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CRC

A cyclic redundancy check (CRC) is an error-detecting code that appends a fixed-size —derived from treating the data as coefficients of a over the GF(2) and performing division by a predefined generator —to blocks of for or , enabling the receiver to verify by recomputing the checksum and comparing it for mismatches indicative of . This method excels at detecting burst errors and single-bit errors with high probability, though it does not correct them nor guarantee detection of all possible error patterns. Invented by American mathematician W. Wesley Peterson in 1961 as a practical application of cyclic codes for reliable , CRC gained widespread adoption due to its computational efficiency using simple bitwise operations like XOR and right shifts, making it suitable for hardware implementation with minimal overhead. Its defining characteristic lies in the mathematical rigor of modulo-2 polynomial arithmetic, which provides robust detection for common transmission impairments like noise-induced bit flips, outperforming simpler checks in real-world scenarios. CRC underpins in diverse applications, including Ethernet frames for local area networks, file formats like and , storage media such as hard disk drives and DVDs, and protocols in wireless communications, ensuring errors from degradation or are flagged without excessive computational burden. Variants like CRC-32, standardized in numerous protocols, balance detection strength against performance, with longer polynomials offering asymptotically better error coverage but beyond certain lengths for practical error rates. While not a cryptographic —vulnerable to intentional tampering—its ubiquity stems from proven empirical effectiveness in non-adversarial environments, as validated through decades of deployment in billions of devices.

Technology and computing

Cyclic redundancy check

A cyclic redundancy check (CRC) is a employed to verify by detecting accidental alterations in transmitted or stored , particularly effective against burst errors in communication channels. It functions as a non-cryptographic that appends a fixed-size to the block, enabling the to recompute and compare it against the received value; a mismatch indicates corruption. Unlike simpler checks, CRC leverages to achieve high detection rates for multiple-bit errors without requiring error correction. The technique was introduced by W. Wesley Peterson in 1961 through the paper "Cyclic Codes for Error Detection," co-authored with D. T. Brown, which formalized its use for reliable data transmission in early computing systems. Peterson's work built on concepts from , emphasizing division in finite fields to generate redundant bits that expose transmission anomalies. Subsequent refinements, including standardized polynomials, expanded its adoption in hardware and protocols by the 1970s. Mathematically, CRC treats the message as a binary polynomial M(x) over the Galois field GF(2), where addition is XOR and coefficients are modulo 2. The sender augments M(x) by shifting left by r bits (degree of the generator polynomial G(x)), yielding x^r M(x), then divides by G(x) using modulo-2 polynomial long division to obtain remainder R(x); the transmitted frame is x^r M(x) + R(x), divisible by G(x). At the receiver, dividing the frame by G(x) should yield zero remainder if no errors occurred; division employs XOR for subtraction, implemented efficiently via shift registers in hardware or bitwise operations in software.
Example modulo-2 division for data 110101 (M=13 [decimal](/page/Decimal)), G=x^3 + x + 1 (1011 [binary](/page/Binary)):
Augmented: 110101000 (shift by [3](/page/3))
Divide:
  1011 | 110101000
        1011
        ------
         0110
         0000 (shift)
         ----
          01101
          1011
          ----
           1100
           1011
           ----
            0101  ([remainder](/page/Remainder), append to data)
This process ensures the frame polynomial is a multiple of G(x). Generator polynomials are irreducible or primitive over GF(2) to maximize error detection; common standards include CRC-16-CCITT (x^{16} + x^{12} + x^5 + 1, hex 0x1021) for protocols like Modbus, and CRC-32-IEEE (x^{32} + x^{26} + x^{23} + x^{22} + x^{16} + x^{12} + x^{11} + x^{10} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1, hex 0x04C11DB7) for Ethernet frames and ZIP files. Selection depends on data length and error patterns; for instance, CRC-32 provides Hamming distance 4 for messages up to 4,096 bits with suitable polynomials. A degree-r CRC detects all single- and double-bit errors (if G(x) has even weight and factor x+1), all burst errors of length ≤ r, and most longer bursts with probability $1 - 2^{-r}. It fails against intentional tampering or specific undetectable error patterns matching multiples of G(x), hence unsuitable for security but ideal for accidental noise in physical layers. Applications span storage (e.g., hard drives, parity), networks ( checks, ), and peripherals (USB packets, ), where hardware accelerators compute CRC in parallel for high throughput. In embedded systems, optimized table-driven methods reduce computation to lookups, supporting real-time integrity in automotive CAN buses and networks. Despite advances in , CRC persists due to its low overhead and proven efficacy in detecting transient faults.

