elliptic

No one knew exactly what inspired her elliptic comments, and her relations had long since given up trying to interpret them.

Suddenly, the sun sprang out of the seawater, looking like a large elliptic ball, and the sky in the east became red.

anthers on back with a pair of spurs at anther-filament junction, dehiscing by introrse-terminal and elliptic pores.

The form of lacing wires used in the last stage of steam turbines is commonly cylindrical lacing wire or elliptic lacing wire.

Some simulation proved that this arithmetic was valid to elliptic and hyperbolic transfer trajectory.

The security of the scheme is set up on the intricate nature of the elliptic curve discrete logarithm problem and the knapsack problem.

With the computer simulation, the elliptic section cylinder lenses have been designed and used to collimate the semi-conductor laser beam .

The bang of elliptic face mights as well design with asymmetry means, bang length can be in brow upper part or stand by hair boundary line.

Fir living in Xinjiang is an ornamental tree species, with graceful crown of a tree, straight trunk, needle leave and long elliptic fruit.

The estimated radius may also be a reasonable approximation if the shape is not too elliptic.

Glumes ovate-elliptic, hirtellous above middle; leaf sheaths about as long or longer than internodes.

This paper deals with the general boundary value problem for nonlinear elliptic equations of second order.

Due to its slanted end surfaces, a sheared round blank often becomes elliptic in shape after an upset forging process.

By means of properties of certain combinations of complete elliptic integrals, the authors provide a negative answer to M.

From magnetic fields in PF coils of EAST expressed as elliptic integrals, the heat load of PF coils during plasma discharge is determined.

Each of linear, circular and elliptic polarization wave can be decomposed two orthogonal linear polarization wave at the same frequency.

The implementation speed of Elliptic Curve Cryptography (ECC) depends on the implementation speed of elliptic curve point multiplication.

The security of the Elliptic Curve Cryptography (ECC) is built upon the difficulty of solving the elliptic curve discrete logarithm problem.

A numerical method for an anisotropic problem in an exterior elliptic domain is presented in this paper.

Using maximum principle and mountain pass lemma to a class of nonlinear elliptic equations, two different positive solutions are obtained.

Branchlets brown, glabrous or with a few hairs near nodes, with scattered small elliptic lenticels.

The security of the system was improved by adopting the elliptic curve zero knowledge proof in the payment protocol.

It is a bit advertent , you look up can be attracted by elliptic smallpox modelling place.

Stipules elliptic-lanceolate to ovate-lanceolate, margin serrate, teeth with a disciform apical gland.

Petiole 4-6 mm; leaf blade elliptic to oblong-elliptic, base cuneate to broadly cuneate, apex obtuse to subrounded and with a retuse tip.

It prompts you for certificate identification and file information and uses RSA or elliptic - curve encryption technology .

Focused on a posteriori error estimates for a finite volume discretization of an elliptic problem.

The science hall of Chinese Geology University presents a novel architectural image, with irregular fiat, facade and an elliptic auditorium.

In every small area on the circle, elliptic, installation, such forms of polygon square, can be shot depending on improving goal level.

Elliptic curve cryptography and bilinear pairing is employed in the agreement, the dishonest node can be detected and identified.

Elliptic Curve Cryptosystem has complicated mathematical background, covering a wide range of algorithms.

We will focus on some problems for the existence and non-existence of solutions for a class of nonlinear elliptic equations.

Many problems in the engineering practice may be transformed as boundary value problems about elliptic differential equations.

Seeds elliptic, base with a sharp, short rostrum, apex retuse, ventral holes furrowed upward from middle to apex.

Since rice seems elliptic, the long axis of ellipse is used to realize fast calculation of head milled rice.

Thus, each theme develops itself in a cavernous room developed on an elliptic plan.

Leaf blade usually elliptic, leathery, apex acute to acuminate, abaxially scattered punctate; bracteoles close to calyx.

Steel tubes. Unalloyed or alloy steel butt welding elliptic caps with quality requirements. Dimensions. Technical delivery conditions.

The sound absorbing bar body can be in a cylinder shape, or an elliptic cylinder shape or a prism shape.

petiole almost absent; leaf blade elliptic, both surfaces softly pilose, base obliquely rounded, margin sharply serrulate, apex obtuse.

