To the editor, recently, a very interesting study on errors in search strategies used in systematic reviews was published in the Journal of the Medical Library Association  OR “chronicintervillositis”[tw] OR “chronic histiocyticintervillositis”[tw] OR “histiocytic intervillositis”[tw] OR“intervillositis”[tw]But it could well be that upon judging the retrieved references, a decision would be made to do the opposite . As it is not easy or even possible to predict whether this will happen, it is, in my view, best to keep these redundancies in, just to be sure.So, as most search strategies will be revisited in the future, be it after a few months or sometimes years, these redundant terms and variations play a role in further developing a search and deciding whether to add or delete terms. As Salvador-Oliván et al. write, these “search errors” do not affect recall or negatively affect information retrieval with respect to either recall or precision. They do play a positive role in the composition of the search strategy.Jan W. Schoones,j.w.schoones@lumc.nl, http://orcid.org/0000-0003-1120-4781, Collection Development Expert/Information Specialist, Walaeus Library, Leiden University Medical Center, P.O. Box 9600 C1-Q, 2300 RC Leiden, The Netherlands


Nature Communications 10.1038/s41467-019-08625-z, published online 08 February 2019Correction to: y-axis were given in the wrong order: ‘800’ at the bottom through to ‘0’ at the top. This has been corrected in both the PDF and HTML versions of the Article.The original version of this Article contained an error in Fig. 2c, in which the numbers on the Also, the Source Data file initially published online was corrupted and was replaced.


G-representation contains a rigid type. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.We prove that for every Bushnell–Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. Moreover, we show that every irreducible smooth  F denote a non-Archimedean local field with finite residue field. Let G be the F-points of a connected reductive algebraic group F. Fix a Haar measure G. Let G, i.e., G-representation and this induces an equivalence between the category G-representations and the category of nondegenerate Let K in G. Then e as its identity element. Let G-representations and let e is self-adjoint, i.e., G  with the Plancherel measure G-representations. Denote by S of Let l et al.  give an G-representations with unitary G-representations.) In complete generality, this is clearly false, as seen, for example, by taking V is any irreducible representation with a It is natural to ask if in fact  induces e, define e is special if G-subquotients. This is equivalent K-signaturoved in , 19. We ras from .K-signature character of a Hermitian G-representation and adapt Vogan’s signature theorem to our setting. The main result of the section is Theorem We give a brief outline of the paper. In Sect. K-character and signature K-character for an admissible G-representation K is a compact open subgroup, and explain an analogue of a formula of Vogan for the signature character [In this section, we recall the definition of the multiplicity entclass1pt{minimaharacter .p-adic group G. A star operation V such thatV is Hermitian if it has a nondegenerate We begin by recalling the idea of the Grothendieck group of modules with nondegenerate Hermitian forms from . As in tK is a compact open subgroup of G. Every smooth irreducible K-representation K-invariant form K-representations. If V is admissible, denote by V, and byV has an invariant Hermitian form, let V is admissible, let V is irreducible, V has a nondegenerate Suppose entclass1pt{minimaU is an U in V with respect to If R is any admissible module, define the hyperbolic form on If V is a smooth admissible U is an V and R is the radical of the form U, we haveK, we have a well defined signature homomorphismFun denotes the set of functions.[L ranges over the irreducible Hermitian The following result expresses the Hermitian form of an admissible module in terms of the Hermitian forms of its composition factors.As in , a betteentclass1pt{minimaV can be written uniquely in K, the K-signature character of V is[V is positive definite fi the coefficient of s in the signature is zero.In this notation, a nondegenerate Hermitian form F. Let F. We will denote by F-points of F-points of P, A) is defined by a standard parabolic F-split torus in the center of P, A), we regard G. Let i.Fix a minimal parabolic subgroup c. As in , §XI.1, G, denote by N-coinvariants). Let M, an open normal subgroup of M. Let R(G) the Grothendieck group of admissible G-representations. We may regard For a standard parabolic subgroup M-representation, and G-action given by right translations.If P, A) is a standard pair, p-adic case, and summarizes [A Langlands datum is a triple P.g on compacta, and it is an intertwining operator with respect to the G-actions.The integral operator Let G-representation. Then there exists a Langlands datum Let Suppose Let G-representations is well known. The p-adic case, see for example [From now on, we restrict to the case of Hermitian representations with respect to the natural star operation  example , §5, is A in G, such thatw in the normalizer Let M be a Levi subgroup of G with the subgroup M.We recall the definitions of the Bernstein center and the inertial equivalence class. Let M is a Levi subgroup and M. Define the relation of equivalence such pairs: G.Consider the set of pairs Define the weaker relation of inertial equivalence on the same pairs: As it is well known, if M of a standard parabolic subgroup P, let The Bernstein center G). From , one dec by \ies, see , §2.4.loc. cit. that for every e such that We call the inertial class mentclasspt{minimasults of , 3.6) saK be a compact open subgroup. The goal is to express the K-signature of an irreducible smooth G-representation in terms of that of tempered modules. A first step is the simpler result for K-multiplicity functions.Let G-representation in G-representations K,Let G-representations, all in Set Suppose Now let p-adic analogue.We summarize the necessary results about the Jantzen filtration and the signature character, see , §3, 2,,2, §14, X. Explicitly, let K, the restriction of K is independent of t.Let X such thatX;n at for all Define a Hermitian form The Hermitian forms [G-representations.K-signature.The radical of the form For small For small Equivalently,K-signature of With the notation as above, the Jantzen filtration Before we get to the signature theorem, we need a lemma which will insure that the inductive algorithm terminates  Let n∈W (see ) such thloc. cit., except that we need to explain in our setting the independence of K, the Bernstein component The proof is the same as in V in terms of the Langlands classification as V is:Write ,m.As in , 3.38, tY which have a nondegenerate Hermitian form, then it holds for Y itself. This is because of Proposition One remarks, as in [V as well, by  denotes the G-representations.Recall that G . The Hecke algebra K-biinvariant functions, where K ranges over a system of good compact open subgroups . For exy–Prasad  filtratiDefine P ranges over the set of parabolic subgroups, (cf. ) Let \doG is semisimple, this is the same notion as that of the rigid quotient from [In the case when ent from .G-representations.The space R(G) is spanned by the Langlands standard modules G. By the Langlands classification, The elliptic representation space is example .G-representations. LetLet  example , §2.2.p-adic groups proved in [The following result is essentially the abstract Selberg principle for roved in .(see [The map K ranges over the compact open subgroups of G and The space We will not need these finer results.More is known about d in the  thatThe (i)M-representation For every standard Levi subgroup (ii)K such that There exists a compact open subgroup Denote by R(G). The following is the classical trace Paley–Wiener theorem.As in , §1.2, a The mapU(G) denote the inverse limit algebraU(G) has the equivalent description as the convolution algebra of essentially compact distributions:U(G) by settingU(G). The Bernstein center U(G). This also gives an action of R(G) via infinitesimal characters (on each irreducible representation), see Sect. We need to look at the action of the Bernstein center. Let is known , see alsis known , the alg[For every idempotent element Here, we think of For every Bernstein component The map The first part is immediate from Theorem R(G) spanned by the irreducible objects in Now suppose that V not in If The restriction of the trace map The category mentclasspt{minimaLet rigid if the composition of the natural mapsWe say that a pair (not necessarily a type) In Sect. We are now in position to prove Theorem Firstly, we can sharpen Theorem G-representation with inertial support V admits a K, the K-signature of V is(The signature theorem II) Let Notice that, in this version, we cannot say that the scalars K-character of every irreducible tempered representation K) of In light of Theorem p-adic groups, we may decompose ).As in , §4.4, d[W is a unitary In particular, we have a natural bijection between the unitary modules for the two algebras. Finally, the correspondenceWe retain the notation from Sect. I be an Iwahori subgroup and Let If I. Since G-conjugacy classes of maximal compact open subgroups of G, the natural mapLet K-type of positive level is rigid.A Moy–Prasad unrefined minimal This is immediate from Proposition K-type of positive level is a Bushnell–Kutzko type. This proves Theorem G-representations which contain a Moy–Prasad type of positive level. By [G-representation contains a level zero type.By [loc. cit. for the necessary structural results from Bruhat–Tits theory, in particular, about parahoric subgroups. Fix an Iwahori subgroup I, and recall the Bruhat decompositionW is the Iwahori–Weyl group and G of w. Let I and W of elements of length zero. Let W generated by the reflections W(J) is a subgroup of W.We now look at the case of level zero types –34. We rA level zero type is a pair w of minimal length in w has finite order in W. Therefore the compact cocenter of w has finite order, the subgroup W is finite. By the Bruhat–Tits fixed point theorem [G and The main result of  is a desEvery level zero type Notice that now the proof of Theorem As shown in , §7.3, tputed in , §7.11. Here The algebra p-adic group G is equivalent with the classification of the unitary dual of the Hecke algebras in Theorem G is adjoint and In other words,  Lusztig  gives a When the cocycle lable by , 25.G. In this case, there exists a unique conjugacy class of maximal compact open subgroups. Let Suppose now that D is a F-algebra, GL) and [If  work of  (for GL( F)) and  (in gene


