text:
JHLT websites:
https://www.jhltonline.org or https://www.editorialmanager.com/JHLT/default.aspx
AudioSlide Presentation on JHLT Editorial Workflow: Presented by Dr. Daniel Goldstein, MD

Aims and scope
The Journal of Heart and Lung Transplantation, the official publication of the International Society for Heart and Lung Transplantation, brings readers cutting-edge clinical practice, basic and translational science in the fields of cardio-pulmonary transplantation, mechanical and biological support of the failing heart, advanced lung disease, pulmonary vascular disease (pulmonary arterial hypertension and CTEPH), extracorporeal support of donor organs, and cell replacement therapy.

Editorial policies
Statements and opinions expressed in the articles and communications herein are those of the author(s) and not necessarily those of the Editor(s)or publisher, and the Editor(s) and publisher disclaim any responsibility or liability for such material. Neither the Editor(s) nor the publisher guarantees, warrants, or endorses any product or service advertised in this publication. Neither do they guarantee any claim made by the manufacturer of such product or service.

Editorial Office Contact
Any question may be directed to our office by e-mail to [email protected] or [email protected] (Mr. David Newcombe, JHLT Managing Editor). In general, an email receives a response within 72 hours. If there is an urgent issue that requires the editor's attention or intervention, it is usually attended to immediately. The journal website can be accessed at https://www.jhltonline.org.
Please do not contact the Editor-in-Chief via direct personal email. These emails will be ignored.

Manuscript submission
The Journal of Heart and Lung Transplantation will accept only online submissions of original manuscripts through the Editorial Manager system at https://www.editorialmanager.com/JHLT/default.aspx. Manuscripts submitted through this online system can easily be tracked by the authors, editors, and reviewers through final disposition. The corresponding author involved with a specific manuscript will receive automatic e-mail notification as the manuscript proceeds through the system. Click the 'Register' link on the toolbar at the top left to input author demographics and set up an account. After registration is complete, a notice will be sent via email indicating a user ID and password. Use this information to log in as an author by choosing the 'Login' link on the toolbar and select 'Submit New Manuscript.' Follow the prompts to complete the submission according to the specifications below. Further questions may be directed to the editorial office by e-mail to [email protected] or directly to the Managing Editor, David Newcombe ([email protected]). The user ID and password can be changed at any time.

Duplicate/Prior/Overlapping Publication or Submission
Manuscripts are accepted for review with the stipulation that they are submitted solely to JHLT. Submission of an article implies that the work described has not been published previously (except in the form of an abstract or as part of a published lecture or academic thesis or as an electronic preprint, see https://www.elsevier.com/postingpolicy), that it is not under consideration for publication elsewhere, that its publication is approved by all authors and tacitly or explicitly by the responsible authorities where the work was carried out, and that, if accepted, it will not be published elsewhere including electronically in the same form, in English or in any other language, without the written consent of the copyright-holder. Authors should disclose prior publication as an abstract or an electronic preprint in the Cover Letter.

Submission Checklist
Please submit your materials as 6 separate files organized as follows and in accordance with following the guidelines below.
1. Cover letter of submission, to include author agreement as described below
2. ICMJE Disclosure form for each listed author (combined into one file)
3. Manuscript (double spaced, combined into one file):
• Title page
• Title of article
• Full name(s), academic degrees, and affiliations of authors, including each author's ORCID#
• Name, email address, and any relevant social media pages (i.e. Twitter handle, Facebook page, Mendeley account page, etc.) of corresponding author
• Reprint and corresponding author byline
• Manuscript file (double spaced), to include:
• Introduction
• Abstract
• Material and Methods (see below), to include:
• Statement of IRB/Ethics board approval
• Statement of compliance with the ISHLT Ethics statement
• Statement of humane animal care
• Statement of informed consent
• Discussion
• Financial conflict of interest statement (to be included as a separate section after the Discussion and before the references)
• Author contributions
• Acknowledgments
• References
• Figure Legends
• Tables (if needed)
• Figures
4. Written releases from all persons/patients whose information or likeness is represented or makes them identifiable
5. Permissions to reproduce material published elsewhere
6. Supplementary Appendix Material (if needed)

Manuscript Categories

1. For Original Science papers we review them in Clinical, Translational or Basic categories. For these papers, we impose a 250-word limit for the structured abstract and a 3000-word limit for the main text and (excluding tables, figures, and references). We will allow an unlimited supplementary file which is designed to be made available in a PDF form and can enhance the availability of more detailed information. The supplementary file should be provided as a WORD document or PDF and must be structured - see below for details. The articles should, in general, average 8 journal pages with no more than a total of 8 figures/tables and 40-50 references.

Meta-analysis studies and systematic reviews are considered clinical research papers. We recommend reviewing the following guidelines for constructing such articles (http://www.ccace.ed.ac.uk/research/software-resources/systematic-reviews-and-meta-analyses).

2. Innovation articles are accepted rarely and reserved typically for major articles that have the potential to substantially advance clinical practice or advance research. Such papers should be no more than 1500 words. The entire double-spaced, typewritten manuscript, including tables, figures, and references should be five to seven pages, and the abstract is not required to be structured and should be no more than four-five sentences.

3. Brief Communication is a unique category that is designed to publish exceptionally novel preliminary information with a high potential to change paradigms and to open new avenues of investigation. Brief Communications must contain substantive information concerning experimental or clinical studies, accompanied by a brief reference list (of no more than ten references) and three display items (Tables/Figures) and no more than 1000 words (not including 150 words for the unstructured abstract). No other subsections are permitted. Supplementary data files are permitted but anything described in such contexts must have a relevant tag in the main paper.

4. Case Anecdotes (which are designed to succinctly describe a novel clinical or surgical direction) and Comments and Opinions (which could represent a reaction to a published article or could be unrelated) should be no more than 500 words (in total main text). Authors are limited to 5 references for the Case Anecdotes, Comments and Opinion category. This communication is retained and indexed in the Cumulated Index Medicus.

5. State-of-the-Art reviews (2500 words) are entertained for publication in the journal only if the article has been invited by the Editor-in-Chief. State-of-the-art reviews must contain a brief abstract of less than 250 words, and ideally, they should contain at least one display item or figure.

6. Perspective pieces (1500 words) acan be a discussion or debate on any ongoing clinical or scholarly issue in the field or a look into the future. Perspectives must contain an unstructured abstract of less than 250 words and they can be unsolicited.

7. A Review Article can be unsolicited. We will consider overviews and topics that are relevant to the readership. The review must be less than 2500 words for the main text. Review articles must have an unstructured abstract of less than 250 words and they should ideally contain one display item.

Study Requirements
1. For observational studies, we recommend using the STROBE criteria which are available at https://www.strobe-statement.org/fileadmin/Strobe/uploads/checklists/STROBE_checklist_v4_combined.pdf
2. A review article must ideally follow the guidelines for a systematic review and meta analyses must be properly coordinated. We recommend reviewing the following guidelines for constructing such articles (http://www.ccace.ed.ac.uk/research/software-resources/systematic-reviews-and-meta-analyses).
3. A Clinical Trial (any research project that prospectively assigns human subjects to intervention and comparison groups to study the cause-and-effect relationship between a medical intervention and a health outcome) must be registered online before the start of patient enrollment. Trials in which the primary goal is to determine pharmacokinetics may be exempt. A list of acceptable registries may be found at http://www.icmje.org. You must indicate the trial registry and its access mechanism at the time of submission. In general, we advise that a list of the investigators, trial protocol, detailed inclusion and inclusion criteria and methodological information be also provided in the supplementary files.

Cover Letter/Author Agreement
The authors must include a cover letter with the manuscript submission that indicates the manuscript has not been published previously and that it is not under consideration for publication elsewhere. It should be stated that all authors agree with the content of the manuscript. The cover letter should concisely summarize the findings of the study and the importance to the field.

Informed consent
Reports describing data obtained from research conducted in human subjects must contain a statement in the Methods section indicating approval by the institutional review board or Ethics board and affirm that informed consent was obtained as required by the study authorizing entity. In some instances we may ask to see the approval certificates.
All submissions to the journal must attest their strict compliance with the ISHLT ethics statement (http://ishlt.org/ishlt/media/documents/ISHLT_Statement_Transplant-Ethics_2022.pdf). In addition, for specific submissions from countries where it is known that transplants may be performed in violation of these guidelines (e.g. transplants involving executed prisoners), a specific statement to this regard is required. The above statements are required to be made within the Methods section.
Authors should remove information from photographs and manuscripts that might identify a patient. Where this is impossible, submissions must be accompanied by a written release from the patient (or any other person whose likeness is represented).
Manuscripts that do not comply with the requirements regarding informed consent will be returned during the processing phase by the Editorial Office.

Humane animal care
All manuscripts reporting experiments using animals must include a statement in the Methods section giving assurance that all animals received humane care in compliance with the 'Principles of Laboratory Animal Care' formulated by the National Society for Medical Research and the 'Guide for the Care and Use of Laboratory Animals' prepared by the Institute of Laboratory Animal Resources and published by the National Institutes of Health (NIH Publication No. 86-23, revised 1996).

Authorship/Author Contributions
Each person designated as author should qualify for authorship. Each author should have participated sufficiently in the work to take public responsibility for the content. The contributions to author section of the manuscript should concisely detail each author's role in the study, to include their study design, data analysis, writing of manuscript, acquisition of funding. This section should come after the Discussion and before the Acknowledgments and More details for such qualification can be found by accessing http://www.icmje.org/recommendations/browse/roles-and-responsibilities/defining-the-role-of-authors-and-contributors.html.
Authors for whom English is not their native language are strongly encouraged to seek the aid of a professional English language medical editing service. JHLT endorses the use of Elsevier Author Services: https://webshop.elsevier.com/

Acknowledgments
Funding acknowledgments and any other notable acknowledgments (i.e., use of services to enhance writing of the manuscript) should be placed in a separate section, to come after Author Contributions and before Disclosures.

CONFLICT OF INTEREST DISCLOSURE AND MANAGEMENT FOR AUTHORS

Introduction
In order to ensure and protect the integrity of the JHLT, the ISHLT has developed these policies on COI as it applies to the JHLT, its Editor-in-Chief, Deputy Editors, Section Editors, Guest/Other Editors, and Editorial Board members/Reviewers, as well as authors of manuscripts submitted to JHLT for consideration for publication. This policy is based upon the foundational work of The International Committee of Medical Journal Editors (ICMJE), which updated its Recommendations for the Conduct, Reporting, Editing, and Publication of Scholarly Work in Medical Journals in December 2019, see http://www.icmje.org/icmje-recommendations.pdf. Further guidance came from the Committee on Publication Ethics, COPE, see https://publicationethics.org/competinginterests and the World Association of Medical Editors, WAME, see http://wame.org/conflict-of-interest-in-peer-reviewed-medical-journals. The JHLT COI principles below are embodied within the ICMJE Recommendations and reflect accepted COI practice for major journals.

