
inversions. The diagram below shows a C major scale. All major intervals, when inverted, become minor intervals. Thus, inversion is a combination of an inversion followed by a transposition. As for
the quality of the interval, perfect remains perfect when inverted, major becomes minor, … Once inverted, they will switch. Inverted Intervals Intervals no larger than an octave are called simple
intervals. But what if the root note is the higher of the two notes? Thus, seconds become sevenths, thirds become sixths, and fourths become fifths. Introductory and intermediate music theory
lessons, exercises, ear trainers, and calculators. For example, in root-position triad C–E–G, the intervals above bass note C are a third and a fifth, giving the figures 53. 9 – 3 = 6, then switch
the “major” to “minor.”. For example, the set C–E♭–E–F♯–G–B♭ has an axis at F, and an axis, a tritone away, at B if the set is listed as F♯–G–B♭–C–E♭–E. Exercises, ear trainers, and augmented
intervals, which share identical pitches in parallel major minor. Their smaller counterparts by subtracting seven from whatever the number is axis is the center around which a melody inverted... Root
is the lowest note distance from C to D is a perfect,! [ citation needed ] add the intervals in example 11–4, then switch the “ ”! Whenever you invert a set of inversion also plays an important
compositional and analytical in... `` pitch axis '' works in the opening two bars even transpositional equivalency ( Doubly diminished intervals when inverted add. Inverted, are still perfect ( this
is the root note is the inversionof a minor 7th }! Map onto themselves under inversion than an octave are called _____ intervals )... Major becomes minor, dimished, and an inverted minor 6 th
transposition is carried out after inversion an is! Chord is in root position if its root is not the lowest note is obviously being counte… a interval. Or nearly identical in musical set theory,
chords in different inversions are considered functionally equivalent chord its! Root inversion, [ clarification needed ] add the intervals by which voice... (! minor 6 th the lesson steps then
explain how to the! Or lowering the top note the new top note the new bottom note or lowering the in! Or augmented/diminished, you must remember the following measure, the root of ;. 4Th becomes a
perfect 5th ) becomes G-C which is a major interval inverts into a minor 7th and become... Are announced in the second case the upper pitch and vice versa, and.. Intervals are intervals bigger than
an octave are called simple intervals. ) inverted!: an inverted interval is always classed as the keynote or root of the notes nearly identical in set. You get something called an “ inverted ” (
turned upside down which a melody inverted... Major/Minor and diminished/augmented quality of a perfect fourth interval larger than an octave e.g than generic intervals be! Analytical notations of
function ; e.g., the notation of octave position determine! Changing the voices above it … interval inversion, [ clarification needed add. Are all a fourth apart we will be inverting intervals works
in key... If you didn ’ T know the Cipher ’ s quality carried out after inversion the bottom note, than! Quality change and intermediate music theory lessons, exercises, ear trainers, and inversions.
To determine the quality you must flip the major/minor and diminished/augmented quality of the dominant spelling and.... If they map onto themselves under inversion when intervals are inverted,
become minor intervals are intervals whose root is... Intervals expand perfect and major intervals become minor and vice versa. ) important. Numbered in the opening two bars diminished, perfects stay
perfect minor 3rd C-perf-8th: A1 Caug1... Root of the interval is always classed as the parent chord of its inversions, fifth octave! Equivalency, octave equivalency and even transpositional
equivalency n't recommend this method though because is! Case the upper note, rather than the bottom and an a note on the bottom and an inverted is! As their smaller counterparts by subtracting seven
from whatever the number nine ) their is! Four perfect intervals, and augmented intervals, chords in different inversions are considered functionally.! Chord symbol to indicate root position or
inversion difference between these two intervals is that in the first.... ) a perfect 4th, a pitch axis '' works in the first measure simultaneously pays to! Become a 6th ( 9 - 4 = 5 ) example: the
inverted distance of a interval!, 1-5, when perfect intervals are inverted, they subtract one, seconds become sevenths, thirds become sixths, and fourths become.... Augmented/Diminished, you must
flip its quality quality with a perfect or minor is... Examples of invertible counterpoint is also known as the parent chord of its lowest notes the... The change of both number size and quality
under inversion perfect ; Study examples... As a category from Measures 14–16, Parry H ( 1897 ) `` Rustington '' note or the note! Refer to all of the most spectacular examples of invertible
counterpoint occurs in the of! A transposition the original lower pitch becomes the upper note, and augmented intervals become diminished relate. A fourth apart out an inversion followed by a
lower-case letter: Cb ) notes appear a. One add up to 9 ( there are 8 notes in a close root-position chord from... Rustington '' counte… a diminished interval occurs when a perfect 5th ) becomes G-C
which is a perfect 4th would... Comes in the finale of Mozart 's Jupiter Symphony in when perfect intervals are inverted, they. [ 6.. Each interval why they are called simple intervals. ) '' works in
the form of interval! And … perfect interval why they are only assumed identical or nearly in. Examples of invertible counterpoint occurs in the keyboard prelude in A♭ major from J.S found the new
number,... The center around which a melody is inverted, become diminished 8 ] to invert an from. Technique in music, involving both variable and constant features 14–16, Parry (! Reversing pairs to
determine intervals. ) no larger than an octave are called “ ”... That results from displacing one pitch by an octave are called simple intervals. ) {... Following pairs position may determine how
many lines and spaces appear to share the axis of (... Keys, are still perfect ( this is called double counterpoint when two voices involved... Be “ inverted ” ( turned upside down pitch that the
interval that... It becomes the opposite perfect intervals. ) C-down- > G = P4 ) put it on quality! A transposition order, we will be inverting intervals works in the first note of major! By flipping
it `` upside-down '', reversing the melody 's contour … all perfect intervals: prime, unison! Sixths, sevenths - … all perfect intervals. ) what if the interval becomes that a... Distinct intervals
in a close root-position chord ( from bottom to top.... To share the axis of symmetry ( or center ) the two notes, when perfect intervals are inverted, they. Are intervals bigger than an octave are
called simple intervals. ) must the! Rd is an important compositional and analytical technique in music, involving both variable constant. ( assuming that microtones are not used ) the inversions are
considered functionally equivalent is in root or... Interval it becomes the lower note and vice versa. ) minor 7th fourths become fifths then switch the major! To all of the interval of inversion
pairs in figure 8 are the. After inversion below, we can see that they are all a fourth apart, when inverted, add to! A combination of three themes and sevenths various intervals, when inverted, it
remains a interval! Technique in music, involving both variable and constant features will determine the distance of major! That specificity comes in the following C-major triads are both in root
position or.! That intervals, and i6/4/3 for second inversion triad ( interval ) = ( inverted interval is made.! Major 9th of changing the voices above it is a major interval into... And octave out
an inversion followed by a lower-case letter: Cb ) the perfect intervals. ) size quality. Are simply intervals which have been turned upside down in example 11–4, then switch the “ major ” “... In
interval quality and interval number under inversion in interval quality and interval number inversion... Themes that can be perfect, major, or augmented/diminished, you must remember the following
triads... Intervals encompass one octave or less sevenths, thirds become sixths, augmented. Note interval name produces a specific pitch or halfway between two notes one add up 9. Exact same pitch is
called a perfect fourth ( the B ♭ create a perfect 4th would...: the original lower pitch becomes the upper note, rather than the note... In parallel major and minor keys, are never major or minor
note above it interval... Unison and the octave, less often at the strings in ascending order..., inversion is: 9 – ( interval ) = 2 { \displaystyle T_ 5! Original lower pitch becomes the upper note,
rather than the bottom note or the note... H ( 1897 ) `` Rustington '' and triple counterpoint when three are involved )... Now been made _____ a combination of three themes the major interval
inverts into a minor 7th is! Root-Position chord ( from bottom to top ) encompass two half steps are major seconds steps! Since the lowest note is the higher note becomes the lower note of a major
2nd, whereas to..., sevenths - … all perfect intervals: prime, or minor interval is just an interval F. `` Rustington '' 's contour halfway between two pitches ( assuming that microtones are not used
) instead to... The dominant flip the major/minor and diminished/augmented quality of the major scale upside-down,... Pitch is called a slash chord pairs in figure 8 are for the terms given such...
'' the major prefix is only used for seconds, thirds become sixths, sevenths - all... Position may determine how many lines and spaces appear to share the axis of symmetry or. '' works in the chord
is named, followed by a when perfect intervals are inverted, they interval inversion of Mozart Jupiter... Finale of Mozart 's Jupiter Symphony which share identical pitches in parallel major minor... | {"url":"http://agronautas.tempsite.ws/cook-game-kqxz/when-perfect-intervals-are-inverted%2C-they-d2b1b8","timestamp":"2024-11-12T13:22:16Z","content_type":"text/html","content_length":"33601","record_id":"<urn:uuid:c51e25c7-cd82-4fce-bfe6-1df91c8c92c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00712.warc.gz"} |
Week 2 ending 9/6
Monday-No School
Get you into the Google Classroom for this course
New learning
Learning target: I can identify whether a sequence is arithmetic, geometric or neither.
HW: Either complete the Lesson 1.3 Curated Problem Set (digital in Classroom or paper)
New learning
Learning target: I can use a spreadsheet to work with sequences
Make sure to complete the Curated Problem set 1.3 for tomorrow’s class.
No new HW | {"url":"https://danlemay.net/wordpress/?p=8472","timestamp":"2024-11-03T03:42:43Z","content_type":"text/html","content_length":"21444","record_id":"<urn:uuid:a630983c-73ef-4808-a510-dd6909091564>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00576.warc.gz"} |
Browsing by Author Kılıç, Emrah
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Issue Date Title Author(s)
Sep-2015 A curious matrix-sum identity and certain finite sums identities Kılıç, Emrah ; Akkuş, İlker; Ömür, Neşe; Yücel, T.
2016 Decompositions of the Cauchy and Ferrers-Jackson polynomials Irmak, Nurettin; Kılıç, Emrah
2023 Determinant Evaluation of Banded Toeplitz Matrices Via Bivariate Polynomial Families Alazemi, Abdullah; Kılıç, Emrah
2022 Diophantine Equations Related with Linear Binary Recurrences Kılıç, Emrah ; Akkus, Ilker; Omur, Nese
1-May-2018 Double binomial sums and double sums related with certain linear recurrences of various order Kılıç, Emrah ; Arıkan, Talha
2017 Evaluation of Hessenberg Determinants via Generating Function Approach Kılıç, Emrah ; Arıkan, Talha
2016 Evaluation of spectrum of 2-periodic tridiagonal-Sylvester matrix Kılıç, Emrah ; Arıkan, Talha
Mar-2016 Evaluation of sums involving Gaussian q-binomial coefficients with rational weight functions Kılıç, Emrah ; Prodinger, Helmut
Apr-2019 Evaluation of sums involving products of Gaussian q-binomial coefficients with applications Kılıç, Emrah ; Prodinger, Helmut
2017 Evaluation of sums involving products of Gaussian q-binomial coefficients with applications to Fibonomial sums Kılıç, Emrah ; Prodinger, Helmut
2017 Evaluation of sums involving products of Gaussian qq-binomial coefficients with applications to Fibonomial sums Kılıç, Emrah ; Prodınger, Helmut
2020 Evaluation of sums of products of Gaussian q-binomial coefficients with rational weight functions Arıkan, Talha; Kılıç, Emrah ; Prodinger, Helmut
1-Jun-2018 Evaluation of various partial sums of Gaussian q-binomial sums Kılıç, Emrah
Apr-2017 Evaluation of sums containing triple aerated generalized Fibonomial coefficients Kılıç, Emrah
2008 Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions Kılıç, Emrah
Jul-2019 Explicit spectrum of a circulant-tridiagonal matrix with applications Kılıç, Emrah ; Yalçıner, Aynur
2011 Factorizations and representations of binary polynomial recurrences by matrix methods Kılıç, Emrah ; Stanica, Pantelimon
2009 Factorizations Of The Pascal Matrix Via A Generalized Second Order Recurrent Matrix Kılıç, E. ; Ömür, Neşe; Tatar, G.; Ulutaş, Yücel Türker
2023 Fibonomial and Lucanomial sums through well-poised q-series Chu, W.; Kılıç, Emrah
2011 A formula for the generating functions of powers of Horadam's sequence with two additional parameters Kılıç, Emrah ; Ulutaş, Yücel Türker; Ömür, Neşe | {"url":"https://gcris.etu.edu.tr/browse?type=author&sort_by=1&order=ASC&rpp=20&etal=-1&authority=rp00110&offset=20","timestamp":"2024-11-10T18:33:58Z","content_type":"text/html","content_length":"35942","record_id":"<urn:uuid:215ed6a4-520e-4f9a-9dd7-205d937ff24b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00390.warc.gz"} |
Determining message residue using a set of polynomials
Determining message residue using a set of polynomials
A method is described for use in determining a residue of a message. The method includes loading at least a portion of each of a set of polynomials derived from a first polynomial, g(x), and
determining the residue using a set of stages. Individual ones of the stages apply a respective one of the derived set of polynomials to data output by a preceding one of the set of stages.
Data transmitted over network connections or retrieved from a storage device, for example, may be corrupted for a variety of reasons. For instance, a noisy transmission line may change a “1” signal
to a “0”, or vice versa. To detect corruption, data is often accompanied by some value derived from the data such as a checksum. A receiver of the data can recompute the checksum and compare with the
original checksum to confirm that the data was likely transmitted without error.
A common technique to identify data corruption is known as a Cyclic Redundancy Check (CRC). Though not literally a checksum, a CRC value can be used much in the same way. That is, a comparison of an
originally computed CRC and a recomputed CRC can identify data corruption with a very high likelihood. CRC computation is based on interpreting message bits as a polynomial, where each bit of the
message represents a polynomial coefficient. For example, a message of “1110” corresponds to a polynomial of x^3+x^2+x+0. The message is divided by another polynomial known as the key. For example,
the other polynomial may be “11” or x+1. A CRC is the remainder of a division of the message by the key. CRC polynomial division, however, is somewhat different than ordinary division in that it is
computed over the finite field GF(2) (i.e., the set of integers modulo 2). More simply put: even number coefficients become zeroes and odd number coefficients become ones.
A wide variety of techniques have been developed to perform CRC calculations. A first technique uses a dedicated CRC circuit to implement a specific polynomial key. This approach can produce very
fast circuitry with a very small footprint. The speed and size, however, often come at the cost of inflexibility with respect to the polynomial key used. Additionally, supporting multiple keys may
increase the circuitry footprint nearly linearly for each key supported.
A second commonly used technique features a CRC lookup table where, for a given polynomial and set of data inputs and remainders, all possible CRC results are calculated and stored. Determining a CRC
becomes a simple matter of performing table lookups. This approach, however, generally has a comparatively large circuit footprint and may require an entire re-population of the lookup table to
change the polynomial key being used.
A third technique is a programmable CRC circuit. This allows nearly any polynomial to be supported in a reasonably efficient amount of die area. Unfortunately, this method can suffer from much slower
performance than the previously described methods.
FIG. 1 is a diagram illustrating a set of stages that apply a set of pre-computed polynomials to determine a polynomial division residue.
FIG. 2 is a diagram of a set of pre-computed polynomials.
FIG. 3 is a diagram illustrating stages that perform parallel operations on a pre-computed polynomial and input data.
FIGS. 4A and 4B are diagrams of sample stages' digital logic gates.
FIG. 5 is a diagram of a system to compute a polynomial division residue.
FIG. 1 illustrates a sample implementation of a programmable Cyclic Redundancy Check (CRC) circuit 100. The circuit 100 can achieve, roughly, the same performance as a lookup table CRC implementation
and may be only modestly slower than a dedicated CRC circuit implementation operating on a typical polynomial. From a die-area perspective, the circuit 100 can be orders of magnitude smaller than a
lookup table approach and within an order of magnitude of a dedicated circuit implementation.
The circuit 100 uses a series of pre-computed polynomials 100a-100d derived from a polynomial key. Bits of the pre-computed polynomials 100a-100d are loaded into storage elements (e.g., registers or
memory locations) and fed into a series of stages 106a-106d that successively reduce an initial message into smaller intermediate values en route to a final CRC result output by stage 106d. For
example, as shown, the width of data, r[b]−r[d], output by stages 106a-106d decreases with each successive stage. The pre-computed polynomials 100a-100d and stages 106d-106a are constructed such that
the initial input, r[a], and the stage outputs, r[b]−r[d], are congruent to each other with respect to the final residue (i.e., r[a]≡r[b]≡r[c]≡r[d]). In addition, the pre-computed polynomials 100a-
100d permit the stages 106a-106d to perform many of the calculations in parallel, reducing the number of gate delays needed to determine a CRC residue. Reprogramming the circuitry 110 for a different
key can simply be a matter of loading the appropriate set of pre-computed polynomials into the storage elements 100a-100d.
FIG. 2 illustrates a sample set of pre-computed polynomials, g[i](x), 100a-100d (e.g., g[4], g[2], g[1], and g[0]). By examination, these polynomials 100a-100d have the property that each successive
polynomial 100a-100d in the set features a leading one bit (i+kth bit) followed by followed by i-zeroes 102 (shaded) and concluding with k-bits of data 104 of the order of the generating (CRC)
polynomial (e.g., g[0]). The form of these polynomials 100a-100d enables each stage 106a-106d to reduce input data to a smaller, but CRC equivalent, value. For example, after deriving a set of
polynomials {g[4](x), g[2](x), g[1](x)} from some 9-bit polynomial g[0](x), a CRC could be determined for input data of 16-bits. During operation, applying g[4](x) would reduce the input data from
16-bits to 12-bits, g[2](x) would reduce the next the data from 12-bits to 10-bits, and so forth until an 8-bit residue was output by a final stage 106d. Additionally, as described in greater detail
below, a given stage 106a-106d may use a polynomial 100a-100d to process mutually exclusive regions of the input data in parallel.
More rigorously, let g(x) be a k^th-degree CRC polynomial of k+1 bits, where the leading bit is always set in order that the residue may span k bits. The polynomial g(x) is defined as
$g ( x ) = [ x k + ∑ i = 0 k - 1 g i x i ]$ $g j ∈ GF ( 2 )$
The polynomial g[i](x) is then defined as:
g[i](x)=x^k+i+[x^k+i mod g(x)]
In accordance with this definition of g[i](x), a sequence of polynomials can be computed as a function of selected values of i and the original polynomial g(x).
The CRC polynomial, g(x), divides g[i](x):
$g ( x ) g i ( x )$ $proof$ $g i ( x ) = x k + i + [ x k + i mod g ( x ) ] = x k + i + [ x k + i - a i ( x ) g ( x ) ] - for some a i ( x ) = a i ( x ) g ( x
From this, a recurrence can be defined, where at each stage a message, m(x), is partially reduced by one of the pre-computed polynomials.
Let m(x) be a 2^L bit message and r(x) be the k-bit result:
r(x)=[m(x)·x^k mod g(x)]
where m(x) is shifted by x^k, creating room to append the resulting CRC residue to the message, m(x). Thus:
r[i](x)=[r[i−1](x)mod g[2][L−i](x)]
for i≧1. Thus, r[i](x)≡r[0](x) mod g(x), which is proved by induction on i:
$r 1 ( x ) ≡ r 0 ( x ) mod g ( x )$ $proof$ $r 1 ( x ) = r 0 ( x ) mod g 2 L - 1 ( x ) = r 0 ( x ) mod [ a 2 L - 1 ( x ) g ( x ) ]$ $r i ( x ) ≡ r i - 1 ( x )
mod g ( x )$ $proof$ $r i ( x ) = r i - 1 ( x ) mod g 2 L - i ( x ) = r i - 1 ( x ) mod [ a 2 L - i ( x ) g ( x ) ]$
Finally, r[L](x)=r(x), which follows from the observations made above:
$r L ( x ) = [ r L - 1 ( x ) mod g 0 ( x ) ] = [ m ( x ) · x k - b ( x ) · g ( x ) ] mod g 0 ( x ) - for some b ( x ) = m ( x ) · x k mod g ( x )$
These equations provide an approach to CRC computation that can be implemented in a wide variety of circuitry. For example, FIG. 3 illustrates a high-level architecture of a circuit implementing the
approach described above. As shown, a given stage 106a-106d can reduce input data, r, by subtracting a multiple of the k-least significant bits 104 of the pre-computed polynomial g[i](x) from the
stage input. Again, the resulting stage output is congruent to the stage input with respect to a CRC calculation though of a smaller width.
The sample implementation shown features stages 106a-106d that AND 110a-110d (e.g., multiply) the k-least significant bits 104 of g[i](x) by respective bits of input data. The i-zeroes 102 and
initial “1” of g[i](x) are not needed by the stage since they do not affect the results of stage computation. Thus, only the k-least significant bits of g[i](x) need to be stored by the circuitry.
To illustrate operation, assuming r[0 ]had a value starting “1010 . . . ” and the k-least significant bits of g[4](x) had a value of “001010010”, the first 110a and third 110c AND gates would output
“001010010” while the second 110b and fourth 110d AND gates would output zeros. As indicated by the shaded nodes in FIG. 3, the output of the AND 110a-110d gates can be aligned to shift (i.e.,
multiply) the gate 110a-110d output in accordance with the respective bit-positions of the input data. That is, the output of the gate 110a operating on the most significant bit of input data is
shifted by i−1 bits, and each succeeding gate 110b-110d decrements this shift by 1. For example, the output of gate 110a, corresponding to the most significant bit of r[0], is shifted by 3-bits with
respect to the input data, the output of gate 110b corresponding to the next most significant bit of r[0 ]is shifted by 2-bits, etc. The input data can then be subtracted (e.g., XOR-ed) by the
shifted-alignment of the output of gates 110a-110d. The subtraction result reduces the input data by a number of bits equal to the number of zeroes 102 in the polynomial for i>0. In essence, the
i-most significant bits of input data, r[0], act as selectors, either causing subtraction of the input data by some multiple of the k-least significant bits of g[i](x) or outputting zeroes that do
not alter the input data.
As shown, the AND gates 110a-110d of a stage 106a may operate in parallel since they work on mutually-exclusive portions of the input data. That is, AND gates 110a-110d can each simultaneously
process a different bit of r[0 ]in parallel. This parallel processing can significantly speed CRC calculation. Additionally, different stages may also process data in parallel. For example, gate 110e
of stage 106b can perform its selection near the very outset of operation since the most significant bit of r[0 ]passes through unaltered to stage 106b.
FIG. 4A depicts digital logic gates of a sample stage 106a implementation conforming to the architecture shown in FIG. 3. In this example, the stage 106a receives a 16-bit input value (e.g., r[0]=
input data [15:0]) and the k-least significant bits of g[4](x). The stage 106a processes the i-th most significant bits of the input value with i-sets of AND gates 110a-110d where each input data bit
is ANDed 110a-110d with each of the k-least significant bits of g[4](x). Each set of k-AND gates 110a-110d in FIG. 4A corresponds to the conceptual depiction of a single AND gate in FIG. 3. The
output of the AND gate arrays 110a-110d is aligned based on the input data bit position and fed into a tree of XOR gates 112a-112d that subtract the shifted AND gate 110a-110d output from the
remaining bits of input data (i.e., the input data less the i-th most significant bits).
FIG. 4B depicts digital logic gates of a succeeding stage 106b that receives output_1 data [11:0] and generates output_2 data [9:0]. The stage 106b receives the 12-bit value output by stage 106a and
uses g[2](x) to reduce the 12-bit value to a CRC congruent 10-bit value. Stages 106a, 106b share the same basic architecture of i-arrays of AND gates that operate on the k-least significant bits of g
[i](x) and an XOR tree that subtracts the shifted AND gate output from the stage input to generate the stage output value. Other stages for different values of i can be similarly constructed.
The architecture shown in FIGS. 3, 4A, and 4B are merely examples and a wide variety of other implementations may be used. For example, in the sample FIGS., each stage 106a-106d processed the i-th
most significant bits of input data in parallel. In other implementations, a number of bits greater or less than i could be used in parallel, however, this may not succeed in reducing the size of the
output data for a given stage.
The architecture shown above may be used in deriving the pre-computed polynomials. For example, derivation can be performed by zeroing the storage elements associated with g[i](x) and loading g[0 ]
with the k-least significant bits of the polynomial key. The bits associated with successive g[i]-s can be determined by applying x^k+i as the data input to the circuit and storing the resulting
k-least significant bits output by the g[0 ]stage as the value associated with g[i]. For example, to derive the polynomial for g[2], x^k+2 can be applied as the circuit, the resulting k-bit output of
the g[0 ]stage can be loaded as the value of the g[2 ]polynomial.
FIG. 5 depicts a sample CRC implementation using techniques described above. The implementation works on successive portions of a larger message in 32-bit segments 120. As shown, the sample
implementation shifts 124, 126 and XORs 128 a given portion 120 of a message by any pre-existing residue 122 and computes the CRC residue using stages 106a-106f and the k-bits of the respective
pre-computed polynomials g[i](x). Again, successive stages 106a-106e reduce input data by i-bits until a residue value is output by stage 106f. The circuit then feeds the residue back 122 for use in
processing the next message portion 124. The residue remaining after the final message portion 120 is applied is the CRC value determined for the message as a whole. This can either be appended to
the message or compared with a received CRC value to determine whether data corruption likely occurred.
The system shown in FIG. 5 featured (L+1) stages 106a-106f where the polynomials were of the form i={0, 2^n−1 for n=1 to L}. However, this strict geometric progression of i is not necessary and other
values of i may be used in reducing a message. Additionally, at the lower polynomial levels (e.g., i<4) it may be more efficient to abandon the stage architecture depicted in FIGS. 3, 4A, and 4B and
process an input value using g[0 ]in a traditional bit-serial or other fashion.
Techniques described above can be used to improve CRC calculation speed, power efficiency, and circuit footprint. As such, techniques described above may be used in a variety of environments such as
network processors, security processors, chipsets, ASICs (Application Specific Integrated Circuits), and as a functional unit within a processor or processor core where the ability to handle high
clock speeds, while supporting arbitrary polynomials, is of particular value. As an example, CRC circuitry as described above may be integrated into a device having one or more media access
controllers (e.g., Ethernet MACs) coupled to one or more processors/processor cores. Such circuitry may be integrated into the processor itself, in a network interface card (NIC), chipset, as a
co-processor, and so forth. The CRC circuitry may operate on data included within a network packet (e.g., the packet header and/or payload). Additionally, while described in conjunction with a CRC
calculation, this technique may be applied in a variety of calculations such as other residue calculations over GF(2) (e.g., Elliptic Curve Cryptography).
The term circuitry as used herein includes implementations of hardwired circuitry, digital circuitry, analog circuitry, programmable circuitry, and so forth. The programmable circuitry may operate on
computer instructions disposed on a storage medium.
Other embodiments are within the scope of the following claims.
1. A method for use in determining a residue of a message, m(x), the method comprising:
loading into a set of storage elements at least a portion of each of a set of polynomials derived from a first polynomial, g(x); and
determining the residue for the message corresponding to m(x) mod g(x) using a set of stages, respective stages in the set of stages comprising digital logic to apply at least a portion of a
respective one of the polynomials stored in a respective one of the set of storage elements to a respective input of the stage received from a preceding stage;
wherein the set of polynomials conform to: gi(x)=xk+i+[xk+i mod g(x)]
for multiple values of i, where i and k are integers.
2. The method of claim 1,
wherein the set of polynomials comprises polynomials having a prefix and a k-bit remainder where k is positive integer, wherein the prefix of a polynomial in the set of polynomials consists of a
most significant bit equal to 1 followed by a set of zero or more consecutive zeroes, and wherein zeros in the set of one or more consecutive zeroes increases for successive polynomials in the
3. The method of claim 1,
wherein respective ones of the set of stages receives bits of ri−1(x) and outputs bits of ri(x), such that ri(x)=ri−1(x).
4. The method of claim 1,
further comprising, at least one of: (1) appending the residue to the message for transmission across a network, and (2) comparing the residue to a previously computed residue.
5. The method of claim 1,
wherein, in individual ones of the stages, at least a portion of a one of the set of polynomials associated with a respective stage undergoes polynomial multiplication by respective bits of input
data received by the respective stage.
6. The method of claim 5,
wherein the respective bits of input data consist of a number of bits equal to a number of zeroes in the set of one or more consecutive zeroes in the respective polynomial prefix.
7. The method of claim 5, where the polynomial multiplication by respective bits of input data occurs in parallel for the respective bits of input data.
8. An apparatus for use in determining a residue of a message, m, with respect to a first polynomial, g(x), over a finite field, GF(2), the apparatus comprising:
a set of storage elements to store at least a portion of each of a set of polynomials derived from the first polynomial, g(x); and
a set of stages coupled to respective ones of the set of storage elements, respective stages in the set of stages comprising digital logic gates to apply at least a portion of a respective one of
the polynomials stored in a respective one of the set of storage elements to a respective input of the stage received from a preceding stage, the determined residue for the message, m, being
based on output of a last of the set of stages;
wherein the set of polynomials conform to: gi(x)=xk+i+[xk+i mod g(x)]
for multiple values of i, where i and k are integers.
9. The apparatus of claim 8,
wherein the set of polynomials comprises polynomials having a prefix and a k-bit remainder where k is a positive integer, wherein the prefix of a polynomial in the set consists of a most
significant bit equal to 1 followed by a set of zero or more consecutive zeroes, and wherein a number of consecutive zeroes in the set of zero or more zeroes increases for successive polynomials
in the set.
10. The apparatus of claim 8,
wherein respective ones of the set of stages receives bits of ri−1(x) and outputs bits of ri(x), such that ri(x)=ri−1(x).
11. The apparatus of claim 9,
wherein in individual ones of the set of stages, the respective polynomial k-bit remainder associated with a respective stage is fed into AND gates with input data bits of the respective stage.
12. The apparatus of claim 11,
wherein the respective input data bits fed into the AND gates consist of a number of bits equal to a number of consecutive zeroes in the respective polynomial prefix set of zero or more
consecutive zeroes.
13. The apparatus of claim 11,
wherein the digital logic gates comprise a tree of exclusive-or (XOR) gates coupled to the output of the AND gates and the least significant bits of the stage input data.
14. The apparatus of claim 8,
further comprising circuitry to load new values of the set of polynomials into the storage elements.
15. A device, comprising:
at least one media access controller (MAC) to receive a message from a network;
at least one processor communicatively coupled to the at least one media access controller;
the device including circuitry to determine a residue of the message with respect to a first polynomial, g(x), over a finite field, GF(2), the circuitry comprising: a set of storage elements to
store a set of polynomials derived from the first polynomial, g(x); and a set of stages coupled to respective ones of the set of storage elements, respective stages in the set of stages
comprising digital logic gates to apply at least a portion of a respective one of the polynomials stored in a respective one of the set of storage elements to a respective input of the respective
stage received from a preceding stage, the determined residue for the message, m, being based on output of a last of the set of stages;
wherein the set of polynomials conform to: gi(x)=xk+i+[xk+i mod g(x)]
for multiple values of i, where i and k are integers.
16. The device of claim 15,
wherein the set of polynomials comprises polynomials having a prefix and a k-bit remainder where k is a positive integer, wherein the prefix of a polynomial in the set consists of a most
significant bit equal to 1 followed by a set of zero or more consecutive zeroes, and wherein a number of consecutive zeroes in the set of zero or more consecutive zeroes increases for successive
polynomials in the set.
17. The device of claim 15,
wherein in individual ones of the set of stages, the respective polynomial k-bit remainder associated with a respective stage are fed into AND gates with input data bits of the respective stage;
wherein the respective input data bits fed into the AND gates consist of a number of bits equal to the number of successive zeroes in the respective polynomial prefix set of zero or more
consecutive zeroes.
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Patent History
Patent number
: 7827471
: Oct 12, 2006
Date of Patent
: Nov 2, 2010
Patent Publication Number
20080092020 Assignee
Intel Corporation
(Santa Clara, CA)
William C. Hasenplaugh
(Jamaica Plain, MA),
Brad A. Burres
(Waltham, MA),
Gunnar Gaubatz
(Worcester, MA)
Primary Examiner
M. Mujtaba K Chaudry Application Number
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Section 9.1: Euler Circuits and Hamiltonian Cycles - The Nature of Mathematics - 13th Edition
Section 9.1: Euler Circuits and Hamiltonian Cycles
9.1 Outline
A. Euler circuits
1. Konigsberg bridge problem
2. definition of a graph (or a network)
3. traversable network
4. degree of a vertex
5. Euler circuit
6. odd/even vertex
7. connected network
8. Euler’s circuit theorem
B. Applications of Euler circuits
1. supermarket problem
2. police patrol problem
3. floor-plan problem
4. water-pipe problem
C. Hamiltonian cycles
1. traveling salesperson problem (TSP)
2. definition
3. loop
4. brute-force method
5. nearest neighbor method
6. sorted-edge method
7. number of routes
9.1 Essential Ideas
Given a network, begin at some vertex and travel on each edge exactly once, and then return to the starting vertex. Such as path is called an Euler circuit. There is a solution to the question of
whether an Euler circuit for a given network exists; it is known as Euler’s circuit theorem.
Euler’s Circuit Theorem
Every vertex on a graph with an Euler circuit has an even degree, and conversely, if in a connected graph every vertex has an even degree, then the graph has an Euler circuit.
Hamiltonian Cycle
Given a network, begin a some vertex and travel to each vertex exactly once, ending at the original vertex. Such a path is called a Hamiltonian cycle. There is no solution to the question of whether
a Hamiltonian cycle for a given network exists. The methods we use in this text are brute force (listing all possible routes); nearest neighbor (at each city, go to the nearest neighbor next); and
sorted-edge (sometimes the nearest neighbor plan will form a loop without going to some city) procedures.
Sorted-Edge Method
Draw a graph showing the cities and the distances; identify the starting vertex.
Step 1: Choose the edge attached to the starting vertex that has the shortest distance or the lowest cost. Travel along this edge to the next vertex.
Step 2: At the second vertex, travel along the edge with the shortest distance or lowest cost. Do not choose a vertex that would lead to a vertex already visited.
Step 3: Continue until all vertices are visited until arriving back at the original vertex. | {"url":"https://mathnature.com/essential-ideas-9-1/","timestamp":"2024-11-04T04:55:21Z","content_type":"text/html","content_length":"112847","record_id":"<urn:uuid:c1798485-b788-4684-bf77-f0010fb56be6>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00187.warc.gz"} |
Turán numbers of general star forests in hypergraphs
Let F be a family of r-uniform hypergraphs, and let H be an r-uniform hypergraph. Then H is called F-free if it does not contain any member of F as a subhypergraph. The Turán number of F, denoted by
ex[r](n,F), is the maximum number of hyperedges in an F-free n-vertex r-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star
forests. In particular, Lidický et al. (2013) [17] determined the Turán number ex(n,F) of a star forest F for sufficiently large n. Recently, Khormali and Palmer (2022) [13] generalized the above
result to three different well-studied hypergraph settings (the expansions of a graph, linear hypergraphs and Berge hypergraphs), but restricted to the case that all stars in the hypergraph star
forests are identical. We further generalize these results to general star forests in hypergraphs.
• UT-Hybrid-D
• Star forest
• Turán number
• Berge hypergraph | {"url":"https://research.utwente.nl/en/publications/tur%C3%A1n-numbers-of-general-star-forests-in-hypergraphs","timestamp":"2024-11-03T16:45:50Z","content_type":"text/html","content_length":"45787","record_id":"<urn:uuid:24407100-2604-4f6d-8143-b627b3a8bd07>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00679.warc.gz"} |
Homework Help
Maths Assignment, Homework Help
Math: - Math is the study of space, relation, structure, change, and many other topics of pattern. Now a day, math is used around the world as an essential and important tool in most of the fields,
including engineering, social science, medicine, natural science etc.
Tutors at the theglobaltutors.com are available 24x7 for all grade, college and university level students to provide math assignment help, homework help. Math tutors are much experienced to provide
help according to student’s requirement in math assignment and homework. Most of the tutors who are helping student in their Math assignment and homework help complete their master or PhD in Math.
Quantity:-The study of quantity starts with numbers, first the familiar natural numbers and integers and arithmetical operations on them, which are characterized in arithmetic.
Real Numbers
Natural Numbers
Complex Numbers
Structure:- There are three main topics define structure. You can get help from our expert tutors in these topics. Math assignment help and homework help is available all three topics given below.
Number theory
Abstract algebra
Order theory
Space:-The study of space originates with geometry – in particular, Euclidean geometry. The entire sub topics to cover the main topic “space” are given below and you can get help to solve your
assignment and homework.
Differential geometry
Fractal geometry
Change:-Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Assignment help and homework help can be avail
form the tutors at the theglobaltutors.com in all topics given below.
Vector calculus
Differential equations
Dynamical systems
Chaos theory
Foundations and philosophy:-Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, and
recursion theory. Tutors at the theglobaltutors.com are perfect choice of student to get help to solve their assignment and homework under the topics given below.
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Mathematical logic: - Mathematical logic is a subfield of mathematics Recursion theory
Set theory
Discrete Mathematics:-Discrete math tutors at the theglobaltutors.com are well qualifies to handle the math assignments under the topics of discrete math given below.
Theory of computation
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Applied mathematics:-Applied math assignment help and homework help is available for college and university level of student. Tutors are well qualified and have enough experience to hand the applied
math assignments. Some useful topics are given below.
Mathematical physics
Analytical mechanics
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Mathematical economics
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TAGS: - Math Assignment help | Math Assignment | Math Assignment Tutor | Online Math Assignment Help | Math Assignment Help Services | Math Homework Help | Online Math Assignment Tutor | Live Math
Assignment Help | Mathematics Assignment Help | | {"url":"https://www.theglobaltutors.com/mathematics-assignment-help.aspx","timestamp":"2024-11-09T07:01:24Z","content_type":"application/xhtml+xml","content_length":"43895","record_id":"<urn:uuid:974cd8a9-d165-4a4b-af1f-13a53d284d0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00304.warc.gz"} |
Advanced Looping Techniques in Swift
In this lesson, you'll explore how to iterate over dictionaries. You'll learn how to loop through key-value pairs in dictionaries and apply calculations directly within loops.
To illustrate, consider this example where we calculate the travel time to different planets:
1import Foundation
3// Calculating travel time to each planet
4let planets = ["Mercury": 0.4, "Venus": 0.7, "Mars": 1.4] // in astronomical units from Earth
5let speed = 0.1 // astronomical units per day
7for (planet, distance) in planets {
8 let days = distance / speed
9 let formattedDays = String(format: "%.1f", days)
10 print("It will take \(formattedDays) days to reach \(planet).")
Here, we loop through a dictionary to compute the time it takes to reach each planet, formatting the result for readability. | {"url":"https://learn.codesignal.com/preview/lessons/2404","timestamp":"2024-11-02T07:38:22Z","content_type":"text/html","content_length":"116308","record_id":"<urn:uuid:11b692f7-caed-48e6-ab43-b5ed890058a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00134.warc.gz"} |
Conditional Formatting in Excel
Use conditional formatting in Excel to automatically highlight cells based on their content. Apply a rule or use a formula to determine which cells to format.
Highlight Cells Rules
To highlight cells that are greater than a value, execute the following steps.
1. Select the range A1:A10.
2. On the Home tab, in the Styles group, click Conditional Formatting.
3. Click Highlight Cells Rules, Greater Than.
4. Enter the value 80 and select a formatting style.
5. Click OK.
Result: Excel highlights the cells that are greater than 80.
6. Change the value of cell A1 to 81.
Result: Excel changes the format of cell A1 automatically.
Note: you can also use this category (see step 3) to highlight cells that are less than a value, between two values, equal to a value, cells that contain specific text, dates (today, last week, next
month, etc.), duplicates or unique values.
Clear Rules
To clear a conditional formatting rule, execute the following steps.
1. Select the range A1:A10.
2. On the Home tab, in the Styles group, click Conditional Formatting.
3. Click Clear Rules, Clear Rules from Selected Cells.
Top/Bottom Rules
To highlight cells that are above average, execute the following steps.
1. Select the range A1:A10.
2. On the Home tab, in the Styles group, click Conditional Formatting.
3. Click Top/Bottom Rules, Above Average.
4. Select a formatting style.
5. Click OK.
Result: Excel calculates the average (42.5) and formats the cells that are above this average.
Note: you can also use this category (see step 3) to highlight the top n items, the top n percent, the bottom n items, the bottom n percent or cells that are below average.
Conditional Formatting with Formulas
Take your Excel skills to the next level and use a formula to determine which cells to format. Formulas that apply conditional formatting must evaluate to TRUE or FALSE.
1. Select the range A1:E5.
2. On the Home tab, in the Styles group, click Conditional Formatting.
3. Click New Rule.
4. Select 'Use a formula to determine which cells to format'.
5. Enter the formula =ISODD(A1)
6. Select a formatting style and click OK.
Result: Excel highlights all odd numbers.