Previous research has indicated that adults have a future-oriented cognitive bias, one illustration of which is their tendency to report more thoughts about the future than the past during mind-wandering. We examined whether children showed a similar bias, and whether there were any developmental changes in the magnitude of such a bias. Children aged 6–7 and 9–10 years, adolescents, and adults completed two tasks in which they could report either past or future thoughts: a mind-wandering task assessing spontaneous past and future thinking and a cued episodic thinking task in which they were free to describe either past or future events. Only adults showed a future-oriented bias in the mind-wandering task. Participants in all groups were much more likely to describe past events in the cue word task, and the proportion of future events described did not change developmentally. However, more than a third of the youngest age group produced no descriptions at all of future events, which was a significantly larger proportion than in any other age groups, and illustrates the difficulty that some children of this age have with future thinking. Our findings indicate that future-oriented bias and developmental changes in such bias may be task-specific. Research on thinking in adults has suggested that there is what has been termed a “future-oriented” or “prospective” bias . Studies with adults suggest that the rated clarity of remembered events might be greater than imagined future events , alongside an adolescent group (aged 14–15) and an adult group. Use of these age groups allowed us to examine the development profile of spontaneous and cued past and future thinking in detail. In addition to the mind-wandering and word-cue tasks, participants completed a task assessing working memory capacity. The latter task was included because of previous suggestions that future-oriented thinking in mind-wandering may make more demands on working memory resources than past-oriented thinking , thirty one 9- to 10-year-olds , twenty three 14- to 15-year-olds  and thirty-seven adults . All participants were tested in the developmental laboratory in the home department of the first author. Two participants from the youngest age group did not contribute data to the word cue task due to equipment error and experimenter error, respectively. One further participant from the youngest age group refused to give any responses during the digit span task. One participant each from the 14- to 15-year-olds, and adult age groups did not contribute data to the mind-wandering data task due to equipment failure. Adult participants were primarily recruited through notices across the university, and were paid £15 (UK) for their participation. Child and adolescent participants were primarily recruited by advertising in social media and through information distributed in schools.There were four groups of participants in the study: forty two 6- to 7-year-olds , and asked to try to match them to specific faces. This was to encourage them to use faces that were not at the ends of the scale. After completing the pre-training, participants began the word cue task.Participants were introduced to the word cue task by the experimenter explaining that they would see some words on a screen and that each time they saw a word they had to think about something that had already happened or something that was going to happen that was related to that word. The experimenter then gave two examples and also demonstrated the ratings that had to be made. The word “ball” came on the screen, and the experimenter described an event s/he had participated in that involved playing in a netball competition and losing a game when the other team scored late in the game. This event was described as “a bit annoying” by the experimenter, because valence was one of the event properties to be reported. The experimenter then said “This is something that has already happened, so I need to touch the Past button.” Getting participants to explicitly report whether the event was in the past or future functioned to prepare them for making similar judgments in the mind-wandering task. The experimenter then showed participants the speed of retrieval scale and explained it to them as a way of reporting how quickly something came to mind. The experimenter selected the tortoise button to match the speed at which the netball event was retrieved. The clarity scale appeared on the next display, and the experimenter explained that this could be used to report how clearly the event looked in your head, selecting the fourth picture on the scale for the netball event. Finally, the emotion scale appeared, and the experimenter chose the third face from the left (“a bit sad”). After this initial demonstration, a second word appeared on the screen, the word “cake”. The experimenter then described an event that was going to happen next week, which was a party for his/her mother for which s/he was going to bake her favorite chocolate cake, and then moved through the screens reporting on this event, pressing the “future” button on the first display. Before moving on to the task itself, descriptions of each of the scales were reiterated.https://corpus.byu.edu/bnc/). The cues were presented in a randomized order and included the following words: book, game, pet, song, friend, bicycle, family, rain, winter, and park.Following this introduction and demonstration, participants completed the task themselves. In each of ten trials, a cue word appeared on the screen  and participants described either an event that had happened or that was going to happen “just one time”. If participants described a repeated or ongoing event, they were reminded that the experimenter wanted them to talk about an event that had happened or was going to happen just one time. Participants then made the speed, clarity, and emotion judgments for each event before moving on to the next cue. Cue words were all nouns; the words that were selected all had neutral or mildly positive emotional significance, an age of acquisition rating between one and four years, and were of high frequency in the British National Corpus , 3 of the 9- to 10-year-olds (10%), and 1 of the adolescents (4%). The majority (8/12) of these participants responded to 9 out of 10 cues, and the lowest number of cues responded to was 6. Only data for the cues for which these participants produced a response were included in the analyses.F = 2.26, p = .09, ηp2 = 0.05. Because of the relatively low numbers of future events described in all groups, we also examined the proportions of participants in each group who produced no future event descriptions at all; 14 (35%) of 6- to-7-year-olds, four (13%) 9- to-10-year-olds, two (9%) 14- to-15-year-old, and 3 (8%) adults never produced descriptions of future events; chi-squared analyses showed that the proportion of participants who produced no future events varied by age group, χ2  = 12.40, p < .01. A greater proportion of 6- to 7-year-olds produced no future events than of either 9- to 10-year-olds, χ2  = 4.51, p = .03, 14- to 15-year-olds, χ2  = 5.33, p = .02, or adults, χ2  = 8.08, p < .01. There was no significant difference between the three older age groups in the proportion of participants who produced no future events .The first analyses examined the proportion of the described events that were from the past versus the future. Figure F = 1.51, p = .22, ηp2 = 0.01, however, there was an interaction between temporal focus and age group, F = 3.95, p = .01, ηp2 = 0.10. The interaction was explored with four paired samples t tests. Future episodes were judged as coming to mind quicker than past episodes in the adult group, t(33) = 2.79, p < .01. There was a marginal effect in the same direction for the adolescent group, t(20) = 2.03, p = .06. There was no difference between past and future speed estimates for either of the child groups . Figure F = 12.26, p < .01, ηp2 = 0.11, with past events reported as being clearer in the mind (M = 4.60) than future events (M = 4.15). Finally, Fig. F = 25.49, p < .01, ηp2 = 0.20.In subsequent analyses, we looked at the other measures in this task as a function of temporal focus and age group; these analyses excluded participants who did not describe any future events. Figure F = 1.04, p = .38, indicating that there was no developmental change in the likelihood that participants were task-focused. To examine mind-wandering, an ANOVA was then conducted on absolute numbers of thoughts with a between-subjects factor of group and a within-subjects factor of temporal focus . There was a main effect of temporal focus, F = 10.92, p < .01, ηp2 = 0.08, with a greater number of future (M = 2.38) than past (M = 1.84) thoughts reported, and an interaction between age group and temporal focus, F = 5.19, p < .01, η2 = 0.11. Subsequent paired t tests showed that only the adult group produced more future than past thoughts, t(34) = 5.56, p < .001. One-way ANOVAs showed that the effect of age on the number of future thoughts reported did not reach significance, F = 2.54, p = .06, but there was a significant effect of age on the number of past thoughts reported, F = 5.30, p < .01, with post hoc tests showing that the adults produced significantly fewer past thoughts than either the 6- to 7-year-olds or the 9- to 10-year-olds, both ps < 0.05.Figure F = 6.43, p < .01, with post hoc Tukey tests showing that the adult group reported a higher percentage of mind-wandering episodes to the future than all of the other groups , who did not differ from each other.A final analysis examined whether there was a developmental change in the percentage of mind-wandering episodes that were to the future . A one-way ANOVA found a significant effect of age, F = 24.18, p < .01, ηp2 = 0.36, and direction, F = 189.65, p < .01, ηp2 = 0.60, but no significant interaction between age and direction. Post hoc Tukey tests revealed that average digit span (the average of forward plus backward) of 9- to 10-year-olds was significantly greater than of 6- to 7-year-olds. Average digit span of 14- to 15-year-olds was significantly greater than of 6- to 7-year-olds and 9- to 10-year-olds. There was no significant difference between adults’ total digit span and that of 14- to 15-year-olds.An ANOVA on scores on the digit span task with a between-subjects factor age group and a within-subjects factor of direction (forward versus backward) revealed a main effect of age, p values > 0.1).The final analyses examined the correlations between measures. We examined, for the sample as a whole, whether there were correlations between the percentage of mind-wandering episodes that were future-oriented, the overall number of mind-wandering reports (summed across past and future), the proportion of future events described in the word cue task, and digit span performance. None of the correlations were significant  of participants’ event descriptions were of past events.Our mind-wandering task was somewhat unusual, in that, for the reasons described in the introduction, mind-wandering was assessed during a task that primarily involved motor skills rather than higher-level cognitive abilities. Under these sorts of circumstances, it would be expected that a future-oriented bias would be particularly large in adults, and indeed they reported around twice as many episodes of future-oriented mind-wandering than past-oriented mind-wandering. Strikingly, though, we did not see this pattern in any of our other groups. Although the overall number of times that participants reported future thoughts did not change developmentally, there was a developmental change in the percentage of mind-wandering episodes that were future-oriented relative to those that were past-oriented. Notably, the age pattern that we observed could not be explained by the differential availability of working memory resources in the groups; we found no relation between the percentage of mind-wandering reports that were future-oriented and working memory capacity, unlike Baird et al. , but repOur findings are the first to suggest that there are developmental changes in future-oriented bias in mind-wandering. However, the fact that none of the age groups other than adults showed a future-oriented bias stands in contrast to the findings of Ye et al.  and ZhanSuch an explanation, though, is at odds with the idea that levels of future-oriented mind-wandering decrease as a function of the extent to which the primary task recruits working memory , we did not ask participants to describe the contents of their mental states, making it impossible to be completely sure that their self-reports accurately reflected the temporal focus of their thoughts. However, we believe it is unlikely that the data could be explained by failure to understand the task. First, children had already been trained to report on the temporal focus of their thoughts (past versus future) in the word cue task, and were able to do this accurately . This is consistent with other studies that indicate that by 6–7-year-old children can reliably report on the deictic status of events  was not found in the child/adolescent groups. Thus, our findings suggest that in the context of spontaneously produced thoughts about other times, the future is less dominant than the past in the minds of children and adolescents than adults. Taken together, these findings indicate that there are some developmental changes in future thinking, and that even adolescents may not resemble adults in the degree to which they focus on the future relative to the past.