Medicine and biology

Colorectal cancer

Colorectal cancer, also known as colon cancer or rectal cancer, is a malignancy that originates in the epithelial cells lining the colon or rectum, often developing from precancerous polyps such as adenomas. These tumors typically progress slowly over years, allowing opportunities for early detection through screening. In 2024, the American Cancer Society estimated 152,810 new cases in the United States, with men accounting for 81,540 and women for 71,270. Globally, it ranks as the third most common cancer and second leading cause of cancer mortality, with incidence rates varying by region due to differences in screening, diet, and lifestyle. Epidemiological data indicate a lifetime of approximately 3.9% for in the U.S., with an age-adjusted incidence rate of 37.1 new cases per 100,000 people annually and a of 12.9 per 100,000. While overall incidence has declined by over 30% in adults aged 50 and older due to screening efforts, rates are rising among younger adults under 50, projected to double by 2030, comprising 10.9% of colon cancers and 22.9% of rectal cancers. This trend, observed in high-income countries, lacks a fully explained cause but correlates with increasing and sedentary behavior. Most cases occur after age 50, with risk escalating thereafter. Key risk factors include hereditary syndromes like Lynch syndrome or , which account for 5-10% of cases; inflammatory bowel diseases such as Crohn's or ; and lifestyle elements like low , diets high in red/processed meats and low in fiber/vegetables, , , and heavy use. Family history of or polyps elevates risk independently of shared or . Type 2 diabetes and prior polyps also contribute, underscoring modifiable and non-modifiable causal pathways. Symptoms often emerge in advanced stages and include , persistent changes in bowel habits ( or ), abdominal cramping, unexplained , , and narrow stools. Early disease is frequently , emphasizing the role of screening. Diagnosis typically involves for visualization and confirmation, supplemented by imaging like scans or blood tests for (CEA) levels. uses TNM classification to assess tumor depth, involvement, and . Treatment is stage-dependent, with surgery as the cornerstone for localized disease, often curative when confined to the bowel. Adjuvant chemotherapy, such as FOLFOX regimens, improves survival in stages III-IV; radiation is used for rectal cancers; and targeted therapies (e.g., anti-EGFR for KRAS wild-type tumors) or immunotherapy (e.g., pembrolizumab for MSI-high tumors) address advanced cases. Five-year survival exceeds 90% for localized disease but drops to 15% for distant metastases. Prevention focuses on screening starting at age 45 for average-risk individuals, via every 10 years or stool-based tests annually, which can detect and remove polyps. Lifestyle modifications—regular exercise, high-fiber diets, limited alcohol, and —reduce incidence by addressing modifiable risks. Aspirin use may lower risk in select high-risk groups, per clinical guidelines.

Religion and denominations

Christian Reformed Church

The Christian Reformed Church in (CRCNA) is a Protestant within the Reformed tradition, comprising over 900 congregations across the and as of the late . Founded in 1857 through the secession of four churches representing 130 families from the [Reformed Church in America](/page/Reformed Church in America), the CRCNA originated from Dutch immigrants who settled in , in 1847 under the leadership of Rev. Albertus C. Van Raalte, seeking to preserve stricter adherence to Reformed doctrines and practices amid perceived liberalization in the parent body. Its theological roots trace to the 16th-century Protestant , particularly the teachings of , emphasizing God's sovereignty, human depravity, and salvation by grace through faith alone. Doctrinally, the CRCNA subscribes to the —the (1561), (1563), and (1618–1619)—as historic Reformed summaries of biblical truth that form the basis for its creeds, preaching, teaching, and governance. These confessions articulate key tenets including the absolute authority of Scripture, the , , and rejection of Arminian views on , as codified against Remonstrant errors at the . The denomination also draws from the broader Reformed emphasis on Christ's lordship over all spheres of life, a perspective advanced by theologian in the late 19th century, which influenced the CRCNA's commitments to Christian education, cultural engagement, and social reform grounded in biblical principles. Governed by a synodical structure where local classes (regional bodies) oversee consistories (local church councils), the CRCNA transitioned from Dutch-language services to English by the early , expanding through post-World War II immigration and missions. It maintains institutions such as and Calvin Theological Seminary to advance Reformed scholarship and pastoral training. Membership grew steadily post-founding, reaching a peak of over 315,000 confessing members in nearly 1,000 churches by 1992, with an average congregation size of about 300. The denomination has navigated internal tensions, including a 1960s debate over women's that permitted deaconesses but led to the departure of over 40,000 members forming separate groups like the Protestant Reformed Churches; more recently, synodical affirmations in 2022–2023 of prohibitions on unrepentant homosexual practice prompted further exits by progressive-leaning congregations, reflecting ongoing divides over biblical amid cultural shifts. Despite declines to under 250,000 members by the , the CRCNA sustains ministries in , , and global partnerships, prioritizing fidelity to its heritage.