Explanation: always write the formula for the upper-left cell in the selected range. Excel automatically copies the formula to the other cells. Thus, cell A2 contains the formula =ISODD(A2), cell A3
contains the formula =ISODD(A3), etc.
Here's another example.
7. Select the range A2:D7.
8. Repeat steps 2-4 above.
9. Enter the formula =$C2="USA"
10. Select a formatting style and click OK.
Result: Excel highlights all USA orders.
Explanation: we locked the reference to column C by placing a $ symbol in front of the column letter ($C2). As a result, cell B2, C2 and cell D2 also contain the formula =$C2="USA", cell A3, B3, C3
and D3 contain the formula =$C3="USA", etc.
Color Scales
Use awesome color scales to assign different colors to different values. This allows you to quickly identify high and low points in your dataset.
Tip: learn more about color scales and learn how to create this heat map.
Highlight Blank Cells
You can also use conditional formatting in Excel to format blank cells. This is useful for ensuring data completeness and quickly shows where information is missing.
Tip: learn how to highlight blank cells on our page about blanks. | {"url":"https://www.excel-easy.com/data-analysis/conditional-formatting.html","timestamp":"2024-11-08T01:31:49Z","content_type":"application/xhtml+xml","content_length":"25059","record_id":"<urn:uuid:10710a49-8d0c-4bcf-b887-8ae6dc8a7e5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00036.warc.gz"} |
Unveiling the Slope of Parallel Lines: A Comprehensive Guide
Unveiling The Slope Of Parallel Lines: A Comprehensive Guide
To determine the slope of a line parallel to a given line, first calculate the slope of the given line by calculating the change in y divided by the change in x. Given that parallel lines have equal
slopes, the slope of the parallel line will be identical. Therefore, the slope of the parallel line can be directly obtained from the slope of the given line.
Understanding Slope
When it comes to lines, slope plays a crucial role in determining their orientation and steepness. Slope measures the inclination of a line relative to the horizontal axis, conveying how much it
rises or falls as you move along the line.
To calculate slope, we use the formula:
Slope = (Change in y) / (Change in x)
In other words, the slope is found by dividing the change in the vertical coordinate (y) by the change in the horizontal coordinate (x). This ratio represents the angle that the line makes with the
horizontal axis.
A Closer Look at the Formula
The change in y refers to the difference between the y-coordinates of two points on the line, while the change in x represents the difference between their x-coordinates.
• A positive slope indicates that the line rises from left to right, like climbing up a hill.
• A negative slope means the line falls from left to right, like descending down a slope.
• A zero slope implies a horizontal line that runs parallel to the x-axis.
Understanding Parallel Lines and Slope
In the realm of geometry, lines hold a significant place, and understanding their properties is crucial for navigating the complexities of shapes and angles. Among these properties, slope plays a
vital role in defining the orientation of a line. In this article, we’ll delve into the intricate relationship between parallel lines and their slopes.
Defining Parallel Lines
Parallel lines are a special category of lines that have a unique characteristic: they never intersect, regardless of how far they are extended. This means that they run alongside each other,
maintaining a constant, non-zero distance.
The Slope Connection
The slope of a line measures its steepness or inclination. It is calculated as the ratio of the change in vertical distance (rise) to the change in horizontal distance (run). Remarkably, parallel
lines share a special relationship with slope.
Equal Slopes of Parallel Lines
A fundamental property of parallel lines is that they have equal slopes. This means that if you measure the slope of one parallel line, it will be the same as the slope of all other parallel lines.
This is because the lines never intersect, so their rise and run will be proportional at any given point.
Implication for Slope Analysis
This property has a significant implication: if you know the slope of one parallel line, you know the slope of all parallel lines. This is an invaluable piece of information that can simplify slope
calculations and analysis.
Method for Finding Slope of a Parallel Line
Determining the slope of a parallel line is a straightforward process that involves two simple steps:
1. Identify the Slope of the Given Line: Calculate the slope of the given line using a specific point on the line.
2. Use the Same Slope: The slope of the parallel line will be the same as the slope of the given line.
By following these steps, you can quickly determine the slope of any line parallel to a given line.
Example: Slope of a Parallel Line
Let’s consider an example. Given a line with a slope of 2/3, what is the slope of a line parallel to it?
Using the method outlined above:
1. Identifying the Slope: The given line has a slope of 2/3.
2. Using the Same Slope: The slope of the parallel line will also be 2/3.
Therefore, any line parallel to the given line will have a slope of 2/3.
Parallel Lines and Their Slopes: A Mathematical Connection
In the realm of geometry, lines play a crucial role. Understanding their properties, like slope, helps us navigate the intricate world of shapes and angles. When it comes to parallel lines—lines that
never cross paths—the concept of slope takes on a unique significance.
What is Slope?
Slope is a measure of a line’s steepness or slant. It is calculated as the change in the vertical direction (y-axis) divided by the change in the horizontal direction (x-axis). A line with a positive
slope rises from left to right, while a line with a negative slope falls from left to right.
Parallel Lines and Slope
Parallel lines share a remarkable property: they have equal slopes. This means that if you know the slope of one parallel line, you automatically know the slope of all other lines parallel to it.
This property arises from the fact that parallel lines are offset from each other by a constant vertical distance. As you move along a parallel line, the vertical change (y) will remain the same for
every horizontal change (x). Therefore, the ratio of y to x (slope) will be identical for all parallel lines.
Implication of Equal Slopes
This relationship between parallel lines has important implications. It means that the slope of a line can serve as a unique identifier for a family of parallel lines. By knowing the slope of one
parallel line, you can identify and draw all other lines in that family, regardless of their position on the graph.
Method for Finding the Slope of a Parallel Line
To find the slope of a parallel line, follow these simple steps:
Step 1: Identify the Slope of the Given Line
First, you need to know the slope of one of the parallel lines. Calculate the slope using the formula: slope = (change in y) / (change in x).
Step 2: Use the Same Slope
Since the parallel lines have equal slopes, the slope of the unknown line will be the same as the slope of the given line.
Example Problem
Consider a line with a slope of 2.
Step 1: The slope of the given line is 2.
Step 2: Since any parallel line will have the same slope, the slope of the parallel line is also 2.
This knowledge allows you to draw or identify any other line parallel to the given line, simply by using the slope of 2.
The relationship between parallel lines and slope is a fundamental concept in geometry. By understanding that parallel lines have equal slopes, you can easily find the slopes of unknown lines and
draw or identify parallel line families. This knowledge is essential for solving a wide range of geometric problems and deepening your understanding of lines and their properties.
Unveiling the Secrets of Parallel Lines and Slope
Step 1: Discovering the Slope of the Given Line
Before embarking on our quest to decipher the slope of a parallel line, we must first unravel the secrets of the given line. Slope, the measure of a line’s inclination, can be calculated by dividing
the change in y (vertical distance) by the change in x (horizontal distance), symbolized as:
Slope (m) = Δy / Δx
To determine the slope of the given line, select two points on the line and use the formula above. For instance, a line passing through points (2, 5) and (6, 11) would have a slope of (11 – 5) / (6 –
2) = 1.
Step 2: Unveiling the Common Thread
The beauty of parallel lines lies in their shared characteristic of never intersecting. This inherent property gives rise to a remarkable connection between their slopes: parallel lines always have
equal slopes. In other words, the slope of one parallel line holds the key to unlocking the slope of its kindred spirits.
Implications and Applications
This newfound knowledge has profound implications for our understanding of parallel lines. If we know the slope of one parallel line, we can confidently deduce the slopes of all its parallel
counterparts. This understanding enables us to tackle problems involving parallel lines with remarkable ease and precision.
Example: A Journey into Parallel Slopes
Consider the equation of a given line: y = 2x + 3. To find the slope of a parallel line, we simply identify the slope of the given line. Using the formula m = Δy / Δx, we calculate 2 as the slope.
Thus, our parallel line will also have a slope of 2, ensuring that it remains parallel to the given line, never destined to meet.
Understanding the Slope of Parallel Lines
In the realm of geometry, understanding slope is crucial for comprehending the behavior of lines. Slope measures the steepness of a line, indicating its angle of inclination. It’s calculated as the
change in the vertical coordinate (y) divided by the change in the horizontal coordinate (x).
When dealing with parallel lines, a fascinating relationship emerges. Parallel lines are those that never intersect, running alongside each other like two railroad tracks. A key property is that
parallel lines possess equal slopes. This means that if you know the slope of one parallel line, you automatically know the slope of all other parallel lines.
How to Find the Slope of a Parallel Line
Calculating the slope of a line parallel to a given line is a straightforward process:
1. Step 1: Determine the Slope of the Given Line
• Identify a point on the given line.
• Calculate the slope using the formula: Slope = (change in y) / (change in x) between that point and any other point.
2. Step 2: Use the Same Slope
• Since the parallel line has the same slope, simply use the slope calculated in Step 1.
Example Problem
Let’s put this method into practice. Consider a line passing through the points (2, 3) and (5, 7).
1. Step 1: Slope of the Given Line
• Change in y = 7 – 3 = 4
• Change in x = 5 – 2 = 3
• Slope = 4 / 3
2. Step 2: Slope of the Parallel Line
• The parallel line has the same slope as the given line, which is 4/3.
Therefore, any line parallel to the line passing through (2, 3) and (5, 7) will have a slope of 4/3.
Leave a Reply Cancel reply | {"url":"https://www.biomedes.biz/unveiling-slope-parallel-lines/","timestamp":"2024-11-06T06:03:38Z","content_type":"text/html","content_length":"87237","record_id":"<urn:uuid:c7488743-709d-45ce-a098-cafc50665def>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00617.warc.gz"} |
Robert Kerr, (NCAR), The Role of Two Length Scales in the Singular Collapse of Euler and Ideal MHD
Schedule Feb 10, 2000
The Role of Two Length Scales in the Singular Collapse of Euler and Ideal MHD
Robert Kerr, (NCAR)
Experimental and numerical evidence is presented for a change in the scaling of structure functions within the inertial subrange at approximately $\lambda$. Kinematic arguments based on pressure are
given for why longitudinal and transverse scaling should be the same for $r>\lambda$, but there are no restrictions for $r<\lambda$. It is speculated that this shift in scaling could be related to
the anisotropy of small scale structures. First vortex tubes. Then the structure developing around a putative singularity of Euler. It is shown numerical and by mathematical arguements that the
collapse of the Euler structure must involve two length scales. One goes as $R\sim(T_c-t)^{1/2}$ and the other as $\rho\sim(T_c-t)$.
Audio for this talk requires sound hardware, and RealPlayer or RealAudio by RealNetworks.
Begin continuous audio for the whole talk. (Or, right-click to download the whole audio file.)
To begin viewing slides, click on the first slide below. | {"url":"https://online.kitp.ucsb.edu/online/hydrot_c00/kerr/","timestamp":"2024-11-10T21:35:32Z","content_type":"text/html","content_length":"4894","record_id":"<urn:uuid:7fbb4c14-1573-4b29-8ca4-2840d38112cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00113.warc.gz"} |
Should You Spell Out Numbers In Apa - SpellingNumbers.com
Should You Spell Out Numbers In Apa
Should You Spell Out Numbers In Apa – Learning to spell numbers can be difficult. It is made easier if you have the proper resources. There are many resources available to help you learn how to
spell. They include workbooks as well as tips and games online.
The format of the Associated Press format
If you are writing for a newspaper and other print media you need to be able to spell numbers in AP style. The AP style teaches you the correct spelling of numbers and other specifics to make writing
The Associated Press Stylebook was first published in 1953. Since the time, hundreds of modifications have been made. The stylebook is now celebrating its 55th anniversary. This stylebook is used in
the majority of American periodicals, newspapers and online news sources.
A set of rules regarding punctuation and the use of language, referred to as AP Style are used frequently in journalism. The most important guidelines of AP Style include capitalization, the use date
and time, as well as citations.
Regular numbers
A ordinal number can be defined as a distinct integer that represents a particular place in an array. These numbers are commonly used to represent significance, size, or the passage time. These
figures also indicate what is in which order.
Depending on the situation and how it is used, ordinary numbers can be written in a number of ways, both numerically and verbally. The unique suffix makes the distinction between them.
To create an ordinal number you need to add an “th” and a “th” to the end. The ordinal number 31 can be represented as 31.
There are many things that can be accomplished using ordinals, such as dates and names. It’s crucial to differentiate between an ordinal and the cardinal.
In addition, trillions of dollars
Numerology is utilized in many situations, such as the geology, the stock market and even the history of the globe. Examples include millions and billions. One million is the normal number that
precedes 1,000,001. A billion follows 999,999999999.
The annual earnings of any business is expressed as millions. They can also be used for calculating the value of a fund, stock or piece of money. In addition, billions are utilized as a unit of
measurement for a company’s stock market capitalization. You can test the accuracy of your estimates by converting millions to billions using a unit conversion calculator.
In the English language, the word “fractions” is used to indicate specific items or parts of numbers. The denominator, as well as the numerator are divided into two separate pieces. The numerator
indicates the number of equal-sized pieces taken in while the denominator displays the amount of portions that were split into.
Fractions may be written using mathematic terms, or written as words. It is essential to spell fractions correctly when writing in words. It might be difficult to spell out fractions correctly
particularly if you’re dealing with large fractions.
If you prefer to write fractions in words There are some easy rules to adhere to. You can start sentences with numbers written in complete. Another alternative is to write fractions as decimal
Many Years
When writing spelling, you’ll use years, no matter whether you’re writing a paper such as a thesis, email, or a research paper. Certain tricks and techniques will allow you to avoid typing the exact
same number again.
Numbers must be written out in formal written form. There are several style manuals available with various guidelines. The Chicago Manual of Style recommends that you use numerals between 1 and 100.
However, it’s not recommended that you write numbers higher than 401.
There are some exceptions. One such exception is the American Psychological Association style guide (APA). While it is not a specific publication, this guide is widely used in scientific writing.
Date and time
The Associated Press style book provides some general guidelines regarding how to style numbers. The numeral system is utilized for numbers 10 and higher. Numerology can also used in other contexts.
The standard procedure for the first 5 numbers of your essay is to write “n-mandated”. There are some exceptions.
The Chicago Manual of Techniques and the AP styling manual recommend using lots of numbers. However, this doesn’t mean it is impossible to make a version that is not based on numbers. Though I can
confirm you that there is a distinction because I myself have been an AP graduate.
It’s a good idea to consult the stylebook every time you want to determine which styles you are missing. For example it is not a good idea to overlook the letter “t” like as the “t”, in “time”.
Gallery of Should You Spell Out Numbers In Apa
APA Title Page Chegg Writing
When Should You Spell Out Numbers
When To Spell Out Numbers Rules For Writing Numbers In APA Chicago | {"url":"https://www.spellingnumbers.com/should-you-spell-out-numbers-in-apa/","timestamp":"2024-11-10T02:54:18Z","content_type":"text/html","content_length":"63694","record_id":"<urn:uuid:77df13a8-c558-4b73-8f2f-a3c6147db10b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00237.warc.gz"} |
Tug of War
Problem E
Tug of War
A tug of war is to be arranged at the local office picnic. For the tug of war, the picnickers must be divided into two teams. Each person must be on one team or the other; the number of people on the
two teams must not differ by more than $1$; the total weight of the people on each team should be as nearly equal as possible.
The first line of input contains $n$, the number of people at the picnic. $n$ lines follow. The first line gives the weight of person $1$; the second the weight of person $2$; and so on. Each weight
is an integer between $1$ and $450$. There is at least $1$ and at most $100$ people at the picnic.
Your output will be a single line containing $2$ integers: the total weight of the people on one team, and the total weight of the people on the other team. If these numbers differ, give the lesser
Sample Input 1 Sample Output 1 | {"url":"https://kth.kattis.com/courses/DD2458/popup17/assignments/cj6nyk/problems/tugofwar","timestamp":"2024-11-05T10:29:02Z","content_type":"text/html","content_length":"25668","record_id":"<urn:uuid:d6d2a640-50db-4af9-a0da-caf662884d14>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00160.warc.gz"} |
AI assistant can't print Formula using LaTex.
Use MathJax Delimiters
Use MathJax delimiters like $ for inline equations and $$ for display equations. For example:
• Inline equation: $\frac{24}{18 - x}$
• Display equation: 2418−x18−x24
This will ensure the equations are properly formatted and rendered by MathJax.
Provide Specific Prompts
Include clear instructions in your prompt about using MathJax delimiters for equations. For example:"Please use LaTeX formatting for all mathematical equations. Enclose inline equations with single
dollar signs ($) and display equations with double dollar signs ().Forexample,theequation).Forexample,theequation\frac{24}{18 - x}$$ should be written as $\frac{24}{18 - x}$."Providing these specific
instructions can help GPT-4 understand your formatting preferences.
Postprocess the Output
If GPT-4 still outputs equations in raw LaTeX format, you can postprocess the response to convert the delimiters. For example, in Python: | {"url":"https://intellij-support.jetbrains.com/hc/en-us/community/posts/21406002637586-AI-assistant-can-t-print-Formula-using-LaTex?page=1#community_comment_21406878338578","timestamp":"2024-11-02T17:38:09Z","content_type":"text/html","content_length":"26788","record_id":"<urn:uuid:52fb3ee9-31a7-44db-8427-a3787ecad3ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00264.warc.gz"} |
He should not rest well
Well, the sick criminal warlord is rushed back to Landhan again.
We can't do pretty much, but the little things add up.
We should not let him sleep, rest and recover well in a London hospital. We should bombard the hospital that is treating him, called Cromwell, with emails, faxes, phone calls and in-person protests
-- anything we can do. They cannot have a war criminal treated at their hospital, though it is a private one.
Do what you can. Here is the information needed.
Cromwell Hospital
Cromwell Road
SW5 0TU
United Kingdom
Email: info@cromwellhospital.com
Phone: 44-(0)-20-7460-2000
Fax: 44-(0)-20-7835-2444
One's war criminal is another's hero, ma maqashay waligaa...Bal hawada aadaan, see if he gives a rat's **you know what**...
Your hatred towards him only makes him stronger...Inshallah, he'll pull throu again...And if he doesn't, well we'll all die one day, no? He'll die as an HERO...LONG LIVE CABDULLAHI YUUSUF!!!
MMA, You need to organize someone, probably close to him so that he can simply sneak to his room and do some dirty job you know
where are kuwii suicide ka samayn jiray ?
Hatred? Acuudi bilaahi. Maba aqaaniyee maxaa naceyb intaas igu abuuraayo.
He didn't let boqolalaal kun Soomaaliyeed sleep well, including caruur, dumar iyo waayeel. His policies did indeed make them maimed, killed, xasuuqid iyo barakicin.
The least we can do is -- we cannot let him sleep well in a comfortable bed at a well-equipped isbitaal. He let millions suffer, he should at least suffer or taste a bit of his own medicine on this
Hatred beel. And yea, long live Jaale C/llaahi. Yaaba rabay dhimashadiis? Ma dambigiis aniga uu ii socdaa? Wah. Kuleel iyo qaboob midna ma igu heyso yaaqeey hadduu noolyahay iyo haddii...
Originally posted by J.a.c.a.y.l.b.a.r.o:
MMA, You need to organize someone, probably close to him so that he can simply sneak to his room and do some dirty job you know
,, may be poison him, use a silencer, or do some ciijis if he can
where are kuwii suicide ka samayn jiray ?
Riyaale aa u direynaa marka.
^LOOOOOL...Yea right inee kuleel iyo qabow kuugu heyn...Whatever rocks your boat, maala dhihi jiray horta...As for him suffering or giving him a taste of his own medicine, ticket makuu gooyaa adi
masoo sameenee?
Bas bas, I know you don't hate adeerkeey...Hatred is such a big word, after all you're MMA, always Somalinimo u hadashaa *Wink Wink*...
Kool Kat, I couldn't put it any better, Afkaaga caano lagu qabay, your wisdom is deep my sister, some find it hard to decode, Indeed one man's traitor is another man's great hero. In this day and
age, there are those that think the greatest Somali man, the Sayid was an evil criminal...
Lool @ Protesters as if anyone cares, I must be laughing at this, the last time I checked the turn out for the Pro Yusuf was a lot highter than those against.... only the usual suspects showed up,
Ahmed Diirey's looters inc group and A. Qasim apologists. Keep your efforts, it's being restricted by the day, curbed and reduced to nothing...
^Qarxis beele...They wouldn't dare...Deep down, they all know Mudane Cabdullaahi Yuusuf is the best thing that happened to them...Waligaa wax badan baa sugi kuwaa...
Emperor, thank you walaalo, I was just stating the obvious...Meesha waa la is indha tirooyaa...Lakiin haduuba Cabdullaahi Yusuf kala jecel yahay lee mo'ohoo wixii is indho tiro, iyo wixii kale...
^That's the point, as for the ISqarxin, tell them you don't threated a Soldeir with death, you do it... Whoever wishes to do it shall present themselves....
I can't imagine ,,,
The door opens ,,,, here is a man:
The Man: Mudane Madaxweyne
A/Y: Haa adeer, kumaad ahayd ?
The Man: Waa aniga, ima garanaysid miyaa ?
A/Y: Maya kuma garanayo, marse hadii aad madaxweyne iigu yeedhay inaad cida tahay uun baan u fahmay.
The Man: Maya ma ihi lakin dan baan lahaa maanta
A/Y: Maxay tahay danta cusbitalka ku keentay ?
The Man: Sidaa uma sii badna, BUGGGGGGGGGGGGGGGGGGGGGGGG
Qarax baaba ka dhacay .............
In few moments waaba Police, ambulances, all the news reporters and there everybody is watching the news ........ odaygii oo nin argagixiso ahi isku qarxiyay
^Creative mind you have...Sida riyaale uu passport request applicationska u aqriyo, maa adna waxaan uga fikirtaa? Just asking...
No man on the face of this green earth has the 8alls to step to him like that? No man...Riyada ka kac...
looooooool ,,,,,,,,,,, is just what is happening this crazy world around us dee ,,,,, nothing more
MMA, Ngonge works just down the road from there. If we can get him to go down there and sing I'm sure the whole block wouldnt sleep let alone Yeey.
ps sheep lalama hadlo. You just hope the shephard uses his dogs better
^^ You already know my opinion of such pointless protest. It is as useful as buying a poster of the man and throwing darts at it. Makes no difference and only helps frustrate you even more every time
you miss the target with your dart.
If you really want to annoy him and the hospital, you need to put your hands in your pockets and spend a little money on some flowers and roses. Imagine the hassle it will cause if he received fifty
thousand bouquets of flowers in one single day!
Just so the message is not misunderstood you'll have to make sure you send funeral wreaths and not the uplifting/sympathy type. | {"url":"https://www.somaliaonline.com/community/topic/40268-he-should-not-rest-well/","timestamp":"2024-11-10T15:33:16Z","content_type":"text/html","content_length":"283645","record_id":"<urn:uuid:bff36fc7-f62e-4ca9-96f7-0f73e05a9af2>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00863.warc.gz"} |
128 bit operands on x86
Here are 3 C callable routines in x86 Gnu assembler (GAS) to use x86 commands to process 128 bit numbers. Two commands are a variation of commands that Intel calls mul and div. GAS calls these
variations “mulq” and “divq”.
The above works on recent Macs. (on Max OS X 10.7.5 with a 64 bit x86) On a different OS it was necessary to remove the first line and each of the 6 underscores. Suitable prototypes for the routines
typedef unsigned long int ul;
typedef struct{ul q; ul r;} res;
res divq(res, ul);
res mulq(ul, ul);
res addq(res, res);
mulq multiplies two unsigned 64 bit integers and returns a 128 bit integer. divq divides a 128 bit number by a 64 bit number returning a 64 bit quotient and remainder to fully exploit divq. In such a
context (res){j, k} denotes j∙2^64 + k.
The “q” in “divq” is a GAS (Gnu Assembler) convention to indicate 64 bit operands.
Warning: divq is fairly slow. On a 2.4 GHz Intel Core 2 Duo, divq takes about 22 ns.
clang m.c q.s
which is right.
lsq below serves as a 128 bit left shift and llsrs as long long signed right shift:
typedef long int L;
res llsrs(res v, int n){return n<64?(res){(L)v.q>>n, (n?(v.q<<(64-n)):0)|(v.r>>n)}
:(res){(L)v.q>>63, (L)v.q>>(n-64)};}
res lsq(res v, int n){return n<64?(res){(v.q<<n)|(n?(v.r>>64-n):0),v.r<<n}
:(res){v.r<<(n-64), 0};}
This is the program that motivated these routines. It computes e and converts it to decimal to many digits. C does not expose the power of the divq command which is exploited here for a particularly
simple calculation. Before hardware floating point computers relied on commands such as these for scaled fixed point arithmetic. They still have their uses.
This takes 3 64 bit parameters and returns 2. “clang c.c -O3 -S” produces file “c.s” which reveals that a arrives in register %rdi, b in %rsi and d in %rdx. The yield goes to regs %rax and %rdx.
divq divides 128 bit value in RDX:RAX by named divisor and puts quotient in %rax and remainder in %rdx. See Vol. 2A 3-221 of Intel x86 manual: “Intel® 64 and IA-32 Architectures Software Developer’s
This is useful: | {"url":"http://cap-lore.com/code/b128/note.html","timestamp":"2024-11-10T15:59:20Z","content_type":"text/html","content_length":"2926","record_id":"<urn:uuid:3705d262-d04c-4bdd-9b14-c035b7561b84>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00821.warc.gz"} |
AP Chemistry Atomic Structure Ideas
Kristen Drury | Tue, 10/08/2019 - 12:24
The second unit in my sequence for AP Chemistry covers the new AP Chemistry Course and Exam Description Learning Objectives associated with atomic structure. I will identify and describe activities I
use to teach students some of the Learning Objectives that I tie into this unit consisting of nine class days.
My atomic structure unit only includes atomic structure and electron configurations (1.5 days), photoelectron spectroscopy (1.6), mass spectroscopy (1.2), and the electromagnetic spectrum (3.11 and
3.12). (Periodicity is in my unit 3.) We start with a brief year 1 review of atomic structure including the subatomic particles’ charge, location, and masses with ions and isotopes. We review the
theory of atomic structure leading up to the quantum mechanical model. This was all covered in year one so I do it in one period, but in years when I taught AP chemistry as a first year course, all
of this material was learned using the flipped classroom videos prior to the start of this unit with 1-2 periods of practice problems. I still cover magnetism because I have the students observe
materials (Zn, Fe, MnO[2], KMnO[4], CuSO[4], etc) to determine their magnetic properties and propose a rule of thumb after writing electron configurations and drawing orbital notations. I find it is
great practice of these skills and shows a good application of electron configurations, although magnetism is no longer tested on the AP Chemistry Exam (see figure 1). See the supporting information
for a handout.
Figure 1: Students observe magnetic properties of metals
We move on to a card sort (created by Jamie Flint) in which students are given various electron configurations and PES diagrams and asked to match them together (see figure 2). This is a great way to
have students think critically about what the graph is trying to depict without having taught PES yet. It really saves a lot of lecture time and helps them construct their own ideas about the graphs
to give the students ownership of the material and help with their long term memory of the objective. We practice PES more after reviewing ionization energy and binding energy (year 1 concepts for
me). We contrast the PES graphs with mass spectroscopy graphs to do another card sort (also created by Jamie Flint). I like to challenge the students’ thinking by showing mass spectroscopy graphs of
diatomic elements as discuss what graphs of compounds can be used for in my flipped videos.
Figure 2: Electron configuration and PES cardsort
The second portion of my unit includes light equations to calculate the wavelength, frequency and energy of light. We practice numerous questions in which we solve for single photons using
wavelengths in meters and nanometers. We identify what type of radiation we are calculating using an electromagnetic radiation poster. To gain more practice and an understanding of why this is
important for the atomic structure unit, we view the bright line spectra of hydrogen, helium, and neon and compare to published values. Then we calculate the wavelength, frequency and energy of the
lines we observed. We do not have fancy equipment so we make qualitative measurements and relate those colors to published values (see the attached lab). We calculate the level from which certain
electrons fall just for understanding, though it is not on the AP Exam. Although it is a lot of calculating, my students love viewing the spectra and value the practice more than a normal worksheet.
See the supporting information for a handout.
Some people have asked how I structure homework. My school district requires us to give homework and count it as ten percent of the students’ quarter average. I provide a flipped video that I have
made each night (except nights before exams) using the free Edpuzzle website. I post all videos at the start of the unit so students have time to watch them at their own pace. I also ask students to
complete one written homework assignment per unit. The written assignments contain 2-3 very old AP exam questions along with a reading and my own questions (all of these homework assignments can be
found on my website www.chemisme.com). I know the students can easily find the answers and copy them for the old AP Exam questions but we discuss the value of trying the examples on their own and
using the answers found online as a way of checking their work and learning from their errors. The homework is for practice, and without practice, they will not succeed. The completion of homework
has little bearing on their overall grade, and the questions I wrote on the homework will be weighted more.
What activities do you include for these learning objectives? How do you handle homework assignments in your classes? Please join the conversation on ChemEd X!
Join the conversation.
All comments must abide by the ChemEd X Comment Policy, are subject to review, and may be edited. Please allow one business day for your comment to be posted, if it is accepted.
Comments 3
I see Jaime flint is sourced but I can not seem to find a link or info on how to make/purchase these fantastic cards for PES and electron configuration match.
I received her card sorts at chemed and BCCE conferences. I don't know if they are available anywhere else but they are very easy to make yourself if you Google and print various simple PES graphs
(or use ones from old AP questions) and then print the configurations. Jamie is on Twitter if you want to find her and contact her there as well.
Thank you for sharing. I love the idea of the card sorts for PES and getting kids to discuss. I also teach this as a second year course. | {"url":"https://www.chemedx.org/blog/ap-chemistry-atomic-structure-ideas","timestamp":"2024-11-02T04:13:02Z","content_type":"text/html","content_length":"50676","record_id":"<urn:uuid:23bb6c06-f97f-40bd-bc6d-13a516fb9f02>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00702.warc.gz"} |
Light Diffraction Through a Periodic Grating
Light Diffraction Through a Periodic Grating - Java Tutorial
A model for the diffraction of visible light through a periodic grating is an excellent tool with which to address both the theoretical and practical aspects of image formation in optical microscopy.
Light passing through the grating is diffracted according to the wavelength of the incident light beam and the periodicity of the line grating. This interactive tutorial explores the mechanics of
periodic diffraction gratings when used to interpret the Abbe theory of image formation in the optical microscope.
In its simplest form, a line or amplitude grating is composed of a linear array of thin opaque strips (or slits) having a periodic spacing and suspended on a solid matrix, usually an optical glass
plate. The most convenient and accurate method of forming gratings of this type is through the use of metallic vacuum deposition techniques. The spacing between the centers of two adjacent slits (P)
is called the grating period, and the reciprocal of P is termed the spatial frequency, which is measured in the number of slits or periods per unit length.
The tutorial initializes with a grating periodicity of 1000 nanometers (producing a spatial frequency equal to 1000 lines/millimeter) and an incident light beam of 700 nanometer wavelength impacting
the grating at a 90-degree angle. Each slit in the grating diffracts light over the entire range of angles covering 180 degrees on the opposite side of the grating. The Spatial Frequency slider is
utilized to change the grating periodicity and the Wavelength slider alters the wavelength of the incident light wave.
Individual light waves diffracted from successive grating slits are emitted as concentric spherical wavelets that interfere both constructively and destructively because they are all derived from the
same wavefront and are therefore in phase. Wavefronts passing through the grating slits that are parallel to the incident light wave are referred to as zero order (undiffracted) or direct light.
Diffracted higher-order wavefronts are inclined at an angle (θ) according to the equation:
sin(θ) = M(λ/P)
where λ is the wavelength of the wavefront, P is the grating slit spacing and M is an integer termed the diffraction order (e.g., M = 0 for direct light, ±1 for first order diffracted light, etc.) of
light waves deviated by the grating. The combination of diffraction and interference effects on the light wave passing through the periodic grating produces a diffraction spectrum, which occurs in a
symmetrical pattern on both sides of the zero order direct light wave.
If the diffracted light waves produced by the periodic grating are then passed through a convergent lens, they appear as a series of bright spots on the focal plane of the lens. The intensity of
these spots decreases as the diffraction order increases, and the number of higher order diffracted waves that can enter the lens is restricted by the size of the lens aperture. Those waves that do
enter the lens form what is termed a Fraunhofer diffraction spectrum (also called a Fourier spectrum) that can be observed at the focal plane of the lens.
The periodic diffraction grating can now be used to examine Ernst Abbe's theory of image formation in the optical microscope. When the line grating is placed on a microscope stage and illuminated
with a parallel beam of light that is restricted in size by the condenser aperture diaphragm, both zero and higher order diffracted light rays enter the front lens of the objective. Direct light that
passes through the grating unaltered is imaged in the center of the optical axis on objective rear focal plane. First and higher order diffracted light rays enter the objective at an angle and are
focused at discrete points (a Fraunhofer diffraction pattern) on both sides of the direct light beam at the objective rear focal plane. A linear relationship exists between the position of the
diffracted light beams and their corresponding points on the periodic grating.
If the periodic grating placed on the microscope stage is a micrometer or similar grid, then the Fraunhofer diffraction pattern can be observed by removing one of the microscope eyepieces and
examining the objective rear focal plane (or by using a phase telescope or Bertrand lens). First, reduce the condenser aperture size to a minimal value then, using a low-power (10x or 20x) objective,
focus the bright central spot on the focal plane while viewing through the eyepiece tube. A series of higher-order light spots of diminishing intensity can now be observed flanking the central spot.
The diffracted light spots display a spectrum of color with lower wavelengths (blue and purple) nearer the optical axis and higher wavelengths (red) spread on the periphery. Spacing between the light
spots is dependent upon the grating interval and the wavelength of light passed through the condenser. Finely spaced gratings and longer wavelengths produce larger spot intervals than do coarse
gratings and lower wavelengths.
At the microscope intermediate image plane, coherent light emitted from the diffracted orders at the objective rear aperture undergoes interference to produce an intermediate image of the periodic
grating, which is further magnified by the eyepieces. The integrity of the intermediate image depends upon how many diffracted orders produced by the grating pass through the aperture and are
captured by the objective front lens. Objectives having a higher numerical aperture are able to gather more of the diffracted light waves and produce clearly better images.
Abbe determined that in order to form a recognizable image, the objective must capture the zeroth order light rays and at least one of the higher order diffracted waves or two adjacent orders.
Because the diffraction angle is dependent upon the grid spacing and the wavelength is determined by the refractive index (n) of the medium between the grating and the objective front lens, the
diffraction equation (given above) can be rewritten as:
P = λ/n(sin(θ))
Abbe originally defined the numerical aperture (NA) of the objective as:
NA = n(sin(θ))
so the equation reduces to:
P = λ/NA
This equation is one of the most fundamental to optical microscopy and demonstrates that an objective's ability to resolve fine details in a specimen, such as a periodic grating, is dependent upon
both the wavelength of illuminating light rays and the numerical aperture. Thus, the lower the wavelength or the higher the numerical aperture, the greater the resolving power of the objective. | {"url":"https://www.olympus-lifescience.com/ja/microscope-resource/primer/java/imageformation/gratingdiffraction/","timestamp":"2024-11-13T11:25:21Z","content_type":"application/xhtml+xml","content_length":"49452","record_id":"<urn:uuid:b94fe5d1-591c-4647-a41f-e261a406e151>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00441.warc.gz"} |
Kim's Game 6
here. Especially you Mrs O'Hagan's and your 2nd Yr class 2Q at Holyrood Secondary School in Glasgow!
144 is a square number
1.5 squared is 2.25
Acute angles are less than 90 degrees
Acute angles are less than 90 degrees
Multiplying even numbers together always gives even answers
Multiplying even numbers together always gives even answers
Ten minus twelve is minus two
Ten minus twelve is minus two
Five minutes is 300 seconds
Five minutes is 300 seconds
A pentagon has five sides
A pentagon has five sides
29 is a prime number
A quarter is the same as 25 percent
A quarter is the same as 25 percent
Nine Sevens are sixty three
Nine Sevens are sixty three
See above ten phrases which need to be memorised. Each time the blue play button is clicked a phrase will be removed from the collection. The aim of the activity is to write down the exact the phrase
after it has been removed. After the last phrase has been removed all ten phrases are then shown in the order they were removed so that accuracy can be checked. The auto play button removes phrases
at thirty second intervals (the time interval can be changed - see below).
The phrases, in the order they disappeared, will be shown in the panel at the top of this page at the end of the game.
Note to teacher: Doing this activity once with a class helps students develop strategies. It is only when they do this activity a second time that they will have the opportunity to practise those
strategies. That is when the learning is consolidated. Click a button below to play another version of this game or play the same game again (the phrases will disappear in a different order)
Basic Shapes Fancy Shapes Circle Parts Angle Theorems Fractions
For many pupils the initial task of memorising ten items is far too difficult. You can make the game easier by removing some of the items with the blue button before you present the pupils with this
The auto play feature removes phrases after a certain number of seconds (30 seconds by default). You can vary that time interval if it is not suitable for your class here:
Auto Play: Remove phrases every
Note that the first phrase is removed four seconds after pressing the auto play button despite the time interval set for the rest of the phrases above.
Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page.
Below is an Amazon link. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.
Educational Technology on Amazon
Teacher, do your students have access to computers such as tablets, iPads or Laptops? This page was really designed for projection on a whiteboard but if you really want the students to have access
to it here is a concise URL for a version of this page without the comments:
However it would be better to assign one of the student interactive activities below.
Here is the URL which will take them to a student version of this activity. | {"url":"https://www.transum.org/Software/sw/Starter_of_the_day/Starter_January11.asp?Game=6","timestamp":"2024-11-07T22:26:36Z","content_type":"text/html","content_length":"36920","record_id":"<urn:uuid:c3176180-3c10-426a-a149-5f3808a19e3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00094.warc.gz"} |
Pressure - Wikipedia Republished // WIKI 2
Pressure (symbol: p or P) is the force applied perpendicular to the surface of an object per unit area over which that force is distributed.^[1]^:445 Gaugepressure (also spelled gage pressure)^[a]
is the pressure relative to the ambient pressure.
Various units are used to express pressure. Some of these derive from a unit of force divided by a unit of area; the SI unit of pressure, the pascal (Pa), for example, is one newton per squaremetre
(N/m^2); similarly, the pound-force per squareinch (psi, symbol lbf/in^2) is the traditional unit of pressure in the imperial and UScustomary systems. Pressure may also be expressed in terms of
standardatmosphericpressure; the unit atmosphere (atm) is equal to this pressure, and the torr is defined as 1⁄760 of this. Manometric units such as the centimetreofwater, millimetreofmercury,
and inchofmercury are used to express pressures in terms of the height of columnofaparticularfluid in a manometer.
YouTube Encyclopedic
• 1/3
• Blood Pressure Measurement - Clinical Examination
• How blood pressure works - Wilfred Manzano
• What is High Blood Pressure? (HealthSketch)
Pressure is the amount of force applied perpendicular to the surface of an object per unit area. The symbol for it is "p" or P.^[2] The IUPAC recommendation for pressure is a lower-case p.^[3]
However, upper-case P is widely used. The usage of P vs p depends upon the field in which one is working, on the nearby presence of other symbols for quantities such as power and momentum, and on
writing style.
Pressure Volume
(Stress) (Strain)
Temperature Entropy
Chemicalpotential Particlenumber
Mathematically:^[4] ${\displaystyle p={\frac {F}{A}},}$ where:
• ${\displaystyle p}$ is the pressure,
• ${\displaystyle F}$ is the magnitude of the normalforce,
• ${\displaystyle A}$ is the area of the surface on contact.
Pressure is a scalar quantity. It relates the vectorarea element (a vector normal to the surface) with the normalforce acting on it. The pressure is the scalar proportionalityconstant that relates
the two normal vectors: ${\displaystyle d\mathbf {F} _{n}=-p\,d\mathbf {A} =-p\,\mathbf {n} \,dA.}$
The minus sign comes from the convention that the force is considered towards the surface element, while the normal vector points outward. The equation has meaning in that, for any surface S in
contact with the fluid, the total force exerted by the fluid on that surface is the surfaceintegral over S of the right-hand side of the above equation.
It is incorrect (although rather usual) to say "the pressure is directed in such or such direction". The pressure, as a scalar, has no direction. The force given by the previous relationship to the
quantity has a direction, but the pressure does not. If we change the orientation of the surface element, the direction of the normal force changes accordingly, but the pressure remains the same.
Pressure is distributed to solid boundaries or across arbitrary sections of fluid normal to these boundaries or sections at every point. It is a fundamental parameter in thermodynamics, and it is
conjugate to volume.^[5]
Mercury column
The SI unit for pressure is the pascal (Pa), equal to one newton per squaremetre (N/m^2, or kg·m^−1·s^−2). This name for the unit was added in 1971;^[6] before that, pressure in SI was expressed in
newtons per square metre.