AbstractVelellavelella) for Ecuador. Twelve individuals were found along different beaches of San Cristóbal and Santa Cruz Islands in Galápagos Archipelago, Ecuador. These sightings may be influenced by El Niño Southern Oscillation events.We present the first official record of the by-the-wind-sailor ( Velellavelella  is a holoplanktonic athecate hydroid (Hydrozoa: Anthoathecata) from the Porpitidae family that is well known as “by-the-wind-sailor” , its common distribution is in the northern hemisphere in the Pacific and Atlantic Oceans, as well as in the Mediterranean Sea . On 28 August 2017, approximately eight individuals of V.velella were spotted on La Lobería beach  at San Cristóbal Island. We photographed one individual that corresponded to a “right-by-the-wind-sailor” due to the direction of its sail , a beach at Santa Cruz Island. Although no exact counts are available, at least four different individuals were photographed  and Physaliaphysalis  washed up along with V.velella (Fig. Southwest orientated, Tortuga Bay is a dissipative beach with a gentle slope exposed to southern swells. These features make Tortuga Bay a trap for drifting organisms when winds hit from the south. In this respect, we also observed several specimens of lla Fig. . These mlla Fig. .V.vellella in Ecuador. We suggest that they are likely uncommon in the Galápagos Archipelago. V.velella arrived at the Galápagos (To the best of our knowledge, this is the first official record of alápagos .


In the last June 2017, Dr. Martin Buchheit alerted the Sport's community with an editorial entitled “Houston, we still have a problem,” , although professional teams spend a lot of money using products without evidences  and psychological variables  which are influenced by both training and non-training stressors  and exogenic factors . Among all endogenic factors, biological sub-factors are the significant predictors of happiness  and a significant predictor of successful performance (Uphill et al., Time will tell, but we must remember that once again, we think this applies to all jobs within social services as we are working with people and impacting on their lives. Therefore, we could say: “Houston we still have a problem” but we could add., “Remember that “We are physicians, but first and foremost, human beings”” Kimura, .JC-G and NT: First Idea; RM-A: Review; CL-P: Coauthor; IJ: Review; JM-A: Writer; DM-J: References; AD: Language; SO: Main director.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Nature Communications 10.1038/s41467-019-10380-0, published online 06 June 2019Correction to: The original version of this Article contained an error in the spelling of the word “adaptation” both in the Title and Abstract, which was incorrectly written as “adaption”. This has been corrected in both the PDF and HTML versions of the Article.


Nature Communications 10.1038/s41467-019-09301-y, published online 27 March 2019.Correction to: In the original version of this Article, the legend in the upper panel of Fig. 2 incorrectly read ‘paternal imprinting’ and should have read ‘maternal imprinting’. This has been corrected in both the PDF and HTML versions of the Article.


The data were used to obtain the limiting apparent molar volumes, limiting apparent molar compressibilities and the corresponding transfer parameters. Limiting apparent molar expansibilities and their second order derivatives and volume interaction coefficients were also estimated. These parameters are discussed in terms of d-sorbitol and co-solute (amino acid or sodium chloride) interactions in aqueous solutions.Apparent molar volumes and apparent molar compressibilities for  In recent years, sugar alcohols, owing to their desirable properties, have found applications in many branches of industry. Properties of polyols, such as good taste, low calorie content and little effect on blood sugar levels have paved the way for the use of these compounds in the production of prepared foods, cosmetics and pharmaceuticals –3. It had-sorbitol, a six-carbon sugar alcohol, is widely used as sweetening agent, plasticizer, as well as capsule and tablet diluent [d-sorbitol, despite having a structure similar to that of its isomer d-mannitol, shows a real difference in solubility in water and in osmotic pressure coefficients [d-sorbitol is a relatively stronger disruptor of the water structure than d-mannitol [Among various polyols,  diluent , 7. It ificients . It has mannitol .d-mannitol and xylitol [d-sorbitol and amino acids. Density and sound velocity data at 298.15 K for some ternary mixtures (amino acid + d-sorbitol + water) have been reported by Jha and Kishore [d-sorbitol solutions for amino acids and they postulated the existence of hydrophilic/polar group interactions in the systems. The same conclusion was reached by Banipal et al. who studied volumetric properties of d-sorbitol in aqueous solutions of l-glycine [To the best of our knowledge, reports on the effects of amino acids on the structure of water modified by sugar alcohol are still scare. Most of the thermodynamic data have been collected for systems with  xylitol –15. So f Kishore . Moreove Kishore . The aut-glycine .d-sorbitol in aqueous amino acid solutions using volumetric and acoustic measurements. The densities at T =  K and sound velocities at T = 298.15 K for d-sorbitol in  mol·kg−1 aqueous solutions of l-alanine, l-cysteine and l-histidine are reported. Additionally, corresponding data for d-sorbitol in aqueous solutions of NaCl was also collected. The paper presents calculations of the apparent molar volumes and the apparent molar compressibilities, as well as their limiting values. The corresponding transfer parameters and limiting apparent molar expansibilities are also estimated. The evaluated parameters were correlated and interpreted in terms of the different types of interactions that can occur in ternary systems. The structure making/breaking tendency of d-sorbitol in aqueous amino acid or NaCl solutions is also analyzed.The present study was aimed at providing some additional data on the properties of d-sorbitol (≥ 0.98), l-alanine (≥ 0.99), l-cysteine (≥ 0.99) and l-histidine (≥ 0.99) were obtained from Carl Roth GMbH + Co. KG, sodium chloride (≥ 0.999) was purchased from POCh and were used without further purification. Table d-sorbitol and all amino acids and NaCl were dried under reduced pressure at 323 K. Deionized, doubly distilled, degassed water with a specific conductance of 1.15 × 10−6 S·cm−1 was used for the preparation of  mol·kg−1 solutions of l-alanine and l-cysteine, whereas in the case of l-histidine only three solutions  mol·kg−1 were made due to its lower solubility. All the solutions were prepared by weight dilution of the stock solution of d-sorbitol using a Mettler Toledo balance with a precision of ± 0.0001 g.−3 kg·m−3.The densities of the ternary mixtures were measured at different temperatures using a digital vibrating-tube analyzer , equipped with a built-in solid-state thermostat that controls the temperature by means of a combination of thermoelectric Peltier elements and an intergrated Pt-100 resistance thermometer with an accuracy of 0.01 K. The measurement cell in the apparatus is made of borosilicate glass. Prior to each series of measurements, the apparatus was calibrated using doubly distilled and degassed water and with dry air at atmospheric pressure (0.1 MPa). The standard uncertainty in the density measurements was within ± 35 × 10−1 using the sound analyzer OPTIME 1.0 from OPTEL (Poland), based on the time of flight method. Measurements are based on the determination of the time that the acoustic signal with a frequency of 8 MHz takes to pass through a sample of known length. The length of the quartz measurement cell was determined with doubly distilled water, with the value 1496.69 m·s−1 used as the sound velocity in pure water at 298.15 K. Temperature during measurements was stabilized by a constant temperature bath (model: PolyScience 8202) at 298.15 ± 0.01 K.The sound velocities were determined with a standard uncertainty of 0.15 m·sd-sorbitol in aqueous l-alanine, l-cysteine, l-histidine and NaCl are reported in Tables d-sorbitol and also with the molality of amino acids or sodium chloride. Irrespective of temperature, the increase in density resulting from increased concentration of d-sorbitol is greatest in the case of solutions in water. The greater the amount of amino acid or NaCl in the solution, the smaller is the increase. Moreover, with increasing temperature the observed effect, i.e. the increase in density with increasing concentration of d-sorbitol, becomes less pronounced. For all the systems studied, the density decreases with an increase in temperature.The experimental values of molality, density and apparent molar volume at various temperatures of the solutions of ϕV of d-sorbitol in the ternary systems were calculated from the densities of the solutions using the following equation:m denotes the molality of solution, d and d0 are densities of solution and solvent, respectively, and M2 is the molar mass of the solute.The corresponding values of the apparent molar volumes d-sorbitol in 0.3 mol·kg−1l-alanine at temperatures between 288.15 and 318.15 K. As seen from the plots, the relationship is linear across the whole d-sorbitol molality range and the whole temperature range studied. Therefore, the limiting apparent molar volumes d-sorbitol in water [d-sorbitol in aqueous 2 and 4 mol·kg−1 NaCl at 298.15 K reported by Jasra and Ahluwalia and the data for  ternary solutions obtained by Jha and Kishore and by Ren et al. [l-alanine in fixed concentrations of aqueous d-sorbitol, not d-sorbitol in fixed concentrations of aqueous l-alanine as in the present study. Moreover, different temperature ranges and distinctly higher concentrations of d-sorbitol were studied. Thus, direct comparison of the results is impossible.Figure in water . As is sn et al. , 17, 21.d-sorbitol is very small and therefore the standard deviations of the experimental slope SV are relatively high. However, a decrease of SV with temperature and co–solute concentration can be observed. The results indicate that increasing temperature and increasing molality of amino acid or NaCl make the solute–solute interactions weaker. Additionally, the values of the experimental slope for the same molality of co-solute and the same temperature follow the following order:d-sorbitol + NaCl + water) ternary system.As can be seen from Tables d- sorbitol are positive and increase with increasing molality of amino acid or sodium chloride, as well as with increasing temperature. The limiting apparent molar expansibilities A0, A1 and A2 are constants. As is seen, the values of the limiting apparent molar expansibilities for all the systems studied are positive. This suggests that the temperature increase causes the liberation of solvent molecules from the solvation layer of d-sorbitol, thereby increasing the total volume of the system. Moreover, the fact that the limiting apparent molar expansibility values decrease with rising temperature indicates that at higher temperatures this effect is weaker.An examination of the data also reveals that the limiting apparent molar volumes of d-sorbitol is a structure breaker in water, as well as in aqueous amino acid and sodium chloride [The calculated values of the second derivate chloride .d-sorbitol, l-alanine, l-cysteine, l-histidine and NaCl. The values of Figure Vint is the intrinsic molar volume, σ is the shrinkage in the volume due to hydrogen bonding between water and the hydrophilic solute and n denotes the number of hydrogen bonding sites [Vint consists of the van der Waals volume and the associated void or empty volume. For dilute solutions, it is reasonable to assume that the intrinsic molar volume is nearly constant with respect to