Canonical Integral Research Articles

Neural Sequence Transformation

AbstractMonte Carlo integration is a technique for numerically estimating a definite integral by stochastically sampling its integrand. These samples can be averaged to make an improved estimate, and the progressive estimates form a sequence that converges to the integral value on the limit. Unfortunately, the sequence of Monte Carlo estimates converges at a rate of O(), where n denotes the sample count, effectively slowing down as more samples are drawn. To overcome this, we can apply sequence transformation, which transforms one converging sequence into another with the goal of accelerating the rate of convergence. However, analytically finding such a transformation for Monte Carlo estimates can be challenging, due to both the stochastic nature of the sequence, and the complexity of the integrand. In this paper, we propose to leverage neural networks to learn sequence transformations that improve the convergence of the progressive estimates of Monte Carlo integration. We demonstrate the effectiveness of our method on several canonical 1D integration problems as well as applications in light transport simulation.

Reproductive developmental transcriptome analysis of Tripidium ravennae (Poaceae)

BackgroundTripidium ravennae is a cold-hardy, diploid species in the sugarcane complex (Poaceae subtribe Saccharinae) with considerable potential as a genetic resource for developing improved bioenergy and ornamental grasses. An improved understanding of the genetic regulation of reproductive processes (e.g., floral induction, inflorescence development, and seed development) will enable future applications of precision breeding and gene editing of floral and seed development. In particular, the ability to silence reproductive processes would allow for developing seedless forms of valuable but potentially invasive plants. The objective of this research was to characterize the gene expression environment of reproductive development in T. ravennae.ResultsDuring the early phases of inflorescence development, multiple key canonical floral integrators and pathways were identified. Annotations of type II subfamily of MADS-box transcription factors, in particular, were over-represented in the GO enrichment analyses and tests for differential expression (FDR p-value < 0.05). The differential expression of floral integrators observed in the early phases of inflorescence development diminished prior to inflorescence determinacy regulation. Differential expression analysis did not identify many unique genes at mid-inflorescence development stages, though typical biological processes involved in plant growth and development expressed abundantly. The increase in inflorescence determinacy regulatory elements and putative homeotic floral development unigenes at mid-inflorescence development coincided with the expression of multiple meiosis annotations and multicellular organism developmental processes. Analysis of seed development identified multiple unigenes involved in oxidative-reductive processes.ConclusionReproduction in grasses is a dynamic system involving the sequential coordination of complex gene regulatory networks and developmental processes. This research identified differentially expressed transcripts associated with floral induction, inflorescence development, and seed development in T. ravennae. These results provide insights into the molecular regulation of reproductive development and provide a foundation for future investigations and analyses, including genome annotation, functional genomics characterization, gene family evolutionary studies, comparative genomics, and precision breeding.

High-mobility group AT-Hook 1 mediates the role of nuclear factor I/X in osteogenic differentiation through activating canonical Wnt signaling.

It was previously reported that the loss of the transcription factor nuclear factor I/X (NFIX) gene in mice impaired endochondral ossification and mineralization in bone. However, the cellular and molecular basis for the defect remains unexplored. In this study, we investigated if and how NFIX regulates osteoblast differentiation. Nfix mRNA was induced during osteogenic and adipogenic differentiation of progenitor cells. Loss-of-function and gain-of-function studies revealed that NFIX induced osteoblast differentiation and impaired adipocyte formation from progenitor cells. RNA-seq and promoter analysis revealed that NFIX transcriptionally stimulated the expression of high-mobility group AT-Hook 1 (HMGA1). We then demonstrated that HMGA1 stimulated osteogenic differentiation of progenitor cells at the expense of adipogenic differentiation. The effect of Nfix siRNA on the differentiation of progenitor cells could be attenuated when HMGA1 was simultaneously overexpressed. Further investigations revealed the stimulatory effect of NFIX and HMGA1 on canonical wingless-type MMTV integration site family (Wnt) signaling. HMGA1 transcriptionally activates the expression of low-density lipoprotein receptor-related protein 5. Finally, in vivo transfection of Nfix siRNA to the marrow of mice reduced osteoblasts and increased fat accumulation in the marrow, and inactivated HMGA1/β-catenin signaling in bone marrow mesenchymal stem cells. This study suggests that HMGA1 plays a role in osteoblast commitment and mediates the function of NFIX through transcriptionally activating canonical Wnt signaling.

