(Ac 10:1)NKJV There was a certain man in Caesarea called Cornelius, a centurion of what was called the Italian Regiment,
(Ac 10:2) a devout man and one who feared God with all his household, who gave alms generously to the people, and prayed to God always.
(Ac 10:3) About the ninth hour of the day he saw clearly in a vision an angel of God coming in and saying to him, "Cornelius!"
(Ac 10:4) And when he observed him, he was afraid, and said, "What is it, lord?" So he said to him, "Your prayers and your alms have come up for a memorial before God.
(Ac 10:5) Now send men to Joppa, and send for Simon whose surname is Peter.

After Peter went and Cornelius, his family, and friends were converted, Peter reported these things to the fellow believers in Jerusalem. Among other things, he mentioned the following.

(Ac 11:11)EMTV And behold, immediately three men stood before the house in which I was, having been sent from Caesarea to me.
(Ac 11:12) And the Spirit said to me to go with them, without wavering. Now these six brothers also went with me, and we entered into the house of the man.
(Ac 11:13) And he related to us how he had seen the angel in his house, standing and saying to him, 'Send men to Joppa, and send for Simon whose surname is Peter,
(Ac 11:14) 'who will speak words to you by which you and all your household shall be saved.'

The passage above mentions explicitly in Ac 11:14 that Cornelius was not saved. He was a devout man who feared God (Ac 10:2), though he was not saved. God did not ignore his prayers and alms, for we read "Your prayers and your alms have come up for a memorial before God" (Ac 10:4). Just as a memorial may help people remember, so also by means of his prayers and alms Cornelius received special attention from the Lord. Cornelius was already praying to the true God before being saved. Since no one can come to the Lord unless the Father draws him (Jn 6:44), the prayers and alms of Cornelius were a response to work that God had already done in him. While Ac 10:4 relates the prayers and alms of Cornelius to the events recorded in Ac 10, God could have saved Cornelius even if he did not pray or give alms. Perhaps the Lord worked compassion and piety in Cornelius before saving him in order to show the church that he was working among the Gentiles, and to help the church begin receiving Gentile believers who were not converts to Judaism. Nonetheless, Ac 10:4 does indicate that prayer and generosity are important in God's eyes.

Under the old covenant there were many who offered sacrifices when they were seeking the Lord. A phrase that is found very often in the books of the law in the context of sacrifices is "a pleasing aroma to the LORD". We could reason that just as a pleasing aroma would get the attention of a man, so also an acceptable sacrifice was getting the attention of the Lord. The phrase "pleasing aroma" appears also in the New Testament with the same Greek words as in the Septuagint, the Greek translation of the Old Testament. The following translation uses the words "fragrant aroma" for the same phrase.

(Php 4:18)EMTV And I have all things and abound. I have been made full, having received from Epaphroditus the things sent from you, a fragrant aroma, an acceptable sacrifice, well pleasing to God.
(Php 4:19) And my God will supply all your need according to His riches in glory in Christ Jesus.

The apostle Paul mentions above that what the Philippians had given was a "fragrant aroma, an acceptable sacrifice, well pleasing to God". The following passage could remind one of the importance of offerings. The statement "May he remember all your offerings ... " precedes "May he grant you your heart's desire ... ".

(Ps 20:1)ESV To the choirmaster. A Psalm of David. May the LORD answer you in the day of trouble! May the name of the God of Jacob protect you!
(Ps 20:2) May he send you help from the sanctuary and give you support from Zion!
(Ps 20:3) May he remember all your offerings and regard with favor your burnt sacrifices! Selah
(Ps 20:4) May he grant you your heart's desire and fulfill all your plans!

If translated literally, 2Th 1:11 is related to Ps 20:4. We read " we also pray always for you, that our God may count you worthy of your calling, and fulfill every desire of goodness and every work of faith, with power" (2Th 1:11)ASV. Though God cannot forget, he can choose to ignore something. The statement "May he remember all your offerings" is a wish that God will not choose to ignore the offerings but rather that he will take them into account favorably when making decisions. The gift of the Philippians to the ministry of the apostle Paul was "an acceptable sacrifice, well pleasing to God" (Php 4:18). Thus, it was something that the Lord would have remembered. So we read the promise that "my God will supply all your need" in the following verse, in Php 4:19. We also read in Ac 10:4 that the prayers and alms of Cornelius were "a memorial before God". Therefore, the Lord was not ignoring them. Prayer is likened to a sacrifice in Ps 141:2: "Let my prayer be set forth before thee as incense; and the lifting up of my hands as the evening sacrifice". Thanksgiving is also mentioned in Ps 50:14 in contrast to the sacrifices of animals: "Will I eat the flesh of bulls, or drink the blood of goats? Offer unto God thanksgiving ... " (Ps 50:13-14). Moreover, as we read in Heb 13:5, when the believer praises the Lord, he offers "the sacrifice of praise".

(1Pt 2:4)Darby To whom coming, a living stone, cast away indeed as worthless by men, but with God chosen, precious,
(1Pt 2:5) yourselves also, as living stones, are being built up a spiritual house, a holy priesthood, to offer spiritual sacrifices acceptable to God by Jesus Christ.

(Heb 13:15)Darby By him therefore let us offer the sacrifice of praise continually to God, that is, the fruit of the lips confessing his name.

Believers "are being built up a spiritual house, a holy priesthood" (1Pt 2:5). Just as the Old Testament priests offered sacrifices, the New Testament believers offer "spiritual sacrifices" (1Pt 2:5). As indicated in the Scripture passages above, giving, prayer, thanksgiving, and praise are seen as offerings. Two other sacrifices mentioned in the New Testament are "the sacrifice and service of your faith" in Php 2:17 and the presentation of one's own body as "a living sacrifice, holy, acceptable unto God" (Ro 12:1).