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We study non-smoothness of the fundamental solution for the Schrödinger equation with a spherically symmetric and super-quadratic potential in the sense that V(x) ≥ C|x|2+ɛ at infinity with constants C > 0 and ɛ > 0. More precisely, we show the fundamental solution E(t, x, y) does not belong to C1 as a function of (t, x, y), which partially solves Yajima's conjecture. [ABSTRACT FROM AUTHOR]

STUDY QUESTION What clinical and laboratory differences emerge from parallel direct comparison of embryos reaching the blastocyst stage between Days 4, 5, 6, and 7 (Days 4–7)? SUMMARY ANSWER Increasing times to blastocyst formation are associated with a worse clinical outcome and perturbations in developmental patterns appear as early as the fertilization stage. WHAT IS KNOWN ALREADY Previous evidence indicates that later times to blastocyst development are associated with a worse clinical outcome. However, the vast majority of these data concern Day 5 and Day 6 blastocysts, while Day 4 and Day 7 blastocysts remain less thoroughly investigated. In addition, studies comparing in parallel the developmental patterns and trajectories of Day 4–7 blastocysts are lacking. This leaves unanswered the question of when and how differences among such embryos emerge. Acquisition of such knowledge would significantly contribute to understanding the relative impact of intrinsic and extrinsic causes of embryo developmental kinetics and competence. STUDY DESIGN, SIZE, DURATION This retrospective study involved time-lapse technology (TLT) monitoring of Day 4 (N = 70), Day 5 (N = 6147), Day 6 (N = 3243), and Day 7 (N = 149) blastocysts generated in 9450 ICSI cycles. Oocyte retrievals were carried out after clomiphene citrate-based minimal ovarian stimulation, between January 2020 and April 2021. PARTICIPANTS/MATERIALS, SETTING, METHODS Couples included in the study presented with different diagnoses, mainly male factor and unexplained infertility. Cases involving cryopreserved gametes or surgically retrieved sperm were excluded. Microinjected oocytes were assessed by a combined TLT-culture system. Day 4–7 blastocyst groups were compared in terms of morphokinetics (pronuclear dynamics, cleavage patterns and timings, and embryo quality) and clinical outcome. Clinically usable blastocysts were cryopreserved and transferred in single vitrified-warmed blastocyst transfers (SVBT). MAIN RESULTS AND THE ROLE OF CHANCE From 19 846 microinjected oocytes, 17 144 zygotes (86.4%) were obtained. Overall, the blastocyst development rate was 56.0%. Rates of blastocysts formation on Days 4, 5, 6, and 7 were 0.7%, 64.0%, 33.8%, and 1.6%, respectively. The average expanded blastocyst development times were 98.4 ± 0.4, 112.4 ± 0.1, 131.6 ± 0.1, and 151.2 ± 0.5 h in the Day 4–7 groups, respectively. Female age was positively associated with longer times to blastocyst development. Rates of both inner cell mass (ICM) and trophectoderm (TE) morphological grade A blastocysts were negatively associated with the day of blastocyst development (P  < 0.0001). The differences in development times and intervals increased progressively until blastocyst expansion (P  < 0.0001 for all development times). Strikingly, such differences were already markedly evident as early as the time of pronuclear fading (tPNf) (20.6 ± 0.3, 22.5 ± 0.0, 24.0 ± 0.0, 25.5 ± 0.3; Days 4–7, respectively; P  < 0.0001). Rates of cleavage anomalies (tri-/multi-chotomous mitosis or rapid cleavage) occurring at the first or second/third division cycles were also positively associated with longer times to blastocyst development. Implantation, ongoing pregnancy, and live birth rates were progressively reduced with increasing blastocyst development times (P  < 0.0001), even after stratification for maternal age. When controlled for female age, male age, number of previous embryo transfer cycles, morphological grade of the ICM and TE, and progesterone supplementation, the probabilities of implantation, clinical, and ongoing pregnancy and live birth were significantly decreased in Day 6 blastocysts in comparison to Day 5 blastocysts. Follow-up data on birth length, weight, and malformations were comparable among the four blastocyst groups. LIMITATIONS, REASONS FOR CAUTION The study is limited by its retrospective design. Having been obtained from a single centre, the data require independent validation. WIDER IMPLICATIONS OF THE FINDINGS This study extends previous data on the relation between time of blastocyst formation and clinical outcome. It also indicates that differences in developmental times and patterns of Day 4–7 blastocysts occur as early as the fertilization stage, possibly dictated by intrinsic gamete-derived factors. STUDY FUNDING/COMPETING INTEREST(S) This study was supported by the participating institutions. The authors have no conflict of interest to declare. TRIAL REGISTRATION NUMBER N/A. [ABSTRACT FROM AUTHOR]

