2,358 research outputs found
The fractional variation and the precise representative of functions
No full textWe continue the study of the fractional variation following the distributional approach developed in the previous works Brue et al. (2021), Comi and Stefani (2019), Comi and Stefani (2019). We provide a general analysis of the distributional space BV alpha,p(R-n) of L-p functions, with p is an element of [1, +infinity], possessing finite fractional variation of order alpha is an element of (0, 1). Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a BV alpha,p function
The fractional variation and the precise representative of BV alpha,p functions
No full textWe continue the study of the fractional variation following the distributional approach developed in the previous works Bruè et al. (2021), Comi and Stefani (2019), Comi and Stefani (2019). We provide a general analysis of the distributional space BVα,p(Rn) of Lp functions, with p∈ [1 , + ∞] , possessing finite fractional variation of order α∈ (0 , 1). Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a BVα,p function
A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
No full textWe continue the study of the space BVα(Rn) of functions with bounded fractional variation in Rn of order α∈ (0 , 1) introduced in our previous work (Comi and Stefani in J Funct Anal 277(10):3373–3435, 2019). After some technical improvements of certain results of Comi and Stefani (2019) which may be of some separated insterest, we deal with the asymptotic behavior of the fractional operators involved as α→ 1 -. We prove that the α-gradient of a W1,p-function converges in Lp to the gradient for all p∈ [1 , + ∞) as α→ 1 -. Moreover, we prove that the fractional α-variation converges to the standard De Giorgi’s variation both pointwise and in the Γ -limit sense as α→ 1 -. Finally, we prove that the fractional β-variation converges to the fractional α-variation both pointwise and in the Γ -limit sense as β→ α- for any given α∈ (0 , 1)
Measures in the dual of BV: perimeter bounds and relations with divergence-measure fields
No full textWe analyze some properties of the measures in the dual of the space BV, by considering (signed) Radon measures satisfying a perimeter bound condition, which means that the absolute value of the measure of a set is controlled by the perimeter of the set itself, and whose total variations also belong to the dual of BV. We exploit and refine the results of Cong Phuc and Torres (2017), in particular exploring the relation with divergence-measure fields and proving the stability of the perimeter bound from sets to BV functions under a suitable approximation of the given measure. As an important tool, we obtain a refinement of Anzellotti-Giaquinta approximation for BV functions, which is of separate interest in itself and, in the context of Anzellotti's pairing theory for divergence-measure fields, implies a new way of approximating lambda-pairings, as well as new bounds for their total variation. These results are also relevant due to their application in the study of weak solutions to the non-parametric prescribed mean curvature equation with measure data, which is explored in a subsequent work
Novel network pharmacology methods for drug mechanism of action identification, pre-clinical drug screening and drug repositioning
No full textThe high rates of failure in oncology drug clinical trials highlight the problems of using pre-clinical data to predict the clinical effects of drugs. Here we present two methodology innovations on network pharmacology modeling. (1) We hypothesize that the gene network associated with cancer outcome in heterogeneous patient populations could serve as a reference for identifying drug effects. We proposed a novel in vivo genetic interaction between genes as ‘synergistic outcome determination’, in a similar way to ‘synthetic lethality’. We scanned above genetic interactions based on microarray profiling for cancer prognosis, and identified a cluster of important yet epigenetically regulated gene modules. By projecting drug sensitivity-associated genes on to this network, we could define a perturbation index for each drug based upon its characteristic perturbation pattern. Finally, by using this index, we significantly discriminated successful drugs from the candidate pool, and revealed the mechanisms of drug combinations. Thus, the prognosis-guided synergistic gene-gene interaction networks could serve as an efficient in silico tool for pre-clinical drug prioritization and rational design of combinatorial therapies. Part of this work was published, and we will present new results on this project. (2) MicroRNAs (miRNAs) play a key role in the regulation of the transcriptome and have been identified as a key mediator in human disease and drug response. we introduced a novel concept, the Context-specific MiRNA activity (CoMi activity), to reflect a miRNA’s regulation effect on a context specific gene set .Using breast cancer as an example, we examined the CoMi activity based on a Gene Ontology (GO) term as context. Interestingly, we found that chemotherapeutic drug treatment can counteract the dis-regulated CoMi activity in the cancer-specific network. For instance, 100% of down-regulated CoMi activities in a “core” breast cancer network contains apoptosis-related GO terms that could be counteracted by Paclitaxel treatment. By defining a Stability Index for in silico drug screening, we found CoMi activity signatures strikingly outperformed the traditional CMAP method or mRNA-based signatures. Thus, the dynamic remodeling of context-specific miRNAs regulation network could reveal the hidden miRNAs that act as key mediators of drug action and facilitate in silico cancer drug screening
Leibniz rules and Gauss–Green formulas in distributional fractional spaces
No full textWe apply the results established in [12] to prove some new fractional Leibniz rules involving BVα,p and Sα,p functions, following the distributional approach adopted in the previous works [8,13,14]. In order to achieve our main results, we revise the elementary properties of the fractional operators involved in the framework of Besov spaces and we rephraze the Kenig–Ponce–Vega Leibniz-type rule in our fractional context. We apply our results to prove the well-posedness of the boundary-value problem for a general 2α-order fractional elliptic operator in divergence form
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