Stable Envelopes Research Articles

Motivic Chern classes and K‐theoretic stable envelopes

We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class is characterized by an axiom system inspired by that of "K-theoretic stable envelopes," recently defined by Okounkov and studied in relation with quantum group actions on the K-theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formulas for the motivic Chern classes of determinantal varieties and more general degeneracy loci.

Stable Automatic Envelope Estimation for Noisy Doppler Ultrasound

Doppler ultrasound technology is widespread in clinical applications and is principally used for blood flow measurements in the heart, arteries, and veins. A commonly extracted parameter is the maximum velocity envelope. However, current methods of extracting it cannot produce stable envelopes in high noise conditions. This can limit clinical and research applications using the technology. In this article, a new method of automatic envelope estimation is presented. The method can handle challenging signals with high levels of noise and variable envelope shapes. Envelopes are extracted from a Doppler spectrogram image generated directly from the Doppler audio signal, making it less device-dependent than existing image-processing methods. The method's performance is assessed using simulated pulsatile flow, a flow phantom, and in vivo ascending aortic flow measurements and is compared with three state-of-the-art methods. The proposed method is the most accurate in noisy conditions, achieving, on average, for phantom data with signal-to-noise ratios (SNRs) below 10 dB, bias and standard deviation of 0.7% and 3.3% lower than the next-best performing method. In addition, a new method for beat segmentation is proposed. When combined, the two proposed methods exhibited the best performance using in vivo data, producing the least number of incorrectly segmented beats and 8.2% more correctly segmented beats than the next best performing method. The ability of the proposed methods to reliably extract timing indices for cardiac cycles across a range of signal quality is of particular significance for research and monitoring applications.

Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the K\"ahler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same cardinality and can be identified so that the restriction matrix of $X$ becomes equal to the restriction matrix of $X'$ after transposition and interchanging the equivariant and K\"ahler parameters of $X$, respectively, with the K\"ahler and equivariant parameters of $X'$. The first examples of pairs of 3d symmetric varieties were constructed in [Rim\'anyi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle $T^*\operatorname{Gr}(k,n)$ to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of $A_{n-1}$-type. In this paper we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities.

Motivic characteristic classes in cohomological Hall algebras

The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are important characteristic classes of singular varieties in cohomology and K theory—and their theory overlaps with the theory of Okounkov's stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes cβo, Cβo parameterized by positive roots β. We prove an identity in a deformed version of Kontsevich-Soibelman's Cohomological (and K-theoretic) Hall Algebra (CoHA, KHA), namely, that a product of exponentials of cβo (or Cβo) classes formally depending on a stability function Z, does not depend on Z. This identity—which encodes infinitely many identities among rational functions in growing number of variables—has the structure of Donaldson-Thomas type quantum dilogarithm identities. Using a wall-crossing argument we present the cβo, Cβo classes as certain commutators in the CoHA, KHA.

Elliptic and K-theoretic stable envelopes and Newton polytopes

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $$K$$ -theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $$K$$ -theoretic stable envelopes and our elliptic stable envelopes. We show that the $$K$$ -theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $${\mathfrak {gl}}_2$$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $$K$$ -theoretic stable envelopes.

Avancées concernant les R-matrices et leurs applications (d’après Maulik-Okounkov, Kang-Kashiwara-Kim-Oh, ...)

R-matrices are the solutions of the Yang-Baxter equation. At the origin of the quantum group theory, they may be interpreted as intertwining operators. Recent advances have been made independently in different directions. Maulik-Okounkov have given a geometric approach to R-matrices with new tools in symplectic geometry, the stable envelopes. Kang-Kashiwara-Kim-Oh proved a conjecture on the categorification of cluster algebras by using R-matrices in a crucial way. Eventually, a better understanding of the action of transfer-matrices obtained from R-matrices led to the proof of several conjectures about the corresponding quantum integrable systems.

Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we show a correspondence of the Gelfand-Tsetlin bases (resp. the standard bases) to the fixed point classes (resp. the stable classes) in the equivariant elliptic cohomology $E_T(X)$ of the cotangent bundle $X$ of the partial flag variety. As a result we obtain a geometric representation of the elliptic quantum group on $E_T(X)$.