Conflict of Interest
An actual or potential conflict of interest occurs when a competing interest (financial, competitive, or otherwise) does or may influence (intentionally or not) the assessment of the validity of the interpretation of data and results. It is recognized that many authors and members of editorial boards have competing interests, and it is often not clear to journal readers, researchers, and editorial board members when a competing interest affects decision-making. Disclosure of all actual or potential conflicts allows all parties to make informed judgments.
The following competing interests or relationships must be disclosed by all Authors, Reviewers, and Editors ("covered individuals"), including when such interests/relationships are those of a spouse, child, or any other individual with whom the covered individual shares income or assets:
• Employment: Employment with an entity having an investment, licensing, or other commercial interest in the subject matter under consideration must be disclosed.
• Leadership Position: Service as an officer or director for an entity having an investment, licensing, or other commercial interest in the subject matter under consideration must be disclosed.
• Consultant/Advisory Role/Expert Testimony: Arrangements with an entity having an investment, licensing, or other commercial interest in the subject matter under consideration must be disclosed if payments are made within the past 24 months or anticipated to be paid in the current or upcoming 12 months, regardless of whether payment was made to the individual or his/her institution. Payments totaling less than $500 per incident are excluded from disclosure requirements. • Stock Ownership/Patents: Any ownership interest including stock, options, and warrants (except when invested in a publicly traded fund not controlled by the covered individual) in any publicly traded or privately held company, and/or any patents generating income in the past 24 months or anticipated to generate income in the current or upcoming 12 months must be disclosed if the company is an entity having an investment, licensing, or other commercial interest in the subject matter under consideration. • Honoraria: Disclosure of honoraria for presentations or other services is required if payments are made by an entity having an investment, licensing, or other commercial interest in the subject matter under consideration and are paid within the past 24 months or anticipated to be paid in the current or upcoming 12 months, regardless of whether payment was made to the individual or his/her institution. Payments totaling less than$500 per incident are excluded from disclosure requirements. Payments totaling less than $500 per incident are excluded from disclosure requirements. • Research Funding: All payments (including in-kind payments) associated with the conduct of research in which the disclosing individual is participating/has participated must be disclosed if the payment is provided by or on behalf of a company having an investment, licensing, or other commercial interest in the subject matter under consideration, and is paid within the past 24 months or anticipated to be paid in the current or upcoming 12 months, regardless of whether payment was made to the individual or his/her institution. • Other Compensation: Trips, travel, gifts, or other in-kind payments, including those not directly related to specific research activities, must be disclosed if received from an entity having an investment, licensing, or other commercial interest in the subject matter under consideration and are paid within the past 24 months or are anticipated to be paid in the current or upcoming 12 months, regardless of whether payment was made to the individual or his/her institution, or a third party on the individual's behalf. Payments totaling less than$500 per incident are excluded from disclosure requirements.

Conflict of interest disclosure for authors
The ISHLT and the JHLT currently abide by the ICMJE Recommendations ("the Uniform Requirements") as they relate to authors. Accordingly:
• All listed authors must complete and submit to JHLT at the time of manuscript submission the ICMJE's standardized disclosure form, which can be found at http://www.icmje.org/disclosure-of-interest. If uncertain as to what might be considered a potential conflict of interest, authors should err on the side of full disclosure of all relationships described above.
• Such disclosure must include actual or potential conflicts of interest related to all aspects of the manuscript. This includes any actual or potential competing interest related to the design and conduct of the study and interpretation of the study data.
• In addition, each manuscript, regardless of its type, must include a financial conflict of interest disclosure statement. This statement should be included AFTER the Discussion Section and BEFORE the References in a separate section. This statement must include all funding sources or acknowledgements of contributions of any material nature pertinent to the manuscript and must describe ALL authors' relationships with companies or relevant entities that make products pertinent to the manuscript. The statement must specify the type of relationships (e.g., consulting, paid speaking, grant support, equity, patents) EACH author has with EACH company. The role(s) of any funding organization(s), if any, in the collection of data, its analysis and interpretation, and/or in the right to approve or disapprove publication of the finished manuscript must be described in this statement as well. This information is of serious concern to the Editorial Office and no manuscripts will move forward for processing if this statement is not provided as described herein.
• Authors must completely spell out all grant funding agency abbreviations.
• If an author has no conflicts to declare, he or she must indicate None.
• Failure to include the disclosure statement at the time of manuscript submission will result in the Editorial Office returning the manuscript, which may be resubmitted provided the disclosure is included with the next submission. Allegations of inappropriate disclosure will be investigated.

Consequences of not disclosing COI
There are serious consequences for any author failing to disclose potential or actual conflicts of interest. These will be determined on a case-by-case basis and may include:
• Ban from publishing in JHLT for from 1 year to indefinitely
• Expression of concern published in JHLT
• Retraction of article
• Instigation of institutional inquiry into scientific misconduct
• Report to Retraction Watch of misconduct

STYLE GUIDELINES
Manuscripts should be organized in the following order:

Title page, Abstract, Introduction, Materials and Methods, Results, Discussion, Author Contributions, Acknowledgments, Financial Disclosure Statement, Figures with Legends, References

Number all pages including figures, references and tables.

Title Page
The title page should include the following:
• Authors and affiliations
• Corresponding author contact information (see above for details)
• Running Title
• List of non-standard abbreviations (i.e. not included in AMA Manual of Style list of commonly used abbreviations)
• Word count

Abstracts
All manuscripts with the exception of Case Anecdotes, Comments and Opinions, MUST contain an abstract of no more than 250 words that briefly summarizes the content. The abstract should be divided into four sections: Background, Methods, Results, and Conclusions. Brief Communications abstracts must be <150 words and are unstructured. Abstracts should contain no references or abbreviations. Manuscripts submitted without an abstract will NOT be considered for publication. Abstracts will not count toward the total manuscript page count. For innovation articles or perspectives we ask for an unstructured narrative abstract limited to 4-5 sentences.

Graphical Abstract (not required for original submission; may be requested for revised manuscripts)
A graphical abstract should allow readers to quickly gain an understanding of the main take-home message of the paper and is intended to encourage browsing, promote interdisciplinary scholarship, and help readers identify more quickly which papers are most relevant to their research interests. Graphical abstracts can be uploaded selecting "Graphical Abstract" when uploading files. Graphical abstracts should be submitted as a separate file in the online submission system. Image size: Please provide an image with a minimum of 531 1328 pixels (h w) or proportionally more. The image should be readable at a size of 5 13 cm using a regular screen resolution of 96 dpi. Preferred file types: TIFF, EPS, PDF or MS Office files. You can view Example Graphical Abstracts on our information site, https://www.elsevier.com/authors/journal-authors/graphical-abstract. We would typically advise that such a graphical abstract could also be used to develop a central figure for your paper and could be included in the main manuscript for more vivid use since graphical abstracts are only slowly coming in vogue. Please note that you should not include the same graphical abstract and central illustration to avoid redundancy but may have both if they show incremental data.

Material and Methods
JHLT will require authors to strickly follow these guidelines for data reporting:

This section should have sufficient detail to permit reproducibility of data. For animal studies: state species, strain, gender of animals used and sample size employed. Describe any surgical procedure and state that the study was performed in accordance National Institutes of Health (NIH) Guide for the Care and Use of Laboratory Animals or equivalent. State the dose, route of administration and frequency of any anesthetic agents.
State the commercial source and catalogue number for all reagents. State how many times each experiment was repeated.
Statistical methods: Describe the statistical methods employed and include measures of variance such as standard deviation, standard error. Individal data points should be included in all graphs and the figure legend should detail if each point reflects an individual biological replicate. For any immunoblots, entire uncut gels should be presented to the reviewers as a supplement.

Graphs and Images: Graphs should show samples size (i.e dot plots) and variation of data, and the figure legends should detail if each point reflects an individual biological replicate. For immunoblots, entire uncut immunoblots should be presented to the reviewers, along with corresponding molecular weight standards, as supplement data. For all images no specific feature should be enhanced, hidden, removed, or moved. For flow cytometric data, the gating strategy for each plot / figure should be shown in a supplement. Additionally, authors should state whether isotype controls, florescence minus one (FMO) or "dump" channels have been employed to define expression levels or identify cell populations of interest.

If irregularities are identified in images, graphs etc., the editors hold the right to inform the author's Institutional Integrity offices (or equivalent) and the funders of the study.

Please note that from January 2021, all revised manuscripts that contain the above image data will be reviewed by a technical editor.

Style, abbreviations, and references
References must contain the full title of the citation. Authors are encouraged to refer to the American Medical Association Manual of Style, 10th Edition as a specific style guide. The title page and abstract should be prepared according to the guidelines listed above. Abbreviations should be used sparingly unless they are commonly accepted such as RNA, CsA, etc. When such a word appears for the first time, it should be written out in full, followed by the abbreviation in parentheses. This abbreviation may then be used subsequently when referring to the same term. All hematologic and clinical chemistry measurements should be reported in the metric system in terms of the International System of Units (SI). Abbreviations for titles of medical periodicals should conform to those used in the latest edition of Index Medicus. (A 'List of Journals Indexed in Index Medicus' - with abbreviations is obtainable from the Superintendent of Documents, US Government Printing Office, Washington, DC 20402 at a modest charge.) References should be listed by number in the body of the text and should be listed in the reference section in the same order in which they appear in the manuscript. Include authors' last names and initials, journal, volume, page, and year. Include the city and the publisher's name for books.

Reference examples
(List all authors when six or fewer; when seven or more, list first three and add et al.) Journal article: Salvatierra O, Vincenti F, Armend W, et al. Four-year experience with donor-specific blood transfusions. Transplant Proc 1983;15:924-6.
Books: Eisen HN. Immunology: an introduction to molecular and cellular principles of the immune response. 5th ed. New York: Harper & Row; 1974. p. 406.
Chapters: Fromer PL. Workshop on congestive heart failure. In: Braunwald E, Mock MB, editors. Congestive heart failure. New York: Grune & Stratton; 1982. p. 1-2.

Tables and figures

Figures (Initial Submission)
For initial submission, figures should be placed on the same page as figure Legends to assist in the review process. A single PDF that contains all the manuscript text and Figures should be used for the initial submission. For accepted manuscripts only, all figures must be submitted in final form ready for publication (this includes all labeling, shading, arrows, and other aspects of the figure). Prepare the figures for reduction in size of at most 50% and ensure that the lettering does not vary in size. The final font size (after reduction) should be 6-8 point. The physical dimensions of the figures should match the dimension of the journal.

Tables
Tables must be submitted as separate WORD or EXCEL files and should not be embedded in the main manuscript file. Tables should be used to organize key information in the manuscript. Tables should be used to make the presentation more concise and not used to duplicate what is already contained in the text.
For insights as to what should be included in uncropped gels in any submitted manuscript for consideration at the JHLT that has western blot of similar data, please refer to the reference image which was developed by one of JHLT's Technical Editors.

Supplementary Material
The Supplemental Material section should be used for non-critical data (Figures, Tables), large data sets and non-essential methods and will be hosted online only. This information can be submitted as a single file, for which the first includes the title and list of authors. The pages should be numbered, a table of contents should be provided, text of methods or other text related details, then figures, tables and references, in that order. Errors made in this file cannot be corrected so care must be taken in developing this file which is typically not copy-edited. The figures and tables should be tagged as "Figure S1, S2, and Table S1, S2" - No data should be provided in the supplement that is not home to a tag in the main manuscript.