concentration. Thus, the observed positive volume change accompanying the transfer of d-sorbitol from water to aqueous amino acid or NaCl solution may result from a decrease in the shrinkage in volume because of solute co-solute interactions. Additionally, the increasing values of the limiting apparent molar volumes of transfer with molality of NaCl or amino acid reveal that the interactions resulting in the dehydration of the sugar alcohol increase with the concentration of co-solute.It has been established by Shahidi, Farrell and Edward that the limiting apparent molar volume of a hydrophilic solute in water can be divided into two terms as:ng sites . Moreoved-sorbitol + NaCl + water) ternary system.As can be seen from Fig. d-sorbitol and l-alanine, l-cysteine, l-histidine and NaCl in aqueous solutions can also be explained on the basis of the co-sphere overlap model [hydrophilic–ionic interactions between hydrophilic groups of solute and ions of co-solute;hydrophobic–ionic interactions between the hydrophobic alkyl chain of solute and ions of co-solute;hydrophilic–hydrophilic interactions between the hydrophilic groups of solute and the hydrophilic groups of co-solute;hydrophobic–hydrophobic interactions between the hydrophobic alkyl chain of solute and the hydrophobic alkyl chain of co-solute;hydrophobic–hydrophilic interactions between the hydrophobic alkyl chain of solute and the hydrophilic groups of co-solute.The interactions between ap model . Accordid-sorbitol in aqueous amino acids all five types of interactions are possible, while in solutions of d-sorbitol in aqueous sodium chloride only two types of interactions, i.e. (1) and (2), can occur. Hydrophilic–ionic interactions and hydrophilic–hydrophilic interactions result in positive values of the limiting apparent molar volumes of transfer, whereas hydrophobic–ionic, hydrophobic–hydrophobic and hydrophobic–hydrophilic interactions result in negative d-sorbitol in aqueous NaCl interactions of type (1) and for d-sorbitol in aqueous amino acids interactions of types (1) and (3) are predominant.It is obvious that in solutions of d-sorbitol versus molalities of l-alanine and NaCl at 298.15 K together with the literature data for d-sorbitol in aqueous l-glycine and sodium chloride [d-sorbitol in aqueous l-cysteine and L- histidine solutions are not shown. As is seen from Figs. d-sorbitol displays highest positive values of the limiting apparent molar volumes of transfer in aqueous l-glycine, the smallest amino acid studied, while lowest values of l-alanine, the most hydrophobic amino acid studied. Comparison of the limiting apparent molar volumes of transfer of d-sorbitol between amino acids of the same alkyl chain length, i.e. l-alanine, l-cysteine and l-histidine, indicates that l-histidine is more hydrophilic than l-cysteine. This results in stronger hydrophilic–ionic and hydrophilic–hydrophilic interactions between d-sorbitol and l-histidine than those between d-sorbitol and l-cysteine, leading to more positive values of the limiting apparent molar volumes of transfer [Figure chloride , 21. Fortransfer .Fig. 4Thd-sorbitol in aqueous sodium chloride are consistent with that in literature. Moreover, the observed values of the limiting apparent molar volumes of transfer for that system are lower than those for (d-sorbitol + l-glycine + water). This is clearly the result of the strong hydrophilic–hydrophilic interactions between d-sorbitol and l-glycine which do not take place between sugar alcohols and NaCl.As mentioned above, the experimental data on Vab and Vabb are the pairwise and triplet interaction coefficients, respectively, and mb is the molality of the co-solute [d-sorbitol in aqueous L-aniline, l-cysteine, L- histidine and NaCl solutions obtained in the present study. As is seen, in all cases the pairwise interaction coefficients are positive while the triplet interaction coefficients are negative. In general, the magnitude of volume interaction coefficients decreases with rising temperature. The fact that the pairwise interaction coefficients are positive and larger in magnitude than the triplet interaction coefficients suggests that the volumetric properties of the studied solutions are determined by the strong interactions between d-sorbitol and the amino acids or sodium chloride and that these interactions dominate over triplet interactions. Moreover, the decrease of volume interaction coefficients with rising temperature indicates that pair-wise interactions, as well as triplet interactions, become weaker at higher temperatures.In line with the proposition of Friedman and Krishnan, according to which the thermodynamic transfer properties of solutes in dilute aqueous solution are determined by solute and co-solute interactions without the consideration of solute–solute interactions, and in conformity with the McMillan–Mayer theory, the limiting apparent molar volume of transfer can be expressed as:o-solute , 27. Tabd-sorbitol in aqueous l-alanine, l-cysteine, L- histidine and NaCl are reported in Tables d-sorbitol and water and intramolecular hydrogen bonds within d-sorbitol molecules themselves become stronger, resulting in a more ordered solution. Moreover, for solutions with the same molality of d-sorbitol, an increase in sound velocity with growing concentration of co-solute is also observed. This suggests that association among the molecules of the solution becomes stronger as the amount of amino acid or NaCl in solution increases.The experimental values of molality, sound velocity and apparent molar compressibility at 298.15 K of the solutions of ϕKS, of d-sorbitol in the ternary systems were calculated from the densities and the sound velocities of the solutions according to the equation:KS and The apparent molar compressibilities, KS was obtained from density and sound velocity values using the Laplace equation:The isentropic compressibility mentclasspt{minimad-sorbitol. Thus, the apparent molar isentropic compressibilities at infinite dilution were obtained by using the method of linear regression of the following relation:d-sorbitol in water obtained by Banipal et al. [d-sorbitol in aqueous amino acid or NaCl solutions is available for comparison purposes.The apparent molar compressibilities were found to increase linearly with the molality of l et al. . As is sd-sorbitol increase (become less negative) with the increase in molality of co-solute. Obviously, this is the result of the fact that when sugar alcohol is dissolved in aqueous amino acid or salt solution, interactions between co-solute and water and between water molecules themselves are disrupted and interactions between solute and co-solute and between solute and water are formed. Thus, with increasing concentration of co-solute, more water molecules are released into the bulk solution and the solution becomes more compressible.The values of the limiting apparent molar compressibilities of d-sorbitol from aqueous to aqueous amino acid or sodium chloride solutions according to the equation:The obtained values of d-sorbitol versus molalities of l-alanine, l-cysteine, L- histidine and NaCl. It can be seen that the l-alanine < l-cysteine < l-histidine < sodium chloride. The results obtained from acoustic studies are in accordance with the results from volumetric studies discussed earlier. They support the conclusion that the hydrophilic–ionic and hydrophilic–hydrophilic interactions in solutions of d-sorbitol in aqueous amino acids are predominant, while the properties of solutions of d-sorbitol in aqueous sodium chloride are determined by hydrophilic–ionic interactions. Moreover, the absence of hydrophobic–hydrophobic and hydrophilic–hydrophobic interactions in the ternary system (d-sorbitol + NaCl + water) results in more positive values of the limiting apparent molar properties of transfer.Figure d-sorbitol from water to aqueous solutions of l-alanine, l-cysteine, l-histidine and NaCl. Positive d-sorbitol in aqueous sodium chloride and the co-domination of hydrophilic–ionic and hydrophilic–hydrophilic interactions in solutions of d-sorbitol in aqueous amino acids. As the obtained values of the limiting apparent molar expansibilities are positive and decrease as the temperature rises, these dominating effects also increase with rising temperature.The limiting apparent molar volumes and the limiting apparent molar compressibilities of transfer have been determined for the transfer of d-sorbitol + NaCl + water) ternary system indicates that interactions between solute and co-solute in this system are stronger than in (d-sorbitol + amino acid + water) mixtures. Moreover, the order of d-sorbitol in aqueous solutions of l-alanine, l-cysteine and l-histidine, i.e. three amino acids with the same alkyl chain length, strongly suggests that the strength of alcohol sugar and amino acid interactions is determined by the hydrophobicity of amino acid.The fact that the highest values of the limiting apparent molar volumes and the limiting apparent molar compressibilities were obtained for the (d-sorbitol and amino acids or sodium chloride.The fact that the pairwise interaction coefficients are positive and of greater magnitude than the triplet interaction coefficients supports the conclusion that the properties of the studied solutions are determined by the strong interactions between


The Fokker–Planck equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This forms a natural generalisation of the transport-of-intensity equation. The x-ray Fokker–Planck equation can simultaneously account for both propagation-based phase contrast, and the diffusive effects of sample-induced small-angle x-ray scattering, when forming an x-ray image of a thin sample. Two derivations are given for the Fokker–Planck equation associated with x-ray imaging, together with a Kramers–Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample micro-structure. These equations may be applied to the forward problem of modelling image formation in the presence of both coherent and diffusive energy transport. They may also be used to formulate associated inverse problems of retrieving the phase shifts due to a sample placed in an x-ray beam, together with the diffusive properties of the sample. The domain of applicability for the Fokker–Planck and Kramers–Moyal equations for paraxial imaging is at least as broad as that of the transport-of-intensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons. A, and the “local converging lens” at C, the nominal focal length f for each lens being much larger than Δ. This is the key situation, of combined coherent and diffusive energy transport in a near-field paraxial imaging setting, that we wish to consider in the present paper. While we focus on the case of hard x rays, the methods considered here will also be applicable to paraxial imaging using visible light, electrons, neutrons etc.Suppose that coherent visible-light plane waves normally illuminate a thin lens made of glass that is slightly cloudy. For such a lens, a large percentage of the light will be coherently focused or defocused upon propagation beyond the lens. Conversely, a small percentage of the light will be diffusely scattered by the random cloudy inclusions within the glass. In the hard-x-ray analogue of this experiment, replace the glass lens with a thin semi-transparent sample whose large-scale projected structure locally resembles an aberrated converging or diverging lens see Fig. , and rep3. One of the most often-employed, and most elementary, means is to first use the projection approximation to quantify the passage of an incident coherent scalar x-ray beam through the sample, and then use the Fresnel diffraction formalism to propagate the resulting exit-surface wave field to the surface of a detector4. The projection approximation assumes that: (i) the radiation is of sufficiently high energy that