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We consider a ratinonal map $\phi$ from m-dimensional projective space to n-dimensional projective space that is a parameterization of m-dimensional variety. Our main goal is to study the (m-1)-dimensional fibers of $\phi$ in relation with the m-th local cohomology modules of Rees algebra of its base ideal.

Abstract Using some recent results of fixed point of weakly contractive mappings on the partially ordered space, the existence and uniqueness of solution for interval fractional delay differential equations (IFDDEs) in the setting of the Caputo generalized Hukuhara fractional differentiability are studied. The dependence of the solution on the order and the initial condition of IFDDE is shown. A new technique is proposed to find the exact solutions of IFDDE by using the solutions of interval integer order delay differential equation. Finally, some examples are given to illustrate the applications of our results.

In this paper, we establish the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy integrodifferential equations by using the fixed point method and the successive approximation method.

In this paper, we consider the existence and uniqueness of solutions of the fractional random differential equations with delay. Moreover, some kind of boundedness of the solution is proven. Finally, the applicability of the theoretical results is illustrated with some real world examples.

We consider the random fuzzy differential equations (RFDEs) with impulses. Using Picard method of successive approximations, we shall prove the existence and uniqueness of solutions to RFDEs with impulses under suitable conditions. Some of the properties of solution of RFDEs with impulses are studied. Finally, an example is presented to illustrate the results.

In this paper, we prove the existence and uniqueness of solution for the fuzzy functional differential equation under generalized Hukuhara derivative via contractive-like mapping principles.

STUDY QUESTION Among couples with infertility and normal total sperm count and motility, can sperm morphology be used as a biomarker to identify couples who benefit more from ICSI over conventional IVF (c-IVF) on fertility outcomes? SUMMARY ANSWER Based on this secondary analysis of a large randomized clinical trial (RCT), sperm morphology has limited value as a biomarker to identify couples who benefit more from ICSI over c-IVF on live birth, ongoing pregnancy, clinical pregnancy or total fertilization failure. WHAT IS KNOWN ALREADY Our recent RCT showed that ICSI did not result in higher live birth rates in couples with normal total sperm count and motility. It is unclear whether sperm morphology can be used as a biomarker to identify couples who benefit more from ICSI over c-IVF in this population. STUDY DESIGN, SIZE, DURATION This was a secondary analysis of an open-label, multi-centre, RCT comparing ICSI versus c-IVF in 1064 couples with infertility and normal total sperm count and motility. In this secondary study, we evaluated the effectiveness of ICSI over c-IVF in relation to sperm morphology. PARTICIPANTS/MATERIALS, SETTING, METHODS Couples were eligible if they had ≤2 previous IVF/ICSI attempts, and the male partner had normal total sperm count and motility according to the fifth edition of the WHO laboratory manual for the examination and processing of human semen. Sperm morphology was measured from samples obtained during the first consultation and data for sperm morphology were available in partners of all participants in this trial. The outcomes of interest were live birth, ongoing pregnancy, clinical pregnancy, and total fertilization failure. We first conducted a logistic regression analysis with an interaction term (sperm morphology as a continuous variable by treatment (ICSI versus c-IVF)) on the four outcomes. We also used restricted cubic spline analysis to evaluate non-linear interaction and plotted the treatment effects of ICSI over c-IVF at different sperm morphology levels and the predicted probability of these outcomes in both ICSI and c-IVF groups. MAIN RESULTS AND THE ROLE OF CHANCE The median proportion of sperm with normal morphology in both groups was 3% (Interquartile range 1–6%). Live birth rates were (184/532) 34.6% for ICSI versus (166/532) 31.2% for c-IVF. No significant interaction was found between sperm morphology and treatment effect of ICSI versus c-IVF on the rates of live birth, ongoing pregnancy, clinical pregnancy, and total fertilization failure (P  = 0.181, 0.153, 0.168, and 0.788 respectively). In the analyses using restricted cubic splines, no evidence of interaction between sperm morphology and the treatment effect was found. Interaction figures showed that the treatment effect of ICSI over c-IVF at different sperm morphology levels was fluctuating around no effect line, and the predicted outcomes for the two groups were mostly overlapping at different sperm morphology levels. LIMITATIONS AND REASONS FOR CAUTION This secondary analysis may be underpowered to detect a difference in treatment effects at different sperm morphology levels due to relatively small number of events at some sperm morphology levels. Moreover, sperm morphology assessment was performed during the first consultation, rather than on the day of randomization. WIDER IMPLICATIONS OF THE FINDINGS In couples with infertility and normal total sperm count and motility, sperm morphology has a limited role as a biomarker to identify couples who benefit more from ICSI over c-IVF on fertility outcomes. STUDY FUNDING/COMPETING INTEREST(S) This study was funded by My Duc Hospital, Ho Chi Minh City, Vietnam. RW was supported by an NHMRC EL Investigator Grant (GNT2009767). LNV has received speaker and conference fees from Merck, grant, speaker, conference fees from Merck Sharpe and Dohme, and speaker, conference, and scientific board fees from Ferring. TMH has received speaker fees from Merck, Merck Sharp Dohme, and Ferring. BWM reports consultancy, travel support and research funding from Merck and consultancy for Organon and Norgine. BWM holds stock from ObsEva. TRIAL REGISTRATION NUMBER NCT03428919 [ABSTRACT FROM AUTHOR]

This study demonstrates the existence and uniqueness of solutions to stochastic fuzzy fractional integral equations and investigates some of their properties. [ABSTRACT FROM AUTHOR]