Higher-Dimensional Algebra VII: Groupoidification

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify - or more precisely, groupoidify - the positive part of the quantum group associated to the quiver.Comment: 67 pages, 14 eps figures; uses undertilde.sty. This is an expanded version of arXiv:0812.486

From plant to the pump : how plant genome research at MU is helping to achieve bioenergy goals

Track II: Transportation and BiofuelsIncludes audio file (16 min.)Part of America's answer to the current energy crisis could be fuels made from plants. Fuel made from plant materials, such as cellulose or corn kernels, not only holds promise of reducing our nation's dependence on foreign sources of energy, but also offers a 'green' alternative to traditional petroleum-based fuels. Researchers are investigating a number of different plants as possible sources of biofuels, with corn, soybean, switchgrass, algae, and sugar cane, being the most popular. No matter the source, the process of converting plant material into fuel will require fundamental knowledge of plant development and growth in response to changing environments. For example, production of cellulosic ethanol requires a genetic understanding of how plants control the composition and structure of their cell walls. A number of faculty in the Interdisciplinary Plant Group at the University of Missouri are working on projects that could help scientists and engineers develop new energy crops. Plant sciences at MU could also lead to other improvements in energy crops, including maximizing their productivity, increasing their resistance to pests and drought, and reducing the need for fertilizers

CONSTITUTIONAL LAW-STATE TAXATION OF GROSS RECEIPTS FROM INTERSTATE COMMERCE

A New York statute imposed a tax of two per cent on the gross receipts of all utilities doing business within the state. The State Tax Commission construed this statute as applicable to the total receipts of petitioner derived from transporting passengers for hire from a point within New York to another point within the same state over a route which passed through New Jersey and Pennsylvania. The state courts affirmed the determination of the commission, and the petitioner appealed. Held, reversed and remanded. The transportation was interstate, and an unapportioned tax on the gross receipts derived therefrom was invalid under the commerce clause, since such a tax made interstate commerce bear more than its fair share of the cost of local government. The tax would be sustained if apportioned according to the percentage of the total mileage which was traversed within the taxing state. Three justices dissented. Central Greyhound Lines,Inc., of New York v. Mealey, 334 U.S. 653, 68 S.Ct. 1260 (1948)

FEDERAL COURTS-DISQUALIFICATION OF DISTRICT JUDGE FOR PREJUDICE- SUFFICIENCY OF AFFIDAVIT

Defendants were indicted in a federal district court for conspiring to organize as the Communist Party of the United States and to advocate overthrowing the government by force or violence in violation of a federal statute. During argument on their motion for a 90-day extension, the judge remarked he thought public policy might require that the matter be given prompt attention . . . when perhaps there may be some more of these fellows up to that sort of thing ; that I am not going to give them anything like 90 days, I am going to tell you right now ; and, in answer to defense counsel\u27s contention that the indictment failed to allege any acts of force or violence, No they want to wait until they get everything set and then the acts will come. Defendants filed timely affidavits of personal bias and prejudice pursuant to section 144 of the new Judicial Code, setting out the above facts as grounds for disqualification of the judge. The district judge refused to disqualify himself and defendants petitioned the United States Court of Appeals for a writ of mandamus requiring him to do so. Held, petitions dismissed. Taken in their context the remarks of the judge did not lend fair support to the charge that he had a personal bias or prejudice against petitioners; the affidavit was therefore legally insufficient. Foster v. Medina, (App. 2d, 1948) 170 F. (2d) 632

CONTRACTS-IMPOSSIBILITY-EFFECT OF CROP FAILURE ON MIDDLEMAN\u27S CONTRACT

Defendant, a produce dealer, offered to sell to plaintiff a specified quantity of Texas New Crop U.S. 1 blackeye peas. When the original offer was made on June 13, 1947, plaintiff asked how defendant could be sure the peas would be \u27\u27No. 1.\u27\u27 Defendant replied that because the locality of Dilley, Texas, had been unusually dry, the pea crop would be sold as dry \u27\u27blackeyes; defendant also stated that he had already made a contract to purchase 7,000 bags of these blackeyes from a certain canning company. On June 16 the contract for 800 bags of peas was consummated, delivery to plaintiff to be made on or before June 30, 1947. A few days later, the entire pea crop in the locality of Dilley was destroyed by a torrential rainfall. In spite of exhaustive efforts to obtain U.S. 1 blackeyes from other parts of Texas, defendant was unable to do so. The trial court found that the contract did not contemplate delivery of peas from any particular locality and that delivery of peas of the quality contracted for from the state of Texas was not made impossible by an act of God. Plaintiff recovered judgment and defendant appealed. Held, reversed. The contract contemplated delivery from a specific crop from a particular locality. The subject matter of the contract had been destroyed by an act of God, discharging defendant\u27s duty to perform. Pearce-Young-Angel Co., Inc. v. Charles R. Allen, Inc., (S.C. 1948) 50 S.E. (2d) 698