Petals free, white, blue, purple, or greenish yellow, sometimes spotted, ovate-elliptic, ovate-orbicular, or obovate, sometimes clawed.

Petals narrowly oblong, elliptic, ligulate, or spatulate, glabrous outside, orange punctate, apex rounded or emarginate.

old branches yellowish brown or grayish brown, glabrous, with sparsely scattered yellowish white narrowly elliptic lenticels.

The invention is applied to the solid laser provided with the elliptic cylinder photospot chamber.

Therefore, the key of the efficient executing elliptic curve cryptogram is the efficient computing of multiple point.

In particular, a conjecture on complete elliptic integrals is proved to be true and improved.

Leaf blade elliptic-oblong or broadly elliptic to orbicular, thickly leathery, margin laxly spinose dentate; calyx and pedicels glabrous.

Based on the concept of two-step recursive interpolation, a new algorithm for interpolating a 3-D elliptic arc has been developed.

Finite field inversion, an important module of elliptic curve cryptosystems and hyper-elliptic curve cryptosystems, is introduced.

12 (11) At least some leaf blades obovate-elliptic, broadest above middle, tertiary veins usually impressed adaxially in dry material.

bracts black or brownish yellow proximally, elliptic or ovate, villous, often glabrescent distally, apex shortly acuminate or acute.

The nonlinear for singularly perturbed problems of elliptic equations with boundary perturbation are considered.

In this dissertation the elliptic curve cryptosystems and fast scalar multiplication algorithms are investigated.

Styles divided to base, free arms strongly recurved in fruit; leaflets elliptic (rarely narrowly elliptic), apex acute to shortly acuminate.

Stipules ovate to ovate-elliptic, margin serrate, teeth with a conical apical gland.

With the traditional symmetric encryption algorithm, the algorithm uses elliptic curve cryptography which is difficult to be broken.

Stress distribution is determined in a thin conductive plate containing an elliptic hole under the external electromagnetic loads.

prickles dense, straight, fine, firm, flat, tapering to an elliptic base, intermixed with glandular hairs on young branches.

To take another example, elliptic geometries with different radii of curvature have isomorphic automorphism groups.

Petals elliptic to suborbicular, entire, slightly concave or forked-lobed with 2 empty anthers, rarely with a nectary.

In chapter two, we prove the existence of nontrivial solutions for the elliptic equation of high order.

The article deals with the discontinuous boundary value problem for nonlinear elliptic complex equations of second order.

Then expanded to linear equations and parabolic equation, elliptic equation, the solution of the problem of hyperbolic equation.

A verifiable vector space secret sharing scheme is proposed, whose security relies on the security of Elliptic Curve Cryptography (ECC).

It is discussed that the numerical approximation of interface problems for elliptic partial differential equations.

The present work deals with oblique derivative problems for second order quasilinear elliptic equations with parabolic degeneracy .

anther ovate-elliptic, connective prominent, narrowly triangular.

upper bracteoles greenish white, flattened, orbicular to broadly elliptic, shorter than perianth.

In this paper, we study the existence of solutions of semilinear elliptic systems with sign-changing weights.

Leaves basal, subdistichous or fasciculate, sessile or petiolate, usually linear to elliptic-lanceolate, conduplicate, base sheathing.

Sepals triangular-elliptic, as long as hypanthium, recurved, margin frequently glandular-dentate, apex acute.

Leaflets leathery, elliptic, secondary and tertiary veins adaxially impressed in dry material; inflorescence a corymbose panicle, glabrous.

fruiting rachis and pedicels tomentose when young, glabrescent, with grayish white elliptic lenticels.

Branches light brown; leaf blade elliptic, apex acuminate, tip entire; inflorescence with shorter branches from base.

Bracts narrowly elliptic or obovate-lanceolate, (20-)25-26 mm, constricted into a stipe at base; female inflorescence loosely racemose.

Leaves alternate, sessile or petiolate, usually elliptic to ovate, sometimes basal leaf solitary and early withered.

Leaves elliptic to elliptic-lanceolate, apex acuminate and mucronate, adaxial surface glabrous.

In this thesis, we concerned with the existence of nontrivial solutions to the strongly indefinite elliptic problems.

The improved system utilizes good properties of elliptic curve cryptography system and adopts zero knowledge proof.

The starch granule in two species of Fritillaria L. was in cycloidal or elliptic-shape.