individual rays within the sample  are unchanged within the volume of the said sample in comparison to the streamlines that would have existed in the absence of the sample, and (ii) both phase and amplitude shifts are accumulated continuously and locally, as one traverses any such ray from the entrance surface of the sample to the exit surface–see e.g. Sec. 2.2 of Paganin4. The Fresnel approximation, which is a consequence of the paraxial approximation, assumes that spherical Huygens-type wavelets can be approximated as having parabolic wave fronts. If the sample-to-detector propagation distance is sufficiently small, and if diffuse scattering is negligible, Fresnel diffraction implies that the transport-of-intensity equation5 (TIE) may be used to model the evolution of image intensity with respect to propagation distance downstream of the sample, for a paraxial quasi-monochromatic beam. The TIE, which is the continuity equation for paraxial wave fields, expresses local energy conservation6. It forms a basis for several imaging techniques in coherent x-ray optics7. Like the formalism of Fresnel diffraction upon which it is based, the TIE models what may be called “coherent energy transport” downstream of a thin sample. That is, at each point in the space downstream of the sample, there is one time-independent energy-flow vector . However, if there is unresolved spatially random micro-structure within the sample11, then, as hinted in our opening paragraph, small-angle x-ray scatter12 (SAXS) will also be present. This augments the previously mentioned coherent energy-flow vector at each point in space, with an ensemble (“SAXS fan”) of energy-flow vectors associated with diffusive energy transport. How can this additional effect be taken into account, in modelling paraxial hard x-ray imaging of thin objects?We make the simplifying assumptions that: (i) magnetic effects such as x-ray circular dichroism in the sample can be ignored; (ii) the sample is non-crystalline, allowing x-ray Bragg-diffraction effects to be ignored; (iii) the sample is sufficiently thin that dynamical x-ray diffraction effects, within the volume of the sample, can also be ignored; (iv) all polarisation-sensitive effects can be ignored. Many well-known means exist, for the quantitative modelling of an x-ray beam interacting with such a sample and subsequently forming a paraxial image as it propagates through space13. The latter equation, which can model flows in radiation and matter wave fields that have both a coherent and a diffusive component, may be obtained in a very general setting as a limit case of the still more general Boltzmann equation of classical statistical mechanics14. The Fokker–Planck equation is used in many fields: examples include Brownian motion14, hydrodynamics15, electron and photon transport in biological tissues17 and homogeneous media18, high-temperature atomic-level disorder in pre-melting surfaces19, gaseous micro-flows20, droplet nucleation21, dilute polymer dynamics22, hot plasmas23, biaxial fluids24, cold turbulent gas jets25 and quark gluon plasmas26. Notable also is the use of the Fokker–Planck equation in the non-imaging context of x-ray kinematical and dynamical diffraction from imperfect crystals with stochastically distributed defects28. An extremely recent application, which is particularly close to that developed in the present paper, uses Fokker–Planck concepts to model the position probability density of a nano-particle in a near-field optical trap29.One answer is provided by a more general equation than the TIE, namely the the Fokker–Planck equation30);Local absorption by the object ;Local lensing by the object which concentrates or rarefies energy density due to local focusing or defocusing (propagation-based Laplacian-type phase contrast31);Local prism-like effects of the object which transversely shift optical energy .The first three effects are associated with the “TIE part” of the Fokker–Planck equation, with the fourth effect associated with the “diffusion equation” part. Of particular interest is the use of an x-ray form of the Fokker–Planck equation for modelling forward problems in paraxial x-ray imaging in a simple manner, e.g. by expressions valid in the near field  the transport-of-intensity equation for coherent energy transport in paraxial monochromatic x-ray beams, with (ii) the diffusion equation for diffusive paraxial energy transport associated with small-angle x-ray scattering within a sample. The second derivation is a microscopic treatment based on first principles, which relies on the crucial assumption that the phase at the exit surface of the object can be written as a sum of (i) a component that varies slowly with respect to the detector pixel size, and (ii) a component that fluctuates many times over one detector pixel and is therefore unresolved. We then give a generalised form of the Fokker–Planck equation, known as the Kramers–Moyal equation, as a means for treating the in-sample SAXS in a more precise manner. Next we sketch some indicative means by which the Fokker–Planck and Kramers–Moyal equations may be used in the forward problem of modelling near-field intensity contrast in coherent x-ray imaging, together with the associated inverse problem of phase retrieval and SAXS determination. We also consider some broader implications of the Fokker–Planck formalism.Here we give two complementary derivations of the x-ray Fokker–Planck equation. The first is phenomenological, based on local conservation of energy in the presence of both coherent and diffusive paraxial x-ray energy transport. The second is a microscopic first-principles analysis. The former derivation is more general, while the latter is more physically illuminating.5 for coherent paraxial x-ray energy transport and then separately consider the diffusion equation for diffusive paraxial x-ray energy transport. We then show that these two equations may be merged, in a manner that conserves energy both locally and globally, to give the Fokker–Planck equation for paraxial x-ray imaging.X-ray beams “flow” in the sense that they stream optical energy as they propagate. As a descriptor of such energy flows, the Fokker–Planck equation may be viewed as a diffusive generalisation of coherent paraxial photon energy transport. Adopting this perspective, below we consider the transport-of-intensity equation5 describes the intensity evolution of a paraxial complex coherent scalar wave field as it propagates. This continuity equation, arising from the parabolic equation of paraxial scalar wave optics, expresses local energy conservation. It states that the negative divergence – which may loosely be spoken of as the “convergence” – of the transverse Poynting vector (energy flow vector) is proportional to the longitudinal rate of change of intensity5:I is the intensity of the monochromatic scalar wave field, ϕ is its phase, k is the wave-number and ∇⊥ is the gradient in  planes perpendicular to the optical axis z. Equation (The transport-of-intensity equationcoherent x-ray energy transport. The conserved (Noether39) current is the transverse Poynting vector, which up to a multiplicative constant is10As previously mentioned, the TIE is a continuity equation expressing local energy conservation under The corresponding constant Noether charge in vacuo through a distance Δ ≥ 0 that is sufficiently small for the Fresnel number to be much greater than unity, a longitudinal finite-difference approximation to the TIE can be used to model the resulting propagation-based phase contrast4:For propagation speculum). Rays passing through A have a positive divergence, hence the intensity at B is reduced compared to that which would have been registered if the rays passing through A had all been parallel to the optical axis z. Similarly, rays passing through C have a negative divergence (positive “convergence”), hence the intensity at D is increased compared to that which would have been obtained had the rays exiting C been parallel to the optical axis. This phase contrast effect is increased with increased propagation distance Δ, reduced x-ray energy (decreased wave-number k) or faster spatial variations in wave-field phase (larger D being slightly shifted from the point x = x0. Concentration or rarefaction of energy density (and hence intensity) may be thought of as due to the lensing term proportional to 31.The underpinning physics is sketched in an imaging context in Fig. ) in Eq. , while tD to describe the effects at distance z = Δ downstream of an illuminated thin object in the plane z = 0. The fact that this coefficient depends on x and y reflects the fact that the SAXS fan will in general vary with position over the nominal exit surface z = 0 of the thin sample. Bearing all of this in mind, the intensity evolution of the beam may then be described by the diffusion equation:If there were only diffusive energy transport on account of unresolved micro-structure in the sample (i.e. SAXS), but no specular refraction, we could instead introduce a diffusion coefficient D, since SAXS-induced diffusive blur has a characteristic transverse width that scales with Δ rather than Δ1/2.Note that a formal dependence on Δ has been added to the diffusion coefficient x illuminates a slab containing SAXS-inducing unresolved micro-structure. This width Δx is assumed to be large compared to the characteristic length scale of the projected micro-structure, but small compared to the characteristic transverse width associated with the macroscopic projected phase and attenuation coefficients of the object. The illuminated region E of the slab, located at transverse coordinate x = x0, results in a smooth SAXS fan of opening angle θ(x0), which smears the propagated intensity over the plane z = Δ. The characteristic width of this blur, which may be considered as a broadening of the local rocking curve due to the sample41, isThe effect shown in one transverse dimension in Fig. θ(x0) of the SAXS fan is taken to completely characterise its diffusive effects at each location x0; a more sophisticated treatment which replaces the single number θ(x0) (or its associated diffusion coefficient) with a hierarchy of tensors, will be given later.Thus, local diffuse scattering by a thin object leads to a transverse smearing of intensity upon propagation. This blur increases with increased propagation distance Δ. At the level at which diffusion is here being considered, the opening angle diffusive x-ray energy transport. Here, the conserved current is given by Fick’s first law as42Like the TIE, the diffusion equation may be viewed as a continuity equation expressing local energy conservation, albeit under F is the fraction of the optical energy converted to SAXS43 when illuminating a thin sample in the plane z = 0 at the point . If 11 and hard44 x-rays) one can replace 1 − F in the above expression by 1. The resulting convection–diffusion equation (forward Kolmogorov equation13) is the Fokker–Planck equation in two transverse dimensions. It models both effects in Fig. If both specular refraction and local-SAXS effects are present simultaneously, we can add the TIE and diffusion equations together, in an energy-preserving manner. This gives the continuity equation:mentclass2pt{minimF is slowly varying with x and y, we obtain the 2 + 1-dimensional Fokker–Planck equation:If the expressions for the coherent current In one transverse dimension, this becomes the one-dimensional Fokker–Planck equation:The corresponding finite-difference forms, which are to be preferred for reasons that have already been outlined, are respectively given by the following pair of equations:D may be considered to be sufficiently slowly varying with respect to transverse coordinates that D commutes with the transverse Laplacian, we may instead work with the simpler finite-difference expressions:We close this section by noting that, if FD plays the role of an effective diffusion coefficientIn the above two finite-difference Fokker–Planck equations, Here we derive the x-ray Fokker–Planck equation from first principles. We consider two length scales of sample-induced wave-field variation. The larger scale corresponds to the length scales in the sample that are sufficiently large for their absorptive and refractive effects to lead to detector-resolvable intensity variations. The smaller length scale corresponds to diffuse scattering from unresolved micro-structure resulting in SAXS. This derivation clarifies the physical mechanisms underpinning both coherent and diffusive x-ray energy transport, in a paraxial x-ray imaging context.