International law and treaties

Convention on the Rights of the Child

The Convention on the Rights of the Child (CRC) is a establishing comprehensive protections for individuals under age 18, encompassing civil, political, . Adopted by the via Resolution 44/25 on November 20, 1989, it opened for signature on January 26, 1990, and entered into force on September 2, 1990, following ratification or accession by 20 states. The document comprises a and 54 articles, defining a as "every human being below the age of eighteen years unless under the law applicable to the child, majority is attained earlier." Core provisions emphasize four guiding principles: non-discrimination (Article 2); the as a primary consideration in actions affecting children (Article 3); the , survival, and development (Article 6); and respect for the views of the child in accordance with their age and maturity (Article 12). Rights include protection from , , and ; access to , , and an adequate ; freedom of expression, thought, and ; and safeguards against involvement in armed conflict or hazardous labor. States parties undertake to respect, protect, and fulfill these rights through legislation, policies, and resource allocation, while permitting reservations compatible with the treaty's object and purpose. As of October 27, 2025, 196 states are parties to the CRC, representing near-universal adherence among UN member states and observers, with , , and the as initial holdouts—though acceded in 2015 and in 2011, leaving the as the sole UN member state that signed but has not ratified. The signed on February 16, 1995, but Senate ratification has stalled due to constitutional concerns, including potential supremacy of over , impacts on parental , and conflicts with domestic policies on issues like juvenile and abortion notification. Implementation is overseen by the Committee on the of the Child, an 18-member expert body elected by states parties, which reviews periodic reports and issues general comments on application. Three optional protocols, adopted in 2000 and 2011, extend obligations: prohibiting under-18 recruitment into armed forces (83 parties as of 2025); criminalizing child sale, prostitution, and pornography (178 parties); and establishing an individual complaints mechanism (50 parties). Critics, including U.S. legal scholars, argue the CRC's emphasis on children's evolving capacities risks eroding parental by or judicial overrides in , potentially conflicting with cultural or religious practices and lacking enforcement mechanisms beyond reporting. Empirical assessments of implementation vary, with data from reports showing progress in areas like school enrollment but persistent gaps in child labor and protection from violence in regions with weak governance.

Publishing and reference works

CRC Press

CRC Press, originally the publishing arm of the Chemical Rubber Company, specializes in scientific, technical, and medical reference works, textbooks, and handbooks. The Chemical Rubber Company was established in , , in 1900 to supply laboratory equipment such as rubber aprons and tubing, with its publishing operations commencing in 1913 alongside the release of the inaugural Handbook of Chemistry and Physics, a compact reference designed to fit in lab coat pockets and promote equipment sales. By the , the company had grown into a multimillion-dollar enterprise, employing 125 staff, maintaining in-house art and photography departments, and issuing approximately 100 titles annually focused on advanced scientific texts and reports. Following periods of expansion and ownership transitions—including acquisition by in 1986 and subsequent sales— was purchased by Group in 2003 for $95 million, integrating it as the primary book division within the Informa-owned publisher. This merger enhanced its portfolio through synergies with imprints like Lewis Publishers (acquired by CRC in 1990), bolstering offerings in , , and biomedical fields. Today, maintains a catalog exceeding thousands of titles, emphasizing peer-reviewed, data-driven resources for researchers, engineers, and academics, with ongoing publication of updated editions of cornerstone references like the of Chemistry and Physics.

