Impacts of inquiry pedagogy on undergraduate students conceptions of the function of proof

Mathematicians and mathematics educators agree that proof is an important tool in mathematics, yet too often undergraduate students see proof as a superficial part of the discipline. While proof is often used by mathematicians to justify that a theorem is true, many times proof is used for another purpose entirely such as to explain why a particular statement is true or to show mathematics students a particular proof technique. This paper reports on a study that used a form of inquiry-based learning (IBL) in an introduction to proof course and measured the beliefs of students in this course about the different functions of proof in mathematics as compared to students in a non-IBL course. It was found that undergraduate students in an introduction to proof course had a more robust understanding of the functions of proof than previous studies would suggest. Additionally, students in the course taught using inquiry pedagogy were more likely to appreciate the communication, intellectual challenge, and providing autonomy functions of proof. It is hypothesized that these results are a response to the pedagogy of the course and the types of student activity that were emphasized.     

On the acceleration of shock waves and concentration of energy

By a series of simple examples related to exact solutions of problems in gas dynamics and magnetohydrodynamics, possible mechanisms of acceleration of shock waves and concentration of energy are elucidated. The acceleration of a shock wave is investigated in the problem of motion of a plane piston at a constant velocity in the case when the initial density of the medium drops in the presence of constant counterpressure. It is shown that in this situation a “blow-up” regime is induced by a shock wave going to infinity in finite time even for limited work of the piston. A simple spherically symmetric solution with a converging shock wave is constructed and shown to lead to the concentration of energy. A general method for solving one-dimensional non-self-similar problems related to matching the equilibrium state to a motion with homogeneous deformation on a shock wave is discussed; this method leads to a solution in quadratures.     

Impact of the cost of the time resource on efficiency of economic processes

The objective of the paper is to simulate behavior of economic agents with a special emphasis on the use of the time resource in economic processes. The case studied is from the fishing industry. With simulations, the study analyzes the behavior of economic agents as they lead their processes through time. The paper estimates the value of time in a fishery operation, and shows that the time cost is essential in clarifying some phenomena in the economic behavior of agents. Among the conclusions is that the difference in economic efficiency between strong and weak ownership is due to difference in time costs. The secondary goal of the paper is to develop a simulation tool for analyzing a range of problems in the fishing industry, ranging from decision situations onboard fishing vessels to serving as a tool for fishery managers and legislators.     

格中两类分配等式的关系研究

针对离散数学经典教材中提出的"交运算对并运算的分配等式和并运算对交运算的分配等式是等价的"这一结论,分析了一种常见的错误证明,通过一个反例说明该结论在一般的格中不一定成立,进一步证明这两个分配等式在且仅在模格中是等价的,并提出利用定义判断一个模格是否是分配格的简便算法.作为一个应用,重新证明了该教材中的一条定理.     

Examples of non-convergence of conjugate descent algorithms with exact line-searches

Examples are given in which quasi-Newton and other conjugate descent algorithms fail to converge to a minimum of the object function. In the first there is convergence to a point where the gradient is infinite; in the second, a region characterized by a fine terraced structure causes the iterates to spiral indefinitely. A modification of the second construction gives convergence to a point where the gradient is non-zero, but the gradient is not continuous at this point. The implication of these examples is discussed.     

A New Definition of A Priori Knowledge: In Search of a Modal Basis

In this paper I will offer a novel understanding of a priori knowledge. My claim is that the sharp distinction that is usually made between a priori and a posteriori knowledge is groundless. It will be argued that a plausible understanding of a priori and a posteriori knowledge has to acknowledge that they are in a constant bootstrapping relationship. It is also crucial that we distinguish between a priori propositions that hold in the actual world and merely possible, non-actual a priori propositions, as we will see when considering cases like Euclidean geometry. Furthermore, contrary to what Kripke seems to suggest, a priori knowledge is intimately connected with metaphysical modality, indeed, grounded in it. The task of a priori reasoning, according to this account, is to delimit the space of metaphysically possible worlds in order for us to be able to determine what is actual.     