ψ be the spatial part of the complex scalar amplitude at the exit surface of a sample. Subsequent Fresnel diffraction gives4PΔ(x) is the Fresnel propagator corresponding to free-space diffraction through a distance Δ ≥ 0, and paraxial radiation is assumed throughout. The squared modulus of the above convolution integral gives the propagated intensity as the following linear combination of contributions from every pair of points x′, x″ in the unpropagated wave field45:We work in one transverse dimension for simplicity, ignore polarisation and assume monochromatic x rays. Let z = 0. Many models exist for incorporating unresolved micro-structure into this sample47. Follow Voronovich47, Nesterets45 and Yashiro et al.49 in decomposing the phase ϕ of ψ asϕs(x) is a phase component that varies slowly with respect to the detector pixel size, and (ii) ϕf(x) is a fast-varying spatially random phase due to the sample micro-structure, which fluctuates many times over one detector pixel and is thus unresolved . Further assume a slowly-varying intensity Is(x), together with x-rays that are of sufficiently high energy for the projection approximation to hold for a given sample. Hence45:The sample is assumed to be thin, normal to the incident plane-wave illumination, and with nominal exit surface d so Eq.  becomes4ϕf(x)52. Denoting this average by an over-line,Average over an ensemble of realisations of the unresolved fast phase-maps ϕf to be spatially statistically stationary over transverse displacements that are small compared to the characteristic length scale over which ϕsfluctuates appreciably, the over-lined expression in Eq. (x′ − x″. Assuming ϕf(x − x′) and ϕf(x − x″) to be Gaussian variables, with zero mean, this correlation function is53x-dependent variance of the ensemble of fast phase maps {ϕf(x)}, and54 as a function of coordinate separation Δx and transverse coordinate x.Assuming the fast phase maps n in Eq.  depends et al.55).The quantity e of Eq.  does not51 in separating out the asymptote term from C is a re-normalised correlation functionτ is the transverse correlation length , if we 59 and the frozen-phonon model of electron diffraction theory. Note also that a non-zero specular-diffraction component  indicates the SAXS-induced speckle to be “partially developed”60; this may be contrasted with the case of “fully developed” SAXS speckle, which is not relevant to our context, in which the specular component would be fully extinguished.Note that a similar bifurcation of energy flows occurs, for similar reasons, in statistical dynamical x-ray diffraction theoryIs(x) to be sufficiently slowly varying that the square-root term in Eq. (Is(x). For the complex exponent in Eq. (ϕs(x) with respect to x; the above equation makes implicit use of the fact that ϕf varies much more quickly than ϕs. Then write PΔ(x′) in Eq. :Assume m in Eq.  may be rt in Eq. , make th) in Eq.  as a FouC in Eq.  as Fourix, integrates over a fan of rays making angles ϕf(x)}. The whole fan of SAXS rays is rotated by the angle x = x0, are integrated over a single pixel. See Fig. 63, while the fan rotation angle 64. The function 43, is proportional to the power spectrum of the ensemble of fast-phase fields 65); this power spectrum  g(cf. Eq. ).d is a diffusive blur kernel given byIf the object-to-detector distance Δ is large enough that the SAXS-induced fan of rays in Fig. d is simply a transverse re-scaling of kx  to a function of the transverse coordinates x over the detector plane. Stated differently, d is the intensity distribution of SAXS, as a function of position x on the detector, due to x rays incident at position x0. If the object-to-detector distance Δ is sufficiently small, two additional approximations can be made.(i)d to be sufficiently narrow for its Fourier transform x to be sufficiently broad that it can be represented by a second-order Taylor expansion in kx. The term in the Taylor series for kx will vanish since C is approximately even in Δx (cf. Eq. (x0-dependent blurring width (this blurring being due to SAXS), and Q is a positive constant. Invoke conservation of energy, which implies that the total scattered intensity, due to an incident x-ray pencil beam of intensity Is(x) and with small thickness Δx, will have the same integrated intensity after scattering from the object; see Fig. Q = 1, so that et al.67). Hence we may approximate Eq. . Given ⊗ in Eq.  is well mate Eq.  as:36\dod depends on x0. We use the convolution notation for clarity, however, to denote such smearing with a position dependent point spread function that results from the position-dependent SAXS. See also the next section, for an analysis that considers the same linear integral transform using a more conventional and explicit notation.Before proceeding, let us back-track a little, and point out that in Eq.  we consi(ii)5 may be applied to the first term on the right of Eq. 37) are sAssume only a small fraction of the incident x rays to be converted to SAXS, so that Upon comparing the final term of Eq.  with theDeff obeys Eq.  and (b);The link between D and L follows from Eq.  and (d);The link between D and θ follows upon eliminating L from relations (a) and (d);The link between D and Deff follows from Eqs (The link between from Eqs  and 41)D and Deffrom Eqs .We close this section with Fig. L is directly proportional to the angular spread θ of the SAXS fan; the blur width L is directly proportional to the sample-to-detector distance Δ; (b) the angular width θ of the SAXS fan is inversely proportional to the transverse length scale l of the projected unresolved micro-structure, etc.All physical dependencies, for the quantities in Fig. θ at each point over the exit surface z = 0 of a thin sample. Here, we show that a less-crude treatment leads to the Kramers–Moyal generalisation of the Fokker–Planck equation, in a form suitable for x-ray near-field imaging of thin objects.The preceding analysis gives only a crude treatment of the angular distribution of the local SAXS fan, which is described merely via its angular spread F of the x-ray beam which is channelled into SAXS. The two-transverse-dimensional forms of Eqs  an anF → 0, Eqs z = 0, whdΩ is an infinitesimal element of solid angle and θx, θy describe angular deflections (scattering angles) in the x, y transverse directions, respectively69. Hence conclude that 66 associated with unresolved micro-structure in the sample, namely a differential scattering cross section x′, y′), as well as upon the scattering angles θx, θy. Thus we have the proportionalityz = 0, superposes a micro-structure-independent near-field Fresnel diffraction pattern that is damped with a decoherence factor45 1 − F, with a diffusely-scattered signal et al.36; this amounts to a measurement of Recall the usual definition of a differential scattering cross section d in Eq.  is propoquations  and 43)\documentz = 0. In these same equations, Δ may be sufficiently large that the SAXS fan x′, y′, 0) over the exit surface z = 0 of the thin object, and considered as a function of coordinates  over the detector in the plane z = Δ—can have a width that is several pixels or more 43), the cf. Fig. . If thesthen Eqs  and (43)θ of all differential scattering cross sections contained in x, y), so that Eq. 43). Thisx′, y′) over the exit surface of the sample to produce an intensity distribution that is a function of detector coordinates  in the plane z = Δ—obeys:We have assumed that each SAXS fan F is a scattering fraction which accounts for the fact that only a certain percentage of the incident beam is converted to SAXS upon passage through the sample.This normalisation condition is appropriate since the multiplicative factor of We have assumed thatr in Eq. . This eld in Eq. . There iet al.67), which arise from terms of progressively higher order in the Taylor expansion of Is about :We have defined the following hierarchy of diffusion tensors  = 0 case of Eq.  and all higher-order diffusion tensors to zero, and assuming rotationally-invariant SAXS fans viaSetting D(3) and all higher-order diffusion tensors to zero, thereby reducing Eq.  may be obtained using Eqs ;;72 from Eschewing the Kramers–Moyal and Fokker–Planck approximations and instead working with Eqs  and 43)43) direcThe e of Eq.  is the c(see Eq. ). The fin in Eq.  if we caThe primary goal of this paper is to give several means by which the Fokker–Planck equation  may be used to model forward and inverse problems of near-field x-ray imaging, in the presence of both phase–amplitude shifts and non-negligible small-angle x-ray scattering from unresolved micro-structure in a thin sample.76, is studied in detail in a companion paper77 that gives a number of analytical expressions based on the finite-difference expression in Eq. , this being the fraction of the incident radiation that is converted to SAXS, may be brought into the second square brackets on the right hand side of Eqs 76, is ste of Eqs , and thee of Eqs , 29) an an76, ise of Eqs , in addi (i) Eqs  and 36)76, is st (i) Eqs  and 43)76, is st (i) Eqs  and 45)76, is stThe above indications, of the forward problem of near-field and intermediate-field x-ray imaging for a thin sample in the presence of both coherent and diffusive scattering, set up Fokker–Planck and Kramers–Moyal equations that can be used to consider the associated inverse problem of reconstructing intensity, phase and diffusion from measured intensity data. It is again beyond the scope of our paper to explore this point further, beyond the indicative suggestions given in the three examples below.72, and the scattering fractions F obtained using the same data. Since both k are known, the only remaining unknown in Eq. .As a first example of inverse problems that may be tackled based on the formalism of this paper, observe that the diffusion-tensor moments may be measured e.g. using the previously-mentioned technique of raster scanning a focused quasi-monochromatic x-ray probe over a lattice of points at the entrance surface of a thin objectn in Eq.  would be82. The technique of x-ray speckle tracking illuminates an object with a known resolved reference speckle field, and then measures one or more distorted forms of that speckle field which arise when a sample is placed in the illuminating speckle beam. The diffuse-scatter signal, considered in the present paper, arises naturally in x-ray speckle tracking – via unresolved sample-induced speckles that should be carefully distinguished from the titular illuminating speckles – and gives useful information that may be extracted using a variety of techniques. These techniques are typically based on the reduction in illuminating-speckle visibility due to SAXS-induced diffusion: see e.g. the review by Zdora83, and references therein. The geometric-flow approach to speckle tracking, which has recently been developed for x-rays87 but has also been very recently applied to visible-light imaging88, takes as a starting-point the following equation:A second application of the formalism of the present paper, is to the field of x-ray speckle trackingIR of a reference speckle image obtained in the absence