CORPORATIONS-POWER OF DIRECTORS TO TRANSFER ALL ASSETS-DISSENTERS\u27 RIGHTS TO APPRAISAL AND PAYMENT

By action of its board of directors, defendant corporation entered into a written extension of a lease of substantially all its assets. This action was not authorized by a majority vote at a shareholders\u27 meeting or by the written consent of the holders of a majority of the shares. Plaintiff, a shareholder of record on the date the lease was made, had no knowledge of the transaction until about three months later, at which time he objected to the making of the lease and demanded payment for his shares as provided in section 44 of the Michigan general corporation act under which defendant was organized. Upon defendant\u27s refusal to comply with this demand plaintiff brought suit, citing section 57 of the act but basing his right to recovery entirely on section 44. Plaintiff alleged that the lack of a shareholders\u27 meeting and timely knowledge on his own part made it impossible for him to comply with the provisions of section 44 which required him to vote against the lease and to file written objection within twenty days of the time the action was taken. The trial court found that the directors owned or controlled voting rights of more than two-thirds of defendant\u27s stock and that the lease divested defendant of substantially all its assets. An order was made for the appointment of appraisers, and defendant appealed. Held, reversed. The extension of the lease by action of the board of directors did not violate section 57, but was lawful and valid, since it was in furtherance of the purposes of the corporation as set out in the articles of incorporation. Since a shareholders\u27 meeting was not necessary under section 57 and was never held, it was impossible for plaintiff to meet the statutory conditions precedent to appraisal and payment under section 44. Pollack v. Adwood Corporation, (Mich. 1948) 32 N.W. (2d) 62

JUDGMENTS-COLLATERAL ESTOPPEL BY A LOWER COURT JUDGMENT WHEN APPEAL THEREFROM IS DISMISSED BECAUSE THE CASE HAS BECOME MOOT

The United States sued defendant in two counts for violation of OPA price regulations. The first count asked for an injunction to restrain further violations while the second sought treble damages for past violations. By agreement of the parties the injunction issue was tried first. The trial court found that there had been no violation of price regulations and dismissed the bill. This phase of the case was appealed by the government as a final order. The appeal was dismissed on the ground that the injunction issue had become moot, the particular commodity having been decontrolled in the interim. The government then sought to prosecute its claim for damages under the second count. Held, the trial court\u27s determination in the injunction branch of the case that there had been no violation of price regulations was conclusive as against the government. It was a final determination by a court of competent jurisdiction of an essential issue actually litigated between the same parties. One judge dissented. United States v. Munsingwear, Inc., (8th Cir. 1949) 178 F. (2d) 204

Noise generated at the tyre/road interface

The work described in this thesis is directed towards the reduction of tyre/road interface noise and embodies a study of the factors involved in its generation. These factors comprise: (a) materials and construction of tyres and road surfaces (b) the spectral distribution of the noise. The importance of this work has become greater with reduction in engine noise. A review of the literature shows what has been achieved so far, and stresses the importance of maintaining other desirable tyre properties such as adhesion in wet conditions. The work has involved an analysis of mechanical factors in tyre construction and the behaviour of road surfaces. Measurements on noise have been carried out under practical conditions and also on replica surfaces in the laboratory, and in addition tests of wet road adhesion have been carried out with a variety of road surfaces. Consideration has been given to the psychological effects of the spectral distribution of noise. A major part of the work under-taken has been the development of a computer program, the results of which have made it possible to design a tyre tread block pattern to give an optimum spectral distribution. Sample tyres built to this design have been subjected to noise measurements and these have been shown to agree closely with the theoretical prediction and other properties of this tyre have proved to be satisfactory

Management of Postsurgical Hyperhidrosis With Direct Current and Tap Water

Background and Purpose. Excessive sweating, known as hyperhidrosis, involves the eccrine sweat glands of the axillae, soles, palms, and/or forehead. The use of iontophoresis to reduce or eliminate excessive sweating has been described since 1952. The purpose of this case report is to describe the use of tap water galvanism (TWG) using direct current (DC) with a patient who had postsurgical hyperhidrosis. Case Description. The patient was a 36-year-old male electrician with traumatic phalangeal amputation and postsurgical development of hyperhidrosis. Tap water galvanism was administered using a DC generator, 2 to 3 times per week for 10 treatments. The patient\u27s hands were individually submerged in 2 containers of tap water with the electrodes immersed directly into the containers. Each hand was treated with 30 minutes of TWG at 12 mA. Hyperhidrosis was measured by a 5-second imprint and subsequent tracing of the left hand placed on dry paper toweling. Outcomes. The patient\u27s hyperhidrosis decreased from the full left palmar pad, with a surface area of 10.3×12.0 cm, to a reduced area of wetness that covered a 2.2-×2.7-cm area. The patient returned to work as an electrician without needing absorbent gloves, which had prevented him from performing electrical work. Discussion. Following use of TWG, the patient\u27s palmar hyperhidrosis returned to normhidrosis

COURTS-VALIDITY OF CONTRACTS RESTRICTING VENUE IN ACTIONS UNDER THE FEDERAL EMPLOYERS\u27 LIABILITY ACT

Petitioner suffered injuries in the course of his duties as an employee of respondent railroad. Subsequently, respondent advanced money to petitioner and the latter agreed in writing that if his claim could not be settled he would sue only in the county or district where he resided at the time of the injury, or in the county or district where the injury was sustained. This agreement restricted petitioner\u27s choice of venue to either a state or federal court sitting in Michigan. Ignoring the contract, petitioner sued in an Illinois court. Respondent then brought suit in the Michigan courts to enjoin the Illinois proceeding and the injunction was granted. On certiorari, held, reversed. The contract restricting venue was void since it conflicted with the provisions of sections 5 and 6 of the Federal Employers\u27 Liability Act. Boyd v. Grand Trunk Western Railroad Company, (U.S. 1949) 70 S.Ct. 26