WNT signaling in pre-granulosa cells is required for ovarian folliculogenesis and female fertility.

ABSTRACTIn mammalian ovaries, immature oocytes are reserved in primordial follicles until their activation for potential ovulation. Precise control of primordial follicle activation (PFA) is essential for reproduction, but how this is achieved is unclear. Here, we show that canonical wingless-type MMTV integration site family (WNT) signaling is pivotal for pre-granulosa cell (pre-GC) activation during PFA. We identified several WNT ligands expressed in pre-GCs that act in an autocrine manner. Inhibition of WNT secretion from pre-GCs/GCs by conditional knockout (cKO) of the wntless (Wls) gene led to female infertility. In Wls cKO mice, GC layer thickness was greatly reduced in growing follicles, which resulted in impaired oocyte growth with both an abnormal, sustained nuclear localization of forkhead box O3 (FOXO3) and reduced phosphorylation of ribosomal protein S6 (RPS6). Constitutive stabilization of β-catenin (CTNNB1) in pre-GCs/GCs induced morphological changes of pre-GCs from a squamous into a cuboidal form, though it did not influence oocyte activation. Our results reveal that canonical WNT signaling plays a permissive role in the transition of pre-GCs to GCs, which is an essential step to support oocyte growth.

Constructing canonical Feynman integrals with intersection theory

Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct dlog-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales.

Sommerfeld Integrals and Their Relation to the Development of Planar Microwave Devices

This paper deals with the mathematical expressions called Sommerfeld integrals. Introduced by A. Sommerfeld in 1909, they are mathematically equivalent to inverse Hankel transforms. The original historical goal of these integrals was to provide an accurate mathematical description of the electromagnetic phenomena involved in long-distance wireless radio and telegraphy. However, their scope was quickly enlarged thanks to the so-called spectral-domain stratified theory, and now they are ubiquitous in the mathematical models associated to many electromagnetic technologies, ranging from EMC lightning modelization and ground penetrating radar to optical and plasmonic integrated devices and going through the familiar microwave and millimeter-wave planar structures using printed circuit technology. In all these areas, Sommerfeld integrals can provide direct evaluations of the involved electromagnetic fields or they can be used as Green's functions in the frame of integral equation formulations. Other disciplines involving stratified media, like seismology and geological prospection, also benefit from these integrals. After discussing the most canonical Sommerfeld integral, appearing in the so-called Sommerfeld identity, this paper reviews three classical structures, namely, the original Sommerfeld problem involving two semi-infinite media, and the microstrip and stripline geometries. It is shown that Sommerfeld integrals provide a unifying treatment of these three problems and that their mathematical features have a direct translation in terms of the physical properties exhibited by the electromagnetic fields that can exist in them.

Integrals whose saddle-point expansions terminate

The saddle-point expansion for integrals with integrand exp(−kf(x)) is a series in powers of 1/k. Usually this series diverges, but there is a family of exponent functions f(x), defining a family of canonical integrals, for which the series terminates and the saddle-point expansion is exact. For this family, the transformation x → X such that f(x) = X 2 possesses a Jacobian that is a polynomial in X, whose coefficients parameterise the canonical integrals.

On the asymptotic properties of a canonical diffraction integral.

We introduce and study a new canonical integral, denoted , depending on two complex parameters α 1 and α 2. It arises from the problem of wave diffraction by a quarter-plane and is heuristically constructed to capture the complex field near the tip and edges. We establish some region of analyticity of this integral in , and derive its rich asymptotic behaviour as |α 1 | and |α 2 | tend to infinity. We also study the decay properties of the function obtained from applying a specific double Cauchy integral operator to this integral. These results allow us to show that this integral shares all of the asymptotic properties expected from the key unknown function G +- arising when the quarter-plane diffraction problem is studied via a two-complex-variables Wiener-Hopf technique (see Assier & Abrahams, SIAM J. Appl. Math., in press). As a result, the integral can be used to mimic the unknown function G +- and to build an efficient 'educated' approximation to the quarter-plane problem.