of a sample, to the distorted/coded/encrypted form of that speckle image, denoted IS, obtained in the presence of a pure-phase-object sample; D⊥ is a transverse displacement-vector field, which distorts the reference speckle image IR into the speckle image IS obtained in the presence of the sample. The above equation, modelled on the continuity equation that has played such a dominant role in the present paper, conserves photon energy both globally and locally. It may be used to reconstruct the speckle displacement D⊥ induced by the phase object, and hence a map of the phase shift ϕ due to the object, in a simple and efficient manner that implicitly rather than explicitly tracks speckles87. Note that the displacement field is related to the phase shift of the object, viak is the wave-number of the x-rays. A Fokker–Planck form of Eq.  and effective diffusion coefficient Deff, is:This relates the intensity m of Eq. , suitablϕ and Deff rather than just ϕ. Of possible relevance in such a context – namely, of separating out the signal due to coherent versus diffusive energy flow – is the fact that Deff will transform as a scalar upon rotation of the sample through 180 degrees , whereas D⊥ will transform as a vector under the same rotation.This incorporates both coherent flow and diffuse flow, into the geometric-flow method for x-ray speckle tracking. It would be interesting to investigate an augmented form of the geometric-flow x-ray speckle-tracking method, which takes Eq.  rather t489. The associated unresolved micro-structure is left arbitrary, and the exit surface of the sample is assumed to correspond to the plane z = 0. Further assume normally-incident unit-intensity illumination by z-directed quasi-monochromatic x-ray plane waves. This single-material approximation, together with the projection approximation4, implies that the x rays at the exit surface of the sample have an intensity As a third and somewhat more detailed example of inverse problems that may be tackled using the formalism outlined in the present paper, consider a static non-magnetic non-crystalline sample that is made of a single material. This sample may and in general will have varying spatial density, with complex refractive indexμ = 2kβ is the linear attenuation coefficient of the material from which the sample is composed, and T is the projected thickness of the sample along the z direction. The fact that7Here n in Eq.  to be trDeff. Equation ] of the single-material sample with projected thickness T, and is an algebraic equation for the unknown effective diffusion coefficient  Deff. Note that the Deff → 0 limit of Eq.  to vanish, we may solve Eq.  and T, as follows. Suppose propagation-based phase contrast images are obtained at two different propagation distances, Δ1 and Δ2. From Eq. (f) in Fig. Here, we have used Eq.  to incorEquation  is a lint of Eq.  is the bolve Eq.  for bothz = 0 of the object and the entrance surface z = Δ of the detector, is not a diffusive medium. Thus the width of the SAXS fans scales linearly with Δ, rather than being proportional to Δ1/2 (as would be the case for true diffusion). Setting D ∝ Δ ensures that the final term on the right side of Eq. 2 = Δ, as required. As is the case throughout the paper, the artifice of a Δ-dependent diffusion coefficient is a price to be paid for the simplicity of being able to work simultaneously with both diffusion concepts and free-space coherent propagation concepts, in the context of the Fokker–Planck and Kramers–Moyal equations for x-ray imaging. To proceed we write the z = Δ1 and the z = Δ2 cases of Eq. . Hence:This scaling results from the already-mentioned fact that the slab of vacuum, in between the exit surface e of Eq.  is propos of Eq. , and thes of Eq.  to elimiet al.7, and may therefore be solved using the same Fourier-transform method. This gives:Equation  is mathex and y, in any convention that transforms ∂/∂x into ikx and ∂/∂y into iky;  are Fourier coordinates corresponding to , and 1 and Δ2, may be combined to yield the projected thickness T of a thin single-material sample, in a manner that is independent of SAXS scattering by the sample . Once T has been so obtained, Eq. . Such a phase/dark-field retrieval approach that incorporates dark-field effects might benefit samples from a range of fields including materials development  and biomedical imaging (e.g. diagnostic imaging of the lungs91).Here, Equation  shows honed, Eq.  can thenThe first contributor to the hierarchy of diffusion tensors is the sample-induced scatter due to unresolved micro-structure, that has been a main theme of the present paper.44 that is now known to be due to unresolved sharp edges92, and which may be viewed as a Young–Maggi–Rubinowicz type boundary wave97 or a Keller-type98 diffracted ray. This is examined in further detail in the previously mentioned companion paper77.The second contributor to the diffusion tensors is an edge-scattering signalDeff to a constant Deff(Δ) that is independent of x, y so as to model penumbral blur due to finite source size in a linear shift-invariant imaging system, and then note that the resulting equation has a solution identical to that recently published by Beltran et al.99.The third contribution to the diffusion tensors is associated with incoherent aberrations  of the imaging system that is used to measure images of the sample. As an example: take Eq. , set DefIn the third example above, and indeed throughout the paper, the effective diffusion coefficient ) wi) wi50)) All three effects will in general be present simultaneously, so that e.g. the SAXS fans associated with sample-induced blurring will need to be convolved with the intensity point-spread function of a given optical imaging system, and similarly with the signal associated with unresolved sharp edges. These interesting complications, which may be summarised in the statement that “partial coherence can come from the source, or the object itself”, are beyond the scope of the present paper.101. From an unsharp-mask image processing perspective, the linear differential operator102I to which it is applied, with a characteristic position-dependent partial-deconvolution transverse length scale equal to w; cf. Eq. , is 102:64\dThis blurring is rotationally symmetric. If this rotational symmetry is not present, we can use a Kramers–Moyal type linear differential operator to effect local blurring with a position-dependent point-spread function, namelyx ≡ ∂x, ∂/∂y ≡∂y. This gives another perspective on the use of derivative operators to effect blurring associated with a position-dependent point spread function. Similar ideas are used in image sharpening via coherent defocus67 and image blurring operations of the convolution type103.The notation above is analogous to the diffusive terms in Eq. , and we 104 would be an interesting avenue for future research, since Wigner functions naturally capture the idea that there can be a distribution of energy flow vectors at each point in space105. See e.g. Eq. (19) of Nugent and Paganin106, which was obtained via a Wigner-function formulation of paraxial optics, and which may be readily manipulated into a Fokker–Planck form given by Eq.  the partial coherence of the illuminating beam and its associated spatio-temporal speckles, and (ii) unresolved speckle associated with sample micro-structure. In the present paper, the spatial width w of the point-spread function (PSF) of the position-sensitive detector—a lower limit on which is given by the pixel size—defines a natural transverse length scale. Speckles much smaller in size than w will be spatially averaged when registering an image. If ergodicity of the associated stochastic process can be assumed, over the area of each pixel, then by definition the spatially-averaged intensity at each pixel location can be replaced with a corresponding ensemble average over many realisations of the ensemble of fields illuminating each pixel. Physically, and as has been used in the present paper, this corresponds to a statistical ensemble of objects, each of which has the same coarse-grained projected complex refractive index distribution, but differing spatially-random micro-structure. This construct permits us to approximate the resulting measured intensity map, with a probability distribution obeying the 2 + 1-dimensional Kramers–Moyal equation, including the special case of the 2 + 1-dimensional Fokker–Planck equation. Refraction and prism terms are associated with the resolved coarse-grained structure of the ensemble of objects and the ensemble of illuminations, with the diffusion term being associated with the unresolved speckles. What is deemed SAXS is a detector-resolution-dependent construct associated with unresolved sample-induced speckles. Thus, the component of the scattered radiation attributed to SAXS is dependent on w, and thereby detector dependent. This last statement ties in with the broader idea that there is a close link between partial coherence and the spatio-temporal resolution of one’s detector—see e.g. Sec. 3.6.3 of Paganin4. It is also closely related to what component of the x-ray energy flow is classed as “coherent energy transport”, and what is classed as “diffusive energy transport”, in the Fokker–Planck and Kramers–Moyal constructions: the finer the spatio-temporal resolution of the intensity detection, the smaller the fraction of the x-ray energy flow that will be classed as diffusive. Both the TIE-type and SAXS-type scattering (resolved and unresolved structures) arise from the same fundamental physics, as is clear in the first-principles derivation of the x-ray Fokker–Planck equation. Again, what length scales ultimately fall into either category is determined by the resolution of the detector.Unresolved speckle lies at the heart of many phenomena observed in imaging using partially coherent optical fields109, electrons110, neutrons111 and x-rays7, there is every expectation that the formalism of the present paper may be utilised for paraxial-imaging scenarios using all of the just-listed radiation and matter wave fields. Extensions of the present paper, to imaging applications beyond those listed previously, include a Fokker–Planck approach to neutron phase contrast imaging112, x-ray magnetic circular dichroism imaging113, x-ray edge-illumination imaging114, x-ray Zernike phase contrast imaging115, neutron grating imaging116, electron out-of-focus contrast imaging (Fresnel contrast imaging)117 and visible-light out-of-focus contrast imaging109. As with the present study, a core motivation for such an approach is its potential simultaneously to model, under the aegis of a single powerful formalism, both deterministic and stochastic aspects of paraxial thin-object imaging using partially coherent radiation and matter wave fields.We close this discussion by emphasising the broader applicability, beyond the margins delineated in the present paper, of the Fokker–Planck and Kramers–Moyal equations to (i) partially coherent paraxial x-ray imaging in particular, and (ii) partially-coherent paraxial imaging more generally. Context may be given to this claim, by the huge domain of applicability of Fokker–Planck and Kramers–Moyal concepts. Of equal relevance is the likelihood that the domain of validity of the paraxial-imaging Fokker–Planck and Kramers–Moyal equations is at least as broad at that of the transport-of-intensity equation which they generalise; and, since the transport-of-intensity equation has been applied to visible lightFokker–Planck and Kramers–Moyal equations were obtained, to model near-field and intermediate-field paraxial quasi-monochromatic x-ray imaging of thin samples exhibiting both phase–amplitude and small-angle-scattering contrast. Two complementary derivations were given. Possible applications, to both forward and inverse imaging problems, were discussed.