REVISED MANUSCRIPTS
Revised manuscripts require the following:
• Point-by-point rebuttal letter. This should include a detailed response to the issues raised by the original review and reference changes in the manuscript to line#, page# and paragraph.
• 250-word Tweet
• Twitter handle for Corresponding and 1st author
• ORCID ID # see https://orcid.orgfor details
• Manuscript text with changes highlighted in yellow
• Clean manuscript text
• Individually uploaded Figures, with the following specifications.
• All figures should be submitted in final form ready for publication (this includes all labeling, shading, arrows, and other aspects of the figure). Prepare the figures for reduction in size of at most 50% and ensure that the lettering does not vary in size. The final font size (after reduction) should be 6-8 point. The physical dimensions of the figures should match the dimension of the journal. Most figures will be reduced to fit one-column width (3 1/4 inches). Computer-generated illustrations and halftones/line/ tones (either color or black & white) are acceptable. Figures should be submitted in a Graphics Format for PCs (.tiff) on Editorial Manager along with the manuscript. Although we welcome color figures, please note that there is an additional charge for color printing that appear in print. There is no fee for color figures that are accepted for publication that will appear in color online but are print in black & white. Please inform us if you wish to have on line only color figures. Uniform lettering, sizing, type of software used, and logical naming for the artwork files is strongly encouraged when submitting electronically. Line graphs and bar charts should be sent in bitmap TIFF files with a minimum resolution of 1200 dpi, or as EPS files, with a preferred line width of 1 pt. (minimum line width of 0.5 pt.). Bitmap images should be sent as TIFF files and not placed within EPS files. Pictures, photographs and micrographs should be sent as Grayscale TIFF files with a minimum resolution of 300 dpi without lettering, and 500 dpi with lettering. Color images should be sent as CMYK-encoded TIFF images. RGB-encoded color images will be difficult to match for color. Graphs produced in software such as Excel or Word should be submitted in black and white with different types of dashed lines for total clarity. Please visit the publisher's website (https://www.elsevier.com/authors/author-schemas/artwork-and-media-instructions) for detailed instructions and visual guidelines. All illustrations should show optimal clarity of detail. All illustrations should be accompanied by a suitable descriptive text legend.
• Tables and other components per initial submission

RESEARCH DATA

Mendeley Data
This journal supports Mendeley Data, enabling you to deposit any research data (including raw and processed data, video, code, software, algorithms, protocols, and methods) associated with your manuscript in a free-to-use, open access repository. Before submitting your article, you can deposit the relevant datasets to Mendeley Data. Please include the DOI of the deposited dataset(s) in your main manuscript file. The datasets will be listed and directly accessible to readers next to your published article online.

Data statement
To foster transparency, we encourage you to state the availability of your data in your submission. This may be a requirement of your funding body or institution. If your data is unavailable to access or unsuitable to post, you will have the opportunity to indicate why during the submission process, for example by stating that the research data is confidential. The statement will appear with your published article on ScienceDirect. For more information, visit the Data Statement page.

MANUSCRIPT REVIEW PROCESS AND OUTCOMES
After submission to the journal, the Managing Editor will assess the manuscript to ensure that it is formatted correctly and meets the submission requirements. Failure to do so will lead to the manuscript being returned to the authors. The manuscript will then be assigned to an appropriate Section Editor based upon subject matter expertise. The Section Editor will then decide whether the manuscript merits external peer review or if it should be rejected without review or "streamlined". Any "streamlined" decision by the Section Editor is passed to the appropriate Deputy Editor who can either agree with the decision or decide that the manuscript should be returned to the Section Editor and sent for external peer review. We aim to make all "streamline" decisions within three days. Manuscripts that the Section Editor determines to send out for external peer review will seek reviews from 2-3 external experts. Some of the external peer reviewers may be Editorial Board Members. External peer reviewers are expected to return their reviews within 12 days. Once the Section Editor has acquired all reviews, the Section Editor will then formulate an opinion of the manuscript and communicate with the appropriate Deputy Editor. The Deputy Editor along with the Editor-in-Chief will then render a decision on the manuscript. We aim to render decisions on external peer reviewed manuscripts within three weeks. Manuscripts that are invited for revision will be given Editorial guidance as to what are the key components that need to be addressed to achieve acceptance. All revised manuscripts will be reviewed by a biostatistical editor to assess the quality of statistical methods and analysis. Upon revision, the authors will require to provide a point-by-point rebuttal to the reviewer issues. Please note, that the anonymous reviewer comments and the authors response will be published as a supplement with all published articles. The Editor is responsible for the final decision regarding acceptance or rejection of articles. The Editor's decision is final. More information on types of peer review can be found at https://www.elsevier.com/reviewers/what-is-peer-review).

Acceptance Rates
Typical acceptance rates for original science articles are in the 10-15% range.

Appeals
Appeals are rarely successful but if the authors wish to challenge a manuscript decision, please email the editorial office with rebuttal letter describing the issues in a point-by-point format.

PUBLISHING POLICIES

Submission of an article implies that the work described has not been published previously (except in the form of an abstract or as part of a published lecture or academic thesis), that it is not under consideration for publication elsewhere, that its publication is approved by all authors and tacitly or explicitly by the responsible authorities where the work was carried out, and that, if accepted, it will not be published elsewhere in the same form in English or in any other language, without the written consent of the copyright holder.
Upon acceptance of an article, authors will be asked to transfer copyright to the International Society for Heart and Lung Transplantation. A letter will be sent to the corresponding author confirming receipt of the manuscript, and a form facilitating transfer of copyright will be provided.
If excerpts from other copyright works are included, the author(s) must obtain written permission from the copyright owners and credit the source(s) in the articles. Copies of letters granting permission must be submitted with the manuscript.

Open access
The journal offers the option of making an article freely available to all via the ScienceDirect platform. To prevent any conflict of interest, an author can only make this choice after receiving notification that the article has been accepted for publication. The fee of US\$3,000 excludes taxes and other potential author fees such as color charges. In some cases, institutions and funding bodies have entered into agreement with Elsevier to meet these fees on behalf of their authors. Details of these agreements are available at https://www.elsevier.com/fundingbodies. Authors of accepted articles, who wish to take advantage of this option, are required to complete and submit the order form (available at https://www.elsevier.com/locate/openaccessform.pdf). Whatever access option is chosen, the authors retain many rights as an author, including the right to post a revised personal version of your article on your own website. More information can be found here: https://www.elsevier.com/authorsrights.

Green open access
Authors can share their research in a variety of different ways and Elsevier has a number of green open access options available. We recommend authors see our green open access page for further information (https://elsevier.com/greenopenaccess). Authors can also self-archive their manuscripts immediately and enable public access from their institution's repository after an embargo period. This is the version that has been accepted for publication and which typically includes author-incorporated changes suggested during submission, peer review and in editor-author communications.

Embargo period
For subscription articles, an appropriate amount of time is needed for journals to deliver value to subscribing customers before an article becomes freely available to the public. Generally, we now make papers available online within 1 week of final acceptance in their non-copyedited form to facilitate dissemination of information with immediacy. In some instances, we may choose, at the Editorial Office discretion, to embargo a paper for a specific issue or time. We typically are open to discussing this with authors.

Permission to Reuse
No part of materials published in The Journal of Heart and Lung Transplantation may be reproduced without written permission of the publisher. For information on requesting permission to reuse The Journal of Heart and Lung Transplantation, click on https://www.elsevier.com/about/our-business/policies/copyright/permissions. Some requests from other STM signatory publishers qualify for gratis reuse. See the STM website for details: http://www.stm-assoc.org/permissions-guidelines/. Please visit the following website for permission forms and for more information about how to obtain permission to reproduce published data: https://www.elsevier.com/about/our-business/policies/copyright/permissions.
Questions about obtaining permission? Contact the Permissions Helpdesk at [email protected] or (+1) 215-239-3867.


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# with sample problems

## Intro

In business calculus, concavity is a word used to describe the shape of a curve. In general, a curve can be either concave up or concave down.

## Concave Up

First, let's figure out how concave up graphs look.

## Learn the concept

A concave up graph is a curve that "opens upward", meaning it resembles the shape $\cup$

You can think of the concave up graph as being able to "hold water", as it resembles the bottom of a cup.

For example, the graph of the function $y=x^2+2$ results in a concave up curve. To view the graph of this function, click here. Notice that the graph opens "up".

## Concave Down

The opposite of concave up graphs, concave down graphs point in the opposite direction.

## Learn the concept

On the other hand, a concave down curve is a curve that "opens downward", meaning it resembles the shape $\cap$

So, a concave down graph is the inverse of a concave up graph. Using the same analogy, unlike the concave up graph, the concave down graph does NOT "hold water", as the water within it would fall down, because it resembles the top part of a cap.

For example, the graph of the function $y=-3x^2+5$ results in a concave down curve. To view the graph, click here. Notice this graph opens "down".

## Both Concave Up & Down

These two examples are always either concave up or concave down. However, a function can be concave up for certain intervals, and concave down for other intervals. This means that the graph can open up, then down, then up, then down, and so forth.

The perfect example of this is the graph of $y=sin(x)$. Click here to view the graph for this function. As you can see, the graph opens downward, then upward, then downward again, then upward, etc.

## A Little Tip

Just as functions can be concave up for some intervals and concave down for others, a function can also not be concave at all. If the second derivative of the function equals $0$ for an interval, then the function does not have concavity in that interval.

## Concavity Using Derivatives

You can easily find whether a function is concave up or down in an interval based on the sign of the second derivative of the function.

## Learn the concept

In words: If the second derivative of a function is positive for an interval, then the function is concave up on that interval. Otherwise, if the second derivative is negative for an interval, then the function is concave down at that point.

In math notation: If $f''(x) > 0$ for $[a,b]$, then $f(x)$ is concave up on $[a,b]$. Otherwise, if $f''(x) < 0$ for $[a,b$], then $f(x)$ is concave down on $[a,b]$

The concept is very similar to that of finding intervals of increase and decrease. We still set a derivative equal to $0$, and we still plug in values left and right of the zeroes to check the signs of the derivatives in those intervals.

The main difference is that instead of working with the first derivative to find intervals of increase and decrease, we work with the second derivative to find intervals of concavity.

## How to Apply

In business calculus, you will be asked to find intervals of concavity for graphs. In other words, this means that you need to find for which intervals a graph is concave up and for which others a graph is concave down.

In general, concavity can only change where the second derivative has a zero, or where it is undefined. Here are the steps to determine concavity for $f(x)$:

• Step 1: Find the second derivative of the original function
• Step 2: If the second derivative only has a numerator (it is not a fraction), set the function equal to $0$
• Step 3: Find the numbers that make the numerator equal to $0$ by solving for $x$ in Step 2
• Step 4: Test one number to the left (less than) and one to the right (greater than) of each zero found in order to find the concavity on both sides
• Step 5: If the second derivative has a denominator, repeat Steps 2, 3 & 4 for the denominator
• Step 6: Based on all the zeros you found (both numerator and denominator) and the signs left and right, write out each interval and its concavity

While this might seem like too many steps, remember the big picture:

## The big picture

To find the intervals of concavity, you need to find the second derivative of the function, determine the $x$ values that make the function equal to $0$ (numerator) and undefined (denominator), and plug in values to the left and to the right of these $x$ values, and look at the sign of the results:

$+ \ \rightarrow$ interval is concave up

$- \ \rightarrow$ interval is concave down

## Sample Problem

Given the function $f(x)=5x^3+30x^2$

Question 1Determine where this function is concave up and concave down

The first step in determining concavity is calculating the second derivative of $f(x)$. So, we differentiate it twice,

$$f'(x)=15x^2+60x$$

$$f''(x)=30x+60$$

Now that we have the second derivative, we want to find concavity at all points of this function. The function can either be always concave up, always concave down, or both concave up and down for different intervals.