Maneuvers of Multi Perspective Media Retrieval

Recently, Learning Machines have achieved a measure of success in the representation of multiple views. Since the effectiveness of data mining methods is highly dependent on the ability to produce data representation, learning multi-visual representation has become a very promising topic with widespread use. It is an emerging data mining guide that looks at multidisciplinary learning to improve overall performance. Multi-view reading is also known as data integration or data integration from multiple feature sets. In general, learning the representation of multiple views is able to learn the informative and cohesive representation that leads to the improvement in the performance of predictors. Therefore, learning multi-view representation has been widely used in many real-world applications including media retrieval, native language processing, video analysis, and a recommendation program. We propose two main stages of learning multidisciplinary representation: (i) alignment of multidisciplinary representation, which aims to capture relationships between different perspectives on content alignment; (ii) a combination of different visual representations, which seeks to combine different aspects learned from many different perspectives into a single integrated representation. Both of these strategies seek to use the relevant information contained in most views to represent the data as a whole. In this project we use the concept of canonical integration analysis to get more details. Encouraged by the success of in-depth reading, in-depth reading representation of multiple theories has attracted a lot of attention in media access due to its ability to read explicit visual representation.

Convenient antiderivatives for differential linear categories

AbstractDifferential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

Roles of Dkk2 in the Linkage from Muscle to Bone during Mechanical Unloading in Mice.

Mechanical unloading simultaneously induces muscle and bone loss, but its mechanisms are not fully understood. The interactions between skeletal muscle and bone have been recently noted. Although canonical wingless-related integration site (Wnt)/β-catenin signaling is crucial for bone metabolism, its roles in the muscle and bone interactions have remained unknown. Here, we performed comprehensive DNA microarray analyses to clarify humoral factors linking muscle to bone in response to mechanical unloading and hypergravity with 3 g in mice. We identified Dickkopf (Dkk) 2, a Wnt/β-catenin signaling inhibitor, as a gene whose expression was increased by hindlimb unloading (HU) and reduced by hypergravity in the soleus muscle of mice. HU significantly elevated serum Dkk2 levels and Dkk2 mRNA levels in the soleus muscle of mice whereas hypergravity significantly decreased those Dkk2 levels. In the simple regression analyses, serum Dkk2 levels were negatively and positively related to trabecular bone mineral density and mRNA levels of receptor activator of nuclear factor-kappa B ligand (RANKL) in the tibia of mice, respectively. Moreover, shear stress significantly suppressed Dkk2 mRNA levels in C2C12 cells, and cyclooxygenase inhibitors significantly antagonized the effects of shear stress on Dkk2 expression. On the other hand, Dkk2 suppressed the mRNA levels of osteogenic genes, alkaline phosphatase activity and mineralization, and it increased RANKL mRNA levels in mouse osteoblasts. In conclusion, we showed that muscle and serum Dkk2 levels are positively and negatively regulated during mechanical unloading and hypergravity in mice, respectively. An increase in Dkk2 expression in the skeletal muscle might contribute to disuse- and microgravity-induced bone and muscle loss.

Mediator Kinase as a Transducer and Therapeutic Target in Oncogenic WNT/β‐ CATENIN Signaling