Industrial and chemical companies

CRC Industries

, Inc. is a global manufacturer and distributor of chemical specialty products focused on , repair, and operations (MRO), including cleaners, lubricants, inhibitors, protective coatings, degreasers, greases, and additives. Headquartered in , the company produces over 1,300 items serving industries such as automotive, marine, electrical, industrial, , , oil and gas, and . The company was founded on October 6, 1958, as Corrosion Reaction Consultants, initially operating from a garage where its first product, CRC —a multi-purpose and protectant—was mixed in 55-gallon drums; this formulation later evolved into the widely recognized CRC 5-56. Over decades, CRC expanded from a single-product startup to a multinational operation with 26 facilities worldwide, emphasizing innovation in and specialty chemical delivery systems. A pivotal development occurred in 1971 with the launch of CRC Brakleen, the first aerosolized brake parts cleaner, which enabled in-place cleaning of automotive components and has since become the category's top-selling product globally. CRC's product portfolio includes trademarked brands such as SmartWasher (parts washing systems), K&W (automotive chemicals), Sta-Lube (greases and lubricants), Marykate (marine products), Weld-Aid (welding aids), and Evapo-Rust (rust removers), distributed across , Central and South America, , , , and . The company maintains ISO 9001:2015 certification for in its manufacturing processes. In 2016, CRC acquired ChemFree Corporation, enhancing its offerings in environmentally focused technologies. Berwind Corporation, a private investment firm, acquired CRC in 1981, supporting its growth into a leader in chemical MRO solutions without public disclosure of financial details. CRC continues to prioritize first-to-market innovations, such as non-chlorinated cleaners and emissions-testing formulas, while serving professional technicians and end-users in demanding environments.

Research and cooperative organizations

Cooperative Research Centres

The Cooperative Research Centres (CRC) Program is an initiative that funds industry-led collaborative partnerships to address national challenges through and technology solutions. Established to bridge the gap between fundamental and application, the program fosters consortia involving universities, public organizations, businesses, and end-users, emphasizing long-term commitments to in sectors such as , , , and resources. Launched in with the first grants awarded in , the has operated for over three decades as a of 's national , supporting more than 100 centres across its history. It evolved from recommendations in the 1990 report Australia's Universities: Issues and Options and subsequent policy frameworks aimed at enhancing economic competitiveness through targeted R&D investment. The program underwent reviews, including expansions in to incorporate shorter-term CRC Projects (CRC-Ps), but maintains its core focus on enduring collaborations. Oversight is provided by Innovation and Science Australia via its CRC Advisory Committee, which evaluates proposals based on criteria like industry relevance, additionality of funding, and potential for . CRCs receive grants typically spanning 7 to 10 years, requiring matched contributions from participants to leverage public funds—often achieving cash and in-kind leverage ratios exceeding 2:1. As of October 2025, 25 active CRCs operate with a total government investment exceeding $1.1 billion (excluding ), covering diverse areas including the CRC (focused on AI-driven diagnostics, funded $50 million from July 2018 to December 2026) and the Zero Net Emissions from CRC (addressing sustainable farming, funded $87 million from July 2021 to June 2031). Recent approvals in 2025 added three new centres: the Solutions for Manufacturing Advanced Regenerative Therapies CRC (SMART CRC), Care Economy CRC, and Additive Manufacturing CRC, backed by $158 million to advance medical therapies, aged care innovations, and advanced materials production, respectively. Independent evaluations affirm the program's efficacy, estimating that CRCs since 2005 yield $5.61 in (GDP) per dollar of government funding through direct outputs like patents, startups, and workforce training, alongside indirect effects such as enhancements. A 2022 impact assessment projected $32.5 billion in lifetime economic benefits from the program, including CO2 abatement and productivity gains, though it notes challenges in measuring long-tail outcomes and ensuring equitable regional participation. Critics, including some stakeholders, argue for streamlined selection processes to reduce administrative burdens, but the program's additionality—eliciting private investments unlikely without public support—remains a key strength per government analyses.

Other research centers

The , established in 1984, is a Washington, D.C.-based that examines funding patterns in the philanthropic and nonprofit sectors, with a focus on regarding donor influence and policy . It produces reports and analyses critiquing what it identifies as left-leaning biases in grantmaking by major foundations, emphasizing empirical data on financial flows to advocacy groups. The Cyber Research Center (CRC) at the at West Point operates as one of three research centers in the Department of and , concentrating on cybersecurity innovations, , and defense technologies tailored to military applications. Established to support cadet-led and faculty-driven projects, it integrates computational methods with strategic defense needs, drawing on funding for applied outputs. The Cultural Research Center (CRC) at conducts nationwide surveys and studies on worldview formation, tracking shifts in American beliefs through quantitative data collection since its inception in the early 2000s. Its research highlights correlations between cultural trends and personal philosophies, often attributing societal changes to declining adherence to biblical principles based on longitudinal polling. The Chesapeake Research Consortium (CRC), formed in 1972, coordinates multidisciplinary environmental research across academic, governmental, and nonprofit partners in the watershed, emphasizing data-driven assessments of and restoration strategies. It facilitates collaborative projects on , , and policy impacts, supported by federal grants and member institutions.