Pore formation in lipid membrane II: Energy landscape under external stress

Lipid membranes are extremely stable envelopes allowing cells to survive in various environments and to maintain desired internal composition. Membrane permeation through formation of transversal pores requires substantial external stress. Practically, pores are usually formed by application of lateral tension or transmembrane voltage. Using the same approach as was used for obtaining continuous trajectory of pore formation in the stress-less membrane in the previous article, we now consider the process of pore formation under the external stress. The waiting time to pore formation proved a non-monotonous function of the lateral tension, dropping from infinity at zero tension to a minimum at the tension of several millinewtons per meter. Transmembrane voltage, on the contrary, caused the waiting time to decrease monotonously. Analysis of pore formation trajectories for several lipid species with different spontaneous curvatures and elastic moduli under various external conditions provided instrumental insights into the mechanisms underlying some experimentally observed phenomena.

Partial flag varieties, stable envelopes, and weight functions

We consider the cotangent bundle T^*\mathfrak F_{\lambda} of a \mathrm {GL}_n partial flag variety, \lambda = (\lambda_1,\dots,\lambda_N), |\lambda|=\sum_i\lambda_i=n , and the torus T=(\mathbb C^\times)^{n+1} equivariant cohomology H^*_T(T^*\mathfrak F_{\lambda}) . In [9], a Yangian module structure was introduced on \bigoplus_{|\lambda|=n} H^*_T(T^*\mathfrak F_{\lambda}) . We identify this Yangian module structure with the Yangian module structure introduced in [5]. This identifies the operators of quantum multiplication by divisors on H^*_T(T^*\mathfrak F_{\lambda}) , described in [9], with the action of the dynamical Hamiltonians from [20], [10], [5]. To construct these identifications we provide a formula for the stable envelope maps, associated with the partial flag varieties and introduced in [9]. The formula is in terms of the Yangian weight functions introduced in [19], c.f. [21], [22], in order to construct q-hypergeometric solutions of qKZ equations.

Nonlinear propagation of terahertz pulses in ammonia vapor

We consider the propagation of terahertz (THz) pulses resonant with the lowest rotational levels in ammonia vapors at room temperature. We demonstrate a quasisimultonic regime of propagation for two-color THz pulses (their lossless simultaneous propagation with equal group velocities and stable envelopes at finite distances, like coupled solitons). The quasisimultons are formed by two pulses synchronized in time, the first one being intense and two humped with an area multiple of 4π , while the another one is weaker, one humped, and with an area much less than π. Two-and one-humped pulses have equal width and copropagate in ammonia vapor at the same group velocity. Moreover, the larger the dip of the two-hump pulse is, the weaker may be the one-humped pulse. The stability of this coupled state is provided by the fact that the two-humped pulse makes the medium transparent for the one-humped pulse. In its turn, the latter prevents the breakup of the former into separate pulses. It is established as well that a multihumped pulse with an odd or even number of humps and a hyperbolic secant pulse can behave like a simulton. In this case, the area of multihumped pulse equals to 2nπ , where n is the number of humps in the powerful pulse structure.

Modulational instability of dust acoustic solitary waves for variable-charge dust grains in an ion beam–dusty plasma

A theoretical study is presented for the nonlinear self-modulation of low-frequency electrostatic dust acoustic (DA) waves propagating in a dusty plasma exposed to an external ion beam. By employing the derivative expansion perturbation method, a nonlinear Schrödinger equation (NLSE) is derived for the electric potential wave amplitude. Both dispersion and nonlinearity coefficients of the NLSE are explicit functions of the carrier wavenumber and of relevant physical parameters background species density and temperature, and dust charging fluctuation. The modulation instability of the NLSE is investigated. It is found that the presence of the ion beam species reduces the stable domain of the DA waves and the stable DA waves are created at higher wavenumber, k. Increasing ion beam temperature results as an increment of the wave dispersion and nonlinearity and the wave frequency. Also it widens the stability domain of the DA waves. Including the dust charge fluctuations changes drastically the domain where one can find stable DA wave envelopes. A brief conclusion is provided related to the application of the current findings in laboratory plasma experiments.

Evolution of supermassive stars as a pathway to black hole formation

Supermassive stars, with masses greater than a million solar masses, are possible progenitors of supermassive black holes in galactic nuclei. Because of their short nuclear burning timescales, such objects can be formed only when matter is able to accumulate at a rate exceeding ~ 1 solar mass/yr. Here we revisit the structure and evolution of rotationally-stabilized supermassive stars, taking into account their continuous accumulation of mass and their thermal relaxation. We show that the outer layers of supermassive stars are not thermally relaxed during much of the star's main sequence lifetime. As a result, they do not resemble n=3 polytropes, as assumed in previous literature, but rather consist of convective (polytropic) cores surrounded by convectively stable envelopes that contain most of the mass. We compute the structures of these envelopes, in which the specific entropy is proportional to the enclosed mass M(R) to the 2/3-power. By matching the envelope solutions to convective cores, we calculate the core mass as a function of time. We estimate the initial black hole masses formed as a result of core-collapse, and their subsequent growth via accretion from the bloated envelopes ("quasistars") that result. The seed black holes formed in this way could have typical masses in the range ~ 10^4-10^5 solar masses, considerably larger than the remnants thought to be left by the demise of Population III stars. Supermassive black holes therefore could have been seeded during an epoch of rapid infall considerably later than the era of Pop III star formation.