Colorectal cancer (CRC) is the second and third most commonly diagnosed cancer in women and men, respectively. Few treatment options exist for advanced stage disease, thus, the development of targeted therapies is crucial for treatment. The canonical wingless related integration site (WNT)/β‐CATENIN pathway is a driver in &gt;90% of CRCs. The mediator kinase module (mediator complex subunit 12 (MED12), mediator complex subunit 13 (MED13), CYCLIN C, and cyclin dependent kinase 8 (CDK8)) is a key node of oncogenic activation within the WNT/β‐CATENIN pathway, since β‐CATENIN activates transcription through its direct physical interaction with MED12. Additionally, CDK8 is an oncoprotein required for β‐CATENIN‐dependent gene activation and CRC growth.The objective of this study was to investigate the effects of genetic (disruption of MED12:CYCLIN C interface) and chemical inhibition of Mediator Kinase activity on CRC cell oncogenic properties and gene regulation. Oncogenic properties were assessed by colony formation and cell proliferation assays. Chromatin immunoprecipitation sequencing and RNA sequencing were used to evaluate gene regulation. Interestingly, genetic but not chemical inhibition led to a significant reduction in oncogenic properties. Superenhancer alterations and differential expression of superenhancer‐associated genes were observed with both genetic and chemical inhibition. However, little overlap was found for down‐regulated superenhancer‐associated genes.Based on the significant reduction in oncogenic properties, genetic disruption of the MED12:CYCLIN C interface appears to be an attractive target for the development of targeted molecular therapies for CRC. The difference in effect on oncogenic properties with chemical versus genetic inhibition is likely explained by the differences in down‐regulated superenhancer‐associated genes. Concordantly, genes involved in the WNT/β‐CATENIN pathway were down‐regulated with genetic but not chemical inhibition.Support or Funding InformationLindsey Barron was supported by a CPRIT postdoctoral training fellowship, the CPRIT Research Training Award (RP170345) and is currently supported by an IRACDA Grant (K12GM111726).

A novel Sprouty4-ERK1/2-Wnt/β-catenin regulatory loop in marrow stromal progenitor cells controls osteogenic and adipogenic differentiation

BackgroundSprouty (SPRY) proteins play critical roles in controlling cell proliferation, differentiation, and survival by inhibiting receptor tyrosine kinase (RTK)-mediated extracellular signal-regulated kinase (ERK) signaling. Recent studies have demonstrated that SPRY4 negatively regulates angiogenesis and tumor growth. However, whether SPRY4 regulates osteogenic and/or adipogenic differentiation of mesenchymal stem cells remains to be explored. ResultsIn this study, we investigated the expression pattern of Spry4 and found that its expression was regulated during the differentiation of mouse marrow stromal progenitor cells and increased in the metaphysis of ovariectomized mice. In vitro loss-of-function and gain-of-function studies demonstrated that SPRY4 inhibited osteogenic differentiation and stimulated adipogenic differentiation of progenitor cells. In vivo experiments showed that silencing of Spry4 in the marrow of C57BL/6 mice blocked fat accumulation and promoted osteoblast differentiation in ovariectomized mice. Mechanistic investigations revealed the inhibitory effect of SPRY4 on canonical wingless-type MMTV integration site (Wnt) signaling and ERK pathway. ERK1/2 was shown to interact with low-density lipoprotein receptor-related protein 6 (LRP6) and activate the canonical Wnt signaling pathway. Inactivation of Wnt signaling attenuated the inhibition of adipogenic differentiation and stimulation of osteogenic differentiation by Spry4 small interfering RNA (siRNA). Finally, promoter study revealed that β-catenin transcriptionally inhibited the expression of Spry4. ConclusionsOur study for the first time suggests that a novel SPRY4-ERK1/2-Wnt/β-catenin regulatory loop exists in marrow stromal progenitor cells and plays a key role in cell fate determination. It also highlights the potential of SPRY4 as a novel therapeutic target for the treatment of metabolic bone disorders such as osteoporosis.

The Effect of Renal Dysfunction on Circulating Sclerostin Level in Patients with Type 2 Diabetes