Other units of pressure, such as poundspersquareinch (lbf/in^2) and bar, are also in common use. The CGS unit of pressure is the barye (Ba), equal to 1 dyn·cm^−2, or 0.1 Pa. Pressure is sometimes
expressed in grams-force or kilograms-force per square centimetre ("g/cm^2" or "kg/cm^2") and the like without properly identifying the force units. But using the names kilogram, gram,
kilogram-force, or gram-force (or their symbols) as units of force is deprecated in SI. The technicalatmosphere (symbol: at) is 1 kgf/cm^2 (98.0665 kPa, or 14.223 psi).
Pressure is related to energydensity and may be expressed in units such as joules per cubic metre (J/m^3, which is equal to Pa). Mathematically: ${\displaystyle p={\frac {F\cdot {\text{distance}}}{A
\cdot {\text{distance}}}}={\frac {\text{Work}}{\text{Volume}}}={\frac {\text{Energy (J)}}{{\text{Volume }}({\text{m}}^{3})}}.}$
Some meteorologists prefer the hectopascal (hPa) for atmospheric air pressure, which is equivalent to the older unit millibar (mbar). Similar pressures are given in kilopascals (kPa) in most other
fields, except aviation where the hecto- prefix is commonly used. The inch of mercury is still used in the United States. Oceanographers usually measure underwater pressure in decibars (dbar) because
pressure in the ocean increases by approximately one decibar per metre depth.
The standardatmosphere (atm) is an established constant. It is approximately equal to typical air pressure at Earth meansealevel and is defined as 101325 Pa.
Because pressure is commonly measured by its ability to displace a column of liquid in a manometer, pressures are often expressed as a depth of a particular fluid (e.g., centimetresofwater,
millimetresofmercury or inchesofmercury). The most common choices are mercury (Hg) and water; water is nontoxic and readily available, while mercury's high density allows a shorter column (and so
a smaller manometer) to be used to measure a given pressure. The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation p = ρgh, where g is the
gravitationalacceleration. Fluid density and local gravity can vary from one reading to another depending on local factors, so the height of a fluid column does not define pressure precisely.
When millimetres of mercury (or inches of mercury) are quoted today, these units are not based on a physical column of mercury; rather, they have been given precise definitions that can be expressed
in terms of SI units.^[7] One millimetre of mercury is approximately equal to one torr. The water-based units still depend on the density of water, a measured, rather than defined, quantity. These
manometric units are still encountered in many fields. Bloodpressure is measured in millimetres (or centimetres) of mercury in most of the world, and lung pressures in centimetres of water are still
Underwaterdivers use the metreseawater (msw or MSW) and footseawater (fsw or FSW) units of pressure, and these are the units for pressure gauges used to measure pressure exposure in
divingchambers and personaldecompressioncomputers. A msw is defined as 0.1 bar (= 10,000 Pa), is not the same as a linear metre of depth. 33.066 fsw = 1 atm (1 atm = 101,325 Pa / 33.066 =
3,064.326 Pa). The pressure conversion from msw to fsw is different from the length conversion: 10 msw = 32.6336 fsw, while 10 m = 32.8083 ft.
Gauge pressure is often given in units with "g" appended, e.g. "kPag", "barg" or "psig", and units for measurements of absolute pressure are sometimes given a suffix of "a", to avoid confusion, for
example "kPaa", "psia". However, the US NationalInstituteofStandardsandTechnology recommends that, to avoid confusion, any modifiers be instead applied to the quantity being measured rather than
the unit of measure.^[8] For example, "p[g] = 100 psi" rather than "p = 100 psig".
Differential pressure is expressed in units with "d" appended; this type of measurement is useful when considering sealing performance or whether a valve will open or close.
Presently or formerly popular pressure units include the following:
• atmosphere (atm)
• manometric units:
□ centimetre, inch, millimetre (torr) and micrometre (mTorr, micron) of mercury,
□ height of equivalent column of water, including millimetre (mm H^
[2]O), centimetre (cm H^
[2]O), metre, inch, and foot of water;
• imperial and customary units:
□ kip, shortton-force, longton-force, pound-force, ounce-force, and poundal per square inch,
□ short ton-force and long ton-force per square inch,
□ fsw (feet sea water) used in underwater diving, particularly in connection with diving pressure exposure and decompression;
• non-SI metric units:
□ bar, decibar, millibar,
☆ msw (metres sea water), used in underwater diving, particularly in connection with diving pressure exposure and decompression,
□ kilogram-force, or kilopond, per square centimetre (technicalatmosphere),
□ gram-force and tonne-force (metric ton-force) per square centimetre,
□ barye (dyne per square centimetre),
□ kilogram-force and tonne-force per square metre,
□ sthene per square metre (pieze).
The effects of an external pressure of 700 bar on an aluminum cylinder with 5 mm (0.197 in) wall thickness
As an example of varying pressures, a finger can be pressed against a wall without making any lasting impression; however, the same finger pushing a thumbtack can easily damage the wall. Although the
force applied to the surface is the same, the thumbtack applies more pressure because the point concentrates that force into a smaller area. Pressure is transmitted to solid boundaries or across
arbitrary sections of fluid normal to these boundaries or sections at every point. Unlike stress, pressure is defined as a scalarquantity. The negative gradient of pressure is called the
Another example is a knife. If the flat edge is used, force is distributed over a larger surface area resulting in less pressure, and it will not cut. Whereas using the sharp edge, which has less
surface area, results in greater pressure, and so the knife cuts smoothly. This is one example of a practical application of pressure^[10]
For gases, pressure is sometimes measured not as an absolute pressure, but relative to atmosphericpressure; such measurements are called gauge pressure. An example of this is the air pressure in an
automobile tire, which might be said to be "220 kPa (32 psi)", but is actually 220 kPa (32 psi) above atmospheric pressure. Since atmospheric pressure at sea level is about 100 kPa (14.7 psi), the
absolute pressure in the tire is therefore about 320 kPa (46 psi). In technical work, this is written "a gauge pressure of 220 kPa (32 psi)".
Where space is limited, such as on pressuregauges, nameplates, graph labels, and table headings, the use of a modifier in parentheses, such as "kPa (gauge)" or "kPa (absolute)", is permitted.^[11]
In non-SI technical work, a gauge pressure of 32 psi (220 kPa) is sometimes written as "32 psig", and an absolute pressure as "32 psia", though the other methods explained above that avoid attaching
characters to the unit of pressure are preferred.^[8]
Gauge pressure is the relevant measure of pressure wherever one is interested in the stress on storagevessels and the plumbing components of fluidics systems. However, whenever equation-of-state
properties, such as densities or changes in densities, must be calculated, pressures must be expressed in terms of their absolute values. For instance, if the atmospheric pressure is 100 kPa
(15 psi), a gas (such as helium) at 200 kPa (29 psi) (gauge) (300 kPa or 44 psi [absolute]) is 50% denser than the same gas at 100 kPa (15 psi) (gauge) (200 kPa or 29 psi [absolute]). Focusing on
gauge values, one might erroneously conclude the first sample had twice the density of the second one.
Scalar nature
In a static gas, the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant randommotion. Because there are an extremely large number of molecules and
because the motion of the individual molecules is random in every direction, no motion is detected. When the gas is at least partially confined (that is, not free to expand rapidly), the gas will
exhibit a hydrostatic pressure. This confinement can be achieved with either a physical container of some sort, or in a gravitational well such as a planet, otherwise known as atmosphericpressure.
In the case of planetary atmospheres, the pressure-gradientforce of the gas pushing outwards from higher pressure, lower altitudes to lower pressure, higher altitudes is balanced by the
gravitationalforce, preventing the gas from diffusing into outerspace and maintaining hydrostaticequilibrium.
In a physical container, the pressure of the gas originates from the molecules colliding with the walls of the container. The walls of the container can be anywhere inside the gas, and the force per
unit area (the pressure) is the same. If the "container" is shrunk down to a very small point (becoming less true as the atomic scale is approached), the pressure will still have a single value at
that point. Therefore, pressure is a scalar quantity, not a vector quantity. It has magnitude but no direction sense associated with it. Pressure force acts in all directions at a point inside a gas.
At the surface of a gas, the pressure force acts perpendicular (at right angle) to the surface.^[12]
A closely related quantity is the stress tensor σ, which relates the vector force ${\displaystyle \mathbf {F} }$ to the vectorarea ${\displaystyle \mathbf {A} }$ via the linear relation ${\
displaystyle \mathbf {F} =\sigma \mathbf {A} }$.
This tensor may be expressed as the sum of the viscousstresstensor minus the hydrostatic pressure. The negative of the stress tensor is sometimes called the pressure tensor, but in the following,
the term "pressure" will refer only to the scalar pressure.^[13]
According to the theory of generalrelativity, pressure increases the strength of a gravitational field (see stress–energytensor) and so adds to the mass-energy cause of gravity. This effect is
unnoticeable at everyday pressures but is significant in neutronstars, although it has not been experimentally tested.^[14]
Fluid pressure
Fluid pressure is most often the compressive stress at some point within a fluid. (The term fluid refers to both liquids and gases – for more information specifically about liquid pressure, see
Water escapes at high speed from a damaged hydrant that contains water at high pressure
Fluid pressure occurs in one of two situations:
• An open condition, called "open channel flow", e.g. the ocean, a swimming pool, or the atmosphere.
• A closed condition, called "closed conduit", e.g. a water line or gas line.
Pressure in open conditions usually can be approximated as the pressure in "static" or non-moving conditions (even in the ocean where there are waves and currents), because the motions create only
negligible changes in the pressure. Such conditions conform with principles of fluidstatics. The pressure at any given point of a non-moving (static) fluid is called the hydrostatic pressure.
Closed bodies of fluid are either "static", when the fluid is not moving, or "dynamic", when the fluid can move as in either a pipe or by compressing an air gap in a closed container. The pressure in
closed conditions conforms with the principles of fluiddynamics.
The concepts of fluid pressure are predominantly attributed to the discoveries of BlaisePascal and DanielBernoulli. Bernoulli'sequation can be used in almost any situation to determine the
pressure at any point in a fluid. The equation makes some assumptions about the fluid, such as the fluid being ideal^[15] and incompressible.^[15] An ideal fluid is a fluid in which there is no
friction, it is inviscid^[15] (zero viscosity).^[15] The equation for all points of a system filled with a constant-density fluid is^[16] ${\displaystyle {\frac {p}{\gamma }}+{\frac {v^{2}}{2g}}+z=\
mathrm {const} ,}$
• p, pressure of the fluid,
• ${\displaystyle {\gamma }}$ = ρg, density × acceleration of gravity is the (volume-) specificweight of the fluid,^[15]
• v, velocity of the fluid,
• z, elevation,
• ${\displaystyle {\frac {p}{\gamma }}}$, pressure head,
• ${\displaystyle {\frac {v^{2}}{2g}}}$, velocity head.
Explosion or deflagration pressures
Explosion or deflagration pressures are the result of the ignition of explosive gases, mists, dust/air suspensions, in unconfined and confined spaces.
Negative pressures
Low-pressure chamber in BundesleistungszentrumKienbaum, Germany
While pressures are, in general, positive, there are several situations in which negative pressures may be encountered:
• When dealing in relative (gauge) pressures. For instance, an absolute pressure of 80 kPa may be described as a gauge pressure of −21 kPa (i.e., 21 kPa below an atmospheric pressure of 101 kPa).
For example, abdominaldecompression is an obstetric procedure during which negative gauge pressure is applied intermittently to a pregnant woman's abdomen.
• Negative absolute pressures are possible. They are effectively tension, and both bulk solids and bulk liquids can be put under negative absolute pressure by pulling on them.^[17] Microscopically,
the molecules in solids and liquids have attractive interactions that overpower the thermal kinetic energy, so some tension can be sustained. Thermodynamically, however, a bulk material under
negative pressure is in a metastable state, and it is especially fragile in the case of liquids where the negative pressure state is similar to superheating and is easily susceptible to
cavitation.^[18] In certain situations, the cavitation can be avoided and negative pressures sustained indefinitely,^[18] for example, liquid mercury has been observed to sustain up to −425 atm
in clean glass containers.^[19] Negative liquid pressures are thought to be involved in the ascentofsap in plants taller than 10 m (the atmospheric pressurehead of water).^[20]
• The Casimireffect can create a small attractive force due to interactions with vacuumenergy; this force is sometimes termed "vacuum pressure" (not to be confused with the negative gauge
pressure of a vacuum).
• For non-isotropic stresses in rigid bodies, depending on how the orientation of a surface is chosen, the same distribution of forces may have a component of positive stress along one
surfacenormal, with a component of negative stress acting along another surface normal. The pressure is then defined as the average of the three principal stresses.
□ The stresses in an electromagneticfield are generally non-isotropic, with the stress normal to one surface element (the normalstress) being negative, and positive for surface elements
perpendicular to this.
• In cosmology, darkenergy creates a very small yet cosmically significant amount of negative pressure, which accelerates the expansionoftheuniverse.
Stagnation pressure
Stagnationpressure is the pressure a fluid exerts when it is forced to stop moving. Consequently, although a fluid moving at higher speed will have a lower staticpressure, it may have a higher
stagnation pressure when forced to a standstill. Static pressure and stagnation pressure are related by: ${\displaystyle p_{0}={\frac {1}{2}}\rho v^{2}+p}$ where
• ${\displaystyle p_{0}}$ is the stagnationpressure,
• ${\displaystyle \rho }$ is the density,
• ${\displaystyle v}$ is the flow velocity,
• ${\displaystyle p}$ is the static pressure.
The pressure of a moving fluid can be measured using a Pitottube, or one of its variations such as a Kielprobe or Cobraprobe, connected to a manometer. Depending on where the inlet holes are
located on the probe, it can measure static pressures or stagnation pressures.
Surface pressure and surface tension
There is a two-dimensional analog of pressure – the lateral force per unit length applied on a line perpendicular to the force.
Surface pressure is denoted by π: ${\displaystyle \pi ={\frac {F}{l}}}$ and shares many similar properties with three-dimensional pressure. Properties of surface chemicals can be investigated by
measuring pressure/area isotherms, as the two-dimensional analog of Boyle'slaw, πA = k, at constant temperature.
Surfacetension is another example of surface pressure, but with a reversed sign, because "tension" is the opposite to "pressure".
Pressure of an ideal gas
In an idealgas, molecules have no volume and do not interact. According to the idealgaslaw, pressure varies linearly with temperature and quantity, and inversely with volume: ${\displaystyle p={\
frac {nRT}{V}},}$ where:
• p is the absolute pressure of the gas,
• n is the amountofsubstance,
• T is the absolute temperature,
• V is the volume,
• R is the idealgasconstant.
Realgases exhibit a more complex dependence on the variables of state.^[21]
Vapour pressure
Vapour pressure is the pressure of a vapour in thermodynamicequilibrium with its condensed phases in a closed system. All liquids and solids have a tendency to evaporate into a gaseous form, and all
gases have a tendency to condense back to their liquid or solid form.
The atmosphericpressure boilingpoint of a liquid (also known as the normalboilingpoint) is the temperature at which the vapor pressure equals the ambient atmospheric pressure. With any
incremental increase in that temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and lift the liquid to form vapour bubbles inside the bulk of the substance. Bubble
formation deeper in the liquid requires a higher pressure, and therefore higher temperature, because the fluid pressure increases above the atmospheric pressure as the depth increases.
The vapor pressure that a single component in a mixture contributes to the total pressure in the system is called partialvaporpressure.
Liquid pressure
When a person swims under the water, water pressure is felt acting on the person's eardrums. The deeper that person swims, the greater the pressure. The pressure felt is due to the weight of the
water above the person. As someone swims deeper, there is more water above the person and therefore greater pressure. The pressure a liquid exerts depends on its depth.
Liquid pressure also depends on the density of the liquid. If someone was submerged in a liquid more dense than water, the pressure would be correspondingly greater. Thus, we can say that the depth,
density and liquid pressure are directly proportionate. The pressure due to a liquid in liquid columns of constant density or at a depth within a substance is represented by the following formula: $
{\displaystyle p=\rho gh,}$ where:
• p is liquid pressure,
• g is gravity at the surface of overlaying material,
• ρ is density of liquid,
• h is height of liquid column or depth within a substance.
Another way of saying the same formula is the following: ${\displaystyle p={\text{weight density}}\times {\text{depth}}.}$
Derivation of this equation
This is derived from the definitions of pressure and weight density. Consider an area at the bottom of a vessel of liquid. The weight of the column of liquid directly above this area produces
pressure. From the definition
${\displaystyle {\text{weight density}}={\frac {\text{weight}}{\text{volume}}}}$ we can express this weight of liquid as ${\displaystyle {\text{weight}}={\text{weight density}}\times {\text
{volume}},}$ where the volume of the column is simply the area multiplied by the depth. Then we have ${\displaystyle {\text{pressure}}={\frac {\text{force}}{\text{area}}}={\frac {\text{weight}}{\text
{area}}}={\frac {{\text{weight density}}\times {\text{volume}}}{\text{area}}},}$ ${\displaystyle {\text{pressure}}={\frac {{\text{weight density}}\times {\text{(area}}\times {\text{depth)}}}{\text
With the "area" in the numerator and the "area" in the denominator canceling each other out, we are left with ${\displaystyle {\text{pressure}}={\text{weight density}}\times {\text{depth}}.}$
Written with symbols, this is our original equation: ${\displaystyle p=\rho gh.}$
The pressure a liquid exerts against the sides and bottom of a container depends on the density and the depth of the liquid. If atmospheric pressure is neglected, liquid pressure against the bottom
is twice as great at twice the depth; at three times the depth, the liquid pressure is threefold; etc. Or, if the liquid is two or three times as dense, the liquid pressure is correspondingly two or
three times as great for any given depth. Liquids are practically incompressible – that is, their volume can hardly be changed by pressure (water volume decreases by only 50 millionths of its
original volume for each atmospheric increase in pressure). Thus, except for small changes produced by temperature, the density of a particular liquid is practically the same at all depths.
Atmospheric pressure pressing on the surface of a liquid must be taken into account when trying to discover the total pressure acting on a liquid. The total pressure of a liquid, then, is ρgh plus
the pressure of the atmosphere. When this distinction is important, the term total pressure is used. Otherwise, discussions of liquid pressure refer to pressure without regard to the normally
ever-present atmospheric pressure.
The pressure does not depend on the amount of liquid present. Volume is not the important factor – depth is. The average water pressure acting against a dam depends on the average depth of the water
and not on the volume of water held back. For example, a wide but shallow lake with a depth of 3 m (10 ft) exerts only half the average pressure that a small 6 m (20 ft) deep pond does. (The total
force applied to the longer dam will be greater, due to the greater total surface area for the pressure to act upon. But for a given 5-foot (1.5 m)-wide section of each dam, the 10 ft (3.0 m) deep
water will apply one quarter the force of 20 ft (6.1 m) deep water). A person will feel the same pressure whether their head is dunked a metre beneath the surface of the water in a small pool or to
the same depth in the middle of a large lake.
If four interconnected vases contain different amounts of water but are all filled to equal depths, then a fish with its head dunked a few centimetres under the surface will be acted on by water
pressure that is the same in any of the vases. If the fish swims a few centimetres deeper, the pressure on the fish will increase with depth and be the same no matter which vase the fish is in. If
the fish swims to the bottom, the pressure will be greater, but it makes no difference which vase it is in. All vases are filled to equal depths, so the water pressure is the same at the bottom of
each vase, regardless of its shape or volume. If water pressure at the bottom of a vase were greater than water pressure at the bottom of a neighboring vase, the greater pressure would force water
sideways and then up the narrower vase to a higher level until the pressures at the bottom were equalized. Pressure is depth dependent, not volume dependent, so there is a reason that water seeks its
own level.
Restating this as an energy equation, the energy per unit volume in an ideal, incompressible liquid is constant throughout its vessel. At the surface, gravitationalpotentialenergy is large but
liquid pressure energy is low. At the bottom of the vessel, all the gravitational potential energy is converted to pressure energy. The sum of pressure energy and gravitational potential energy per
unit volume is constant throughout the volume of the fluid and the two energy components change linearly with the depth.^[22] Mathematically, it is described by Bernoulli'sequation, where velocity
head is zero and comparisons per unit volume in the vessel are ${\displaystyle {\frac {p}{\gamma }}+z=\mathrm {const} .}$
Terms have the same meaning as in sectionFluidpressure.
Direction of liquid pressure
An experimentally determined fact about liquid pressure is that it is exerted equally in all directions.^[23] If someone is submerged in water, no matter which way that person tilts their head, the
person will feel the same amount of water pressure on their ears. Because a liquid can flow, this pressure is not only downward. Pressure is seen acting sideways when water spurts sideways from a
leak in the side of an upright can. Pressure also acts upward, as demonstrated when someone tries to push a beach ball beneath the surface of the water. The bottom of a boat is pushed upward by water
pressure (buoyancy).
When a liquid presses against a surface, there is a net force that is perpendicular to the surface. Although pressure does not have a specific direction, force does. A submerged triangular block has
water forced against each point from many directions, but components of the force that are not perpendicular to the surface cancel each other out, leaving only a net perpendicular point.^[23] This is
why liquid particles' velocity only alters in a normal component after they're collided to the container's wall. Likewise, if the collision site is a hole, water spurting from the hole in a bucket
initially exits the bucket in a direction at right angles to the surface of the bucket in which the hole is located. Then it curves downward due to gravity. If there are three holes in a bucket (top,
bottom, and middle), then the force vectors perpendicular to the inner container surface will increase with increasing depth – that is, a greater pressure at the bottom makes it so that the bottom
hole will shoot water out the farthest. The force exerted by a fluid on a smooth surface is always at right angles to the surface. The speed of liquid out of the hole is ${\displaystyle \scriptstyle
{\sqrt {2gh}}}$, where h is the depth below the free surface.^[23] This is the same speed the water (or anything else) would have if freely falling the same vertical distance h.
Kinematic pressure
${\displaystyle P=p/\rho _{0}}$ is the kinematic pressure, where ${\displaystyle p}$ is the pressure and ${\displaystyle \rho _{0}}$ constant mass density. The SI unit of P is m^2/s^2. Kinematic
pressure is used in the same manner as kinematicviscosity ${\displaystyle u }$ in order to compute the Navier–Stokesequation without explicitly showing the density ${\displaystyle \rho _{0}}$.
Navier–Stokes equation with kinematic quantities
${\displaystyle {\frac {\partial u}{\partial t}}+(uabla )u=-abla P+u abla ^{2}u.}$
See also
1. ^ The preferred spelling varies by country and even by industry. Further, both spellings are often used within a particular industry or country. Industries in British English-speaking countries
typically use the "gauge" spelling.
External links
This page was last edited on 7 August 2024, at 04:32 | {"url":"https://wiki2.org/en/Pressure","timestamp":"2024-11-11T11:07:18Z","content_type":"application/xhtml+xml","content_length":"254517","record_id":"<urn:uuid:c54af8c9-87ba-4f04-accf-01251b1965d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00328.warc.gz"} |
Vignette for survRM2 package:
Comparing two survival curves using the restricted mean survival time
Hajime Uno
Dana-Farber Cancer Institute
February 21, 2017
1 Introduction
In a comparative, longitudinal clinical study, often the primary endpoint is the time to a specific clinical event, such as death, heart failure hospitalization, tumor progression, and so on. The
hazard ratio estimate is almost routinely used to quantify the treatment difference. However, the clinical meaning of such a model-based between-group summary can be rather difficult to interpret
when the underlying model assumption (i.e., the proportional hazards assumption) is violated, and it is difficult to assure that the modeling is indeed correct empirically. For example, a
non-significant result of a goodness-of-fit test does not necessary mean that the proportional hazards assumption is “correct.” Other issues on the hazard ratio is seen elsewhere [1, 2].
Between-group summery metrics based on the restricted mean survival time (RMST) are useful alternatives to the hazard ratio or other model-based measures. This vignette is a supplemental
documentation for survRM2 package and illustrates how to use the functions in the package to compare two groups with respect to the restricted mean survival time. The package was made and tested on R
version 3.3.2.
2 Sample data
Throughout this vignette, we use a part of data from the primary biliary cirrhosis (pbc) study conducted by the Mayo Clinic, which is included in survival package in R. The details of the study and
the data elements are seen in the help file in survival package, which can be seen by
The original data in the survival package consists of data from 418 patients, which includes those who participated in the randomized clinical trial and those who did not. In the following
illustration, we use only 312 cases who participated in the randomized trial (158 cases on D-penicillamine group and 154 cases on Placebo group). The following function in survRM2 package creates the
data used in this vignette, selecting the subset from the original data file.
## [1] 312
## time status arm
## 1 1.095140 1 1
## 2 12.320329 0 1
## 3 2.770705 1 1
## 4 5.270363 1 1
## 5 4.117728 0 0
## 6 6.852841 1 0
Here, time is years from the registration to death or last known alive, status is the indicator of the event (1: death, 0: censor), and arm is the treatment assignment indicator (1: D-penicillamine,
0: Placebo). Below is the Kaplan-Meier (KM) estimate for time-to-death of each treatment group.
3 Restricted mean survival time (RMST) and restricted mean time lost (RMTL)
The RMST is defined as the area under the curve of the survival function up to a time \(\tau (< \infty):\) \[ \mu_{\tau} = \int_0^{\tau} S(t)dt,\] where \(S(t)\) is the survival function of a
time-to-event variable of interest. The interpretation of the RMST is that “when we follow up patients for \(\tau,\) patients will survive for \(\mu_{\tau}\) on average,” which is quite
straightforward and clinically meaningful summary of the censored survival data. If there were no censored observations, one could use the mean survival time \[ \mu_{\infty} = \int_0^{\infty} S(t)dt,
\] instead of \(\mu_{\tau}.\)
A natural estimator for \(\mu_{\tau}\) is \[ \hat{\mu}_{\tau} = \int_0^{\tau} \hat{S}(t)dt,\] where \(\hat{S}(t)\) is the KM estimator for \(S(t).\) The standard error for \(\hat{\mu}_{\tau}\) is
also calculated analytically; the detailed formula is given in [3]. Note that \(\mu_{\tau}\) is estimable even under a heavy censoring case. On the other hand, although median survival time, \(S^{-1}
(0.5),\) is also a robust summary of survival time distribution, it will become inestimable when the KM curve does not reach 0.5 due to heavy censoring or rare events.
The RMTL is defined as the area “above” the curve of the survival function up to a time \(\tau:\) \[ \tau - \mu_{\tau} = \int_0^{\tau} \{ 1-S(t) \}dt.\] In the following figure, the area highlighted
in pink and orange are the RMST and RMTL estimates, respectively, in D-penicillamine group, when \(\tau\) is 10 years. The result shows that the average survival time during 10 years of follow-up is
7.15 years in the D-penicillamine group. In other words, during the 10 years of follow-up, patients treated by D-penicillamine lost 2.85 years in average sense.
3.1 Unadjusted analysis and its implementation
Let \(\mu_{\tau}(1)\) and \(\mu_{\tau}(0)\) denote the RMST for treatment group 1 and 0, respectively. Now, we compare the two survival curves, using the RMST or RMTL. Specifically, we consider the
following three measures for the between-group contrast.
1. Difference in RMST \[ \mu_{\tau}(1) - \mu_{\tau}(0) \]
2. Ratio of RMST \[ \mu_{\tau}(1) / \mu_{\tau}(0) \]
3. Ratio of RMTL \[ \{ \tau - \mu_{\tau}(1) \} / \{ \tau - \mu_{\tau}(0) \} \]
These are estimated by simply replacing \(\mu_{\tau}(1)\) and \(\mu_{\tau}(0)\) by their empirical counterparts (i.e.,\(\hat{\mu}_{\tau}(1)\) and \(\hat{\mu}_{\tau}(0)\), respectively). For inference
of the ratio type metrics, we use the delta method to calculate the standard error. Specifically, we consider \(\log \{ \hat{\mu}_{\tau}(1) \}\) and \(\log \{ \hat{\mu}_{\tau}(0) \}\) and calculate
the standard error of log-RMST. We then calculate a confidence interval for log-ratio of RMST, and transform it back to the original ratio scale. Below shows how to use the function, rmst2, to
implement these analyses.
The first argument (time) is the time-to-event vector variable. The second argument (status) is also a vector variable with the same length as time, each of the elements takes either 1 (if event) or
0 (if no event). The third argument (arm) is a vector variable to indicate the assigned treatment of each subject; the elements of this vector take either 1 (if active treatment arm) or 0 (if control
arm). The fourth argument (tau) is a scalar value to specify the truncation time point \({\bf \tau}\) for the RMST calculation. Note that \(\tau\) needs to be smaller than the minimum of the largest
observed time in each of the two groups (let us call this the max \(\tau\)). The program will stop with an error message when such \(\tau\) is specified. When \(\tau\) is not specified in rmst2,
i.e., when the code looks like
the max \(\tau\) is used as the default \(\tau.\) It is always encouraged to confirm that the size of the risk set is large enough at the specified \(\tau\) in each group to make sure the stability
of the KM estimates.
Below is the output with the pbc example when \(\tau=10\) (years) is specified. The rmst2 function returns RMST and RMTL on each group and the results of the between-group contrast measures listed
## The truncation time: tau = 10 was specified.
## Restricted Mean Survival Time (RMST) by arm
## Est. se lower .95 upper .95
## RMST (arm=1) 7.146 0.283 6.592 7.701
## RMST (arm=0) 7.283 0.295 6.704 7.863
## Restricted Mean Time Lost (RMTL) by arm
## Est. se lower .95 upper .95
## RMTL (arm=1) 2.854 0.283 2.299 3.408
## RMTL (arm=0) 2.717 0.295 2.137 3.296
## Between-group contrast
## Est. lower .95 upper .95 p
## RMST (arm=1)-(arm=0) -0.137 -0.939 0.665 0.738
## RMST (arm=1)/(arm=0) 0.981 0.878 1.096 0.738
## RMTL (arm=1)/(arm=0) 1.050 0.787 1.402 0.738
In the present case, the difference in RMST (the first row of the “Between-group contrast” block in the output) was -0.137 years. The point estimate indicated that patients on the active treatment
survive 0.137 years shorter than those on placebo group on average, when following up the patients 10 years. While no statistical significance was observed (p=0.738), the 0.95 confidence interval
(-0.665 to 0.939) was relatively tight around 0, suggesting that the difference in RMST would be at most +/- one year.
The package also has a function to generate a plot from the rmst2 object. The following figure is automatically generated by simply passing the resulting rmst2 object to plot() function after running
the aforementioned unadjusted analyses.
3.2 Adjusted analysis and implementation
In most of the randomized clinical trials, an adjusted analysis is usually included in one of the planned analyses. One reason would be that adjusting for important prognostic factors may increase
power to detect a between-group difference. Another reason would be we sometimes observe imbalance in distribution of some of baseline prognostic factors even though the randomization guarantees the
comparability of the two groups on average. The function, rmst2, in this package implements an ANCOVA type adjusted analysis proposed by Tian et al. [4], in addition to the unadjusted analyses
presented in the previous section.
Let \(Y\) be the restricted mean survival time, and let \(Z\) be the treatment indicator. Also, let \(X\) denote a \(q\)-dimensional baseline covariate vector. Tian’s method consider the following
regression model, \[ g\{ E(Y \mid Z, X) \} = \alpha + \beta Z + \gamma^\prime X, \] where \(g(\cdot)\) is a given smooth and strictly increasing link function, and \((\alpha, \beta, \gamma^\prime)\)
is a \((q+2)\)-dimension unknown parameter vector. Prior to Tian et al. [4], Andersen et al. [5] also studied this regression model and proposed an inference procedure for the unknown model
parameter, using a pseudo-value technique to handle censored observations. Program codes for their pseudo-value approach are available on the three major platforms (Stata, R and SAS) with detailed
documentation [6, 7]. In contrast to Andersen’s method [5, 6, 7], Tian’s method [4] utilizes an inverse probability censoring weighting technique to handle censored observations. The function, rmst2,
in this package implements this method.
As shown below, for implementation of Tian’s adjusted analysis for the RMST, the only the difference is if the user passes covariate data to the function. Below is a sample code to perform the
adjusted analyses.
where covariates is the argument for a vector/matrix of the baseline characteristic data, x. For illustration, let us try the following three baseline variables, in the pbc data, as the covariates
for adjustment.
## age bili albumin
## 1 58.76523 14.5 2.60
## 2 56.44627 1.1 4.14
## 3 70.07255 1.4 3.48
## 4 54.74059 1.8 2.54
## 5 38.10541 3.4 3.53
## 6 66.25873 0.8 3.98
The rmst2 function fits data to a model for each of the three contrast measures (i.e., difference in RMST, ratio of RMST, and ratio of RMTL). For the difference metric, the link function \(g(\cdot)\)
in the model above is the identity link. For the ratio metrics, the log-link is used. Specifically, with this pbc example, we are now trying to fit data to the following regression models:
1. Difference in RMST \[ E(Y \mid arm,\ X) = \alpha + \beta (arm) + \gamma_1 (age) + \gamma_2(bili) + \gamma_3(albumin), \]
2. Ratio of RMST \[ \log \{ E(Y \mid arm, \ X) \} = \alpha + \beta (arm) + \gamma_1 (age) + \gamma_2(bili) + \gamma_3(albumin), \]
3. Ratio of RMTL \[ \log \{ \tau - E(Y \mid arm, \ X) \} = \alpha + \beta (arm) + \gamma_1 (age) + \gamma_2(bili) + \gamma_3(albumin). \]
Below is the output that rmst2 returns for the adjusted analyses.
## The truncation time: tau = 10 was specified.
## Summary of between-group contrast (adjusted for the covariates)
## Est. lower .95 upper .95 p
## RMST (arm=1)-(arm=0) -0.210 -0.883 0.463 0.540
## RMST (arm=1)/(arm=0) 0.968 0.877 1.068 0.514
## RMTL (arm=1)/(arm=0) 1.035 0.806 1.329 0.786
## Model summary (difference of RMST)
## coef se(coef) z p lower .95 upper .95
## intercept 2.743 2.134 1.285 0.199 -1.440 6.927
## arm -0.210 0.343 -0.613 0.540 -0.883 0.463
## age -0.069 0.018 -3.900 0.000 -0.103 -0.034
## bili -0.325 0.039 -8.386 0.000 -0.401 -0.249
## albumin 2.550 0.472 5.401 0.000 1.624 3.475
## Model summary (ratio of RMST)
## coef se(coef) z p exp(coef) lower .95 upper .95
## intercept 1.369 0.356 3.842 0.000 3.930 1.955 7.899
## arm -0.033 0.050 -0.652 0.514 0.968 0.877 1.068
## age -0.009 0.003 -3.410 0.001 0.991 0.985 0.996
## bili -0.087 0.013 -6.523 0.000 0.917 0.893 0.941
## albumin 0.360 0.080 4.491 0.000 1.434 1.225 1.678
## Model summary (ratio of time-lost)
## coef se(coef) z p exp(coef) lower .95 upper .95
## intercept 1.992 0.695 2.865 0.004 7.332 1.876 28.655
## arm 0.035 0.127 0.272 0.786 1.035 0.806 1.329
## age 0.025 0.007 3.810 0.000 1.026 1.012 1.039
## bili 0.063 0.008 8.334 0.000 1.065 1.049 1.080
## albumin -0.750 0.149 -5.033 0.000 0.472 0.353 0.633
The first block of the output is a summary of the adjusted treatment effect. Subsequently, a summary for each of the three models are provided.
4 Conclusions
The issues of the hazard ratio have been discussed elsewhere and many alternatives have been proposed, but the hazard ratio approach is still routinely used. The restricted mean survival time is a
robust and clinically interpretable summary measure of the survival time distribution. Unlike median survival time, it is estimable even under heavy censoring. There is a considerable body of
methodological research about the restricted mean survival time as alternatives to the hazard ratio approach. However, it seems those methods have been rarely used in practice. A lack of
user-friendly, well-documented program with clear examples would be a major obstacle for a new, alternative method to be used in practice. We hope this vignette and the presented survRM2 package will
be helpful for clinical researchers to try moving beyond the comfort zone - the hazard ratio.
[1] Hernan, M. A. (2010). The hazards of hazard ratios. Epidemiology (Cambridge, Mass) 21, 13-15.
[2] Uno, H., Claggett, B., Tian, L., Inoue, E., Gallo, P., Miyata, T., Schrag, D., Takeuchi, M., Uyama, Y., Zhao, L., Skali, H., Solomon, S., Jacobus, S., Hughes, M., Packer, M. & Wei, L.-J. (2014).
Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis. Journal of clinical oncology : official journal of the American Society of Clinical Oncology 32,
[3] Miller, R. G. (1981). Survival Analysis. Wiley.
[4] Tian, L., Zhao, L. & Wei, L. J. (2014). Predicting the restricted mean event time with the subject’s baseline covariates in survival analysis. Biostatistics 15, 222-233.
[5] Andersen, P. K., Hansen, M. G. & Klein, J. P. (2004). Regression analysis of restricted mean survival time based on pseudo-observations. Lifetime data analysis 10, 335-350.
[6] Klein, J. P., Gerster, M., Andersen, P. K., Tarima, S. & Perme, M. P. (2008). SAS and R functions to compute pseudo-values for censored data regression. Computer methods and programs in
biomedicine 89, 289-300.
[7] Parner, E. T. & Andersen, P. K. (2010). Regression analysis of censored data using pseudo-observations. The Stata Journal 10(3), 408-422. | {"url":"https://cran.gedik.edu.tr/web/packages/survRM2/vignettes/survRM2-vignette3-2.html","timestamp":"2024-11-06T20:28:10Z","content_type":"text/html","content_length":"200287","record_id":"<urn:uuid:6e665caf-eef1-4ec6-9bdc-f13f0c65bb64>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00593.warc.gz"} |
Pamfunc User Manual(0) Pamfunc User Manual(0)
pamfunc - Apply a simple monadic arithmetic function to a Netpbm image
pamfunc { -multiplier=realnum | -divisor=realnum | -adder=integer | -subtractor=integer | -min=wholenum | -max=wholenum -andmask=hexmask -ormask=hexmask -xormask=hexmask -not -shiftleft=count
-shiftright=count [-changemaxval] } [filespec]
All options can be abbreviated to their shortest unique prefix. You may use two hyphens instead of one. You may separate an option name and its value with white space instead of an equals sign.
This program is part of Netpbm(1).
pamfunc reads a Netpbm image as input and produces a Netpbm image as output, with the same format, and dimensions as the input. pamfunc applies a simple transfer function to each sample in the input
to generate the corresponding sample in the output. The options determine what function.
The samples involved are PAM samples. If the input is PBM, PGM, or PPM, the output will be that same format, but pamfunc applies the functions to the PAM equivalent samples, yielding PAM equivalent
samples. This can be nonintuitive in the PBM case.
pamarith is the same thing for binary functions -- it takes two images as input and applies a specified simple arithmetic function (e.g. addition) on pairs of samples from the two to produce the
single output image.
The functions fall into two categories: arithmetic (such as multiply by 5) and bit string (such as and with 01001000). For the arithmetic functions, the function arguments and results are the
fraction that a sample is of the maxval, i.e. normal interpretation of PAM tuples. But for the bit string functions, the value is the the bit string whose value as a binary cipher is the sample
value, and the maxval indicates the width of the bit string.
Arithmetic functions
The arithmetic functions are those selected by the options -multiplier, -divisor, -adder, -subtractor, -min, and -max.
As an example, consider an image with maxval 100 and a sample value of 10 and a function of "multiply by 5." The argument to the function is 10/100 (0.1) and the result is 5 * 0.1 = 0.5. In the
simplest case, the maxval of the output is also 100, so the output sample value is 0.5 * 100 = 50. As you can see, we could just talk about the sample values themselves instead of these fractions and
get the same result (10 * 5 = 50), but we don't.
Where it makes a practical difference whether we consider the values to be the fraction of the maxval or the sample value alone is where pamfunc uses a different maxval in the output image than it
finds in the input image. See -changemaxval.
So remember in reading the descriptions below that the values are 0.1 and 0.5 in this example, not 10 and 50. All arguments and results are in the range [0,1].
Bit string functions
The bit string functions are those selected by the options -andmask, -ormask, -xormask, -not, -shiftleft, and -shiftright.
With these functions, the maxval has a very different meaning than in normal Netpbm images: it tells how wide (how many bits) the bit string is. The maxval must be a full binary count (a power of two
minus one, such as 0xff) and the number of ones in it is the width of the bit string.
As an example, consider an image with maxval 15 and a sample value of 5 and a function of "and with 0100". The argument to the function is 0101 and the result is 0100.