We set the second derivative equal to $0$, and solve for $x$

$$30x+60=0$$

First, we subtract $60$ from both sides,

$$30x=-60$$

And then we divide by $30$ on both sides,

$$x=-2$$

This means that this function has a zero at $x=-2$. Therefore, we need to test for concavity to both the left and right of $-2$. Let's pick $-5$ and $1$ for left and right values, respectively. We build a table to help us calculate the second derivatives at these values:

$x$ $f''(x)$ $+$ or $-$
$-5$ $30(-5)+60=-90$ $-$
$1$ $30(1)+60=90$ $+$

As per our table, when $x=-5$ (left of the zero), the second derivative is negative. Also, when $x=1$ (right of the zero), the second derivative is positive. That gives us our final answer:

$in \ (-\infty,-2) \ \rightarrow \ f(x) \ is \ concave \ down$

$in \ (-2,+\infty) \ \rightarrow \ f(x) \ is \ concave \ up$

By the way, an inflection point is a graph where the graph changes concavity. Therefore, there is an inflection point at $x=-2$.

## Lesson review

• Graphs of curves can either be concave up or concave down
• Concave up graphs open upward, and have the shape $\cup$
• Concave down graphs open downward, with the shape $\cap$
• To determine the concavity of a graph, find the second derivative of the given function and find the values that make it $0$ or undefined. Then check for the sign of the second derivative in all intervals
• If $f''(x) > 0$, the graph is concave up on the interval. Else, if $f''(x)<0$, the graph is concave down on the interval

Liked this lesson? If so, you will love our complete business calculus course.

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# What is the upward velocity?

1. Sep 1, 2011

### bionut

1. The problem statement, all variables and given/known data

A long jumper rises 0.5 meters during the flight phase of his jump. His forward velocity is 8 ms. What is the upward velocity?

2. Relevant equations

3. The attempt at a solution

Vv=Vsin$\vartheta$

So $\vartheta$ is the unkown

Okay, I did a skectch to work out basic trig, with 8m/s as my H and 0.5m as my 0
using sin$\vartheta$=o/h, =0.5/8 = 3

Then VV=8Sin3 = 0.38 m/s (which is wrong, but where did I go wrong??

2. Sep 1, 2011

### bp_psy

Re: Projectile

Is this the exact wording ? If it is then I don't think you have enough info to solve it.

This does not make sense. You try to find an angle between a velocity and distance when you need an angle between velocities. you also assume forward velocity to be the total velocity not just the vertical component.

3. Sep 1, 2011

### lewando

Re: Projectile

What is preventing you from using one of the standard kinematic equations? Hint: you know vy, ay, y, and y0.

4. Sep 1, 2011

### bionut

Re: Projectile

Hi, yea the exact words are "A long jumper rises 0.5 meters during the flight phase of his jump. His forward velocity is 8 ms. What is the upward velocity? " I assumed because its asking for upward velocity = Vertical component which is Vsin(theata) so Theta is the unknown?

The next questions after that asks "A long jumper rises 0.5 meters during the flight phase of his jump. His forward velocity is 8 ms. What is the angle of take off?"

So beacuse off this I assume I have the wrong idea regarding the first part; hence I need to not find the angle to calculate the upward velocity... if so Im confused...

5. Sep 1, 2011

### bionut

Re: Projectile

oh thanks, so if I use 0=Vi^2 + 2ad

0=Vi^2 + (2 x 9.81 X .05)
Vi^2=9.81
v=3.13

Would I be correct??

6. Sep 1, 2011

### lewando

Re: Projectile

Isolate the activity in the +y direction from the activity in the +x direction. Now ignore the activity in the +x direction. What's left is the jumper is jumping up 0.5m.

7. Sep 1, 2011

### lewando

Re: Projectile

Yes, in m/s.

8. Sep 1, 2011

### bionut

Re: Projectile

Hi again... the second part to the question "A long jumper rises 0.5 meters during the flight phase of his jump. His forward velocity is 8 ms. What is the angle of take off?"

Okay, I did a skectch to work out basic trig, with 8m/s as my H and 0.5m as my 0
using sinϑ=o/h, =0.5/8 = 3
but it should be 21.6 .... do you know where I went wrong??

9. Sep 2, 2011

### lewando

Re: Projectile

bp_psy has pointed out that your vectors are not consistent. Since you now know Vy and Vx is given, work with them. If you want to use sinϑ=o/h, then you need to compute the magnitude of the Vx + Vy combined. Or you could just run with tanϑ=o/a.

Last edited: Sep 2, 2011


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# spfilteR: Spatial Filtering with Eigenvectors

## Introduction

The spfilteR package provides tools to implement the eigenvector-based spatial filtering (ESF) approach put forth by Griffith (2003) in linear and generalized linear regression models. It allows users to obtain eigenvectors from a transformed connectivity matrix and supports the supervised and unsupervised creation of a synthetic spatial filter. For the unsupervised approach, eigenvectors can be selected based on either i) the maximization of model fit, ii) the minimization of residual autocorrelation, iii) the statistical significance of residual autocorrelation, or iv) the significance level of candidate eigenvectors. Alternatively, all eigenvectors in the search set may be included so that no selection takes place. While the functions provide a high flexibility, only a minimum of code is required to implement the unsupervised ESF approach.

This vignette solely focuses on the workflow to illustrate the functionality of the spfilteR package. For a discussion on the ESF methodology, interested readers may be referred to other sources, e.g., Griffith, Chun, and Li (2019) or Tiefelsdorf and Griffith (2007).

## Installation

A stable release version of the spfilteR package is available on CRAN and the latest development version can be downloaded from GitHub.

# CRAN release
install.packages("spfilteR")

# GitHub
library(devtools)
devtools::install_github("sjuhl/spfilteR")

## Getting Started

Together with the main functions, the package contains an artificial dataset (fakedataset) and a connectivity matrix (W) connecting the 100 cross-sections on a regular grid by using the rook criterion of adjacency. For illustrative purposes, the following examples utilize the fake dataset.

# load the package
library(spfilteR)

data("fakedata")

# take a look at the connectivity matrix and the variables
dim(W)
#> [1] 100 100

#>   ID indicator count       x1        x2         x3  negative negcount
#> 1  1         0     0 11.29514  5.527426  6.6406409 -2.338103        9
#> 2  2         0     4 13.92705  7.047870  4.0049350 -2.048737        3
#> 3  3         1     1 17.31431 10.525834 -0.7022595 -1.868379        5
#> 4  4         1     4 15.21778  7.817901  1.5948141 -2.737914        2
#> 5  5         1     3 15.56298  7.903947 -1.2708973 -1.117427        2
#> 6  6         1     2 17.43941 11.254842 -1.4158475  1.577442        2

Besides an ID variable, the dataset contains a binary indicator variable, two count variables, and four continuous variables.

### Identifying Spatial Autocorrelation

In a first step, the function MI.vec() checks for the presence of spatial autocorrelation in the continuous variables by means of Moran’s I (see also Cliff and Ord 1981, 1972). The function further allows users to define the alternative hypothesis in order to obtain p-values.

# select continuous variables
cont <- cbind(fakedataset$x1, fakedataset$x2, fakedataset$x3, fakedataset$negative)
colnames(cont) <- c("x1", "x2", "x3", "negative")

# Moran test of spatial autocorrelation
(I <- MI.vec(x = cont, W = W, alternative = 'greater'))
#>                     I          EI        VarI         zI           pI
#> x1        0.311178214 -0.01010101 0.005344193  4.3948248 5.543107e-06 ***
#> x2        0.168757531 -0.01010101 0.005344193  2.4466317 7.209904e-03  **
#> x3        0.009112739 -0.01010101 0.005344193  0.2628276 3.963417e-01
#> negative -0.244064424 -0.01010101 0.005344193 -3.2004192 9.993139e-01

The output suggests that the variables x1 and x2 are positively autocorrelated at conventional levels of statistical significance. Moreover, the standardized value of Moran’s I (zI) indicates that the variable negative is negatively autocorrelated. We can use the function MI.vec() and specify alternative = 'lower' to assess the significance of the negative autocorrelation:

X <- cbind(1, fakedataset$x2) # OLS regression ols.resid <- resid(lm(y ~ X)) # Moran test of residual spatial autocorrelation MI.resid(resid = ols.resid, W = W, alternative = 'greater') #> I EI VarI zI pI #> 1 0.350568 -0.01010101 0.005344193 4.933643 4.035495e-07 *** The results show that the residuals are significantly autocorrelated which violates the assumption of uncorrelated errors. In order to resolve this problem of spatial autocorrelation in regression residuals, the function lmFilter() estimates a spatially filtered linear regression model using an unsupervised stepwise regression to identify relevant eigenvectors derived from the transformed connectivity matrix. Below, the unsupervised eigenvector search algorithm selects eigenvectors based on the reduction in residual autocorrelation. The output is a class spfilter object. # ESF model (lm.esf <- lmFilter(y = y, x = X, W = W, objfn = 'MI', positive = TRUE, ideal.setsize = TRUE, tol = .2)) #> 8 out of 22 candidate eigenvectors selected summary(lm.esf, EV = TRUE) #> #> - Spatial Filtering with Eigenvectors (Linear Model) - #> #> Coefficients (OLS): #> Estimate SE p-value #> (Intercept) 9.2169316 0.66572188 5.088489e-24 *** #> beta_1 0.9947113 0.07995667 2.932913e-21 *** #> #> Adjusted R-squared: #> Initial Filtered #> 0.4673945 0.7257458 #> #> Filtered for positive spatial autocorrelation #> 8 out of 22 candidate eigenvectors selected #> Objective Function: "MI" #> #> Summary of selected eigenvectors: #> Estimate SE p-value partialR2 VIF MI #> ev_13 -9.580545 1.447552 2.550912e-09 0.23208263 1.005781 0.6302019 *** #> ev_4 -3.239103 1.481047 3.133258e-02 0.02873360 1.049729 0.9257835 * #> ev_10 -5.529712 1.453592 2.589498e-04 0.07901543 1.013360 0.7303271 *** #> ev_2 4.885225 1.444218 1.064466e-03 0.06058390 1.001660 1.0004147 ** #> ev_9 3.458918 1.469630 2.076982e-02 0.03041775 1.034209 0.7638378 * #> ev_20 4.188058 1.445305 4.720290e-03 0.04411345 1.002998 0.4539879 ** #> ev_5 -2.863850 1.452437 5.170840e-02 0.02055372 1.011899 0.8968632 . #> ev_19 3.805433 1.443517 9.873617e-03 0.03671349 1.000798 0.4615722 ** #> #> Moran's I (Residuals): #> Observed Expected Variance z p-value #> Initial 0.35056797 -0.01192610 0.01207299 3.2990854 0.0004850019 *** #> Filtered -0.03446177 -0.07716672 0.04880722 0.1933019 0.4233612531 While the print method shows that 8 eigenvectors were selected from the candidate set consisting of 22 eigenvectors, the summary method provides additional information. Besides the ordinary least squares (OLS) parameter estimates, the output shows that the ESF model filters for positive spatial autocorrelation using the minimization of residual autocorrelation as objective function during the eigenvector search. A comparison between the filtered and the nonspatial OLS model with respect to model fit and residual autocorrelation is also provided. Since the option EV is set to TRUE, the summary method also displays information on the selected eigenvectors. As the results show, the ESF model successfully removes the spatial pattern from model residuals. The plot method allows for an easy visualization of the results. The graph displays the Moran coefficient associated with each of the eigenvectors. The shaded area signifies the candidate set and selected eigenvectors are illustrated by black dots. plot(lm.esf) Moreover, lmFilter() also allows users to select eigenvectors based on alternative selection criteria: ### Alternative selection criteria: # maximization of model fit lmFilter(y = y, x = X, W = W, objfn = 'R2', positive = TRUE, ideal.setsize = TRUE) #> 11 out of 22 candidate eigenvectors selected # significance of residual autocorrelation lmFilter(y = y, x = X, W = W, objfn = 'pMI', sig = .1, bonferroni = FALSE, positive = TRUE, ideal.setsize = TRUE) #> 3 out of 22 candidate eigenvectors selected # significance of eigenvectors lmFilter(y = y, x = X, W = W, objfn = 'p', sig = .1, bonferroni = TRUE, positive = TRUE, ideal.setsize = TRUE) #> 3 out of 22 candidate eigenvectors selected # all eigenvectors in the candidate set lmFilter(y = y, x = X, W = W, objfn = 'all', positive = TRUE, ideal.setsize = TRUE) #> 22 out of 22 candidate eigenvectors selected If users wish to select eigenvectors based on individual selection criteria, they can obtain the eigenvectors using the function getEVs() and perform a supervised selection procedure using the basic stats::lm() command. ### Extension to Generalized Linear Models The ESF approach outlined above easily extends to generalized linear models (GLMs) as well (see also Griffith, Chun, and Li 2019). Therefore, the spfilteR package contains the function glmFilter() which uses maximum likelihood estimation (MLE) to fit a spatially filtered GLM and performs an unsupervised eigenvector search based on alternative objective functions. Except for minor differences, glmFilter() works similar to the lmFilter() function discussed above. The option model defines the model that needs to be estimated. Currently, unsupervised spatial filtering is possible for logit, probit, and poisson models. Moreover, the option optim.method specifies the optimization method used to maximize the log-likelihood function. Finally, resid.type allows users to define the type of residuals used to calculate Moran’s I and boot.MI is an integer specifying the number of bootstrap permutations to obtain the variance of Moran’s I. # define outcome variables y.bin <- fakedataset$indicator
y.count <- fakedataset\$count