Background and aim:Sclerostin, a soluble canonical Wingless integration site signaling inhibitor, is formed by osteocytes and is vital for bone physiology. This work aimed to appraise the role of sclerostin level in early detection of renal osteodystrophy in patients with type 2 diabetes mellitus (T2DM).Method:This cross sectional study was done on 75 diabetic patients with different stages of chronic kidney disease (CKD) in the Internal Medicine department, Zagazig University hospital.The patients are classified into 3 groups according to glomerular filtration rate(GFR),group 1(stage 1,2 CKD),group 2(stage 3 CKD),group 3(stage 4,5 CKD). The patients underwent history taking, examination and laboratory studies including routine investigations and assessment of serum sclerostin. Results: The mean age of the patients was 51.07 ± 4.40 years. Males represented 61.3% of them. Mean Sclerostin of them was 5.42 ± 4.53 ng/ml. There are significant differences between groups with different CKD stages regarding sclerostin, hemoglobin, serum creatinine, blood urea, eGFR, phosphorus and parathyroid hormone. There is significant positive correlation between serum sclerostin and all of serum creatinine, blood urea, phosphorus and parathyroid hormone. There is significant negative correlation between it and eGFR. On multivariate regression analysis, it was found that age, serum creatinine, blood urea, eGFR and albumin/creatinine ratio (ACR) significantly associated with serum level.Conclusion: Sclerostin level is negatively correlated with eGFR in diabetic patients with impaired renal function. Serum sclerostin levels increase in diabetic patients starting from CKD-G3 stage. Age, serum creatinine, blood urea, eGFR and albumin/creatinine ratio (ACR) significantly associated with its level.

Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards

The authors have recently introduced the class of topological billiards. Topological billiards are glued from elementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classical linearly and quadratically integrable geodesic flows on tori and spheres are Liouville equivalent to appropriate topological billiards. Moreover, the linear and quadratic integrals of the geodesic flows reduce to a single canonical linear integral and a single canonical quadratic integral on the billiard. These results are obtained within the framework of the Fomenko–Zieschang theory of the classification of integrable systems.

Autosomal-Recessive Mutations in MESD Cause Osteogenesis Imperfecta

Osteogenesis imperfecta (OI) comprises a genetically heterogeneous group of skeletal fragility diseases. Here, we report on five independent families with a progressively deforming type of OI, in whom we identified four homozygous truncation or frameshift mutations in MESD. Affected individuals had recurrent fractures and at least one had oligodontia. MESD encodes an endoplasmic reticulum (ER) chaperone protein for the canonical Wingless-related integration site (WNT) signaling receptors LRP5 and LRP6. Because complete absence of MESD causes embryonic lethality in mice, we hypothesized that the OI-associated mutations are hypomorphic alleles since these mutations occur downstream of the chaperone activity domain but upstream of ER-retention domain. This would be consistent with the clinical phenotypes of skeletal fragility and oligodontia in persons deficient for LRP5 and LRP6, respectively. When we expressed wild-type (WT) and mutant MESD in HEK293T cells, we detected WT MESD in cell lysate but not in conditioned medium, whereas the converse was true for mutant MESD. We observed that both WT and mutant MESD retained the ability to chaperone LRP5. Thus, OI-associated MESD mutations produce hypomorphic alleles whose failure to remain within the ER significantly reduces but does not completely eliminate LRP5 and LRP6 trafficking. Since these individuals have no eye abnormalities (which occur in individuals completely lacking LRP5) and have neither limb nor brain patterning defects (both of which occur in mice completely lacking LRP6), we infer that bone mass accrual and dental patterning are more sensitive to reduced canonical WNT signaling than are other developmental processes. Biologic agents that can increase LRP5 and LRP6-mediated WNT signaling could benefit individuals with MESD-associated OI.

BMP9 directly induces rapid GSK3-β phosphorylation in a Wnt-independent manner through class I PI3K-Akt axis in osteoblasts.