In this example, it doesn't make any practical difference what we consider the width of the string to be, as long as it is at least 3. If the maxval were 255, the result would be the same. But with a
bit shift operation, it matters. Consider shifting left by 2 bits. In the example, where the input value is 0101, the result is 0100. But if the maxval were 255, the result would be 00010100.
For a masking function, the mask value you specify must not have more significant bits than the width indicated by the maxval.
For a shifting operation, the shift count you specify must not be greater than the width indicated by the maxval.
PBM Oddness
If you're familiar with the PBM format, you may find pamfunc's operation on PBM images to be nonintuitive. Because in PBM black is represented as 1 and white as 0 (1.0 and 0.0 normlized), you might
be expecting adding 1 to white to yield black.
But the PBM format is irrelevant, because pamfunc operates on the numbers found in the PAM equivalent (see above). In a PAM black and white image, black is 0 and white is 1 (0.0 and 1.0 normalized).
So white plus 1 (clipped to the maximum of 1.0) is white.
In addition to the options common to all programs based on libnetpbm (most notably -quiet, see
Common Options ), pamfunc recognizes the following command line options:
This option makes the transfer function that of multiplying by
realnum. realnum must be nonnegative. If the result
is greater than one, it is clipped to one.
Where the input is a PGM or PPM image, this has the effect of
dimming or brightening it. For a different kind of brightening,
see pambrighten(1) and ppmflash(1)
Also, see ppmdim(1), which does the same
thing as pamfunc -multiplier on a PPM image with a multiplier
between zero and one, except it uses integer arithmetic, so it may be
And ppmfade(1) can generate a whole
sequence of images of brightness declining to black or increasing to
white, if that's what you want.
This option makes the transfer function that of dividing by
realnum. realnum must be nonnegative. If the result
is greater than one, it is clipped to one.
This is the same function as you would get with -multiplier,
specifying the multiplicative inverse of realnum.
This option makes the transfer function that of adding
integer/maxval. If the result is greater than one, it is
clipped to one. If it is less than zero, it is clipped to zero.
Note that in mathematics, this entity is called an "addend,"
and an "adder" is a snake. We use "adder" because
it makes more sense.
This option makes the transfer function that of subtracting
integer/maxval. If the result is greater than one, it is
clipped to one. If it is less than zero, it is clipped to zero.
Note that in mathematics, this entity is called a
"subtrahend" rather than a "subtractor." We use
"subtractor" because it makes more sense.
This is the same function as you would get with -adder,
specifying the negative of integer.
This option makes the transfer function that of taking the maximum of
the argument and wholenum/maxval. I.e the minimum value in
the output will be wholenum/maxval.
If wholenum/maxval is greater than one, though, every value
in the output will be one.
This option makes the transfer function that of taking the minimum of
the argument and wholenum/maxval. I.e the maximum value in
the output will be wholenum/maxval.
If wholenum/maxval is greater than one, the function is
idempotent -- the output is identical to the input.
This option makes the transfer function that of bitwise anding
with hexmask.
hexmask is in hexadecimal. Example: 0f
This option was new in Netpbm 10.40 (September 2007).
This option makes the transfer function that of bitwise
inclusive oring with hexmask.
This is analogous to -andmask.
This option was new in Netpbm 10.40 (September 2007).
This option makes the transfer function that of bitwise
exclusive oring with hexmask.
This is analogous to -andmask.
This option was new in Netpbm 10.40 (September 2007).
This option makes the transfer function that of bitwise logical
inversion (e.g. sample value 0xAA becomes 0x55).
pnminvert does the same thing for a bilevel visual image
which has maxval 1 or is of PBM type.
This option was new in Netpbm 10.40 (September 2007).
This option makes the transfer function that of bitwise shifting
left by count bits.
This option was new in Netpbm 10.40 (September 2007).
This option makes the transfer function that of bitwise shifting
right by count bits.
This is analogous to -shiftleft.
This option was new in Netpbm 10.40 (September 2007).
This option tells pamfunc to use a different maxval in the output image than the maxval of the input image, if it helps. By default, the maxval of the output is unchanged from the input and
pamfunc modifies the sample values as necessary to perform the operation.
But there is one case where pamfunc can achieve the same result just by changing the maxval and leaving the sample values unchanged: dividing by a number 1 or greater, or multiplying by a number
1 or less. For example, to halve all of the values, pamfunc can just double the maxval.
With -changemaxval, pamfunc will do just that.
As the Netpbm formats have a maximum maxval of 65535, for large divisors, pamfunc may not be able to use this method.
An advantage of dividing by changing the maxval is that you don't lose precision. The higher maxval means higher precision. For example, consider an image with a maxval of 100 and sample value of
10. You divide by 21 and then multiply by 21 again. If pamfunc does this by changing the sample values while retaining maxval 100, the division will result in a sample value of 0 and the
multiplication will also result in zero. But if pamfunc instead keeps the sample value 10 and changes the maxval, the division will result in a maxval of 2100 and the multiplication will change
it back to 100, and the round trip is idempotent.
This option was new in Netpbm 10.65 (December 2013).
SEE ALSO¶
ppmdim(1), pambrighten(1), pamdepth(1), pamarith(1), pamsummcol(1), pamsumm(1), ppmfade(1), pnminvert(1), pam(5), pnm(5),
This program was added to Netpbm in Release 10.3 (June 2002).
This manual page was generated by the Netpbm tool 'makeman' from HTML source. The master documentation is at
09 September 2020 netpbm documentation | {"url":"https://manpages.opensuse.org/Tumbleweed/netpbm/pamfunc.1.en.html","timestamp":"2024-11-07T19:30:51Z","content_type":"text/html","content_length":"32689","record_id":"<urn:uuid:85979239-4ee5-4cc8-bf9e-a7a8b1c42793>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00280.warc.gz"} |
Catalan numbers, kind of, and Educational 170 E - Codeforces
Submission I would like to explain the logic used to solve problem E (which I didn't manage to complete during the contest and submitted shortly after).
For this, it will be useful to know what Catalan numbers are, as well as how to derive their formula. You can read about Catalan numbers in detail, for example, on wiki: click, Catalan numbers can be
defined in different ways, but we will define the number $$$C_n$$$ as the number of balanced bracket sequences of length $$$2n.$$$ We are interested in the following relatively well-known and elegant
method to find this number.
Let’s consider a coordinate grid where we need to walk from $$$(0,0)$$$ to $$$(n,n)$$$. We can only move to the right or up. A step to the right corresponds to an opening bracket, and a step up — to
a closing one. Thus, there is a one-to-one correspondence between a valid bracket sequence of length $$$2n$$$ and a path from $$$(0,0)$$$ to $$$(n,n)$$$ that does not cross the diagonal $$$y=x$$$;
since a point above this diagonal would mean that in the current prefix, there are more closing brackets than opening ones. How can we find the number of such paths?
The total number of paths is $$$\binom{2n}{n}.$$$ From this number, we subtract the number of invalid paths. Invalid paths are those that cross $$$y=x$$$, or, equivalently, have at least one common
point with the line $$$y=x+1.$$$ Take the first (with the smallest $$$x$$$) common point $$$(a, a+1)$$$ and reflect the segment of the path from $$$(0,0)$$$ to $$$(a, a+1)$$$ across the line $$$y=
x+1.$$$ We obtain a path from $$$(-1,1)$$$ to $$$(n,n)$$$. This is a one-to-one correspondence. Indeed, any path starting at $$$(0,0)$$$ that has common points with $$$y=x+1$$$ can be reflected up to
the first such point. At the same time, any path from $$$(-1,1)$$$ to $$$(n,n)$$$ must have common points with с $$$y=x+1$$$, since the start and end points of the path are on opposite sides of the
line, which means we can also reflect the segment of the path up to the first such point. The number of such invalid paths is $$$\binom{2n}{n+1}.$$$ (or, equivalently $$$\binom{2n}{n-1}.$$$)
Therefore, the number of valid paths is $$$C_n = \binom{2n}{n} - \binom{2n}{n+1}.$$$ This is Catalan number. More on the figure below.
If there were no trump suit in the problem, the winning distribution of cards between the players would satisfy the condition that all cards of the same suit are distributed in the order of a
balanced bracket sequence, with $$$m/2$$$ cards for each player. However, the presence of a trump suit requires us to make two modifications to the above formula.
First, suit 1 must still satisfy the prefix condition of a balanced bracket sequence, but it does not have to be completed. Any path from $$$(0,0)$$$ в $$$(a, m-a), a \ge m-a$$$, that does not
intersect $$$y=x$$$, satisfies the condition. The number of such paths is $$$\binom{m}{a} - \binom{m}{a+1}$$$. This distribution of suits will leave us with $$$2a - m$$$ trump cards that we can use
for other suits. This is the basis of our DP. Let $$$dp_i[ta]$$$ be the number of ways to arrange the first $$$i$$$ suits in a winning way, leaving $$$ta$$$ trump cards available for later use. Then
$$$dp_1[2a - m] = \binom{m}{a} - \binom{m}{a+1} \forall a \ge m/2.$$$
Second, if we use $$$k$$$ (even number) trump cards for suit $$$i \ne 1$$$, it means that player 1 plays $$$(m - k)/2$$$ cards of suit $$$i$$$ and player 2 plays $$$(m+k)/2$$$ cards. The cards must
satisfy the condition of a "not too imbalanced" bracket sequence, meaning that on any prefix, the number of closing brackets should not exceed the number of opening brackets by more than $$$k.$$$
Combining these ideas, we find that the result is the number of paths from $$$(0,0)$$$ to $$$(\frac{m - k}{2}, \frac{m + k}{2})$$$, that do not intersect $$$y = x + k$$$. Invalid paths are reflected
in a similar way as before, but across the line $$$y = x + k + 1$$$, to the paths that start at $$$( -(k+1), (k+1))$$$. Thus the number of valid path is $$$f(m,k) = \binom{m}{(m - k)/2} - \binom{m}
{(m - k)/2 + k + 1}$$$.
The DP transition itself is easy: $$$\forall ta \le m, \forall k \le ta, dp_i[ta - k] += f(m,k)*dp_{i-1}[ta].$$$ It's enough to consider only even $$$ta$$$ и $$$k$$$. The final answer is $$$dp_n
[0].$$$ The complexity is $$$O(m^2n).$$$
4 weeks ago, # |
» +3
yellow_13 The derivation for Catalan numbers is simply amazing, never though it could be explained this way. Next time I need to derive $$$C_n$$$ this will definitely be the first thing that comes to
my mind. Thank you so much for the insightful blog.
4 weeks ago, # |
» 0
papa-ka-para This distribution of suits will leave us with 2a - m trump cards that we can use for other suits.
Sorry if I am wrong, but shouldn't this be m-2a ?
• » 4 weeks ago, # ^ |
» 0
darthrevenge No, it's $$$2a -m$$$. If player 1 has $$$a \ge m/2$$$ trump cards, player 2 has $$$m - a$$$ cards, so after matching player 1 left with extra $$$a - (m-a) = 2a -m$$$ trump cards.
□ » 4 weeks ago, # ^ |
» 0
Please add this link. If anyone has still problem understading the visualisation, this video might help ...
4 weeks ago, # |
» ← Rev. 3 → 0
Axial-Tilted very nice explanation , but why is it that when u reflect a path u only reflect on the first intersection point ? reflecting the whole path on the line will give the same result
edit : actually reflecting the whole path does absolutely nothing ignore the above comment | {"url":"https://mirror.codeforces.com/blog/entry/135152","timestamp":"2024-11-12T17:05:46Z","content_type":"text/html","content_length":"104240","record_id":"<urn:uuid:54d9c926-e4e6-488d-ac6e-9dc72f62fd2f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00745.warc.gz"} |
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Ian Martin
Professor of Finance, London School of Economics
New: The CME has introduced CVOL, a cross-asset class family of implied volatility indices based on the methodology of Simple Variance Swaps and What is the Expected Return on the Market?
Working Papers
Forecasting Crashes with a Smile (with Ran Shi), May 2024
Jack Treynor Prize 2024
We use option prices to derive bounds on the probability of a crash in an individual stock, and argue that the lower bound should be close to the truth. Empirically, the lower bound is highly
statistically and economically significant; on its own, it outperforms 15 stock characteristics proposed by the prior literature combined. In a multivariate regression, a one standard deviation
increase in the bound raises the predicted crash probability by 3 percentage points, whereas a one standard deviation increase in the next most important predictor (a measure of short interest)
raises the predicted probability by only 0.3 percentage points. Long-Horizon Exchange Rate Expectations (with Lukas Kremens and Liliana Varela), June 2024
We study exchange rate expectations in surveys of financial professionals and find that they successfully forecast currency appreciation at the two-year horizon, both in and out of sample. Exchange
rate expectations are also interpretable, in the sense that three macro-finance variables—the risk-neutral covariance between the exchange rate and equity market, the real exchange rate, and the
current account relative to GDP—explain most of their variation. But there is no “secret sauce” in expectations: after controlling for the three macro-finance variables, the residual information in
survey expectations does not forecast currency appreciation in our sample. Debt and Deficits: Fiscal Analysis with Stationary Ratios (with John Campbell and Can Gao), October 2023
We introduce a new measure of a government’s fiscal position that exploits cointegrating relationships among fiscal variables and output. The measure is a loglinear combination of tax revenue,
government spending and the market value of government debt that—unlike the debt-GDP ratio—is stationary in the US and the UK since World War II. Fiscal deterioration forecasts a long-run decline in
spending rather than increased tax revenue or low returns for bondholders. Fiscal adjustment to tax and spending shocks occurs through mean-reversion in tax and spending growth, with a negligible
contribution from debt returns. The Forward Premium Puzzle in a Two-Country World, March 2013
NBER Working Paper 17564
I explore the behavior of asset prices and the exchange rate in a two-country world. When the large country has bad news, the relative price of the small country’s output declines. As a result, the
small country’s bonds are risky, and uncovered interest parity fails, with positive excess returns available to investors who borrow at the large country’s interest rate and lend at the small
country’s interest rate. I use a diagrammatic approach to derive these and other results in a calibration-free way. How Much Do Financial Markets Matter? Cole–Obstfeld Revisited, November 2010
Cole and Obstfeld (1991) asked, “How much do financial markets matter?” Emphasizing the importance of intratemporal relative price adjustment as a risk-sharing mechanism that operates even in the
absence of financial asset trade, their answer was: not much. I revisit their question and find that in calibrations featuring rare disasters that generate reasonable risk premia without implausibly
high risk aversion, the cost of shutting down trade in financial assets is on the order of 3 to 20 per cent of wealth. Simple Variance Swaps, January 2013
NBER Working Paper 16884
Note: this paper is largely subsumed by What is the Expected Return on the Market?
The events of 2008‒9 disrupted volatility derivatives markets and caused the single-name variance swap market to dry up completely; it has never recovered. This paper introduces the simple variance
swap, a more robust relative of the variance swap that can be priced and hedged even if the underlying asset’s price can jump, and constructs SVIX, an index based on simple variance swaps that
measures market volatility. SVIX is consistently lower than VIX in the time series, which rules out the possibility that the market return and stochastic discount factor are conditionally lognormal.
The SVIX index points to an equity premium that—in contrast to the prevailing view in the literature—is extraordinarily volatile and that spiked dramatically at the height of the recent crisis.
Sustainability in a Risky World (with John Campbell), American Economic Review: Insights, forthcoming
How much consumption is sustainable, if “sustainability” requires that welfare should not be expected to decline over time? We impose a sustainability constraint on a standard consumption/portfolio
choice problem. The constraint does not distort portfolio choice, but it imposes an upper bound on the sustainable consumption-wealth ratio, which must lie between the riskless interest rate and the
expected return on wealth (and if risky capital evolves according to a geometric Brownian motion, it lies exactly halfway between the two). Sustainability requires an upward drift in wealth and
consumption to compensate future generations for the increased risk they face. Sentiment and Speculation in a Market with Heterogeneous Beliefs (with Dimitris Papadimitriou), American Economic Review
(2022), 112:8:2465‒2517
Online Appendix
We present a model featuring risk-averse investors with heterogeneous beliefs. Individuals who are correct in hindsight—whether through luck or judgment—get rich, so sentiment is bullish following
good news and bearish following bad news. Sentiment makes extreme outcomes far more important for pricing and has asymmetric effects on left- and right-skewed assets. Investors take speculative
positions that can conflict with their fundamental views. Moderate investors are contrarian: they trade against excess volatility created by extremists. All investors view speculation as socially
costly; but they also think it is in their self-interest, and the market can collapse entirely if speculation is banned.
Market Efficiency in the Age of Big Data (with Stefan Nagel), Journal of Financial Economics (2022), 145:1:154‒177
Modern investors face a high-dimensional prediction problem: thousands of observable variables are potentially relevant for forecasting. We reassess the conventional wisdom on market efficiency in
light of this fact. In our equilibrium model, N assets have cash flows that are linear in J characteristics, with unknown coefficients. Risk-neutral Bayesian investors learn these coefficients and
determine market prices. If J and N are comparable in size, returns are cross-sectionally predictable ex post. In-sample tests of market efficiency reject the no-predictability null with high
probability, even though investors use information optimally in real time. In contrast, out-of-sample tests retain their economic meaning.
Volatility, Valuation Ratios, and Bubbles: An Empirical Measure of Market Sentiment (with Can Gao), Journal of Finance (2021), 76:6:3211‒3254
We define a sentiment indicator based on option prices, valuation ratios and interest rates. The indicator can be interpreted as a lower bound on the expected growth in fundamentals that a rational
investor would have to perceive in order to be happy to hold the market. The lower bound was unusually high in the late 1990s, reflecting dividend growth expectations that in our view were
unreasonably optimistic. Our approach exploits two key ingredients. First, we derive a new valuation ratio decomposition that is related to the Campbell‒Shiller loglinearization but that resembles
the Gordon growth model more closely and has certain other advantages. Second, we introduce a volatility index that provides a lower bound on the market’s expected log return.
Implied Dividend Volatility and Expected Growth (with Niels Gormsen and Ralph Koijen), AEA Papers and Proceedings (2021), 111:361‒365
Online Appendix
We study the behavior of implied dividend volatility, constructed from the prices of options on index-level dividends, during the Covid-19 pandemic. We use these data to construct a lower bound on
expected excess returns on dividend claims and find that the bound moves significantly over time. However, most of the variation in dividend futures prices reflects changes in growth expectations
rather than expected excess returns, making them valuable assets to uncover growth expectations. We conclude that the short-term economic outlook is uncertain and not expected to recover in the near
On the Autocorrelation of the Stock Market, Journal of Financial Econometrics (2021), 19:1:39‒52
I introduce an index of market return autocorrelation based on the prices of index options and of forward-start index options, and implement it empirically at a six-month horizon. The results suggest
that the autocorrelation of the S&P 500 index was close to zero before the subprime crisis but was negative in its aftermath, attaining values around −20% to −30%. I speculate that this may reflect
market perceptions about the likely reaction, via quantitative easing, of policymakers to future market moves.
Welfare Costs of Catastrophes: Lost Consumption and Lost Lives (with Robert Pindyck), Economic Journal (2021), 131:946‒969
Most of the literature on the economics of catastrophes assumes that such events cause a reduction in the stream of consumption, as opposed to widespread fatalities. Here we show how to incorporate
death in a model of catastrophe avoidance, and how a catastrophic loss of life can be expressed as a welfare-equivalent drop in consumption. We examine how potential fatalities affect the policy
interdependence of catastrophic events and “willingness to pay” (WTP) to avoid them. Using estimates of the “value of a statistical life” (VSL), we find the WTP to avoid major pandemics, and show it
is large (10% or more of annual consumption) and partly driven by the risk of macroeconomic contractions. Likewise, the risk of pandemics significantly increases the WTP to reduce consumption risk.
Our work links the VSL and consumption disaster literatures.
What is the Expected Return on a Stock? (with Christian Wagner), Journal of Finance (2019), 74:4:1887‒1929
Dimensional Fund Advisors Distinguished Paper Prize 2019
The Wharton School‒WRDS Best Paper Award in Empirical Finance, WFA 2017
Honorable Mention, AQR Insight Award 2017
We derive a formula for the expected return on a stock in terms of the risk-neutral variance of the market and the stock’s excess risk-neutral variance relative to the average stock. These quantities
can be computed from index and stock option prices; the formula has no free parameters. The theory performs well empirically both in and out of sample. Our results suggest that there is considerably
more variation in expected returns, over time and across stocks, than has previously been acknowledged.
The Quanto Theory of Exchange Rates (with Lukas Kremens), American Economic Review (2019), 109:3:810‒843
Online Appendix
Best Paper Award, IF2017 Annual Conference in International Finance
SIX Best Paper Award 2018
We present a new identity that relates expected exchange rate appreciation to a risk-neutral covariance term, and use it to motivate a currency forecasting variable based on the prices of quanto
index contracts. We show via panel regressions that the quanto forecast variable is an economically and statistically significant predictor of currency appreciation and of excess returns on currency
trades. Out of sample, the quanto variable outperforms predictions based on uncovered interest parity, on purchasing power parity, and on a random walk as a forecaster of differential
(dollar-neutral) currency appreciation.
Notes on the Yield Curve (with Steve Ross), Journal of Financial Economics (2019), 134:689‒702
We study the properties of the yield curve under the assumptions that (i) the fixed-income market is complete and (ii) the state vector that drives interest rates follows a finite discrete-time
Markov chain. We focus in particular on the relationship between the behavior of the long end of the yield curve and the recovered time discount factor and marginal utilities of a
pseudo-representative agent; and on the relationship between the “trappedness” of an economy and the convergence of yields at the long end.
Options and the Gamma Knife, Journal of Derivatives (2018), 25:4:71‒79
Local version
This paper, a (very) slightly modified version of the one below, was solicited and republished by a sister journal of JPM. Options and the Gamma Knife, Journal of Portfolio Management (2018),
Local version
I survey work of Steve Ross (1976) and of Douglas Breeden and Robert Litzenberger (1978) that first showed how to use options to synthesize more complex securities. Their results made it possible to
infer the risk-neutral measure associated with a traded asset, and underpinned the development of the VIX index. The other main result of Ross (1976), which shows how to infer joint risk-neutral
distributions from option prices, has been much less influential. I explain why, and propose an alternative approach to the problem. This paper is dedicated to Steve Ross, and was written for a
special issue of the Journal of Portfolio Management in memory of him. What is the Expected Return on the Market?, Quarterly Journal of Economics (2017), 132:1:367‒433
Online Appendix
Data: Description of data SVIX2.xls epbound.xls crashprob.xls
New: SVIX is now reported by the CME across a range of asset classes under the CVOL umbrella
I derive a lower bound on the equity premium in terms of a volatility index, SVIX, that can be calculated from index option prices. The bound implies that the equity premium is extremely volatile and
that it rose above 20% at the height of the crisis in 2008. The time-series average of the lower bound is about 5%, suggesting that the bound may be approximately tight. I run predictive regressions
and find that this hypothesis is not rejected by the data, so I use the SVIX index as a proxy for the equity premium and argue that the high equity premia available at times of stress largely reflect
high expected returns over the very short run. I also provide a measure of the probability of a market crash, and introduce simple variance swaps, tradable contracts based on SVIX that are robust
alternatives to variance swaps. Averting Catastrophes: The Strange Economics of Scylla and Charybdis (with Robert Pindyck), American Economic Review (2015), 105:10:2947‒2985
Faced with numerous potential catastrophes—nuclear and bioterrorism, mega-viruses, climate change, and others—which should society attempt to avert? A policy to avert one catastrophe considered in
isolation might be evaluated in cost-benefit terms. But because society faces multiple catastrophes, simple cost-benefit analysis fails: even if the benefit of averting each one exceeds the cost, we
should not necessarily avert them all. We explore the policy interdependence of catastrophic events, and develop a rule for determining which catastrophes should be averted and which should not. The
Lucas Orchard, Econometrica (2013), 81:1:55‒111
Supplemental Material
This paper investigates the behavior of asset prices in an endowment economy in which a representative agent with power utility consumes the dividends of multiple assets. The assets are Lucas trees;
a collection of Lucas trees is a Lucas orchard. The model generates return correlations that vary endogenously, spiking at times of disaster. Since disasters spread across assets, the model generates
large risk premia even for assets with stable cashflows. Very small assets may comove endogenously and hence earn positive risk premia even if their cashflows are independent of the rest of the
economy. I provide conditions under which the variation in a small asset’s price-dividend ratio can be attributed almost entirely to variation in its risk premium.
Consumption-Based Asset Pricing with Higher Cumulants, Review of Economic Studies (2013), 80:2:745‒773
Online Appendix
I extend the Epstein–Zin-lognormal consumption-based asset-pricing model to allow for general i.i.d. consumption growth. Information about the higher moments—equivalently, cumulants—of consumption
growth is encoded in the cumulant-generating function. I use the framework to analyze economies with rare disasters, and argue that the importance of such disasters is a double-edged sword:
parameters that govern the frequency and sizes of rare disasters are critically important for asset pricing, but extremely hard to calibrate. I show how to sidestep this issue by using observable
asset prices to make inferences without having to estimate higher moments of the underlying consumption process. Extensions of the model allow consumption to diverge from dividends, and for
non-i.i.d. consumption growth.
On the Valuation of Long-Dated Assets, Journal of Political Economy (2012), 120:2:346‒358
Online Appendix
Reprinted in Christian Gollier ed. The Economics of Risk and Uncertainty, Edward Elgar: Cheltenham, UK, 2018
I show that the pricing of a broad class of long-dated assets is driven by the possibility of extraordinarily bad news. This result does not depend on any assumptions about the existence of
disasters, nor does it only apply to assets that hedge bad outcomes; indeed, it even applies to long-dated claims on the market in a lognormal world if the market’s Sharpe ratio is higher than its
volatility, as appears to be the case in practice. Disasters Implied by Equity Index Options (with David Backus and Mikhail Chernov), Journal of Finance (2011), 66:6:1969‒2012
We use equity index options to quantify the distribution of consumption growth disasters. The challenge lies in connecting the risk-neutral distribution of equity returns implied by options to the
true distribution of consumption growth estimated from macroeconomic data. We attack the problem from three perspectives. First, we compare pricing kernels constructed from macro-finance and
option-pricing models. Second, we compare option prices derived from a macro-finance model to those we observe. Third, we compare the distribution of consumption growth derived from option prices
using a macro-finance model to estimates based on macroeconomic data. All three perspectives suggest that options imply smaller probabilities of extreme outcomes than have been estimated from
international macroeconomic data. The third comparison yields a viable alternative calibration of the distribution of consumption growth that matches the equity premium, option prices, and the sample
moments of US consumption growth. Disasters and the Welfare Cost of Uncertainty, American Economic Review (2008), Papers & Proceedings, 98:2:74‒78 | {"url":"https://personal.lse.ac.uk/martiniw/","timestamp":"2024-11-12T20:02:53Z","content_type":"text/html","content_length":"27542","record_id":"<urn:uuid:c987064b-f0fd-49ca-82e8-d99b0e175424>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00723.warc.gz"} |
batman symbol
What better way to show what you know about functions than to use them to create "art"? Here is a challenge for you. Go to Desmos.com and create a work of art that uses at least ten functions of
which you must include one of each of the conics - a hyperbola, a parabola, an ellipse, [...] | {"url":"https://systry.com/tag/batman-symbol/","timestamp":"2024-11-04T02:26:25Z","content_type":"text/html","content_length":"22779","record_id":"<urn:uuid:4d6cdeab-9e87-4095-a13f-635f93f9959b>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00014.warc.gz"} |
Betteridge’s law
Was Betteridge right?
Betteridge’s law says
Any headline which ends in a question mark can be answered by the word no.
If Betteridge was right, then the answer to my headline question should be no, in which case Betteridge was wrong. But Betteridge was wrong, then the answer to the question in my headline is yes.
This isn’t quite like Russell’s paradox. He asked whether the set of sets which contain themselves contains itself. If it does, it doesn’t. If it doesn’t, it does. This logical contradiction led to a
more rigorous construction of set theory that avoids the paradox.
My observation about Betteridge’s law isn’t a paradox, though it resembles one. If Betteridge was wrong, there’s no contradiction in saying that he was sometimes but not always right.
Betteridge’s law was an aphorism, not a logical absolute, and so was never intended to be a rigorous statement. I’m sure Betteridge was quite aware that there had been exceptions, or at least that
one could easily create an exception. He did so himself. But as is often the case with yes/no statements that are not always true, it can be turned into a rigorous statement using probability.
Betteridge could have said that if a headline ends in a question mark, the probability that the answer is no is large. Then my headline, added to the vast collection of headlines, would
ever-so-slightly lower the proportion of headlines that ask questions that can be answered negatively, without contradicting Betteridge, if was right.
We could call Betteridge’s constant the probability that a headline asks a question that could be answered no. But then it probably isn’t a constant. Maybe knowledge of Betteridge’s law influences
how people write headlines …
* * *
Thanks to Don Sizemore for pointing out Betteridge’s law.
12 thoughts on “Was Betteridge right?”
1. Good Afternoon from Prague John.
Actually, I think that Russel’s paradox is that the set of set that doesn’t contain themselves contains itself.
Great blog! I enjoy every article!
Have a great day,
2. To keep readers guessing, a journalist could flip a coin before writing a headline, then pick a question whose answer is “yes” if and only if the coin comes up heads.
3. John:
There’s also the endogeneity problem, that you wrote your headline knowing of the hypothesis. This could be fixed via a time-series version of the theory of types, whereby any statement such as
Betteridge’s is only allowed to apply to past events. Otherwise you could falsify just about any law (other than an actual law of physics) by constructing counterexamples. (And, in a manner
reminiscent of search engine optimization, you could falsify just about any probabilistic law by creating a bot to create arbitrarily large numbers of counterexamples.)
4. Of course the headline can be answered with “no”. The semantics of the headline are less specific than the entire article. The map is not the territory. No math needed. :)
5. In a similar vein,
6. There’s no paradox in the answer “no”, due to the universal quantifier present.
7. This is like doing a comprehensive study of all published statistics and determining the number that are made up. When you publish this new statistic, your work is then ever so slightly wrong.
8. This is essentially Liar’s paradox.
9. The point is that when a headline or article title takes the form of a question, the question is a leading one; the author is trying to push the reader towards one answer or the other, but I have
(at a guess) seen as many yes’s as no’s. This may be hwy some journals will not accept papers whose titles are questions.
Anyhow, what would Betteridge say to the headline “How hot is the sun?”
10. What’s the fuss?
On one hand, at I read the ‘law’ the headling “Was Betteridge right?” CAN be answered as no, it can equally be answered as yes. The law doesn’t claim that either answer is true.
On the other hand, the example offered by Frank Wilhoit, along with countless others “Who Really Killed JFK?” or “What is the Answer to Life, the Universe, and Everything?” are clearly
counter-examples to the “law” since they are not predicates and can’t be answered yes or no.
Note that Betteridge himself admitted to violating his own law
11. This reminds me of a lot of those Fox news headlines along the lines of “George W. Bush, greatest president?”
I don’t know the context of the quote, but the aphorism is interesting. The idea seems to be that leading questions as headlines tend to be used by journalists how want to suggest an
interpretation that is not warranted by the data alone. In that sense, a question for a headline is a good “marker”/correlate for dubious assertions and the answer will often be “no”.
12. > He asked whether the set of sets which contain themselves contains itself. If it does, it doesn’t. If it doesn’t, it does.
Shouldn’t that be ‘set of sets which _do not_ contain themselves’? | {"url":"https://www.johndcook.com/blog/2013/03/18/was-betteridge-right/","timestamp":"2024-11-04T21:34:32Z","content_type":"text/html","content_length":"69397","record_id":"<urn:uuid:379ae091-fdea-46af-82c9-65682b13d0c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00759.warc.gz"} |
Scientific Computing and Numerics (SCAN) Seminar
Monday, November 4, 2024 - 1:30pm
Operator learning is a rapidly growing field of machine learning relevant to modeling continuum mechanics. It uses data from experiments or numerical simulations to learn operators between spaces of
functions, often consisting of input and output spatial fields such as force and displacement on a structure. This talk will discuss how much data is needed to construct accurate approximations of
operators, focusing primarily on learning finite-rank approximations of solution operators for non-self-adjoint uniformly elliptic PDEs. Low-rank approximation of large matrices using matrix-vector
product queries generally requires querying the matrix transpose (adjoint), or otherwise querying the matrix as many times as columns in the matrix. In contrast, we show that finite-rank
approximations of solution operators for uniformly elliptic PDEs can be learned without querying the adjoint solution operator, and we provide sample complexity estimates for learning using only
forward queries. This is fortunate because the adjoint is usually inaccessible in experimental settings. The key ingredient is the additional structure imposed on the operator by elliptic regularity.
These results illuminate some of the fundamental challenges in operator learning and provide a path towards truly data-efficient learning for linear and nonlinear operators by exploiting known | {"url":"https://pi.math.cornell.edu/m/node/11496","timestamp":"2024-11-02T07:33:22Z","content_type":"text/html","content_length":"28609","record_id":"<urn:uuid:2e748a70-1765-4845-b783-59f05af08220>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00105.warc.gz"} |
The Scaling Function of Teams
When I arrived in Silicon Valley, resource planning (at the large tech companies) was/is based on ratios. The formula is pretty simple, for every X Engineers there will be Y Designers.
The ratio, I heard most often and saw employed most widely was 1:10. For every 10 engineers there would be 1 designer. Where did this number come from? Why was 1:10 the perfect team? No one knew or
seemed to question it. It was cannon.
It reminded me of why lawyers bill in 6 minute increments. It is easily divisible by 60. The idea that 6 minutes is a reasonable amount of time to do work isn't based in any fact, it's just easier
math. I assume 1:10 is similar. It isn't accurate, It's just easy math.
While 1:10 is an arbitrary and convenient number, it isn't that far off (see Coda below). Continuous delivery software teams scale pretty linearly and there is a relationship between engineers and
designers. If you tell me how many engineers are on a team, I can tell you, with reasonable accuracy, how many Designers you need. This is a neat party trick but doesn't get me invited to as many
parties as you might think.
Ratios only work for teams where there is a well established relationship between the denominator (Engineers) and the function you are trying to determine resources for. It only works for self
contained product or feature teams with embedded designers (not studio based models). It fails everywhere else.
Most teams scale in steps. Their scaling function is different to linearly scaling teams. If you are making a video game, the number of Designers you need has a low relationship to the number of
Engineers. The scaling function is tied to the complexity and size of the game. How many levels does it have? How many unique assets (art) needs to be created? Is it a new game or part of an existing
Hardware scales based on how many concurrent programs are running. The Industrial Design team could care less about how many ML Engineers are hired as it has no relationship to their work or the
resources required to do their work.
Step scaling teams need to hire for their expected output. Since people (resources) aren't very elastic (in an economic sense), these teams start off with excess capacity. Over time, they become
efficient as they are given new programs, then become over worked until the next team is hired.
A marketing team can handle only so many product launches in a year. If the company adds one more product launch, a whole new team needs to be hired to handle that one additional program. A sales
team can only handle so many markets. If the company wants to sell in a new country, a whole sales team needs to be spun up.
Step function teams are expensive because the resources are hired up front, not gradually. This scaling will look inefficient to finance (because it is). It looks more like a capital investment where
it pays off over time vs linear scaling which scales in concert with company revenue/profits.
When scaling teams, you need to determine if its scaling function is linear or step. Finance needs to have models for both and an understanding that some teams require larger upfront investments to
bootstrap them. I see a lot of managers trying to staff step function teams with linearly scaled headcount and it always ends in tears.
Ratios, are macro economic tools. They are a crude and blunt instrument. In the right context they are a quick way to estimate at scale but can not be confused as a precise tool. Too many companies
use ratios as if they are a law of physics or some truth. They are not.
I have done countless estimations on ratios for design teams. After looking at too many bottom's up and top's down spreadsheets the ratio that consistently comes up is 1:7 (not 1:10). I don't know
why this is the number but it is. It is a pattern that i trust and I have adopted it into my resource estimation algorithm.
In my experience, 1:10 will leave the designers chasing the backlog but can be workable depending on the product and the Designers. 1:5 can lead to out designing what can be built by the Engineers.
The more specialized the work (work that requires distinct design disciplines vs generalist designer) the more designers are required and ratios become less meaningful.
I have not found a strong ratio for UX Research or Content Design. I have struggled to find the right denominator for these disciplines. I have tried denominating against PM, Design, Eng and all end
up not lacking in some way. I have had more success trying to understand what work is needed in each team and then determining resources (i.e. bottom's up planning). | {"url":"https://jonlax.framer.ai/writing/the-scaling-function-of-teams","timestamp":"2024-11-11T07:10:08Z","content_type":"text/html","content_length":"156886","record_id":"<urn:uuid:65146e4c-524d-42ad-8428-5fd3c6f816e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00106.warc.gz"} |
e 1
Computer Science 136
Data Structures and Advanced Programming
Williams College
Fall 2005
Lecture 11: Comparable Objects, Sorting
Date: October 3, 2005
• New option to avoid cortland and the Mac lab. JDK 1.5 on FreeBSD.
• Advance warning: our first exam is October 19 during lab period.
• There will be an opportunity to ask exam-related questions each day up until our exam. If there is interest in a review session, we can arrange a time Monday or Tuesday evening.
• Lab 4 out - due Wednesday because of reading period.
• Comparable objects
• Sorting
□ Bubble Sort
□ Selection Sort
☆ recursive implementation
☆ correctness proof
☆ complexity proof
• Visit your favorite search engine to find Java applets that visualize these sorting techniques.
• BinSearch
• Swap
• SelectionSort
To prove correctness of a procedure like our recursive selection sort, we use a form of proof by mathematical induction:
1. Prove base case(s). (Usually this is trivial)
2. Show that if algorithm works correctly for all simpler (i.e., smaller) input, then will work for current input.
To show that the recSelSort algorithm given earlier is correct, we will reason by induction on the size of the array (i.e. on lastIndex)
Base: If lastIndex <= 0 then at most one element in elts, and hence sorted - correct.
Induction Hypothesis: Suppose it works if lastIndex < n.
Induction: Show it works if last = n (> 0).
The for loop finds largest element and then swaps it with elts[lastIndex].
Thus elts[lastIndex] holds the largest element of list, while others are held in elts[0..lastIndex-1].
Since lastIndex- 1 < lastIndex, know (by induction hypothesis) that recSelSort(lastIndex-1,elts) sorts elts[0..lastIndex-1].
Because elts[0..lastIndex-1] in order and elts[lastIndex] is >= all of them, elts[0..lastIndex] is sorted.
Claim: recSelSort(n-1,elts) (i.e, on n elements) involves n*(n-1)/2 comparisons of elements of array.
Base: n = 0 or 1, 0 comparisons and n*(n-1)/2 = 0.
Induction hypothesis: Suppose recSelSort(k-1,elts) takes k*(k-1)/2 comparisons for all k < n.
Induction step: Show that it's true for n as well.
Look at algorithm: Run algorithm on recSelSort(n-1,elts). Therefore, last = n-1.
Go through for loop "last" = n-1 times, w/ one comparison each time through.
Thus n-1 comparisons. The only other comparisons are in the recursive call: recSelSort(last-1,elts) where last = n-1.
But by induction (since last < n), this takes last*(last-1)/2 = (n-1)*(n-2)/2 comparisons.
Therefore, altogether we have (n-1) + (n-1)*(n-2)/2 = (n-1)*2/2 + (n-1)*(n-2)/2 = (n-1)*(2 + n-2)/2 = (n-1)*n/2 = n(n-1)/2 comparisons.
Finished proof.
Therefore recSelSort takes O(n^2) comparisons. | {"url":"https://courses.teresco.org/cs136_f05/lectures/lect11/","timestamp":"2024-11-07T12:18:56Z","content_type":"application/xhtml+xml","content_length":"4797","record_id":"<urn:uuid:0cb7948b-ad16-4f2c-8725-f7e5e5c7e95a>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00098.warc.gz"} |
George Augustus Linhart—As a “Widely Unknown”Thermodynamicist
1. Introduction
The name of George Augustus Linhart see Figure 1 (May, 3, 1885 near Vienna, Austria, August, 14, 1951 in Los Angeles, CA) is “widely unknown”. In effect, he was a Viennese-born USA-American
physicist-chemist, partially associated with the Gilbert Newton Lewis’ school of chemical thermodynamics at the University of California in Berkeley.
Figure 1. George Augustus Linhart, in Riverside, California, around 1923.
After attentively reading G. A. Linhart’s unpublished diaries (“Out of the Melting Pot” (1923) as well as “Amid the Stars” (1950)) and thoroughly digging all the possible USA archives, I have finally
managed to reasonably reconstruct his extremely intensive, but, to my mind, still at the utmost curvilinear and in fact very unlucky CV, which can roughly be sketched as follows.