# ESF logit model
(logit.esf <- glmFilter(y = y.bin, x = NULL, W = W, objfn = 'p', model = 'logit',
optim.method = 'BFGS', sig = .1, bonferroni = FALSE,
positive = TRUE, ideal.setsize = FALSE, alpha = .25,
resid.type = 'deviance', boot.MI = 100))
#> 5 out of 31 candidate eigenvectors selected

# ESF probit model
(probit.esf <- glmFilter(y = y.bin, x = NULL, W = W, objfn = 'p', model = 'probit',
optim.method = 'BFGS', sig = .1, bonferroni = FALSE,
positive = TRUE, ideal.setsize = FALSE, alpha = .25,
resid.type = 'deviance', boot.MI = 100))
#> 5 out of 31 candidate eigenvectors selected

# ESF poisson model
(poisson.esf <- glmFilter(y = y.count, x = NULL, W = W, objfn = 'BIC', model = 'poisson',
optim.method = 'BFGS', positive = TRUE, ideal.setsize = FALSE,
alpha = .25, resid.type = 'deviance', boot.MI = 100))
#> 7 out of 31 candidate eigenvectors selected

Again, if users wish to define individual selection criteria or fit a GLM currently not implemented in glmFilter(), they can obtain the eigenvectors using the getEVs() command and perform supervised eigenvector selection using the standard stats::glm() function.

## References

Cliff, Andrew D, and John K. Ord. 1972. “Testing for Spatial Autocorrelation Among Regression Residuals.” Geographical Analysis 4 (3): 267–84. https://doi.org/10.1111/j.1538-4632.1972.tb00475.x.
———. 1981. Spatial Processes: Models & Applications. Advances in Spatial Science. London: Pion.
Dray, Stéphane. 2011. “A New Perspective about Moran’s Coefficient: Spatial Autocorrelation as a Linear Regression Problem.” Geographical Analysis 43 (2): 127–41. https://doi.org/10.1111/j.1538-4632.2011.00811.x.
Griffith, Daniel A. 2003. Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization. Advances in Spatial Science. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-540-24806-4.
Griffith, Daniel A., Yongwan Chun, and Bin Li. 2019. Spatial Regression Analysis Using Eigenvector Spatial Filtering. Spatial Econometrics and Spatial Statistics. London: Elsevier. https://doi.org/10.1016/B978-0-12-815043-6.09990-0.
Tiefelsdorf, Michael, and Daniel A Griffith. 2007. “Semiparametric Filtering of Spatial Autocorrelation: The Eigenvector Approach.” Environment and Planning A: Economy and Space 39 (5): 1193–1221. https://doi.org/10.1068/a37378.


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Article

# Heat transport of the quasi-one-dimensional alternating spin chain material (CH_{3})_{2}NH_{2}CuCl_{3}

Physical review. B, Condensed matter (Impact Factor: 3.66). 10/2011; 84. DOI: 10.1103/PhysRevB.84.134429
Source: arXiv

ABSTRACT We report a study of the low-temperature heat transport in the
quasi-one-dimensional S = 1/2 alternating antiferromagnetic-ferromagnetic chain
compound (CH_{3})_{2}NH_{2}CuCl_{3}. Both the temperature and magnetic-field
dependencies of thermal conductivity are very complicated, pointing to the
important role of spin excitations. It is found that magnetic excitations act
mainly as the phonon scatterers in a broad temperature region from 0.3 to 30 K.
In magnetic fields, the thermal conductivity show drastic changes, particularly
at the field-induced transitions from the low-field N\'{e}el state to the
spin-gapped state, the field-induced magnetic ordered state, and the spin
polarized state. In high fields, the phonon conductivity is significantly
enhanced because of the weakening of spin fluctuations.

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##### Article: Low-temperature thermal conductivity of antiferromagnetic S = 1/2 chain material CuCl_2$\cdot$2((CH_3)_2SO)
[Hide abstract]
ABSTRACT: We study the heat transport of S = 1/2 chain compound CuCl_2$\cdot$2((CH_3)_2SO) along the b axis (vertical to the chain direction) at very low temperatures. The zero-field thermal conductivity (\kappa) shows a distinct kink at about 0.9 K, which is related to the long-range antiferromagnetic (AF) transition. With applying magnetic field along the c axis, \kappa(H) curves also show distinct changes at the phase boundaries between the AF and the high-field disordered states. These results indicate a strong spin-phonon interaction and the magnetic excitations play a role in the b-axis heat transport as phonon scatterers.
Journal of Applied Physics 10/2013; 115(17). · 2.19 Impact Factor
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##### Article: Phonon-glass-like behavior of magnetic origin in single-crystal Tb2Ti2O7
[Hide abstract]
ABSTRACT: We report a study on the thermal conductivity (κ) of Tb2Ti2O7 single crystals at low temperatures. It is found that in zero field this material has an extremely low phonon thermal conductivity in a broad temperature range. The mean free path of phonons is even smaller than that of amorphous materials and is 3–4 orders of magnitude smaller than the sample size at 0.3 K. The strong spin fluctuation of the spin-liquid state is discussed to be the reason of the strong phonon scattering. The magnetic-field dependence of κ and comparison with Y2Ti2O7 and TbYTi2O7 confirm the magnetic origin of this phonon-glass-like behavior.
Physical review. B, Condensed matter 06/2013; 87(21). · 3.66 Impact Factor
• ##### Article: Magnetically originated phonon-glass-like behavior in Tb_2Ti_2O_7 single crystal
[Hide abstract]
ABSTRACT: We report a study on the thermal conductivity (\kappa) of Tb_2Ti_2O_7 single crystals at low temperatures. It is found that in zero field this material has an extremely low phonon thermal conductivity in a broad temperature range. The mean free path of phonons is even smaller than that of amorphous materials and is 3--4 orders of magnitude smaller than the sample size at 0.3 K. The strong spin fluctuation of the spin-liquid state is discussed to be the reason of the strong phonon scattering. The magnetic-field dependence of \kappa and comparison with Y_2Ti_2O_7 and TbYTi_2O_7 confirm the magnetic origin of this phonon-glass-like behavior.
05/2013;


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• Browse all
Measurement of the cross section for production of $b b^-$ bar $X$, decaying to muons in $pp$ collisions at $\sqrt{s}=7$ TeV

The collaboration
JHEP 1206 (2012) 110, 2012

Abstract (data abstract)
CERN-LHC. Measurement of the inclusive cross section for b-quark-antiquark pairs from an inclusive dimuon final state in proton proton collisions at a centre of mass energy of 7 TeV. The data sample has a total integrated luminosity of 27.9 pb-1. Results are presented from selecting pairs of muons each with a pseudorapidity value in the range -2.1 to 2.1 and transverse momentum greater than either 4 GeV or 6 GeV.

• #### Table 1

Data from P 11 (C=Preprint)

10.17182/hepdata.58906.v1/t1

The measured integrated cross sections. The second systematic error is due to the luminosity uncertainty.


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• Corpus ID: 238743919

# On Convergence of Training Loss Without Reaching Stationary Points

@article{Zhang2021OnCO,
title={On Convergence of Training Loss Without Reaching Stationary Points},
author={Jingzhao Zhang and Haochuan Li and Suvrit Sra and Ali Jadbabaie},
journal={ArXiv},
year={2021},
volume={abs/2110.06256}
}
It is a well-known fact that nonconvex optimization is computationally intractable in the worst case. As a result, theoretical analysis of optimization algorithms such as gradient descent often focuses on local convergence to stationary points where the gradient norm is zero or negligible. In this work, we examine the disconnect between the existing theoretical analysis of gradient-based algorithms and actual practice. Specifically, we provide numerical evidence that in large-scale neural…

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ResNet strikes back: An improved training procedure in timm
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ArXiv
• 2021
This paper re-evaluate the performance of the vanilla ResNet-50 when trained with a procedure that integrates such advances, and shares competitive training settings and pre-trained models in the timm open-source library, with the hope that they will serve as better baselines for future work.


text:
# 8.1: Exponents and Roots

Before defining the next family of functions, the exponential functions, we will need to discuss exponent notation in detail. As we shall see, exponents can be used to describe not only powers (such as $$5^2$$ and $$2^3$$), but also roots (such as square roots - $$\sqrt{2}$$ and cube roots - $$\sqrt[3]{2}$$ ). Along the way, we’ll define higher roots and develop a few of their properties. More detailed work with roots will then be taken up in the next chapter.

## Integer Exponents

Recall that use of a positive integer exponent is simply a shorthand for repeated multiplication. For example,

$5^2 = 5 \cdot 5 \label{1}$

and

$2^3 = 2 \cdot 2 \cdot 2. \label{2}$

In general, $$b^n$$ stands for the quantity $$b$$ multiplied by itself $$n$$ times. With this definition, the following Laws of Exponents hold.

Laws of Exponents

1. $$b^{r}b^s = b^{r+s}$$
2. $$\frac{b^r}{b^s} = b^{r−s}$$
3. $$(b^r)^s=b^{rs}$$

The Laws of Exponents are illustrated by the following examples.

Example $$\PageIndex{3}$$

1. $$2^{3}2^2 = (2 \cdot 2 \cdot 2)(2 \cdot 2) = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5 = 2^{3+2}$$
2. $$\frac{2^4}{2^2} = \frac{2 \cdot 2 \cdot 2 \cdot 2}{2 \cdot 2} = 2 \cdot 2 =2^2 = 2^{4−2}$$
3. $$(2^3)^2 = (2^3)(2^3) = (2 \cdot 2 \cdot 2)(2 \cdot 2 \cdot 2) = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^6 = 2^{3 \cdot 2}$$

Note that the second law only makes sense for $$r > s$$, since otherwise the exponent $$r − s$$ would be negative or 0. But actually, it turns out that we can create definitions for negative exponents and the 0 exponent, and consequently remove this restriction.