Leaf blade broadly elliptic, similar to pear leaves, margin serrulate or serrate; pyrenees oblong, endocarp stony.

stipules membranous; leaflets lanceolate, ovate, or broadly ovate to elliptic, margin dentate.

Seeds obovoid-elliptic, base rostrate, apex rounded or retuse, ventral holes broadly obovate-elliptic, nearly to apex.

leaf blade elliptic-ovate, apex acute or slightly obtuse, base gradually shortly attenuate or truncate to briefly cuneate.

Stem erect. Cauline leaves alternate or whorled, sessile, linear-lanceolate to elliptic-lanceolate.

Leaf blade narrowly oblong, rarely lanceolate or elliptic, abaxial surface with whitish indumentum.

Rootstock moniliform; leaflets obovate-elliptic to elliptic, not pinnatifid.

Seeds straight, curved, or horseshoe-shaped, surface reticulate-striate with hexagonal or elliptic pits.

Leaves oblong or elliptic, or lanceolate, prominently veined; outer tepals reniform, spreading in fruit.

Petals pale yellow, elliptic, with a narrowly inflexed apex.

Membership card after binding main equipment: 2sets Shanghai you juan elliptic binding package on this computer.

robust upright shrub of mountains of northern India having oblong-elliptic leaves and pale lilac or white malodorous flowers.

In Chapter 3, we discuss the existence of the positive solution to singular quasilinear elliptic equation with Dirchlet boundary condition.

old branches dark brown, with yellowish white or grayish white elliptic to rounded lenticels and semicircular leaf scars.

Laminar fluid flow and heat transfer in twisted elliptic tubes is simulated by the dual-velocity algorithm.

Leaves ovate-lanceolate or rarely elliptic to broadly ovate, base cuneate to rounded, apex acute to apiculate-obtuse or rounded.

For the elliptic cylindrical surface which is the trunk of the blade, the geodesic and non-geodesic differential equations are deduced.

Leaf blade elliptic, rarely narrowly elliptic, abaxially tomentose or glabrescent, margin entire or with inconspicuous, gland-tipped teeth.

leaf blade linear to elliptic-ovate, subplicate veined, with conspicuous, transverse veins between main, longitudinal veins, glabrous.

The finite field arithmetic, elliptic curve cryptography (ECC) and the finite field multiplier are investigated in this thesis.

Basal leaves linear, linear-elliptic, or rarely oblong; corolla tubular-campanulate to funnelform.

Leaf blade mainly elliptic, margin entire or obscurely crenulate; capsule lobes never bending backward.

Sepals long elliptic, lanceolate, or oblanceolate, apex acute, acuminate, or attenuate.

Leaves elliptic or ovate-lanceolate, usually glaucous, veinlets conspicuous and raised.

stigmas terminal or decurrent on inner face of styles, or sessile on disk, circular to elliptic and radiating.

Potato tuber for plants for circular or elliptic shape, structure, by cuticular layer and layer, the pulp and pulp within four parts.

Sepals entire, rarely apiculate, elliptic to narrowly oblong or obovate-spatulate; leaf apex acute or rarely obtuse.

Corolla lobes yellowish; labellum white with yellowish margin, narrowly elliptic.

Leaves deeply divided, lobes oblong- lanceolate or narrowly elliptic- lanceolate.

The multi-scalar multiplication with two scalars is the most time consuming operations in elliptic curve cryptography(ECC).

and the dirt storage tank is in a horizontal elliptic cylinder shape.

Leaf blade leathery, broadly obovate or broadly elliptic, rounded or obcordate, netting veins obvious.

anthers large, elliptic, extrorse, connectives broader.

Shrubs. Leaf blade oblong, obovate, or oblong-elliptic, apex rounded, obtuse, or rarely acute, never acuminate.

Bracts entire at apex; young leaf blade gland-tipped or dentate only distally, elliptic to broadly elliptic and serrate in fruit.

We first get the nonexistence result for a related elliptic systems of nonincreasing positive solutions.

Leaf blade on flowering shoots elliptic-lanceolate or ovate-lanceolate, base cuneate, apex acuminate.

Leaf blade oblong to rarely elliptic or lanceolate, glabrous; peduncles and pedicels not brown pubescent.

Corona lobes elliptic, fleshy, apex acute, reaching middle of stamens, with minute, basal, adaxial scale.