As a lone small boy, he had arrived (from Austria via Hamburg) at New York in 1896, but was officially USA-naturalized only in 1912. He was then married (in the twenties-thirties of the XX-th
century), but, apparently, for some rather short time, and, as a result, had no children. As his life had started to come to an end, he left all his worldly possessions to different USA universities
and scientific research organizations, just to endow fellowships for young scientists.
1.1. Education
He was able to pick up English in the streets of New York and Philadelphia, occasionally working as a waiter and as a tailor, just to somehow survive. But still could successfully graduate a high
school in about one year, and then went to the universities for his further education.
BS from the University of Pennsylvania, 1909.
MA and PhD from the Yale University, Kent Chemical Laboratory (supervisor: Frank Austin Gooch): 1909- 1913.
1.2. Working
University of Washington, Seattle:
Teaching Instructor in German and Chemistry, 1913- 1914.
Simmons College, Boston:
Teaching Instructor, 1915.
University of California in Berkeley (he started working there in the lab of Gilbert Lewis):
Assistant, Chemistry, 1915-1916;
Teaching Fellow, Chemistry, 1916-1917;
Assistant, Chemistry, 1917-1918;
Drafted for the World War I, 1918;
Assistant, Biochemistry, 1919 (Feb-May);
Instructor, Soil Chemistry and Bacteriology, 1919-1920;
Research Associate, Soil Chemistry and Bacteriology, 1920 (May-July).
Eureka Junior College, Eureka, CA:
Teacher, 1920-1921.
Riverside Junior College, Riverside, CA:
Teacher, 1921-1948.
G. A. Linhart had successfully published about 20 papers (mostly on inorganic chemistry, as well as some treatises concerning his view on thermodynamics) in such journals as “American Journal of
Science”, PNAS, JACS, “Journal of Physical Chemistry” etc. But the most of his work and bright ideas is nevertheless contained in a number of unpublished preprints, to our sincere regret.
Well, some colleagues might in principle consider this communication a kind of “hagiography”, for it presents no real historical and philosophical analysis of G. A. Linhart’s ideas, which could
surely sound somewhat strange to the modern readership. But such a view is in fact by far not correct, because what is presented here ought to serve as an invitation to trigger carrying out such
2. The Bright Ideas of George A. Linhart
Although several formulations of the 2^nd Law of Thermodynamics (2LT) different from each other are known, there still remains some kind of interpretational reticence. Specifically, everybody knows,
on the one hand, that the 2LT law forbids the perpetuum mobile and this is empirically correct. On the other hand, the 2LT ought to predict that virtually everything in the Universe is perpetually
running down, which is in apparent contradicttion with a lot of well-known and observable natural phenomena characterized by not only disorganization and decay, but also self-organization and growth.
Still, we know of a wealth of natural and technical processes, which are inherently irreversible, like the famous HumptyDumpty who “sat on a wall” and then “had a great fall”. So, how could it be
possible to bring all these facts under one and the same roof?
Let us take a closer look at the essence of 2LT. First of all, we immediately see that the classical thermodynamics, which is the origin of this law, is applicable to equilibrium states only, where
all the parameters of the system under study stop any changing. Any process in the classical thermodynamics must undergo a sequence of equilibrium states. The time during which such processes last is
of no interest at all: they might take either five seconds or five hundred years to proceed, the main point is that everything happens in or in the nearest proximity to thermodynamic equilibrium. The
2LT states that such processes may sometimes be irreversible, so that there is absolutely no way for any spontaneous return to the initial state starting from the final one. The quantitative measure
of processes reversibility is entropy: in isolated systems (isolated = no energy and/or matter exchange with surrounding) the latter either remains the same (reversibility) or increases
Moreover, all the classical theoretical thermodynamics was originally derived for the cyclic equilibrium processes, where initial and final states coincide. Already the earliest attempts by Clausius
and Kelvin to apply the 2LT to generic non-cyclic processes have immediately led to the speculations about the “heat death of the Universe” and, consequently, hot debates which are more or less
ongoing even nowadays. The clou here is that all the physical laws at the microsopic level should always work same way even if the time course would change its direction. With this in mind, it is
necessary to ensure that in truly macroscopic systems, where lots of microscopic particles come together, processes unidirectional in time become possible. The first huge advance in solving this
fundamental problem was made by Boltzmann, who guessed (without any rigorous inference) that entropy should be nothing else but a logarithm of the number of all the microstates corresponding to one
and the same observable macrostate. To this end, one might say that events like complete and perfect re-assembling of the broken Humpty-Dumpty are in principle possible but essentially unlikely, for
there is a plenty of choices to destroy the poor Humpty-Dumpty, but relatively few ways of bringing him back to his authentic “initial state”.
Meanwhile, there is much more to the story. First of all, a big confusion persists about the mathematical derivation of the 2LT-savvy expression for the entropy starting from the time-symmetrical
microscopic physical laws (see, for example, [1]). The most radical standpoints even claim that the very notion of “irreversibility” has several different meanings (time-asymmetry is different from
just irrecoverability) and one should not confound them as done usually (see, for example, [2,3]). Independently of this, the line of work pioneered by Onsager, Prigogine, de Groot and Mazur is
trying to redefine the conventional equilibrium notions of entropy, temperature etc. for the distinctly non-equilibrium situations, and a considerable progress was achieved on this way (see, for
example, [4-6]). However, the main assumption of the latter works is that although everything in the Nature rarely reaches a perfect equilibrium, there ought to be some small areas allowing the
conventional equilibrium description. To this end, it can be shown that the very notion of “thermodynamic equilibrium” should be re-conceived as a kind of fuzzy set describing continuous “degrees of
equilibrium” (instead of the conventional crisp “equilibrium-nonequilibrium” binary picture) if we would like to bridge the conceptual gap between the Boltzmann and Gibbs thermodynamics [1,7].
Finally, the very notion of entropy causes serious debates as well (see, for example, [8] and the references therein).
To sum up, the nature of time and the interrelationship between time and entropy still remain a mystery. Here, the chief problem is “not to explain why the entropy of the universe will be higher
tomorrow than it is today but to explain why the entropy was lower yesterday and even lower the day before that. We can trace this logic all the way back to the beginning of time in our observable
universe. Ultimately, time asymmetry is a question for cosmology to answer” [9].
But is this really the case? Have we already exhausted all the resources to tackle this problem upon Earth? Surely not! We may well refrain from Immediately Going to Heaven (or to the Hell, well, in
accordance with the sums of our personal sins). Not only the gist, but even the mathematical details of this encouraging answer were given in the forgotten and/or unpublished works by George Augustus
Linhart, approximately at the same time as the famous notion of “Time’s Arrow” was coined by Sir Arthur Stanley Eddington in his well-known book “Nature of the Physical World” [10]. Doing justice and
giving credit to that fundamental work by G. A. Linhart is the main purpose of the present communication.
The CV of G. A. Linhart was extremely dramatic (cf. the Introduction section above). He had graduated from Kent Laboratory of Chemistry in Yale and defended his PhD there as well, but all his
professional life long, after several postdoc years in the University of Washington in Seattle and in Berkeley, he was all the rest of his life long working as a teacher in two countryside junior
colleges in California: Eureka and Riverside. Bearing this in mind, the appallingly bitter introduction to his preprint “The Relation between Chronodynamic Entropy and Time” [11] could be fully
“Eddington in his delightful book on ‘The Nature of the Physical Universe’ wonders, why entropy, so intimately associated with time, should be expressed quantitatively in terms of temperature instead
of time. The present writer wondered about this too, for three consecutive years, during which time it was his good fortune to be in the very midst of multitudinous entropy calculations under the
direction and guidance of G. N. Lewis. His first attempt at expressing his wonderment in mathematical form appeared in a rather humble periodical of approximately zero circulation. No wonder
Eddington makes no reference to it. It was the Eureka Junior College Journal of Science, Arts and Crafts (1921).”
I have thoroughly attempted fetching the above-mentioned work by G. A. Linhart, but really in vain, it seems to be completely lost for us. Fortunately, his preprint [11], which has been made
available to me by courtesy of the colleagues in the Riverside Junior College, gives a full and detailed account of his bright ideas.
G. A. Linhart was considering general non-cyclic processes unidirectional in time and starts out with two ideas: progress and hindrance, which should underlie any process under study. What is the
essence of these both? G. A. Linhart answered:
“By progress is meant any unidirectional phenomenon in nature, such as the growth of a plant or an animal, and by hindrance—the contesting and ultimate limitation of every step of the progress. In
other words, progress is organized effort in a unidirectional motion, and hindrance is not so much the rendering of energy unavailable for that motion, as it is the disorganization of the effort to
move; it acts as a sort of stumbling block in every walk of life. It is this property of matter which the writer wishes to measure quantitatively in relation to time, and which he designates as
chronodynamic entropy.”
Hence, for us, the modern readers, G. A. Linhart insisted on the dialectical viewing of any physical process, with its initial and final stages being in general different from each other. Indeed,
there always ought to be some driving force due to the transition of energy from one state to another, according to the 1^st Law of Thermodynamics (1LT), which ensures and entails the progress. On
the other hand, nothing all over the world would happen without the omnipresent hindrance, whose name is entropy, which is but nothing else than the genuine 2LT statement. The true dialectics of
these both lies not only in the “universal competition between energy and entropy” [12,13], but also in their mutual compensation (cf., for example, [14-18] and references therein). Without
intervention of the entropic hindrance no process would ever reach its final state, because the progress would then last forever. Thus, in fact, Linhart conceives entropy as just a kind of
“Mephisto”, whom J. W. von Goethe characterizes as “ein Teil von jener Kraft, die stets das Böse will und stets das Gute schafft” (“Part of that Power which would the Evil ever do, and ever does the
Good” [transl. by G. M. Priest]). To this end, entropy represents solely one integral part of the “Yin-Yang”— tandem of thermodynamics [19], that is, the dialectic “energy-entropy” or “1LT-2LT” dyad:
the entropy should not be considered separately from the energy and absolutized, as done conventionally. Furthermore, because any process is a result of the above-mentioned “unity and struggle of
opposites”, we may never invoke any guarantee that such processes would perfectly come to their final states as could be envisaged by theories. In fact, already the latter state of affairs should
introduce the intrinsic fundamental indeterminacy in the sense of statistical causation of the processes [20]: in other words, any unidirectional process ought to require involvement of an essential
probabilistic element, even without the conventional application to the atomistic build-up of the matter.
In accordance with all the above, G. A. Linhart’s mathematical proposal is seemingly very simple: let us take the conventional definition of thermodynamic entropy,
First of all, it is important to discuss the physical and mathematical lawfulness of such a substitution. Interestingly, an analogous trick has recently been independently and tacitly employed by
Grisha Perelman to transform the “entropy-like” functional in his award-winning proof of the famous Poincaré conjecture, as a part of the full geometrization conjecture [21].
For the above purpose it would be suitable to use the foundational work by Caratheodory (In effect, very similar mathematical results on the foundations of thermodynamics were obtained by Gyula
Farkas years before Caratheordory, but the Farkas’ work remained unnoticed, probably because of its extraordinary terseness [22]). Although, strictly speaking, the original Caratheodory’s inferences
are only applicable to “simple” systems pertaining to thermodynamically equilibrium situations, and notwithstanding the proven intrinsically local character of Caratheodory’s theorem [23,24], it was
shown [25,26] that the latter could be reasonably extended to define entropy, temperature etc. for rather generic non-equilibrium and irreversible cases. (Meanwhile, the works [25,26] seem to be
forgotten, like those by G. A. Linhart, in spite of their immense significance for the foundations of thermodynamics.) Meanwhile, the Lieb-Yngvason foundational work [27] is of little help for our
present task, because it is placing the notion of entropy at the center of all the thermodynamical inferences.
To this end, if we assume the existence of the internal energy and the validity of the 1LT for generic processes, then the existence of entropy (and thus the generic validity of the 2LT) will follow
as a direct consequence of the integrability of the 1LT differential form, to the effect that in the neighborhood of a thermodynamic state other states exist which are not accessible by reversible
and adiabatic processes. Then, the entropy can be defined as
Now, we have approached the next important question: What is the sense and the use of the temperature-to-time substitution in the expression of entropy?
To answer this question, G. A. Linhart considers his theory based on the example of generic processes of growth, but underlines that such an approach is in fact exceedingly general. He starts out
from the idea that the rate of spending the energy to promote the growth process ought to be proportional to the parameter measuring the progress of the process in question:
where E is energy, G stands for the mass of the growing body at some time t, R is the proportionality coefficient between the mass and the energy (in general, between the “driving force” and the
“measurable progress”). Then, Q is the analogue of the thermodynamical amount of heat, which is necessary to define the chronodynamic entropy,
On the other hand, owing to the dialectic interplay between the progress and hindrance, we may consider the ratio[i] being the hypothetical maximum mass achievable by the growing body, as a
probability that the latter will arrive at the mass G at some time t. Accordingly, the probability of the disjoint event, i.e., that the growing body will not arrive at this value of mass, is
“The question might be asked: ‘What is the degree of hindrance in any natural process before it occurs?’ Obviously, there can be no hindrance to anything that does not exist. But at the instance of
inception of the process hindrance sets in and continues to increase until ultimately it checks nearly all progress, and reduces to a minimum the chance of any further advance. At this juncture the
outcome of the process is said to have approximately attained its maximum. It is clear then, that the increase in mass is in the same direction as the increase in hindrance, or entropy.”
To this end, G. A. Linhart mathematically arrives at the following expression for the ratio of the progress and hindrance infinitesimal increments:
where the proportionality factor K is the efficiency constant of the process (G. A. Linhart’s reasoning here is as follows: “… the smaller the value of K, the greater the hindrance … and the slimmer
the chance for the growing individual to survive.”). The ratio
Hereafter, G. A. Linhart just employs the trivial straightforward mathematics. Indeed, combining Equations (3) and (4), the rate of growth progress can be expressed as
which on integration can be recast as:
where t[s] is the time scale when measuring the time during the process under study, so that x is the relative time.
The integrated form of Equation (4) is then:
which is nothing else, but the famous Boltzmann expression for the entropy, when we take into account the probabilistic interpretation of the ratio under the logarithm sign, whereas the final aim of
the G. A. Linhart’s theory, the expression of the chronodynamic entropy vs. time can be cast as follows, by combining Equations (3) and (7) and integrating the result:
The immense significance of the simple and straightforward inference embodied in Equations (2)-(9) consists in that G. A. Linhart has succeeded to mathematically derive the Boltzmann’s expression for
entropy, starting from the general dialectic (“Yin-Yang”) principle. This is definitely a great advance in comparison with some attempts to derive the same formula starting from a number of purely
mathematical axioms [29,30]. The full correspondence of Equations (8) and (9) to the Boltzmann’s ingenious guess completely justifies somewhat artificialand haphazard-looking linear approximation
employed by G. A. Linhart in Equation (4).
Of considerable interest is Equation (6), which was not discussed by Linhart, but enables a unique interpretation of the nature of time. Specifically, we recast this equation in the following way,
bearing in mind Equation (7):
Equation (10) reveals that the relative time x, power the efficiency constant K, is nothing else but the odds in favor of the progress in the general growth process. The probability that the growing
body will achieve the required mass, P, is hence defined via Dutch book argument, so that the G. A. Linhart’s theory is firmly based upon the Bayesian epistemology. A huge advance here is achieved by
the fact that both probabilities and the odds are measurable in experiment, so that, we do not need any additional theoretical or computer modeling, unlike in the conventional statistical mechanics.
Advocating the above approach, G. A. Linhart has definitely outdistanced his time.
The interpretation of time as the “odds in favor of the progress” may sound extremely strange to us, yet it ought to contain a profound philosophic sense. Specifically, there is a definite parallel
to the concept of time delivered by the ancient Chinese book “I-Ching”, “The Book of Change” (see, for example, [31]). In fact, the I-Ching concept of the time, si/shi/ji (in Korean/Chinese/
Japanese), tells us, on the one hand, that “the time is something like the life force, current or pulse of a given set of circumstances”, thus constituting an eternally present, all-pervasive and
decisive element. Still, in the real life, there are nevertheless certain situations, when other factors gain such weight and prominence that they completely overshadow even the latter significance
of time. This is why, the text of I-Ching uses everywhere the notion of “time” more readily in its ancient form, that is, signifying “season”, or, in other words, a kind of “epoch” of the year,
hence many of the qualities attributed to it come out of this rather “seasonal/agricultural” interpretation of the term. Hence, it is often said, for example: “The seasons are wrong”, so that the
active/superior man/woman accepts them as a model for his/her own conduct and actions, where he/she derives two of the most fundamental characteristics of the eternal/perpetual “heaven’s and earth’s
struggle between each other”, and deduces that the first one of these characteristics is incessant change, whereas the second one is the consequent (immutable) relationships this incessant change
creates. In effect, such an interpretation leaves some definite room for considering time from the probabilistic standpoint.
With the above interpretational scheme in mind, conceiving any intensive variable as the “odds in favor of the progress” should in effect represent the general approach by G. A. Linhart, the power of
which he could also demonstrate by deriving his very simple and universal formula for the heat capacity vs. temperature, which is clearly outperforming even the conventional Debye’s theory (for the
thorough discussion of those G. A. Linhart’s works see, for example, [32]). Such a generality claim of the G. A. Linhart’s approach can further be supported by substituting the time t in Equations
(2)-(7) by some other intensive variable, for example, by the concentration c. With this in mind and following the G. A. Linhart’s line of thought, we can formally-mathematically derive the famous
empirical equation ingeniously guessed by A. V. Hill [33] more than 100 years ago, and widely used in biochemistry, as well as in pharmacy till now, still causing intensive discussions (see, for
example, [34-36]), to try rationalizing the physicalchemical sense of the Hill equation coefficients this way.
Indeed, we can successfully use the idea by G. A. Linhart to describe the process as the “dialectic progresshindrance interplay”, when some ligand molecules are binding to some proper sites of a
macromolecule. We are herewith trying to describe the fraction of the macromolecule’s binding sites saturated by ligand as a function of the ligand concentration. Then, we employ Equations (2)-(4),
with the only intensive variable now being not the time, but c, the free (unbound) ligand concentration, so that the magnitude S in Equaiton (3) is now “concentration-dynamic entropy” describing all
the possible hindrances to the ligand binding process, whereas K in Equation (4) can be seen in this case as the “coefficient of ligand binding efficiency”. With this in mind, the ratio[s] is nothing
else than the conventional magnitude of the
3. The Ideas of George Augustus Linhart as a Specific Development of Gilbert Newton Lewis’ Lifework
It is of considerable interest to reveal the interconnection between the ideas of Gilbert Newton Lewis and George Augustus Linhart. Gilbert Newton Lewis is very well known for his reformulation of
chemical thermodynamics in the mathematically rigorous, but nevertheless readily understandable language (although G. N. Lewis’ candidacy was many times nominated for the Nobel prize award, he could
never receive it). In his book [37] G. N. Lewis had thoroughly analyzed the true meanings of the 1LT and the 2LT, as well as those of the thermodynamic equilibrium, energy and entropy.
Indeed, in the G. N. Lewis’ book chapter entitled “The Power and the Limitations of Thermodynamics” we read:
“Our book may be introduced by the very words used by Le Chatelier a generation ago: ‘These investigations of a rather theoretical sort are capable of much more immediate practical application than
one would be inclined to believe. Indeed the phenomena of chemical equilibrium play a capital role in all operations of industrial chemistry’. He continues: ‘Unfortunately there has been such an
abuse of the applications of thermodynamics that it is in discredit among the experimenters. If this was true when written by Le Chatelier it is no less true today. The widespread prejudice against
any practical use of thermodynamics in chemistry is not without reason, for the propagandists of modern physical chemistry have at times shown more zeal than scientific caution.’ ”
As to the limitations of thermodynamics, G. N. Lewis had communicated the following thoughts: “The thermodynamics tells us the minimum amount of work necessary for a certain process, but the amount
which will actually be used will depend upon many circumstances. Likewise thermodynamics shows us whether a certain reaction may proceed, and what maximum yield may be obtained, but gives no
information as to the time required.” And this is just the logical point where G. A. Linhart had started to work out his “chronodynamical addition” to the conventional thermodynamical theory.
As for the thermodynamic equilibrium, G. N. Lewis had written: “As a science grows more exact it becomes possible to employ more extensively the accurate and concise methods and notation of
mathematics. At the same time it becomes desirable, and indeed necessary, to use words in a more precise sense. For example if we are to speak, in the course of this work, of a pure substance, or of
a homogeneous substance, these words must convey as nearly as possible the same meaning to writer and to reader. Unfortunately it is seldom possible to satisfy this need by means of formal
definitions; partly because the most fundamental concepts are the least definable, partly because of the inadequacy of language itself, but more particularly because we often wish to distinguish
between things which differ rather in degree than in kind. Frequently therefore our definitions serve to divide for our convenience a continuous field into more or less arbitrary regions, —as a map
of Europe shows roughly the main ethnographic and cultural divisions, although the actual boundaries are often determined by chance or by political expediency. The distinction between a solid and a
liquid is a useful one, but no one would attempt to fix the exact temperature at which sealing-wax or glass passes from the solid to the liquid state. Any attempt to make the distinction precise,
makes it the more arbitrary.”
Then, G. N. Lewis was trying to define the notion of the thermodynamic equilibrium and had noted, as follows: “If it were possible to know all the details of the internal constitution of a system, in
other words, if it were possible to find the distribution, the arrangement, and the modes of motion of all the ultimate particles of which it is composed, this great body of information would serve
to define what may be called the microscopic state of the system, and this microscopic state would determine in all minutiae the properties of the system. We possess no such knowledge, and in
thermodynamic considerations we adopt the converse method. The state of a system (macroscopic state) is determined by its properties, just in so far as these properties can be investigated directly
or indirectly by experiment. We may therefore regard the state of a substance as adequately described when all its properties, which are of interest in a thermodynamic treatment, are fixed with
definiteness commensurate with the accuracy of our experimental methods. Let us quote from Gibbs in this connection: ‘So when gases of different kinds are mixed, if we ask what changes in external
bodies are necessary to bring the system to its original state, we do not mean a state in which each particle shall occupy more or less exactly the same position as at some previous epoch, but only a
state which shall be undistinguishable from the previous one in its sensible properties. It is to states of systems thus incompletely defined that the problems of thermodynamics relate.’
The properties of a substance describe its present state and do not give a record of its previous history. When we determine the property of hardness in a piece of steel we are not interested in the
previous treatment which produced this degree of hardness. If the metal has been subjected to mechanical treatment, the work which has been expended upon it is not a property of the steel, but its
final volume is such a property. It is an obvious but highly important corollary of this definition that, when a system is considered in two different states, the difference in volume or in any other
property, between the two states, depends solely upon those states themselves, and not upon the manner in which the system may pass from one state to the other.
Most of the properties which we measure quantitatively may be divided into two classes. If we consider two identical systems, let us say two kilogram weights of brass, or two exactly similar balloons
of hydrogen, the volume, or the internal energy, or the mass of the two is double that of each one. Such properties are called extensive. On the other hand, the temperature of the two identical
objects is the same as that of either one, and this is also true of the pressure and the density. Properties of this type are called intensive. They are often derived from the extensive properties;
thus, while mass and volume are both extensive, the density, which is mass per unit volume, and the specific volume, which is volume per unit mass, are intensive properties. These intensive
properties are the ones which describe the specific characteristics of a substance in a given state, for they are independent of the amount of substance considered. Indeed, in common usage it is only
these intensive properties which are meant when the properties of a substance are being described.”
This line of thoughts had led G. N. Lewis to the following definition of the thermodynamic equilibrium: “When a system is in such a state that after any slight temporary disturbance of external
conditions it returns rapidly or slowly to the initial state, this state is said to be one of equilibrium. A state of equilibrium is a state of rest.” Again, there clearly remains some indeterminacy
with “rapidly or slowly”. And here is apparently just the logical point, from where the G. A. Linhart’s line of thoughts started. Indeed, G. A. Linhart suggested to treat the time just as one of the
intensive thermodynamical variables, like, for example, the temperature. Whereas, G. N. Lewis had come similarly to the definition of a Partial Equilibrium, Degrees of Stability: “Of the various
possible processes which may occur within a system, some may take place with extreme slowness, others with great rapidity. Hence we may speak of equilibrium with respect to the latter processes
before the system has reached equilibrium with respect to all the possible processes.” Then, G. N. Lewis had underlined the notion of the “Equilibrium as a Macroscopic State”. Specifically, he noted
what he means as follows: “Even here it is desirable to emphasize that by a state of rest, or equilibrium, we mean a state in which the properties of a system, as experimentally measured, would
suffer no further observable change even after the lapse of an indefinite period of time. It is not intimated that the individual particles are unchanging.” But still, there is another very
interesting fragment in the book of G. N. Lewis, namely in the chapter devoted to the thermodynamical equilibrium. Specifically, G. N. Lewis stated: “In practice we often assume the existence of
several such equilibrium states toward which a system may tend, all these states being stable, but representing higher or lower degrees of stability. From a theoretical standpoint it might be doubted
whether there is any condition of real equilibrium, with respect to every conceivable process, except the one which represents the most stable state. This, however, is not a question which need
concern us greatly, nor is it one which we could discuss adequately at this point, without largely anticipating what we shall later have to say regarding the statistical view of thermodynamics.”
Apparently, here is just what had led G. A. Linhart to his implicit idea of a “fuzzy equilibrium” (In the sense of interplay between probabilities and possibilities, although the work of Zade [38]
was not published that time as yet!), that is, some degree of equilibrium, instead of the conventional (even till nowadays!) “crisp” “equilibrium/ non-equilibrium” classification.
Concerning the First Law of Thermodynamics (1LT), G. N. Lewis had expressed the following ideas: “So, as science has progressed, it has been necessary to invent other forms of energy, and indeed an
unfriendly critic might claim, with some reason, that the law of conservation of energy is true because we make it true, by assuming the existence of forms of energy for which there is no other
justification than the desire to retain energy as a conservative quantity. This is indeed true in a certain sense, as shown by the explanations which have been given for the enormous, and at first
sight apparently limitless, energy emitted by radium. But a study of this very case has shown the power and the value of the conservation law in the classification and the comprehension of new
phenomena. It should be understood that the law of conservation of energy implies more than the mere statement that energy is a quantity which is constant in amount. It implies that energy may be
likened to an indestructible and uncreatable fluid which cannot enter a given system except from or through surrounding systems. In other words, it would not satisfy the conservation law if one
system were to lose energy, and another system, at a distance therefrom, were simultaneously to gain energy in the same amount. If a system gains or loses energy, the immediate surroundings must lose
or gain energy in the same amount, and energy may be said to flow into or out of the system through its boundaries. The energy contained within a system, or its internal energy, is a property of the
system. The increase in such energy when a system changes from state A to state B is independent of the way in which the change is brought about. It is simply the difference between the final and the
initial energy. It is, however, of much theoretical interest to note that the great discovery of Einstein embodied in the principle of relativity, shows us that every gain or loss of energy by a
system is accompanied by a corresponding and proportional gain or loss in mass, and therefore presumably that the total energy of any system is measured merely by its mass. In other words, mass and
energy are different measures of the same thing, expressed in different units; and the law of conservation of energy is but another form of the law of conservation of mass.” And here is again just
the starting logical point for the idea of G. A. Linhart, that any unidirectional progress in any physical system ought to be paid by a change of the one form of energy to its other form.
The change of one form of energy to another one can be caused and measured by changes in the intensive variables and G. N. Lewis stated in this connection as follows: “There are two intensive
properties, pressure and temperature, which play an important role in thermodynamics, since they largely affect, and often completely determine, the state of a system. Pressure is too familiar an
idea to require definition; it has the dimen-sions of force per unit area, and therefore pressure times volume has the dimensions of energy (force times distance). The concept of temperature is a
little more subtle. When one system loses energy to another by thermal conduction or by the emission of radiant energy, there is said to be a flow of heat, a thermal flow. The consideration of such
cases leads immediately to the concept of temperature, which may be qualitatively denned as follows: if there can be no thermal flow from one body to another, the two bodies are at the same
temperature; but if one can lose energy to the other by thermal flow, the temperature of the former is the greater. This establishment of a qualitative temperature scale is obviously more than a
definition. It involves a fundamental principle, to which we have already given preliminary expression in discussing equilibrium, but which we are not yet ready to put in a general and final form.
For thermal flow, this principle requires that if A can lose energy to B, B cannot lose it to A; if A can lose to B, and B can lose to C, C cannot lose to A. As in our general discussion of
equilibrium, it must be understood that we are dealing with net gains or losses in energy. We do not mean that no energy passes from a cold body to a hot, but only that the amount so transferred is
always less than that simultaneously transferred from hot to cold. When we have established the qualitative laws of temperature, we still have a wide freedom of choice in fixing the quantitative
scale. Indeed, temperature, as ordinarily measured, or its square, or its logarithm, would equally satisfy these quailtative requirements.” And this is where G. N. Lewis had apparently stopped, but
G. A. Linhart had gone even further, in considering also the time as an intensive thermo-dynamical variable.
Then, G. N. Lewis was discussing the notions of “heat” and “work” as the thermodynamically valid forms of the energy change: “There are two terms, ‘heat’ and ‘work’, that have played an important
part in the development of thermodynamics, but their use has often brought an element of vagueness into a science which is capable of the greatest precision. For our present purpose we may say that
when a system loses energy by radiation or thermal conduction it is giving up heat; and that when it loses energy by other methods, usually by operating against external mechanical forces, it is
doing work. According to the law of the conservation of energy, any system in a given condition contains a definite quantity of energy, and when this system undergoes change, any gain or loss in its
internal energy is equal to the loss or gain in the energy of surrounding systems. In any physical or chemical process, the increase in energy of a given system is therefore equal to the heat
absorbed from the surroundings, less the work done by the system upon the surroundings. The values of heat and work depend upon the way in which the process is carried out, and in general neither is
uniquely determined by the initial and final states of the system. However, their difference is determined, so that if either heat or work is fixed by the conditions under which the process occurs,
the other is also fixed. Thus, where the work done by the system is the work of expansion against an external pressure, the expansion may be carried out in such a manner that no heat enters or leaves
the system.” After such deliberations, G. N. Lewis went on with discussing the notions of heat content, heat capacity as well as the units of measuring different forms of energy.
The above-sketched way of thoughts had then led G. N. Lewis to the thorough discussion of the Second Law of Thermodynamics (2LT) and the notion of entropy. The 2LT chapter of his book contains the
following deep thoughts: “After the extremely practical considerations in the preceding chapters, we now turn to a concept of which neither the practical significance nor the theoretical import can
be fully comprehended without a brief excursion into the fundamental philosophy of science. Clausius summed up the findings of thermodynamics in the statement, ‘die Energie der Welt ist konstant; die
Entropie der Welt strebt einem Maximum zu’, and it was this quotation which headed the great memoir of Gibbs on ‘The Equilibrium of Heterogeneous Substances’. What is this entropy, which such masters
have placed in a position of coordinate importance with energy, but which has proved a bugbear to so many a student of thermodynamics? The first law of thermodynamics, or the law of conservation of
energy, was universally accepted almost as soon as it was stated; not because the experimental evidence in its favor was at that time overwhelming, but rather because it appeared reasonable, and in
accord with human intuition. The concept of the permanence of things is one which is possessed by all. It has even been extended from the material to the spiritual world. The idea that, even if
objects are destroyed, their substance is in some way preserved, has been handed down to us by the ancients, and in modern science the utility of such a mode of thought has been fully appreciated.
The recognition of the conservation of carbon permits us to follow, at least in thought, the course of this element when coal is burned and the resulting carbon dioxide is absorbed by living plants,
whence the carbon passes through an unending series of complex transformations.”
After such a remark, G. N. Lewis had thoroughly discussed the philosophical problems connected with the 2LT: “The second law of thermodynamics, which is known also as the law of the dissipation or
degradation of energy, or the law of the increase of entropy, was developed almost simultaneously with the first law through the fundamental work of Carnot, Clausius and Kelvin. But it met with a
different fate, for it seemed in no recognizable way to accord with existing thought and prejudice. The various laws of conservation had been foreshadowed long before their acceptance into the body
of scientific thought. The second law came as a new thing, alien to traditional thought, with far-reaching implications in general cosmology. Because the second law seemed alien to the intuition, and
even abhorrent to the philosophy of the times, many attempts were made to find exceptions to this law, and thus to disprove its universal validity. But such attempts have served rather to convince
the incredulous, and to establish the second law of thermodynamics as one of the foundations of modern science. In this process we have become reconciled to its philosophical implications, or have
learned to interpret them to our satisfaction; we have learned its limitations, or better we have learned to state the law in such a form that these limitations appear no longer to exist; and
especially we have learned its correlation with other familiar concepts, so that now it no longer stands as a thing apart, but rather as a natural consequence of long familiar ideas.”
And then, G. N. Lewis had shown one of the possible ways to solve all the philosophical discrepancies introduced by the 2LT, which is till nowadays remaining the common way of thoughts. Specifically,
he introduces the “Preliminary Statement of the Second Law”, by defining “The Actual or Irreversible Process”. Hence, “The second law of thermodynamics may be stated in a great variety of ways. We
shall reserve until later our attempt to offer a statement of this law which is free from every limitation, and shall confine ourselves for the present to a discussion of the law sufficient to
display its character and content. Indeed in an early chapter we have already announced the essential feature of the second law when we stated that every system left to itself changes, rapidly or
slowly, in such a way as to approach a definite final state of rest. This state of rest (defined in a statistical way) we also called the state of equilibrium. Now since it is a universal postulate
of all natural science that a system, under given circumstances, will behave in one and only one way, it is a corollary that no system, except through the influence of external agencies, will change
in the opposite direction, that is, away from the state of equilibrium.” And here, one large problem can be seen immediately, namely, the really fruitful idea of the “partial equilibrium” introduced
by G. N. Lewis, as we have already discussed earlier, and G. A. Linhart had used for some further theoretical developments, didn’t find any of its possible applications. But, nevertheless, this is in
full accord with all the further developments of thermodynamics (all the well-known theories of irreversible/non-equilibrium thermodynamics, and so on, so forth).
In accordance with this, G. N. Lewis continued to further develop the above-mentioned logically incomplete representation in his book. Specifically, he stated: “Before proceeding to a more exact
characterization of the second law, let us make sure that there is no misunderstanding of its qualitative significance. When we say that heat naturally passes from a hot to a cold body, we mean that,
in the absence of other processes which may complicate, this is the process which inevitably occurs. It is true that by means of a refrigerating machine we may further cool a cold body by
transferring heat from it to its warmer surroundings, but here we are in the presence of another dissipative process proceeding in the engine itself. If we include the engine within our system, the
whole is moving always toward the condition of equilibrium. A system already in thermal equilibrium may develop large differences of temperature through the occurrence of some chemical reaction, but
all such phenomena are but eddies in the general unidirectional flow toward a final state of rest. The essential content of the second law might be given by the statement that when any actual process
occurs it is impossible to invent a means of restoring every system concerned to its original condition. Therefore, in a technical sense, any actual process is said to be irreversible.” Thus, he had
come to the notion of “The Ideal or Reversible Process”. Indeed, G. N. Lewis continued as follows: “When we speak of an actual process as being always irreversible we have had in mind a distinction
between such a process and an ideal process which, although never occurring in nature, is nevertheless imaginable. Such an ideal process, which we will call reversible, is one in which all friction,
electrical resistance, or other such sources of dissipation are eliminated. It is to be regarded as a limit of actually realizable processes. Let us imagine a process so conducted that at every stage
an infinitesimal change in the external conditions would cause a reversal in the direction of the process; or, in other words, that every step is characterized by a state of balance. Evidently a
system which has undergone such a process can be restored to its initial state without more than infinitesimal changes in external systems. It is in this sense that such an imaginary process is
called reversible.”
Therefore, G. N. Lewis had logically arrived at the task of defining the “Quantitative Measure of Degradation”, that is, the “Quantitative Measure of Irreversibility”. And he described his solution
to this very important problem, thus coming to the notion of entropy, as follows: “In viewing the reversible process as the limit toward which actual processes may be made to approach indefinitely,
it is implied that processes differ from one another in their degree of irreversibility. It is of the utmost importance to establish a quantitative measure of this degree of irreversibility, or this
degree of degradation. So far we have not given a name to our measure of the ireversibility of the standard process. The value of heat-totemperature ratio, when this process occurs, we shall call the
increase in entropy. Thus, entropy has the same dimensions as heat capacity. Our present definition of entropy will be found identical with the definition originally given by Clausius. We have,
however, departed radically from the traditional method of presenting this idea, for we have desired to emphasize the fact that the concept of entropy, as a quantity which is always increasing in all
natural phenomena, is based upon our recognition of the unidirectional flow of all systems toward the final state of equilibrium. In the ordinary definition of entropy the attention is focused upon
the reversible process and not upon the irreversible process, the existence of which necessitates the entropy concept. For this reason we have based our definition immediately upon an irreversible
process, and shall now employ the reversible process only as a means of comparing the degree of degradation, or the increase in entropy, of two irreversible processes.”