Negative exponents, as well as the 0 exponent, are simply defined in such a way that the Laws of Exponents will work for all integer exponents.

• For the 0 exponent, the first law implies that $$b^{0}b^1 = b^{0+1}$$, and therefore $$b^{0}b = b$$. If $$b \ne 0$$, we can divide both sides by $$b$$ to obtain $$b^{0} = 1$$ (there is one exception: $$0^0$$ is not defined).
• For negative exponents, the second law implies that $b^{−n} = b^{0−n} = \frac{b^0}{b^n} = \frac{1}{b^n}$

provided that $$b \ne 0$$. For example, $$2^{−3} = \frac{1}{2^3} = \frac{1}{8}$$, and $$2^{−4} = \frac{1}{2^4} = \frac{1}{16}$$. Therefore, negative exponents and the 0 exponent are defined as follows:

Definition $$\PageIndex{4}$$

$$b^{−n} = \frac{1}{b^n}$$ and $$b^0 = 1$$

provided that $$b \ne 0$$.

Example $$\PageIndex{5}$$

a) $$4^{−3} = \frac{1}{4^3} = \frac{1}{64}$$

b) $$6^0 = 1$$

c) $$(\frac{1}{5})^{−2} = \frac{1}{(\frac{1}{5})^2} = \frac{1}{\frac{1}{25}} = 25$$

We now have $$b^n$$ defined for all integers n, in such a way that the Laws of Exponents hold. It may be surprising to learn that we can likewise define expressions using rational exponents, such as $$2^{\frac{1}{3}}$$, in a consistent manner. Before doing so, however, we’ll need to take a detour and define roots.

## Roots

Square Roots: Let’s begin by defining the square root of a real number. We’ve used the square root in many sections in this text, so it should be a familiar concept. Nevertheless, in this section we’ll look at square roots in more detail.

Definition $$\PageIndex{6}$$

Given a real number a, a “square root of a” is a number x such that $$x^2 = a$$.

For example, 3 is a square root of 9 since $$3^2 = 9$$. Likewise, 4 is a square root of 16 since $$(−4)^2 = 16$$. In a sense, taking a square root is the “opposite” of squaring, so the definition of square root must be intimately connected with the graph of $$y = x^2$$, the squaring function. We investigate square roots in more detail by looking for solutions of the equation

$$x^2 = a$$. (7)

There are three cases, each depending on the value and sign of a. In each case, the graph of the left-hand side of $$x^2 = a$$ is the parabola shown in Figures 1(a), (b), and (c).

• Case I: a < 0

The graph of the right-hand side of $$x^2 = a$$ is a horizontal line located a units below the x-axis. Hence, the graphs of $$y = x^2$$ and y = a do not intersect and the equation $$x^2 = a$$ has no real solutions. This case is shown in Figure 1(a). It follows that a negative number has no square root.

• Case II: a = 0

The graph of the right-hand side of $$x^2 = 0$$ is a horizontal line that coincides with the x-axis. The graph of $$y = x^2$$ intersects the graph of y = 0 at one point, at

the vertex of the parabola. Thus, the only solution of $$x^2 = 0$$ is x = 0, as seen in Figure 2(b). The solution is the square root of 0, and is denoted $$\sqrt{0}$$, so it follows $$\sqrt{0} = 0$$.

• Case III: a > 0

The graph of the right-hand side of $$x^2 = a$$ is a horizontal line located a units above the x-axis. The graphs of $$y = x^2$$ and y = a have two points of intersection, and therefore the equation $$x^2 = a$$ has two real solutions, as shown in Figure 1(c). The solutions of $$x^2 = a$$ are $$x = \pm \sqrt{a}$$. Note that we have two notations, one that calls for the positive solution and a second that calls for the negative solution.

Let’s look at some examples.

Example $$\PageIndex{8}$$

What are the solutions of $$x^2 = −5$$?

The graph of the left-hand side of $$x^2 = −5$$ is the parabola depicted in Figure 1(a). The graph of the right-hand side of $$x^2 = −5$$ is a horizontal line located 5 units below the x-axis. Thus, the graphs do not intersect and the equation $$x^2 = −5$$ has no real solutions.

You can also reason as follows. We’re asked to find a solution of $$x^2 = −5$$, so you must find a number whose square equals −5. However, whenever you square a real number, the result is always nonnegative (zero or positive). It is not possible to square a real number and get −5.

Note that this also means that it is not possible to take the square root of a negative number. That is, $$\sqrt{−5}$$ is not a real number.

Example $$\PageIndex{9}$$

What are the solutions of $$x^2 = 0$$?

There is only one solution, namely x = 0. Note that this means that $$\sqrt{0} = 0$$.

Example $$\PageIndex{10}$$

What are the solutions of $$x^2 = 25$$?

The graph of the left-hand side of $$x^2 = 25$$ is the parabola depicted in Figure 1(c). The graph of the right-hand side of $$x^2 = 25$$ is a horizontal line located 25 units above the x-axis. The graphs will intersect in two points, so the equation $$x^2 = 25$$ has two real solutions.

The solutions of $$x^2 = 25$$ are called square roots of 25 and are written $$x = \pm \sqrt{25}$$. In this case, we can simplify further and write $$x = \pm 5$$.

It is extremely important to note the symmetry in Figure 1(c) and note that we have two real solutions, one negative and one positive. Thus, we need two notations, one for the positive square root of 25 and one for the negative square root 25.

Note that $$(5)^2 = 25$$, so x = 5 is the positive solution of $$x^2 = 25$$. For the positive solution, we use the notation

$$\sqrt{25} = 5$$.

This is pronounced “the positive square root of 25 is 5.”

On the other hand, note that $$(−5)^2 = 25$$, so x = −5 is the negative solution of $$x^2 = 25$$. For the negative solution, we use the notation

$$−\sqrt{25} = −5$$.

This is pronounced “the negative square root of 25 is −5.”

This discussion leads to the following detailed summary.

SUMMARY: SQUARE ROOT

The solutions of $$x^2 = a$$ are called “square roots of a.”

• Case I: a < 0. The equation $$x^2 = a$$ has no real solutions.
• Case II: a = 0. The equation $$x^2 = a$$ has one real solution, namely x = 0. Thus, $$\sqrt{0} = 0$$.
• Case III: a > 0. The equation $$x^2 = a$$ has two real solutions, $$x = \pm \sqrt{a}$$. The notation $$\sqrt{a}$$ calls for the positive square root of a, that is, the positive solution of $$x^2 = a$$. The notation $$−\sqrt{a}$$ calls for the negative square root of a, that is, the negative solution of $$x^2 = a$$.

Cube Roots: Let’s move on to the definition of cube roots.

Definition $$\PageIndex{11}$$

Given a real number a, a “cube root of a” is a number x such that $$x^3 = a$$.

For example, 2 is a cube root of 8 since $$2^3 = 8$$. Likewise, −4 is a cube root of −64 since $$(−4)^3 = −64$$. Thus, taking the cube root is the “opposite” of cubing, so the definition of cube root must be closely connected to the graph of $$y = x^3$$, the cubing function. Therefore, we look for solutions of

$$x^3 = a$$. (12)

Because of the shape of the graph of $$y = x^3$$, there is only one case to consider. The graph of the left-hand side of $$x^3 = a$$ is shown in Figure 2. The graph of the right-hand side of $$x^3 = a$$ is a horizontal line, located a units above, on, or below the x-axis, depending on the sign and value of a. Regardless of the location of the horizontal line y = a, there will only be one point of intersection, as shown in Figure 2.

A detailed summary of cube roots follows.

SUMMARY: CUBE ROOT

The solutions of $$x^3 = a$$ are called the “cube roots of a.” Whether a is negative, zero, or positive makes no difference. There is exactly one real solution, namely $$x = \sqrt[3]{a}$$.

Let’s look at some examples.

Example $$\PageIndex{13}$$

What are the solutions of $$x^3 = 8$$?

The graph of the left-hand side of $$x^3 = 8$$ is the cubic polynomial shown in Figure 2. The graph of the right-hand side of $$x^3 = 8$$ is a horizontal line located 8 units above the x-axis. The graphs have one point of intersection, so the equation $$x^3 = 8$$ has exactly one real solution.

The solutions of $$x^3 = 8$$ are called “cube roots of 8.” As shown from the graph, there is exactly one real solution of $$x^3 = 8$$, namely $$x = \sqrt[3]{8}$$. Now since $$(2)^3 = 8$$, it follows that x = 2 is a real solution of $$x^3 = 8$$. Consequently, the cube root of 8 is 2, and we write

$$\sqrt[3]{8} = 2$$.

Note that in the case of cube root, there is no need for the two notations we saw in the square root case (one for the positive square root, one for the negative square root). This is because there is only one real cube root. Thus, the notation $$\sqrt[3]{8}$$ is pronounced “the cube root of 8.”

Example $$\PageIndex{14}$$

What are the solutions of $$x^3 = 0$$?

There is only one solution of $$x^3 = 0$$, namely x = 0. This means that $$\sqrt[3]{0} = 0$$.

Example $$\PageIndex{15}$$

What are the solutions of $$x^3 = −8$$?

The graph of the left-hand side of $$x^3 = −8$$ is the cubic polynomial shown in Figure 2. The graph of the right-hand side of $$x^3 = −8$$ is a horizontal line located 8 units below the x-axis. The graphs have only one point of intersection, so the equation $$x^3 = −8$$ has exactly one real solution, denoted $$x = \sqrt[3]{−8}$$. Now since $$(−2)^3 = −8$$, it follows that x = −2 is a real solution of $$x^3 = −8$$. Consequently, the cube root of −8 is −2, and we write

$$\sqrt[3]{−8} = −2$$.

Again, because there is only one real solution of $$x^3 = −8$$, the notation $$\sqrt[3]{−8}$$ is pronounced “the cube root of −8.” Note that, unlike the square root of a negative number, the cube root of a negative number is allowed.

Higher Roots: The previous discussions generalize easily to higher roots, such as fourth roots, fifth roots, sixth roots, etc.

Definition $$\PageIndex{16}$$

Given a real number a and a positive integer n, an “$$n^{th}$$ root of a” is a number x such that $$x^n = a$$.

For example, 2 is a $$6^{th}$$ root of 64 since $$2^6 = 64$$ and −3 is a fifth root of −243 since $$(−3)^5 = −243$$.

The case of even roots (i.e., when n is even) closely parallels the case of square roots. That’s because when the exponent n is even, the graph of $$y = x^n$$ closely resembles that of $$y = x^2$$. For example, observe the case for fourth roots shown in Figures 3(a), (b), and (c).

The discussion for even $$n^{th}$$ roots closely parallels that presented in the introduction of square roots, so without further ado, we go straight to the summary.

SUMMARY: EVEN $$N^{th}$$ ROOT

If n is a positive even integer, then the solutions of $$x^n = a$$ are called “$$n^{th}$$ roots of a.”