Ray chaos and eigenrays

In a realistic range-dependent deep ocean environment that gives rise to ray chaos, a complete set of eigenrays at long ranges is computed using a numerical technique based on the shooting method. The numerical results support and illustrate the known connection between ray chaos and eigenrays, namely, that the number of eigenrays grows exponentially, on average, with increasing range. It is further found that groups of chaotic eigenrays tend to form ‘‘clusters’’ having stable envelopes.

Modeling issues for predicting sound propagation at moderate frequencies in bottom-limited sound channels

Consider sound propagation in the moderate frequency range (depth to wavelength ratio, H/λ, between 20 and 100) in shallow water environments with bottom roughness. Measurements in the Straits of Florida have shown remarkable phase stability together with rapid (minutes-to-hours) unstable amplitude fluctuations that nevertheless yield stable envelopes when averaged over many hours. What physical effects should be contained in an acoustic model in order to predict the observed behavior? It is suggested that a full-wave model that is capable of multiple rough bottom forward scattering and includes temporally slowly varying range-dependent sound-speed profiles will yield observed results, when used together with coherent signal processing techniques. A broadband PE model will illustrate the assertion, at least qualitatively. Absolute level prediction is quite another matter that requires more extensive environmental inputs.

Event-based architecture for diagnosis in control advisory systems

Artificial intelligence research on diagnostic systems is focused mainly on static problems in which the object of interest is assumed to be in a stable condition that will remain stable until the diagnostic agent has completed its deliberation. Control systems, on the other hand, cannot afford the luxury of ‘suspended time’; faults must be detected, isolated and reconfiguration implemented within one computation cycle to prevent departure from stable control envelopes. A new class of system is emerging from recent programs directed toward vehicle operator aids for fighter aircraft, submarines and helicopters. These systems are neither static off-line aids nor real-time controllers; instead they are control advisory systems which span the time scales of both regimes. These systems interface with controllers to interpret the error codes and to conduct tests and implement reconfigurations. On the other hand, these systems also interact with the vehicle operator to prioritize their activity consistent with the operator's goals and to recommend diagnostic/emergency procedures. Pilot's Associate (PA), a complex system of software and cockpit displays currently being developed to aid Air Force fighter pilots, is used to exemplify a control advisor and to display the requirements for real-time equipment diagnosis in such a system. It is shown that the required functionality challenges the capabilities of current diagnostic algorithms. The PA diagnostic architecture is presented. This architecture is centred around the concept of a diagnostic event which begins when abnormal data are detected, and which is treated as an interruptable computational process which may exist in conjunction with other event processes in multiple fault situations. The architecture solves a number of problems such as interruptability, and integrating the different time-scales of the pilot and the local system controllers, and the architecture provides a means for fusing statistical fault detection methods such as Kalman filters with artificial intelligence techniques such as model-based reasoning and classification. The result is an approach which exploits the strengths of each technique and provides a mechanism for automated reasoning using both quantitative and qualitative information.

Construction of protocellular structures under simulated primitive earth conditions.

We have developed experimental approaches for the construction of protocellular structures under simulated primitive earth conditions and studied their formation and characteristics. Three types of envelopes; protein envelopes, lipid envelopes, and lipid-protein envelopes are considered as candidates for protocellular structures. Simple protein envelopes and lipid envelopes are presumed to have originated at an early stage of chemical evolution, interaction mutually and then evolved into more complex envelopes composed of both lipids and proteins. Three kinds of protein envelopes were constructed in situ from amino acids under simulated primitive earth conditions such as a fresh water tide pool, a warm sea, and a submarine hydrothermal vent. One protein envelope was formed from a mixture of amino acid amides at 80 degrees C using multiple hydration-dehydration cycles. Marigranules, protein envelope structures, were produced from mixtures of glycine and acidic, basic and aromatic amino acids at 105 degrees C in a modified sea medium enriched with essential transition elements. Thermostable microspheres were also formed from a mixture of glycine, alanine, valine, and aspartic acid at 250 degrees C and above. The microspheres did not form at lower temperatures and consist of silicates and peptide-like polymers containing imide bonds and amino acid residues enriched in valine. Amphiphilic proteins with molecular weights of 2000 were necessary for the formation of the protein envelopes. Stable lipid envelopes were formed from different dialkyl phospholipids and fatty acids. Large, stable, lipid-protein envelopes were formed from egg lecithin and the solubilized marigranules. Polycations such as polylysine and polyhistidine, or basic proteins such as lysozyme and cytochrome c also stabilized lipid-protein envelopes.

