Over 90% of cancer-related mortality worldwide is due to metastatic disease since the dynamic tumor microenvironment poses huge challenges in preventing the spread of metastatic cancer. Introducing the advent of advance biomaterials and their swift evolution, this review highlights the great potential of innovative biomaterials to proficiently tackle the metastatic tumor environment. Focusing on four distinct categories of biomaterials systems, action mechanism of biomaterials utilized in anti-tumor therapy is explained in detail: 1. Nanoplatforms sensitive to biochemical cues including pH, redox, and enzymes are known as stimuli-responsive nanoplatforms that react according their environment, 2. Smart nanoplatforms changing their morphology to penetrate impermeable physical barriers at tumor site, 3. Ingenious biomaterial participating in tumor normalization, and 4. Nanoplatforms with real-time theranostic capabilities due to innate feedback-loop mechanism. Ingeniously structured biomaterials with extensive evidence in preclinical efficacy encourage their inclusion in metastatic tumor therapy however, their utilization in medical settings is prevented due to various challenges; impractical manufacturing cost, regulatory and safety issues as well as large-scale production are major challenges restraining their widespread use. A concrete framework is proposed in this review to accelerate the biomaterial structure standardization process, following the GMP and other regulatory guidelines with the aim of implementing biomaterial-based tumor diagnostics and therapies. Since incorporating advancing technologies in tumor therapy such AI-driven, autonomous biomaterial structure or patient-specific tumor models would enable confront the proliferating metastatic tumor cases.

The recession cone and recession function are very important research objects in Convex Analysis. They have extensive applications in the optimization theory. Firstly, we study the properties of the recession cone and recession function. The positive homogeneity and subadditivity of recession function are mainly discussed. And the different methods are considered to prove these properties. Secondly, we discuss the unboundedness of the convex sets and convex functions by using recession cone and recession function.

Usually in the study of limit problems, will encounter more complex problems, in this paper, we discuss how to use the concept of equivalent infinitesimal better limit operation. At the same time, in the process of research, we re-explore the proof of Taylor's formula, and find that some functions have a similar expansion form to Taylor's formula, that is, 'fractional expansion'. It is also found that after the linear combination of Taylor expansion and fractional expansion, the obtained expansion is more accurate, which helps us to have a better understanding of the approximation of function expansion.