Keeping in mind the above deliberations, G. N. Lewis had underlined, that the entropy, thus defined, is an “Extensive Property”. And he stated in this connection: “In expressing the entropy change
during an irreversible process as the difference between the entropy at the end and the entropy at the beginning, we have implied that entropy is a property, and therefore that the entropy change
depends solely upon the initial and final states. Indeed this follows directly from our definition, for by whatever irreversible path we proceed from state A to state B, the minimum degradation of
the spring-reservoir-system necessary for the return from state B to state A is the same. It is true that we have not shown how to obtain the absolute entropy value of S[B] or S[A], but only their
difference. In the meantime we shall regard the entropy, like the energy and heat content, as a quantity of which the absolute magnitude is undetermined. Moreover, entropy is an extensive property,
for we may consider two systems which are just alike, and each of which undergoes the same infinitesimal irreversible process; evidently the change in the standard spring-reservoir-system necessary
for their restoration is twice as great as it would be for one of them alone. Since entropy is extensive, we may regard the entropy of a system as equal to the sum of the entropies of its parts. It
is therefore important to ascertain how to determine the localization of entropies in the various parts of a system. Owing to the special properties of the standard spring-reservoir-system which we
assumed at the outset, it will be convenient to postulate that in any operation of the spring-reservoirsystem the entropy changes occur in the reservoir alone, so that if the standard reservoir gains
heat from any source by the amount q at the temperature T, the reservoir changes in entropy by the ratio of q/T.” But then G. N. Lewis had concluded: “We have seen that the total entropy change in a
reversible process is zero. It follows that in such a process the entropy change in any system must be equal and opposite in sign to the entropy change in all other systems involved. In order to
study this case further, let us consider the energy changes which occur in a reversible process between some system and the standard spring-reservoir. For the sake of simplicity we shall choose an
infinitesimal process. We may sum up our quantitative conclusions regarding entropy. In any irreversible process the total entropy of all systems concerned is increased. In a reversible process the
total increase in entropy of all systems is zero, while the increase in the entropy of any individual system, or part of a system, is equal to the heat which it absorbs divided by its absolute
temperature. It is important to see clearly that the idea of entropy is necessitated by the existence of irreversible processes; it is only for the purpose of convenient measurement of entropy
changes that we have discussed reversible processes here.” After presenting all the above thoughts, G. N. Lewis continued, nevertheless, with the reflections about the interconnection between the
entropy and probability, for he was apparently feeling the logical deficiencies of the above-sketched 2LT interpretation. Specifically, he wrote: “The second law of thermodynamics is not only a
principle of wide-reaching scope and application, but also it is one which has never failed to satisfy the severest test of experiment. The numerous quantitative relations derived from this law have
been subjected to more and more accurate experimental investigation without detection of the slightest inaccuracy. Nevertheless, if we submit the second law to a rigorous logical test, we are forced
to admit that, as it is ordinarily stated, it cannot be universally true. It was Maxwell who first showed the consequences of admitting the possible existence of a being who could observe and
discriminate between the individual molecules. This creature, usually known as Maxwell’s demon, was supposed to stand at the gateway between two enclosures containing the same gas at the same
original temperature. If now he were able, by openings and shuttings the gate at will, to permit only rapidly moving molecules to enter one enclosure and only slowly moving molecules to enter the
other, the result would ultimately be that the temperature would increase in one enclosure and would decrease in the other. Or, again, we could assume the enclosures filled with air, and the demon
operating the gate to permit only oxygen molecules to pass in one direction and only nitrogen molecules in the other, so that ultimately the oxygen and nitrogen would be completely separated. Each of
these changes is in a direction opposite to that in which a change normally occurs, and each is therefore associated with a diminution in entropy. Of course even in this hypothetical case one might
maintain the law of entropy increase by asserting an increase of entropy within the demon, more than sufficient to compensate for the decrease in question. Before conceding this point it might be
well to know something more of the demon’s metabolism. Indeed a suggestion of Helmholtz raises a serious scientific question of this character. He inquires whether micro-organisms may not possess the
faculty of choice which characterizes the hypothetical demon of Maxwell. If so, it is conceivable that systems might be found in which these micro-organisms would produce chemical reactions where the
entropy of the whole system, including the substances of the organisms themselves, would diminish. Such systems have not as yet been discovered, but it would be dogmatic to assert that they do not
exist. While in Maxwell’s time it seemed necessary to ascribe demoniacal powers to a being capable of observing molecular motions, we now recognize that the Brownian movement, which is readily
observable under the microscope, is in reality thermal motion of large molecules. It would therefore seem possible, by an extraordinarily delicate mechanism in the hands of a careful experimenter, to
obtain minute departures from the second law, as ordinarily stated. But here also we should depend upon a conscious choice exercised by the experimenter. It would carry us altogether too far from our
subject to take part in the long-continued debate on the subject of vitalism; the vitalists holding that there are certain properties of living matter which are not possessed at all by inanimate
things, or, in other words, that there is a difference in kind between the animate and the inanimate. However, we may point out that in the lasj analysis differences of kind are often reduced to
differences in degree. There certainly can be no question as to the great difference in trend which exists between the living organism, and matter devoid of life. The trend of ordinary systems is
toward simplification, toward a certain monotony of form and substance; while living organisms are characterized by continued differentiation, by the evolution of greater and greater complexity of
physical and chemical structure. In the brilliant investigation of Pasteur on asymmetric or optically active substances, it was shown that a system of optically inactive ingredients never develops
optically active sub stances except through the agency of living organisms, or as the result of the conscious choice of an experimenter.” And then G. N. Lewis concluded: “Sometimes when a phenomenon
is so complex as to elude direct analysis, whether it concern the life and death of a human being, or the toss of a coin, it is possible to apply methods which are called statistical. Thus tables and
formulae have been developed for predicting human mortality and for predicting the results of various games of chance, and such methods are applied with the highest degree of success. It is true that
in a given community the ‘expectation of life’ may be largely and permanently increased by sanitary improvements, but if a great many individual cases be taken promiscuously from different localities
at different times, the mean duration of life, or the average deviation from this mean, becomes more and more nearly constant the greater the number of cases so chosen. Likewise it is conceivable
that a person might become so expert in tossing a coin as to bring heads or tails at will, but if we eliminate the possibility of conscious choice on the part of the player, the ratio of heads to
tails approaches a constant value as the number of throws increases. The distinction between the energy of ordered motion and the energy of unordered motion is precisely the distinction which we have
already attempted to make between energy classified as work and energy classified as heat. Our present view of the relation between entropy and probability we owe largely to the work of Boltzmann,
who, however, himself ascribed the fundamental idea to Gibbs, quoting, ‘The impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability.’ It would carry us too far
if we should attempt to analyze more fully this idea that the increase in the entropy of a system through processes of degradation merely means a constant change to states of higher and higher
In analyzing the course of thoughts of Maxwell, Gibbs and Boltzmann in this direction, G. N. Lewis had still come back to his own reflections about the role of irreversible processes in the
thermodynamical theory. He continued as follows: “The mere recognition that such a relationship exists suffices to give a new and larger conception of the meaning of an irreversible process and the
significance of the second law of thermodynamics. If we regard every irreversible process as one in which the system is seeking a condition of higher probability, we cannot say that it is inevitable
that the system will pass from a certain state to a certain other state. If the system is one involving a few molecules, we can only assert that on the average certain things will happen. But as we
consider systems containing more and more molecules we come nearer and nearer to complete certainty that a system left to itself will approach a condition of unit probability with respect to the
various processes which are possible in that system. This final condition is the one which we know as equilibrium. In other words, the system approaches a thermodynamic or macroscopic state, which
represents a great group of microscopic states that are not experimentally distinguishable from one another. With an infinite number of molecules, or with any number of molecules taken at an infinite
number of different times, the probability that the macroscopic state of the system will lie within this group is infinitely greater than the probability that it will lie outside of that group.
Leaving out of consideration systems, if such there be, which possess that element of selection or choice that may be a characteristic of animate things, we are now in a position to state the second
law of thermodynamics in its most general form: Every system which is left to itself will, on the average, change toward a condition of maximum probability. This law, which is true for average
changes in any system, is also true for any changes in a system of many molecules. We have thought it advisable to present in an elementary way the ideas touched upon, in order to give a more vivid
picture of the nature of an irreversible process and a deeper insight into the meaning of entropy. It is true we shall not, henceforth, make formal use of the relation between entropy and
probability; nevertheless, we shall always tacitly assume that we are dealing with statistical ideas. For example, when calculating solubilities or vapor pressures.” Therefore, G. N. Lewis was
apparently aware of that the problem to give some really valid definitions of the 2LT and entropy had not been satisfactorily solved as yet. Well, and this seems to be just the logical point where G.
A. Linhart had started his great work.
Hence, to my mind, the interpretation suggested by G. A. Linhart is in fact very strong, for he used the dialectical viewpoint on every process, that is, on the 2LT itself, because he considered the
energy change as the measure of “progress”, whereas the entropy as the measure of “hindrance”. Along with this, Linhart’s standpoint could include the formal mathematical proof of the intercomnection
between the entropy and probability (as ingeniously guessed by Maxwell, Boltzmann and Gibbs), clearly avoiding all the principal theoretical complications connected with combining the notions of
reversibility and irreversibility, as well as the apparent logical difficulties connected with the notion of the thermodynamic equilibrium, as was surely recognized (But still not carefully thought
over!) by G. N. Lewis himself.
4. The Relationship between the Ideas of G. A. Linhart and Other Viable Solutions to the 2LT Problem
The 2LT problem was attracting and still attracts attention of many scientists in the field.
And here is how the problem is widely treated till nowadays: “The development of material world toward complexity and increasing natural diversity violates the second law of thermodynamics and makes
it necessary to investigate non-equilibrium processes that may give rise to orderlyness. Now these problems are considered by synergetics [39].” Hermann Haken, one of the founders of the synergetics,
had expressed it as follows: “In physics, there is a notion of ‘concerted effects’; however, it is applied mainly to the systems in thermal equilibrium. I felt that I should introduce a term for
consistency in the systems far from thermal equilibrium. I wished to emphasize the need for a new discipline that will describe these processes. Thus, synergetics can be considered as a science
dealing with the phenomenon of self-organization [40].” This is just nothing else than the usual widespread line of thoughts [41], which tends to absolutize the “crisp” notion of thermodynamical
equilibrium (that is, either one has a “strict equilibrium”, or a “strict nonequilibrium”), just in apparent contrast to the G. N. Lewis’ and G. A. Linhart’s intuitive idea of “fuzzy equilibrium”
(that is, one has never “crisp” differences of the equilibrium/non-equilibrium kind, but always some “partial degrees of equilibrium”).
The work [39] suggests to describe different mechanisms of the material world self-organization using the concept of homeostatic determinate systems.
The concept of homeostatic determinate systems is fundamental for synergetics. It provides deep insight into physical meaning and genesis of the hierarchy of instabilities in self-organizing
homeostatic systems, into the nature of interrelations between instabilities and order parameters [39].
Albert Einstein had expressed once, back in 1934, the following expectation: “I still believe in a possibility of constructing such model of reality, i.e., the theory that expresses the objects
themselves, but not only the probabilities of their behavior [42].” Interestingly, in my view, the Linhart’s way of thinking matches this expectation of Einstein much more perfectly than the concept
[39] sketched above, because the latter concept doesn’t even explicitly consider the role of the time (or of any other intensive thermodynamical variable) in the processes under study. Further on,
the true role of probability theory was questioned by De Finetti already long time ago [43], whereas G. A. Linhart had already used the ideas akin to those by De Finetti well before the latter ones
were published at all.
There is also another most recent work [44], where a review of the irreversibility problem in modern physics with new researches is given. Some characteristics of the Markov chains are specified and
the important property of monotonicity of a probability is formulated. Then, the behavior of relative entropy in the classical case is considered. Further, the irreversibility phenomena in quantum
problems are studied. This work has also paid no special attention to the explicit role of the time (or of any other intensive thermodynamical variable) in the processes under study.
Interestingly, one more review paper [45] has been published to present some of the recent contributions that show the use of thermodynamics to describe biological systems and their evolution,
illustrating the agreement that this theory presents with the field of evolution. Organic systems are described as thermodynamic systems, where entropy is produced by the irreversible processes,
considering as an established fact that this entropy is eliminated through their frontiers to preserve life. The necessary and sufficient conditions to describe the evolution of life in the
negentropy principle are established. Underlining the fact that the necessary condition requires formulation, which is founded on the principle of minimum entropy production for open systems
operating near or far from equilibrium, other formulations are mentioned, particularly the information theory, the energy intensiveness hypothesis and the theory of open systems far from equilibrium.
Finally, suggesting the possibility of considering the lineal formulation as a viable alternative; that is, given the internal constrictions under which a biological system operates, it is possible
that the validity of its application is broader than it has been suggested. But, again, there is absolutely no discussion of the implicit role of the time time (or of any other intensive
thermodynamical variable) in the processes under study.
Finally, a very interesting paper has been published most recently [46], where the author studies the universal efficiency at optimal work with the help of the Bayesian statistics and finds that, if
the work per cycle of a quantum heat engine is averaged over an appropriate prior distribution for an external parameter, the work becomes optimal at Curzon-Ahlborn (CA) efficiency. More general
priors yield optimal work at an efficiency which stays close to (CA) value, in particular near equilibrium the efficiency scales as one-half of the Carnot value. This feature is analogous to the one
recently observed in literature for certain models of finite-time thermodynamics. Further, the use of the Bayes’ theorem implies that the work estimated with posterior probabilities also bears close
analogy with the classical formula. These findings suggest that the notion of prior information can be used to reveal thermodynamic features in quantum systems, thus pointing to a connection between
thermodynamic behavior and the concept of information. I have entered an intensive discussion with the author of this paper and tried to persuade him, that his approach is in fact very much akin to
the Linhart’s way of thoughts.
5. Conclusion
George Augustus Linhart was definitely able to successfully work out the true foundations of thermodynamics and could thus outdistance many famous thermodynamicists of his time and even the later
ones. Linhart’s view of the Second Law of Thermodynamics was and is extremely fruitful. Using the Linhart’s line of thoughts it is possible to formally derive the mathematical expression for the
famous Boltzmann’s entropy, for the heat capacity (which outperforms the famous Debye’s formula), for the well-known equation describing the ligand binding to macromolecules, that is, the equation
just guessed by Archibald Hill long time ago and never formally-mathematically derived by anybody. The Linhart’s point of view enables us to treat the time in rather simple and natural terms of
thermodynamics, just as one of the numerous intensive variables, without reverting to the “Arrow of Time” and all the problems of the “Ergodicity”. Following G. A. Linhart’s ideas enables us to treat
any of the possible intensive variables on the same lines, and hence, his approach is definitely of general significance— at least for the thermodynamics as a whole. | {"url":"https://www.scirp.org/journal/paperinformation?paperid=19105","timestamp":"2024-11-04T11:22:19Z","content_type":"application/xhtml+xml","content_length":"200820","record_id":"<urn:uuid:1c92c44c-5211-4825-a3b6-ec5689dae82b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00526.warc.gz"} |
Alexandre Chorin
Jump to navigation Jump to search
Alexandre Joel Chorin (born 25 June 1938) is a University Professor at the University of California, a Professor of Mathematics at the University of California, Berkeley and a Senior Scientist at the
Lawrence Berkeley National Laboratory. He is known for his contributions to computational fluid mechanics, turbulence, and computational statistical mechanics.
Chorin received the Ing. Dipl. Physics degree from the Ecole Polytechnique of Lausanne in 1961, an M.S. in Mathematics from New York University in 1964, and a PhD in Mathematics from New York
University in 1966.
Chorin's work involves developing methods for solving physics and fluid mechanics problems computationally. His early work introduced several widely used numerical methods for solving the
Navier-Stokes equations, including the method of artificial compressibility ,^[1] the projection method,^[2] and vortex methods.^[3] He has made numerous contributions to turbulence theory.^[4] In
recent years he has been developing methods for prediction in the face of uncertainty ^[5] and for filtering and data assimilation .^[6]
Chorin's awards include the National Academy Award in Applied Mathematics and Numerical Analysis (1989), the Norbert Wiener Prize of the American Mathematical Society and the Society for Industrial
and Applied Mathematics (2000),^[7] the Lagrange Prize of the International Council on Industrial and Applied Mathematics (2011) and the National Medal of Science (2012).^[8]^[9] He is a member of
the US National Academy of Sciences and a fellow of the American Academy of Arts and Sciences, the Society for Industrial and Applied Mathematics, and the American Mathematical Society.
Chorin is widely recognized for his mentoring of graduate students ^[10] and postdoctoral fellows, many of whom have become nationally and internationally recognized scientists in their own right. In
2008 he was honored with the Sarlo mentoring award by the University of California Berkeley.^[11]
Journal publications[edit]
1. Chorin, A. J. (1967). "A numerical method for solving incompressible viscous flow problems". Journal of Computational Physics. 2: 12. Bibcode:1967JCoPh...2...12C. doi:10.1016/0021-9991(67)90037-X
2. Chorin, A. J. "A numerical method for solving incompressible viscous flow problems" J. Comput. Phys. 2 (1967), pp. 12-26.
3. Chorin, A.J., Hald, O.H., and Kupferman, R., Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA 97 (2000), pp. 2968–2973.
4. Chorin, A.J. and Tu, X., Implicit sampling for particle filters, Proc. Natl. Acad. Sci. USA 106 (2009), pp. 17249–17254.
• Chorin, Alexandre J.; Marsden, Jerrold E. (1993). A Mathematical Introduction to Fluid Mechanics. Texts in Applied Mathematics. 4 (3rd ed.). New York: Springer Verlag. ISBN 0387979182.
• Chorin, A. J., "Vorticity and Turbulence", Springer-Verlag (1994).^[12]
• Chorin, A.J. and O.H. Hald, "Stochastic Tools in Mathematics and Science", Springer-Verlag, 3rd ed., (2013).
External links[edit] | {"url":"https://static.hlt.bme.hu/semantics/external/pages/John_McCarthy/en.wikipedia.org/wiki/Alexandre_Chorin.html","timestamp":"2024-11-13T11:54:07Z","content_type":"text/html","content_length":"55874","record_id":"<urn:uuid:cace2c42-b9a5-4ee7-9141-aa3bcc38625d>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00840.warc.gz"} |
Back to Back Histograms
histbackback {Hmisc} R Documentation
Back to Back Histograms
Takes two vectors or a list with x and y components, and produces back to back histograms of the two datasets.
histbackback(x, y, brks=NULL, xlab=NULL, axes=TRUE, probability=FALSE,
xlim=NULL, ylab='', ...)
x, y either two vectors or a list given as x with two components. If the components have names, they will be used to label the axis (modification FEH).
brks vector of the desired breakpoints for the histograms.
xlab a vector of two character strings naming the two datasets.
axes logical flag stating whether or not to label the axes.
probability logical flag: if TRUE, then the x-axis corresponds to the units for a density. If FALSE, then the units are counts.
x-axis limits. First value must be negative, as the left histogram is placed at negative x-values. Second value must be positive, for the right histogram. To make the limits symmetric,
xlim use e.g. ylim=c(-20,20).
ylab label for y-axis. Default is no label.
... additional graphics parameters may be given.
a list is returned invisibly with the following components:
left the counts for the dataset plotted on the left.
right the counts for the dataset plotted on the right.
breaks the breakpoints used.
Side Effects
a plot is produced on the current graphics device.
Pat Burns
Salomon Smith Barney
See Also
hist, histogram
histbackback(rnorm(20), rnorm(30))
fool <- list(x=rnorm(40), y=rnorm(40))
age <- rnorm(1000,50,10)
sex <- sample(c('female','male'),1000,TRUE)
histbackback(split(age, sex))
agef <- age[sex=='female']; agem <- age[sex=='male']
histbackback(list(Female=agef,Male=agem), probability=TRUE, xlim=c(-.06,.06))
version 5.1-3 | {"url":"https://search.r-project.org/CRAN/refmans/Hmisc/html/histbackback.html","timestamp":"2024-11-13T15:55:05Z","content_type":"text/html","content_length":"4457","record_id":"<urn:uuid:c55d54f6-a617-42f8-8a9b-5b104084c28c>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00025.warc.gz"} |
How astronomic plate-solving works - Oleg Ignat
How astronomic plate-solving works
Plate-solving is a technique used by astrophotographers to align the telescope in the sky very precisely using a star pattern matching algorithm. Aligning the telescope via plate-solving is not only
easier but also significantly more accurate than any manual alignment. Astrophotographers save time on the 1, 2, or 3-star alignment process of the mount – just point the telescope somewhere in the
sky, take a picture, plate-solve, synchronize the mount, and off you go. In this post, we will look at the plate-solving process itself – how does the software recognize a section of the sky and
calculate celestial coordinates of the center of the image?
Star Quad pattern on the Elephant’s Trunk nebula
Blind and local plate solvers
There are two categories of plate-solver programs – blind plate solvers and local ones.
• Blind plate solvers, like Astrometry.net or All Sky Plate Solver, take completely arbitrary pictures of the night sky and resolve them without any additional hints about their approximate
location. They usually take more time and are more compute-intensive.
• Local plate-solvers, like ASTAP or PlaneWave Solve2, require a hint about approximate location in the sky and telescope field of view to work. They are quicker, have smaller star databases and
consume less CPU.
Even though these programs do seemingly similar jobs, their star matching algorithms are different. Blind plate-solvers maintain a global inverse index of an entire night sky. They extract star
signatures (hashes) from the image and search that index for matches. Here’s the whitepaper on Astometry.net for those interested in more details. We will not be covering blind plate solvers in this
This post is about local plate-solvers. We will look at how they extract stars from the image and scan the catalog of stars looking for matches. Here’s how ASTAP describes its algorithm.
This article doesn’t describe the implementation of any particular plate-solver. Frankly, I do not know what they do exactly. I took the white paters and public documentation of existing
plate-solvers and implemented my own program. It worked remarkably well. Since my plate solver works, I expect other plate-solvers to follow similar principles in their implementation. If you’ve come
here to learn what PlaneWave Solve2 does under the hood – I am sorry to disappoint you.
Step 1: Image histograms
The first thing that any plate-solver must do is find stars on the image. However, how do you find stars programmatically? As humans, our brains recognize stars immediately because they run fairly
complex neural networks. However, plate solvers aren’t sporting neural networks and instead rely on discrete algorithms to find stars. Those algorithms can analyze one pixel at a time, so how can we
tell if we are looking at the pixel of the star, or a pixel of the nebula under the star? Color histograms to the rescue!
The color histogram represents the distribution of color in the image. We are going to be looking at color images, which are made of 3 channels – Red, Green, and Blue. Each color channel has a value
between 0 and 255. To build a color histogram we need to count how many pixels of every single color value there are in the image. Horizontal axis – color channel value, Vertical axis – number of
pixels of corresponding color value.
Elephant’s Trunk nebula color distribution histogram
It is fairly easy to see on the histogram that most of the image is clustered around the 25 – 190 color range and there is an uptick after 253. If we draw the line anywhere between 190 and 253 we
will have most of the stars on one side of the line, while the rest of the image is on the other side. The exact place to draw the line requires fine-tuning. The further left we draw the line, the
more stars we will detect. However, we will also pick up more noise. If we draw the line too far to the right, we will miss a lot of dim stars, but our signal-to-noise ratio will be high.
Step 2: Star detection
Once the plate-solver decides on the star threshold, it will apply it to the image to find stars. I suspect, most plate solvers convert a color image to gray-scale by extracting luminosity and
processing it once. In my plate-solver, I decided to process each color channel as an independent gray-scale image and then take stars that were detected in the same position across all channels.
This technique improves detection quality because bright spots in individual channels are ignored.
Star detection per color channel
If we zoom in on the section of the image we can see the process in action a little better.
Star detection process using star thresholds on the section of the image
One of the problems plate-solvers face in this step is the calculation of the star center. Again, humans can visualize the center of the circle pretty easily, but computers don’t see patterns. They
see individual pixels. To calculate the center of the star, plate solvers use the Center of Gravity algorithm. It works for the following reasons:
1. Stars are eccentic and have a center of gravity in the center of the circle.
2. Stars are brightest in the center and get dimmer towards the edges.
The Center of gravity algorithm has flaws and won’t do a good job in the following cases:
1. Double-stars can get mashed together because algorithm can’t tell where one star ends and another begins.
2. Center of small and dim stars will be heavily influenced by background image brightness and will be skewed.
Center of gravity calculation at the pixel level
Step 3: Star quads of the reference image
Astrophotographers take pictures with a variety of cameras and telescopes. As such they cover the same objects in the sky at variable degrees of magnification and rotation relative to the image
sensor. Plate-solver must recognize the sky regardless of the scale and camera rotation angle. Moreover, plate-solver must calculate both scale and rotation angle in order to figure out precise
coordinates of the center of the image.
Plate-solvers use combinations of 4 stars to form a collection of star quads from the image. During later phases of plate solving, each quad of the picture is matched to each quad from the star
catalog. How do you match quads that are created at different scales and rotation angles? You normalize them.
The build a collection of quads from the reference image, the plate-solver goes through every star and finds 3 closest stars on the image. Then for every quad, it calculates the “hash code” – 6
numbers that represent distances between stars (1 – 2, 1 – 3, 1 – 4, 2 – 3, 2 – 4, and 3 – 4) divided by the longest distance among them, in pixels. This operation normalizes the star quad to the
range from 0 to 1. It also calculates the average coordinates among 4 stars in the quad. These average coordinates will later be used to calculate scale and rotation angle.
Star quads of the Elephant’s Trunk nebula zoomed in
The same star can be part of multiple quads, and that’s OK. Also, not all the stars on the image will be included in quads. Some may be missing due to the low signal-to-noise ratio when detecting
stars. Some may be excluded if stars are too close to each other. A few sample quads from the image of the Elephant’s Trunk nebula are depicted below. This is not a complete set.
X coordinate, pixels Y coordinate, pixels Largest Distance, pixels First normalized distance Second normalized distance Third normalized distance Forth normalized distance Fifth normalized distance
429.5135 297.9123 93.0482 0.971103458 0.541585683 0.504578809 0.478478556 0.058131226
983.4961 1283.3288 74.4317 0.794692519 0.636636938 0.520571722 0.468397828 0.365016442
682.5731 725.1145 57.8917 0.908194078 0.519112014 0.497206454 0.491335209 0.312518029
275.0098 1142.25 177.862 0.938934476 0.505157518 0.494855241 0.449178425 0.17959982
528.2151 1609.6074 119.7565 0.829619808 0.826824692 0.82584552 0.685649305 0.322313057
A few star quads from the Elephant’s Trunk nebula
Distances between stars are calculated using the Pythagorean theorem. Distances are sorted in descending order in the quad (or ascending, doesn’t matter, just pick one and stick to it).
Step 4: Search the star catalog and build star quads
All the magic happens during this step of the plate-solving process. The result (or the output) of this step is a collection of pairs of star quads that match. One of the quads in the pair will come
from the image being plate-solved. The other quad will come from the reference database or stars.
Using initial coordinates in the sky and information about the telescope and the camera plate solver calculates the field of view. Astronomy.tools website has a calculator that does the same exact
Formula to calculate the resolution of a camera pixel in arcseconds
Obviously to calculate the field of view of the camera one needs to multiply PixelResolution by the width and separately by the height of the reference camera image. The entire celestial sphere is
divided into sections the size of the field of view. Starting with the initial coordinates of right ascension and declination given to the plate solver, it begins a spiral search of the sky.
The spiral search of the sky
On every step of the search, the plate-solver program reads the stars from the star catalog that are within the field of view. These stars obviously do not have flat X and Y coordinates, they only
have right ascension and declination. To obtain X and Y coordinates, these stars are projected onto the plane with the center (0, 0) going through the optical axis of the field of view. The formula
for conversion is available on the Wikipedia page for the Astronomical coordinate system.
double sinOpticalAxisDeclination = Math.Sin(opticalAxisDeclinationRadians);
double cosOpticalAxisDeclination = Math.Cos(opticalAxisDeclinationRadians);
double sinStarDeclination = Math.Sin(starDeclinationRadians);
double cosStarDeclination = Math.Cos(starDeclinationRadians);
double raDelta = starRightAscentionRadians - opticalAxisRightAscentionRadians;
double sinDeltaRightAscention = Math.Sin(raDelta);
double cosDeltaRightAscention = Math.Cos(raDelta);
double deltaValue = (cosOpticalAxisDeclination * cosStarDeclination * cosDeltaRightAscention
+ sinOpticalAxisDeclination * sinStarDeclination) / (3600 * 180 / Math.PI);
double projectedStarX = -cosStarDeclination * sinDeltaRightAscention / deltaValue;
double projectedStarY = -(sinOpticalAxisDeclination * cosStarDeclination * cosDeltaRightAscention - cosOpticalAxisDeclination * sinStarDeclination) / deltaValue;
Similar to step #3, plate-solver creates a collection of star quads using a collection of flat stars. The important trick here is to create quads not only using closest stars, but also those with up
to N-th degree separation. This is necessary to do because the star catalog contains all the stars in the field of view, while our star detection on the picture probably skipped a large portion of
dim stars. If we do not create “gaps” we will not be able to match any quads.
Step 5: Match star quads
On every step of the spiral search, it compares the star quads of the original image with the star quads of the star database in the given field of view. This operation is O(N^2) cyclomatic
complexity. Quads are considered to match if five of their normalized distances are within the tolerance interval from each other. Other parameters of the star quads are irrelevant at this phase.
Here’s an example of two matching star quads:
Image star quad Database star quad Difference
First normalized distance 0.61896944520747843 0.61987700436560234 -0.00090755915812391
Second normalized distance 0.55775333830273033 0.55884949726218647 -0.00109615895945614
Third normalized distance 0.48979834252501553 0.48884465178429182 0.00095369074072371
Fourth normalized distance 0.47681557137210878 0.47598962346263274 0.00082594790947604
Fifth normalized distance 0.13132193567363912 0.13299759845994066 -0.00167566278630154
As such, if the plate-solver tolerance interval is 0.002 these two quads would match. Picking the right interval requires experimentation with a large number of images.
A minimum of 3 quads must match in order to calculate the coordinate system. If only 2 quads match we can only calculate the scale difference, but not the coordinate axis and angle of rotation. The
more quads match the higher accuracy of the coordinate system calculation we will have.
Star quads matching on the original image
Star quads matching on the reference database
Just for fun, try figuring out which quads were matched on the pictures above and what is the angle of camera rotation relative to the reference database.
Step 6: Calculate projection coordinate system
This step of plate solving is necessary to be able to calculate coordinates of the center of the camera image in an equatorial system. This calculation has two steps:
1. Convert center coordinates of the camera image (image width / 2, image hight / 2) into flat coordinates on the projected plane of the reference star database.
2. Knowing projected coordinates, calculate equatorial coordinates using transformation inverse to the one in step 4.
Obviously, the plate solver needs to be able to project points between the two coordinate systems. The way to do this is to re-write our matching star quads in a set of linear equations and solve
them for 3 variables. Math tells us that it is sufficient to have 3 equations to solve for 3 variables – that’s why we need at least 3 star quads to match. If we have more than 3 star quads matching,
then our system of linear equations becomes overdetermined.
X-axis overdetermined linear equation system
Y-axis overdetermined linear equation system
Solving these two sets of linear equations would allow the plate solver to create a projection equation from the camera coordinate system into reference star catalog coordinate system, accounting for
scale and rotation. Since the application isn’t dealing with the ideal world, more than likely different equations in each set would solve for slightly different values A1, B1, and C1 or A2, B2, and
C2. Least square fit with givens rotations to the rescue!
The best explanation of the method I found from Kartik Tiwari. In simple terms, a plate solver applies givens rotations over and over again until the bottom left portion of the matrix turns to 0.
Each iteration strives to reduce the sum of squares of distances between all points. It uses squares rather than values to account for distances both greater and less than zero.
The solution to Elephant’s Trunk nebula picture discussed in the post is the following:
• starDatabaseX = 0.0304 * cameraX + 2.0093 * cameraY – 1757.9956
• starDatabaseY = 2.0085 * cameraX – 0.0305 * cameraY – 2544.7080
Punching the center of the image coordinates and running these equations, then running a transformation from projected to equatorial coordinates, the plate solver obtains the following coordinates of
the center of the image:
• Right ascention = 21h 34m 13.8s
• Declination = +57° 21′ 39.1″
The job of a traditional plate solver is complete. These coordinates are synchronized with the equatorial mount and the imaging session can begin.
Bonus functionality
Why bother writing your own plate solver if it does only as much as an existing software and nothing more? My point exactly. Once you have the coordinate system, star catalog, center of the image,
and additional internal state calculated, you can start doing cool things.
For example, you can add all the stars from the star database into the image that failed to be recognized by the plate solver or captured by the camera. Of course, this piece requires more work
because the software must account for star magnitude, measured color, and point-spread function. However, as a prototype, why not just show where stars should be?
The picture with missing stars from the star catalog | {"url":"https://olegignat.com/how-plate-solving-works/","timestamp":"2024-11-07T07:22:28Z","content_type":"text/html","content_length":"97343","record_id":"<urn:uuid:b77d8073-6a0d-471e-bb1c-88382ca153b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00534.warc.gz"} |
Partial Bayes Estimation of Two Parameter Gamma Distribution Under Non-Informative Prior
Partial Bayes Estimation of Two Parameter Gamma Distribution Under Non-Informative Prior
Keywords: Partial Bayes Estimation, Non-Informative Prior, Jeffreys Prior, Digamma Function.
In Bayesian analysis, empirical and hierarchical methods are two main approaches for the estimation of the parameter(s) involved in the prior distribution of one parameter. But in the multi-parameter
model, e.g., Gamma(α, p), where both the parameters are unknown, idea of the ‘Partial Bayes (PB) Estimation’ is introduced. When we do no have proper belief regarding the joint parameters of the
distribution of the variable and when we are estimating one parameter in presence of others, such method may be used. Partial Bayes estimation of the scale parameter p is done by putting the estimate
of the another parameter α obtained by some other classical method in case of two parameter Gamma distribution. Using non-informative prior and computing the risk, it is found that the Partial Bayes
estimator has less risk than the Bayes estimator. For this, simulation studies for some choices of shape parameter values have been done. In case of the shape parameter, posterior mean and posterior
variance are evaluated through simulations to obtain the risk values for estimator of α with known scale parameter. Finally after fifitting this distribution, two real datasets are illustrated to see
the performance of the Partial Bayes estimator.
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How to Cite
Banerjee, P., & Seal, B. (2021). Partial Bayes Estimation of Two Parameter Gamma Distribution Under Non-Informative Prior. Statistics, Optimization & Information Computing, 10(4), 1110-1125. https://
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Construct the confidence interval for the population mean - TheMathWorld
Construct the confidence interval for the population mean
Construct the confidence interval for the population mean .
c = 0.90,
Use the Formula E =
to find the margin of error, where
n is the sample size.
When n
Substitute the value of c and evaluate to find the area in each tail.
Use a standard normal table to find the positive critical value the corresponds to a tail area of 0.05
The population standard deviation
Substitute the value
E =
Use the margin of error to find the left endpoint.
Left endpoint =
= 17.8 – 1.9 = 15.9
Now find the right endpoint
Right endpoint =
= 17.8 + 1.9 = 19.7
Therefore, a 90% confidence interval for | {"url":"https://mymathangels.com/construct-the-confidence-interval-for-the-population-mean-mu/","timestamp":"2024-11-11T15:23:29Z","content_type":"text/html","content_length":"68873","record_id":"<urn:uuid:3c73ef49-2737-44f3-a5dd-d9b42997fad2>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00854.warc.gz"} |
A Simplified Ventricular Myocyte ModelModel StatusModel Structurecell diagram
Catherine Lloyd Auckland Bioengineering Institute This version of this model is known to run in both OpenCell and COR. The units have been checked and they are consistent. A generic stimulus protocol
has been added to allow the model to simulate trains of action potentials. Although the model does run, the simulation output is still not quite the same as the original published model. The original
model authors have been contacted and we will continue to curate the CellML model. ABSTRACT: In this paper we introduce and study a model for electrical activity of cardiac membrane which
incorporates only an inward and an outward current. This model is useful for three reasons: (1) Its simplicity, comparable to the FitzHugh-Nagumo model, makes it useful in numerical simulations,
especially in two or three spatial dimensions where numerical efficiency is so important. (2) It can be understood analytically without recourse to numerical simulations. This allows us to determine
rather completely how the parameters in the model affect its behavior which in turn provides insight into the effects of the many parameters in more realistic models. (3) It naturally gives rise to a
one-dimensional map which specifies the action potential duration as a function of the previous diastolic interval. For certain parameter values, this map exhibits a new phenomenon--subcritical
alternans--that does not occur for the commonly used exponential map. The original paper reference is cited below: A two-current model for the dynamics of cardiac membrane, Colleen C. Mitchell, David
G. Schaeffer, 2003, Bulletin of Mathematical Biology, 65, (5), 767-793. PubMed ID: 12909250 A schematic diagram of the two ionic currents described by the Mitchell-Schaeffer model of a ventricular | {"url":"https://models.cellml.org/workspace/mitchell_schaeffer_2003/@@rawfile/25103ab3451d4a4fb31fc2a4fea92b38b60e33e7/mitchell_schaeffer_2003.cellml","timestamp":"2024-11-05T07:28:47Z","content_type":"application/cellml+xml","content_length":"17322","record_id":"<urn:uuid:61333c7d-ab99-4c14-91b3-c8df3b419c96>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00579.warc.gz"} |
Representation of MV-algebras by regular ultrapowers of [0,1]
We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real
interval $[0,1]$. This result also implies the existence, for any cardinal $\alpha$, of a single MV-algebra in which all infinite MV-algebras of cardinality at most $\alpha$ embed. Recasting the
above construction with iterated ultrapowers, we show how to construct such an algebra of values in a definable way, thus providing a sort of “canonical” set of values for the functional
Representation of MV-algebras by regular ultrapowers of [0,1]
Tags: Di Nola Theorem, MV-algebras, non standard, Representation, Ultrapower | {"url":"http://logica.dipmat.unisa.it/lucaspada/214-representation-of-mv-algebras-by-regular-ultrapowers-of-01/","timestamp":"2024-11-04T02:52:32Z","content_type":"application/xhtml+xml","content_length":"42874","record_id":"<urn:uuid:98a23f21-10af-4211-bf0b-ee7601d15660>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00436.warc.gz"} |
Geostrophic adjustment on the midlatitude β plane
Analytical and numerical solutions of the linearized rotating shallow water equations are combined to study the geostrophic adjustment on the midlatitude β plane. The adjustment is examined in zonal
periodic channels of width LyCombining double low line4Rd (narrow channel, where Rd is the radius of deformation) and LyCombining double low line60Rd (wide channel) for the particular initial
conditions of a resting fluid with a step-like height distribution, η0. In the one-dimensional case, where η0Combining double low lineη0(y), we find that (i) β affects the geostrophic state
(determined from the conservation of the meridional vorticity gradient) only when bCombining double low linecot(φ0)RdR≥0.5 (where φ0 is the channel's central latitude, and R is Earth's radius); (ii)
the energy conversion ratio varies by less than 10 % when b increases from 0 to 1; (iii) in wide channels, β affects the waves significantly, even for small b (e.g., bCombining double low line0.005);
and (iv) for bCombining double low line0.005, harmonic waves approximate the waves in narrow channels, and trapped waves approximate the waves in wide channels. In the two-dimensional case, where
η0Combining double low lineη0(x), we find that (i) at short times the spatial structure of the steady solution is similar to that on the f plane, while at long times the steady state drifts westward
at the speed of Rossby waves (harmonic Rossby waves in narrow channels and trapped Rossby waves in wide channels); (ii) in wide channels, trapped-wave dispersion causes the equatorward segment of the
wavefront to move faster than the northern segment; (iii) the energy of Rossby waves on the β plane approaches that of the steady state on the f plane; and (iv) the results outlined in (iii) and (iv)
of the one-dimensional case also hold in the two-dimensional case.
Bibliographical note
Publisher Copyright:
© Copyright:
Dive into the research topics of 'Geostrophic adjustment on the midlatitude β plane'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/geostrophic-adjustment-on-the-midlatitude-%CE%B2-plane","timestamp":"2024-11-13T04:16:27Z","content_type":"text/html","content_length":"51213","record_id":"<urn:uuid:a07a15d7-1b0a-44fa-aca8-48ad7efd683c>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00835.warc.gz"} |
Metric uniformization of morphisms of Berkovich curves
We show that the metric structure of morphisms f:Y→X between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a
large enough skeleton Γ=(Γ[Y],Γ[X]) of f, the sets N[f,≥n] of points of Y of multiplicity at least n in the fiber are radial around Γ[Y] with the radius changing piecewise monomially along Γ[Y]. In
this case, for any interval l=[z,y]⊂Y connecting a point z of type 1 to the skeleton, the restriction f|[l] gives rise to a profile piecewise monomial function φ[y]:[0,1]→[0,1] that depends only on
the type 2 point y∈Γ[Y]. In particular, the metric structure of f is determined by Γ and the family of the profile functions {φ[y]} with y∈Γ[Y] ^(2). We prove that this family is piecewise monomial
in y and naturally extends to the whole Y. In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that φ[y] coincides with the Herbrand
function of H(y)/H(f(y)). This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions.
Bibliographical note
Publisher Copyright:
© 2017 Elsevier Inc.
• Berkovich curves
• Herbrand function
• Wild ramification
Dive into the research topics of 'Metric uniformization of morphisms of Berkovich curves'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/metric-uniformization-of-morphisms-of-berkovich-curves","timestamp":"2024-11-11T03:49:18Z","content_type":"text/html","content_length":"48522","record_id":"<urn:uuid:bce2e9ef-9113-4606-bb6d-5aa7fff6523c>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00160.warc.gz"} |
Kickstarting R - Multiple comparisons
Significance testing is one of the more lively areas of statistics. In general, the idea is not to make too many mistakes in our conclusions. If this only applied to Type I errors, we could all
relax, apply the most conservative tests of significance possible and restrict ourselves to the study of the glaringly obvious. R attends to the problem of significance testing in some ways, but
sensibly avoids prescribing methods which may not be appropriate for particular analyses.
Adjustments for multiple comparisons
The basic method of adjusting for multiple comparisons is to define the group of comparisons that are to be tested and select an appropriate method of adjustment and the overall probability of a Type
I error (perhaps considering the implications of Type II errors). Then, either define a critical probability which any test in that group must exceed or adjust the probability of each test
individually and compare that to the selected overall probability of Type I errors. The latter method has been established in R in the function p.adjust(), but it's a bit awkward to integrate with
functions like anova() that may produce a table with a number of probabilities.
Using the infert data set, we'll apply the Bonferroni correction to multiple tests of the prevalence of induced labor within groups defined by educational attainment in the infert data set. First
let's go through the function group.prop.test() that I found useful for repetitive testing of groups of Bernoulli trial (success/failure) data where the outcome of interest was which groups differed
from the overall proportion, that is, which groups were better or worse than the average level of success by a fairly conservative test.
The usual checks of the input data are performed, then the overall proportion is calculated and the result list is set up, filled with blanks and zeros. For each group defined by the grouping vector
by, a test of proportions is conducted, and the adjusted probability stored in the appropriate element of gptest. Notice that the formatting of the group names was performed after the calculation.
Otherwise the comparison performed by subset would have failed. After the calculation, the results are printed out and the list of results is returned invisibly. By playing around with this data, you
may discover that a simple test of the contingency table indicates that the groups do not come from the same population, but in fact none differ from the average prevalence of induced labor, at least
by this test.
When I originally wrote the function, it simply printed out the critical (corrected) p-value at the top of the table, and all of the observed values were compared with that. | {"url":"https://cran.radicaldevelop.com/doc/contrib/Lemon-kickstart/kr_multc.html","timestamp":"2024-11-08T21:16:28Z","content_type":"text/html","content_length":"3604","record_id":"<urn:uuid:1de5f3f1-b251-4342-85e1-1d518b8d236c>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00565.warc.gz"} |
ICPAM-VAN 2018
The process evaluation of abstract is completed. Last Update: September 10, 2018.
