Neoadjuvant chemoradiotherapy (nCRT) followed by total mesorectal excision (TME) surgery is an accepted standard of care for many patients with locally advanced rectal cancer (LARC). Recently published prospective clinical trials tried to address the value of total neoadjuvant therapy (TNT) in high risk groups. Here, we present T4/N2 rectal cancer's clinical outcomes after long-course chemotherapy. This is a retrospective chart review study of patients with clinical high-risk (T4 and / or N2) LARC who were diagnosed between January 2007 and December 2019 and received nCRT followed by surgery. Demographic, clinical and pathological data were collected. Overall survival (OS) was calculated using the Kaplan-Meier method and recurrence was calculated using cumulative incidence with competing risk. Predictive factors were identified using Cox regression and competing risk multivariable analyses. A sub-group analysis for T4 and N2 disease was undertaken for survival and local recurrence outcomes. 331 patients with T4 (n=113) and/or N2 (n=271) LARC rectal cancer were identified. Median follow-up was 1.8 years. 3 years relapse free survival (RFS), in the whole group, was 61.2% which is probably driven by a poor distant metastasis free survival (DMFS) of 65% while the locoregional recurrence (LRR) was only 8%. 3 -year OS improved from 84%, in the whole cohort, to 87% in those who completed adjuvant chemotherapy (ACT). In patients with T4 versus N2 disease, the R0 status was 85.8% and 96.7% and the 5-year LRR was 16.1% and 7.5%, respectively. On multivariable analysis, LRR and OS rates in patients with T4 or N2 were correlated with R status (P < 0.015). ACT improves survival in patients with high-risk rectal cancer (T4 or N2). Obtaining an R0 resection remains an important predictor of local control and survival. Therefore, patients with margin-threatening disease could potentially benefit from further intensification of neoadjuvant treatment by the addition of neoadjuvant chemotherapy.

X-Ray computed tomography is a non-destructive method that is used, among many applications, to study the size, shape, 3D structures and interconnections of pores in shale. We use phase retrieval methods to deal with the “edge enhancement” effect caused by phase shift. The process of phase retrieval can be described by the transport-of-intensity equation (TIE). But this is an ill-posed problem. The existing methods focus on phase retrieval in the frequency domain. To tackle the ill-posedness, we propose a new method whose main idea is to solve this problem in space domain with a regularization technique. We study a synthetic shale model and simulate the projection data. Then we apply three methods to retrieve the phase: conventional method in frequency domain, direct solving method and iterative Tikhonov regularization method in space domain. Finally, we use the standard filtered back-projection (FBP) method to present the outcome. By analyzing the results, we find advantages of the new method: more stability and fewer artifacts under noise perturbations. The study shows that relative errors of the new method are nearly 1% of that of the traditional method based on frequency domain, and hence the new method is promising for the practical data processing.

This paper is devoted to a well-posedness analysis of elliptic variational–hemivariational inequalities in Banach spaces. The differential operator associated with the variational–hemivariational inequality is assumed to be strongly monotone of a general order, in contrast to that in the majority of existing references on this subject where the differential operator is assumed to be strongly monotone of order 2. Moreover, the solution existence is proved with an approach more accessible to applied mathematicians and engineers, instead of through an abstract surjectivity result for pseudomonotone operators in existing references. Equivalent minimization principles are established for certain variational–hemivariational inequalities, which are valuable for developing efficient numerical algorithms. The theoretical results are applied to the analysis of a mixed hemivariational inequality in the study of a generalized Newtonian fluid flow problem involving a nonsmooth slip boundary condition of friction type. Existence and uniqueness of both the velocity and pressure unknowns are shown for the mixed hemivariational inequalities.

Mosić and Djordjević introduced the notation of the gDMP inverse for a linear operator on Hilbert space in Mosić and Djordjević (J Spectr Theory 8(2):555–573, 2018) by considering generalized Drazin inverse with the Moore-Penrose inverse. This article introduces two new classes of inverses: GD1 (generalized Drazin and inner) inverse and 1GD (inner and generalized Drazin) inverse for Banach space operators. The existence and uniqueness of the GD1 (also 1GD) inverse are discussed along with some properties through core-quasinilpotent decomposition and closed range decomposition operator. We further establish a few explicit representations of the GD1 inverse and their interconnections with generalized Drazin inverse. In addition, we discuss a few properties of GD1 (also 1GD) inverse through binary relation.

We consider the nonlinear hyperbolic‐type inequality under the Dirichlet‐type boundary condition where , and . An optimal Fujita‐type result is obtained for this problem. Namely, we prove that when belongs to a certain functional space , and or , then there exists no global weak solution, while global solutions exist for some when and . The obtained result shows an interesting phenomenon of discontinuity of the Fujita critical exponent, jumping from to as reaches the value from above.

Mosic and Djordjevic introduced the notation of the gDMP inverse for Hilbert space operators in [J. Spectr. Theory, 8(2):555-573, 2018] by considering generalized Drazin inverse with the Moore-Penrose inverse. This paper introduces two new classes of inverses: GD1 (generalized Drazin and inner) inverse and 1GD (inner and generalized Drazin) inverse for Hilbert space operators. The existence and uniqueness of the GD1 (also 1GD) inverse are discussed, along with some properties through core-quasinilpotent decomposition and closed range decomposition operator. We further establish a few explicit representations of the GD1 inverse and their interconnections with generalized Drazin inverse. In addition, we discuss a few properties of GD1 (also 1GD) inverse through binary relation.