STUDY QUESTION Does maternal ageing impact early and late morphokinetic and cellular processes of human blastocyst formation? SUMMARY ANSWER Maternal ageing significantly affects pronuclear size and intra- and extra-nuclear dynamics during fertilization, dysregulates cell polarity during compaction, and reduces blastocoel expansion. WHAT IS KNOWN ALREADY In ART, advanced maternal age (AMA) affects oocyte yield, fertilization, and overall developmental competence. However, with the exception of chromosome segregation errors occurring during oocyte meiosis, the molecular and biochemical mechanisms responsible for AMA-related subfertility and reduced embryo developmental competence remain unclear. In particular, studies reporting morphokinetics and cellular alterations during the fertilization and pre-implantation period in women of AMA remain limited. STUDY DESIGN, SIZE, DURATION A total of 2058 fertilized oocytes were stratified by maternal age according to the Society for Assisted Reproductive Technology classification (<35, 35–37, 38–40, 41–42, and >42 years) and retrospectively analysed. AMA effects were assessed in relation to: embryo morphokinetics and morphological alterations; and the presence and distribution of cell polarity markers—Yes-associated protein (YAP) and protein kinase C-ζ (PKC-ζ)—involved in blastocyst morphogenesis. PARTICIPANTS/MATERIALS, SETTING, METHODS A total of 1050 cycles from 1050 patients met the inclusion criteria and were analysed. Microinjected oocytes were assessed using a time-lapse culture system. Immature oocytes at oocyte retrieval and mature oocytes not suitable for time-lapse monitoring, owing to an excess of residual corona cells or inadequate orientation for correct observation, were not analysed. Phenomena relevant to meiotic resumption, pronuclear dynamics, cytoplasmic/cortical modifications, cleavage patterns and embryo quality were annotated and compared among groups. Furthermore, 20 human embryos donated for research by consenting couples were used for immunofluorescence. MAIN RESULTS AND THE ROLE OF CHANCE Static microscopic observation revealed that blastocyst formation and expansion were impaired in the 41–42 and >42-year groups (P  <   0.0001). The morphological grades of the inner cell mass and trophectoderm were poorer in the >42-year group than those in the <35-year group (P  =   0.0022 and P  <   0.0001, respectively). Time-lapse microscopic observation revealed a reduction in nucleolus precursor body alignment in female pronuclei in the 41–42 and >42-year groups (P  =   0.0010). Female pronuclear area decreased and asynchronous pronuclear breakdown increased in the >42-year group (P  =   0.0027 and P  <   0.0122, respectively). Developmental speed at cleavage stage, incidence of irregularity of first cleavage, type and duration of blastomere movement, and number of multinucleated cells were comparable among age groups. Delayed embryonic compaction and an increased number of extruded blastomeres were observed in the >42-year group (P  =   0.0002 and P  =   0.0047, respectively). Blastulation and blastocyst expansion were also delayed in the 41–42 and >42-year groups (P  < 0.0001 for both). YAP positivity rate in the outer cells of morulae and embryo PKC-ζ immunoflourescence decreased in the >42-year group (P  <   0.0001 for both). LIMITATIONS, REASONS FOR CAUTION At the cellular level, the investigation was limited to cell polarity markers. Cell components of other developmental pathways should be studied in relation to AMA. WIDER IMPLICATIONS OF THE FINDINGS The study indicates that maternal ageing affects the key functions of embryo morphogenesis, irrespective of the well-established influence on the fidelity of oocyte meiosis. STUDY FUNDING/COMPETING INTEREST(S) This study was supported by the participating institutions. The authors have no conflicts of interest to declare. TRIAL REGISTRATION NUMBER N/A. [ABSTRACT FROM AUTHOR]

Purpose: This study aimed to examine the embryonic development of human 4‐cell stage embryos after warming with fatty acids (FAs) and to assess the pregnancy outcomes after single vitrified‐warmed cleavage stage embryo transfers (SVCTs). Methods: Experimental study: A total of 217 discarded, vitrified human 4‐cell stage embryos donated for research by consenting couples were used. The embryos were warmed using the fatty acid (FA)‐supplemented solutions (FA group) or nonsupplemented solutions (control group). The developmental rate, morphokinetics, and outgrowth competence were analyzed. Clinical study: The treatment records of women undergoing SVCT in natural cycles between April and September 2022 were retrospectively analyzed (April–June 2022, control group; July–September 2022, FA group). Results: Experimental study: The rate of morphologically good blastocysts was significantly higher in the FA group than in the control group (p = 0.0302). The morphokinetics during cleavage, morula, and blastocyst stages were comparable between the groups. The outgrowth was significantly increased in the FA group (p = 0.0438). Clinical study: The rates of implantation, clinical pregnancy, and ongoing pregnancy after SVCTs were significantly increased in the FA group (p = 0.0223–0.0281). Conclusions: Fatty acid‐supplemented warming solutions effectively improve embryo development to the blastocyst stage and pregnancy outcomes after SVCTs. [ABSTRACT FROM AUTHOR]

Purpose: Since 1986, the Japan Society of Obstetrics and Gynecology assisted reproductive technology (ART) registry system has collected data on national ART use and outcomes trends in Japan. Herein, we describe the characteristics and outcomes of ART cycles registered during 2020 and compare the results with those from 2019. Methods and Results: In 2020, 621 ART facilities participated in the registration. The total number of registered cycles was 449 900, and there were 60 381 live births, which decreased from the previous year (1.79% and 0.36% decrease, respectively). The number of freeze‐all in vitro fertilization (IVF) and intracytoplasmic sperm injection (ICSI) cycles increased in 2020, and the number of neonates born was 2282 for IVF‐embryo transfer (ET) cycles and 2596 for ICSI cycles, which had decreased from the previous year. Frozen–thawed ET (FET) cycles had slightly increased from 2019 (0.04%). In 2020, 215 285 FET cycles were conducted, resulting in 76 196 pregnancies and 55 503 neonates. Single ET was performed in 81.6% of fresh transfers and 85.1% of frozen–thawed cycles, respectively, resulting in over 97% singleton pregnancies/livebirths rates. Conclusion: Despite the COVID‐19 pandemic during 2020, the overall number of ART cycles and neonates born demonstrated only a slight decrease in 2020 compared with 2019. [ABSTRACT FROM AUTHOR]