Bone morphogenetic protein (BMP)9 has been reported to be the most potent BMP to induce bone formation. However, the details of BMP9-transduced intracellular signaling remain ambiguous. Here, we have investigated signal transduction mechanisms of BMP9 in comparison to BMP2, another potent inducer of bone formation, in osteoblasts. In a mouse osteoblast cell line, BMP9 induced higher mRNA levels of alkaline phosphatase (ALP) and runt-related transcription factor 2 (Runx2) than BMP2 within 2 h. Unlike BMP2, BMP9 induced rapid phosphorylation of glycogen synthase kinase 3-β (GSK3-β) and protein kinase B (Akt) and increased the cellular protein content of β-catenin. BMP9 moderately increased mRNA levels of several canonical Wingless-related integration site to lower degrees than BMP2. Furthermore, BMP9-induced GSK3-β phosphorylation was not inhibited by pretreatment with actinomycin D, cycloheximide, or Brefeldin A, indicating it is independent of Wnt protein secretion. BMP9-induced GSK3-β phosphorylation was abrogated by Akt or class I PI3K-specific inhibitors. Moreover, inactivation of GSK3-β by LiCl did not further promote ALP and Runx2 mRNA induction by BMP9 as significantly as that by BMP2. Notably, BMP9-induced GSK3-β phosphorylation was inhibited by small interfering RNA against endoglin and GIPC PDZ domain-containing family, member 1. Taken together, our present findings have indicated that BMP9 directly activates GSK3β-β-catenin signaling pathway through class I PI3K-Akt Axis in osteoblasts, which may be essential for the potent osteoinductive activity of BMP9.-Eiraku, N., Chiba, N., Nakamura, T., Amir, M. S., Seong, C.-H., Ohnishi, T., Kusuyama, J., Noguchi, K., Matsuguchi, T. BMP9 directly induces rapid GSK3-β phosphorylation in a Wnt-independent manner through class I PI3K-Akt axis in osteoblasts.

Hormonal and spatial control of SUMOylation in the human and mouse adrenal cortex.

SUMOylation is a highly conserved and dynamic post-translational mechanism primarily affecting nuclear programs for adapting organisms to stressful challenges. Alteration of SUMOylation cycles leads to severe developmental and homeostatic defects and malignancy, but signals coordinating SUMOylation are still unidentified. The adrenal cortex is a zonated endocrine gland that controls body homeostasis and stress response. Here, we show that in human and in mouse adrenals, SUMOylation follows a decreasing centripetal gradient that mirrors cortical differentiation flow and delimits highly and weakly SUMOylated steroidogenic compartments, overlapping glomerulosa, and fasciculata zones. Activation of PKA signaling by acute hormonal treatment, mouse genetic engineering, or in Carney complex results in repression of small ubiquitin-like modifier (SUMO) conjugation in the inner cortex by coordinating expression of SUMO pathway inducers and repressors. Conversely, genetic activation of canonical wingless-related integration site signaling maintains high SUMOylation potential in the outer neoplastic cortex. Thus, SUMOylation is tightly regulated by signaling pathways that orchestrate adrenal zonation and diseases.-Dumontet, T., Sahut-Barnola, I., Dufour, D., Lefrançois-Martinez, A.-M., Berthon, A., Montanier, N., Ragazzon, B., Djari, C., Pointud, J.-C., Roucher-Boulez, F., Batisse-Lignier, M., Tauveron, I., Bertherat, J., Val, P., Martinez, A. Hormonal and spatial control of SUMOylation in the human and mouse adrenal cortex.

The reduction of the degree of integrals of hamiltonian systems with the help of billiards

In the theory of integrable Hamiltonian systems with two degrees of freedom there are widely known integrable systems whose integrals have a high degree, namely 3 and 4: the Kovalevskaya system and its generalizations - the Kovalevskaya - Yahya system and the Kovalevskaya system on the Lie algebra so(4), Goryachev-Chaplygin-Sretensky, Sokolov and Dullin-Matveyev. The article shows that using integrable billiards bounded by arcs of confocal quadrics decreases the degree of integrals 3 and 4 of these systems fo some isoenergy 3-surfaces. Moreover, the integrals of degree 3 and 4 reduce to the same canonical quadratic integral on billiards.

On the Approximate Evaluation of Some Oscillatory Integrals

To determine the photon emission or absorption probability for a diatomic system in the context of the semiclassical approximation it is necessary to calculate the characteristic canonical oscillatory integral which has one or more saddle points. Integrals like that appear in a whole range of physical problems, e.g., the atom–atom and atom–surface scattering and various optical phenomena. A uniform approximation of the integral, based on the stationary phase method is proposed, where the integral with several saddle points is replaced by a sum of integrals each having only one or at most two real saddle points and is easily soluble. In this way we formally reduce the codimension in canonical integrals of “elementary catastrophes” with codimensions greater than 1. The validity of the proposed method was tested on examples of integrals with three saddle points (“cusp” catastrophe) and four saddle points (“swallow-tail” catastrophe).