Acoustic reciprocal transmission experiments, Florida Straits

An acoustic approach for the measurement of range- and depth-averaged current and temperature is successfully demonstrated for the bottom-limited acoustic transmission of the Florida Straits. The shallow water environment prohibits the use of deep ocean tomography methods since there are usually no separable arrivals associated with eigenrays. Instead, multipath groupings of arrivals are shown to have stable envelopes if averaged sufficiently long to eliminate effects of interference. The shape of the envelope is controlled to some extent by the source–receiver depth and has characteristics which can be used for very precise travel time measurements. Ray models predict a unique relation between ray height above the bottom and arrival time allowing for unambiguous inversion. Two three-point reciprocal transmission experiments were conducted in the Florida Straits (ranges approx. 25 and 45 km). Two-way channel pulse responses are analyzed to give travel times which are then inverted to give depth averages of temperature and current; the depth and extent of the average is determined by ray models. The method is shown to be precise in resolving variations in temperature of 0.002 °C and current of 2 cm/s. The spatially averaged tomographic results are compared with point measurements of current and temperature from current meter moorings along the path of propagation.

Tomographic observation of depth averaged current and temperature in the Florida Straits

A tomographic approach for the measurement of range and depth averaged current and temperature is successfully demonstrated for the bottom limited acoustic transmission of the Florida Straits. The shallow water environment prohibits the use of deep ocean tomography methods since there are usually no observable separable arrivals associated with eigenrays. Instead, multipath groupings of arrivals are shown to have stable envelopes if averaged sufficiently long to eliminate effects of interference. The shape of the envelope is controlled to some extent by the source receiver depth and has characteristics which can be used for very precise travel time measurements. Ray models predict a unique relation between ray height above the bottom and arrival time allowing for unambiguous inversion. Two three‐point reciprocal transmission experiments were conducted in the Florida Straits (ranges approximately 25 and 45 km). Two‐way channel pulse responses are analyzed to give travel times which are then inverted to give depth averages of temperature and current and the depth of the average is determined by ray models. The method is shown to be precise in resolving variations in temperature of 0.02 °C and current of 2 cm/s. The spatially averaged tomographic results are compared with point measurements of current and temperature from current meter moorings along the path of propagation.

Thermoproteales: A novel type of extremely thermoacidophilic anaerobic archaebacteria isolated from Icelandic solfataras

Summary Two types of anaerobic extremely thermoacidophilic bacteria have been isolated from more than 50% of the samples collected in solfataric fields throughout Iceland. They both possess highly stable cell envelopes with a characteristic subunit structure, are usually rods of variable length (about 1 to more than 50 μ ) without septa, but can give rise to spheric bodies which exist either attached to the rods or in free state. Under certain conditions, truly branched filaments are frequently observed. The similarities in morphology and growth characteristics indicate that the two types are related to each other. In view of its variable appearance the larger organism (diameter about 0.4 μ m) was called Thermoproteus tenax . Thermoproteus tenax forms H 2 S and CO 2 from elemental sulfur and organic substrates, like glucose, ethanol, malate or formamide. The absence of murein, the presence of polyisoprenoid ether lipids, the complete resistance against the antibiotics rifampicin, streptomycin, vancomycin and chloramphenicol and the component pattern and rifampicin insensitivity of the DNA-dependent RNA polymerase are evidence that Thermoproteus tenax belongs to the archaebacteria. It represents a novel order of the thermoacidophilic branch of this urkingdom.

Enzyme envelopes on colloidal particles

A new method has been developed for the preparation of insoluble proteins. The method involves the adsorption of protein as a monolayer onto colloidal silica particles, followed by intermolecular crosslinking with glutaraldehyde. Insoluble bovine trypsin prepared by this method was obtained in about 99% yield with 80% retention of esterase activity. This activity was only partially inhibited by protein inhibitors of trypsin (M.W. 9,000–28,000). Proteolytic activity was approximately 17% that of native trypsin. These particles are easily sedimented, but are nonetheless readily dispersible to form sols. The method appears to be of general applicability for the preparation of insoluble proteins as stable envelopes on colloidal particles.