Balanced projectives and Axiom 3

The splitting length of a mixed Abelian group G is defined as the smallest positive integer n such that \( \mathop \otimes \limits^n G \) splits. The task of determining the splitting length of mixed Abelian groups was formulated by Irwin, Khabbaz, and Rayna. In this paper, a criterion for determining whether \( \mathop \otimes \limits^n G \) splits for countable mixed Abelian groups G of torsion-free rank 1 is found.

Solution of the fourth and fifth boundary value problems for a one-dimensional second-order parabolic equation in a curvilinear region

The inverse problems of finding the free term and the coefficient of u(x,t) in a parabolic equation is considered. The Fredholm property of the linear inverse problem of finding the right-hand side of a special form is proved, along with global existence, uniqueness, and stability theorems for its solutions. A uniqueness theorem is established for the nonlinear inverse problem of determining the coefficient under restrictions in the form of inequalities containing no smallness conditions. The proof is carried out by a method of a priori estimates, with the use of the maximum principle for parabolic and elliptic equations. A connection between the uniqueness of the solution of the inverse problem and the completeness of a certain system of functions is established.

Умножение на абелевой группе G - это гомоморфизм $$\mu: G\otimes G\rightarrow G$$. Абелева группа G называется MT-группой, если любое умноженеие на ее периодической части однозначно продолжается до умножения на G. MT-группы изучались во многих работах по теории аддитивных групп колец, но вопрос об их строении остается открытым. В настоящней работе для MT-группы G рассматривается сервантная вполне характеристическая подгруппа $$G^*_\Lambda$$, одно из основных свойств которой заключается в том, что подгруппа $$\bigcap\limits_{p \in \Lambda (G)}pG^*_\Lambda$$ является ниль-идеалом в любом кольце с аддитивной группой G (здесь $$\Lambda(G)$$ - множество всех простых чисел p, для которых p-примарная компонента группы G отлична от нуля). Показано, что для любой MT-группы G либо $$G=G^*_\Lambda$$, либо факторгруппа $$G/G^*_\Lambda$$ несчетна.

Background : The teaching of differential equations is dominated by an excessively algebraised analytic tradition. For this reason, studies that contribute to conceptualising mathematical objects associated with the differential equation are important, particularly the existence and uniqueness theorem. Objectives : From its genesis, the objective is to analyse the nature of this knowledge, its epistemology from practice. We give an account of the variational arguments , i.e. , based on practices focused on the study of change, with a predictive purpose, which allows obtaining the desired result on the differential equation: demonstrating the existence of a unique solution. Design : A documentary analysis is carried out from the Socioepistemological Theory of the works that marked the construction of this mathematical knowledge. Setting and Participants : Being a documentary-cut study, we did not have participants stricto sensu . Data collection and analysis : Our observation unit includes mathematical works as primary and secondary sources involved in constructing the theorem: its postulations, search for hypotheses and proofs. Results : A reconstruction of the theorem is offered, which from the arguments, characterises some practices that helped in the construction of mathematical objects. Conclusions : We conclude that the bounded variation, as a particular way of using change, contributed to the search or establishment of conditions for the interpretation of the solution of equations and to obtain a unique solution to the differential equation, contributions that should be key for implementations of learning situations.

An existence theorem is proved for closed convex surfaces whose principal radii of curvature regarded as functions of the unit normal vectorn satisfy the equation $$R_1 R_2 + \Phi (R_1 + R_2 ,R_1 ,R_2 ,n) + c \cdot n = \varphi (n),$$ wheree is an unknown constant vector being sought along with the surface. This equation has been studied by Miranda, who proved a variety of existence and uniqueness theorems about its solutions. The results obtained lead to a certain simplification in the hypotheses imposed on the functions Φ andϕ in these theorems.

We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups A and B for which the homomorphism group Hom(A, B) is fully idempotent. We show that if B is a torsion group or a mixed split group and if at least one of the groups A or B is divisible then the full idempotence of the homomorphism group implies its regularity. If at least one of the groups A or B is a reduced torsion-free group and their homomorphism groups are nonzero then the group is not fully idempotent. The study of fully idempotent groups Hom(A, A) comes down to reduced mixed groups A with dense elementary torsion part.

A mixed problem is considered for a system of partial differential equations modeling the process of adsorption dynamics. An existence and uniqueness theorem is proved for this problem, and the solution properties are investigated. The inverse problem is posed, involving the determination of the system coefficient given additional information about the solution. A uniqueness theorem is proved for the solution of the inverse problem.

We consider a free boundary problem for the Navier-Stokes equation in $\mathbb R^n$ ($n \geqq 2$). We prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data. The problem we consider in this paper was already treated by V.~Solonnikov [15]. But, recently in [10] we proved an $L_p$-$L_q$ maximal regularity theorem for the Stokes equation with Neumann boundary condition which is a linearized version of the free boundary problem for the Navier-Stokes equation treated in this paper. Our proof is based on this theorem. Therefore our solution is obtained in the space $W^{2,1}_{q,p}$ ($2 < p < \infty$ and $n < q < \infty$) while a solution in [15] is in $W^{2,1}_q = W^{2,1}_{q,q}$ ($n < q < \infty$). Moreover, our proof of the global in time existence theorem is much simpler than [15], because in [10] we established a maximal regularity theorem on the whole time interval $(0, \infty)$ with exponential stability. The results obtained in this paper were already announced in Shibata-Shimizu [11].