• Case I: a < 0. The equation $$x^n = a$$ has no real solutions.
• Case II: a = 0. The equation $$x^n = a$$ has exactly one real solution, namely x = 0. Thus, $$\sqrt[n]{0} = 0$$.
• Case III: a > 0. The equation $$x^n =a$$ has two real solutions, $$x = \pm \sqrt[n]{a}$$. The notation $$\sqrt[n]{a}$$ calls for the positive $$n^{th}$$ root of a, that is, the positive solution of $$x^n = a$$. The notation $$−\sqrt[n]{a}$$ calls for the negative $$n^{th}$$ root of a, that is, the negative solution of $$x^n = a$$.

Likewise, the case of odd roots (i.e., when n is odd) closely parallels the case of cube roots. That’s because when the exponent n is odd, the graph of $$y = x^n$$ closely resembles that of $$y = x^3$$. For example, observe the case for fifth roots shown in Figure 4.

The discussion of odd $$n^{th}$$ roots closely parallels the introduction of cube roots which we discussed earlier. So, without further ado, we proceed straight to the summary.

SUMMARY: ODD $$N^{th}$$ ROOT

If n is a positive odd integer, then the solutions of $$x^n = a$$ are called the “$$n^{th}$$ roots of a.” Whether a is negative, zero, or positive makes no difference. There is exactly one real solution of $$x^n = a$$, denoted $$x = \sqrt[n]{a}$$.

Remark 17. The symbols $$\sqrt{}$$ and $$\sqrt[n]{}$$ for square root and $$n^{th}$$ root, respectively, are also called radicals.

We’ll close this section with a few more examples.

Example $$\PageIndex{18}$$

What are the solutions of $$x^4 = 16$$?

The graph of the left-hand side of $$x^4 = 16$$ is the quartic polynomial shown in Figure 3(c). The graph of the right-hand side of $$x^4 = 16$$ is a horizontal line, located 16 units above the x-axis. The graphs will intersect in two points, so the equation $$x^4 = 16$$ has two real solutions.

The solutions of $$x^4 = 16$$ are called fourth roots of 16 and are written $$x = \pm \sqrt[4]{16}$$. It is extremely important to note the symmetry in Figure 3(c) and note that we have two real solutions of $$x^4 = 16$$, one of which is negative and the other positive. Hence, we need two notations, one for the positive fourth root of 16 and one for the negative fourth root of 16.

Note that $$2^4 = 16$$, so x = 2 is the positive real solution of $$x^4 = 16$$. For this positive solution, we use the notation

$$\sqrt[4]{16} = 2$$.

This is pronounced “the positive fourth root of 16 is 2.”

On the other hand, note that $$(−2)^4 = 16$$, so x = −2 is the negative real solution of $$x^4 = 16$$. For this negative solution, we use the notation

$$−\sqrt[4]{16} = −2$$. (19)

This is pronounced “the negative fourth root of 16 is −2.”

Example $$\PageIndex{19}$$

What are the solutions of $$x^5 = −32$$?

The graph of the left-hand side of $$x^5 = −32$$ is the quintic polynomial pictured in Figure 4. The graph of the right-hand side of $$x^5 = −32$$ is a horizontal line, located 32 units below the x-axis. The graphs have one point of intersection, so the equation $$x^5 = −32$$ has exactly one real solution.

The solutions of $$x^5 = −32$$ are called “fifth roots of −32.” As shown from the graph, there is exactly one real solution of $$x^5 = −32$$, namely $$x = \sqrt[5]{−32}$$. Now since $$(−2)^5 = −32$$, it follows that x = −2 is a solution of $$x^5 = −32$$. Consequently, the fifth root of −32 is −2, and we write

$$\sqrt[5]{−32} = −2$$.

Because there is only one real solution, the notation $$\sqrt[5]{−32}$$ is pronounced “the fifth root of −32.” Again, unlike the square root or fourth root of a negative number, the fifth root of a negative number is allowed.

Not all roots simplify to rational numbers. If that were the case, it would not even be necessary to implement radical notation. Consider the following example.

EXAMPL $$\PageIndex{20}$$

Find all real solutions of the equation $$x^2 = 7$$, both graphically and algebraically, and compare your results.

We could easily sketch rough graphs of $$y = x^2$$ and y = 7 by hand, but let’s seek a higher level of accuracy by asking the graphing calculator to handle this task.

• Load the equation $$y = x^2$$ and y = 7 into Y1 and Y2 in the calculator’s Y= menu, respectively. This is shown in Figure 5(a).
• Use the intersect utility on the graphing calculator to find the coordinates of the points of intersection. The x-coordinates of these points, shown in Figure 5(b) and (c), are the solutions to the equation $$x^2 = 7$$.

Guidelines for Reporting Graphing Calculator Solutions. Recall the standard method for reporting graphing calculator results on your homework:

• Copy the image from your viewing window onto your homework paper. Label and scale each axis with xmin, xmax, ymin, and ymax, then label each graph with its equation, as shown in Figure 6.
• Drop dashed vertical lines from each point of intersection to the x-axis. Shade and label your solutions on the x-axis.

Hence, the approximate solutions are $$x \approx −2.645751$$ or $$x \approx 2.6457513$$.

On the other hand, to find analytic solutions of $$x^2 = 7$$, we simply take plus or minus the square root of 7.

$$x^2 = 7$$

$$\sqrt{x} = \pm 7$$

To compare these exact solutions with the approximate solutions found by using the graphing calculator, use a calculator to compute $$\pm \sqrt{7}$$, as shown in Figure 7.

Note that these approximations of $$−\sqrt{7}$$ and $$\sqrt{7}$$ agree quite nicely with the solutions found using the graphing calculator’s intersect utility and reported in Figure 6.

Both $$−\sqrt{7}$$ and $$\sqrt{7}$$ are examples of irrational numbers, that is, numbers that cannot be expressed in the form $$\frac{p}{q}$$, where p and q are integers.

## Rational Exponents

As with the definition of negative and zero exponents, discussed earlier in this section, it turns out that rational exponents can be defined in such a way that the Laws of Exponents will still apply (and in fact, there’s only one way to do it).

The third law gives us a hint on how to define rational exponents. For example, suppose that we want to define $$2^{\frac{1}{3}}$$. Then by the third law,

$$(2^{\frac{1}{3}})^3 = 2^{\frac{1}{3} \cdot 3} = 2^1 = 2$$,

so, by taking cube roots of both sides, we must define $$2^{\frac{1}{3}}$$. by the formula

$$2^{\frac{1}{3}} = \sqrt[3]{2}$$.

The same argument shows that if n is any odd positive integer, then $$2^{\frac{1}{n}}$$ must be defined by the formula

$$2^{\frac{1}{n}} = \sqrt[n]{2}$$.

However, for an even integer n, there appears to be a choice. Suppose that we want to define $$2^{\frac{1}{2}}$$. Then

$$(2^{\frac{1}{2}})^2 = 2^{\frac{1}{2} \cdot 2} = 2^1 = 2$$,

so,

$$2^{\frac{1}{2}} = \sqrt{2}$$.

However, the negative choice for the exponent $$\frac{1}{2}$$ leads to problems, because then certain expressions are not defined. For example, it would follow from the third law that

$$(2^{\frac{1}{2}})^\frac{1}{2} = −\sqrt{−\sqrt{2}}$$.

But $$−\sqrt{2}$$ is negative, so $$\sqrt{−\sqrt{2}}$$ is not defined. Therefore, it only makes sense to use the positive choice. Thus, for all n, even and odd, $$2^{\frac{1}{n}}$$ is defined by the formula

$$2^{\frac{1}{n}} = \sqrt[n]{2}$$.

In a similar manner, for a general positive rational $$\frac{m}{n}$$ the third law implies that

$$2^{\frac{m}{n}} = (2^m)^{\frac{1}{n}} = \sqrt[n]{2^m}$$

But also,

$$2^{\frac{m}{n}} = (2^{\frac{1}{n}})^m = (\sqrt[n]{2})^m$$

Thus,

$$2^{\frac{m}{n}} = \sqrt[n]{2^m} = (\sqrt[n]{2})^m$$

Finally, negative rational exponents are defined in the usual manner for negative exponents:

$$2^{−\frac{m}{n}} = \frac{1}{2^{\frac{m}{n}}}$$

More generally, here is the final general definition. With this definition, the Laws of Exponents hold for all rational exponents.

Definition $$\PageIndex{22}$$

For a positive rational exponent $$\frac{m}{n}$$, and b > 0

$$b^{\frac{m}{n}} = \sqrt[n]{b^m} = (\sqrt[n]{b})^m$$ (23)

For a negative rational exponent $$−\frac{m}{n}$$,

$$b^{−\frac{m}{n}} = \frac{1}{b^{\frac{m}{n}}}$$ (24)

Remark 25. For b < 0, the same definitions make sense only when n is odd. For example $$(−2)^{\frac{1}{4}}$$ is not defined.

Example $$\PageIndex{26}$$

Compute the exact values of

(a) $$4^{\frac{5}{2}}$$

(b) $$64^{\frac{2}{3}}$$

(c) $$81^{−\frac{3}{4}}$$

(a) $$4^{\frac{5}{2}} = (4^{\frac{1}{2}})^5 = (\sqrt{4})^5 = 2^5 = 32$$

(b) $$64^{\frac{2}{3}} = (64^\frac{1}{3})^2 = (\sqrt[3]{64})^2 = 4^2 = 16$$

(c) $$81^{−\frac{3}{4}} = \frac{1}{81^{\frac{3}{4}}} = \frac{1}{(81^{\frac{1}{4}})^3} = \frac{1}{3^3} = \frac{1}{27}$$

Example $$\PageIndex{27}$$

Simplify the following expressions, and write them in the form $$x^r$$:

(a) $$x^{\frac{2}{3}}x^{\frac{1}{4}}$$

(b) $$\frac{x^{\frac{2}{3}}}{x^{\frac{1}{4}}}$$

(c) $$(x^{−\frac{2}{3}})^{\frac{1}{4}}$$

(a) $$x^{\frac{2}{3}}x^{\frac{1}{4}} = x^{\frac{2}{3}+\frac{1}{4}} = x^{\frac{11}{12}}$$

(b) $$\frac{x^{\frac{2}{3}}}{x^{\frac{1}{4}}} = x^{\frac{2}{3}−\frac{1}{4}} = x^{\frac{5}{12}}$$

(c) $$(x^{−\frac{2}{3}})^{\frac{1}{4}} = x^{−\frac{2}{3} \cdot \frac{1}{4}} = x^{−\frac{1}{6}}$$

Example $$\PageIndex{28}$$

Use rational exponents to simplify $$\sqrt[5]{\sqrt{x}}$$, and write it as a single radical.

$$\sqrt[5]{\sqrt{x}} = (\sqrt{x})^{\frac{1}{5}} = (x^{\frac{1}{2}})^{\frac{1}{5}} = (x^{\frac{1}{2} \cdot \frac{1}{5}} = x^{\frac{1}{10}} = \sqrt[10]{x}$$

Example $$\PageIndex{29}$$

Use a calculator to approximate $$2^{\frac{5}{8}}$$.

## Irrational Exponents

What about irrational exponents? Is there a way to define numbers like $$2^{\sqrt{2}}$$ and $$3^{\pi}$$? It turns out that the answer is yes. While a rigorous definition of $$b^s$$ when s is irrational is beyond the scope of this book, it’s not hard to see how one could proceed to find a value for such a number. For example, if we want to compute the value of $$2^{\sqrt{2}}$$, we can start with rational approximations for $$\sqrt{2}$$. Since $$\sqrt{2}$$ = 1.41421356237310 . . ., the successive powers

$$2^1$$, $$2^{1.4}$$, $$2^{1.41}$$, $$2^{1.414}$$, $$2^{1.4142}$$, $$2^{1.41421}$$, $$2^{1.414213}$$, $$2^{1.4142135}$$, $$2^{1.41421356}$$, $$2^{1.414213562}$$, $$2^{1.4142135623}$$, . . .

should be closer and closer approximations to the desired value of $$2^{\sqrt{2}}$$.