Geographic and institutional places

Colleges and universities

Crowley's Ridge College is a private two-year liberal arts college affiliated with the , located in . Founded in , it enrolls approximately 200 students and emphasizes a integrated with academics, offering associate degrees in fields such as arts and sciences, , and , and . The institution maintains small class sizes, with a student-to-faculty ratio of 9:1, and provides on-campus housing and athletics programs including and . Cosumnes River College (CRC) is a public in , part of the Los Rios Community College District. Established on September 14, 1970, it initially served 2,150 students and has grown to offer over 80 degree and certificate programs in areas like , , health sciences, and liberal arts, with an annual enrollment exceeding 15,000 students as of recent data. The campus spans 100 acres and includes facilities such as the Visual and and agricultural demonstration areas, supporting pathways to four-year universities and . CRC also operates an extension center in Elk Grove to expand access in southern Sacramento County. Cooperating Raleigh Colleges (CRC) functions as a consortium uniting seven independent private colleges and universities in Raleigh, North Carolina, including Meredith College, Peace University, and Shaw University. Formed in 1968, it enables cross-registration for students across member institutions, promotes collaborative academic and cultural events, and shares resources like library access to enhance educational opportunities without merging administrative structures. The consortium supports approximately 20,000 students collectively through initiatives such as joint faculty development and emergency mutual aid protocols.

Other locations

The (CRC) is a medium-security state operated by the of and , located at 5th Street and Western Avenue in Norco, . Established as a correctional facility after prior use as a and naval , it houses approximately 2,800 inmates focused on programs including vocational , academics, and substance abuse treatment. In August 2025, CDCR announced its closure by fall 2026 to address declining inmate populations and achieve cost savings, impacting about 1,200 staff positions. Camp Red Cloud (CRC) was a in , , , positioned between and the . Operational from the post-Korean War era, it functioned as the headquarters for the 2nd Infantry Division's sustainment brigade and supported logistical operations until its deactivation in October 2018 under the U.S.- Land Partnership Plan, which relocated forces southward. The closure ceremony marked the end of its role in forward-deployed readiness.

Sports organizations

Football clubs

C.R.C. Football Club, based in , , was established in 1974 by John Obie to unite local black footballers amid social challenges including racial abuse. The club fields teams in the Rugby and District Sunday Football League, with its first team competing in Division 1 as of 2024. It has secured 12 league titles, multiple cup victories, a 60-game unbeaten streak in the early , and reached the FA Sunday Cup quarter-finals twice during that period; in 1995, under manager Steve Vassell, it achieved a by winning all four leagues and associated cups in a single season. A veterans team, introduced in 2021, holds the current Veterans League championship. CRC Football Club, a grassroots outfit from , , was founded in 2021 and competes in the Football Association of Penang (FAP) Division 1 League. The team includes players such as Yong Kuong Yong, a Malaysian with prior experience in higher divisions. In , CRC Rocas do Vouga maintains an active presence in lower-tier competitions, with match results and player statistics tracked on major sports databases.

Cycling and other sports

Chain Reaction Cycling (CRC) is a recreational cycling club based in Virginia Beach, Virginia, focused on organizing and promoting non-competitive road cycling events for its members. The CRC Cycling Team operates as a professional cycling outfit in Durango, Colorado, with an emphasis on fostering sportsmanship and providing competitive opportunities for developing athletes. Charlottesville Racing Club (CRC), located in Charlottesville, Virginia, functions as a multi-disciplinary cycling organization encompassing road racing, mountain biking, cyclocross, gravel events, and triathlons. Wyre Forest Cycling Road Club (CRC), a member-affiliated group under in the , supports over 300 riders across disciplines including and social rides, with programs like Go-Ride for youth participation. CRC/ Cycling has fielded competitive teams in events such as the Reston , where rider Panzarella achieved a second-place finish in the Men's Category 4 race on June 26, 2011.

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