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\bigl\vert x(t)\bigr\vert \leq \mathrm{arcsin} \biggl(\frac{P+|c|+k_{3}}{-b} \biggr)-k_{2} \leq\frac{\pi}{2}-k_{2}
(7)
\bigl\vert x(t)\bigr\vert \leq\frac{P+|c|+k_{3}}{-b}+k_{2},
(8)

If further possibly discontinuous nonlinearities or multivalued maps are implemented into the right-hand sides of given differential equations or inclusions whose growth has, for instance, a superlinear character sufficiently far from the origin, then Theorem3 in [ 22 ] does not any longer apply. Having, however, to our disposal explicit estimates of solutions like ( Dona Michi Men Cowboy Genuine Cowhide Leather Plain Square Toe Rodeo Western Boots Black Black UfxRet
) or ( GUESS Womens Halsey Wide Calf Closed Toe Knee High Fashion Boots Black wmN0Y2sY
) and their derivatives, we can formulate criteria such that the implemented new terms can behave in an arbitrary way outside of the domains characterized by these estimates. In this way, all the results under our consideration can be naturally extended.

This will be therefore our main aim of the present paper. Of course, for obtaining the explicit estimates of solutions and their derivatives, we should study, unlike in [ 22 ], exclusively the second-order Dirichlet boundary value problems. For the sake of brevity we will prove only one main theorem for the scalar problem. Nevertheless, since all the related proofs of all the cases under our consideration are quite analogous and differ just by the technical details, we can also present the related solvability criteria and some solutions estimates.

On the other hand, we will discuss in detail the advantages and disadvantages of the usage of one-term vs. complete linear differential operator for the associated Green’s functions.

\left . \textstyle\begin{array}{l} x''(t)+a x'(t) +b x(t)\in P(t)+F_{1}(x(t))+F_{2}(x'(t))-c\operatorname {Sgn} x'(t), \\ x(0)=x_{0},\qquad x(T)=x_{T}, \end{array}\displaystyle \right \}
(9)
\Gamma_{\varphi}:= \bigl\{ (x,y) \in X \times Y \mid y \in\varphi(x) \bigr\} .

A multivalued mapping \varphi\colon X \multimap Y is called upper semicontinuous (u.s.c.) if, for each open U \subset Y , the set \{x \in X \mid\varphi(x) \subset U \} is open in X .

Lemma 1

(, , Proposition I.3.15 in [ Nike Kids Air Huarache Run GS Fashion Sneakers Gym Blue/Obsidian 8ucf4xPICm
])

\varphi\colon X \multimap Y ..., \Gamma _{\varphi} X \times Y .

Lemma 2

(, , Proposition I.3.16 in [ CHFSO Womens Trendy Round Toe Elastic Low Top Slip On Low Heel Platform Sneakers Gold mvHHX2d
])

\varphi\colon X \multimap Y \varphi(X) \subset K , K \subset Y , \Gamma_{\varphi} . ...

Let Y be a separable metric space and (\Omega, \mathcal{U}, \mu) be a measurable space, i.e. a set Ω equipped with a σ -algebra \mathcal{U} of subsets and a countably additive measure μ on \mathcal{U} . For our needs here, Ω will be a bounded domain in \mathbb{R}^{k} , equipped with the Lebesgue measure. A multivalued mapping \varphi\colon\Omega\multimap Y is called measurable if \{\omega\in\Omega\mid\varphi(\omega ) \subset V \} \in\mathcal{U} , for each open set V \subset Y .

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Medtech manufacturers must compete by providing value beyond their products

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Slowing of Innovation

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\begin{array}{*{20}l} XZY^{T} = A, \end{array}
(48)

where Z is an m × n matrix of unknowns, X is the M × m feature matrix of the row entities, Y is the N × n is the feature matrix of the column entities. And, A is the M × N binary association matrix between the row and column entities.

\begin{array}{@{}rcl@{}} X^{T}XZY^{T}Y = X^{T}AY \end{array}
(49)
\begin{array}{@{}rcl@{}} ZY^{T}Y = \left(X^{T}X\right)^{-1}X^{T}AY\\ \Rightarrow \hat{Z} = \left(X^{T}X\right)^{-1}X^{T}AY\left(Y^{T}Y\right)^{-1} \end{array}
(50)
\begin{array}{*{20}l} \underset{W,H}{\text{min}} \quad\varphi = \|Z-WH^{T}\|_{2,1} + \lambda_{1} \|W\|_{2,1} + \lambda_{2} \|H\|_{2,1}\\ \text{such that},\quad W\geq 0, H\geq 0 \end{array}
(51)

This a modified non-negative matrix factorization (NMF) problem; only difference is the usage of the 2,1 norms instead of 2 norms in the loss function and the regularizers.

We can also solve the Stable Robust IMC optimization problem by solving the two problems mentioned above. It is demonstrated in Algorithm 4.

We prepared a sparse association matrix by extracting the lincRNA-disease association dataset from the LncRNADisease [ 4 ] with sparsity indx 0.22%. LincRNA expression dataset was obtained from the co-expression based association study [ 7 ]. Finally, we cataloged 8194 lincRNAs and 2148 human disease phenotypes and the resulting association matrix contains 46,934 associations among these two entities. We followed a standard naming of the disease phenotypes by OMIM identification numbers. We extracted top-5 OMIM phenotypes matching the human disease names using OMIM API [ 16 ].

The features of LincRNAs consist of four groups of information: (i) expression profiles, (ii) transcriptor factor binding sites (TFBS), (iii) functional annotations and (iv) single nucleotide polymorphism (SNP) information. The RNA-seq expression profiles of the 8194 lincRNAs on 22 human tissues were collected from the Human BodyMap Project 2.0 [ 3 ]. The expression scores were measured in FPKM (Fragments Per Kilobase of exons per Million Fragments mapped) unit. Then, TFBS information about the lincRNAs in our study with 120 transcription factors were obtained from ChIP-base dataset [ 17 ]. Linc2GO is a public data repository containing functional annotations of lincRNAs [ 18 ]. There are three different types of functions cataloged in the Lin2GO dataset: gene ontology biological process (GO BP), gene ontology molecular function (GO MF) and KEGG pathways. The 8194 lincRNAs with the functional annotation together make a sparse matrix with sparsity index 0.11%. We performed singular value decomposition on the matrix to compute and use the leading 100 singular vectors in our study as part of the features of the lincRNAs. We extracted links among 368,494 SNPs and the lincRNAs from our study from the lncRNASNP dataset [ Chemistry® FLO3 Womens Shoes Flip Flops Slipon Flat Sandals Slipper Blue xj3FP7Gey
]. Again, the SNP-lincRNA association matrix turned out to a sparse matrix with the sparsity index 0.0077%. Therefore, we performed singular value decomposition on the matrix to compute and use the leading 100 singular vectors. Finally, we performed a filtering on all the four groups of features of the lincRNAs in our study. We found that 6540 out of the initial 8194 lincRNAs have data from all the four groups of featureset. Therefore, our final lincRNA feature matrix ( X in our study) has 6540 rows (lincRNAs) and 342 columns (features).

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