In fact, using more advanced mathematical theory (ultimately based on the actual construction of the real number system), it can be shown that these powers approach a single real number, and we define $$2^{\sqrt{2}}$$ to be that number. Using your calculator, you can observe this convergence and obtain an approximation by computing the powers above.

The last value in the table in Figure 9(a) is a correct approximation of $$2^{\sqrt{2}}$$ to 10 digits of accuracy. Your calculator will obtain this same approximation when you ask it to compute $$2^{\sqrt{2}}$$ directly (see Figure 9(b)).

In a similar manner, $$b^s$$ can be defined for any irrational exponent s and any b > 0. Combined with the earlier work in this section, it follows that $$b^s$$ is defined for every real exponent s.

## Exercise

In Exercises 1-12, compute the exact value.

Exercise $$\PageIndex{1}$$

$$3^{−5}$$

$$\frac{1}{243}$$

Exercise $$\PageIndex{2}$$

$$4^2$$

Exercise $$\PageIndex{3}$$

$$(\frac{3}{2})^3$$

$$\frac{27}{8}$$

Exercise $$\PageIndex{4}$$

$$(\frac{2}{3})^1$$

Exercise $$\PageIndex{5}$$

$$6^{−2}$$

$$\frac{1}{36}$$

Exercise $$\PageIndex{6}$$

$$4^{−3}$$

Exercise $$\PageIndex{7}$$

$$(\frac{2}{3})^{−3}$$

$$\frac{27}{8}$$

Exercise $$\PageIndex{8}$$

$$(\frac{1}{3})^{−3}$$

Exercise $$\PageIndex{9}$$

$$7^1$$

7

Exercise $$\PageIndex{10}$$

$$(\frac{3}{2})^{−4}$$

Exercise $$\PageIndex{11}$$

$$(\frac{5}{6})^3$$

$$\frac{125}{216}$$

Exercise $$\PageIndex{12}$$

$$3^2$$

In Exercises 13-24, perform each of the following tasks for the given equation.

1. Load the left- and right-hand sides of the given equation into Y1 and Y2, respectively. Adjust the WINDOW parameters until all points of intersection (if any) are visible in your viewing window. Use the intersect utility in the CALC menu to determine the coordinates of any points of intersection.
2. Make a copy of the image in your viewing window on your homework paper. Label and scale each axis with xmin, xmax, ymin, and ymax. Label each graph with its equation. Drop dashed vertical lines from each point of intersection to the x-axis, then shade and label each solution of the given equation on the x-axis. Remember to draw all lines with a ruler.
3. Solve each problem algebraically. Use a calculator to approximate any radicals and compare these solutions with those found in parts (1) and (2).

Exercise $$\PageIndex{13}$$

$$x^2 = 5$$

$$x = \pm \sqrt{5}$$

Exercise $$\PageIndex{14}$$

$$x^2 = 7$$

Exercise $$\PageIndex{15}$$

$$x^2 = −7$$

No real solutions.

Exercise $$\PageIndex{16}$$

$$x^2 = −3$$

Exercise $$\PageIndex{17}$$

$$x^3 = −6$$

$$x = \sqrt[3]{−6}$$

Exercise $$\PageIndex{18}$$

$$x^3 = −4$$

Exercise $$\PageIndex{19}$$

$$x^4 = 4$$

$$x = \pm \sqrt[4]{4}$$

Exercise $$\PageIndex{20}$$

$$x^4 = −7$$

Exercise $$\PageIndex{21}$$

$$x^5 = 8$$

$$x = \sqrt[5]{8}$$

Exercise $$\PageIndex{22}$$

$$x^5 = 4$$

Exercise $$\PageIndex{23}$$

$$x^6 = −5$$

No real solutions.

Exercise $$\PageIndex{24}$$

$$x^6 = 9$$

In Exercises 25-40, simplify the given radical expression.

Exercise $$\PageIndex{25}$$

$$\sqrt{49}$$

7

Exercise $$\PageIndex{26}$$

$$\sqrt{121}$$

Exercise $$\PageIndex{27}$$

$$\sqrt{−36}$$

Not a real number.

Exercise $$\PageIndex{28}$$

$$\sqrt{−100}$$

Exercise $$\PageIndex{29}$$

$$\sqrt[3]{−1}$$

3

Exercise $$\PageIndex{30}$$

$$\sqrt[3]{−1}$$

Exercise $$\PageIndex{31}$$

$$\sqrt[3]{−125}$$

−5

Exercise $$\PageIndex{32}$$

$$\sqrt[3]{64}$$

Exercise $$\PageIndex{33}$$

$$\sqrt[4]{−16}$$

Not a real number.

Exercise $$\PageIndex{34}$$

$$\sqrt[4]{81}$$

Exercise $$\PageIndex{35}$$

$$\sqrt[4]{16}$$

2

Exercise $$\PageIndex{36}$$

$$\sqrt[3]{−625}$$

Exercise $$\PageIndex{37}$$

$$\sqrt[5]{−32}$$

−2

Exercise $$\PageIndex{38}$$

$$\sqrt[5]{243}$$

Exercise $$\PageIndex{39}$$

$$\sqrt[5]{1024}$$

4

Exercise $$\PageIndex{40}$$

$$\sqrt[5]{−3125}$$

Exercise $$\PageIndex{41}$$

Compare and contrast $$\sqrt{(−2)^2}$$ and $$(\sqrt{−2})^2$$.

$$\sqrt{(−2)^2} = 2$$, while $$(\sqrt{−2})^2$$ is not a real number.

Exercise $$\PageIndex{42}$$

Compare and contrast $$\sqrt[4]{(−3)^4}$$ and $$(\sqrt[4]{−3})^4$$.

Exercise $$\PageIndex{43}$$

Compare and contrast $$\sqrt[3]{(−5)^3}$$ and $$(\sqrt[3]{−5})^3$$.

Both equal −5.

Exercise $$\PageIndex{44}$$

Compare and contrast $$\sqrt[5]{(−2)^5}$$ and $$(\sqrt[5]{−2})^5$$.

In Exercises 45-56, compute the exact value.

Exercise $$\PageIndex{45}$$

$$25^{−\frac{3}{2}}$$

$$\frac{1}{125}$$

Exercise $$\PageIndex{46}$$

$$16^{−\frac{5}{4}}$$

Exercise $$\PageIndex{47}$$

$$8^{\frac{4}{3}}$$

16

Exercise $$\PageIndex{48}$$

$$625^{−\frac{3}{4}}$$

Exercise $$\PageIndex{49}$$

$$16^{\frac{3}{2}}$$

64

Exercise $$\PageIndex{50}$$

$$64^{\frac{2}{3}}$$

Exercise $$\PageIndex{51}$$

$$27^{\frac{2}{3}}$$

9

Exercise $$\PageIndex{52}$$

$$625^{\frac{3}{4}}$$

Exercise $$\PageIndex{53}$$

$$256^{\frac{5}{4}}$$

1024

Exercise $$\PageIndex{54}$$

$$4^{−\frac{3}{2}}$$

Exercise $$\PageIndex{55}$$

$$256^{−\frac{3}{4}}$$

$$\frac{1}{64}$$

Exercise $$\PageIndex{56}$$

$$81^{−\frac{5}{4}}$$

In Exercises 57-64, simplify the product, and write your answer in the form $$x^r$$.

Exercise $$\PageIndex{57}$$

$$x^{\frac{5}{4}}x^{\frac{5}{4}}$$

$$x^{\frac{5}{2}}$$

Exercise $$\PageIndex{58}$$

$$x^{\frac{5}{3}}x^{−\frac{5}{4}}$$

Exercise $$\PageIndex{59}$$

$$x^{−\frac{1}{3}}x^{\frac{5}{2}}$$

$$x^{\frac{13}{6}}$$

Exercise $$\PageIndex{60}$$

$$x^{−\frac{3}{5}}x^{\frac{3}{2}}$$

Exercise $$\PageIndex{61}$$

$$x^{\frac{4}{5}}x^{−\frac{4}{3}}$$

$$x^{−\frac{8}{15}}$$

Exercise $$\PageIndex{62}$$

$$x^{−\frac{5}{4}}x^{\frac{1}{2}}$$

Exercise $$\PageIndex{63}$$

$$x^{−\frac{2}{5}}x^{−\frac{3}{2}}$$

$$x^{−\frac{19}{10}}$$

Exercise $$\PageIndex{64}$$

$$x^{−\frac{5}{4}}x^{\frac{5}{2}}$$

In Exercises 65-72, simplify the quotient, and write your answer in the form $$x^r$$.

Exercise $$\PageIndex{65}$$

$$\frac{x^{−\frac{5}{4}}}{x^{\frac{1}{5}}}$$

$$x^{−\frac{29}{20}}$$

Exercise $$\PageIndex{66}$$

$$\frac{x^{−\frac{2}{3}}}{x^{\frac{1}{4}}}$$

Exercise $$\PageIndex{67}$$

$$\frac{x^{−\frac{1}{2}}}{x^{−\frac{3}{5}}}$$

$$x^{\frac{1}{10}}$$

Exercise $$\PageIndex{68}$$

$$\frac{x^{−\frac{5}{2}}}{x^{\frac{5}{2}}}$$

Exercise $$\PageIndex{69}$$

$$\frac{x^{\frac{3}{5}}}{x^{−\frac{1}{4}}}$$

$$x^{\frac{17}{20}}$$

Exercise $$\PageIndex{70}$$

$$\frac{x^{\frac{1}{3}}}{x^{−\frac{1}{2}}}$$

Exercise $$\PageIndex{71}$$

$$\frac{x^{−\frac{5}{4}}}{x^{\frac{2}{3}}}$$

$$x^{−\frac{23}{12}}$$

Exercise $$\PageIndex{72}$$

$$\frac{x^{\frac{1}{3}}}{x^{\frac{1}{2}}}$$

In Exercises 73-80, simplify the expression, and write your answer in the form $$x^r$$.

Exercise $$\PageIndex{73}$$

$$(x^{\frac{1}{2}})^{\frac{4}{3}}$$

$$x^{\frac{2}{3}}$$

Exercise $$\PageIndex{74}$$

$$(x^{−\frac{1}{2}})^{−\frac{1}{2}}$$

Exercise $$\PageIndex{75}$$

$$(x^{−\frac{5}{4}})^{\frac{1}{2}}$$

$$x^{−\frac{5}{8}}$$

Exercise $$\PageIndex{76}$$

$$(x^{−\frac{1}{5}})^{−\frac{3}{2}}$$

Exercise $$\PageIndex{77}$$

$$(x^{−\frac{1}{2}})^{\frac{3}{2}}$$

$$x^{−\frac{3}{4}}$$

Exercise $$\PageIndex{78}$$

$$(x^{−\frac{1}{3}})^{−\frac{1}{2}}$$

Exercise $$\PageIndex{79}$$

$$(x^{\frac{1}{5}})^{−\frac{1}{2}}$$

$$x^{−\frac{1}{10}}$$
Exercise $$\PageIndex{80}$$
$$(x^{\frac{2}{5}})^{−\frac{1}{5}}$$