BackgroundKenya is one of the countries in sub-Saharan Africa that is not able to feed its population sufficiently and it, therefore, relies on outside assistance. Many food security projects have been funded by both the Kenyan government and other development partners in an effort to mitigate against food insecurity. Unfortunately, as revealed by assessment reports, such projects leave little impact after the end of funding.Context and purpose of the studyThis study evaluated factors affecting the sustainability of community food security projects funded by various organizations between 2005 and 2009 in the Karai and Ndeiya divisions of Kiambu County, Kenya. This study was necessary because among the literature reviewed, no other study had been done on the sustainability of community food security projects. An evaluation research design was adopted and a purposive sampling method was used to select key informants from stakeholder organizations and project groups. Data were collected using face-to-face interviews with ten key informants (community-based organizations, faith-based organizations, financial institutions and the government of Kenya) and focus group discussions with twenty groups (ten women’s groups, four men’s groups and six mixed groups) that had benefitted from the funded projects. The data collected were analyzed using the chi-square test at the 95% confidence interval level.Results and main findingsThe findings revealed that the sustainability of community food security projects is affected by group members’ participation, rainfall patterns, leadership, management and funding levels.Conclusions and recommendationsBased on the outcome of the study, we conclude that food security projects are not sustainable. The recommendation is that group members (the beneficiaries) must participate in project planning and implementation for purposes of ownership and sustainability. Farmers need to be empowered with knowledge on irrigation and off-season intensive farming of high-value crops. We are very optimistic that this recommendation will be useful for program planners, donors, policymakers, implementers and stakeholders in project design and for funding sustainable community food projects. A further study should be done on the role of stakeholders in project sustainability.

The focus of this paper is on student interpretation and usage of the existence and uniqueness theorems for first-order ordinary differential equations. The inherent structure of the theorems is made explicit by the introduction of a framework of layers concepts–conditions–connectives–conclusions, and we discuss the manners in which students’ understanding of these layers construct their perception of the theorems as a whole.

Considerably much work has been done on Backward Stochastic Differential Equations (BSDEs) in continuous-time with deterministic terminal horizon or stopping times. Various new models have been introduced in these years in order to generalize BSDEs to solve new practical financial problems.&#13;\nOne strand is focused on discrete-time models. Backward Stochastic Difference Equations (also called BSDEs if no ambiguity) on discrete-time finite-state space were introduced by Cohen and Elliott (2010a). The associated theory required only weak assumptions. In the first topic of this thesis, properties of non-linear expectations defined using the discrete-time finite-state BSDEs were studied. A converse comparison theorem was established. Properties of risk measures defined by non-linear expectations, especially the representation theorems, were given. Then the theory of BSDEs was applied to optimal design of dynamic risk measures. Another strand is about a general random terminal time, which is not necessarily a stopping time. The motivation of this model is a financial problem of hedging of defaultable contingent claims, where BSDEs with stopping times are not applicable. In the second topic of this thesis, discrete-time finite-state BSDEs under progressively enlarged filtration were considered. Martingale representation&#13;\ntheorem, existence and uniqueness theorem and comparison theorem were established. Application to nonlinear expectations was also explored. Using the theory of BSDEs, the explicit solution for optimal design of dynamic default risk measures was obtained.&#13;\nIn recent work on continuous-time BSDEs under progressively enlarged filtration, the reference filtration is generated by Brownian motions. In order to deal with cases with jumps, in the third topic of this thesis, a general reference filtration with predictable representation property and an initial time with immersion property were considered. The martingale representation theorem for square-integrable martingales under progressively enlarged filtration was established. Then the existence and uniqueness theorem of BSDEs under enlarged filtration using Lipschitz continuity of the driver was proved. Conditions for a comparison theorem were also presented. Finally applications to nonlinear expectations and hedging of defaultable contingent claims on Brownian-Poisson setting were explored.

Two simple initial value problems are given which illustrate the need to apply standard existence and uniqueness theorems prior to generating numerical solutions to ordinary differential equations.

Our main result is an existence and uniqueness theorem for Steiner triple systems which associates to every such system a binary code — called the "carrier" — which depends only on the order of the system and its 2-rank. When the Steiner triple system is of 2-rank less than the number of points of the system, the carrier organizes all the information necessary to construct directly all systems of the given order and $2$-rank from Steiner triple systems of a specified smaller order. The carriers are an easily understood, two-parameter family of binary codes related to the Hamming codes. We also discuss Steiner quadruple systems and prove an analogous existence and uniqueness theorem; in this case the binary code (corresponding to the carrier in the triple system case) is the dual of the code obtained from a first-order Reed-Muller code by repeating it a certain specified number of times. Some particularly intriguing possible enumerations and some general open problems are discussed. We also present applications of this coding-theoretic classification to the theory of triple and quadruple systems giving, for example, a direct proof of the fact that all triple systems are derived provided those of full 2-rank are and showing that whenever there are resolvable quadruple systems on $u$ and on $v$ points there is a resolvable quadruple system on $uv$ points. The methods used in both the classification and the applications make it abundantly clear why the number of triple and quadruple systems grows in such a staggering way and why a triple system that extends to a quadruple system has, generally, many such extensions.

We prove an existence and uniqueness theorem of global solutions for wave equations with “scalar nonlinearities”. Our paper is a generalization of the work of R. W. Dickey [6].