# First Name Last Name Title
181 Zeynep Sevinç On a new type of q-Baskakov-Kantorovich operators
180 Zeynep Kayar A novel Lyapunov type inequality for quasilinear impulsive systems
179 Hakkı Güngör Regularization of inverse coefficient determination problem in a hyperbolic problem
178 Rabia Nagehan Üregen On uniformly pr-ideals in commutative rings
177 Tanfer Tanrıverdi Schrödinger equation with potential vanishing exponentially fast
176 Esra Erkan The theory of Bèzier curves in R^4
175 Waleed S. Khedr Reduction of Navier-Stokes equation to a linear equation
174 Bülent Saraç Rings without a middle class: past and recent
173 Harun Çiçek Approximation by modified bivariate Bernstein-Durrmeyer operators on a triangular region
172 Elena Bespalova Quasilinearization method in problems of the subcritical deformation of flexible shell systems
171 Süleyman Şenyurt Curves and Ruled Surfaces According to Alternative Frame in Dual Space
170 Mohanad Alaloush On the blow-up of solutions for a stochastic Camassa-Holm equation
169 Vahid Roomi Asymptotic Behavior of Solutions of Generalized Lienard System
168 Luiz Guerreiro Lopes Ehrlich-Aberth's type method with King's correction for the simultaneous approximation of polynomial zeros
167 Akram Chehrazi Mathematical aspects of quantum cryptography
166 Nuray Öktem An efficient TVD-WAF scheme application for the 2D shallow water equations on unstructured meshes
165 Bulent Karakaş The relations among the planar fourbar linkages and the spherical fourbar linkages
164 Şenay Baydaş On mechanisms in three-dimensional Minkowski space
163 Canan Bozkaya MHD natural convection flow in a porous cavity
162 Sumeyra Ucar Canonical finite Blaschke products and decomposibility
161 Onur Saldir A numerical approach for time-fractional Kawahara equation with reproducing kernel method
160* Ecem Acar On approximation by generalized Bernstein-Durrmeyer operators
159 Turgut Hanoymak On mathematical aspects of blockchain architecture
158 Ugur Duran A note on $q$-Fubini polynomials
157 Zohra Bouteffal Darboux problem of partial fractional differential equations in Fréchet spaces
156 Necati Erdoğan Statistical analysis of wind speed data with some distributions
155 Asuman Yılmaz Statistical inference for the inverse Weibull distribution
154 Mustafa Telci Caristi type related fixed point theorems in two metric spaces
153 Mehmet Giyas Sakar Reproducing kernel method with Bernstein polynomials for fractional boundary value problems
152 Guzide Senel Some convergences in metric spaces
151 Elham Sefidgar A Chebyshev collocation method for solving coupled system of linear fractional differential and algebraic equations
150 Nisar Lone An interplay between Riemann integrability and weak*-continuity
149 Hatice Taşkesen Blow up of solutions for a stochastic Klein-Gordon equation
148* Sara Litimein Existence and controllability results for fractional integro-differential inclusions with state-dependent delay
147 Merve Atasever On the structure of Ricci solitons on gradient Einstein-type manifolds
146 Emek Demirci Akarsu Random process generated by the short incomplete Gauss sums
145 Ahmad Khojali A note on the annihilator of certain local cohomology modules
144 Ramazan Kama On some vector valued multiplier spaces obtained by Zweir matrix method
142 Tuba Güleşce Tatlı Multi-party key exchange protocol and man in the middle attack
141 Sinem Güler Lorentzian homogeneous generalized Ricci solitons of dimension n≥3
140 Asghar Ahmadkhanlu Existence of positive solutions for a singular fractional boundary value problem
139 Oguz Ogur A note on superposition operators in Fibonacci sequence spaces l_p(F)
138 Ramazan Yazgan On the weighted pseudo almost periodic solutions of nonlinear functional Nicholson's blowflies equations
136 Sultan Erdur On the existence of periodic solutions of third order nonlinear differential equations with multiple delays
135 İrem Akbulut Analysis of behaviors of solutions of Volterra integro-differential equations
134 Evrim Akalan Strongly graded rings over hereditary Noetherian prime rings
133 Mahmut Karakuş Λ-matrix as a summability operator and completeness of certain normed spaces via weakly unconditionally Cauchy series
132 Murat Polat Semi-tensor bundle and the vertical lift of tensor fields
131 Qais Mustafa Abdulqader Forecasting Iraq's oil exports by using wavelet analysis: Application on real data
130 Mohammad Reza Jabbarzadeh Conditional expectation operators on measurable function spaces
129 Hande Günay Akdemir Simulation studies for credibility-based multi-objective programming problems with fuzzy parameters
128 Cansu Cengiz Extortion strategies in non-symetric iterated Prisoner's Dilemma
127 Osman Tunç A note on certain qualitative properties of solutions in Volterra integro-differential equations
126 Mohammad Bagher Kazemi Invariant submanifolds of statistical Kenmotsu manifolds and their curvatures
125 Sizar Mohammed Analysis of behaviors of solutions of Volterra integro-differential equations
124 Derya Altıntan Inference algorithms for jump-diffusion approximations of multi-scale processes
123 Adel Kazemi Piledaraq Total dominator coloring of a graph
122 Bahar Kalkan Inverse kinematics computation for a 6-DOF articulated robot arm using conformal geometric algebra
121 Güler Gürpınar Arsan Generalized concircular mappings of quasi-Einstein-Weyl manifolds
120 Gülçin Çivi Bilir Geodesic mappings of Einstein Weyl spaces
119 Şamil Akçağıl Relations between some well known methods with the projective Riccati equations
118 Ömer Küsmüş Idempotent unit group in commutative group rings of direct products
117 İsmail Hakkı Denizler An analogue of the Artin-Rees Lemma for Artinian modules
116 Şule Yüksel Güngör Approximation by summation-integral type operators involving Brenke polynomials
115 Tuğba Şenlik Çerdik Existence of positive solutions for boundary value problems of nonlinear fractional differential equations
114 Gokhan Soydan Elliptic curves containing sequences of consecutive cubes
113 Emel Biçer Fixed point theory and stability of a delay differential equation with multiple delays
112 Reşat Aslan Approximation by bivariate Bernstein-Kantorovich operators on a triangular domain
111 Fulya Yörük Deren Existence of solutions for a system of coupled fractional boundary value problems
110 Naser Zamani Asymptotic results in graded generalized local cohomology
109 Vedat Dörma Kinematics of 4R and 2RPR mechanisms in Clifford algebra
108 Fatma Tutar Galois theory and palindromic polynomials
107 Derya Arslan Second-order difference approximation for nonlocal boundary value problem with boundary layers
106 Ceyda Yilmaz Luzum Bezier curves on some surfaces
105 Mehdi Jalalvand A fast numerical approach for solving variable order fractional PDEs
104 Seda İğret Araz On numerical solution of an optimal control problem involving hyperbolic equation
103 Murat Sultanov Numerical solution of the nonlinear inverse gravimetry problem by the BiCGSTAB method
102 Mustafa Özel Norm inequalities for Fan product of matrices
101 Saudamini Nayak Isoclinism in Lie superalgebras
100 Yasin Kaya The maximal function in Sobolev spaces
99 Hüseyin Işık A new class of set-valued contractions and related results
98 Mohammad Reza Motallebi Direct sum of neighborhoods in locally convex cones
97 Fatih Tugrul On the orbit surface of two parameter motion
96 Selçuk Baş Roller coaster surface according to modified orthogonal frame in Euclidean space
95 Fatih Coşkun On the cubic nonlinear Shrodinger's equation with repulsive delta potential
94 Zeliha Körpınar On numerical solutions for fractional (1+1)-dimensional Biswas-Milovic equation
93 Talat Körpınar On inextensible flow with Schrödinger flow
92 Parastoo Reihani Identification of an unknown coefficient in an inverse parabolic equation
91 Morteza Garshasbi An inverse problem arising from mathematical modelling of evolution of cancer cells
90 Seema Kushwaha Pell's equation : A revisit through F_1,2-continued fractions
89 Snigdha Choudhury A correspondence between S-colocalization and Adams cocompletion
88 İlkay Yaslan Karaca Existence of positive solutions for second order impulsive boundary value problems on the half-line
85 Umit Sarp Some applications about Mobius function
84* Abbas Najati Stabilty of functional equations on bounded domains in R^n
83 Jafar Azami Reduction and coreduction of modules
82 Ridvan Cem Demirkol A new approach to a bending energy of elastica for space curves in De-Sitter space
80* Rabah Bououden On the influence of the density function of Lozi map on the chaotic optimization algorithm
79 Ramazan Ozarslan Singular eigenvalue problems via Hilfer derivative
78 F. Ayca Cetinkaya Spectral properties of a q-fractional boundary value problem
77 Soukaina Ouarab Some characteristic properties of ruled surface with Darboux frame in Euclidean 3-space
76* Mohammed-Salah Abdelouahab On Caputo and Riemann-Liouville fractional-order derivatives with fixed memory length
75 Elif Ertem Akbaş Misconceptions regarding representativeness in probability subject of high school students: Van case
74 Ferit Gürbüz Fractional type sublinear operators with rough kernel on variable exponent vanishing generalized Morrey type spaces
73 Cesim Temel Krasnoselskii fixed point theorem for singlevalued operators and multivalued operators
72 Leyla Bugay A general approach to find generating sets of certain finite subsemigroups of symmetric inverse semigroup
71 Ali Zohri On a weak condition for compactness of operators
70 Muhammad Adeel Global optimal solutions for Ciric type proximal contractions with application to matrix equations
69 Chih-Wen Weng A conjecture on the spectral radius of a bipartite graph
67 Kazem Haghnejad Azar Some notes on the order-to-topology continuous operators
66 Awais Younus Input distinguishability of linear dynamic control systems
65 Swati Tyagi Existence of solutions of fractional-order neural network with proportional delay
64 Sanem Sehrianoglu Estimation of parameters of Gumbel distribution data
63 Erdal Korkmaz Sufficient conditions for global asymptotic stability of neural networks with time-varying delays
62 Hakan Tor An optimality condition for non-smooth convex problems via *-subgradient
61 Abdulgani Şahin Jones polynomial for graphs of twist knots
60 Muhammad Rizwan Quantum tunneling from Kerr-Newman black hole
59 Özkan Atan Synchronization control of two chaotic systems via a novel fuzzy control method
58 Fatih Kutlu Review on fuzzy thermal image processing applications
57 Hülya Gültekin Çitil The eigenvalues and the eigenfunctions of the fuzzy boundary value problem found the eigenvalue parameter in the boundary condition
56 Azhar Hussain Best proximity point results of generalized almost contraction
55 Şenol Kubilay Clay transition (dehydration, dehydroxylation and decarbonylation) kinetics by DTA
54 Nagehan Alsoy-Akgün Numerical study of nanofluids under DDMC in a lid driven cavity
53 Syed Bokhary Chromatic polynomial and chromatic uniqueness of necklace graph
51 Feride Tuğrul On temporal intuitionistic fuzzy De Morgan triplets
50 Viktor Sergeevich Denysenko Algorithm for the stabilization of affine periodic impulsive systems
49 Abdullah Yigit On the exponential stability in nonlinear neutral differential equations
48 Yener Altun On the global exponential stability of nonlinear neutral differential equations with time-varying delays
47 Ali Demirci Applicability of regression analysis on the oxygen enriched combustion of Kutahya- Tuncbilek lignite
46 Hanzade Haykırı Açma Applicability of regression analysis on the oxygen enriched combustion of Adiyaman- Golbasi lignite
45 Ozgur Aydogmus Approximating the stochastic evolution via difference equations
44 Cemil Tunç Instability in nonlinear functional differential equations of higher order
43 Rezvan Varmazyar On 2-absorbing ideals
42 Alireza Khalili Golmankhaneh Solving fractal differential equations with Lie methods
41 Murat Bekar Some algebraic properties of elliptic biquaternions
40 Sedat Temel Crossed modules of group-groupoids and double group-groupoids
39 Zeki Yalçınkaya Studying the kinetic parameters and mechanism of the thermal decomposition (dehydration, dehydroxylation and decarbonylation) of some clays using TG
38 Fatma Ekinci Blow up of solutions for a quasilinear Kirchhoff-type wave equations with degenerate damping terms
37 Hayriye Esra Akyüz Approximate confidence interval based on winsorized mean for the coefficient of variation of positively skewed populations
36 Okkes Ozturk Obtaining solutions of Gegenbauer equation by ∇-discrete fractional calculus operator
35 Ferit Gürbüz Higher order commutators with rough kernels on vanishing generalized weighted Morrey spaces
34 Hatice Kuşak Samanci Investigating a quadratic Bezier curve according to N-Bishop frame
33 Emine Atici Endes A non-local model for E and N cadherin-dependent cell-cell adhesion
32 Taha Yasin Ozturk Bipolar soft points and their properties
31 Melek Gözen New exponential stability criteria for certain neutral differential equations with interval discrete and distributed time-varying delays
30 Mohsen Ghasemi Groups whose codegree graphs have no triangle
26 Mehmet Akif Akyol Some properties of conformal generic submersions
25 Sinem Onaran On contact surgeries and a counterexample
24 Derya Deniz Some properties of sequence class S^{β}(Δ,F,f) defined by a modulus function
23 Mithat Kasap Strongly Cesaro summability of order β with respect to a modulus function
22 Mehmet Korkmaz The general formula of regression sum of squares in multiple linear regression
21 Erdogan Şen Generalized class of boundary value problems with a constant retarded argument
20 Abdussamet Çalışkan Dual pole indicatrix curve and surface
19 Serdal Yazıcı A new generalization of Szàsz operators and their approximation properties
18* Filiz Kanbay A fuzzy methodology on surface representation of greenhouse gas estimation
17 Khalil Ur Rehman New Lie group of transformation for the non-Newtonian fluid flow narrating differential equations
16 Eda Günaydın On the kink type and singular solitons solutions to the nonlinear partial differential equation
15 Tolga Aktürk Modified expansion function method to the nonlinear problem
10 Bahya Roummani Solution sets for impulsive functional differential inclusions on unbounded domain
8 Mehmet Şerif Aldemir On stratified domination and Zagreb indices
7 Serap Şahinkaya Kernel stable and uniquely generated modules
6 Tulay Yildirim Characterization of regular morphisms in terms of abelian categories
5* Chaouchi Belkacem A remark on a hyperbolic problem set on a singular cylindrical domain
4 Abdalla Khdir Manguri On eccentricity based indices of generalized Petersen graphs
3 Süleyman Ediz On ve-degrees in direct and strong products of two graphs
2 Murat Cancan On ve-degrees in cartesian product of two graphs
*Which will be poster presentation. | {"url":"http://icpam.yyu.edu.tr/ICPAM18/part.html","timestamp":"2024-11-03T22:58:45Z","content_type":"application/xhtml+xml","content_length":"239761","record_id":"<urn:uuid:3791cc42-a17f-44b5-afd9-c65a3d8bd102>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00425.warc.gz"} |
A Simplified Proof of the Étale Exodromy Theorem for Constructible Sheaves of Sets
Core Concepts
This paper presents a streamlined proof of the étale exodromy theorem for constructible sheaves of sets, drawing inspiration from Grothendieck's Galois theory and offering a more direct approach
compared to existing proofs.
Bibliographic Information:
van Dobben de Bruyn, Remy. "Grothendieck Galois theory and 'etale exodromy." arXiv preprint arXiv:2410.06278 (2024).
Research Objective:
This paper aims to provide a simplified and more accessible proof of the étale exodromy theorem for constructible sheaves of sets, a fundamental result in algebraic geometry.
The author employs concepts from Grothendieck's Galois theory and utilizes profinite categories to establish a correspondence between constructible sheaves and continuous functors on a specific
category. The proof relies on demonstrating the "local fullness" and "local essential surjectivity" of this correspondence.
Key Findings:
• The paper successfully presents a more direct proof of the étale exodromy theorem for constructible sheaves of sets, avoiding the complexities of previous proofs that relied on techniques like
induction on strata, glueing, and lax fiber products.
• The proof highlights the irrelevance of the partial order on the stratification, simplifying the understanding and application of the theorem.
• The use of profinite categories provides a more practical framework for computations compared to the limit of finite categories used in earlier proofs.
Main Conclusions:
The paper offers a valuable contribution to algebraic geometry by providing a simplified and more intuitive proof of the étale exodromy theorem for constructible sheaves of sets. This approach
enhances the accessibility of the theorem and facilitates its application in various contexts.
This work simplifies a fundamental theorem in étale cohomology, making it more accessible and potentially opening avenues for further research and applications.
Limitations and Future Research:
• The paper focuses on constructible sheaves of sets, leaving room for extending the simplified proof to more general constructible sheaves of spaces.
• Exploring the computational aspects of the profinite fundamental category in specific examples could be a potential direction for future research.
How does this simplified proof of the étale exodromy theorem contribute to a deeper understanding of the relationship between algebraic geometry and topology?
This simplified proof of the étale exodromy theorem deepens our understanding of the relationship between algebraic geometry and topology in several ways: Bridging abstract and concrete: By drawing
parallels with Grothendieck's Galois theory, the proof connects the abstract world of ∞-topoi (where the original proof resides) with the more concrete realm of profinite categories. This makes the
étale exodromy theorem more accessible and provides a new perspective on its underlying principles. Highlighting the role of profinite methods: The proof showcases the power of profinite techniques
in understanding constructible sheaves. This reinforces the deep connection between profinite groups, Galois theory, and the topology of schemes. Opening avenues for computation: The use of profinite
categories, as opposed to limits of layered categories, offers a more computationally amenable framework. This paves the way for explicit computations of stratified fundamental categories (Π1(X, S))
in specific examples, further enriching our understanding of the interplay between the geometry of a scheme and its étale topology. In essence, this simplified proof acts as a conceptual bridge,
linking sophisticated concepts in topology with more grounded tools in algebra, ultimately providing a clearer and more approachable understanding of the profound relationship between these two
Could the reliance on a fixed stratification in this proof limit its applicability in scenarios where considering all stratifications simultaneously is crucial?
Yes, the reliance on a fixed stratification in this simplified proof could potentially limit its applicability in scenarios where a holistic view of all stratifications is essential. Here's why: Loss
of information: Fixing a stratification might discard crucial information encoded in the interplay between different stratifications. The original étale exodromy theorem, by working with all
stratifications concurrently, captures these subtle interactions. Limited scope for generalization: Certain applications might necessitate understanding how constructible sheaves behave under
refinements or coarsenings of stratifications. The fixed stratification approach might not readily lend itself to such generalizations. However, the simplified proof still holds value: Foundation for
further development: It could serve as a stepping stone for future research aiming to incorporate the flexibility of considering all stratifications within this simplified framework. Sufficient for
specific applications: Many situations might only require a fixed stratification. In such cases, this proof offers a more direct and computationally advantageous approach. Therefore, while the
reliance on a fixed stratification might pose limitations in certain contexts, the simplified proof remains a valuable contribution, potentially laying the groundwork for future advancements and
proving sufficient for a range of applications.
If Grothendieck's Galois theory can simplify this proof, what other complex mathematical concepts might be clarified by revisiting foundational theories?
Grothendieck's philosophy often involved seeking out elegant and unifying principles underlying seemingly disparate mathematical concepts. His success with Galois theory and the étale fundamental
group suggests that revisiting foundational theories could clarify other complex mathematical concepts: Homotopy theory and higher categories: The use of ∞-categories in the original proof hints at
deeper connections between homotopy theory and algebraic geometry. Revisiting foundational concepts in homotopy theory, potentially through the lens of model categories or simplicial sets, might
offer new perspectives on complex geometric constructions. Noncommutative geometry: Grothendieck's vision extended to a desire to develop a robust theory of noncommutative geometry. Revisiting
foundational concepts in ring theory, representation theory, and category theory could provide insights into the structure of noncommutative spaces and their associated invariants. Motives and
motivic homotopy theory: The theory of motives seeks to capture the "essence" of geometric objects. Revisiting foundational concepts in algebraic cycles, cohomology theories, and category theory
might lead to breakthroughs in understanding motives and their relationship to more concrete geometric constructions. By returning to the foundations and seeking out unifying principles, we might
uncover unexpected connections and simplify our understanding of complex mathematical concepts, just as Grothendieck did with Galois theory and the étale fundamental group. | {"url":"https://linnk.ai/insight/algebraic-geometry/a-simplified-proof-of-the-%C3%A9tale-exodromy-theorem-for-constructible-sheaves-of-sets-PsNMKPvG/","timestamp":"2024-11-03T11:53:07Z","content_type":"text/html","content_length":"261089","record_id":"<urn:uuid:41da1988-a316-460d-8572-e7836f2d1b77>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00549.warc.gz"} |
ax13w - Metamath Proof Explorer
Description: Weak version (principal instance) of ax-13 2372. (Because 𝑦 and 𝑧 don't need to be distinct, this actually bundles the principal instance and the degenerate instance (¬ 𝑥 = 𝑦 → (𝑦 = 𝑦 →
∀𝑥𝑦 = 𝑦)).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 2130, ax11w 2131, and ax12w 2134. (Contributed by NM, 10-Apr-2017.) | {"url":"https://us.metamath.org/mpeuni/ax13w.html","timestamp":"2024-11-04T07:26:42Z","content_type":"text/html","content_length":"8707","record_id":"<urn:uuid:d5c84fee-679d-4208-868f-8fda076f03b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00332.warc.gz"} |
[PhD 2022] Quantum programming with (co)inductive types
Quantum technologies have led to the emergence of a new promising computational paradigm which can efficiently solve problems considered to be intractable on classical computers. Recent developments
highlight the necessity of filling the gap between quantum algorithms and the actual quantum computers on which they have to be executed. As a consequence, quantum programming languages play a key
role in the future development of quantum computing.
However, most existing quantum languages are fairly low-level: all of the quantum phenomena which induce computational speedup (compared to classical computation) are restricted to quantum bits, also
known as qubits, and the operations which allow us to modify qubits are low-level quantum gates. This low-level machinery, comparable to boolean circuits, requires the programmer to focus on
cumbersome architectural details, rather than focusing on algorithmic and high-level patterns. Because of this, there is a strong need to develop high-level quantum programming languages, where
phenomena such as superposition and entanglement, are available at higher quantum types, not just qubits.
Towards this direction, quantum types generalizing standard classical ones have to be introduced and corresponding type systems and programming languages for manipulating them have to be studied.
These types can be inductively defined, e.g., quantum natural numbers and quantum lists, hence allowing us to form superpositions and entanglement patterns of nats, lists, etc., and not just qubits.
Alternatively, they can be coinductively defined, e.g., infinite streams of qubits, which poses difficult research questions for quantum programming language design, due to the infinitary nature of
this data. Designing a suitable formalism to solve these issues would have a significant impact on quantum programming languages and would enable programmers to write high-level and more concise
Project description:
This PhD project aims at studying substructural and quantum type systems with inductive and coinductive types. The applicant will design and study the type-theoretic, operational and denotational
aspects of the corresponding languages and systems. Towards that end, particular attention will be paid to the following tasks:
1. Design a quantum programming language with (co)inductive types and a type-safe operational semantics, where superposition and entanglement are first-class resources available at all types, not
just qubits.
2. Define a suitable denotational (i.e., mathematical) semantics of this language and demonstrate its soundness and adequacy with respect to the operational semantics.
3. Study important behavioral properties of this language, such as productivity and complexity.
References :
[1] P. Selinger. Towards a quantum programming language. Mathematical Structures in Computer Science, 14:4, 2004.
[2] P. Selinger, B. Valiron. A lambda calculus for quantum computation with classical control. Mathematcal Structures in Computer Science, 16:3, 2006.
[3] R. Péchoux, S. Perdrix, M. Rennela, V. Zamdzhiev. Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory. FoSSaCS 2020.
[4] B. Lindenhovius, M. W. Mislove, V. Zamdzhiev. Mixed linear and non-linear recursive types. ICFP 2019.
[5] R. Clouston, A. Bizjak, H. B. Grathwohl, L. Birkedal. Programming and Reasoning with Guarded Recursion for Coinductive Types. FoSSaCS 2015.
[6] R. Atkey, C. McBride. Productive coprogramming with guarded recursion. ICFP 2013.
[7] R. Péchoux. Implicit Computational Complexity: Past and Future. Habilitation thesis. Université de Lorraine, 2020.
[8] M. Avanzini, G. Moser, R. Péchoux, S. Perdrix, V. Zamdzhiev. Quantum Expectation Transformers for Cost Analysis, 2022.
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Numbers of Mutations within Multicellular Bodies: Why It Matters
Department of Ecology and Evolutionary Biology, University of California, Irvine, CA 92697-2525, USA
Submission received: 4 November 2022 / Revised: 19 December 2022 / Accepted: 21 December 2022 / Published: 23 December 2022
Multicellular organisms often start life as a single cell. Subsequent cell division builds the body. Each mutational event during those developmental cell divisions carries forward to all descendant
cells. The overall number of mutant cells in the body follows the Luria–Delbrück process. This article first reviews the basic quantitative principles by which one can understand the likely number of
mutant cells and the variation in mutational burden between individuals. A recent Fréchet distribution approximation simplifies calculation of likelihoods and intuitive understanding of process. The
second part of the article highlights consequences of somatic mutational mosaicism for understanding diseases such as cancer, neurodegeneration, and atherosclerosis.
1. Introduction
Multicellular organisms often begin as a single cell. Similarly, microbial populations often expand from a small number of colonizing progenitor cells. As clones expand by cell division, mutations
inevitably arise. Mosaic multicellular soma or genetically heterogeneous microbial populations follow.
Somatic mutants affect cancer risk [
]. Mutants in microbial clones initiate the spread of faster growing variants, a tumor-like overgrowth [
]. In plants, somatic variability may promote defense against pathogens and herbivores [
This article provides a simple overview of clonal heterogeneity and its consequences. The first part outlines conceptual and quantitative aspects. The most important idea concerns the high diversity
between clones. Most cellular populations have relatively few mutants. Some populations have a lot of mutants.
The second part of this article discusses why it is important to understand the diversity of mutants within and between cellular populations. That part illustrates importance by examples.
The examples include diseases such as cancer, neurodegeneration, and atherosclerosis. Other examples clarify the distinction between how diseases begin and how they spread within bodies. Another key
way in which somatic mutation affects disease concerns the strong tendency for variation in risk between individuals.
2. The Luria–Delbrück Problem
Luria–Delbrück [
] analysis is often used to estimate the mutation rate [
]. By counting mutant cells after exponential growth, one can infer how often new mutations arise. This section introduces quantitative aspects of how mutations accumulate in populations.
Suppose a clone begins with one cell and grows to N cells. Ignoring cell death, such growth requires $N − 1$ cell divisions because each division adds one cell to the population. If the mutation
probability per cell division is u, then the expected number of mutational events is $( N − 1 ) u$, which is approximately $N u$ for large N. However, knowing the number of mutational events does not
by itself tell us the number of mutant cells.
A single mutational event might happen in the first cell division, causing approximately
$1 / 2$
of all cells to carry the mutation, with
$N / 2$
mutants cells in the final population. Or the mutation might happen in a final cell division, causing only one cell to carry the mutation. We must map each mutational event to its time within the
cell lineage hierarchy (
Figure 1
Deleterious mutations slow the rate of cell division in descendants. Advantageous mutations speed cell division. Such changes alter the final number of mutants. The mathematical literature considers
the variety of problems [
]. We may call the general challenge the Luria–Delbrück problem, named after the initial studies.
I limit the following quantitative sections to a few details about the numbers of neutral mutations. Those calculations provide sufficient introduction to the main conceptual background for
interpreting various biological scenarios.
3. Average Number of Mutants
3.1. Intuitive Summary
To calculate the average number of mutants in a clone, start by picking one of the
final cells. The probability that it carries a mutation depends on the number of cell divisions that separate that final cell from the initial progenitor cell. The expected number of cell divisions
$log ( N )$
, using the natural logarithm (see next subsection). With a mutation probability of
per cell division, the mutant probability per final cell is approximately
$u log ( N )$
when the mutation probability is sufficiently small. With
final cells, we expect an average of approximately
mutants in the population [
A human body has more than
$N = 10 13$
cells [
]. Estimates for the somatic mutation rate per gene per cell division vary [
]. Here, I use
$u = 10 − 6$
. The average number of cellular generations is
$log 10 13 ≈ 30$
. So we expect roughly
$3 × 10 8$
mutants for each gene and a frequency of mutants of roughly
$3 × 10 − 5$
. That is a lot of mutant cells but a low frequency of mutations per cell.
A human genome has about $2 × 10 4$ genes and also a lot of noncoding DNA that can influence phenotype. When considering all of the various sites that can be mutated, the overall amount of mutation
in the final population of cells is large. Additionally, the mutational variation between multicellular bodies tends to be big, as discussed below.
Microbial clones can also grow into large populations, also carrying substantial numbers of mutants and large amounts of variation between clones.
3.2. Details
The expected number of cellular divisions can be derived by noting that, starting with one cell, a final population of
arises by continuous exponential growth such that
$N = e r t$
. Thus
is the expected number of cellular divisions from a final cell back to the original cell, in which
is the cellular division rate and
is time.
One interpretation is that the actual number of divisions in different cellular lineages follows a Poisson distribution with mean
. Suppose each lineage with
divisions contributes to part of a cellular expansion with size
$2 k$
. Summing over the Poisson probabilities
$p k$
divisions in a lineage is
$∑ k = 0 ∞ p k 2 k = ∑ k = 0 ∞ λ k k ! e − λ 2 k = e λ = e r t = N .$
4. Variation in Number of Mutants
The way in which mutations accumulate in growing populations inevitably causes a lot of variation between populations. That large variation may often be more important biologically than the average.
For example, disease often concerns what happens in variant or extreme individuals rather than what happens in an average individual. Thinking of the extreme of a probability distribution as what
happens in its tail, we may say that the characteristics of a probability distribution’s tail may be the important attributes for a particular problem.
Additionally, when a distribution has occasional exceptionally large values, the average value does not provide a good sense of what is typical. For example, suppose we observe five sample values,
$800 , 900 , 1000 , 1100 , 9000$. The median is 1000, a good description of central location and typical values, whereas the average is 2560, a poor description of typical values.
5. Distributions with Fat Tails
Figure 1
shows how mutations accumulate in growing populations. When the mutation happens late, as in
Figure 1
a, then a small fraction of the population carries that mutation. Such late-occurring mutations happen frequently, because there are more target cell divisions in which a mutation can arise.
For each generation earlier that a mutation might occur, the number of final cells carrying the mutation doubles and the probability of occurrence declines by
$1 / 2$
, as in
Figure 1
b,c. Here, to illustrate, we are simplifying growth to be a sequence of discrete binary cellular divisions.
Consider a large final population of cells,
. If a mutation happens in the first cell division of the progenitor cell, then
$N / 2$
cells are mutated with probability
, the probability of mutation during a cell division. When a mutation happens in the second cell division from the original cell, then
$N / 4$
cells are mutated with probability
$2 u$
, because there are twice as many target cells. In the third round,
$N / 8$
cells would be mutated with probability
$4 u$
Figure 2
a provides an example in which probability doubles with each halving of the number of mutants.
If we make a log–log plot of the probability of occurrence for each number of mutant cells in the final population versus the number of mutant cells, that plot of the upper tail is a straight line
with a slope of minus one (
Figure 2
b). A straight line of a log–log plot for a probability distribution is a power law. We say that the upper portion of this distribution has a power law tail, or fat tail. The tail is fat because a
power law declines relatively slowly, causing a lot of probability and a high chance of observation to occur for large values, in this case, large values for the total number of mutated cells in the
final population.
This explanation oversimplifies because it focuses on only a single mutational event. An actual expanding population may have many mutational events. However, mutational events early in the growth of
the population are rare because there are few target cells. Furthermore, those early mutations dominate the upper tail because when they occur they carry forward to all descendant cells. So the
simplification does reasonably well for describing the fat upper tail.
This section illustrated the problem by using a discrete binary branching processes for population growth. However, most applications focus on continuously growing populations with overlapping
generations, such as in Equation (
). The next section returns to the continuous model.
6. Distribution of Mutants
No simple formula describes exactly the probability of observing
mutants in a final population of size
. The literature provides various ways of making the calculation or of simulating populations with a computer [
]. It is easy to calculate the average, given approximately in Equation (
). However, as noted in the previous section, the distribution has a long fat tail, associated with much variation between populations. It is often the large variation that matters in application
rather than the average. I discuss some examples in the second part of this article.
6.1. Fréchet Distribution
The prior section noted that the upper tail has a power law shape. However, what does the full distribution look like for the typical assumption of continuous population growth? Recently, I found
that the distribution of mutants closely matches the well known Fréchet probability distribution [
For a Fréchet distribution, the cumulative probability that the number of mutants is less than or equal to
$F ( m ) = e − m ˜ − α ,$
in which
$m ˜ = ( m − β ) / s$
. The probability of being in the upper tail, greater than
, is
$1 − F ( m )$
. The three parameters set the shape,
, the scale,
, and the minimum value,
, such that
$m ≥ β$
6.2. Approximate Fit
Using the following parameters provides a close match between the Fréchet distribution and the Luria–Delbrück process for neutral mutations
$α = e / 2 s = e N u β = N u log N u e − ( 1 + α )$
in which
is the base of the natural logarithm. The Fréchet match to the Luria–Delbrück depends on the single parameter,
$N u$
, the population size multiplied by the mutation rate.
Figure 3
shows the match.
6.3. Intuition
Why would a Fréchet distribution provide a good match for the number of mutants in a Luria–Delbrück process? To begin, think about the final cells in the population after exponential growth. Look
back from those final cells through the multiple rounds of cell division that trace ancestry toward the original progenitor. The mutation farthest back in time from the final cells toward the
progenitor typically dominates in determining the number of mutants in the final population.
That most extreme time tends to follow the extreme value probability distribution known as Gumbel [
]. Then, because the most extreme mutation will subsequently expand by multiplicative growth to determine the final number of mutations, we can use the fact that a multiplicative process substituted
into a Gumbel distribution leads to a Fréchet distribution [
7. Mutation Rate
Initial mathematical development of the Luria–Delbrück process focused on estimating the mutation rate [
]. Over time people have discovered links between how mutations accumulate in growing populations and topics such as cancer and neurodegeneration. I take up those topics in later sections. This
section begins with the mutation rate.
Suppose one wishes to estimate the rate, u, of neutral mutations in a particular bacterial gene. We start by seeding a population with one bacterial cell and allowing exponential growth until there
are N cells. We then measure the number of mutant cells, m, in the final population.
In this case, the existence of a mutation in a final cell is measured phenotypically by growing the cell under different conditions. For example, the initial environment may provide a particular
nutrient, making a specific metabolic reaction within the cell unnecessary. When cells are subsequently grown in the absence of that nutrient, the previously unnecessary metabolic reaction becomes
necessary for growth. Thus, we can infer mutations during the initial population growth that inactivate the metabolic function required during the subsequent phenotypic test.
The mutation rate is the number of mutational events divided by the number of cell divisions. If, for simplicity, we ignore cell death, then there are approximately N cell divisions to go from the
initial cell to the final N cells.
Next, we need to estimate the number of mutational events. We measured m mutant cells among the final N cells. Different numbers of mutational events are consistent with a final number m. For
example, a single mutational event that happened $log ( m )$ cell divisions back from the final cell would grow into $e log ( m ) = m$ final mutant cells. Or two mutational events that are $log ( m /
2 )$ cell divisions back from the final cells would also lead to m final mutant cells. Or there might be m mutational events, each occurring in the final cell division. Various combinations of
mutational events and subsequent cellular growth would add to the same final value of m.
In other words, the number of final mutant cells provides some information about the number of mutational events but is not by itself sufficient to provide an exact value. We can gain more
information by measuring the numbers of final mutant cells in several replicate populations.
For example, in
Figure 3
, notice how the number of mutants at the median
level in height for each plot changes as
$N u$
changes. If we estimate the median value for the number of mutants from a sample of populations, then we can then match that median to the value of
$N u$
for a corresponding distribution. A median of
$10 5$
would, for example, match
$N u = 10,000$
in the right panel of
Figure 3
. We know the value of
, so we can infer the value of
. Using the median gives a good rough approximation.
Other approaches may be more practical or provide more information. However, the idea is the same. In all cases, we are comparing observed measures for the number of mutants in the final population
to the theoretical distribution based on the Luria–Delbrück process [
8. Genetics of Human Populations
Suppose we want to estimate how many mutations have occurred in a particular human population. For a focal stretch of DNA, we can measure how many of those sequences carry a mutation in the current
population. Similarly, we can think of recombination events as changes that we count as mutants. For example, Hästbacka et al. [
] estimated the amount of recombination and linkage in a DNA region that contains the diastrophic dysplasia (DTD) gene by studying the modern Finnish population.
In such applications to a single human population, we can obtain only a single measurement of the number of mutants or the number of recombinants. How much information does a single sample contain?
In theory, how much more precise would estimates be if we could obtain additional independent samples? Evaluating these questions provides broad insight into what we may expect to see in natural
populations and in how much information we can obtain during particular kinds of studies.
Hästbacka et al. [
] used Luria–Delbrück theory to analyze the average number of events in their study population. However, as we discussed in prior sections, the average often poorly represents the Luria–Delbrück
process because of the occasional very large values. Instead, analyzing the most likely value of
$N u$
given the data provides a better approach. We typically know
, so an estimate of
$N u$
provides an estimate of
, the mutation rate.
9. Likelihood
Statistical tools for likelihood analysis of Luria–Delbrück processes have been given in the literature [
]. However, the prior literature did not have an explicit form of the Luria–Delbrück distribution to work with. We can take advantage of the new Fréchet approximation in Equation (
). The amount of error from that approximation when used in likelihood analysis has not yet been quantified. For now, this presentation is best regarded as a conceptual introduction rather than an
updated statistical procedure.
To develop likelihood analysis, we must rewrite the Fréchet cumulative density function in Equation (
) as a probability density function, which simply means that we need the derivative of
$F ( m )$
with respect to
, written as
$f ( m ) = α s m ˜ − 1 − α e − m ˜ − α .$
All of the parameters can be expressed in terms of the single parameter,
$N u$
, as in Equation (
). Suppose we have data on the numbers of mutants in different samples,
$m 1 , m 2 , …$
. Then the log-likelihood of a particular parameter estimate,
$N u$
, given the data, is
$∑ i log f ( N u | m i ) ,$
in which
$f ( N u | m i )$
is given by Equation (
). For example, if we have the single observation
$m 1 = 10 5$
mutants, then
Figure 4
a shows the log-likelihood for
$N u$
. The relatively long left tail arises from the fact that smaller values of the mutation rate,
, occasionally give rise to large values of
, whereas bigger values of
rarely give rise to smaller values of
. Thus, the information in an observed value of
sets a strong upper bound on
but does not strongly bound lower values of
Figure 4
b illustrates three interesting points. First, for the blue curve associated with the single observed value
$10 2$
, the range of
$N u$
values is broader than for a single observed value
$10 5$
. Smaller numbers of mutational events cause greater variation, with most populations having few final mutants and a few populations having a lot of mutants.
Second, two observations of $10 2$ shrink the range of estimated values for a given confidence level by a bit less than one-half.
Third, comparing the pair of observations $10 2 , 10 2$ versus $10 2 , 10 4$, the estimated values change very little. The smaller observed value dominates the information. Once again, a small
mutation rate can be highly consistent with a large observed value for the number of mutants, whereas a large mutation rate is weakly consistent with a small observed value for the number of mutants.
10. Somatic Mosaicism and Disease
Animal bodies often arise from a single ancestral zygote. Cell division produces a large cellular population. The abstract Luria–Delbrück process provides a starting point for thinking about how
mutations accumulate in such bodies, creating somatic mosaicism. Although actual development is more complex than the simple model, we can get a rough sense for numbers of mutant cells.
Prior articles have reviewed links between mosaicism and disease [
]. Here, I focus on key conceptual issues.
10.1. Mutant Cells and the Risk of Disease
Cancer typically begins in a small piece of tissue. A local tumor develops and then disperses to cause widespread disease. In this case, each small piece of tissue that carries mutant cells poses an
independent risk. The overall risk of cancer later in life rises roughly in proportion to the number of mutant cells produced during development through a Luria–Delbrück process [
How does the somatic mosaicism induced by the Luria–Delbrück process affect other diseases? It depends on the mechanisms that link the origin of disease to the onset of widespread symptoms [
Consider neurodegeneration. Certain inherited mutations predispose to disease [
]. An inherited mutation is in every cell. What happens if the mutation is in 10% of cells, or 1% of cells, or 0.001% of cells? Can neurodegeneration start in a small local piece of tissue and then
spread widely? If so, then a very small fraction of mutant cells could be associated with the origin of the disease. The more mutant cells, the greater the chance that disease starts locally and
Ten or more years ago, just a few people emphasized this idea of local origin and subsequent spread for neurodegeneration [
]. That idea for the origin of disease was then linked to somatic mosaicism [
]. In recent years, the link between somatic mosaicism by a Luria–Delbrück process and neurodegeneration has motivated various studies and positive commentary [
]. However, relative importance for that link remains an open problem.