Back of upper lemma facing away from rachis; spikelets thinly biconvex, oblong-elliptic.

leaf blade usually sword-shaped to elliptic-lanceolate, veins truly parallel from base, lateral veins absent.

The hollow part section of an arm part is also formed in an elliptic shape.

The structure and the slightly elliptic form noticeably enhance the thrill of the intimate massage.

bracts caducous, narrowly ovate-elliptic or narrowly elliptic, pubescent at apex, ciliate.

filaments subulate; anthers cylindric to broadly elliptic, with or without an unlike appendage.

Plants dioecious; bracts elliptic; male flowers distinctly stalked.

Filaments glabrous; corolla yellow, pink, or red to deep crimson, sometimes bicolored; leaves obovate, broadly elliptic to narrowly oblong.

leaf blade narrowly to broadly elliptic, rolled longitudinally in bud, glabrous or hairy, margin entire.

Samarium oblong-elliptic, with a seed surrounded by wing.

epicalyx segments long elliptic, slightly shorter or longer than sepals, abaxially sparsely sericeous-strigose, apex rounded or acute.

Capsule dark brown to blackish brown, broadly ellipsoid to ovoid; inner perianth segments ovate to elliptic.

The leaves were elliptic and leaf mesophyll tissue was distinctively differentiated.

leaf blade ovate to elliptic, venation primarily pinnate, base cordate, margin entire, not peltate.

A class of nonlinear singularly perturbed problems for elliptic equations are considered.

Ovary strigose; leaves lanceolate or elliptic-lanceolate, adaxial surface not wax-coated.

Finite difference methods for solving elliptic, parabolic and hyperbolic PDEs.

central European upright shrub having elliptic leaves and upright clusters of lilac or deep violet flowers.

The haemocytes of Chlamys farreri were round, elliptic or shuttle when they were observed with scanning electron microscopy.

Leaves narrowly elliptic or ovate-lanceolate, apex short acuminate, base attenuate.

Leaf blade ovate, elliptic to oblong, margin distally crenulate, proximally entire.

In this paper, a domain decomposition method is presented for the rotated Q1 finite element discretion of second order elliptic problem.

Seed elliptic, base gradually tapering, apex rounded, ventral holes furrowed upward from base to apex.

Leaf blade elliptic-ovate to elliptic-lanceolate, margin biserrate and lobed, apex acute to acuminate.

and present a new fast algorithm for point multiplication on elliptic curves using factorial expansions.

Melon shape is elliptic, flat leather town green, brown, yellow, white, the skin has a variety of markings.

This type of elliptic table also adopted French saber shape table leg design, full of grace and romantic feeling.

epicalyx segments oblong-elliptic, slightly shorter than sepals, apex obtuse.

epicalyx segments lanceolate or elliptic, shorter than or nearly equaling sepals, abaxially white tomentose and pilose, apex acute.

leaf blade green, elliptic, ovate, or ovate-lanceolate, membranous or papery, margin entire.

stipules absent; leaf blade elliptic or broadly to narrowly ovate, leathery, margin serrate.

Anther appendage shortly stalked, purple-red, vesicular, elliptic, slightly shorter than or subequaling anther.

This paper presents a new fast scalar multiplication algorithm on elliptic curve cryptography.

Time Average of Bilinears. Energy Flow in Plane Waves. Circular and Elliptic Polarization. Phase and Group Velocities. Dispersion.

in the two - body case , the approximate equations of relative motion on elliptic orbits are derived.

Leaf blade ovate to elliptic-ovate, margin usually not lobed, irregularly doubly serrate; fruit oblong or ovoid-oblong, rarely subglobose.

this idea was perfectly conscious and deliberate, the analogy with elliptic functions guided me.

Leaves narrowly elliptic or lanceolate to ovate-oblong, base angustate to rounded, apex acute to obtuse.

Leaf blade ovate-elliptic, without basal glands; fruiting pedicel less than 10 mm.

Leaf blade elliptic to suborbicular, abaxially subglabrous, apex with a membranous acumen ca. as long as or longer than blade.

Central leaflet obovate-elliptic; style conspicuous.

old branches purplish red or deep brown, glabrous, with yellowish brown elliptic lenticels.