Optimal detection of a recurring tuple of reference fragments in a quasiperiodic sequence

An a posteriori (off-line) approach to solving the problem of maximum-likelihood detection of a recurring tuple containing reference fragments in a numerical quasiperiodic sequence is studied. The case is analyzed where (1) the total number of fragments in a sequence is unknown; (2) the index of a sequence term corresponding to the beginning of a fragment is a deterministic (not random) value; (3) a sequence distorted by additive uncorrelated Gaussian noise is available for observation. It is shown that the problem under consideration is reduced to testing a set of simple hypotheses about the mean of a random Gaussian vector. The cardinality of this totality grows exponentially as the vector dimension (i.e., the length of the sequence under study) increases. It is established that searching for a maximum-likelihood hypothesis is equivalent to finding arguments that yield a maximum for an auxiliary objective function. It is shown that maximizing the objective function reduces to solving a special optimization problem, which is proved to be solvable in polynomial time. An exact algorithm for solving this problem, which underlies the optimal (maximum-likelihood) detection algorithm for a recurring tuple, is substantiated. The kernel of the exact algorithm is an algorithm for solving a special (basic) optimization problem. Results of numerical simulations are presented.     

The problem of obtaining prescribed distributions of gas parameters

Using one-dimensional cylindrically and spherically symmetric flows as examples, the following problem is investigated in which a prescribed density distribution in a gas is obtained: given a known gas flow (background flow), it is required to continuously attach another, unknown gas flow whose density distribution at a fixed instant of time is described by some previously given function. It is shown that this problem is a characteristic Cauchy problem of standard type for which there is a valid analogue of the Kovalevskaya theorem, provided that the input data are analytic. Other problems of the same type are considered: to ensure that the unknown flow will have a prescribed gas velocity distribution and the prescribed density of the gas in the unknown flow is strictly greater than that of the gas in the background flow (flow with discontinuity). On the assumption that the input data to these problems are analytic, the existence and uniqueness of solutions are proved, in fact—of piecewise analytic solutions. The theorems proved are extended to the case of non-one-dimensional flows.     

On the solution of certain integral equations of mixed problems of plate bending theory

The problem of the bending of a Kirchhoff-Love plate in the shape of a strip under the impression of a thin linear rigid inclusion fastened at one of the edges of the plate when the other edge of the plate is rigidly clamped is considered. The problem is reduced by a Fourier integral transform to the solution of a convolution-type integral equation of the first kind in a finite segment with a regular kernel. The exact inversion of the principal part of the corresponding integral operator is constructed in the class of functions with non-integrable singularities on the segment edges. An effective asymptotic solution is given for the integral equation under investigation in this class of functions in the whole range of variation of the characteristic parameter λ. The results obtained are verified numerically. Analogous integral equations were examined in /1, 2/. The mode of investigation is similar to that proposed in /3/.     

On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals—III: Clusters of saddle points

It is shown how the recently developed Hadamard expansion procedure can be applied to the hyperasymptotic evaluation of Laplace-type integrals containing a large variable when the phase function has a cluster of close-lying saddle points. The modification to this procedure that is required when the saddles in the cluster coalesce to form a single higher-order saddle is discussed. An example is also considered in which there is both a coalescence of saddles and a Stokes phenomenon as the phase of the large variable is allowed to vary. Numerical examples are given to illustrate the accuracy that can be obtained with this new procedure.     

Semigroups of right quotients of a semigroup which is a semilattice of groups

Let S be a semigroup with zero which is a semilattice of groups. In [6], McMorris showed that the semigroup of quotients Q=Q(S) corresponding to the filter of “dense” right ideals of the semigroup S is also a semilattice of groups. He accomplished this by noting that Q is a regular semigroup in which all idempotents are central, an equivalent formulation of a semilattice of groups. In this paper we develop the semigroup of quotients Q corresponding to an arbitrary right quotient filter on S (as defined herein) and note the above result in this more general setting by explicitly constructing a semigroup which is isomorphic to Q. We also see that the underlying semilattice for Q in this case is isomorphic to a semigroup of quotients of the original semilattice for the semigroup S.     

The problem of waves of discrete spectrum in shear flows with sign - constant curvature of the profile

A Fourier transformation method is used to solve the Cauchy problem for a plane-parallel flow with a velocity profile with unchanging sign of the curvature. It is shown that in the case of an ideal fluid a perturbation forms (a residue at the pole of the integrand in the inverse Fourier transformation) in spite of the impossibility of the existence of a discrete eigenmode (Rayleigh's theorem), resembling a decaying eigenmode. This differs from the eigenmode in the fact that the residue in the critical layer is discontinuous. The total perturbation field appears to be continuous, since the discontinuity of the residue is compensated by a discontinuity in another component of the total field (non-harmonic with respect to time), namely by the integral along the edges of the cut. When the viscosity of the medium is taken into account, the pole of the integrand is displaced by a small amount proportional to (v is the kinematic viscosity), the residue at the pole becomes a continuous function and corresponds to the decaying eigenmode for the Orr-Sommerfeld equation. When the viscous times are proportional to , the differences between the solutions of the Cauchy problem for the viscous and inviscid media become small.     