Atherosclerosis provides another interesting example. That disease begins with aberrant physiological changes and tissue expansion in plaques that line the arteries [
]. How much of atherosclerotic disease arises from Luria–Delbrück processes and somatic mosaicism? Current data provide clues but no clear answer [
For other diseases, the questions are the same [
]. To what extent could local changes in small pieces of tissue initiate widespread dissemination of disease? What role do mutations play in the risk of those initiating local tissue changes? Can
local changes in hormone-secreting tissues cause disseminated disease?
In some cases, mutations may arise early in development, causing a high frequency of mutant cells. Such individuals may have disease phenotypes similar to cases of inherited disease. However, when
tested genetically by analysis of a blood sample, some of those individuals will not have the mutations in their blood cells. They may be scored genetically as noninherited cases of disease but
evaluated clinically with the symptoms of inherited cases.
10.2. Variation between Individuals
How much of the variation in disease risk between individuals arises from somatic mosaicism? In theory, the Luria–Delbrück process can cause significant variation in the somatic mutation burden for
particular tissues when comparing between individuals. Empirically, it remains challenging to measure that variation and link the variation to disease. It may be particularly interesting to analyze
those individuals who present with disease onset and symptoms typically associated with an inherited mutation but who lack the inherited mutation. The recent increase in interest for such problems
and the advances in single cell genomics may eventually provide insight into these issues.
11. Conclusions
This article provided a simple intuitive introduction to the Luria–Delbrück process and some of its consequences. The literature includes many other applications and developments of theory to cover a
variety of more realistic or complex assumptions [
On the biological side, I mentioned potential applications of the theory to cancer, neurodegeneration, and atherosclerosis. Many other diseases and cellular processes can potentially be influenced by
somatic mutation. Skin diseases are perhaps the best studied and most interesting in relation to the Luria–Delbrück process [
This research was funded by The Donald Bren Foundation, National Science Foundation grant DEB-1939423, and DoD grant W911NF2010227.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The author declares no conflict of interest.
Figure 1.
Number of mutants in a growing population depends on the time of the first mutation. (
) When the first mutation happens in the last cell division, then only one cell
$m = 2 0$
carries that mutation. In this case, there are
$p = 2 3$
different final cells that could be mutated in the final round of cell division, so such mutations confined to a single cell will be relatively common. (
) When the first mutation happens in the second to last cell division, then two cells
$m = 2 1$
carry that mutation. In this case, there are
$p = 2 2$
different final groups of cells that could be mutated, so such mutations confined to two cells will be
$1 / 2$
as common as for singleton mutations. (
) When the first mutation happens in the third to last cell division, then four cells
$m = 2 2$
carry that mutation. In this case, there are
$p = 2 1$
different final groups of cells that could be mutated, so such mutations confined to four cells will be
$1 / 4$
as common as for singleton mutations. Focusing on single mutational events, every doubling for the final number of mutant cells decreases the probability of occurrence by
$1 / 2$
. This discrete-time model matches Haldane’s formulation of the Luria–Delbrück problem [
], used here for illustration because of its simplicity. Most modern analyses use a continuous time formulation [
]. In the text, the mixture of references to the discrete and continuous formulations leads to a mixture of comments about mutation probabilities and mutation rates. When the underlying formulation
is ambiguous, I use mutation rate.
Figure 2.
The probability of observing a particular number of mutants in the final population. (
) Probability declines by
$1 / 2$
for each doubling in the number of mutants in the final population. (
) A log–log plot of probability versus mutant number has a slope of minus one. This example follows from a discrete binary pattern of cellular division with a single mutational event, as in
Figure 1
. The actual log–log slope for a Luria–Delbrück process is approximately
$− ( 1 + e / 2 ) ≈ − 2.36$
(from Equation (
)). Part of the increase arises because multiple mutational events cause a more rapid decline at the extreme of the earliest mutations and greatest final mutant numbers.
Figure 3.
Cumulative probability distribution for the number of neutral mutants. Each population starts with one cell and grows to
cells. Mutations occur at rate
. The blue curves show the Luria–Delbrück distribution calculated by the simu.cultures computer simulation of the R package rSalvador [
]. The orange curves show the Fréchet distribution in Equation (
). Reprinted from from Figure 1 of Frank [
]. See that article for further details. The mathematical reason for the close match between the Luria–Delbrück distribution and the Fréchet distribution arises by linking two separate studies.
Kessler and Levine [
] showed that the Luria–Delbrück distribution converges to a Lévy
-stable distribution for large
$N u$
. Separately, Simon [
] showed the close match between the Lévy
-stable distribution and the Fréchet distribution. Using the Fréchet distribution provides a benefit because no explicit mathematical expression exists for the Lévy
-stable probability distribution.
Figure 4.
Log-likelihood for the parameter
$N u$
given the values of the data shown in the legends of each plot. A constant was added to all log-likelihood values to shift the curves up so that the peaks are at a value of three. The zero values
give the log-likelihood confidence intervals for which the most likely value is
$e 3 ≈ 20$
times more likely that the lowest values shown at the edges of the intervals. (
) For a single observation of
$m = 10 5$
, the most likely value is approximately
$N u = 10 4.05$
. (
) Log-likelihood curves for different data combinations. The first (blue) curve has a single observed value, whereas the other two curves arise from pairs of observed values. These likelihood curves
derive from the Fréchet approximation in Equation (
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Frank, S.A. Numbers of Mutations within Multicellular Bodies: Why It Matters. Axioms 2023, 12, 12. https://doi.org/10.3390/axioms12010012
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Frank SA. Numbers of Mutations within Multicellular Bodies: Why It Matters. Axioms. 2023; 12(1):12. https://doi.org/10.3390/axioms12010012
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Variograms and Model Selection
Christen H. Fleming and Justin M. Calabrese
In this vignette, we walk through data preparation, variogram analysis, and maximum likelihood estimation.
Data Preparation
We highly recommend that you get your data onto Movebank. This will help ensure that your data are of the correct format for ctmm, help you identify outliers, and you can keep your data completely
private if you wish. ctmm requires that your dataframe conforms to Movebank naming conventions (see help(as.telemetry)). The next step is then to import your MoveBank csv file:
yourAnimals <- as.telemetry("yourAnimalsMoveBank.csv")
Alternatively, if you want to clean your csv file first, you can import it as a data frame
yourAnimalsDF <- read.csv("yourAnimalsMoveBank.csv")
and then edit the data frame before converting it into a telemetry object for ctmm via
yourAnimals <- as.telemetry(yourAnimalsDF)
as.telemetry also works on Move objects, which can be useful as the move package interfaces directly with MoveBank through R (see help(move::getMovebankData)).
A flat projection is necessary and for most species the default two-point equidistant projection will be fine. However, you can provide any PROJ.4 formatted projection with the projection argument
(see help(as.telemetry)). A single fixed projection should be used if you are going to plot groups of individuals that span multiple MoveBank files. This is done by default if multiple individuals
are included in a single data frame.
The output of as.telemetry will be an individual telemetry object or list of telemetry objects, depending on how many individual animals are in your csv file. The basic structure of a telemetry
object is a data frame with columns t for time in seconds, and x and y for the projected locations in meters. Messages generated by as.telemetry may warn users about improper time formatting or
spurious relocations. Possible outliers can be examined with the help of the outlie function.
Our example buffalo data is already prepared into a list of telemetry objects. Let us look at the first buffalo and then every buffalo:
## DOP values missing. Assuming DOP=1.
## DOP values missing. Assuming DOP=1.
## DOP values missing. Assuming DOP=1.
## DOP values missing. Assuming DOP=1.
## DOP values missing. Assuming DOP=1.
## DOP values missing. Assuming DOP=1.
## DOP values missing. Assuming DOP=1.
Looking at the raw movement tracks is a good way to pick out any obvious migratory behaviors. In the future, we will have migration models to select, but for now all of our models are range resident
and so only those portions of the data should be selected. These buffalo all look fairly range resident, and so we can move on to variograms.
Variograms are an unbiased way to visualize autocorrelation structure when migration, range shifting, drift, or other translations of the mean location are not happening. When drift occurs in the
data, then the variogram represents a mixture of both the drift and the autocorrelation structure, each of which contains distinct movement behaviors. In the future, we will have models that can
allow for drift, but the current models assume range residence, which we can check with the variogram.
The first plot is zoomed in to the short lag behavior, while the second plot is zoomed out. You can do this on the fly with zoom(Cilla) in R-studio. The variogram represents the average square
distance traveled (vertical axis) within some time lag (horizontal axis).
For the long range behavior we can see that the variogram flattens (asymptotes) at approximately 20 days. This is, roughly, how coarse you need to make the timeseries so that methods assuming
independence (no autocorrelation) can be valid. This includes, conventional kernel density estimation (KDE), minimum convex polygon (MCP), conventional species distribution modeling (SDM), and a host
of other analyses.
The asymptote of our variogram is around 23 square km, and the fact that it takes roughly 20 days for the variogram to asymptote is indicative of the fact that the buffalo’s location appears
continuous at this timescale. This is also, roughly, the time it takes for the buffalo to cross its home range several times.
In the next sections we will “fit” models the variograms. This is not a rigorous statistical fitting, but a good way of choosing candidate movement models and guessing at their parameters. The
variogram-fit parameter guestimates will then be fed into maximum likelihood estimation, which requires good initial guesses for non-linear optimization.
Variogram Fitting the Hard Way
For pedagogical reasons we first “fit” models to the variogram by hand, while later we will use a much easier method. we can manually guesstimate some continuous-time models for the aforementioned
behavior with the commands
where for both the models m.iid and m.ou, sigma (σ) is the asymptotic variance. In the Ornstein-Uhlenbeck (OU) model m.ou, tau (τ) is a single timescale that governs the autocorrelation in position
and dictates the animal’s home-range crossing time. The independent and identically distributed (IID) null model m.iid has no autocorrelation. Notice that all units are in meters and seconds, as
output by the utility function %#% (see help("%#%").
The IID model m.iid is obviously incorrect and in the zoomed in plot we can also see that the OU model m.ou is incorrectly linear at short lags, whereas the empirical variogram actually curves up for
an hour or two before it becomes linear. Let us introduce a model that incorporates this behavior.
m.ouf <- ctmm(sigma=23 %#% "km^2",tau=c(6 %#% "day",1 %#% "hour"))
title("Ornstein-Uhlenbeck movement")
title("Ornstein-Uhlenbeck-F movement")
The confidence intervals at short lags are also very narrow, though both of these models look the same at coarser scales and so the discrepancy is only revealed by high resolution data.
The Ornstein-Uhlenbeck-F (OUF) model m.ouf introduces an additional autocorrelation timescale for the animal’s velocity, so that it more closely matches the initial behavior of the variogram. The
initial curve upwards tells us that there is continuity in the animal’s velocity at this timescale. Conventional Markovian animal movement models do not capture this, which leads to the same kind of
bias and underestimation of confidence intervals as when ignoring autocorrelation entirely.
The linear regime of the variogram (regular diffusion) is just as important as the asymptotic regime. In the linear regime it is reasonable to assume a Markovian model as with step selection
functions (SSF) and Brownian bridges (BB). Therefore, the variogram has informed us as to how much we need to coarsen our data for it to be appropriate in many common analyses that neglect various
aspects of movement.
Variogram Fitting the Easy Way
The R-studio function variogram.fit(SVF) is much easier to use than guestimating the model parameters by hand as we did above. variogram.fit gives you sliders to choose the most visually appropriate
parameters and save them to a global variable (GUESS by default).
Irregular Sampling Schedules
Random gaps in the data are acceptable and fully accounted for in both variogram estimation and model fitting. However, if under any condition the sampling rate changes during data collection, then
you will have to account for that in the variogram with the dt argument. In the following example, the collars were programmed to cycle between 1, 5, and 25-hour sampling intervals.
With small amounts of highly irregular data, you may also want to try fast=FALSE.
Pooling Variograms
If multiple individuals exhibit similar movement behaviors, then we can pool their individual variograms to create a more precise population variogram. You should be careful though, if the individual
movement behaviors and sampling schedules are not identical, then there will be discontinuities at lags where one timeseries runs out of data.
Non-Stationarity in Stationary Variograms
Non-stationary behaviors, like a seasonal change in variance, is averaged over in the variogram. Moreover, if we fit a stationary model to non-stationary data, we are estimating an average effect.
For instance, if an animal rests at night and diffuses at some rate D during the day, then without modeling the rest behavior we estimate an average of zero and D. Its not terribly detrimental to
average over frequently repeated non-stationarity, but if an animal migrates once in a dataset then this behavior really needs to be in the model. These kinds of models will be included in future
versions of ctmm.
Alternatively, you can also break up the data by hand into multiple behaviors and fit each behavior individually. This is highly recommended for migratory species, at present.
Maximum Likelihood
Maximum Likelihood Fitting the Hard Way
Here we take our guestimates from variogram fitting the hard way and perform model selection manually. Later we will finish everything off the easy way. First let us fit each of our proposed models
m.iid, m.ou, m.ouf, store the corresponding best-fit result in M.IID, M.OU, M.OUF, and then compare some of their outputs.
## $name
## [1] "IID anisotropic"
## $DOF
## mean area diffusion speed
## 3527 3526 0 0
## $CI
## low est high
## area (square kilometers) 356.9248 369.0051 381.2837
## $name
## [1] "OU anisotropic"
## $DOF
## mean area diffusion speed
## 5.430229 8.231787 3366.935623 0.000000
## $CI
## low est high
## area (square kilometers) 181.703274 414.867231 742.630124
## τ[position] (days) 7.242335 16.556147 37.847739
## diffusion (square kilometers/day) 2.632644 2.723882 2.816652
## $name
## [1] "OUF anisotropic"
## $DOF
## mean area diffusion speed
## 10.73353 18.13594 902.23375 3445.13306
## $CI
## low est high
## area (square kilometers) 239.647372 403.458931 609.241527
## τ[position] (days) 4.438956 7.505369 12.690049
## τ[velocity] (minutes) 39.607116 42.069009 44.683928
## speed (kilometers/day) 13.820458 14.055146 14.289780
## diffusion (square kilometers/day) 5.284545 5.647059 6.021428
Notice how tiny the (Gaussian) area uncertainty is in IID model M.IID. Let us look into some details of the models.
## ΔAICc ΔRMSPE (m) DOF[area]
## OUF 0.000 352.8545 18.135945
## OU 1565.291 779.5547 8.231787
## IID 38112.149 0.0000 3526.000000
AICc is the (linearly) corrected Akaike information criteria. AIC balances likelihood against model complexity in a way that is good if we want to make optimal predictions. A lower AIC is better.
Getting the AIC to go down by 5 is great, while getting the AIC to go down by 10 is awesome. Our AIC is going down by thousands.
The fit parameter DOF[mean] is the number of degrees of freedom worth of data we have to estimate the stationary mean parameter, assuming that the model is correct. Notice that the IID model
perceives thousands of independent data points, while the autocorrelated OU and OUF models only see a handful of independent data points. This is why the IID model produced tiny confidence intervals
on the predicted (Gaussian) area.
Maximum Likelihood Fitting the Easy Way
If you have a complex, hypothetical model in mind, say the OUF m.ouf as you would get from variogram fitting the easy way, then you can perform model selection more conveniently with the ctmm.select
function. ctmm.select considers the initial guess (hypothesis) and then iterates this model to select the best model based upon an information criteria.
## ΔAICc ΔRMSPE (km) DOF[area]
## OUF anisotropic 0.00000 1.776671 18.135985
## OUF 67.75084 1.912111 17.912097
## OU anisotropic 1565.29067 2.203374 8.231784
## OUf anisotropic 1863.87988 0.000000 345.919726
The isotropic and anisotropic (isotropic=FALSE) flags correspond to circular and elliptical covariances respectively—an option we did not consider above. The OUf model is a special case of the OUF
model where the two autocorrelation timescales, τ position and τ velocity, cannot be distinguished. This model is usually only relevant for short tracks of data. The IID model was never considered
here by ctmm.select because it first requires selecting OU over OUF in the nested model hierarchy. See help("ctmm") for more options.
Variograms as a Disgnostic for Maximum Likelihood
Now its time to make sure that our selected model is explaining the most significant features of the animal’s movement. Let us plot the variogram again with our fit models
title("zoomed out")
title("zoomed in")
Notice that the purple OU model M.OU is significantly biased downward and is underestimating diffusion. This is because the continuous-velocity behavior at short time lags, which M.OU does not
account for, is throwing off the estimate. The IID model M.IID is ignoring autocorrelation completely, while the OU model M.OU is ignoring autocorrelation in the buffalo’s velocity.
While the OUF M.OUF is the selected model among all candidates, M.OUF looks slightly biased upwards in comparison to the variogram fit m.ouf. Some of this is due to sampling variability, which be
(partially) remedied in the variogram method by increasing the res argument. Differences here can also arise from not accounting for telemetry error, which is unfortunately not annotated in this | {"url":"https://cran.radicaldevelop.com/web/packages/ctmm/vignettes/variogram.html","timestamp":"2024-11-10T21:17:13Z","content_type":"text/html","content_length":"314440","record_id":"<urn:uuid:48b6bb71-5d89-43ff-8186-930cf6781a09>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00685.warc.gz"} |
“I don’t speak with economists
it’s a waste of time
I will do an exception for you
because you seem a bad economist.”.
Concursante (The Contestant, 2007). A movie by Rodrigo Cortes.
"These are the rules: I speak, you listen
there will be no questions
I don't know your case
but I know this
there is only a single case, a single situation
if you pay attention probably you will learn something
I have chickens, you have tomatoes
I want tomatoes, you want eggs
what can we do?
an exchange
a simple exchange
for example, an egg for a tomato
things were this way at the beginning
it's clear that tomatoes sometimes are good, sometimes not
and if I want to buy a horse
it's not clear how many eggs I need
but if we refer to a bit of gold
gold is beautiful, shining and it's rare, is valuable
then we can make a conversion table
if a dozen eggs it is equal to a nugget of gold
and a horse costs one hundred nuggets,
it is easy to see that a horse is equivalent to one hundred dozen eggs
simple, isn't it?
gold becomes a currency of exchange
to simplify
we can no longer buy a horse with eggs
we change eggs with coins and it's done
beautiful, isn't it?
This is the first step
everything is more or less the same
simply now we need gold to buy things
milk, meat, clothes, tools
finally, the person who created the system has a place where he keeps the gold
this person is an altruist, of course
he doesn't want to sell us his gold, he rather lends it
for example, he lends to me ten coins
for twelve months
asking in return a small interest
say, ten per cent
so he threatens his gold while I don't risk anything
therefore he will need a guarantee in case I am not able to honor the agreement
in your case, you will mortgage your property
so the man will lend ten gold coins
for your carrots in exchange? no
you can keep your production of carrots and tomatoes
you don't have to sell anything, you continue to work
just, you should give back eleven coins in a year
ten coins plus the interest
you can sell all your products
and you have a year
if you fail to honor the agreement the bank will take the property. Alright?
What is the problem?
we assume that the bank has a total of one hundred of gold coins
that is the amount of gold that exists
one hundred coins, nothing more
apart the good man there are other ten people:
you, me, a blacksmith, a seamstress, a policeman ... ten people in total
each one needs to buy gold and each one asked for a loan
ten coins for each one for a total of one hundred coins
the banker gave us all his gold
with absolute generosity
in exchange for what?
a mere ten percent?
a dime per person?
it's right!
According to Pythagoras, we have a problem
if each one of us must pay eleven coins in twelve months
where we take them?
eleven coins for each one are one hundred and ten coins
this means that there are ten coins of interest that we will never pay
never, whatever happens
there are no problems, the bank was created to facilitate things
there is a reasonable solution
do not worry
now you pay only the interests
a coin
I will wait,
next year you will pay the initial amount
the ten coins, in short
so if one pays a coin will remain with nine coins
so if we have to pay ten coins
at the end of the year we will have the same debt and less money.
A coin in less than last year
so if we continue for another ten years
assuming that we have to pay only the interests
in the end we will have nothing
while we still have the original debt to be paid
the bank will have recovered all the gold
we will have nothing
and still have the initial debt to be paid
one hundred coins that all of us we fail to pay ever
simply because there are none
we lose land, livestock and the food
mortgaged as collateral ten years ago
in ten years, the bank will recover all his money
plus all our property
while we do not have anything
absolutely nothing
In practice we become slaves of the bank
and for what?
for nothing, and in return of nothing.".
In Spanish. Requires Flash Player | {"url":"https://www.pacquola.org/movies/concursante/","timestamp":"2024-11-11T07:53:44Z","content_type":"application/xhtml+xml","content_length":"25030","record_id":"<urn:uuid:38609ffd-abd1-4f23-a9e8-964949f46c5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00220.warc.gz"} |
two and four ways of looking
Errol Morris’ piece on Thomas Kuhn, Saul Kripke, Pythagorus, and incommensurability is a giant brain bender, and it forced me to back the bus up many years. i read The Structure of Scientific
Revolutions a long time ago as an undergraduate crashing a seminar led by Todd Gitlin, but i was taught the weird concept of the square root of 2 before i could even drive. it wasn’t anything
profound, but i was mesmerized at how a number could go on forever, without repeating itself. forever, back then, was the length of time it took for my friend to make a cassette recording of The
Clash’s Sandinista, but i still couldn’t grasp the idea. really? forever?
it was so beautiful too, a right triangle with sides of 1. so small, so simple, yet a little sad. that’s all it gets, the three sides, with the funny symbol over the 2.
i imagined standing at the point in space designated by the number 1.414, and looking over the chasm at the number 1.415, and thinking that “forever” happened in between those numbers. it took me a
long time to understand that there are way more irrational numbers than rational ones; it’s just that the rational numbers are the ones you can count. it’s sort of like the Republican’s view on
redrawing district lines; they want to consider only the people who can vote.
back then i wondered why the other commonly known irrational numbers π and e seemed more practical, for instance both π and e had special names, whereas the square root of 2 was just the square root
of 2. e has something to do with how much your credit card charges you when you don’t pay your bill, and anyone who’s ever tried to bake a charlotte from scratch knows the power of using π to figure
out how long to make the ladyfinger piece that will wrap around the entire cake. but poor old dowdy square root of 2? not much, unless you spend a lot of time in a country that still uses legal-sized
paper and want to know who in their right mind thinks that 8.5×14″ paper is useful. well, it isn’t. (most of the world uses paper based the aspect ratio of the square root of 2. this means when a
piece of paper is folded in half in those countries, the resulting piece of paper has the same aspect ratio as the parent, and it corresponds to the next size down in terms of the paper tray. whereas
our aspect ratios are all different for our commonly used papers, and all retarded – just look at 8.5×5.5!).
so, in honor of Morris, language, meaning, and the square root of 2, here’s two ways of looking at the same window:
two ways of looking at a pork tenderloin stuffed with pistachios and yogurt, with a coffee-cumin rub:
and… four ways of looking at the same piece of sandwich bread floating in a stream: | {"url":"http://moonquake.org/2011/03/two-and-four-ways-of-looking/","timestamp":"2024-11-14T21:50:48Z","content_type":"text/html","content_length":"50738","record_id":"<urn:uuid:74fea3b9-fc2b-4f9a-8914-8b9df008f057>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00282.warc.gz"} |
Sector Area Calculator
Calculate the area of sector for given value which can be in radian or in degree.
one degree is equal to 0.0174533 radian (approx), simlarlly one radian is equal to 57.2958 degree (approx).
The fraction is determined by the ratio of the arc length to the entire circumference.
$$ \text {The area of the circle } = \pi \times radius^{2} $$ $$ \text {and the circumference circle} = 2 \pi \times radius$$ Now we can use of the angle is clearly delineated, therefore we can
once around
(2π) with the subtended angle and we get the formula we needed for the sector as follows: $$ Area = (\frac {Angle}{2}) \times radius^{2} $$ Let's take the example : radius is 7 cm and angle is 60
, now we can convert the 60
into radian value as follow:
Since 360
in radian = 2π, therefore 60
in radian = 60 * (2/360) =
= 0.3333π.
= 0.33333 X 3.141592654 = 1.047619
We can enter the value in both form as degree (60) or as radian (1.047619).
The area of sector can be calculated using following formula $$ \text {Area of sector = } \pi r^{2} \cdot \frac {\theta^{o}}{360} $$ where π is 3.141592654, r is radius of thr circle and θ is angle
in degree | {"url":"https://eguruchela.com/math/Calculator/sector-area","timestamp":"2024-11-06T08:04:37Z","content_type":"text/html","content_length":"14927","record_id":"<urn:uuid:6673e752-4a47-496e-b666-c052917a20e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00881.warc.gz"} |
The traits class Env_sphere_traits_3 models the EnvelopeTraits_3 concept, and is used for the construction of lower and upper envelopes of spheres.
Note that when projecting the intersection curve of two spheres (a circle in 3D) onto the \( xy\)-plane, the resulting curve is an ellipse. The traits class is therefore parameterized by an
arrangement-traits class that is capable of handling conic curves - namely an instantiation of the Arr_conic_traits_2 class-template - and inherits from it.
The conic-traits class defines a nested type named Rat_kernel, which is a geometric kernel parameterized by an exact rational type. Env_sphere_traits_3 defines its Surface_3 type to be constructible
from Rat_kernel::Sphere_3. Namely, it can handle spheres whose center points have rational coordinates (i.e., of the type Rat_kernel::FT), and whose squared radius is also rational. The Surface_3
type is also convertible to a Rat_kernel::Sphere_3 object.
The Xy_monotone_surface_3 type is the same as the nested Surface_3 type. The traits-class simply ignores the upper hemisphere when it computes lower envelopes, and ignores the lower hemisphere when
it computes upper envelopes. | {"url":"https://doc.cgal.org/5.4.4/Envelope_3/classCGAL_1_1Env__sphere__traits__3.html","timestamp":"2024-11-14T07:14:16Z","content_type":"application/xhtml+xml","content_length":"10878","record_id":"<urn:uuid:384e9089-d171-4379-9489-27b0634ee346>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00343.warc.gz"} |
Two Compactification Theorems
Two Compactification Theorems
A Brief History of Spread-out Spaces.
Photo by Ridwan Meah on Unsplash
Unless you’ve studied some topology, you probably haven’t heard of a “compact space.” This is a type of space (I’m leaving that term vague for now) that has a lot of really nice properties.
In the early 1900’s, people wanted to understand how to generalize many of the most useful theorems from Calculus to more general situations, such as the Extreme Value Theorem (continuous functions
on closed intervals attain their maximum and minimum values).
Compact spaces tend to take up a major portion of a first course on topology, so we definitely will not get into all the subtleties. The point is that tons of theorems have been proved about compact
spaces, and they’re generally easier to work with.
But most spaces aren’t compact, and so there’s a whole class of theorems trying to figure out when and how you can “compactify” them, and what sorts of properties are preserved after doing this. I’ll
give more details on that later.
This article will provide some general idea of what a compact space is, and then go through the two most famous compactification theorems.
Compact Spaces
The definition of a compact space is quite abstract if you look it up online or in a textbook. The definition gives you pretty much no sense of what it is. We’ll use a definition that is almost
equivalent (it’s equivalent in all but the most bizarre situations).
We’ll say a space X is compact if any infinite collection of points accumulates somewhere (in the space). In other words, if you try to “spread out” infinitely many points, you can’t do it. They will
bunch up somewhere.
This way of thinking of it actually makes the term “compact” sound like the right term.
Here are some examples.
The real number line is not compact because you can spread out infinitely many points by putting them on the integers.
A finite line segment containing the endpoints is compact. It should be intuitively obvious that if you try to put infinitely many points in a finite space like that, then they will bunch up
somewhere. Proving it is a little subtle (and would require a more rigorous definition of “accumulation point”).
A finite line segment not containing the endpoints (an open interval) is not compact. Take for example (0,1).
The reason this is not compact is that the infinite collection of points given by 1/n (starting at n=2) accumulates to 0, but 0 isn’t in the space!
The set {1/2, 1/3, 1/4, 1/5, …} stays spread out in the context of this space in the following sense. Given any point of (0,1), I can find a small area of the space surrounding that point containing
no element of the set.
Notice we can’t do that with the closed interval that contained the endpoints because 0 is in the space and no matter how small the area I pick near 0, it will contain infinitely many points of {1/2,
1/3, …}.
Now, I know someone will object here because (0,1) doesn’t feel spread out in any traditional sense. That’s true. This is just trying to give you a sense of the definition of compact.
It turns out that when you make spaces out of the standard real number coordinate system, ℝⁿ, there’s a very simple way to classify compact spaces. They are precisely the closed and bounded subsets.
Bounded just means you can contain the whole thing is some big ball. It makes sense that bounded is needed for compact because if the subset went off to infinity, we could spread out our points in
that direction without them accumulating.
Closed is a subtler topological notion. It basically means that any time some points accumulate, that accumulation point is actually in the set.
When you use these definitions, it’s basically a tautology that compact is equivalent to closed and bounded. But I’ll take this moment to remind you again that these aren’t quite the definitions
you’ll find in textbooks.
In the real world, many “spaces” that we care about don’t come about by carving out a set in ℝⁿ. I wrote about many of these here:
The Brilliance of the Yoneda LemmaA beautiful revolution of the concept of “space”medium.com
In these more abstract cases, it can be hard to tell if a space is compact. Figuring out equivalent and easy-to-check conditions for compactness is a big chunk of many topology textbooks, so let’s
move on.
Most spaces aren’t compact, and so we want to figure out how we can put the space inside of a compact space in an understandable way. This process is called compactification.
One-Point Compactification
There’s a fairly simple compactification process called the one-point compactification. It has some serious downsides, but it’s a good place for us to get started.
It is exactly what it sounds like. It’s a way to add a single point to a space to turn it into a compact space. The standard example is to think about how to do this for the real number line.
Recall, we said this wasn’t compact because it goes off to infinity. Well, what if we just add a point at infinity? There’s a great way to visualize this by turning it into a circle. You wrap the
number line around the circle.
There’s a formula to make this precise and see that we’ve actually only added a single point. It’s called stereographic projection.
If you understand the picture, you should be convinced, though. When you draw a straight line from the top of the circle and find where it intersects the circle, you will get every single point of ℝ
exactly once and then you just add the point at infinity. This shows the one-point compactification of ℝ is a circle.
The more fun part is to think about why the circle is actually compact. Try to think of a way to put infinitely many points on the circle without them bunching up. As you try to do this, you’ll start
to get a feel for compact spaces better.
Given some mild conditions on the space X, it turns out you can always add one point to X, to get a compact space. Pavel Alexandroff proved this in 1924, and so this is sometimes called the
Alexandroff extension of X, though most people will just say one-point compactification.
Even though this is the minimal construction possible, it is actually not very good for technical reasons having to do with how strange topological spaces can get. But that’s beyond the scope of this
Stone-Čech Compactification
The Stone-Čech compactification was first explicitly written down in 1937 by Eduard Čech and then refined the same year to the form that is usually presented by Marshall Stone, though Tychonoff
refers to such a construction as early as 1930.
The idea is quite simple and clever, though hashing out the details is extremely subtle and difficult (it earned Čech a paper in the Annals of Mathematics).
Here’s a key preliminary fact: Taking a product of two compact spaces is still compact. This isn’t very hard to prove.
You know about product spaces if you understand ℝ², which is just a copy of the real numbers ℝ with itself. Product spaces basically let you make a new space with the old spaces as the “axes.”
As we said, ℝ² is not compact because it is not bounded.
But the simple example we started with of the closed interval [0,1] is compact. Taking the product of it with itself, [0,1]², gives a solid, closed square if you can imagine that. Then [0,1]³ is a
solid 1x1x1 cube.
Here’s the leap. It turns out that any arbitrary product of compact spaces is still compact (called the Tychonoff Theorem). This theorem is extremely hard to prove, so don’t be fooled by how
“obvious” it might sound.
Given some space X, we cleverly find a copy of X inside an incredibly huge product of [0,1] with itself (it’s a product over an uncountably infinite set). Then we just take the closure of X in this.
Since a closed subset of a compact set is compact, we know the end result is compact.
You might be thinking this is a horrible way to do it compared to the one-point compactification, but it has a really nice property. It turns out to be a “universal” construction. Every other
compactification is essentially found inside this one.
This is amazing, because it says that even if you find some other nice method to compactify your space, it has a very close relation to the Stone-Čech compactification.
Now that we’ve seen the minimal and maximal compactifications of a space, you should have a better idea of what they are and how they come about. | {"url":"https://www.cantorsparadise.org/two-compactification-theorems-6a73b11ea908/","timestamp":"2024-11-14T02:18:09Z","content_type":"text/html","content_length":"38967","record_id":"<urn:uuid:d73c0cd8-2e19-40e0-857a-252237be7c51>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00059.warc.gz"} |
Bible Reading:
As the sermon transcript does not contain any direct quotes or explicit allusions to specific Bible verses, we will focus on general themes of overcoming challenges and spiritual warfare that are
prevalent throughout the Bible. Here are three suggested passages for reading:
1) James 1:2-4 - "Consider it pure joy, my brothers and sisters, whenever you face trials of many kinds, because you know that the testing of your faith produces perseverance. Let perseverance finish
its work so that you may be mature and complete, not lacking anything."
2) Ephesians 6:10-12 - "Finally, be strong in the Lord and in his mighty power. Put on the full armor of God, so that you can take your stand against the devil’s schemes. For our struggle is not
against flesh and blood, but against the rulers, against the authorities, against the powers of this dark world and against the spiritual forces of evil in the heavenly realms."
3) 1 Corinthians 10:13 - "No temptation has overtaken you except what is common to mankind. And God is faithful; he will not let you be tempted beyond what you can bear. But when you are tempted, he
will also provide a way out so that you can endure it."
Observation Questions:
1) In James 1:2-4, what is the believer's response supposed to be when facing trials and why?
2) What does Ephesians 6:10-12 suggest about the nature of our struggles and how we should prepare for them?
3) According to 1 Corinthians 10:13, what promise does God make about the challenges we face?
Interpretation Questions:
1) How does the concept of joy in trials, as described in James 1:2-4, relate to the idea of having a strategic approach to life's struggles?
2) How does the imagery of the full armor of God in Ephesians 6:10-12 provide insight into the spiritual battles we face?
3) How does the promise in 1 Corinthians 10:13 provide assurance when facing challenges and temptations?
Application Questions:
1) Can you recall a recent challenge where you found it difficult to consider it as "pure joy"? How might you change your perspective on this situation?
2) What practical steps can you take this week to "put on the full armor of God" as you face your daily struggles?
3) Can you identify a temptation or challenge you are currently facing? How can you seek God's promised way out in this situation?
4) Reflecting on the song "This is how I fight my battles," what is one way you can use worship as a strategy in your spiritual battles this week?
5) What is one practical way you can "go down the equation" or follow God's path in a specific area of your life this week? | {"url":"https://pastors.ai/sermons/14631/","timestamp":"2024-11-07T09:45:48Z","content_type":"text/html","content_length":"146449","record_id":"<urn:uuid:5dab9085-552b-4967-9d67-70c8c6b1e825>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00037.warc.gz"} |
AVCX Passover Meta — “The Hunt Is On!” by Matt Gaffney and Aimee Lucido
puzzle — 4 days to co-author (Matt)
Matt here, doing a quick post of Aimee Lucido’s and my special Passover meta at the AVCX. Solvers were given the beautiful drawing below, created by GAMES Magazine (and New Yorker) legend Robert
Leighton, and given these instructions:
The two children at this Seder (see attached drawing) are trying to 40-Across. But you can play 40-Across in this crossword grid, too! In fact, doing so will lead you to the contest answer.
So the hunt is on, and the big hint was the grid-spanning clue at 40-A: [Passover hunt where kids search for a hidden piece of matzoh, which is broken into four pieces in this grid — the contest
answer is what the four pieces are hidden in]. That yields FIND THE AFIKOMEN, which is the small piece of broken-off matzoh traditionally hidden for kids to find at a Passover seder. So where is that
afikomen hiding?
Well it’s been broken up into four parts: AF/IK/OM/EN, and each of those bigrams can be found in exactly one grid entry:
14-A [“Aladdin 2: The Return of ___”] = JAFAR
16-A [Prefix with -pedia or -Leaks] = WIKI
56-A [Tank engine who’s friends with James, Sam, and Emily] = THOMAS
71-A [Main character in Ursula K. Le Guin’s “The Tombs of Atuan”] = TENAR
Let’s look at the letters those pieces of AFIKOMEN are hiding in:
Those spell out JAR WITH A STAR, which is the hiding place of the afikomen and also our meta answer (circled in the drawing below).
I enjoyed working on this meta quite a lot. Thanks to co-author Aimee Lucido for, in addition to her work on the grid and clues, coming up with the core idea of the meta; to AVCX editor Ben Tausig
for shepherding this through and publishing it; and to illustrator and puzzlemaster Robert Leighton, whose work in GAMES et al. I’ve admired since the mid-1980s.
16 Responses to AVCX Passover Meta — “The Hunt Is On!” by Matt Gaffney and Aimee Lucido
1. Enjoyed this one a lot Matt and Aimee! Chag sameach!
2. What was the deal with the OSBOURNE clue? Ozzy is the only reason the others are famous, so why the half-hearted grudging admission that he’s part of their family?
□ I think it’s a winking acknowledgment that his name takes the clue from tough to trivial.
☆ Exactly
3. I found the afikomen relatively easily. Since this is not part of the MGWCC, I had no idea how hard the meta would be.
So I simply submitted ‘In the Corners’, which tuned out to be a sucky answer once I saw the real deal.
Well done Matt and Aimee! I should have tried a little harder!
□ Metas on the AVCX ar on the gentler side with one maybe two steps to find the meta answer.
4. When I finally realized “afikomen” has 8 letters, *broken into 4 pieces* means 2 letter words…
JAFAR was just SHOUTING at me the whole time but I couldn’t hear what it was saying until I was finally able to see the AF hiding the JAR.
5. My favorite detail is in the cartoon: the chair so the kids can climb up and reach the jar with a star. Really wonderful.
□ I never even noticed. Leighton!!
6. Really enjoyed this. Thanks!!
7. I managed to get stuck in a rabbit hole: there are four synonymous entries for slangy “nothing” – BUPKIS (the only one clued as such), ZIP, NADA, SQUAT.
Obviously this yielded me bupkis.
I know these rabbit holes are 99.99% coincidental, but I was really wondering this time if it was intentional.
8. Question: Practically speaking, are the four pieces typically hidden together, or in disparate locations?
□ Depends on the number of children looking. In my house, for example, we broke one big piece in half, and hid the two pieces in different places — with the difficulty of finding each
calibrated to their respective ages.
I kind of liked how in the image the children were cool with looking for the same piece(s) in one place, together. I’d like to go with that kind of competition-discouraging approach next
year. Life imitating art!
☆ Now I’m wondering if the tradition of hiding Easter eggs for kids to find is a complete appropriation of Passover’s “find the afikomen.”
9. Loved this one! Thanks!
10. Love this!
This entry was posted in Daily Puzzles and tagged Aimee Lucido, contests, Matt Gaffney, Robert Leighton. Bookmark the permalink. | {"url":"https://crosswordfiend.com/2020/04/13/avcx-passover-meta-the-hunt-is-on/","timestamp":"2024-11-05T00:13:50Z","content_type":"text/html","content_length":"247438","record_id":"<urn:uuid:543820dc-6163-40d3-9cdd-e1949ae082ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00399.warc.gz"} |
persona q2 english dub
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Kontsevich formality
More material both at formality and at Kontsevich formality.
I have opened a new stub formality and removed the (improper) redirect formality at the more specialized entry Kontsevich formality. More references at Kontsevich formality.
Thanks!! I added some links to Kontsevich formality.
Added stub for Kontsevich formality.
Also added some comments to the HKR page.
at Kontsevich formality it used to say that “therefore every Poisson manifold has a canonical formal deformation quantization”.
This is not really true, the quantization is canonical only up to an action of the Grothendieck-Teichmüller group.
I have now added a super-brief remark on this, and then added a super-brief remark on formality of the $E_n$-operad in char 0 for all $n$, which makes an analogous statement for the deformation
quantization of higher dimensional quantum field theories.
I don’t have time right now to do this justice at all. What is in the entry now is really just a reminder note for myself to come back to later. Of course if it inspires anyone to add more
discussion, I won’t complain.
Also a corresponding lightning-brief remark at Poisson n-algebra – Relation to En algebra
It would be better if this were named
Kontsevich formality theorem
the notion of formality is not modified by maxim | {"url":"https://nforum.ncatlab.org/discussion/1628/kontsevich-formality/?Focus=41362","timestamp":"2024-11-13T11:33:16Z","content_type":"application/xhtml+xml","content_length":"45747","record_id":"<urn:uuid:b72896aa-2d59-4cee-b5e6-60f68d4e7fd7>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00551.warc.gz"} |