Calyx teeth obsolete. Petals white, elliptic to oblong, apex acuminate, inflexed.

double shadow elliptic imagery and double wall superimposed imagery are normally used.

leaflets obovate-orbicular, sometimes broadly elliptic, leathery, abaxially glaucous, adaxially green and shiny, margin thickened.

Anther appendage shortly stalked, purple-red, vesicular, elliptic, subequaling anther.

Outer leaves deep reddish brown when dry, not scarious; bracts elliptic to oblong, longer than pedicel.

Abaxial wing of capsule broadly elliptic to orbicular; leaf blade shiny, somewhat thicker.

fructiferous stipe and fruit stipe glabrous, often with elliptic lenticels.

orchid having large elliptic to obovate fleshy leaves and fragrant pink-and-white flowers dotted with red.

Branchlets red; leaflets narrowly oblong, narrowly ovate, or narrowly obovate-elliptic, abaxially and on inflorescences sparsely pubescent.

Stem leaves alternate, sessile, ovate to elliptic-oblong, margin double dentate.

Calyx teeth obsolete. Petals yellowish green to yellow, elliptic, apex incurved.

proximal bracts leaflike, distal ones elliptic or lanceolate to narrowly linear.

Corolla tube funnelform, shorter than or subequaling lobes; leaves mostly broadly or narrowly ovate, rhomboid, or elliptic.

In this paper, we provide a detailed estimate for the modeling error of elliptic problems with highlyoscillatory coe? cents.

Two types of the mathematical models of Spatial Variogram are inferred based on the elliptic distribution function.

Leaf blade elliptic to ovate, base rounded to broadly cuneate, rarely subcordate, short strigose; plant with white or yellowish hairs.

Stem leaf blades elliptic-lanceolate. Corolla obconic -tubular.

Leaf blade ovate, broadly ovate, or elliptic-ovate, abaxially tomentose, base rounded or cordate, lobes shortly acuminate at apex.

Leaf blade oblong or oblong-elliptic, margin lacking ciliate hairs; corolla pink or light purple.

bracts and bracteoles elliptic, smaller than flowers, margin entire or serrate.

Leaves rhombic-ovate or rhombic-elliptic, base broadly cuneate or suborbicular.

The product specifications are as follows: square, long rectangular, round, elliptic, Edan type, and other specifications.

Semiopenloop microstrip resonators are utilized to realize the quasi elliptic function response.

Several convertible signcryption schemes with semantic security based on elliptic curve cryptosystem are proposed.

Sepals ovate, oblong, or elliptic, ascending or spreading, base of lateral pair not saccate.

glands 5, pale brown, rounded [transversely elliptic], entire. Male flowers many, exserted from involucre.

anthers 2-celled, parallel or oblique, elliptic, ovate, cordate, or bifurcate, dehiscing longitudinally or occasionally poricidally.

hilum transversely elliptic, large; embryo curved.

Branchlets black-brown, terete, glabrous or sparsely acicular hirsute, with elliptic lenticels.

Branchlets brown, glabrous, with scattered elliptic lenticels, grayish brown on following year.

Leaf blade broadly elliptic, densely pubescent abaxially, especially along midrib and veins; inflorescence glabrous or subglabrous.

A dual finite fields algorithm for elliptic curve cryptosystem (ECC) is presented.

Leaves sessile; leaf blade fiddle-shaped-elliptic, base auriculate.

In Chapter 2, we prove the existence of ground state solutions to quasilinear elliptic equations.

Samara oblong-elliptic to oblong-obovate; branchlets without a corky layer.

Female flowers: outer perianth lobes broadly ovate, inner ones elliptic.

stipules of cauline leaves broadly ovate or ovate-elliptic, free part longer than adnate part.

Leaf margin irregularly toothed, usually with large, elliptic, triangular, or narrowly triangular teeth (to 1 cm).

First-year juvenile branchlets brown, gray downy; leaf blade lanceolate or elliptic-lanceolate, base rounded.

Elliptic curve cryptography (ECC) is an important subject in the field of public key cryptography .

Petals free, anterior one narrower, all blue or purple, oblong, obovate, or elliptic.

Leaf blade oblong-elliptic to broadly ovate-elliptic. Corolla bright yellow.

The graphic unit includes line, ellipse, elliptic arc, circle, circular arc.

Leaf blade ovate to narrowly elliptic-ovate, base sagittate or hastate to shallowly cordate, basal lobes bluntly angular.

