Abstract:
The present work is devoted to define a generalized Green’s function solution for the dual-phase-lag model in homogeneous materials in a unified manner .The high-order mixed derivative with respect to space and time which reflect the lagging behavior is treated in special manner in the dual-phase-lag heat equation in order to construct a general solution applicable to wide range of dual-phase-lag heat transfer problems of general initial-boundary conditions using Green’s function solution method. Also, the Green’s function for a finite medium subjected to arbitrary heat source and arbitrary initial and boundary conditions is constructed. Finally, four examples of different physical situations are analyzed in order to illustrate the accuracy and potentialities of the proposed unified method. The obtained results show good agreement with works of [1-4].

Abstract:
Adult Still’s disease is a relatively rare form of rheumatoid arthritis with systemic inflammatory features. The prevalence is around 1.5 cases per 100,000 - 1000,000. In the current case we display a 30-year-old male patient with refractory adult still’s disease who suffered recurrent attacks of fever 39.5°C, arthritis in proximal interphalangeal joints (PIPs), wrists, tempromandibular joints (TMJs), knees and ankles, stitching chest pain, dyspnea, erythematous rash over the trunk, sore throat, weight loss (15 Kilograms in 4 months). The patients’ disease remained uncontrolled despite of synthetic disease modifying anti-rheumatic drugs and repeated intramuscular corticosteroid injections. Laboratory workup revealed erythrocyte sedimentation rate (ESR) of 95, C-reactive protein (CRP) of 100 mg/L, hemoglobin 10.5 gm%, leukocytosis 12,000/microlitre, mild elevation of liver function tests and dyslipidemia. Serology revealed negative rheumatoid factor, anti-nuclear antibody titre of 1:80, elevated serum ferritin 4000 micrograms/litre. The patient was started on rituximab (375 mg/m^{2}), prednisolone 20 mg/day and selective Cox-2 inhibitor. Follow up for over three months following the completion of his pulse therapy, revealed no relapse of fever or fatigue, with morning stiffness of 5 - 10 minutes, VAS of 3, DAS28 of 3.8, HAQDI of 0.62, ESR 23, CRP 4.9, Hb 12.5 gm%, leucocytic count 9000/microlitre, the dose of prednisolone was successfully reduced to a dose of 5 mg/day orally. Conclusion: Anti-CD20 therapy successfully controlled systemic and articular refractory disease with sustained efficacy over a follow up period of up to 24 weeks.

Abstract:
We propose a motion
planning gap-based algorithms for mobile robots in an unknown environment for
exploration purposes. The results are locally optimal and sufficient to
navigate and explore the environment. In contrast with the traditional
roadmap-based algorithms, our proposed algorithm is designed to use minimal
sensory data instead of costly ones. Therefore, we adopt a dynamic data
structure called Gap Navigation Trees (GNT), which keeps track of the depth
discontinuities (gaps) of the local environment. It is incrementally
constructed as the robot which navigates the environment. Upon exploring the
whole environment, the resulting final data structure exemplifies the roadmap
required for further processing. To avoid infinite cycles, we propose to use
landmarks. Similar to traditional roadmap techniques, the resulting algorithm
can serve key applications such as exploration and target finding. The
simulation results endorse this conclusion. However, our solution is cost
effective, when compared to traditional roadmap systems, which makes it more
attractive to use in some applications such as search and rescue in hazardous
environments.

A study was performed along 30 km
distance of the coast of South England to estimate the size of falling blocks
and their extent using RocFall and other computer programmes. In addition to
computer analysis, observational analysis was used to identify many factors
affecting the size and the end point of falling rocks. The analysis was based along
the coastal Chalk cliffs from Brighton to Eastbourne. It involved measurements
of dip and dip direction, classification and characteristics of rocks as well
as noting the history of previous rock fall in relation to the extent and size.
Nine areas were examined and reported according to their location except for
Brighton which was divided according to the cliff protection methods used. For
each area a cliff profile was drawn and information about its height, cliff
sloping angle, geological formations, and type of cliff protection used, if
any, were listed. Types of common failures were also identified from
observation during site reconnaissance and from performed failure analysis
using Dips programme. As a result, falling block size and rocks end point, to
the worst case, could be estimated and judgment about the efficiency of
protection methods used in protected areas has been made.

Abstract:
New proofs and improvements of three classical theorems in analysis are presented. Although these theorems are well-known, and have been extensively investigated over the years, it seems that new light can be shed on them. We first present a quantitative necessarily and sufficient condition for a function to be uniformly continuous, and as a by-product we obtain explicitly the optimal delta for the given epsilon. The uniform continuity of a continuous function defined on a compact metric space follows as a simple consequence. We proceed with the extreme value theorem and present a ``programmer's proof'', a proof which does not use the costume argument of proving boundedness first. We finish with the intermediate value theorem, which is generalized to a class of discontinuous functions, and, in addition, the meaning of the intermediate value property is re-examined and a fixed point theorem for (very) discontinuous functions is established. At the end we discuss briefly in which sense the proofs are constructive.

Abstract:
This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple topological argument. The proof is elementary in nature and does not use any kind of integration (neither complex nor real). In addition, it is also independent of the fact that the roots of an analytic function are isolated. The proof is based on either the Banach or Brouwer fixed point theorems. In particular, this shows that one can obtain a proof of the FTA (albeit indirect) which is based on the Brouwer fixed point theorem, an aim which was not reached in the past and later the possibility to achieve it was questioned. We close this note with a simple generalization of the FTA. A short review of certain issues related to the OMT and the FTA is also included.

Abstract:
We study one-sided substitution subshifts, and how they can be represented using Bratteli-Vershik systems. In particular we focus on minimal recognizable substitutions such that the generated one-sided substitution subshift contains only one non-shift-invertible element (a branch point), and we call these substitutions quasi-invertible. We give an algorithm to check whether a substitution is quasi-invertible, and show that any substitution with a rational Perron value is orbit equivalent to a quasi-invertible substitution. If the quasi-invertible substitution is left proper, then its subshift is equal to a substitution subshift where the original branch point is the new substitution fixed point. We use these results to prove that any reasonable quasi-invertible substitution subshift has a Bratteli-Vershik representation. We also give an example of a pair of substitutions whose 2-sided subshifts are topologically conjugate, while their 1-sided subshifts are not.

Abstract:
This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to treating this equation is considered too. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.

Abstract:
We show that character analysis using Fourier series is possible, at least when a mathematical character is considered. Previous approaches to character analysis are somewhat not in the spirit of harmonic analysis.

Abstract:
We study stationary ordered Bratteli diagrams and give necessary and sufficient conditions for these orders to generate a continuous Vershik map. We apply this to finding adic representations for one sided substitution subshifts. We give an algorithm to find the branch points of a substitution, which have to be mapped to the minimal elements of such an ordering. We find adic representations for substitutions with one branch point, and also substitutions all of whose branch points are fixed.