Abstract: Systems and methods for identifying asymptotes in approximated geometric forms are provided. One disclosed method includes identifying a set of data points that represent an approximated geometric form. The data points may be organized into segments. The method may further include determining a visible range of the geometric form to display. The method may further include looping through successive segments of the approximated geometric form, and on each loop, for a current segment, making a decision whether to draw the current segment based upon a prediction of whether the current segment traverses an asymptote within the visible range. The method may further include displaying on a graphical user interface of a computing device, a graph of the segments of the geometric form in the visible range, the graph not including those segments that were decided not be drawn.

Abstract: A problem checker architecture that monitors user progress during a problem-solving process and assists the user through the process (e.g., when requested) using common human methods of solving the problem. Assistance can be in the form of detecting errors during the process, and providing context-sensitive help information when the user gets stuck or makes a mistake. The problem checker can walk the user through the process of solving a math problem one step at a time allowing the user to learn to solve math problems according to a number of different methods. Rather than simply calculating and displaying the answer, the problem checker allows the user to attempt to solve math problems, providing direction only when asked and correction only when required. The problem checker can recognize multiple solution methods for many common math problems and guide the user to the solution via any of the methods.

Abstract: The present invention extends to methods, systems, and computer program products for protected mode scheduling of operations. Protected mode (e.g., user mode) scheduling can facilitate the development of programming frameworks that better reflect the requirements of the workloads through the use of workload-specific execution abstractions. In addition, the ability to define scheduling policies tuned to the characteristics of the hardware resources available and the workload requirements has the potential of better system scaling characteristics. Further, protected mode scheduling decentralizes the scheduling responsibility by moving significant portions of scheduling functionality from supervisor mode (e.g., kernel mode) to an application.

Abstract: A piece of software running on either a computer or a calculator for receiving a mathematical expression is described. The mathematical expression is evaluated to obtain its symbolic result. If the symbolic result can be resolved to its numeric result, the software executes to obtain the numeric result. The software displays contemporaneously the symbolic result and the numeric result.

Abstract: A centralized view and management of stored variables for calculators and pieces of software emulating calculators is provided. Users can glance at the status bar and know what variables are stored. A single click then brings up a window within which a list of the stored variables and their values is shown. In this window users can clear individual variables or the entire variable list. Users can be reminded that their calculations contain stored variables by displaying the stored variable names in one or more different colors so that errors can be easily detected and corrected.

Abstract: Systems and methods for use in handwriting recognition in computer algebra are provided. One disclosed method includes receiving handwriting input from a user via a handwriting input device, the handwriting input representing a mathematical expression. The method further includes, at a recognizer, processing the handwriting input to recognize a plurality of candidates and ranking the plurality of candidates to form initial candidate data. The method may further include, at an application program, scanning the plurality of candidates for segments that match application-level criteria, and adjusting a rank of one or more of the plurality of candidates based on the matching, to form a processed candidate list. The method may further include displaying the processed candidate list via a graphical user interface.

Abstract: Mathematical expressions and functions are graphed using a computer implemented method that automatically determines an appropriate graph range. One implementation of the method begins with determining an initial variable seed range. A number of points within this seed range are randomly selected. The method determines whether the selected points correspond to the mathematical expression (i.e., whether they produce valid results when the expression is evaluated at each point). When the selected points produce valid results, the points can be used to generate and display a graph of the mathematical expression on a user display device, with a graph range based on the seed range. If not, the variable seed range can be expanded until a predetermined number of the selected points correspond to the mathematical expression, so that a graph of the mathematical expression can be generated and displayed.

Abstract: Systems and methods for identifying asymptotes in approximated geometric forms are provided. One disclosed method includes identifying a set of data points that represent an approximated geometric form. The data points may be organized into segments. The method may further include determining a visible range of the geometric form to display. The method may further include looping through successive segments of the approximated geometric form, and on each loop, for a current segment, making a decision whether to draw the current segment based upon a prediction of whether the current segment traverses an asymptote within the visible range. The method may further include displaying on a graphical user interface of a computing device, a graph of the segments of the geometric form in the visible range, the graph not including those segments that were decided not be drawn.

Abstract: Systems and methods for progressively parsing user input of a mathematical expression are provided. One disclosed method includes looping through characters in an input string, and on each loop, extracting a next token from the input string and determining a current grammar context based on the token or tokens extracted thus far. If it is determined that the current grammar context matches a predetermined condition, then the method may include modifying the tokens extracted from the input string in a predetermined manner associated with the predetermined condition. A parsing result may be obtained based on the modified tokens. The parsing result may be converted to a modified input string.