Analysis of models for calculating the motion of solids of revolution of minimum resistance in soil media

The solution of problems of searching for the optimal shape of a body when it penetrates into dense media is considered using local interaction models (LIMs) and Grigoryan's model of a soil medium in an axisymmetric formulation. A new LIM is obtained that is improved by taking account of the non-linear compressibility and shear strength in the analytical solution of a problem on the expansion of a spherical cavity. The applicability of an LIM that is quadratic with respect to the velocity in determining the forces resisting penetration of sharp bodies into soft soil is justified theoretically and experimentally and the violation of the conditions for the model to be applicable in the case of blunt bodies is established. It is shown that a solution taking account of non-linear flow effects in a two-dimensional formulation enables both the shape as well as power and kinematic characteristics of the optimal blunt bodies as they pass through soil media to be improved considerably. The ratio of the finite depths of penetration of solids of revolution into soft ground taking account of internal friction is estimated by the ratio of the coefficients in the Rankine–Resal formulae.     

关于两种混沌映射的有限乘积性质

首先在一般度量空间上给出有限积映射是Li-Yorke混沌的一个判据,并且用反倒展示:当有限积映射是Li-Yorke混沌时,未必一定存在因子映射是Li-Yorke混沌的.然后,利用上述判据,在[0,1]N上证明有限积映射有不可数scrsmbled集的一个充要条件.进而,推出关于有限积映射为Li-Yorke 混沌的一组等价...     

Exact solutions of the mixed axisymmetric problem of the torsion of an elastic space containing a spherical crack

The problem of the torsion of an elastic space, weakened by a spherical crack, is reduced to a system of paired summation equations in first-order associated Legendre functions. It is assumed that the load, applied to the crack surface, can also be represented in the form of a series in associated Legendre functions. Using special differential operators, this system is reduced to permitting an exact elementary solution of a system of equations in Legendre polynomials. Two examples are given. The solution is compared with a known result in the literature. The problem of the effect of curvature of the surface on the stress intensity factor is investigated.     

Application of moment equations to the mathematical simulation of gas microflows

An approach to the simulation of moderately rarefied gas flows in a transition zone is developed. The applicability of the regularized Grad 13-moment (R13) equations to the numerical simulation of a transition flow between the continual and free-molecular gas flow regimes is explored. For the R13 equations, a numerical method is proposed that is a higher order accurate version of Godunov’s explicit method. A numerical procedure for implementing solid-wall boundary conditions is developed. One- and two-dimensional test problems are solved, including the shock wave structure and the Poiseuille flow in a plane channel. The possibility of applying the R13 equations to the simulation of plane channel and jet flows in a transition regime is explored. To this end, the flow in a square cavity generated by the motion of one of the walls is studied and the operation of the Knudsen pump is analyzed.     

Structure of a Linear Array of Uniform Vortices

The shape and properties of an infinite steady linear array of uniform vortices are calculated. A nonlinear singular integrodifferential equation is obtained for the shapes, which is solved numerically by Newton's method and Euler continuation to give a one parameter family of shapes as size over separation is varied. The kinematic properties and energy of the array are obtained. It is found that there exists an array of maximum area, for given separation, which also possesses minimum energy in accordance with a general argument of Kelvin. A simple model based on elliptical vortices is constructed, which reproduces the qualitative kinematic properties and is quantitatively quite accurate. Continuation of the numerical solution past the array of maximum area leads to a limit of finite, lens shaped, touching vortices. This array is also shown to be limit of a finite amplitude bifurcation of a vortex sheet of finite thickness. The stability of the array to two dimensional subharmonic and superharmonic disturbances is considered. General arguments, based on ideas of Kelvin, are given to show that the array is stable to superharmonic disturbances if the area is less than the maximum and otherwise unstable, and that it is always unstable to subharmonic disturbances, of which the pairing instability is a special case. It is verified by direct calculation in an Appendix that hollow vortices, whose shapes can be determined analytically in closed form, are unstable to the pairing instability whatever their size. Some speculations are made about the possible relevance of the results to the observed properties of organized structures in the turbulent mixing layer.     

合成材料的组成

对工程上所需要的特定材料,若能在理论上预言它并能依理论结果制而即得,乃为工程使用所盼.为此目的,本文即要研究一个合适的理论,选择一些物质组合起来.使作成的合成材料与人们事先所要求的“目标材料”相合.文内前三部分给出理论,第四部分给出结果.并对工程上给定的两种“目标材料”做了理论计算,依计算结果制造了合成材料,且对其做了实验.其结果是满意的.