We study noncoercive nonlinear variational–hemivariational inequalities that encompass semicoercive nonlinear monotone variational inequalities and pseudomonotone variational inequalities in reflexive Banach spaces, respectively, hemivariational inequalities in function spaces. We present existence and approximation results. Our approach consists in a double regularization: we combine a Browder–Tikhonov regularization with regularization tools of nondifferentiable optimization to smooth the jumps in the hemivariational term. As application, we treat a noncoercive unilateral contact problem in continuum mechanics with nonmonotone friction.

Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In this paper, we study different generalized inverses of tensors over a commutative ring and a noncommutative ring. Several numerical examples are provided in support of the theoretical results. We also propose algorithms for computing the inner inverses, the Moore-Penrose inverse, and weighted Moore-Penrose inverse of tensors over a noncommutative ring. The prowess of some of the results is demonstrated by applying these ideas to solve an image deblurring problem.

This chapter characterizes the components of the Earth’s magnetic field, but only to the extent needed for the book. An insight into the constituents of crustal geomagnetic field research is given.

This chapter studies graphical demonstrations of locally supported Haar mollifier decorrelation for special test examples, namely the Marmousi-model and an area of the Bavarian Molasse Basin.

This series of studies is devoted to developing and employing mathematical methods along with quantum algorithms for solving moving boundary value problems which occur in heat and mass transfer problems. In this particular study we develop mathematical framework where we utilize special functions and Harrow-Hassidim-Lloyd (HHL) quantum algorithm for finding exact and approximate solutions of Generalized Heat Equation with moving boundaries and as examples we consider plane and spherical cases. In spherical case the Generalized Heat Equation is reduced to linear moving boundary value problem with discontinuous coefficients and solved exactly. In plane case we use collocation method for approximate solution of Inverse Two-Phase Stefan problem. Experimental verification of suggested mathematical framework has been tested for modeling arcing phenomena in composite electrical contacts with AgCdO (90%) and Ni (10%). HHL algorithm was applied and run on IBM Q with Qiskit and the code is openly available on https://github.com/users/Schrodinger-cat-kz/projects/2.

This chapter discusses inverse magnetometry as an ill-posed problem in dipole reflected nomenclature. It is mentioned that all criteria of Hadamard’s classification (existence, uniqueness, and stability) are violated for terrestrial data. Consequently, inverse magnetometry is considered to be “too ill-posed” in order to use regularization techniques other than mollifier regularization.

We consider optimization problems for a class of convex functions on H x H introduced by Simon Fitzpatrick, where H is a real Hilbert space. We show that the minimization problem of Fitzpatrick functions can be transformed from solving of the correspondent differential inclusions (d.i) on H x H, to solving simplified d. i. on H. By using the idea of optimization of Fitzpatrick functions we introduce a numerical algorithm for solving convex smooth optimization problems by reducing the number of the independent variables. We present a comparative study with numerical examples. Finally, we show that Fitzpatrick functions are closely related to Lyapunov functions.

We consider Dirichlet boundary value problem where the elliptic equation is driven by a (p, q)-Laplacian with weights and that contains a convection term (i.e., it depends on the solution and its gradient). The notion of (p, q)-Laplacian with weights is considered for the first time. We present an existence result whose proof is based on the theory of pseudomonotone operators. An example illustrates its applicability.

This chapter reviews results of potential theory as far as they are significant for the understanding of dipole potential based magnetometry. The various questions and problems of dipole-oriented magnetometry are made scientifically available.

Background Despite level 1 evidence demonstrating the equivalence of single-fraction radiotherapy (sfrt) and multiple-fraction radiotherapy (mfrt) for the palliation of painful bone metastases, sfrt remains underused. In 2015, to encourage the sustainable use of palliative radiation oncology resources, CancerCare Manitoba disseminated, to each radiation oncologist in Manitoba, guidelines from Choosing Wisely Canada (cwc) that recommend sfrt. We assessed whether dissemination of the guidelines influenced sfrt use in Manitoba in 2016, and we identified factors associated with mfrt. Methods All patients treated with palliative radiotherapy for bone metastasis in Manitoba from 1 January 2016 to 31 December 2016 were identified from the provincial radiotherapy database. Patient, treatment, and disease characteristics were extracted from the electronic medical record and tabulated by fractionation schedule. Univariable and multivariable logistic regression analyses were performed to identify risk factors associated with mfrt. Results In 2016, 807 patients (mean age: 70 years; range: 35-96 years) received palliative radiotherapy for bone metastasis, with 69% of the patients having uncomplicated bone metastasis. The most common primary malignancies were prostate (27.1%), lung (20.6%), and breast cancer (15.9%). In 62% of cases, mfrt was used-a proportion that was unchanged from 2015. On multivariable analysis, a gastrointestinal [odds ratio (or): 5.3] or lung primary (or: 3.3), complicated bone metastasis (or: 4.3), and treatment at a subsidiary site (or: 4.4) increased the odds of mfrt use. Conclusions Dissemination of cwc recommendations alone did not increase sfrt use by radiation oncologists in 2016. A more comprehensive knowledge translation effort is therefore warranted and is now underway to encourage increased uptake of sfrt in Manitoba.