Abstract: A problem checker architecture that monitors user progress during a problem-solving process and assists the user through the process (e.g., when requested) using common human methods of solving the problem. Assistance can be in the form of detecting errors during the process, and providing context-sensitive help information when the user gets stuck or makes a mistake. The problem checker can walk the user through the process of solving a math problem one step at a time allowing the user to learn to solve math problems according to a number of different methods. Rather than simply calculating and displaying the answer, the problem checker allows the user to attempt to solve math problems, providing direction only when asked and correction only when required. The problem checker can recognize multiple solution methods for many common math problems and guide the user to the solution via any of the methods.

Abstract: Architecture for a word processing application that facilitates operating on mathematical symbols, expressions, and/or equations input to a word processing document, and returning results back to the document. User input to the document in the form of math symbols, expressions or equations is transformed into a format for processing by a math engine. The engine returns one or more operations to the user that can be performed on the input, including calculating mathematical solutions, graphing equations and viewing steps to solving math problems. A user interface allows the user choose from the possible operations and to interactively manipulate input and graphs in the word application. The results can be inserted directly into the document and also be graded automatically.

Abstract: A problem generator that takes an input as a math problem, analyzes the math problem, and intelligently spawns similar example problem types. The output is a set of math problems based on the conditions set during analysis and customization. For example, if the original problem deals with linear equations, this will be detected during analysis and used to spawn other linear equations as problems. Moreover, if the answer to the original problem is in integer format, so will the answers to the spawned problems. A customizable UI is designed to allow further customization of problem conditions to generate an accurate set of problems based on the initial input. Problem generator templates can be created, shared and modified for distribution and/or future use. Additionally, problem generation APIs can be extended for external code to automate and consume generated math problems.

Abstract: A system is disclosed for generating and displaying intermediate steps in an educational computer algebra system. The system combines templates with an algorithm to generate intermediate steps and explanatory text which may be displayed to the user.

Abstract: A Web crawler, search engine, browser program, or other client application dynamically accesses data by using “friendly” Uniform Resource Locaters (URLs) that do not require query parameters or other non-intuitive coding. A friendly URL includes a static URL that appears to identify a static resource, such as a Hyper Text Markup Language document. A friendly URL can be a link or entered in a browser program's address field. A data type in the friendly URL is mapped to a data source that dynamically accesses data associated with an intuitive data key in the friendly URL. The data key refers to a specific document, and/or is a search term. A query URL is constructed with the data key, and a data source identifier that preferably refers to a database function and is mapped to the data type. The resulting dynamically accessed data are communicated back to the requesting client application.

Abstract: Mathematical expressions and functions are graphed using a computer implemented method that automatically determines an appropriate graph range. One implementation of the method begins with determining an initial variable seed range. A number of points within this seed range are randomly selected. The method determines whether the selected points correspond to the mathematical expression (i.e., whether they produce valid results when the expression is evaluated at each point). When the selected points produce valid results, the points can be used to generate and display a graph of the mathematical expression on a user display device, with a graph range based on the seed range. If not, the variable seed range can be expanded until a predetermined number of the selected points correspond to the mathematical expression, so that a graph of the mathematical expression can be generated and displayed.

Abstract: A sensor for detecting hydrogen peroxide, comprising an element exhibiting piezoelectric properties having a metal-oxide-containing coating, the metal-oxide having a divalent or tetravalent state.

Abstract: In a computing system, evaluating a mathematical expression in presented hierarchically according to the rules of precedence of operations, initial operations at the bottom of the hierarchy may yield values too large to be calculated conventionally, even if the ultimate value of the expression may represent a calculable value. The mathematical expression is evaluated top down to determine if portions of the mathematical expression are re-expressible to simplify the expression and avoid initial or intermediate calculations that would prevent calculation of the mathematical expression. Portions of the original mathematical expression are re-expressible by substituting known mathematical identities, applying arithmetic rules, or treating a portion of the mathematical expression as a variable on which other operations are performed. Once the mathematical expression has been re-expressed, the mathematical expression is simplified, and, if possible, its value calculated.

Abstract: A piece of software running on either a computer or a calculator for receiving a mathematical expression is described. The mathematical expression is evaluated to obtain its symbolic result. If the symbolic result can be resolved to its numeric result, the software executes to obtain the numeric result. The software displays contemporaneously the symbolic result and the numeric result.

Abstract: A centralized view and management of stored variables for calculators and pieces of software emulating calculators is provided. Users can glance at the status bar and know what variables are stored. A single click then brings up a window within which a list of the stored variables and their values is shown. In this window users can clear individual variables or the entire variable list. Users can be reminded that their calculations contain stored variables by displaying the stored variable names in one or more different colors so that errors can be easily detected and corrected.

Abstract: A verb conjugating system allows a user to input a form of a verb and display the verb forms. The verb conjugating system allows the user to input the infinitive form or non-infinitive forms of a verb. When a user inputs a non-infinitive form of a verb, the verb conjugating system identifies a corresponding base form of the verb. The verb conjugating system then uses the base form to retrieve and display the verb forms for the verb. The verb conjugating system may highlight the non-infinitive form of the verb within the displayed verb forms to assist the user in locating the verb form of interest.