Seeds elliptic, flat, base and apex subrounded, ventral holes furrowed from base to apex.

Current year branchlets and terminal buds glabrous; leaf blade elliptic, oblong-elliptic, or obovate-oblong.

Branchlets and stems with yellowish elliptic lenticels, densely rust colored pubescent.

Leaf blade obovate, broadly elliptic, or orbicular, base rounded or obtuse.

Petals lilac, rarely white, oblong-elliptic, shorter than sepals.

Calyx teeth obsolete. Petals white, long-elliptic, apex inflexed.

Leaf blade papery or subleathery, narrowly elliptic or elliptic-lanceolate, netting veins not obvious.

leaf blade linear or elliptic-linear, entire or obscurely denticulate.

anthers dehiscing by introrse-terminal elliptic pores.

Specifies an Elliptic Curve Diffie-Hellman (ECDH) public key for use with the ECDiffieHellmanCng class.

In this paper, some new conclusions about the properties of elliptic paraboloid are introduced.

leaf blade narrowly elliptic to almost needle-shaped, margin entire to incised.

Calyx teeth obsolete. Petals white or yellow, ovate or elliptic, apex incurved.

Sepals early deciduous, yellow-white, elliptic, pubescent abaxially.

Leaf blade elliptic to narrowly ovate; corolla whitish to yellow, rarely pale blue.

Tendrils palmately branched. Compound cyme. Seeds spheroid; hilum elliptic at back, surface with netlike ornamentations .

Anthers ovoid-elliptic, poorly developed in female flowers.

The process of elliptic parts with CNC lathe was taken as an example to explain the reasons of random errors and the ways to eliminate them.

Using bilinear pairings of the elliptic curves or hyperelliptic curvers, presented an ID-based proxy blind signature scheme.

the detailed structure of angle brace, spring plate, elliptic hole, support, connecting plate, beveling and manhole etc.

Leaf blade ovate, broadly ovate, or narrowly elliptic-ovate, abaxially tomentose, base rounded or cordate, lobes apically shortly acuminate.

In the paper, based on the FPGA chip, the realization and optimized design of the elliptic curve cryptography are studied.

Leaf blade obovate-elliptic, glabrous from beginning.

Seeds obovoid, apex rounded and retuse, chalazal knot ovate-elliptic, ventral holes obelliptic.

Petals reddish, narrowly ovate or elliptic, subequaling sepals, apex mucronate.

ligule entire; leaf blade oblong, elliptic, or lanceolate.

Branches brown to gray, glabrous, with pale elliptic lenticels.

Staminode limb broadly obovate or broadly elliptic, entire, emarginate, or 2-lobed, yellow barbate.

A relative motion model, considering the J_2 perturbation and suitable for motion in general elliptic orbit, was used.

Optional Shapes: Circular, doughnut, waisted , elliptic and square, etc.

Leaf blade elliptic to oblong-elliptic, margin distinctly triangular-serrulate; corolla tube ca. 4 mm.

In this paper , an actual method is made to fulfil high - speed precision grinding of elliptic parts based on PMAC time - based control .

Elliptic Curve Cryptography (ECC) has the highest safety strength of private key per bit in the Public-Key Cryptography recently.

bracts purple or purple-blue, elliptic, pinnatipartite, apex somewhat obtuse.

leaf blade elliptic, ovate-elliptic, or elliptic-lanceolate, margin serrate, with 1 gland at serrate tip.

Floral bracts elliptic, early deciduous.

Trees or shrubs. Leaves cordate or ovate-elliptic, entire or serrate, sometimes palmately lobed.

Leaf blade broadly ovate to elliptic, dark brown when dry; bracts awl-shaped.

Leaves elliptic or elliptic-lanceolate, apex caudate; bracteoles spiny, 2-parted at base; pseudostaminode serrulate at apex.

Results The shape of cysts were elliptic, irregular and string shape.

lobes lanceolate to oblong or elliptic, entire to serrate or incised.

Leaflets ovate or elliptic, apex acute or shortly acuminate.

Leaflets ovate to elliptic, apex acute or shortly acuminate.

Tendrils bifurcate; leaves oval or ovate-elliptic.

The relative analysis and resolving usually lead to the elliptic integral which is difficult to be resolved.