The aim of present work is to study some kinds of well-posedness for a class of generalized variational-hemivariational inequality problems involving set-valued operators. Some systematic approaches are presented to establish some equivalence theorems between several classes of well-posedness for the inequality problems and some corresponding metric characterizations, which generalize many known results. Finally, the well-posedness for a class of generalized mixed equilibrium problems is also considered. Nowadays, well-posedness has been drawing great attention in the field of optimization problems and related problems such as variational inequalities, hemivariational inequalities, fixed point problems, equilibrium problems, and inclusion problems see , 23, 33), 3323, 3As an important and useful generalization of variational inequality, hemivariational inequality, which was first studied by Panagiotopoulos , has a gϕ-quasimonotonicity and obtained some existence theorems when the constrained set is nonempty, bounded (or unbounded), closed, and convex in a reflexive Banach space. Hereafter, Wangkeeree and Preechasilp tonicity , a resultonicity , and somtonicity .K be a nonempty, closed, and convex subset of a real Banach space X with its dual u in the direction v. In this paper, we discuss the following generalized variational-hemivariational inequality (GVHVI):Let Find If Find g and He .If Find nd Huang .If Find  The existence of solutions to this inequality was studied by Wangkeeree and Preechasilp [Now, let us consider some particular cases of GVHVI. echasilp .α-well-posed GVHVI in the generalized sense. In particular, we present equivalence results on weak α-well-posedness for GVHVI, which were considered by few authors.Inspired by previous works, we study the well-posedness for GVHVI, which generalizes many known works. Under relatively weak conditions, we establish some equivalence results and some metric characterizations for the strong and weak This paper is organized as follows. In Sect. R, N be the sets of real numbers, nonnegative real numbers, and natural numbers, respectively. Let X be a real Banach space with norm X. Let X with weak topology.Let K be a nonempty subset of X. A function (i)K if for all finite subsets convex on (ii)u if for any sequence (weakly) upper semicontinuous (u.s.c. for short) at (iii)u, if for any sequence f is said to be (weakly) u.s.c. (l.s.c.) on K if f is (weakly) u.s.c. (l.s.c.) at all (weakly) lower semicontinuous (l.s.c. for short) at Let K be a nonempty subset of X. An operator Let (i)lower semicontinuous (l.s.c.) at (ii)F to every line segment of K is lower semicontinuous with respect to the weak topology in lower hemicontinuous (l.h.c.) if the restriction of A set-valued operator A set-valued operator S be a nonempty subset of X. The measure μ of noncompactness for the set S is defined byLet φ at the point φ at Now, let us recall the definitions of the Clarke generalized directional derivative and generalized gradient for a locally Lipschitz function X→R (see ). The Clis u.s.c. as a function ofand, as a function ofvalone, is Lipschitz of ranknearuonXand satisfies(ii)the gradientis a nonempty, convex, and weakly∗compact subset ofbounded by a Lipschitz constantnearx;(iii)for everywe haveα-approximating sequences and α-well-posedness for GVHVI. Let We end this section with the notions of several classes of K is an α-approximating sequence for GVHVI if there exist A sequence α-well-posed if it has a unique solution u and every α-approximating sequence u. In particular, if GVHVI is said to be strongly  α-well-posed in the generalized sense if the solution set Γ of GVHVI is nonempty and every α-approximating sequence GVHVI is said to be strongly  α-well-posedness  implies weak α-well-posedness , but the converse is not true in general.Strong α-well-posed.In this section, we establish metric characterizations and derive some conditions under which GVHVI is strongly (weakly) For any Assume that: (i)Kis a nonempty closed subset of a real Banach spaceX;(ii)is continuous;(iii)is u.s.c. with respect to the first variable;(iv)is such thatwheneverThen, for everythe setis closed inX.X. Then Let Assume that: (i)Kis a nonempty convex subset of a real Banach spaceX;(ii)is l.h.c. and monotone;(iii)is convex with respect to the second variable;(iv)is convex withfor allandThenfor allF, we obtainWe first show that K is convex, for any F is l.h.c.). Taking the limit as Now we show that The following result is a consequence of Lemmas Assume that: (i)Kis a nonempty closed convex subset of a real Banach spaceX;(ii)is l.h.c. and monotone;(iii)is continuous;(iv)is u.s.c. with respect to the first variable and convex with respect to the second variable;(v)is continuous and convex withfor allandThenis closed inXfor allMoreover, that is, GVHVI is equivalent to the following problem:Findsuch thatGVHVI is stronglyα-well-posed if and only if Γ is nonempty andg. □The proof is similar to that of Theorem 4.3 in  by the aAssume that all the assumptions of Theoremare satisfied. Then GVHVI is stronglyα-well-posed if and only ifα-well-posed. Then GVHVI has a unique solution α-approximating sequences of GVHVI, and thusSuppose that GVHVI is strongly ove that  holds. C part of , arguingK be an α-approximating sequence for GVHVI. Then, there exist F we obtain that, for every Conversely, assume that condition  holds. Londition  we deducIt follows from Theorem u is unique. If there exists another solution Finally, we prove that the solution Assume that: (i)Kis a nonempty closed convex subset of a real reflexive Banach spaceX;(ii)is l.h.c. and monotone;(iii)is compact;(iv)is weakly u.s.c. with respect to the first variable and convex with respect to the second variable;(v)for allwheneveras(vi)is a continuous and convex functional withfor allandThen GVHVI is weaklyα-well-posed if and only if GVHVI has a unique solution and there existssuch thatis nonempty and bounded.α-approximating sequence for GVHVI. Then, there exist X. Indeed, since X.The necessity is obvious. We now prove the sufficiency. Let X is reflexive, we can choose a subsequence of F we obtain thatα-well-posed by the uniqueness of the solution to problem GVHVI. This completes the proof. □Since the Banach space K is bounded or that there exists In the theorem, condition (v) can be found in , and the1n0. See , 36, 39 α-posedness in the generalized sense.Next, we give some equivalence results for the strong Assume that all the assumptions of Theoremare satisfied. Then GVHVI is stronglyα-well-posed in the generalized sense if and only if Γ is nonempty compact andwherewithg. □The proof is similar to that of Theorem 5.1 in  by the aAssume that all the assumptions of Theoremare satisfied. Then GVHVI is stronglyα-well-posed in the generalized sense if and only ifg. □The proof is similar to that of Theorem 3.2 in [Assume that all the assumptions of Theoremare satisfied. Then GVHVI is weaklyα-well-posed in the generalized sense if and only if there existssuch thatis nonempty and bounded.g. □The proof is similar to that of Theorem In this section, we consider the following generalized mixed equilibrium problem (GMEP):Find found in .η-monotonicity ..η-monotoF is said to be η-monotone if there exists a function Let F is monotone.If ∈X, then  becomesFor any We can obtain similar results.Assume that all the assumptions of Theoremare satisfied and, in addition, is continuous onwithfor anyand affine with respect to the first variable. Letbe such that: (i)for all(ii)for allis u.s.c.,(iii)for allis convex.Thenis closed inXfor allMoreover, that is, GMEP is equivalent to the following problem:Findsuch thatAssume that all the assumptions of Theoremare satisfied and, in addition, is continuous onwithfor anyand affine with respect to the first variable. Letbe such that: (i)for all(ii)for allis u.s.c.,(iii)for allis convex.Then GMVHVI is stronglyα-well-posed if and only ifAssume that all the assumptions of Theoremare satisfied and, in addition, is continuous onwithfor anyand affine with respect to the first variable. Letbe such that: (i)for all(ii)for allis weakly u.s.c.,(iii)for allis convex.Then GMEP is weaklyα-well-posed if and only if GMEP has a unique solution and there existssuch thatis nonempty and bounded.Assume that all the assumptions of Theoremare satisfied and, in addition, is continuous onwithfor anyand affine with respect to the first variable. Letis such that: (i)for all(ii)for allis u.s.c.,(iii)for allis convex.Then GMEP is stronglyα-well-posed in the generalized sense if and only ifAssume that all the assumptions of Theoremare satisfied and, in addition, is continuous onwithfor anyand affine with respect to the first variable. Letbe such that: (i)for all(ii)for allis weakly u.s.c.,(iii)for allis convex.Then GMEP is weaklyα-well-posed in the generalized sense if and only if there existssuch thatis nonempty and bounded.F, T, g, J (see Theorems α-well-posed GVHVI in the generalized sense. Our results generalize and improve many known results and can be applied to many other problems.In this paper, inspired by the previous works, we study the well-posedness for GVHVI. Under relatively weak conditions for the data


We estimate the rate of convergence of such operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness, establish the Voronovskaja type asymptotic formula for the bivariate λ-Bernstein–Kantorovich operators, as well as give some examples and their graphs to show the effect of convergence.In this paper, we introduce a family of  In 1912, Bernstein [λ-Bernstein operators with parameter f. These operators, which they called λ-Bernstein operators, are defined as follows:Recently, Cai et al.  proposedentclass1pt{minimaλ-Bernstein–Kantorovich operators asIn , Cai intentclass1pt{minimaB-continuous functions was established by Badea et al. in [q-integers, studied the uniform convergence theorem and the degree of approximation of these operators. Very recently, Kajla and Miclăuş [As we know, the generalized Boolean sum operators  to , we obtamentclasspt{minimaλ-Bernstein–Kantorovich operators with GBS operators. Consider the function λ-Bernstein–Kantorovich type for B-continuous and B-differentiable functions by using the mixed modulus of smoothness, as well as obtain the Voronovskaja type asymptotic formula for bivariate λ-Bernstein–Kantorovich operators.In this paper, we deduce the rate of convergence of GBS operators of bivariate tensor product of